`quanscient`

Module This is the python binding for the 'PYSIMCORE' c++ library.

## Functions

`abs`

`def abs( input: expression ) ‑> quanscient.expression`

This returns an expression that is the absolute value of the input expression.

###### Example

```
>>> expr = abs(-2.5)
>>> expr.print();
Expression size is 1x1
@ row 0, col 0 :
2.5
```

`acos`

`def acos( input: expression ) ‑> quanscient.expression`

This returns an expression that is the or of input. The output expression is in `radians`

.

###### Example

```
>>> expr = acos(0)
>>> expr.print(); # in radians
Expression size is 1x1
@ row 0, col 0 :
1.5708
>>>
>>> (expr * 180/getpi()).print(); # in degrees
Expression size is 1x1
@ row 0, col 0 :
90
```

###### See Also

`sin()`

, `cos()`

, `tan()`

, `asin()`

, `atan()`

`adapt`

`def adapt( verbosity: int = 0 ) ‑> bool`

This function is used to perform a h/p/hp adaptation according to the defined h/p/hp adaptivity settings. To define the h-adaptivity use function `mesh.setadaptivity()`

. To define a p-adaptivity for a field use function `field.setorder()`

.

The function returns True if the mesh or any field order was changed by the adaption and returns False if no changes were made.

###### Example

```
>>> all = 1
>>> q = shape("quadrangle", all, [0,0,0, 1,0,0, 1.2,1,0, 0,1,0], [5,5,5,5])
>>> mymesh = mesh([q])
>>> x = field("x"); y = field("y")
>>> criterion = 1 + sin(10*x)*sin(10*y)
>>>
>>> mymesh.setadaptivity(criterion, 0, 5)
>>>
>>> for i in range(5):
... criterion.write(all, f"criterion_{100+i}.vtk", 1)
... adapt(1)
```

###### See Also

`mesh.setadaptivity()`

, `field.setorder()`

, `alladapt()`

`alladapt`

`def alladapt( verbosity: int = 0 ) ‑> bool`

This is a collective MPI operation and hence must be called by all ranks. It replaces the `adapt()`

function in the DDM framework.

###### Example

```
>>> ...
>>> alladapt()
```

###### See Also

`allcomputecohomology`

`def allcomputecohomology( meshfile: str, physregstocut: List[int], subdomains: List[int] = [] ) ‑> List[int]`

`allcomputesparameters`

`def allcomputesparameters( portsphysregs: List[int], E: field, Esources: List[expression], Esols: List[vec], integrationorder: int = 5 ) ‑> List[List[float]]`

`allfinalize`

`def allfinalize() ‑> None`

`allgather`

`def allgather( *args, **kwargs ) ‑> None`

`allinitialize`

`def allinitialize( verbosity: int = 1 ) ‑> None`

`allintegrate`

`def allintegrate( physreg: int, expr: expression, integrationorder: int ) ‑> float`

`allmeasuredistance`

`def allmeasuredistance( a: vec, b: vec, c: vec ) ‑> float`

`allpartition`

`def allpartition( meshfile: str, skinphysreg: int = -1 ) ‑> str`

This is a collective MPI operation and hence must be called by all ranks. This function partitions the requested mesh into a number of parts equal to the number of ranks. All parts are saved to disk and the part file name for each rank is returned. The GMSH API must be available to partition into more than one part. A physical region containing the global skin can be created with the last argument.

This function will be deprecated. Use `mesh.partition()`

instead.

###### Example

```
>>> initialize()
>>> partfilename = allpartition("disk.msh")
>>> finalize()
```

`allsolve`

`def allsolve( relrestol: float, maxnumit: int, nltol: float, maxnumnlit: int, relaxvalue: float, formuls: List[formulation], verbosity: int = 1 ) ‑> int`

This is a collective MPI operation and hence must be called by all ranks. It solves across all the ranks a nonlinear problem with a fixed-point iteration.

###### Example

```
>>> ...
>>> allsolve(1e-8, 500, 1e-4, 1.0, [electrostatics])
```

###### See Also

`andpositive`

`def andpositive( exprs: List[expression] ) ‑> quanscient.expression`

This returns an expression whose value is 1 for all evaluation points where the value of all the input expressions is larger or equal to zero. Otherwise, its value is -1.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x=field("x"); y=field("y"); z=field("z")
>>> expr = andpositive([x,y]) # At points, where x>=0 and y>=0, value is 1, otherwise -1.
>>> expr.write(vol, "andpositive.vtk", 1)
```

###### See Also

`array1x1`

`def array1x1( arg0: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11###### Example

```
>>> myarray = array1x1(1)
```

`array1x2`

`def array1x2( arg0: expression, arg1: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11** arg1** :

`expression`

: term12###### Example

```
>>> myarray = array1x2(1,2)
```

`array1x3`

`def array1x3( arg0: expression, arg1: expression, arg2: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11** arg1** :

`expression`

: term12** arg2** :

`expression`

: term13###### Example

```
>>> myarray = array1x3(1,2,3)
```

`array2x1`

`def array2x1( arg0: expression, arg1: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11** arg1** :

`expression`

: term21###### Example

```
>>> myarray = array2x1(1, 2)
```

`array2x2`

`def array2x2( arg0: expression, arg1: expression, arg2: expression, arg3: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11** arg1** :

`expression`

: term12** arg2** :

`expression`

: term21** arg3** :

`expression`

: term22###### Example

```
>>> myarray = array2x2(1,2, 3,4)
```

`array2x3`

`def array2x3( arg0: expression, arg1: expression, arg2: expression, arg3: expression, arg4: expression, arg5: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11** arg1** :

`expression`

: term12** arg2** :

`expression`

: term13** arg3** :

`expression`

: term21** arg4** :

`expression`

: term22** arg5** :

`expression`

: term23###### Example

```
>>> myarray = array2x3(1,2,3, 4,5,6)
```

`array3x1`

`def array3x1( arg0: expression, arg1: expression, arg2: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11** arg1** :

`expression`

: term21** arg2** :

`expression`

: term31###### Example

```
>>> myarray = array3x1(1, 2, 3)
```

`array3x2`

`def array3x2( arg0: expression, arg1: expression, arg2: expression, arg3: expression, arg4: expression, arg5: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11** arg1** :

`expression`

: term12** arg2** :

`expression`

: term21** arg3** :

`expression`

: term22** arg4** :

`expression`

: term31** arg5** :

`expression`

: term32###### Example

```
>>> myarray = array3x2(1,2, 3,4, 5,6)
```

`array3x3`

`def array3x3( arg0: expression, arg1: expression, arg2: expression, arg3: expression, arg4: expression, arg5: expression, arg6: expression, arg7: expression, arg8: expression ) ‑> quanscient.expression`

This defines a vector or matrix operation of size . The array is populated in a row-major way.

###### Parameters

** arg0** :

`expression`

: term11** arg1** :

`expression`

: term12** arg2** :

`expression`

: term13** arg3** :

`expression`

: term21** arg4** :

`expression`

: term22** arg5** :

`expression`

: term23** arg6** :

`expression`

: term31** arg7** :

`expression`

: term32** arg8** :

`expression`

: term33###### Example

```
>>> myarray = array3x3(1,2,3, 4,5,6, 7,8,9)
```

`asin`

`def asin( input: expression ) ‑> quanscient.expression`

This returns an expression that is the or of input. The output expression is in `radians`

.

###### Example

```
>>> expr = asin(sqrt(0.5))
>>> expr.print(); # in radians
Expression size is 1x1
@ row 0, col 0 :
0.785398
>>>
>>> (expr * 180/getpi()).print(); # in degrees
Expression size is 1x1
@ row 0, col 0 :
45
```

###### See Also

`sin()`

, `cos()`

, `tan()`

, `acos()`

, `atan()`

`atan`

`def atan( input: expression ) ‑> quanscient.expression`

This returns an expression that is the or of input. The output expression is in `radians`

.

###### Example

```
>>> expr = atan(0.57735)
>>> expr.print(); # in radians
Expression size is 1x1
@ row 0, col 0 :
0.523599
>>>
>>> (expr * 180/getpi()).print(); # in degrees
Expression size is 1x1
@ row 0, col 0 :
30
```

###### See Also

`sin()`

, `cos()`

, `tan()`

, `asin()`

, `acos()`

`barrier`

`def barrier() ‑> None`

This is a collective MPI operation (must be called by all ranks). Processing will be waiting until all ranks have reached this call.

###### Example

```
>>> import quanscient as qs
>>> qs.barrier()
```

###### Notes

If this function is not called by all the ranks, then the other ranks that call `barrier()`

will wait indefinitely.

`broadcast`

`def broadcast( *args, **kwargs ) ‑> None`

`cn`

`def cn( n: float ) ‑> quanscient.expression`

`comp`

`def comp( selectedcomp: int, input: expression ) ‑> quanscient.expression`

This returns the selected component of a column vector expression. For a column vector expression, `selectedcomp`

is 0 for the first, 1 for the second component and 2 for the third component respectively. For a matrix expression, the whole corresponding row is returned in the form of an expression. Thus, if `selectedcomp`

, for example is 5, then an expression containing the entries of the fifth row of the matrix is returned.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> u.setvalue(vol, array3x1(10,20,30))
>>> vecexpr = 2*(u+u)
>>> comp(0, vecexpr).write(vol, "comp.vtk", 1)
```

###### See Also

`complexdivision`

`def complexdivision( a: List[expression], b: List[expression] ) ‑> List[quanscient.expression]`

`complexinverse`

`def complexinverse( a: List[expression] ) ‑> List[quanscient.expression]`

`complexproduct`

`def complexproduct( a: List[expression], b: List[expression] ) ‑> List[quanscient.expression]`

`compx`

`def compx( input: expression ) ‑> quanscient.expression`

This returns the first or `x`

component of a column vector expression. For a matrix expression, an expression containing the entries of the first row of the matrix is returned. This is equivalent to setting `selectedcomp=0`

in `comp()`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> u.setvalue(vol, array3x1(10,20,30))
>>> vecexpr = 2*(u+u)
>>> compx(vecexpr).write(vol, "compx.vtk", 1)
```

###### See Also

`compy`

`def compy( input: expression ) ‑> quanscient.expression`

This returns the second or `y`

component of a column vector expression. For a matrix expression, an expression containing the entries of the second row of the matrix is returned. This is equivalent to setting `selectedcomp=1`

in `comp()`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> u.setvalue(vol, array3x1(10,20,30))
>>> vecexpr = 2*(u+u)
>>> compx(vecexpr).write(vol, "compx.vtk", 1)
```

###### See Also

`compz`

`def compz( input: expression ) ‑> quanscient.expression`

This returns the third or `z`

component of a column vector expression. For a matrix expression, an expression containing the entries of the third row of the matrix is returned. This is equivalent to setting `selectedcomp=2`

in `comp()`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> u.setvalue(vol, array3x1(10,20,30))
>>> vecexpr = 2*(u+u)
>>> compx(vecexpr).write(vol, "compx.vtk", 1)
```

###### See Also

`continuitycondition`

`def continuitycondition( *args, **kwargs ) ‑> List[quanscient.integration]`

This returns the formulation terms required to enforce field continuity.

###### Examples

**Example 1**: `continuitycondition(gamma1:int, gamma2:int. u1:field, u2:field, errorifnotfound:true)`

```
>>> ...
>>> u1 = field("h1xyz")
>>> u2 = field("h1xyz")
>>>
>>> elasticity = formulation()
>>> ...
>>> elasticity += continuitycondition(gamma1, gamma2, u1, u2)
```

This returns the formulation terms required to enforce between boundary region and (with , meshes can be non-matching). In case is larger than () the boolean flag must be set to false.

**Example 2:** `continuitycondition(gamma:int, gamma2:int, u1:field, u2:field, rotcent:List[double], rotangz:double, angzmod:double, factor:double)`

```
>>> ...
>>> # Rotor-stator interface
>>> rotorside=11; statorside=12
>>> ...
>>> # Rotor rotation around z axis
>>> alpha = 30.0
>>> ...
>>> az = field("h1")
>>> ...
>>> magentostatics = formulation()
>>> ...
>>> magnetostatics += continuitycondition(statorside, rotorside, az, az, [0,0,0], alpha, 45.0, -1.0)
```

This returns the formulation terms required to enforce field continuity across an degrees slice of a rotor-stator interface where the rotor geometry is rotated by around the axis with rotation center at . This situation arises for example in electric motor simulations when (anti)periodicity can be considered and thus only a slice of the entire 360 degrees needs to be simulated. Use a factor of for antiperiodicity. Boundary is the rotor-stator interface on the (non-moving) stator side while the boundary is the interface on the rotor side. In the unrotated position the bottom boundary of the stator and rotor slice must be aligned with the axis.

The condition is based on a Lagrange multiplier of the same type and the same harmonic content as the field and . The mortar finite element method is used to link the unknown field on and so that there is no restriction on the mesh used for both regions.

###### See Also

`cos`

`def cos( input: expression ) ‑> quanscient.expression`

This returns an expression that is the of input. The input expression is in `radians`

.

###### Example

```
>>> expr = cos(getpi())
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
-1
>>>
>>> expr = cos(1)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
0.540302
```

###### See Also

`sin()`

, `tan()`

, `asin()`

, `acos()`

, `atan()`

`count`

`def count() ‑> int`

This returns the number of processes in the universe.

###### Example

```
>>> import quanscient as qs
>>> numranks = qs.count()
```

###### See Also

`crossproduct`

`def crossproduct( a: expression, b: expression ) ‑> quanscient.expression`

This computes the cross-product of two vector expressions. The returned expression is a vector.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; top=3
>>> E = field("hcurl")
>>> n = normal(vol)
>>> cp = crossproduct(E, n)
```

`curl`

`def curl( input: expression ) ‑> quanscient.expression`

This computes the curl of a vector expression. The returned expression is a vector.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1);
>>> curl(u).write(vol, "curlu.vtk", 1)
```

###### See Also

`d`

`def d( expr: expression, var: expression, flds: List[field], deltas: List[expression] ) ‑> quanscient.expression`

`dbtoneper`

`def dbtoneper( toconvert: expression ) ‑> quanscient.expression`

This converts a value to .

###### Parameters

** toconvert** :

`expression`

: expression value in units.###### Example

```
>>> neperattenuation = dbtoneper(100.0)
```

`determinant`

`def determinant( input: expression ) ‑> quanscient.expression`

This returns the determinant of a square matrix.

###### Example

```
>>> matexpr = expression(3,3, [1,2,3, 6,5,4, 8,9,7])
>>> detmat = determinant(matexpr)
>>> detmat.print()
Expression size is 1x1
@ row 0, col 0 :
21
```

###### See Also

`detjac`

`def detjac() ‑> quanscient.expression`

`div`

`def div( input: expression ) ‑> quanscient.expression`

This computes the divergence of a vector expression. The returned expression is a scalar.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1);
>>> div(u).write(vol, "divu.vtk", 1)
```

###### See Also

`dof`

`def dof( *args, **kwargs ) ‑> quanscient.expression`

This declares an unknown field (*dof* for degree of freedom). The dofs are defined only on the region `physreg`

which when not provided is set to the element integration region.

###### Examples

**Example 1**: `dof(input:expression)`

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
```

**Example 2**: `dof(input:expression, physreg:int)`

```
>>> ...
>>> projection += integral(vol, dof(v, vol)*tf(v) - 2*tf(v))
```

###### See Also

`doubledotproduct`

`def doubledotproduct( a: expression, b: expression ) ‑> quanscient.expression`

This computes the double-dot product of two matrix expressions. The returned expression is a scalar.

###### Example

```
>>> a = array2x2(1,2, 3,4)
>>> b = array2x2(11,12, 13, 14)
>>> addotb = doubledotproduct(a, b)
>>> addotb.print()
Expression size is 1x1
@ row 0, col 0 :
130
```

`dt`

`def dt( *args, **kwargs ) ‑> quanscient.expression`

This returns the first-order time derivative expression.

###### Examples

**Example 1**: `dt(input:expression)`

```
>>> ...
>>> setfundamentalfrequency(50)
>>> vmh = field("h1", [2,3])
>>> vmh.setorder(vol, 1)
>>> dt(vmh).write(vol, "dtv.vtk", 1)
```

**Example 2**: `dt(input:expression, initdt:double, initdtdt)`

This gives the transient approximation of the first-order time derivative of a space-independent expression. The initial values must be provided when using generalized alpha (`genalpha`

) and are ignored otherwise.

```
>>> ...
>>> dtapprox = dt(t()*t(), 0, 2)
```

###### See Also

`dtdt`

`def dtdt( *args, **kwargs ) ‑> quanscient.expression`

This returns the second-order time derivative expression.

###### Examples

**Example 1**: `dtdt(input:expression)`

```
>>> ...
>>> setfundamentalfrequency(50)
>>> vmh = field("h1", [2,3])
>>> vmh.setorder(vol, 1)
>>> dtdt(vmh).write(vol, "dtdtv.vtk", 1)
```

**Example 2**: `dtdt(input:expression, initdt:double, initdtdt:double)`

This gives the transient approximation of the second-order time derivative of a space-independent expression. The initial values must be provided when using generalized alpha (`genalpha`

) and are ignored otherwise.

```
>>> ...
>>> dtapprox = dtdt(t()*t(), 0, 2)
```

###### See Also

`dtdtdt`

`def dtdtdt( input: expression ) ‑> quanscient.expression`

This returns the third-order time derivative expression.

###### Example

```
>>> ...
>>> setfundamentalfrequency(50)
>>> vmh = field("h1", [2,3])
>>> vmh.setorder(vol, 1)
>>> dtdtdt(vmh).write(vol, "dtdtdtv.vtk", 1)
```

###### See Also

`dtdtdtdt`

`def dtdtdtdt( input: expression ) ‑> quanscient.expression`

This returns the fourth-order time derivative expression.

###### Example

```
>>> ...
>>> setfundamentalfrequency(50)
>>> vmh = field("h1", [2,3])
>>> vmh.setorder(vol, 1)
>>> dtdtdtdt(vmh).write(vol, "dtdtdtdtv.vtk", 1)
```

###### See Also

`dx`

`def dx( input: expression ) ‑> quanscient.expression`

This returns the space derivative expression.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> dx(v).write(vol, "dxv.vtk", 1)
```

###### See Also

`dy`

`def dy( input: expression ) ‑> quanscient.expression`

This returns the space derivative expression.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> dy(v).write(vol, "dyv.vtk", 1)
```

###### See Also

`dz`

`def dz( input: expression ) ‑> quanscient.expression`

This returns the space derivative expression.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> dz(v).write(vol, "dzv.vtk", 1)
```

###### See Also

`eigenport`

`def eigenport( portphysreg: int, modetype: str, numeigenvalues: int, targetneffc: float, E: field, mu: expression, eps: expression, ddmrelrestol: float, ddmmaxnumit: int, eigentol: float = 1e-06, eigenmaxnumits: int = 1000, integrationorder: int = 5, verbosity: int = 1 ) ‑> List[Tuple[quanscient.expression, quanscient.expression]]`

`elasticwavespeed`

`def elasticwavespeed( *args, **kwargs ) ‑> List[quanscient.expression]`

`elementwiseproduct`

`def elementwiseproduct( a: expression, b: expression ) ‑> quanscient.expression`

This computes the element-wise product of two matrix expressions. The returned expression has the same size as the two input expressions.

###### Example

```
>>> a = array2x2(1,2, 3,4)
>>> b = array2x2(11,12, 13, 14)
>>> aelwb = elementwiseproduct(a, b)
>>> aelwb.print()
Expression size is 2x2
@ row 0, col 0 :
11
@ row 0, col 1 :
24
@ row 1, col 0 :
39
@ row 1, col 1 :
56
```

`emwavespeed`

`def emwavespeed( mur: expression, epsr: expression ) ‑> quanscient.expression`

`entry`

`def entry( row: int, col: int, input: expression ) ‑> quanscient.expression`

This gets the `(row, col)`

entry in the vector or matrix expression.

###### Parameters

** row** :

`int`

: Row from which the entry is requested.** col** :

`int`

: Column from which the entry is requested.** input** :

`expression`

: Vector or matrix input expression.###### Example

```
>>> u = array3x2(1,2, 3,4, 5,6)
>>> arrayentry_row2col0 = entry(2, 0, u) # entry from third row (index=2), first column (index=0)
>>> arrayentry_row2col0.print()
Expression size is 1x1
@ row 0, col 0 :
5
```

`evaluate`

`def evaluate( toevaluate: expression ) ‑> List[float]`

`exchange`

`def exchange( *args, **kwargs ) ‑> None`

`exp`

`def exp( input: expression ) ‑> quanscient.expression`

This returns an exponential function of base .

###### Example

```
>>> expr = exp(2)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
7.38906
```

`eye`

`def eye( size: int ) ‑> quanscient.expression`

This returns a `size x size`

identity matrix

###### Parameters

** size** :

`int`

: Size of the identity matrix.###### Example

```
>>> II = eye(2)
>>> II.print()
Expression size is 2x2
@ row 0, col 0 :
1
@ row 0, col 1 :
0
@ row 1, col 0 :
0
@ row 1, col 1 :
1
```

`fieldorder`

`def fieldorder( input: field, alpha: float = -1.0, absthres: float = 0.0 ) ‑> quanscient.expression`

This returns an expression whose value is the interpolation order on each element for the provided field. The value is a constant on each element. When argument is set, the value returned is the lowest order required to include of the total shape function coefficient weight. An additional optional argument can be set to provide a minimum total weight below which the lowest possible field order is returned.

###### Parameters

** input** :

`field`

: The field object whose interpolation order is requested.** alpha** :

`double`

: ** absthres** :

`double`

: ###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1; sur = 2; top = 3
>>> v = field("h1")
>>> v.setorder(vol, 2)
>>> fo = fieldorder(v)
>>> fo.write(vol, "fieldorder.vtk", 1)
```

`finalize`

`def finalize() ‑> None`

`gather`

`def gather( *args, **kwargs ) ‑> None`

`getcirculationports`

`def getcirculationports( harmonicnumbers: List[int] = [] ) ‑> Tuple[List[quanscient.port], List[quanscient.expression]]`

`getcirculationsources`

`def getcirculationsources( circulations: List[expression], inittimederivatives: List[List[float]] = [] ) ‑> List[quanscient.expression]`

`getdimension`

`def getdimension( physreg: int ) ‑> int`

This returns the x, y and z mesh dimensions.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> dims = mymesh.getdimensions()
```

`getepsilon0`

`def getepsilon0() ‑> float`

Returns the value of vacuum permittivity .

###### Example

```
>>> eps0 = getepsilon0()
8.854187812813e-12
```

`getextrusiondata`

`def getextrusiondata() ‑> List[quanscient.expression]`

This gives the relative depth in the extruded layer, the extrusion normal and tangents and . This is useful when creating Perfectly Matched Layers (PMLs).

###### Example

We use the `disk.msh`

for the example here.

```
>>> vol=1; sur=2; top=3; circle=4 # physical regions defined in disk.msh
>>> mymesh = mesh()
>>>
>>> # predefine extrusion
>>> volextruded=5; bndextruded=6; # physical regions that will be utilized in extrusion
>>> mymesh.extrude(volextruded, bndextruded, sur, [0.1,0.05])
>>> mymesh.load("disk.msh") # extrusion is performed when the mesh is loaded.
>>> mymesh.write("diskpml.msh")
>>> pmldata = getextrusiondata()
```

###### See Also

`getharmonic`

`def getharmonic( harmnum: int, input: expression, numfftharms: int = -1 ) ‑> quanscient.expression`

This returns a single harmonic from a multi-harmonic expression. Set a positive last argument to use an FFT to compute the harmonic. The returned expression is on harmonic 1.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>> v = field("h1", [2])
>>> v.setorder(vol, 1)
>>> v.harmonic(2).setvalue(vol, 1)
>>> constcomp = getharmonic(1, abs(v), 10)
>>> constcomp.write(vol, "constcomp.pos", 1)
```

`getmaxnumthreads`

`def getmaxnumthreads() ‑> int`

Returns the maximum number of threads allowed.

###### Example

```
>>> setmaxnumthreads(2)
>>> mnt = getmaxnumthreads()
2
```

###### See Also

`getmu0`

`def getmu0() ‑> float`

Returns the value of vacuum permeability .

###### Example

```
>>> mu0 = getmu0()
1.2566370621219e-06
```

`getpi`

`def getpi() ‑> float`

Returns value of .

###### Example

```
>>> pi = getpi()
3.141592653589793
```

`getrandom`

`def getrandom() ‑> float`

Returns a random value uniformly distributed between 0.0 and 1.0.

###### Example

```
>>> rnd = getrandom()
```

`getrank`

`def getrank() ‑> int`

This returns the rank of the current process.

###### Example

```
>>> import quanscient as qs
>>> rank = qs.getrank()
```

###### See Also

`getsstkomegamodelconstants`

`def getsstkomegamodelconstants() ‑> List[float]`

`getsubversion`

`def getsubversion() ‑> int`

`gettime`

`def gettime() ‑> float`

This gets the value of the time variable *t*.

###### Example

```
>>> settime(1e-3)
>>> gettime()
0.001
```

###### See Also

`gettotalforce`

`def gettotalforce( *args, **kwargs ) ‑> List[float]`

This returns the components of the total magnetostatic/electrostatic force acting on a given region. In the axisymmetric case zero and components are returned and the component includes a factor to provide the force acting on the corresponding 3D shape. Units are " per unit depth" in 2D and in 3D and 2D axisymmetry.

###### Examples

**Example 1**: `gettotalforce(physreg:int, EorH:expression, epsilonormu:expression, extraintegrationorder:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> phi = field("h1")
>>> phi.setorder(vol, 2)
>>>
>>> mu0 = 4 * getpi() * 1e-7
>>> mu = parameter()
>>> mu.setvalue(vol, mu0)
>>> totalforce = gettotalforce(vol, -grad(phi), mu)
```

**Example 2**: `gettotalforce(physreg:int, meshdeform:expression, EorH:expression, epsilonormu:expression, extraintegrationorder:int=0)`

This is similar to the above function but the total force is computed on the mesh deformed by the field `u`

.

```
>>> ...
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> totalforce = gettotalforce(vol, u, -grad(phi), mu)
```

###### See Also

`getturbulentpropertiessstkomegamodel`

`def getturbulentpropertiessstkomegamodel( v: expression, rho: expression, viscosity: expression, walldistance: expression, kp: expression, omegap: expression, logomega: expression ) ‑> List[quanscient.expression]`

`getversion`

`def getversion() ‑> int`

`getversionname`

`def getversionname() ‑> str`

`getx`

`def getx() ‑> quanscient.field`

Returns the coordinate.

###### Example

An expression for distance can be calculated as follows:

```
>>> x = getx()
>>> y = gety()
>>> z = getz()
>>> d = sqrt(x*x+y*y+z*z)
```

###### See Also

`gety`

`def gety() ‑> quanscient.field`

Returns the coordinate.

###### Example

In CFD applications, a parabolic inlet velocity profile can be prescribed as follows:

```
>>> y = gety() # y coordinate
>>> U = 0.3 # maximum velocity
>>> h = 0.41 # height of the domain
>>> u = 4*U*y(h-y)/(h*h) # parabolic inlet velocity profile expression
```

###### See Also

`getz`

`def getz() ‑> quanscient.field`

Returns the coordinate.

###### Example

An expression for distance can be calculated as follows:

```
>>> x = getx()
>>> y = gety()
>>> z = getz()
>>> d = sqrt(x*x+y*y+z*z)
```

###### See Also

`grad`

`def grad( input: expression ) ‑> quanscient.expression`

For a scalar input expression, this is mathematically treated as the gradient of a scalar () and the output is a column vector with one entry per space derivative. For a vector input expression, this is mathematically treated as the gradient of a vector () and the output has one row per component of the input and one column per space derivative.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1);
>>> grad(v).write(vol, "gradv.vtk", 1)
```

###### See Also

`greenlagrangestrain`

`def greenlagrangestrain( input: expression ) ‑> quanscient.expression`

This defines the (**nonlinear**) Green-Lagrange strains in Voigt form . The input can either be the displacement field or its gradient.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> glstrain = greenlagrangestrain(u)
>>> glstrain.print()
```

`grouptimesteps`

`def grouptimesteps( *args, **kwargs ) ‑> None`

This writes a .pvd ParaView file to group a set of .vtu files that are time solutions at the time values provided in `timevals`

.

###### Examples

**Example 1**: `grouptimesteps(filename::str, filestogroup:List[str], timevals:List[double])`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> v.write(vol, "v1.vtu", 1)
>>> v.write(vol, "v2.vtu", 1)
>>> grouptimesteps("v.pvd", ["v1.vtu", "v2.vtu"], [0.0, 10.0])
```

**Example 2**: `grouptimesteps(filename::str, fielprefix:str, timevals:List[double])`

This is similar to the previous function except that the full list of file names to group does not have to be provided. The file names are constructed from the file prefix with an appended integer starting from 'firstint' by steps of 1. The filenames are ended with .vtu.

```
>>> ...
>>> grouptimesteps("v.pvd", "v", [0.0, 10.0])
```

`harm`

`def harm( harmnum: int, input: expression, numfftharms: int = -1 ) ‑> quanscient.expression`

`ifpositive`

`def ifpositive( condexpr: expression, trueexpr: expression, falseexpr: expression ) ‑> quanscient.expression`

This returns a conditional expression. The expression value is `trueexpr`

for all evaluation points where `condexpr`

is larger or equal to zero. Otherwise, its value is `falseexpr`

.

###### Parameters

** condexpr** :

`expression`

: Expression that specifies the conditional argument.** trueexpr** :

`expression`

: Expression value at points where `condexpr`

evaluates to True.** falseexpr** :

`expression`

: Expression value at points where `condexpr`

evaluates to False.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1
>>> x=field("x"); y=field("y"); z=field("z")
>>> expr = ifpositive(x+y, 1, -1) # At points, where x+y>=0, value is 1, otherwise -1.
>>> expr.write(vol, "ifpositive.vtk", 1)
```

###### See Also

`initialize`

`def initialize() ‑> None`

`initializelogs`

`def initializelogs( format: logformat ) ‑> None`

INTERNAL -- do not call in script

Initializes the structured logging system

###### Parameters

** format** :

`logformat`

: Format for the logs. One of `logformat.plain`

or `logformat.json`

###### Raises

None

`insertrank`

`def insertrank( name: str ) ‑> str`

`integral`

`def integral( *args, **kwargs ) ‑> quanscient.integration`

`inverse`

`def inverse( input: expression ) ‑> quanscient.expression`

This returns the inverse of a square matrix.

###### Example

```
>>> matexpr = array2x2(1,2, 3,4)
>>> invmat = inverse(matexpr)
>>> invmat.print()
Expression size is 2x2
@ row 0, col 0 :
-2
@ row 0, col 1 :
1
@ row 1, col 0 :
1.5
@ row 1, col 1 :
-0.5
>>>
>>> matexpr = expression(3,3, [1,2,3, 6,5,4, 8,9,7])
>>> invmat = inverse(matexpr)
```

###### See Also

`invjac`

`def invjac( *args, **kwargs ) ‑> quanscient.expression`

`isavailable`

`def isavailable() ‑> bool`

`isdefined`

`def isdefined( physreg: int ) ‑> bool`

This checks if a physical region is defined.

###### Parameters

** physreg** :

`int`

: Physical region identifier.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3; circle=4;
>>> isdefined(sur) # returns True as the physical region sur=2 is defined
True
>>>
>>> isdefined(5) # returns False as the mesh loaded has no physical region 5
>>> False
```

###### See Also

`isempty()`

, `isinside()`

, `istouching()`

`isempty`

`def isempty( physreg: int ) ‑> bool`

This checks if a physical region is empty.

###### Parameters

** physreg** :

`int`

: Physical region identifier.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3; circle=4;
>>> isempty(sur)
False
```

###### See Also

`isdefined()`

, `isinside()`

, `istouching()`

`isinside`

`def isinside( physregtocheck: int, physreg: int ) ‑> bool`

This checks if a physical region is fully included in another region.

###### Parameters

** physregtocheck** :

`int`

: Physical region to check.** physreg** :

`int`

: Physical region with which `physregtocheck`

is checked.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3; circle=4;
>>> isinside(sur, vol)
True
>>>
>>> isinside(vol, sur)
False
```

###### See Also

`isdefined()`

, `isempty()`

, `istouching()`

`istouching`

`def istouching( physregtocheck: int, physreg: int ) ‑> bool`

This checks if a region is touching another region.

###### Parameters

** physregtocheck** :

`int`

: Physical region to check.** physreg** :

`int`

: Physical region with which `physregtocheck`

is checked.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3; circle=4;
>>> istouching(sur, top)
True
```

###### See Also

`isdefined()`

, `isempty()`

, `isinside()`

`jac`

`def jac( *args, **kwargs ) ‑> quanscient.expression`

`linspace`

`def linspace( a: float, b: float, num: int ) ‑> List[float]`

This gives a vector of `num`

equally spaced values from `a`

to `b`

. The space between each values is calculated as:

###### Example

```
>>> vals = linspace(0.5, 2.5, 5)
>>> printvector(vals)
0.5 1 1.5 2 2.5
```

###### See Also

`loadshape`

`def loadshape( meshfile: str ) ‑> List[List[quanscient.shape]]`

This function loads a mesh file to shapes. The output holds a shape for every physical region of dimension *d* (0D, 1D, 2D, 3D) defined in the mesh file. The loaded shapes can be edited (extruded, deformed, ...) and grouped with other shapes to create a new mesh. Note that the usage of loaded shapes might be more limited than other shapes.

###### Example

```
>>> diskshapes = loadshape("disk.msh")
>>>
>>> # Add a thin slice on top of the disk (diskshapes[2][1] is the top face of the disk)
>>> thinslice = diskshapes[2][1].extrude(5, 0.02, 2)
>>>
>>> mymesh = mesh([diskshapes[2][0], diskshapes[2][1], diskshapes[3][0], thinslice])
>>> mymesh.write("editeddisk.msh")
```

`loadvector`

`def loadvector( filename: str, delimiter: str = ',', sizeincluded: bool = False ) ‑> List[float]`

This loads from a file a list/vector. The `delimiter`

specfies the character separating each entry in the file. The `sizeincluded`

must be set to `True`

if the first number in the file is list/vector length integer.

###### Parameters

** filename** :

`str`

: The name of the file containing the entries of a list/vector.** delimiter** :

`str`

, default=`','`

: Character separating each entry of the list/vector when writing to a file. E.g ',' or '\n'.** sizeincluded** :

`bool`

, default=`False`

: Must be set to `True`

if the first number in the file is list/vector length integer.###### Example

```
>>> v = [2.4, 3.14, -0.1]
>>> writevector("vecvals.txt", v, '\n', True)
>>> vloaded = loadvector("vecvals.txt", \n', True)
>>> vloaded
[2.4, 3.14, -0.1]
```

###### See Also

`log`

`def log( input: expression ) ‑> quanscient.expression`

This returns an expression that is the natural logarithm of the input expression.

###### Example

```
>>> expr = log(2);
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
0.693147
```

`logspace`

`def logspace( a: float, b: float, num: int, basis: float = 10.0 ) ‑> List[float]`

This returns the `basis`

to the power of each of the `num`

values in the linspace.

###### Example

```
>>> vals = logspace(1, 3, 3) # resulting linspace: 1, 2, 3
>>> printvector(vals) # 10^1, 10^2, 10^3
10 100 1000
```

###### See Also

`makeharmonic`

`def makeharmonic( *args, **kwargs ) ‑> quanscient.expression`

`max`

`def max( *args, **kwargs ) ‑> quanscient.expression`

This returns an expression whose value is the maximum of the two input arguments.

###### Parameters

** a** :

`expression/field/parameter`

: First of the two input arguments.** b** :

`expression/field/parameter`

: Second of the two input arguments.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x=field("x"); y=field("y"); z=field("z")
>>> max(x,y).write(vol, "max.pos", 1)
```

###### See Also

`meshsize`

`def meshsize( integrationorder: int ) ‑> quanscient.expression`

This returns an expression whose value is the length/area/volume for each 1D/2D/3D mesh element respectively. The value is constant on each mesh element.

###### Parameters

** integrationorder** :

`int`

: Determines the accuracy of mesh size calculated. Higher the number, the better the accuracy. Integration order can be negative.###### Raises

`RuntimeError`

: If integration order < .

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1; sur = 2; top = 3
>>> h = meshsize(0)
>>> h.write(top, "meshsize.vtk", 1)
```

`min`

`def min( *args, **kwargs ) ‑> quanscient.expression`

This returns an expression whose value is the minimum of the two input arguments.

###### Parameters

** a** :

`expression/field/parameter`

: First of the two input arguments.** b** :

`expression/field/parameter`

: Second of the two input arguments.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x=field("x"); y=field("y"); z=field("z")
>>> min(x,y).write(vol, "min.pos", 1)
```

###### See Also

`mod`

`def mod( input: expression, modval: float ) ‑> quanscient.expression`

This is a modulo function. This returns an expression equal to the remainder resulting from the division of `input`

by `modval`

.

###### Example

```
>>> expr = mod(10, 9)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
1
>>> expr = mod(99, 100)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
99
>>> expr = mod(2.55, 0.6)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
0.15
```

`moveharmonic`

`def moveharmonic( origharms: List[int], destharms: List[int], input: expression, numfftharms: int = -1 ) ‑> quanscient.expression`

This returns an expression equal to the input expression with a selected and moved harmonic content. Set a positive last argument to use an FFT to compute the harmonics of the input expression.

###### Example

```
>>> ...
>>> movedharm = moveharmonic([1,2], [5,3], 11+v)
>>> movedharm.write("vol", "movedharm.pos", 1)
```

`norm`

`def norm( arg0: expression ) ‑> quanscient.expression`

This gives the norm of an expression input.

###### Example

```
>>> myvector = array3x1(1,2,3)
>>> normL2 = norm(myvector)
>>> normL2.print()
Expression size is 1x1
@ row 0, col 0 :
3.74166
```

`normal`

`def normal( *args, **kwargs ) ‑> quanscient.expression`

This defines a normal vector with unit norm. If a physical region is provided as an argument then the normal points out of it. if no physical region is provided then the normal can be flipped depending on the element orientation in the mesh.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3 # physical regions
>>> normal(vol).write(sur, "normal.vtk", 1)
```

`on`

`def on( *args, **kwargs ) ‑> quanscient.expression`

This function allows to use fields, unknown *dof* fields or general expressions across physical regions with possibly non- matching meshes by evaluating the expression argument using a (x, y, z) coordinate interpolation. It makes it straightforward to setup the **mortar finite element method** to enforce general relations, such as field equality , at the interface of non-matching meshes. This can for example be achieved with a Lagrange multiplier such that

holds for any appropriate field , and . The example below illustrates the formulation terms needed to implement the Lagrange multiplier between interfaces and .

###### Examples

**Example 1**: `on(physreg:int, expr:expression, errorifnotfound:bool=True)`

`physreg`

is the physical region across which the expression `expr`

is evaluated. The `expr`

argument can be fields or *dof* fields or any general expressions.

```
>>> ...
>>> u1=field("h1xyz"); u2=field("h1xyz"); lambda=field("h1xyz")
>>> ...
>>> elasticity = formulation()
>>> ...
>>> elasticity += integral(gamma1, dof(lambda)*tf(u1) )
>>> elasticity += integral(gamma2, -on(gamma1, dof(lambda)) * tf(u2) )
>>> elasticity += integral(gamma1, (dof(u1) - on(gamma2, dof(u2)))*tf(lambda) )
```

When setting the flag `errorifnotfound=False`

, any point in without a relative in (and vice versa) does not contribute to the assembled matrix. The default value is True for which an error is raised if a point in is without a relative in (and vice versa).

The case where there is no unknown *dof* term in the expression argument is described below:

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>> x=field("x"); y=field("y"); z=field("z"); v=field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(top, dof(v)*tf(v) - on(vol, dz(z))*tf(v))
>>> projection.solve()
>>> v.write(top, "dzz.pos", 1)
```

With the *on* function, the (x, y, z) coordinates corresponding to each Gauss point of the integral are first calculated then the *dz(z)* expression is evaluated through interpolation at these (x, y, z) coordinates on region *vol*. With *on* here *dz(z)* is correctly evaluated as because the *z*-derivative calculation is performed on the volume region *vol*. Without the *on* operator the *z*-derivative would be wrongly calculated on the *top* face of the disk (a plane perpendicular to the *z*-axis).

If a requested interpolation point cannot be found (because it is outside of *physreg* or because the interpolation algorithm fails to converge, as can happen on curved 3D elements) then an error occurs unless `errorifnotfound`

is set to False. In the latter case, the value returned at any non-found coordinate is zero, without raising an error.

**Example 2**: `on(physreg:int, expr:coordshift, expr:expression, errorifnotfound:bool=True)`

This is similar to the previous example but here the (x, y, z) coordinates at which to interpolate the expression are shifted by (x+*compx(coordshift)*, y+*compy(coordshift)*, z+*compz(coordshift)*).

```
>>> ...
>>> projection += integral(top, dof(v)*tf(v) - on(vol, array3x1(2*x,2*y,2*z), dz(z))*tf(v))
```

`orpositive`

`def orpositive( exprs: List[expression] ) ‑> quanscient.expression`

This returns an expression whose value is 1 for all evaluation points where at least one input expression has a value larger or equal to zero. Otherwise, its value is -1.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x=field("x"); y=field("y"); z=field("z")
>>> expr = orpositive([x,y]) # At points, where x>=0 or y>=0, value is 1, otherwise -1.
>>> expr.write(vol, "orpositive.vtk", 1)
```

###### See Also

`periodicitycondition`

`def periodicitycondition( gamma1: int, gamma2: int, u: field, dat1: List[float], dat2: List[float], factor: float, lagmultorder: int = 0 ) ‑> List[quanscient.integration]`

This returns the formulation terms required to enforce on field a rotation or translation periodic condition between boundary region and (meshes can be non-conforming). A factor different than can be provided to scale the field on (use for antiperiodicity).

###### Example

```
>>> ...
>>> u = field("h1xyz")
>>> ...
>>> elasticity = formulation()
>>> ...
>>> # In case gamma2 is gamma1 rotated by (ax, ay, az) degrees around first the x, then y and then z-axis.
>>> # Rotation center is (cx, cy, cz)
>>> cx=0; cy=0; cz=0
>>> ax=0; ay=0; az=60
>>>
>>> elasticity += periodicitycondition(gamma1, gamma2, u, [cx,cy,cz], [ax,ay,az], 1.0)
>>>
>>> # In case gamma2 is gamma1 translated by a distance d in direction (nx, ny, nz)
>>> d=0.8
>>> nx=1; ny=0; nz=0
>>> elasticity += peridocitycondition(gamma1, gamma2, u, [nx,ny,nz], [d], 1.0)
```

The condition is based on a Lagrange multiplier of the same type and the same harmonic content as the field and . The mortar finite element method is used to link the unknown field on and so that there is no restriction on the mesh used for both regions.

More advanced periodic conditions can be implemented easily using `on()`

function.

###### See Also

`pow`

`def pow( base: expression, exponent: expression ) ‑> quanscient.expression`

This is a power function. This returns an expression equal to `base`

to the power `exponent`

:

###### Example

```
>>> expr = pow(2, 5)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
32
```

`predefinedacousticradiation`

`def predefinedacousticradiation( dofp: expression, tfp: expression, soundspeed: expression, neperattenuation: expression ) ‑> quanscient.expression`

This function defines the equation for the Sommerfeld acoustic radiation condition

which forces the outgoing pressure waves at infinity: a pressure field of the form

propagating in the direction perpendicular to the truncation boundary indeed satisfies the Sommerfeld radiation condition since

Zero artificial wave reflection at the truncation boundary happens only if it is perpendicular to the outgoing waves. In practical applications however the truncation boundary is not at an infinite distance from the acoustic source and the wave amplitude is not constant and thus, some level of artificial reflection cannot be avoided. To minimize this effect the truncation boundary should be placed as far as possible from the acoustic source (at least a few wavelengths away).

An acoustic attenuation value can be provided (in ) in case of harmonic problems. For convenience use the function `dbtoneper()`

to convert attenuation values to .

###### Example

```
>>> ...
>>> acoustics += integral(sur, predefinedacousticradiation(dof(p), tf(p), 340, dbtoner(500)))
```

`predefinedacousticstructureinteraction`

`def predefinedacousticstructureinteraction( dofp: expression, tfp: expression, dofu: expression, tfu: expression, soundspeed: expression, fluiddensity: expression, normal: expression, neperattenuation: expression, scaling: float = 1.0 ) ‑> quanscient.expression`

This function defines the bi-directional coupling for acoustic-structure interaction at the medium interface. Field is the acoustic pressure and field is the mechanical displacement. Calling the normal to the interface pointing out of the solid region, the bi-directional coupling is obtained by adding the fluid pressure loading to the structure

as well as linking the structure acceleration to the fluid pressure normal derivative using Newton's law:

To have a good matrix conditioning a scaling factor (e.g ) can be provided. In this case, the pressure source is divided by and, to compensate, the pressure force is multiplied by . This leads to the correct membrane deflection but the pressure field is divided by the scaling factor.

An acoustic attenuation value can be provided (in ) in case of harmonic problems. For convenience use the function `dbtoneper()`

to convert attenuation values to .

###### Example

```
>>> ...
>>> u = field("h1xy", [2,3])
>>> u.setorder(sur, 2)
>>> acoustics += integral(left, predefinedacousticstructureinteraction(dof(p), tf(p), dof(u), tf(u), 340, 1.2, array2x1(1,0), dbtoneper(500), 1e10)
```

`predefinedacousticwave`

`def predefinedacousticwave( *args, **kwargs ) ‑> List[Tuple[quanscient.expression, int, quanscient.preconditioner]]`

This function defines the equation for (linear) acoustic wave propagation:

An acoustic attenuation value can be provided (in ) in case of harmonic problems. For convenience use the function `dbtoneper()`

to convert attenuation values to .

###### Parameters

** dofp** :

`expression`

: dof of the acoustic pressure field.** tfp** :

`expression`

: test function of the acoustic pressure field.soundspeed: speed of sound in . neperattenuation: attenuation in . ** pmlterms** :

`List[expression]`

: List of pml terms.** precondtype** :

`str`

: Type of precondition. The default is an empty string "".###### Examples

**Example 1**: `predefinedacousticwave(dofp:expression, tfp:expression, soundspeed:expression, neperattenuation:expression, precondtype:str="")`

In the illustrative example below, a highly-attenuated acoustic wave propogation in a rectangular 2D box is simulated.

```
>>> sur=1; left=2; wall=3
>>> h=10e-3; l=50e-3
>>> q = shape("quadrangle", sur, [0,0,0, l,0,0, l,h,0, 0,h,0], [250,50,250,50])
>>> ll = q.getsons()[3]
>>> ll.setphysicalregion(left)
>>> lwall = shape("union", wall, [q.getsons()[0], q.getsons()[1], q.getsons()[2]])
>>>
>>> mymesh = mesh([q, ll, lwall])
>>>
>>> setfundamentalfrequency(40e3)
>>>
>>> # Wave propogation requires both the in-phase (2) and quadrature (3) harmonics:
>>> p=field("h1", [2,3]); y=field("y")
>>> p.setorder(sur, 2)
>>> # In-phase only pressure source
>>> p.harmonic(2).setconstraint(left, y*(h-y)/(h*h/4))
>>> p.harmonic(3).setconstraint(left, 0)
>>> p.setconstraint(wall)
>>>
>>> acoustics = formulation()
>>> acoustics += integral(sur, predefinedacousticwave(dof(p), tf(p), 340, dbtoneper(500)))
>>> acoustics.solve()
>>> p.write(sur, "p.vtu", 2)
>>>
>>> # Write 50 timesteps for a time visualization
>>> # p.write(sur, "p.vtu", 2, 50)
```

**Example 2**: `predefinedacousticwave(dofp:expression, tfp:expression, soundspeed:expression, neperattenuation:expression, pmlterms:List[expression], precondtype:str="")`

This is the same as the previous example but with PML boundary conditions.

```
>>> ...
>>> pmlterms = [detDr, detDi, Dr, Di, invDr, invDi]
>>> acoustics += integral(sur, predefinedacousticwave(dof(p), tf(p), 340, 0, pmlterms))
```

`predefinedadvectiondiffusion`

`def predefinedadvectiondiffusion( doff: expression, tff: expression, v: expression, alpha: expression, beta: expression, gamma: expression, isdivvzero: bool = True ) ‑> quanscient.expression`

This defines the weak formulation for the generalized advection-diffusion equation:

where is the scalar quantity of interest and is the velocity that the quantity is moving with. With and set to unit, the classical advection-diffusion equation with diffusivity tensor is obtained. Set `isdivvzero`

to True if is zero (for incompressible flows).

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> T = field("h1")
>>> v = field("h1xyz")
>>> T.setorder(vol, 1)
>>> v.setorder(vol, 1)
>>> advdiff = formulation()
>>> advdiff += integral(vol, predefinedadvectiondiffusion(dof(T), tf(T), v, 1e-4, 1.0, 1.0, True))
```

###### See Also

`predefinedaml`

`def predefinedaml( c: expression, shifted: bool = False ) ‑> List[quanscient.expression]`

`predefinedboxpml`

`def predefinedboxpml( pmlreg: int, innerreg: int, c: expression, shifted: bool = False ) ‑> List[quanscient.expression]`

This is a collective MPI operation and hence must be called by all the ranks. This function returns

where is the PML transformation matrix for a square box in a square box. A hyperbolic or shifted hyperbolic PML can be selected with the last argument.

###### Example

```
>>> ...
>>> k = 2*getpi()*freq/c # wave number
>>> Dterms = predefinedboxpml(pmlreg, innerreg, k)
```

`predefineddiffusion`

`def predefineddiffusion( doff: expression, tff: expression, alpha: expression, beta: expression ) ‑> quanscient.expression`

This defines the weak formulation for the generalized diffusion equation:

where is the scalar quantity of interest. With set to unit, the classical diffusion equation with diffusivity tensor is obtained.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; top=3
>>>
>>> # Temperature field in [K]:
>>> T = field("h1")
>>> T.setorder(vol, 1)
>>> T.setconstraint(top, 298)
>>>
>>> # Material properties:
>>> k = 237.0 # thermal conductivity of aluminium [W/mK]
>>> cp = 897.0 # heat capacity of aluminium [J/kgK]
>>> rho = 2700 # density of aluminium [Kg/m^3]
>>>
>>> heatequation = formulation()
>>> heatequation += integral(vol, predefineddiffusion(dof(T), tf(T), k, rho*cp))
```

###### See Also

`predefinedadvectiondiffusion()`

`predefinedelasticity`

`def predefinedelasticity( *args, **kwargs ) ‑> quanscient.expression`

This defines a classical linear elasticity formulation.

###### Examples

**Example 1**: `predefinedelasticity(dofu:expression, tfu:expression, Eyoung:expression, nupoisson:Expression, myoption:str="")`

This defines a classical linear **isotropic** elasticity formulation whose strong form is:

where,

- is the displacement field vector
- is the Cauchy stress tensor in Voigt notation
- is the strain tensor in Voigt notation
- is the order elasticity/stiffness tensor
- is the volumetric body force vector
- is the mass density

This is used when the material is isotropic. *u* is the mechanical displacement, *Eyoung* is the Young's modulus [Pa] and *nupoisson* is the Poisson's ratio. In 2D the option string must be either set to "planestrain" or "planestress" for a plane strain or plane stress assumption respectively.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e9, 0.3))
>>>
>>> # elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)); # 2300 is mass density
>>>
>>> # Atmospheric pressure load (volumetric force) on top face deformed by field u (might require a nonlinear iteration)
>>> elasticity += integral(top, u, -normal(vol)*1e5 * tf(u))
>>>
>>> elasticity.solve()
>>> u.write(top, "u.vtk", 2)
```

**Example 2**: `predefinedelasticity(dofu:expression, tfu:expression, elasticitymatrix:expression, myoption:str="")`

This extends the previous definition (Example 1) to general **anisotropic** materials. The elasticity matrix [Pa] must be provided such that it relates the stress and strain in Voigt notation.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> # It is enough to only provide the lower triangular part of the elasticity matrix as it is symmetric
>>> H = expression(6,6, [195e9, 36e9,195e9, 64e9,64e9,166e9, 0,0,0,80e9, 0,0,0,0,80e9, 0,0,0,0,0,51e9])
>>>
>>> elasticity = formulation()
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), H))
>>>
>>> # elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)); # 2300 is mass density
>>>
>>> # Atmospheric pressure load (volumetric force) on top face deformed by field u (might require a nonlinear iteration)
>>> elasticity += integral(top, u, -normal(vol)*1e5 * tf(u))
>>>
>>> elasticity.solve()
>>> u.write(top, "u.vtk", 2)
```

**Example 3**: `predefinedelasticity(dofu:expression, tfu:expression, u:field, Eyoung:expression, nupoisson:expression, prestress:expression, myoption:str="")`

This defines an **isotropic** linear elasticity formulation with **geometric nonlinearity** taken into account (full-Lagrangian formulation using the Green-Lagrange strain tensor). Problems with large displacements and rotations can be simulated with this equation but **strains must always remain small**. Buckling, snap-through and the likes or eigenvalues of prestressed structures can be simulated with the above equation in combination with a nonlinear iteration loop.

The prestress vector [Pa] must be provided in Voigt notation . Set the prestress expressiom to for no prestress.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>>
>>> ... # boundary conditions
>>>
>>> prestress = expression(6,1, [10e6,0,0,0,0,0])
>>>
>>> elasticity = formulation()
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), u, 150e9, 0.3, prestress))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)); # 2300 is mass density
>>>
>>> ... # nonlinear iteration loop
```

**Example 4**: `predefinedelasticity(dofu:expression, tfu:expression, u:field, elasticitymatrix:expression, prestress:expression, myoption:str="")`

This extends the previous definition (Example 3) to general **anisotropic** materials. The elasticity matrix [Pa] must be provided such that it relates the stress and strain in Voigt notation. Similarly, the prestress vector [Pa] must be also provided in Voigt notation .

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>>
>>> ... # boundary conditions
>>>
>>> # It is enough to only provide the lower triangular part of the elasticity matrix as it is symmetric
>>> H = expression(6,6, [195e9, 36e9,195e9, 64e9,64e9,166e9, 0,0,0,80e9, 0,0,0,0,80e9, 0,0,0,0,0,51e9])
>>>
>>> prestress = expression(6,1, [10e6,0,0,0,0,0])
>>>
>>> elasticity = formulation()
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), u, H, prestress))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)); # 2300 is mass density
>>>
>>> ... # nonlinear iteration loop
```

`predefinedelasticwave`

`def predefinedelasticwave( *args, **kwargs ) ‑> List[Tuple[quanscient.expression, int, quanscient.preconditioner]]`

`predefinedelectrostaticforce`

`def predefinedelectrostaticforce( input: expression, E: expression, epsilon: expression ) ‑> quanscient.expression`

This function defines the weak formulation term for electrostatic forces. The first argument is the mechanical displacement test function or its gradient, the second is the electric field expression and the third argument is the electric permittivity ( must be a scalar).

Let us call [] the electrostatic Maxwell stress tensor:

where is the electric permittivity, is the electric field and is the identity matrix. The electrostatic force density is N/m^3 so that the loading for a mechanical problem can be obtained by adding the following term:

where is the mechanical displacement. The term can be rewritten in the form that is provided by this function:

where is the infinitesimal strain tensor. This is identical to what is obtained using the virtual work principle. For details refer to *'Domain decomposition techniques for the nonlinear, steady state, finite element simulation of MEMS ultrasonic transducer arrays'*, page 40.

In this function, a region should be provided to the test function argument to compute the force only for the degrees of freedom associated to that specific region (in the example below with *tf(u, top)* the force only acts on the surface region 'top'. In any case, a **correct force calculation requires including** in the integration domain all elements in the region where the force acts and in the **element layer around it** (in the example below 'vol' includes all volume elements touching surface 'top').

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; top=3
>>> v=field("h1"); u=field("h1xyz")
>>> v.setorder(vol,1)
>>> u.setorder(vol,2)
>>>
>>> elasticity = formulation()
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e9, 0.3))
>>> elasticity += integral(vol, predefinedelectrostaticforce(tf(u,top), -grad(v), 8.854e-12))
```

###### See Also

`predefinedmagnetostaticforce()`

`predefinedelectrostatics`

`def predefinedelectrostatics( dofv: expression, tfv: expression, epsilon: expression, precondtype: str = '' ) ‑> Tuple[quanscient.expression, quanscient.preconditioner]`

`predefinedemwave`

`def predefinedemwave( *args, **kwargs ) ‑> Tuple[quanscient.expression, quanscient.preconditioner]`

This defines the equation for (linear) electromagnetic wave propagation:

where is the electric field, is the magnetic permeability, is the electric permiitivity and is the electric conductivity. The real and imaginary parts of each material property can be provided.

###### Parameters

** dofE** :

`expression`

: dof of the electric field.** tfE** :

`expression`

: test function of the electric field.** mur** :

`expression`

: real part of the magnetic permeability .** mui** :

`expression`

: imaginary part of the magnetic permeability .** epsr** :

`expression`

: real part of the electric permittivity .** epsi** :

`expression`

: imaginary part of the electric permittivity .** sigr** :

`expression`

: real part of the electric conductivity .** sigi** :

`expression`

: imaginary part of the electric conductivity .** pmlterms** :

`List[expression]`

: List of pml terms.** precondtype** :

`str`

: Type of precondition. The default is an empty string "".###### Examples

**Example 1**: `predefinedemwave(dofE:expression, tfE:expression, mur:expression, mui:expression, epsr:expression, epsi:expression, sigr:expression, sigi:expression, precondtype:str="")`

```
>>> ...
>>> E = field("hcurl", [2,3])
>>> ...
>>> maxwell += integral(sur, predefinedemwave(dof(E), tf(E), mu0,0, epsilon0,0, 0,0))
```

**Example 2**: `predefinedemwave(dofE:expression, tfE:expression, mur:expression, mui:expression, epsr:expression, epsi:expression, sigr:expression, sigi:expression, pmlterms:List[expression], precondtype:str="")`

This is the same as the previous example but with PML boundary conditions.

```
>>> ...
>>> pmlterms = [detDr, detDi, Dr, Di, invDr, invDi]
>>> maxwell += integral(sur, predefinedemwave(dof(E), tf(E), mu0,0, epsilon0,0, 0,0, pmlterms))
```

`predefinedlinearpoissonwalldistance`

`def predefinedlinearpoissonwalldistance( physreg: int, wallreg: int, interpolationorder: int = 1, relresddmtol: float = 1e-12, maxnumddmit: int = 500, relerrnltol: float = 1e-06, maxnumnlit: int = 100, project: bool = True, verbosity: int = 1 ) ‑> quanscient.expression`

This function calculates and returns distance values from the wall based on the linear Poisson wall distance equation:

where is the approximate wall distance function. The distance is calculated for the physical region *physreg* from the walls defined in *wallreg* argument.

More information can be found in *Computations of Wall Distances Based on Differential Equations Paul G. Tucker, Chris L. Rumsey, Philippe R. Spalart, Robert E. Bartels, and Robert T. Biedron AIAA Journal 2005 43:3, 539-549, https://doi.org/10.2514/1.8626 .*

The system is linear and hence a linear solver is used. After calculating the distance function , a better approximation of distance is obtained as follows:

This wall distance uses an above-zero limiter during calculation. Thus, to ensure that the distance value obtained is smooth, set the *project* argument to True. This will solve a projection of the distance values at the end and return a smooth solution.

**Excerpts from the above paper**: "The derivation of the above formula for assumes extensive (infinite) coordinates in the non-normal wall directions. Hence, the distance is only accurate close to walls. However, turbulence models only need 'd' accurate close to walls."

###### Example

```
>>> mymesh = mesh("2D_Flatplate_35x25.msh")
>>>
>>> # physical regions
>>> fluid=1, inlet=2, outlet=3, top=4, bot_upstream=5, plate=6
>>>
>>> linearpoisson_wd = predefinedlinearpoissonwalldistance(fluid, wall);
```

###### See Also

`predefinedreciprocalwalldistance()`

, `predefinednonlinearpoissonwalldistance()`

`predefinedmagnetostaticforce`

`def predefinedmagnetostaticforce( input: expression, H: expression, mu: expression ) ‑> quanscient.expression`

This function defines the weak formulation term for magnetostatic forces. The first argument ist the mechanical displacement test function or its gradient, the second is the magnetic field expression and the third argument is the magnetic permeability ( must be a scalar).

Let us call [] the magnetostatic Maxwell stress tensor:

where is the magnetic permeability, is the magnetic field and is the identity matrix. The magnetostatic force density is N/m^3 so that the loading for a mechanical problem can be obtained by adding the following term:

where is the mechanical displacement. The term can be rewritten in the form that is provided by this function:

where is the infinitesimal strain tensor. This is identical to what is obtained using the virtual work principle. For details refer to *'Domain decomposition techniques for the nonlinear, steady state, finite element simulation of MEMS ultrasonic transducer arrays'*, page 40.

In this function, a region should be provided to the test function argument to compute the force only for the degrees of freedom associated to that specific region (in the example below with *tf(u, top)* the force only acts on the surface region 'top'. In any case, a **correct force calculation requires including** in the integration domain all elements in the region where the force acts and in the **element layer around it** (in the example below 'vol' includes all volume elements touching surface 'top').

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; top=3
>>> phi=field("h1"); u=field("h1xyz")
>>> phi.setorder(vol,1)
>>> u.setorder(vol,2)
>>>
>>> elasticity = formulation()
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e9, 0.3))
>>> elasticity += integral(vol, predefinedelectrostaticforce(tf(u,top), -grad(phi), 4*getpi()*1e-7))
```

###### See Also

`predefinedelectrostaticforce()`

`predefinedmagnetostatics`

`def predefinedmagnetostatics( *args, **kwargs ) ‑> Tuple[quanscient.expression, quanscient.preconditioner]`

`predefinednavierstokes`

`def predefinednavierstokes( dofv: expression, tfv: expression, v: expression, dofp: expression, tfp: expression, mu: expression, rho: expression, dtrho: expression, gradrho: expression, includetimederivs: bool = False, isdensityconstant: bool = True, isviscosityconstant: bool = True, precondtype: str = '' ) ‑> Tuple[quanscient.expression, quanscient.preconditioner]`

This defines the weak formulation for the general (nonlinear) flow of Newtonian fluids:

where,

- is the fluid density
- is the dynamic viscosity of the fluid
- is the pressure
- is the flow velocity

The formulation is provided in a form leading to a quadratic (Newton) convergence when solved iteratively in a loop. This formulation is only valid to simulate laminar as well as turbulent flows. Using it to simulate turbulent flows leads to a so- called **DNS** method (direct numerical simulation). DNS does not require any turbulence model since it takes into account the whole range of spatial and temporal scales of the turbulence. Therefore, it requires a spatial and time refinement that for industrial applications typically exceeds the computing power of the most advanced supercomputers. As an alternative, RANS and LES method can be used for turbulent flow simulation.

The transition from a laminar to a turbulent flow is linked to a threshold value of the Reynolds number. For a flow in pipes typical Reynolds number below which the flow is laminar is about .

Arguments `dtrho`

and `gradrho`

are respectively the time derivative and the gradient of the density while `includetimederivs`

gives the option to include or not the time-derivative terms in the formulation. In case the density constant argument is set to True, the fluid is supposed incompressible and the Navier-Stokes equations are further simplified since the divergence of the velocity is zero. If the viscosity **in space** (it does not have to be constant in time) the constant viscosity argument can be set to True. By default, the density and viscosity are supposed constant and the time-derivative terms are not included. Please note that to simulate the Stokes flow the **LBB condition has to be satisfied**. This is achieved by using nodal (h1) type shape functions with an interpolation order of at least one higher for the velocity field than for the pressure field. Alternatively, an additional isotropic diffusive term or other stabilization techniques can be used to overcome the LBB limitation.

###### Example

```
>>> mymesh = mesh("microvalve.msh")
>>> fluid = 2 # physical region
>>>
>>> v=field("h1xy"); p=field("h1")
>>> v.setorder(fluid, 2)
>>> p.setorder(fluid, 1) # Satisfies the LBB condition
>>>
>>> laminar = formulation()
>>> laminar += integral(fluid, predefinednavierstokes(dof(v), tf(v), v, dof(p), tf(p), 8.9e-4, 1000, 0, 0))
```

###### See Also

`predefinednavierstokescrosswindstabilization`

`def predefinednavierstokescrosswindstabilization( dofv: expression, tfv: expression, v: expression, p: expression, diffusivity: expression, rho: expression, gradv: List[expression], vorder: int ) ‑> quanscient.expression`

`predefinednavierstokesstreamlinestabilization`

`def predefinednavierstokesstreamlinestabilization( dofv: expression, tfv: expression, v: expression, dofp: expression, tfp: expression, diffusivity: expression, rho: expression, gradv: List[expression], vorder: int, pspg: bool, lsic: bool ) ‑> quanscient.expression`

`predefinednonlinearpoissonwalldistance`

`def predefinednonlinearpoissonwalldistance( physreg: int, wallreg: int, poissonparameter: int, interpolationorder: int = 1, relresddmtol: float = 1e-12, maxnumddmit: int = 500, relerrnltol: float = 1e-06, maxnumnlit: int = 100, project: bool = True, verbosity: int = 1 ) ‑> quanscient.expression`

This function calculates and returns distance values from the wall based on a generic p-Posison wall distance equation:

where is the approximate wall distance function and is the Poisson parameter. The distance is calculated for the physical region *physreg* from the walls defined in *wallreg* argument. The Poisson parameter must be larger than or equal to 2. Higher the parameter better the distance field approximation. The term represents an apparent diffusion coefficient. When , the equation reduces to the linear Poisson wall distance. See `predefinedlinearpoissonwalldistance()`

.

More information can be found in *Wall-Distance Calculation for Turbulence Modelling, J. C. Bakker, Delft University of Technology. http://samofar.eu/wp-content/uploads/2018/10/Bakker_Jelle_BSc-thesis_2018.pdf .*

The above system is non-linear and hence an iterative Newton solver is used. After calculating the distance function , a better approximation of distance is obtained as follows:

This wall distance uses an above-zero limiter during calculation. Thus, to ensure that the distance value obtained is smooth, set the *project* argument to True. This will solve a projection of the distance values at the end and return a smooth solution.

###### Example

```
>>> mymesh = mesh("2D_Flatplate_35x25.msh")
>>>
>>> # physical regions
>>> fluid=1, inlet=2, outlet=3, top=4, bot_upstream=5, plate=6
>>>
>>> nonlinearpoisson_wd = predefinednonlinearpoissonwalldistance(fluid, wall, p=4);
```

###### See Also

`predefinedreciprocalwalldistance()`

, `predefinedlinearpoissonwalldistance()`

`predefinedreciprocalwalldistance`

`def predefinedreciprocalwalldistance( physreg: int, wallreg: int, reflength: float, smoothpar: float = 0.5, interpolationorder: int = 1, relresddmtol: float = 1e-12, maxnumddmit: int = 500, relerrnltol: float = 1e-06, maxnumnlit: int = 100, verbosity: int = 1 ) ‑> quanscient.expression`

This function calculates and returns distance values from the wall based on the reciprocal wall distance (G=1/d) equation:

The distance is calculated for the physical region *physreg* from the walls defined in *wallreg* argument.

More information can be found in *Fares, E., and W. Schröder. "A differential equation for approximate wall distance." International journal for numerical methods in fluids 39.8 (2002): 743-762.*

**Excerpts from the above paper**: "The desired smoothing is controlled by the value of the smoothing parameter . The larger the value means a stronger smoothing at sharp edges (but also a large deviation from exact distances). The value of the wall boundary condition influences the smoothing too.

Reference length is relevant in the definition of the initial and boundary conditions. For geometries with just one-sided wall, does not play a role- since the solution is the exact distance for all and . This formulation promises an enhancement of turbulence models at strongly curved surfaces."

Smaller and larger allow for better approximations of distances although at times it can be difficult to obtain convergence. In such cases, lowering the improves.

###### Example

```
>>> mymesh = mesh("2D_Flatplate_35x25.msh")
>>>
>>> # physical regions
>>> fluid=1, inlet=2, outlet=3, top=4, bot_upstream=5, plate=6
>>>
>>> reciprocal_wd = predefinedreciprocalwalldistance(fluid, wall, Lref=0.15);
```

###### See Also

`predefinedlinearpoissonwalldistance()`

, `predefinednonlinearpoissonwalldistance()`

`predefinedslipwall`

`def predefinedslipwall( physreg: int, dofv: expression, tfv: expression, dofp: expression, tfp: expression, diffusivity: expression, rho: expression, vorder: int ) ‑> quanscient.expression`

`predefinedstabilization`

`def predefinedstabilization( stabtype: str, delta: expression, f: expression, v: expression, diffusivity: expression, residual: expression ) ‑> quanscient.expression`

This function defines the isotropic, streamline anisotropic, crosswind, crosswind shockwave, streamline Petrov_Galerkin and streamline upwind Petrov-Galerkin stabilization methods for the advection-diffusion problem:

where is the scalar quantity of interest, is the velocity that the quantity is moving with and is the diffusivity tensor.

A characteristic number of advection-diffusion problems is the Peclet number:

where is the length of each mesh element. It quantifies the relative importance of advective and diffusive transport rates. When the Peclet number is large () the problem is dominated by faster advection (higher advection transport) and prone to spurious oscillations in the solution. Although lowering the Peclet number can be achieved by refining the mesh, a classical alternative is to add stabilization terms to the original equation. A proper choice of stabilization should remove oscillations while changing the original problem as little as possible. In the most simple method proposed (isotropic diffusion), the diffusivity is artificially increased to lower the Peclet number. The more advanced method proposed attempts to add artificial diffusion only where it is needed. In the crosswind shockwave, SPG and SUPG methods the residual of the advection-diffusion equation is used to quantify the local amount of diffusion to add. The terms provided by the proposed stabilization methods have the following form:

- isotropic diffusion:
- streamline anisotropic diffusion
- crosswind diffusion:
- crosswind shockwave:
- streamline Petrov-Galerkin (SPG):
- streamline upwind Petrov-Galerkin (SUPG):

where is the test function associated with field and .

To understand the effect of the crosswind diffusion one can notice that for a 2D flow in the direction only, tensor becomes

and the artificial diffusion is only added at places where has a component in the direction perpendicular to the flow.

**How to use the predefined stabilization methods:** Due to the large amount of artificial diffusion added by the isotropic diffusion method it should only be considered as a fallback option. In practice, a pair of one streamline and one crosswind method should be used with the smallest possible tuning factor . If the problem allows, SUPG should be preferred over SPG and crosswind shockwave should be preferred over the crosswind because the amount of diffusion added tends to be lower.

###### Examples

The different stabilization methods are defined for the following simulation setup:

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> c = field("h1")
>>> v = field("h1xyz")
>>> c.setorder(vol, 1)
>>> v.setorder(vol, 1)
>>>
>>> # Diffusivity alpha (can be a tensor)
>>> alpha = expression(0.001)
>>>
>>> advdiff = formulation()
>>> advdiff += integral(vol, predefinedadvectiondiffusion(dof(c), tf(c), v, alpha, 1.0, 1.0))
>>>
>>> # Tuning factor (stablization parameter)
>>> delta = 0.5
>>>
>>> # isotropic diffusion
>>> advdiff += integral(vol, predefinedstablization("iso", delta, c, v, 0.0, 0.0))
>>>
>>> # streamline anisotropic diffusion
>>> advdiff += integral(vol, predefinedstabilization("aniso", delta, c, v, 0.0, 0.0))
>>>
>>> # crosswind diffusion
>>> advdiff += integral(vol, predefinedstabilization("cw", delta, c, v, 0.0, 0.0))
```

The following residual-based stabilizations require the strong-form residual. Neglecting the second-order space-derivative still leads to a good residual approximation.

```
>>> # The flow is supposed incompressible, i.e div(v) = 0
>>> dofresidual = dt(dof(c)) + v*grad(dof(c)) # residual at current iteration
>>> residual = dt(c) + v*grad(c) # residual at previous iteration
>>>
>>> # crosswind shockwave diffusion: residual at previous iteration must be considered
>>> advdiff += integral(vol, predefinedstabilization("cws", delta, c, v, alpha, residual))
>>>
>>> # streamline Petrov-Galerkin diffusion
>>> advdiff += integral(vol, predefinedstabilization("spg", delta, c, v, alpha, dofresidual))
>>>
>>> # streamline upwind Petrov-Galerkin diffusion
>>> advdiff += integral(vol, predefinedstabilization("supg", delta, c, v, alpha, dofresidual))
```

`predefinedstabilizednavierstokes`

`def predefinedstabilizednavierstokes( dofv: expression, tfv: expression, v: expression, dofp: expression, tfp: expression, p: expression, mu: expression, rho: expression, dtrho: expression, gradrho: expression, includetimederivs: bool, isdensityconstant: bool, isviscosityconstant: bool, precondtype: str, supg: bool, pspg: bool, lsic: bool, cwnd: bool, vorder: int, gradv: List[expression] ) ‑> Tuple[quanscient.expression, quanscient.preconditioner]`

`predefinedstokes`

`def predefinedstokes( dofv: expression, tfv: expression, dofp: expression, tfp: expression, mu: expression, rho: expression, dtrho: expression, gradrho: expression, includetimederivs: bool = False, isdensityconstant: bool = True, isviscosityconstant: bool = True, precondtype: str = '' ) ‑> Tuple[quanscient.expression, quanscient.preconditioner]`

This defines the weak formulation for the Stokes (**creeping**) flow, a linear form of Navier-Stokes where the advective term is ignored as the inertial forces are smaller compared to the viscous forces:

where,

- is the fluid density
- is the dynamic viscosity of the fluid
- is the pressure
- is the flow velocity

This formulation is only valid to simulate the flow of Newtonian fluids (air, water, ...) with a very small Reynolds number ():

where is the characteristic length of the flow. Low flow velocities, high viscosities or small dimensions can lead to a valid Stokes flow approximation. Flows in microscale devices such as microvalves are also good candidates for Stokes flow simulations.

Arguments `dtrho`

and `gradrho`

are respectively the time derivative and the gradient of the density while `includetimederivs`

gives the option to include or not the time-derivative terms in the formulation. In case the density constant argument is set to True, the fluid is supposed incompressible and the Navier-Stokes equations are further simplified since the divergence of the velocity is zero. If the viscosity **in space** (it does not have to be constant in time) the constant viscosity argument can be set to True. By default, the density and viscosity are supposed constant and the time-derivative terms are not included. Please note that to simulate the Stokes flow the **LBB condition has to be satisfied**. This is achieved by using nodal (h1) type shape functions with an interpolation order of at least one higher for the velocity field than for the pressure field. Alternatively, an additional isotropic diffusive term or other stabilization techniques can be used to over the LBB limitation.

###### Example

```
>>> mymesh = mesh("microvalve.msh")
>>> fluid = 2 # physical region
>>>
>>> v=field("h1xy"); p=field("h1")
>>> v.setorder(fluid, 2)
>>> p.setorder(fluid, 1) # Satisfies the LBB condition
>>>
>>> stokesflow = formulation()
>>> stokesflow += integral(fluid, predefinedstokes(dof(v), tf(v), dof(p), tf(p), 8.9e-4, 1000, 0, 0))
```

###### See Also

`predefinedstreamlinestabilizationparameter`

`def predefinedstreamlinestabilizationparameter( v: expression, diffusivity: expression ) ‑> quanscient.expression`

`predefinedturbulencecrosswindstabilization`

`def predefinedturbulencecrosswindstabilization( v: expression, rho: expression, dofk: expression, tfk: expression, kp: expression, gradk: expression, productionk: expression, dissipationknodof: expression, diffusivityk: expression, korder: int, dofepsomega: expression, tfepsomega: expression, epsomegap: expression, gradepsomega: expression, productionepsomega: expression, dissipationepsomeganodof: expression, diffusivityepsomega: expression, epsomegaorder: int, fv1: expression, cdkomega: expression ) ‑> quanscient.expression`

`predefinedturbulencemodelsstkomega`

`def predefinedturbulencemodelsstkomega( v: expression, rho: expression, viscosity: expression, walldistance: expression, dofk: expression, tfk: expression, kp: expression, gradk: expression, korder: int, dofomega: expression, tfomega: expression, omegap: expression, logomega: expression, gradomega: expression, omegaorder: int, stabsupgkomega: bool, stabcwdkomega: bool ) ‑> quanscient.expression`

`predefinedturbulencestreamlinestabilization`

`def predefinedturbulencestreamlinestabilization( v: expression, rho: expression, dofk: expression, tfk: expression, gradk: expression, productionk: expression, dissipationk: expression, diffusivityk: expression, korder: int, dofepsomega: expression, tfepsomega: expression, gradepsomega: expression, productionepsomega: expression, dissipationepsomega: expression, diffusivityepsomega: expression, epsomegaorder: int, fv1: expression, cdkomega: expression ) ‑> quanscient.expression`

`printonrank`

`def printonrank( rank: int, toprint: str ) ‑> None`

`printsparameters`

`def printsparameters( Sparams: List[List[float]], polar: bool = True ) ‑> None`

`printtotalforce`

`def printtotalforce( *args, **kwargs ) ‑> List[float]`

This prints the total force and its unit. The total force value is returned.

###### Examples

**Example 1**: `printtotalforce(physreg:int, EorH:expression, epsilonormu:expression, extraintegrationorder:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> phi = field("h1")
>>> phi.setorder(vol, 2)
>>>
>>> mu0 = 4 * getpi() * 1e-7
>>> mu = parameter()
>>> mu.setvalue(vol, mu0)
>>> printtotalforce(vol, -grad(phi), mu)
```

**Example 2**: `printtotalforce(physreg:int, meshdeform:expression, EorH:expression, epsilonormu:expression, extraintegrationorder:int=0)`

This is similar to the above function but the total force is computed and returned on the mesh deformed by the field `u`

.

```
>>> ...
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> printtotalforce(vol, u, -grad(phi), mu)
```

###### See Also

`printvector`

`def printvector( *args, **kwargs ) ‑> None`

This prints of the vector as well as its values.

###### Parameters

** input** :

` List[double/int/bool]`

: A list of double/int/bool elements.###### Example

```
>>> v = [2.4, 3.14, -0.1]
>>> printvector(v)
Vector size is 3
2.4 3.14 -0.1
```

`printversion`

`def printversion() ‑> None`

`receive`

`def receive( *args, **kwargs ) ‑> None`

`rectangularport`

`def rectangularport( portphysreg: int, modetype: str, mmode: int, nmode: int, mu: expression, eps: expression, cxynodecoords: List[List[float]], integrationorder: int = 5 ) ‑> Tuple[quanscient.expression, quanscient.expression]`

`scatter`

`def scatter( *args, **kwargs ) ‑> None`

`scatterwrite`

`def scatterwrite( filename: str, xcoords: List[float], ycoords: List[float], zcoords: List[float], compxevals: List[float], compyevals: List[float] = [], compzevals: List[float] = [] ) ‑> None`

This writes to the output file a scalar or vector values at given coordinates. If atleast one of the `compyevals`

or `compzevals`

is not empty then the values saved are vectors and not scalars. For scalars, only `compxevals`

must be provided.

###### Raises

`RuntimeError`

: if length of all the list arguments are not identical.

###### Example

```
>>> Define coordinates of three points: (0.0,0.0,0.0), (1.0,1.0,0.0) and (2.0,2.0,0.0)
>>> coordx = [0.0, 1.0, 2.0]
>>> coordy = [0.0, 1.0, 2.0]
>>> coordz = [0.0, 0.0, 0.0]
>>>
>>> vals = [10, 20, 30]
>>> scatterwrite("scalarvalues.vtk", coordx, coordy, coordz, vals)
>>>
>>> xvals = [10, 20, 30]
>>> yvals = [40, 50, 60]
>>> scatterwrite("vectorvalues.vtk", coordx, coordy, coordz, xvals, yvals)
```

`selectall`

`def selectall() ‑> int`

Returns a new or an existing physical region that covers the entire domain.

###### Returns

`int`

: Physical region that covers the entire domain.

###### Example

```
>>> rega = 1; regb = 2
>>> qa = shape("quadrangle", rega, {0,0,0, 1,0,0, 1,1,0, 0,1,0}, {5,5,5,5})
>>> qb = shape("quadrangle", regb, {1,0,0, 2,0,0, 2,1,0, 1,1,0}, {5,5,5,5})
>>> mymesh = mesh([qa, qb])
>>> wholedomain = selectall()
>>> mymesh.write("mesh.msh")
```

###### See Also

`selectunion()`

, `selectintersection()`

, `selectnooverlap()`

, `shape`

, `mesh`

`selectintersection`

`def selectintersection( physregs: List[int], intersectdim: int ) ‑> int`

Returns a new or an existing physical region that is the intersection of physical regions passed. The `intersectdim`

argument determines the dimensional data from the intersection that would be utilized in subsequent operations such as setting constraints on a physical region or writing a field or expression to the physical region. For, `intersectdim=3`

: from the intersection, only volumes are used in subsequent operations. `intersectdim=2`

: from the intersection, only surfaces are used in subsequent operations. `intersectdim=1`

: from the intersection, only lines are is used in subsequent operations. `intersectdim=0`

: from the intersection, only points are is used in subsequent operations. This is useful in isolating only those dimensional data that might be necessary and ignoring the others arising from the intersection. Note that, the intersected region itself is not affected by `intersectdim`

. This argument only determines which dimensional data is utilized when the intersected physical region is used in calculations or operations.

###### Parameters

** physregs** :

`list`

: List of physical regions** intersectdim** :

`int`

: 0, 1, 2 or 3. This determines which dimensional data from the intersected physical region is returned.###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3 # physical regions
>>>
>>> # Influence of <code>intersectdim</code>
>>> intersectdim = 3
>>> intersectedreg = selectintersection([vol, vol], intersectdim)
>>> expression(1).write(intersectedreg, "out_3D.vtk", 1) # uses only volume from the intersectedreg
>>>
>>> intersectdim = 2
>>> intersectedreg = selectintersection([vol, vol], intersectdim)
>>> expression(1).write(intersectedreg, "out_2D.vtk", 1) # uses only surfaces from the intersectedreg
>>>
>>> intersectdim = 1
>>> intersectedreg = selectintersection([vol, vol], intersectdim)
>>> expression(1).write(intersectedreg, "out_1D.vtk", 1) # uses only lines from the intersectedreg
>>>
>>> intersectdim = 0
>>> intersectedreg = selectintersection([vol, vol], intersectdim)
>>> expression(1).write(intersectedreg, "out_0D.vtk", 1) # uses only points from the intersectedreg
```

###### See Also

`selectunion()`

, `selectall()`

, `selectnooverlap()`

`selectnooverlap`

`def selectnooverlap() ‑> int`

Returns a new or an existing physical region that covers no-overlap domain in case of overlap DDM and the entire domain otherwise.

###### See Also

`selectunion()`

, `selectintersection()`

, `selectall()`

`selectunion`

`def selectunion( physregs: List[int] ) ‑> int`

Returns a new or an existing physical region that is the union of physical regions passed.

###### Parameters

** physregs** :

`list`

: List of physical regions###### Example

```
>>> mymesh = mesh("disk.msh")
>>> sur=2; top=3 # physical regions
>>> surandtop = selectunion([2,3]) # unioned physical region
```

###### See Also

`selectintersection()`

, `selectall()`

, `selectnooverlap()`

`send`

`def send( *args, **kwargs ) ‑> None`

`setaxisymmetry`

`def setaxisymmetry() ‑> None`

This call should be placed at the very beginning of the code. After the call everything will be solved assuming axisymmetry (works for 2D meshes in the xy plane only). All equations should be written in their 3D form.

Please note that in order to correctly take into account the cylindrical coordinate change, the appropriate space derivative operators should be used. For example, the *gradient of a vector* operator required in the mechanical strain calculation to compute the gradient of mechanical displacement should not be defined manually using `dx()`

, `dy()`

and `dz()`

space derivatives. The `grad()`

operator should instead be called on the mechanical displacement vector.

###### Raises

`RuntimeError`

: If the function is called after loading a mesh.

###### Example

```
>>> setaxisymmetry()
```

`setdata`

`def setdata( invec: vec ) ‑> None`

`setfundamentalfrequency`

`def setfundamentalfrequency( f: float ) ‑> None`

This defines the fundamental frequency (in ) required for multi-harmonic problems.

###### Example

```
>>> setfundamentalfrequency(50)
```

`setmaxnumthreads`

`def setmaxnumthreads( mnt: int ) ‑> None`

Sets the maximum number of threads allowed.

###### Parameters

** mnt** :

`int`

: input value specifying the maximum number of threads allowed.###### Example

```
>>> setmaxnumthreads(2)
>>> mnt = getmaxnumthreads()
2
```

###### See Also

`setphysicalregionshift`

`def setphysicalregionshift( shiftamount: int ) ‑> None`

This shifts the physical region numbers by `shiftamount`

x (1 + physical region dimension) when loading a mesh.

###### Example

In the example the point/line/face/volume (0D/1D/2D/3D) physical region numbers will be shifted by 1000/2000/3000/4000 when a mesh is loaded.

```
>>> setphysicalregionshift(1000)
```

`settime`

`def settime( t: float ) ‑> None`

This sets the time variable *t*.

###### Example

```
>>> settime(1e-3)
```

###### See Also

`settimederivative`

`def settimederivative( *args, **kwargs ) ‑> None`

This allows us to provide the time derivative vectors to the universe.

###### Examples

**Example 1*: `settimederivative(dtx:vec)`

This provides the first-order time derivate vector to the universe and removes the second-order time derivative vector.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> v.setconstraint(sur)
>>>
>>> poisson = formulation()
>>> poisson += integral(vol, grad(dof(v))*grad(tf(v)))
>>> solt1=vec(poisson); solt2=vec(poisson)
>>> dtsol = 0.1 * (solt2 - solt1)
>>> settimederivative(dtsol)
>>> dt(v).write(vol, "dtv.pos", 1)
```

**Example 2*: `settimederivative(dtx:vec, dtdtx:vec)`

This provides the first and second-order time derivative vectors to the universe.

```
>>> ...
>>> settimederivative(dtsol, vec(poisson))
>>> dtdt(v).write(vol, "dtdtv.pos", 1)
```

`sin`

`def sin( input: expression ) ‑> quanscient.expression`

This returns an expression that is the of input. The input expression is in `radians`

.

###### Example

```
>>> expr = sin(getpi()/2)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
1
>>>
>>> expr = sin(2)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
0.909297
```

###### See Also

`cos()`

, `tan()`

, `asin()`

, `acos()`

, `atan()`

`sn`

`def sn( n: float ) ‑> quanscient.expression`

`solve`

`def solve( *args, **kwargs ) ‑> quanscient.vec`

This function solves an algebraic problem. This function can solve both nonlinear and linear systems. Nonlinear problems are solved with a fixed-point iteration. Linear problems can be solved with both direct and iterative solvers. Depending on the algebraic problem and the solver needed, the overloaded solve function can be called with different numbers and types of arguments as shown in the examples below.

###### Examples

**Example 1**: `solve(A:mat, b:vec, soltype:str="lu", diagscaling:bool=False) -> quanscient.vec`

This solves a linear algebraic problem with a (possibly reused) LU or Cholesky factorization by calling the mumps parallel direct solver via PETSC. The matrix can be diagonally scaled for improved conditioning (especially in multiphysics problems). In the case of diagonal scaling the matrix is modified after the call.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>>
>>> v=field("h1"); x=field("x")
>>> v.setorder(vol, 1)
>>>
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - x*tf(v)) # linear system
>>>
>>> projection.generate()
>>> sol = solve(projection.A(), projection.b()) # mumps direct solver
```

**Example 2**: `solve(A:mat, b:List[vec], soltype:str="lu") -> List[vec]`

This is same as the previous example but allows us to efficiently solve for multiple right-hand side vectors .

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>>
>>> v=field("h1"); x=field("x")
>>> v.setorder(vol, 1)
>>>
>>> projection = formulation()
>>> projection += integral(vol, (dof(v) - x)*tf(v)) # linear system
>>>
>>> projection.generate()
>>> b0 = projection.b()
>>> b1 = 2 * b0
>>> b2 = 3 * b0
>>>
>>> sol = solve(projection.A(), [b0, b1, b2]) # mumps direct solver
>>>
>>> # solution for b0
>>> v.setdata(vol, sol[0])
>>> v.write(vol, "sol0.pos", 1)
>>>
>>> # solution for b1
>>> v.setdata(vol, sol[1])
>>> v.write(vol, "sol1.pos", 1)
>>>
>>> # solution for b2
>>> v.setdata(vol, sol[2])
>>> v.write(vol, "sol2.pos", 1)
```

**Example 3**: `solve(A:mat, b:vec, sol:vec, relrestol:double, maxnumit:int, soltype:str="bicgstab", precondtype:str="sor", verbosity:int=1, diagscaling:bool=False)`

This solves a linear algebraic problem with a preconditioned *(ilu, sor, gamg)* iterative solver *(gmres or bicgstab)*. Vector *sol* is used as an initial guess and holds the solution at the end of the call. Values *relrestol* and *maxnumit* give the relative residual tolerance and the maximum number of iterations to be performed by the iterative solver. The matrix can be diagonally scaled for improved conditioning (especially in multiphysics problems). In the case of diagonal scaling the matrix is modified after the call.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>>
>>> v=field("h1"); x=field("x")
>>> v.setorder(vol, 1)
>>>
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - x*tf(v)) # linear system
>>>
>>> projection.generate()
>>>
>>> initsol = vec(projection)
>>> solve(projection.A(), projection.b(), initsol, 1e-8, 200) # iterative solver
>>> v.setdata(vol, initsol)
>>> print(f"Max solution value is {v.max(vol, 5)[0]}")
Max solution value is 1.013415
```

**Example 4**: `solve(nltol:double, maxnumnlit:int, realxvalue:double, formuls:List[formulation], verbosity:int=1) -> int`

This solves a nonlinear problem with a fixed point iteration. A relaxation value can be provided with `relaxvalue`

argument. Usually, a relaxation value less than (under-relaxation) is used to avoid divergence of a solution.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>>
>>> v=field("h1");
>>> v.setorder(vol, 1)
>>> v.setconstraint(sur, 0)
>>>
>>> electrostatics = formulation()
>>> electrostatics += integral(vol, grad(dof(v))*grad(tf(v)) + v*tf(v) )
>>>
>>> solve(1e-4, 100, 1.0, [electrostatics])
```

###### See Also

`sqrt`

`def sqrt( input: expression ) ‑> quanscient.expression`

This returns an expression that is the square root of the input expression.

###### Example

```
>>> expr = sqrt(2)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
1.41421
```

`strain`

`def strain( input: expression ) ‑> quanscient.expression`

This defines the (**linear**) engineering strains in Voigt form . The input can either be the displacement field or its gradient.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> engstrain = strain(u)
>>> engstrain.print()
```

`sum`

`def sum( *args, **kwargs ) ‑> None`

`t`

`def t() ‑> quanscient.expression`

This gives the time variable in the form of an expression. The evaluation gives a value equal to `gettime()`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setconstraint(vol, sin(2*t()))
```

###### See Also

`tan`

`def tan( input: expression ) ‑> quanscient.expression`

This returns an expression that is the of input. The input expression is in `radians`

.

###### Example

```
>>> expr = tan(getpi()/4)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
1
>>>
>>> expr = tan(1)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 :
1.55741
```

###### See Also

`sin()`

, `cos()`

, `asin()`

, `acos()`

, `atan()`

`tangent`

`def tangent() ‑> quanscient.expression`

This defines a tangent vector with unit norm.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> top=3
>>> tangent().write(top, "tangent.vtk", 1)
```

`tf`

`def tf( *args, **kwargs ) ‑> quanscient.expression`

This declares a test function field. The test functions are defined only on the region `physreg`

which when not provided is set to the element integration region.

###### Examples

**Example 1**: `tf(input:expression)`

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
```

**Example 2**: `tf(input:expression, physreg:int)`

```
>>> ...
>>> projection += integral(vol, dof(v)*tf(v, vol) - 2*tf(v))
```

###### See Also

`trace`

`def trace( a: expression ) ‑> quanscient.expression`

This computes the trace of a square matrix expression. The returned expression is a scalar.

###### Example

```
>>> a = array2x2(1,2, 3,4)
>>> tracea = trace(a)
>>> tracea.print()
Expression size is 1x1
@ row 0, col 0 :
5
```

`transpose`

`def transpose( input: expression ) ‑> quanscient.expression`

This returns an expression that is the transpose of a vector or matrix expression.

###### Example

```
>>> colvec = array3x1(1,2,3)
>>> rowvec = transpose(colvec)
>>> rowvec.print()
Expression size is 1x3
@ row 0, col 0 :
1
@ row 0, col 1 :
2
@ row 0, col 2 :
3
>>> matexpr = expression(3,3, [1,2,3, 4,5,6, 7,8,9])
>>> transposed = transpose(matexpr)
```

`vonmises`

`def vonmises( stress: expression ) ‑> quanscient.expression`

This returns the von Mises stress expression corresponding to the 3D stress tensor provided as argument. The stress tensor should be provided in Voigt form .

For 2D plane stress problems all related components of the stress tensor are . For plane strain problems do not forget the term $\sigma_{zz} = \nu \cdot (\sigma_{xx} + \sigma_{yy}).

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> u.setconstraint(sur)
>>>
>>> # Material properties
>>> E = 150e9
>>> nu = 0.3
>>>
>>> # Elasticity matrix for isotropic materials
>>> H = expression(6,6, [1-nu,nu,nu,0,0,0, nu,1-nu,nu,0,0,0, nu,nu,1-nu,0,0,0, 0,0,0,0.5*(1-2*nu),0,0, 0,0,0,0,0.5*(1-2*nu),0, 0,0,0,0,0,0.5*(1-2*nu)])
>>> H = H * E/((1+nu)*(1-2*nu))
>>>
>>> elasticity = formulation()
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), H))
>>>
>>> # Atmospheric pressure load (volumetric force) on top face deformed by field u (might require a nonlinear iteration)
>>> elasticity += integral(top, u, -normal(vol)*1e5 * tf(u))
>>>
>>> elasticity.solve()
>>>
>>> cauchystress = H * strain(u)
>>> vonmisesstress = vonmises(cauchystress)
>>>
>>> vonmisesstress.write(vol, "vonmises.vtk", 1)
>>> maxvonmises = vonmisesstress.max(vol, 5)[0]
>>> maxvonmises
```

`wallfunction`

`def wallfunction( fluidphysreg: int, flds: List[field], exprs: List[expression], wftype: str ) ‑> List[quanscient.expression]`

`writecsvfile`

`def writecsvfile( filename: str, header: str, values: List[str] ) ‑> None`

`writeshapefunctions`

`def writeshapefunctions( filename: str, sftypename: str, elementtypenumber: int, maxorder: int, allorientations: bool = False ) ‑> None`

This writes to file all shape functions up to a requested order. It is a convenient tool to visualize the shape functions.

###### Example

```
>>> writeshapefunctions("sf.pos", "hcurl", 2, 2)
```

`writevector`

`def writevector( filename: str, towrite: List[float], delimiter: str = ',', writesize: bool = False ) ‑> None`

This writes all the entries of a list/vector to the file `filename`

with the requested delimiter. The size of the vector can also be written at the beginning of a file if requested.

###### Parameters

** filename** :

`str`

: The name of the file to which the entries of a list/vector are written.** towrite** :

`List`

: The list/vector containing the entries that needs to be written to a file.** delimiter** :

`str`

, default=`','`

: Character separating each entry of the list/vector when writing to a file. E.g ',' or '\n'.** writesize** :

`bool`

, default=`False`

: If True, the size of the list/vector is written at the beginning of the file.###### Example

```
>>> v = [2.4,3.14,-0.1]
>>> writevector("vecvals.txt", v)
2.4,3.14,-0.1
>>>
>>> writevector("vecvals.txt", v, ' ')
2.4 3.14 -0.1
>>>
>>> writevector("vecvals.txt", v, '\n', True)
3
2.4
3.14
-0.1
```

###### See Also

`zienkiewiczzhu`

`def zienkiewiczzhu( input: expression ) ‑> quanscient.expression`

This defines a Zienkiewicz-Zhu type error indicator for the argument expression. The value of the returned expression is constant over each element. It equals the maximum of the argument expression value jump between that element and any neighbour. In the below example, the *zienkiewiczzhu(grad(v))* expression quantifies the discontinuity of the field derivative. For a non-scalar arguments the function is applied to each entry and the norm is returned.

###### Example

```
>>> sur = 1
>>> q = shape("quadrangle", sur, [0,0,0, 5,0,0, 5,1,0, 0,1,0], [10,3,10,3])
>>> mymesh = mesh([q])
>>>
>>> v=field("h1"); x=field("x"); y=field("y")
>>> v.setorder(sur, 1)
>>>
>>> criterion = zienkiewiczzhu(grad(v))
>>> # Target max criterion is 0.05
>>> maxcrit = ifpositive(criterion-0.05, 1, 0)
>>> mymesh.setadaptivity(maxcrit, 0, 3)
>>>
>>> for i in range(10):
... fct = sin(3*x)/(x*x+1)*sin(getpi()*y)
... v.setvalue(sur, fct)
... v.write(sur, f"v{100+i}.vtk", 1)
... fieldorder(v).write(sur, f"fieldorder{100+i}.vtk", 1)
... criterion.write(sur, f"zienkiewiczzhu{100+i}.vtk", 1)
... relL2err = sqrt(pow(v-fct,2)).integrate(sur,5) / pow(fct,2).integrate(sur,5)
... adapt(2)
```

## Classes

`densemat`

`class densemat( **kwargs )`

The `densemat`

object stores a row-major array of doubles that corresponds to a dense matrix. For storing an array of integers, see `indexmat`

object.

#### Examples

There are several ways of instantiating a `densemat`

object. They are listed below:

**Example 1**: `densemat(numberofrows:int, numberofcolumns:int)`

The following creates a matrix with 2 rows and 3 columns. The entries may be undefined.

```
>>> B = densemat(2,3)
```

**Example 2**: `densemat(numberofrows:int, numberofcolumns:int, initvalue:double)`

This creates a matrix with 2 rows and 3 columns. All entries are assigned the value `initvalue`

.

```
>>> B = densemat(2,3, 12)
>>> B.print()
Matrix size is 2x3
12 12 12
12 12 12
```

**Example 3**: `densemat(numberofrows:int, numberofcolumns:int, valvec:List[double])`

This creates a matrix with 2 rows and 3 columns. The entries are assigned the values of `valvec`

. The length of `valvec`

is expected to be equal to the total count of entries in the matrix. So for creating a matrix of size , length of `valvec`

must be 6.

```
>>> B = densemat(2,3, [1,2,3,4,5,6])
>>> B.print()
Matrix size is 2x3
1 2 3
4 5 6
```

**Example 4**: `densemat(numberofrows:int, numberofcolumns:int, init:double, step:double)`

This creates a matrix with 2 rows and 3 columns. The first entry is assigned the value `init`

and the consecutive entries are assigned values that increase by steps of `step`

.

```
>>> B = densemat(2,3, 0, 1)
>>> B.print()
Matrix size is 2x3
0 1 2
3 4 5
```

**Example 5**: `densemat(input:List[densemat])`

This creates a matrix that is the vertical concatenation of `input`

matrices. Since, the concatenation occurs vertically, the number of columns in all the input matrices must match.

```
>>> A = densemat(2,3, 0)
>>> B = densemat(1,3, 2)
>>> AB = densemat([A,B])
>>> AB.print()
Matrix size is 3x3
0 0 0
0 0 0
2 2 2
```

#### Methods

`count`

`def count( self ) ‑> int`

This counts and returns the total number of entries in the dense matrix.

###### Example

```
>>> B = densemat(2,3)
>>> B.count()
6
```

`countcolumns`

`def countcolumns( self ) ‑> int`

This counts and returns the number of columns in the dense matrix.

###### Example

```
>>> B = densemat(2,3)
>>> B.countcolumns()
3
```

`countrows`

`def countrows( self ) ‑> int`

This counts and returns the number of rows in the dense matrix.

###### Example

```
>>> B = densemat(2,3)
>>> B.countrows()
2
```

`print`

`def print( self ) ‑> None`

This prints the entries of the dense matrix.

###### Example

```
>>> B = densemat(2,3, 0,1)
>>> B.print()
Matrix size is 2x3
0 1 2
3 4 5
```

`printsize`

`def printsize( self ) ‑> None`

This prints the size of the dense matrix.

###### Example

```
>>> B = densemat(2,3)
>>> B.printsize()
Matrix size is 2x3
```

`eigenvalue`

`class eigenvalue( **kwargs )`

The eigenvalue object allows us to solve classical, generalized and polynomial eigenvalue problems. The computation is done by SLEPc, a scalable library for eigenvalue problem computation.

#### Examples

**Example 1:** `eigenvalue(A: mat)`

This defines a classical eigenvalue problem

**Example2.1:** `eigenvalue(A:mat, B:mat)`

This defines a generalized eigenvalue problem . Undamped mechanical resonance modes and resonance frequencies can be calculated with this since an **undamped mechanical problem** can be written in the form

which for a harmonic excitation at angular frequency can be rewritten as

so that the generalized eigen value is equal to . To visualize the resonance frequencies of all calculated undamped modes the method `eigenvalue.printeigenfrequencies()`

can be called.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Direct eigen solver (works only for non-DDM simulation)
>>> elasticity.generate()
>>> eig = eigenvalue(elasticity.K(), elasticity.M())
>>> eig.compute(5, 0)
>>>
>>> eig.printeigenfrequencies()
>>>
>>> eigenvalue_real = eig.geteigenvaluerealpart()
>>> eigenvalue_imag = eig.geteigenvalueimaginarypart()
>>>
>>> eigenvector_real = eig.geteigenvectorrealpart()
>>> eigenvector_imag = eig.geteigenvectorimaginarypart()
```

**Example2.2:** `eigenvalue(form: formulation)`

This is same as the example 2.1 but the eigen solution is obtained iteratively. This can be used for both non-DDM and DDM simulation case setup.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Iterative eigen solver
>>> eig = eigenvalue(elasticity)
>>> eig.settolerance(1e-6, 1000)
>>> eig.allcompute(1e-6, 1000, 5, 0)
>>>
>>> eig.printeigenfrequencies()
>>>
>>> eigenvalue_real = eig.geteigenvaluerealpart()
>>> eigenvalue_imag = eig.geteigenvalueimaginarypart()
>>>
>>> eigenvector_real = eig.geteigenvectorrealpart()
>>> eigenvector_imag = eig.geteigenvectorimaginarypart()
```

**Example 3:** `eigenvalue(K: mat, C:mat, M:mat)`

This defined a second-order polynomial eigenvalue problem which allows getting the resonance modes and resonance frequencies for **damped mechanical problems**. The input arguments are respectively the mechanical stiffness, damping matrix and mass matrix. A second-order polynomial eigenvalue problem attempts to find a solution of the form

which corresponds to a damped oscillation at frequency with a damping ratio

In the case of proportional damping (if and only if is symmetric) the oscillation of the undamped system is at . The undamped oscillation frequency can then be calculated as

To visualize all relevant resonance information for the computed eigenvalues the method `eigenvalue.printeigenfrequencies()`

can be called.

**Example 4:** `eigenvalue(inmats: List[mat])`

This defines an arbitraray order polynomial eigenvalue problem.

#### Methods

`allcompute`

`def allcompute( self, relrestol: float, maxnumit: int, numeigenvaluestocompute: int, targeteigenvaluemagnitude: float = 0.0, verbosity: int = 1 ) ‑> None`

This is an iterative eigen solver that attempts to compute the first `numeigenvaluestocompute`

eigenvalues whose magnitude is closest to a target magnitude ( by default). There is no guarantee that SLEPc will return the exact number of eigenvalues requested. This can be used on both non-DDM and DDM simulation setup.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Iterative eigen solver
>>> eig = eigenvalue(elasticity)
>>> eig.settolerance(1e-6, 1000)
>>> eig.allcompute(1e-6, 1000, 5, 0)
```

###### See Also

`compute`

`def compute( self, numeigenvaluestocompute: int, targeteigenvaluemagnitude: float = 0.0 ) ‑> None`

This is a direct eigen solver that attempts to compute the first `numeigenvaluestocompute`

eigenvalues whose magnitude is closest to a target magnitude ( by default). There is no guarantee that SLEPc will return the exact number of eigenvalues requested. This should be used only for non-DDM simulations, i.e when using single node count.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Direct eigen solver (works only for non-DDM simulation)
>>> elasticity.generate()
>>> eig = eigenvalue(elasticity.K(), elasticity.M())
>>> eig.compute(5, 0)
```

###### See Also

`count`

`def count( self ) ‑> int`

This gets the number of eigenvalues found by SLEPc.

`geteigenvalueimaginarypart`

`def geteigenvalueimaginarypart( self ) ‑> List[float]`

This gets the imaginary part of all eigenvalues found.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Iterative eigen solver
>>> eig = eigenvalue(elasticity)
>>> eig.settolerance(1e-6, 1000)
>>> eig.allcompute(1e-6, 1000, 5, 0)
>>>
>>> eig.printeigenfrequencies()
>>>
>>> eigenvalue_real = eig.geteigenvaluerealpart()
>>> eigenvalue_imag = eig.geteigenvalueimaginarypart()
>>>
>>> eigenvector_real = eig.geteigenvectorrealpart()
>>> eigenvector_imag = eig.geteigenvectorimaginarypart()
```

`geteigenvaluerealpart`

`def geteigenvaluerealpart( self ) ‑> List[float]`

This gets the real part of all eigenvalues found.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Iterative eigen solver
>>> eig = eigenvalue(elasticity)
>>> eig.settolerance(1e-6, 1000)
>>> eig.allcompute(1e-6, 1000, 5, 0)
>>>
>>> eig.printeigenfrequencies()
>>>
>>> eigenvalue_real = eig.geteigenvaluerealpart()
>>> eigenvalue_imag = eig.geteigenvalueimaginarypart()
>>>
>>> eigenvector_real = eig.geteigenvectorrealpart()
>>> eigenvector_imag = eig.geteigenvectorimaginarypart()
```

`geteigenvectorimaginarypart`

`def geteigenvectorimaginarypart( self ) ‑> List[quanscient.vec]`

This gets the imaginary part of all eigenvectors found.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Iterative eigen solver
>>> eig = eigenvalue(elasticity)
>>> eig.settolerance(1e-6, 1000)
>>> eig.allcompute(1e-6, 1000, 5, 0)
>>>
>>> eig.printeigenfrequencies()
>>>
>>> eigenvalue_real = eig.geteigenvaluerealpart()
>>> eigenvalue_imag = eig.geteigenvalueimaginarypart()
>>>
>>> eigenvector_real = eig.geteigenvectorrealpart()
>>> eigenvector_imag = eig.geteigenvectorimaginarypart()
```

`geteigenvectorrealpart`

`def geteigenvectorrealpart( self ) ‑> List[quanscient.vec]`

This gets the real part of all eigenvectors found.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Iterative eigen solver
>>> eig = eigenvalue(elasticity)
>>> eig.settolerance(1e-6, 1000)
>>> eig.allcompute(1e-6, 1000, 5, 0)
>>>
>>> eig.printeigenfrequencies()
>>>
>>> eigenvalue_real = eig.geteigenvaluerealpart()
>>> eigenvalue_imag = eig.geteigenvalueimaginarypart()
>>>
>>> eigenvector_real = eig.geteigenvectorrealpart()
>>> eigenvector_imag = eig.geteigenvectorimaginarypart()
```

`printeigenfrequencies`

`def printeigenfrequencies( self ) ‑> None`

This method provides a convenient way to print the eigenfrequencies associated with all eigenvalues calculated for a mechanical resonance problem. In case a generalized eigenvalue problem is used to calculate the resonance modes of an undamped mechanical problem, this method displays the resonance frequency of each calculated resonance mode. In case a second-order polynomial eigenvalue problem is used to calculate the resonance modes of a damped mechanical problem this function displays not only the **damped resonance frequency** of each resonance mode but also the **undamped resonance frequency** (only valid in case of proportional damping), the **bandwidth**, the **damping ratio** and the **quality factor**.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Direct eigen solver (works only for non-DDM simulation)
>>> elasticity.generate()
>>> eig = eigenvalue(elasticity.K(), elasticity.M())
>>> eig.compute(5, 0)
>>>
>>> eig.printeigenfrequencies()
```

`printeigenvalues`

`def printeigenvalues( self ) ‑> None`

This prints the eigenvalues found.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Direct eigen solver (works only for non-DDM simulation)
>>> elasticity.generate()
>>> eig = eigenvalue(elasticity.K(), elasticity.M())
>>> eig.compute(5, 0)
>>>
>>> eig.printeigenvalues()
```

`settolerance`

`def settolerance( self, reltol: float, maxnumits: int ) ‑> None`

This sets the tolerance and maximum number of iterations for the iterative eigen solver. The `settolerance`

should be called only if `eigenvalue.allcompute()`

is used to solve the eigen problem.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>>
>>> u = field("h1xyz")
>>> u.setorder(vol, 2)
>>> u.setconstraint(sur)
>>>
>>> elasticity = formulation()
>>>
>>> elasticity += integral(vol, predefinedelasticity(dof(u), tf(u), 150e-, 0.3))
>>> elasticity += integral(vol, -2300*dtdt(dof(u)) * tf(u)) # 2300 is mass density
>>>
>>> # Iterative eigen solver
>>> eig = eigenvalue(elasticity)
>>> eig.settolerance(1e-6, 1000)
>>> eig.allcompute(1e-6, 1000, 5, 0)
```

`expression`

`class expression( **kwargs )`

The expression object holds a mathematical expression made of operators (such as +, -, *, /), fields, parameters, square operators, abs operators and so on.

#### Examples

An empty expression object can be created as:

```
>>> myexpression = expression()
```

An expression object can be a scalar:

```
>>> myexpression = expression(2)
>>> myexpression.print()
Expression size is 1x1
@ row 0, col 0 :
2
```

An expression object can also be a vector or a 2D array. For this three arguments are required. The first and second arguments specify the number of rows and the number of columns respectively. The expression object is filled with input expressions provided as a list in the third argument. The general syntax is `expression(numrows:int, numcols:int, input:List[expression]`

```
>>> myexpression = expression(1,3, [1,2,3])
>>> myexpression.print()
Expression size is 1x3
@ row 0, col 0 : 1
@ row 0, col 1 : 2
@ row 0, col 2 : 3
```

In a 2D array expression, the inputs are set in row-major order. In the example below, the entry at the index pair (1,0) in the created expression is set to and the entry (1,2) to .

```
>>> myexpression = expression(3,3, [1,2,3, 4,5,6, 7,8,9]) # creates a 3x3 sized expression array
>>> myexpression.at(1,0).evaluate()
4.0
>>> myexpression.at(1,2).evaluate()
6.0
```

A symmetric expression array can also be created by only providing the input list corresponding to the lower triangular part:

```
>>> myexpression = expression(3,3, [1,2,3, 4,5,6])
>>> myexpression.print()
```

A diagonal expression array can also be created by only providing the input list corresponding to the diagonal elements:

```
>>> myexpression = expression(3,3, [1,2,3])
>>> myexpression.print()
```

Note that to create a symmetric or diagonal expression array, the size must correspond to a square array (number of rows = number of columns).

The expression input can also be made of fields. For example:

```
>>> mymesh = mesh("disk.msh")
>>> v = field("h1")
>>> myexpression = expression(2,3, [12,v,v*(1-v), 3,14-v,0])
```

An expression array object can be obtained from the row-wise and column-wise concatenation of input expressions using the syntax `expression(input:[List[List[expression]]`

. Every element in the argument input (i.e. input[0], input[1], ..) is concatenated column-wise with others. Every expression in input[i] (i.e input[i][0], input[i][1], ..) is concatenated row-wise with the other expressions in that List.

```
>>> mymesh = mesh("disk.msh")
>>> v = field("h1")
>>> blockleft = expression(3,1, [1,2*v,3])
>>> blockrighttop = expression(1,2, [4,5])
>>> blockrightbottom = expression(2,2, [6,7,8,9])
>>> exprconcatenated = expression([[blockleft], [blockrighttop,blockrightbottom]])
>>> exprconcatenated.print()
Expression size is 3x3
@ row 0, col 0 : 1
@ row 0, col 1 : 4
@ row 0, col 2 : 5
@ row 1, col 0 : field * 2
@ row 1, col 1 : 6
@ row 1, col 2 : 7
@ row 2, col 0 : 3
@ row 2, col 1 : 8
@ row 2, col 2 : 9
```

It is also useful in many cases to create a conditional expression. It takes the form `expression(condexpr:expression, exprtrue:expression, exprfalse:expression)`

. If the first argument is greater than equal to zero then the expression is equal to the expression provided in the second argument. If smaller than zero, then it is equal to the expression in the third argument.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>> x = field("x"); y = field("y")
>>> expr = expression(x+y, 2*x, 0)
>>> expr.write(top, "conditionalexpr.vtk", 1)
>>> expr.print()
Expression size is 1x1
@ row 0, col 0 : (x + y) ? x * 2, 0)
```

An expression can be used to define an algebraic relation between two variables as in the example below:

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>> C = field("h1") # temperature field in Celsius
>>> F = field("h1") # temperature field in Fahrenheit
>>> F = (9.0/5.0)*C + 32 # Relation for converting Celsius to Fahrenheit
>>> F.write(top, "F.vtk", 1)
```

In the above example, an algebraic relation between two variables was already known allowing us to create an expression that is continuous. However, if the data is from an experiment, they are usually not continuous. As an application example, if measurements of a material stiffness (Young's modulus ) have been performed for a set of temperatures , then only a discrete data set exists between variable and . A discrete data set can be converted to a continuous function ( as a function of ) using cubic splines which allows us to interpolate at any value in the measured discrete temperature range. Using this spline object an expression can be defined that provides cubic spline interpolation of Young's modulus in the measured discrete temperature range. Refer to the `spline`

object for more details.

```
>>> discrete data set from measurement
>>> temperature = [273,300,320,340]
>>> youngsmodulus = [5e9,4e9,5e9,1e9]
>>>
>>> spline object: allows interpolation in the measured data range
>>> spl = spline(temperature, youngsmodulus)
>>>
>>> # expression interpolating E at temperature 310.
>>> # 'spl' holds the interpolation function information.
>>> E = expression(spl, 310)
>>> E.print()
Expression size is 1x1
@ row 0, col 0 :
spline(310)
>>>
>>> # expression interpolating E for space-dependent temperature profile.
>>> mymesh = mesh("disk.msh")
>>> T = field("h1") # space dependent temperature profile
>>> E = expression(spl, T) # space dependent Young's modulus
>>> E.write(top, "E.vtk", 1)
```

The expression object is much more versatile. Say, we have to define an electric supply voltage profile in time that is:

- 0V for time before 1 sec.
- increases linearly from 0V to 1V for time range [1,3] sec.
- 1V for time after 3 sec. This can be created as shown below in the example. The following creates a conditional expression for the intervals defined in the first argument for the time variable t(). In the below example, the defined interval is [1.0,3.0]. This provides information in three intervals:
- interval 1: from to 1.0
- interval 2: between 1.0 to 3.0
- interval 3: from 3.0 to The second argument holds three expressions, each valid in the sequence of the respective interval defined above. The third argument specifies the variable input (time in this case). Printing the expression object provides insight into the conditional expression created with these inputs.

```
>>> vsupply = expression([1.0,3.0], [0, 0.5*(t()-1), 1.0], t())
>>> vsupply.print()
Expression size is 1x1
@ row 0, col 0 : ((t + -3) ? 1, ((t + -1) ? (t + -1) * 0.5, 0))
```

TODO: CUSTOM EXPRESSION

#### Methods

`allintegrate`

`def allintegrate( *args, **kwargs ) ‑> float`

This is a collective MPI operation and hence must be called by all ranks. This integrates an expression across all the DDM ranks.

###### Examples

...

```
>>> integralvalue = myexpression.allintegrate(vol, 4)
>>>
>>> # allintegrate on a mesh deformed by field 'u'
>>> integralvalueondeformedmesh = myexpression.allintegrate(vol, u, 4)
```

`allinterpolate`

`def allinterpolate( *args, **kwargs ) ‑> List[float]`

This is a collective MPI operation and hence must be called by all ranks. Its functionality is as described in `expression.interpolate()`

but considers the physical region partitioned across the DDM ranks. The xyz coordinate argument must be the same for all ranks.

###### Examples

```
>>> ...
>>> interpolated = array3x1(x,y,z).allinterpolate (vol, {0.5,0.6,0.05})
>>>
>>> # allinterpolate on a mesh deformed by field 'u'
>>> interpolated = array3x1(x,y,z).allinterpolate (vol, u, {0.5,0.6,0.05})
```

`allmax`

`def allmax( *args, **kwargs ) ‑> List[float]`

This is a collective MPI operation and hence must be called by all ranks. It computes the max value across all the DDM ranks. The evaluation of the max value can be restricted to a box or evaluated on a deformed mesh similar to as described in `expression.max()`

.

###### See Also

`expression.max()`

, `expression.allmin()`

`allmin`

`def allmin( *args, **kwargs ) ‑> List[float]`

This is a collective MPI operation and hence must be called by all ranks. It computes the min value across all the DDM ranks. The evaluation of the min value can be restricted to a box or evaluated on a deformed mesh similar to as described in `expression.min()`

.

###### See Also

`expression.min()`

, `expression.allmax()`

`at`

`def at( self, row: int, col: int ) ‑> quanscient.expression`

This returns the entry at the requested row and column.

###### Example

```
>>> myexpression = expression(2,2, [1,2, 3,4])
>>> myexpression.at(0,1)
2
```

`atbarycenter`

`def atbarycenter( self, physreg: int, onefield: field ) ‑> quanscient.vec`

This outputs a `vec`

object whose structure is based on the field argument `onefield`

and which contains the expression evaluated at the barycenter of each **reference** element of physical region `physreg`

. The barycenter of the reference element might not be identical to the barycenter of the actual element in the mesh (for curved elements, for general quadrangles, hexahedra and prisms). The evaluation at barycenter is constant on each mesh element.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x = field("x"); f = field("one")
>>>
>>> # Evaluating the expression
>>> (12*x).write(vol, "expression.vtk", 1)
>>>
>>>> # Evaluating the same expression at barycenter
>>> myvec = (12*x).atbarycenter(vol, f)
>>> f.setdata(vol, myvec)
>>> f.write(vol, "barycentervalues.vtk", 1)
```

`countcolumns`

`def countcolumns( self ) ‑> int`

This counts the number of columns in an expression.

###### Example

```
>>> myexpression = expression(2,1, [0,1])
>>> myexpression.countcolumns()
1
```

`countrows`

`def countrows( self ) ‑> int`

This counts the number of rows in an expression.

###### Example

```
>>> myexpression = expression(2,1, [0,1])
>>> myexpression.countrows()
2
```

`evaluate`

`def evaluate( self ) ‑> float`

This evaluates a scalar, space-independent expression.

###### Example

```
>>> settime(0.5)
>>> expr = 2*abs(-5*t())+3
>>> expr.evaluate()
8.0
```

`getcolumn`

`def getcolumn( self, colnum: int ) ‑> quanscient.expression`

This returns for a matrix expression the column corresponding to the specified input index `colnum`

.

###### Example

```
>>> myexpression = expression(2,2, [0,1, 2,3])
>>> subexpr = myexpression.getcolumn(0)
>>> subexpr.print()
Expression size is 2x1
@ row 0, col 0 : 0
@ row 0, col 1 : 2
```

`getrow`

`def getrow( self, rownum: int ) ‑> quanscient.expression`

This returns for a matrix expression the row corresponding to the specified input index `rownum`

.

###### Example

```
>>> myexpression = expression(2,2, [0,1, 2,3])
>>> subexpr = myexpression.getrow(1)
>>> subexpr.print()
Expression size is 1x2
@ row 0, col 0 : 2
@ row 0, col 1 : 3
```

`integrate`

`def integrate( *args, **kwargs ) ‑> float`

This integrates an expression over the physical region `physreg`

. The integration is exact up to the order of polynomials specified in the argument `integrationorder`

. Integrate `expression(1)`

to calculate volume/area/ length. For axisymmetric problems, the value returned is the integral of the requested expression times the coordinate change Jacobian. In the case of axisymmetry, the volume/area/length of the 3D shape corresponding to the physical region on which to integrate can be obtained by integrating `expression(1)`

and multiplying the output by .

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> myexpression = expression(12.0)
>>> integralvalue = myexpression.integrate(vol, 4)
>>>
>>> # integrate on a mesh deformed by field 'u'
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> integralvalueondeformedmesh = myexpression.integrate(vol, u, 4)
```

`interpolate`

`def interpolate( *args, **kwargs ) ‑> None`

This interpolates the expression at a single point whose [x,y,z] coordinate is provided as an argument. The flattened interpolated expression values are returned if the point was found in the elements of the physical region `physreg`

. If not found an empty list is returned. An expression can also be interpolated on a deformed mesh by passing its corresponding field.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x=field("x"); y=field("y"); z=field("z")
>>> xyzcoord = [0.5,0.6,0.05]
>>>
>>> interpolated = array3x1(x,y,z).interpolate(vol, xyzcoord)
>>> interpolated
[0.5, 0.6, 0.05]
>>>
>>> # interpolation on the mesh deformed by field 'u'
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> interpolated = array3x1(x,y,z).interpolate(vol, u, xyzcoord)
```

`isscalar`

`def isscalar( self ) ‑> bool`

This returns True if the expression is a scalar (i.e. has a single row and column).

###### Examples

```
>>> myexpression = expression(12.0)
>>> myexpression.isscalar()
True
>>> myexpression = expression(2,2, [0,1, 2,3])
>>> myexpression.isscalar()
False
```

`iszero`

`def iszero( self ) ‑> bool`

This returns True if all the entries in the expression is zero, otherwise False.

###### Examples

```
>>> myexpression = expression(12.0)
>>> myexpression.iszero()
False
>>> myexpression = expression(0.0)
>>> myexpression.iszero()
True
>>> myexpression = expression(2,2, [0,0, 0,0])
>>> myexpression.iszero()
True
>>> myexpression = expression(2,2, [1,0, 0,0])
>>> myexpression.iszero()
False
```

`max`

`def max( *args, **kwargs ) ‑> List[float]`

This gives the max value of the expression over the geometric region `physreg`

. The max value is obtained by splitting all elements `refinement`

times in each direction. Increasing the refinement will thus lead to a more accurate max value, but at an increased computational cost. The max value is exact when the refinement nodes added to the elements correspond to the position of max. For a first-order nodal shape function interpolation, on a mesh that is not curved, the max is always exact to machine precision.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x = field("x")
>>> maxdata = (2*x).max(vol, 1)
>>> maxdata[0]
2.0
```

The search of the max value can be restricted to a box delimited by the last argument whose form is [xboxmin,xboxmax, yboxmin, yboxmin, zboxmax, zboxmin]. The output returned is a list of the form [maxvalue, xcoordmax, ycoordmax, zcoordmax] or an empty list if the physical region argument is empty or is not in the box provided. If the argument defining the box is not provided, then the whole geometric region is considered for evaluating the max value.

```
>>> maxdatainbox = (2*x).max(vol, 5, [-2,0, -2,2, -2,2])
```

The max value can also be evaluated on the geometry deformed by a field (possibly a curved mesh). The max location and the delimiting box are on the undeformed mesh.

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> maxdataondeformedmesh = (2*x).max(vol, u, 1)
```

###### See Also

`min`

`def min( *args, **kwargs ) ‑> List[float]`

This gives the min value of the expression over the geometric region `physreg`

. The min value is obtained by splitting all elements `refinement`

times in each direction. Increasing the refinement will thus lead to a more accurate min value, but at an increased computational cost. The min value is exact when the refinement nodes added to the elements correspond to the position of min. For a first-order nodal shape function interpolation, on a mesh that is not curved, the min is always exact to machine precision.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x = field("x")
>>> mindata = (2*x).min(vol, 1)
>>> mindata[0]
-2.0
```

The search of the min value can be restricted to a box delimited by the last argument whose form is [xboxmin,xboxmax, yboxmin, yboxmin, zboxmax, zboxmin]. The output returned is a list of the form [maxvalue, xcoordmax, ycoordmax, zcoordmax] or an empty list if the physical region argument is empty or is not in the box provided. If the argument defining the box is not provided, then the whole geometric region is considered for evaluating the min value.

```
>>> mindatainbox = (2*x).min(vol, 5, [-2,0, -2,2, -2,2])
```

The min value can also be evaluated on the geometry deformed by a field (possibly a curved mesh). The min location and the delimiting box are on the undeformed mesh.

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> mindataondeformedmesh = (2*x).min(vol, u, 1)
```

###### See Also

`print`

`def print( self ) ‑> None`

This prints the expression to the console.

###### Example

```
>>> myexpression = expression(2,2, [0,1, 2,3])
>>> myexpression.print()
Expression size is 2x2
@ row 0, col 0 : 0
@ row 0, col 1 : 1
@ row 1, col 0 : 2
@ row 1, col 1 : 3
```

`reordercolumns`

`def reordercolumns( self, neworder: List[int] ) ‑> None`

This reorders the columns of a matrix expression in the specified order `neworder`

.

###### Example

```
>>> expr = expression(2,2, [0,1, 2,3])
0 1
2 3
>>> expr.reordercolumns([1,0])
1 0
3 2
```

`reorderrows`

`def reorderrows( self, neworder: List[int] ) ‑> None`

This reorders the rows of a matrix expression in the specified order `neworder`

.

###### Example

```
>>> expr = expression(2,2, [0,1, 2,3])
0 1
2 3
>>> expr.reorderrows([1,0])
2 3
0 1
```

`resize`

`def resize( self, numrows: int, numcols: int ) ‑> quanscient.expression`

This resizes an expression. Any newly created expression entry is set to zero.

###### Example

```
>>> myexpression = expression(2,2, [1,2, 3,4])
>>> resizedexpr = myexpression.resize(1,3)
>>> resizedexpr.print()
Expression size is 1x3
@ row 0, col 0 : 1
@ row 0, col 1 : 2
@ row 0, col 2 : 0
```

`reuseit`

`def reuseit( self, istobereused: bool = True ) ‑> None`

In case an expression appears multiple times, say in a formulation, and requires much time to compute, then the expression can be reused by calling this method and setting `istobereused=True`

. With this, the expression is computed only once to assemble a formulation block and reused as long as its value remains changed.

###### Example

```
>>> myexpression = expression(12.0)
>>> myexpression.resuseit() # myexpression.resuseit(True)
```

`rotate`

`def rotate( self, ax: float, ay: float, az: float, leftop: str = 'default', rightop: str = 'default' ) ‑> None`

`streamline`

`def streamline( self, physreg: int, filename: str, startcoords: List[float], stepsize: float, downstreamonly: bool = False ) ‑> None`

This follows and writes to disk all paths tangent to the expression vector that are starting at a set of points whose , and coordinates are provided in `startcoords`

. These coordinates can for example be obtained via `.getcoords()`

on a shape object. A fourth-order Runge-Kutta algorithm is used. The `stepsize`

argument is related to the distance between two vector direction updates; decrease it to more accurately follow the paths. The paths will be followed as long as they remain in the physical region `physreg`

. In case the vector norm is zero somewhere on the paths or a path is a closed loop then the function might enter an **infinite loop** and never return. To use this function on closed loops (for example to get magnetic field lines of a permanent magnet) a solution is to break the loops by excluding the permanent magnet domain from the physical region (`selectexclusion`

) function can be called for that) and set the starting coordinates on the boundary of the magnet.

###### Example

```
>>> vol = 1
>>> mymesh = mesh("disk.msh")
>>> x=field("x"); y=field("y"); z=field("z")
>>> startcoords = 12*3*[0.05] # list of 36 elements with each element value being 0.05
>>> for i in range(0,12):
... startcoords[3*i+0] += 0.1 + 0.05*i
>>> array3x1(y+2*x, -y+2*x, 0).streamline(vol, "streamlines.vtk", startcoords, 1.0/100.0)
```

`write`

`def write( *args, **kwargs ) ‑> None`

This evaluates an expression in the physical region `physreg`

and writes it to the file `filename`

. The `lagrangeorder`

is the order of interpolation for evaluation of the expression.

###### Examples

```
>>> # setup
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> v = field("h1", [1,2,3])
>>> u.setorder(vol, 1)
>>> v.setorder(vol, 1)
>>>
>>> # interpolation order for writing an expression
>>> (1e8*u).write(vol, "uorder1.vtk", 1) # interpolation order is 1
>>> (1e8*u).write(vol, "uorder3.vtk", 3) # interpolation order is 3
```

In the example below, an additional integer input is passed in the second argument. The here means that the expression is treated as multi-harmonic, nonlinear in time variable and an FFT is performed to get the first harmonics. All harmonics whose magnitude is above a threshold are saved with '_harm i' extension (except for time-constant harmonic).

```
>>> abs(v).write(vol, 10, "order1.vtk", 1) # interpolation order is 1
>>> (u*u).write(vol, 10, "order3.vtk", 3) # interpolation order is 3
```

In the example below, an additional integer input is instead passed as the last argument posterior to the interpolation order argument. This represents that `numtimesteps`

(default=-1). For a positive value of , the multi-harmonic expression is saved at equidistant timesteps in the fundamental period and can then be visualized in time.

```
>>> (1e8*u).write(vol, "uintime.vtk", 2, 50)
```

The expressions can also be evaluated and written on a mesh deformed by a field. If field 'v' is the deformed mesh, then:

```
>>> (1e8*u).write(vol, v, "uorder1.vtk", 1)
>>> (u*u).write(vol, 10, v, "order3.vtk", 3)
>>> (1e8*u).write(vol, v, "uintime.vtk", 2, 50)
```

`field`

`class field( **kwargs )`

The field object holds the information of the finite element fields. The field object itself only holds a pointer to a 'rawfield' object.

#### Examples

**Example 1:** `field(fieldtypename:str)`

```
>>> mymesh = mesh("disk.h")
>>> v = field("h1")
```

This creates a field object with the specified shape functions. The full list of shape functions available are:

- Nodal shape functions "h1" e.g. for electrostatic potential and acoustic or fluid pressure.
- Two-components nodal shape functions "h1xy" e.g. for 2D mechanical displacements and 2D fluid velocity.
- Three-components nodal shape functions "h1xyz" e.g. for 3D mechanical displacements and 3D fluid velocity.
- Nedelec's edge shape functions "hcurl" e.g. for the electric field in the E-formulation of electromagnetic wave propagation (here order 0 is allowed).
- "one", one0", one1", one2", one3" (trailing "xy" or "xyz" allowed) shape functions have a single shape function equal to a constant one on respectively an n, 0, 1, 2, 3-dimensional element (n is the geometry dimension).
- "h1d", "h1d0", "h1d1", "h1d2", "h1d3" (trailing "xy" or "xyz" allowed) shape functions are elementwise-"h1" shape functions that allow storing fields that are fully discontinuous between elements.

Additionally, types "x", "y" and "z" can be used to define the x, y and z coordinate fields.

```
>>> mymesh = mesh("disk.h")
>>> x = field("x")
>>> y = field("y")
>>> z = field("z")
```

**Example 2:** `field(fieldtypename:str, harmonicnumbers:List[int])`

```
>>> mymesh = mesh("disk.msh")
>>> v = field("h1", [1,4,5,6])
```

Consider the infinite Fourier series of a field that is periodic in time:

where is the time variable, is the space variable and is the fundamental frequency of the periodic field. The coefficients only depend on the space variable, not on the time variables which have now moved to the sines and cosines. In the example above, field is a multi-harmonic "h1" type field that includes harmonic fields: the , , and fields in the truncated Fourier series above. All other harmonics in the infinite Fourier series are supposed to equal zero so that the field can be rewritten as:

This is the truncated multi-harmonic representation of field (which must be periodic in time). The following can be used to get the harmonic from field . It can then be used like any other field.

```
>>> v4 = v.harmonic(4)
```

**Example 3:** `field(fieldtypename:str, spantree:spanningtree)`

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>> spantree = spanningtree([sur, top])
>>> a = field("hcurl", spantree)
```

This adds the spanning tree input argument needed when the field has to be gauged. (e.g. for the magnetic vector potential formulation of the magnetostatic problem in 3D).

**Example 3:** `field(fieldtypename:str, harmonicnumbers:List[int], spantree:spanningtree)`

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>> spantree = spanningtree([sur, top])
>>> aharmonic = field("hcurl", [2,3], spantree)
```

This adds the spanning tree input argument needed when a field has to be gauged.

#### Methods

`allintegrate`

`def allintegrate( *args, **kwargs ) ‑> float`

`allinterpolate`

`def allinterpolate( *args, **kwargs ) ‑> List[float]`

`allmax`

`def allmax( *args, **kwargs ) ‑> List[float]`

`allmin`

`def allmin( *args, **kwargs ) ‑> List[float]`

`atbarycenter`

`def atbarycenter( self, physreg: int, onefield: field ) ‑> quanscient.vec`

This outputs a `vec`

object whose structure is based on the field argument `onefield`

and which contains the field evaluated at the barycenter of each **reference** element of physical region `physreg`

. The barycenter of the reference element might not be identical to the barycenter of the actual element in the mesh (for curved elements, for general quadrangles, hexahedra and prisms). The evaluation at barycenter is constant on each mesh element.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1"); f = field("one")
>>> v.setorder(vol, 1)
>>>
>>> # Evaluating the field
>>> v.write(vol, "expression.vtk", 1)
>>>
>>>> # Evaluating the same field at barycenter
>>> myvec = v.atbarycenter(vol, f)
>>> f.setdata(vol, myvec)
>>> f.write(vol, "barycentervalues.vtk", 1)
```

`automaticupdate`

`def automaticupdate( self, updateit: bool ) ‑> None`

`comp`

`def comp( self, component: int ) ‑> quanscient.field`

This gets the , or component of a field with subfields.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz")
>>> ux = u.comp(0)
>>> uy = u.comp(1)
>>> uz = u.comp(2)
```

`compx`

`def compx( self ) ‑> quanscient.field`

This gets the component of a field with multiple subfields.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz")
>>> ux = u.compx()
```

`compy`

`def compy( self ) ‑> quanscient.field`

This gets the component of a field with multiple subfields.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz")
>>> uy = u.compy()
```

`compz`

`def compz( self ) ‑> quanscient.field`

This gets the component of a field with multiple subfields.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz")
>>> uz = u.compz()
```

`cos`

`def cos( self, freqindex: int ) ‑> quanscient.field`

This gets the "h1xyz" type field that is the harmonic at `freqindex`

times the fundamental frequency in field .

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz", [1,2,3,4,5])
>>> uc = u.cos(0) # gets the harmonic 1
```

`countcomponents`

`def countcomponents( self ) ‑> int`

This returns the number of components in the field.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> E = field("hcurl")
>>> numcomp = E.countcomponents()
>>> numcomp
3
```

`getharmonics`

`def getharmonics( self ) ‑> List[int]`

This returns the list of harmonics of the field object.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> v = field("h1", [1,4,5,6])
>>> myharms = v.getharmonics()
>>> myharms
[1, 4, 5, 6]
```

`getnodalvalues`

`def getnodalvalues( self, nodenumbers: indexmat ) ‑> quanscient.densemat`

This gets the values of a "h1" type field at a set of `nodenumbers`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v.setorder(vol, 1)
>>> nodenums = indexmat(5,1, [0,1,2,3,4])
>>> outvals = v.getnodalvalues(nodenums) # returns a densemat
>>> outvals.print()
```

`harmonic`

`def harmonic( *args, **kwargs ) ‑> quanscient.field`

This gets a "h1xyz" type field that includes the `harmonicnumber(s)`

.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz", [1,2,3])
>>> u2 = u.harmonic(2) # gets harmonic 2 of field u
>>> u23 = u.harmonic([1,3]) # gets harmonics 1 and 3 of field u
```

This gets a "h1xyz" type field that includes the `harmonicnumber(s)`

.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz", [1,2,3])
>>> u2 = u.harmonic(2) # gets harmonic 2 of field u
>>> u23 = u.harmonic([1,3]) # gets harmonics 1 and 3 of field u
```

`integrate`

`def integrate( *args, **kwargs ) ‑> float`

`interpolate`

`def interpolate( *args, **kwargs ) ‑> None`

`loadraw`

`def loadraw( self, filename: str, isbinary: bool = False ) ‑> List[float]`

This loads the .slz file created with the `writeraw`

method. If the .slz file was written in the binary format then `isbinary`

argument must be set to *True* else to *False*. The same mesh must be used when loading with `loadraw`

as the one that was used during the corresponding `writeraw`

call.

###### Example

```
>>> vol = 1
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz")
>>> u.loadraw("v.slz.gz", True)
```

`max`

`def max( *args, **kwargs ) ‑> List[float]`

`min`

`def min( *args, **kwargs ) ‑> List[float]`

`noautomaticupdate`

`def noautomaticupdate( self ) ‑> None`

After this call, the field and all its subfields will not have their value automatically updated after hp-adaptivity. If the automatic update is not needed then this call is recommended to avoid a possible costly field value update.

###### Example

```
>>> ...
>>> v.noautomaticupdate()
```

`print`

`def print( self ) ‑> None`

This prints the field name.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> v = field("h1")
>>> v.setname("velocity")
>>> v.print()
velocity
```

`printharmonics`

`def printharmonics( self ) ‑> None`

This prints a string showing the harmonics in the field.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> v = field("h1", [1,4,5,6])
>>> v.printharmonics()
+vc0*cos(0*pif0t) +vs2*sin(4*pif0t) +vc2*cos(4*pif0t) +vs3*sin(6*pif0t)
```

`setcohomologysources`

`def setcohomologysources( self, cutvalues: List[float] ) ‑> None`

This method assigns cohomology coefficients to the field. The field value is reset to zero on the cohomology regions before the coefficients are added on their respective regions.

###### Example

```
>>> mymesh = mesh()
>>> mymesh.setcohomologycuts([chreg1, chreg2])
>>> mymesh.load("disk.msh")
>>> v = field("hcurl")
>>> ...
>>> v.setcohomologysources([100, 50])
>>> v.write(chreg1, "v.pos", 1)
```

`setconditionalconstraint`

`def setconditionalconstraint( self, physreg: int, condexpr: expression, valexpr: expression ) ‑> None`

This forces the field value (i.e. Dirichlet constraint) on the region `physreg`

to a value `valexpr`

for all node-associated degrees of freedom for which the condition `condexpr`

evaluates to greater than or equal to zero at the nodes. This should only be used for fields with "h1" type functions.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; top=3
>>> v=field("h1"); x=field("x"); y=field("y")
>>> v.setorder(vol, 1)
>>> v.setconditionalconstraint(vol, x+y, 12)
>>>
>>> form = formulation()
>>> form += integral(vol, dof(v)*tf(v) - 1*tf(v))
>>> form.generate()
>>> sol = solve(form.A(), form.b()) # returns a vec object
>>> v.setdata(vol, sol)
>>> v.write(top, "v.vtk", 1)
```

`setconstraint`

`def setconstraint( *args, **kwargs ) ‑> None`

This forces the field value (i.e. Dirichlet condition) on the region `physreg`

to `input`

expression. An extra int argument `extraintegrationdegree`

can be used to increase or decrease the default integration order when computing the projection of the expression on the field. Increasing it can give a more accurate computation of the expression but might take longer. The default integration order is equal to "*field order* ". **Dirichlet constraints have priority over conditional constraints and gauge conditions**. Defining any of these on a Dirichlet constrained region has no effect.

###### Examples

**Example 1:** `field.setconstraint(physreg:int, input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> w = field("h1")
>>> v.setconstraint(vol, 12+w*w)
```

This forces the field value (i.e Dirichlet constraint) on region *vol* to expression .

**Example 2:** `field.setconstraint(physreg:int, meshdeform:expression, input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> u = field("h1xyz")
>>> v.setconstraint(vol, u, expression(12))
```

This forces the field value on region *vol* to expression but on a mesh deformed by `meshdeform`

.

**Example 3:** `field.setconstraint(physreg:int, input:List[expression], input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1", [2,3])
>>> v.setconstraint(vol, [1,0]) # sets 1 for harmonic 2, 0 for harmonic 3
```

This sets a Dirichlet constraint with the given value for each field harmonic.

**Example 4:** `field.setconstraint(physreg:int, meshdeform:expression, input:List[expression], extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1", [2,3])
>>> u = field("h1xyz")
>>> v.setconstraint(vol, u, [1,0]) # sets 1 for harmonic 2, 0 for harmonic 3
```

This sets a Dirichlet constraint for each field harmonic with the given expression computed on a mesh deformed by `meshdeform`

.

**Example 5:** `field.setconstraint(physreg:int, numfftharms:int, input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v1 = field("h1", [2,3])
>>> v2 = field("h1", [1,4,5])
>>> v2.setconstraint(vol, 5, v1*v1)
>>> v2.write(vol, "v2.vtk", 1)
```

This calls an FFT for the calculation required for nonlinear multi-harmonic expressions. The FFT is computed at `numfftharms`

timesteps.

**Example 6:** `field.setconstraint(physreg:int, numfftharms:int, meshdeform:expression, input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v1 = field("h1", [2,3])
>>> v2 = field("h1", [1,4,5])
>>> u = field("h1xyz")
>>> v2.setconstraint(vol, 5, u, v1*v1)
```

This calls an FFT for the calculation and the expression is evaluated on a mesh deformed by `meshdeform`

.

**Example 7:** `field.setconstraint(physreg:int)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setconstraint(vol)
```

This forces the field value (i.e. Dirichlet condition) on region *vol* to .

`setdata`

`def setdata( *args, **kwargs ) ‑> None`

This either sets or adds the data in the vector to the field. If the argument `op`

is "set", then the vector data is set and if it is "add" then the vector data is added to the existing field values.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1"); w = field("h1")
>>> v.setorder(vol, 1)
>>>
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> projection.generate()
>>> sol = solve(projection.A(), projection.b())
>>> v.setdata(vol, sol)
```

`setgauge`

`def setgauge( self, physreg: int ) ‑> None`

This sets a gauge condition on region`physreg`

. It must be used e.g. for the magnetic vector potential formulation of the magnetostatic problem in 3D since otherwise, the algebraic system to solve is singular. It is only defined for edge shape functions ("hcurl"). Its effect is to constrain to zero all degrees of freedom corresponding to:

- gradient type shape functions.
- lowest order edge-shape functions for all edges on the spanning tree provided.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3
>>> spantree = spanningtree([sur, top])
>>> a = field("hcurl", spantree)
>>> a.setgauge(vol)
```

`setname`

`def setname( self, name: str ) ‑> None`

This gives a name to the field. Useful when printing expressions including fields.

```
>>> mymesh = mesh("disk.msh")
>>> v = field("h1")
>>> v.setname("velocity")
>>> v.print()
velocity
```

`setnodalvalues`

`def setnodalvalues( self, nodenumbers: indexmat, values: densemat ) ‑> None`

This sets the values of a "h1" type field at a set of `nodenumbers`

to `values`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> nodenums = indexmat(5,1, [0,1,2,3,4])
>>> nodevals = densemat(5,1, [10,11,12,13,14])
>>> v.setnodalvalues(nodenums, nodevals)
```

`setorder`

`def setorder( *args, **kwargs ) ‑> None`

This sets the specified interpolation order of the field object.

###### Examples

**Example 1:** `field.setorder(physreg:int, interpolorder:int)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 3)
```

This sets the interpolation order to on the physical region 'vol'. When using different interpolation orders on different physical regions for a given field it is only allowed to set the interpolation orders in a decreasing way. i.e starting with the physical region with the highest order and ending with the physical region with the lowest order. This is required to enforce field continuity and is due to the fact the interpolation order on the interface between multiple physical regions must be the one of lowest touching region.

**Example 2:** `field.setorder(criterion:expression, loworder:int, highorder:int)`

```
>>> all=1; inn=2; out=3
>>> n = 20
>>> q0 = shape("quadrangle", all, [0,0,0, 1,0,0, 1,0.3,0, 0,0.3,0], [n,n,n,n])
>>> q1 = shape("quadrangle", all, [1,0,0, 2,0,0, 2,0.3,0, 1,0.3,0], [n,n,n,n])
>>> linein = q0.getsons()[3]
>>> linein.setphysicalregion(inn)
>>> lineout = q1.getsons()[0]
>>> lineout.setphysicalregion(out)
>>> mymesh = mesh([q0, q1, linein, lineout])
>>>
>>> v = field("h1")
>>> v.setname("v")
>>> v.setorder(all, 1)
>>> v.setorder(norm(grad(v)), 1, 5)
>>> v.setconstraint(inn, 1)
>>> v.setconstraint(out)
>>>
>>> electrostatics = formulation()
>>> electrostatics += integral(all, 8.854e-12 * grad(dof(v))*grad(tf(v)))
>>>
>>> for i in range(5):
... electrostatics.solve()
... v.write(all, f"v{100+i}.pos", 5)
... (-grad(v)).write(all, f"E{100+i}.pos", 5)
... fieldorder(v).write(all, f"vorder{100+i}.pos", 1)
...
... adapt(2)
```

In the above example, the field interpolation order will be adapted on each mesh element (of the entire geometry) based on the value of a **positive** criterion (p-adaptivity). The max range of the criterion is split into a number of intervals equal to the number of orders in range 'loworder' to 'highorder'. All intervals have the same size. The barycenter value of the criterion on each mesh element is considered to select the interval, and therefore the corresponding interpolation order to assign to the field on each element. As an example, for a criterion with the highest value of 900 over the entire domain and a low/high order requested of 1/3 the field on elements with criterion values in range 0 to 300, 300 to 600, 600 to 900 will be assigned order 1, 2, 3 respectively.

**Example 3:** `field.setorder(targeterror:double, loworder:int, highorder:int, absthres:double)`

```
>>> sur = 1
>>> q = shape("quadrangle", sur, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [20,20,20,20])
>>> mymesh = mesh([q])
>>> v=field("h1xy"); x=field("x"); y=field("y")
>>> v.setorder(sur, 1)
>>> v.setorder(1e-5, 1, 5, 0.001)
>>> for i in range(5):
... v.setvalue(sur, array2x1(0,cos(10*x*y)))
... adapt()
... v.write(sur, f"v{i}.vtu", 5)
... fieldorder(v).write(sur, f"fov{i}.vtu", 1)
```

The field interpolation order will be adapted on each mesh element (of the entire geometry) based on a criterion measuring the Legendre expansion decay. The target error gives the fraction of the total shape function weight that does not need to be captured. The low order is used on all elements where the total weight is lower than the absolute threshold provided.

`setport`

`def setport( self, physreg: int, primal: port, dual: port ) ‑> None`

This function associates a primal-dual pair of ports to the field on the requested physical region. As a side effect, it lowers the field order on that region to the minimum possible. **Ports** have priority over Dirichlet constraints, conditional constraints and gauge conditions}. Defining any of these on a port region has no effect.

Ports that have been associated to a field with a *setport* call and unassociated ports are visible to a formulation only if they appear in a port relation (in the example below: *electrokinetic += I - 1.0* ). The primal and dual of associated ports are always made visible together even if only of them appears in a port relation. Unassociated ports are not connected to the weak form terms: the primal can be used as the lumped field value on the associated region while the dual can be used as the total contribution over that region of the Neumann term in the formulation. The field value is considered constant by the formulation over the region of each associated port visible to it.

To illustrate the meaning of the dual port let us consider the below DC current flow simulation example code. The strong form to solve is

where is the electric conductivity and is the electric potential field. The corresponding weak form is

which after integration by parts can be rewritten as

where is the boundary of , is the unit normal pointing outward from and . The Neumann term is the second term of the weak formulation. The dual port in the below example, therefore equals the total current flowing through the electrode and thus can be used to impose a total current source condition on the electrode. More details about the associated mathematics can be found in the paper *'Coupling of local and global quantities in various finite element formulations and their application to electrostatics, magnetostatics and magnetodyanmics'*, Dular et al.

###### Example

```
>>> sur=1; left=2; right=3
>>> q = shape("quadrangle", sur, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [10,10,10,10])
>>> ll = q.getsons()[3]
>>> ll.setphysicalregion(left)
>>> rl = q.getsons()[1]
>>> rl.setphysicalregion(right)
>>> mymesh = mesh([q, ll, rl])
>>>
>>> v = field("h1")
>>> y = field("y")
>>> v.setorder(sur, 2)
>>>
>>> # Electric conductivity increasing with the y-coordinate
>>> sigma = expression(0.01*(1+2*y))
>>>
>>> # Ground the right electrode
>>> v.setconstraint(right)
>>>
>>> V = port() # primal port
>>> I = port() # dual port
>>> v.setport(left, V, I)
>>> # The dual port holds the global Neumann term on the port region.
>>> # For an electrokinetic formulation this equals the total current.
>>>
>>> electrokinetic = formulation()
>>> # Set a 1A current flowing in through the left electrode with the port relation I - 1.0 = 0:
>>> elctrokinetic += I - 1.0 # port relation
>>> # Define the weak formulation for the DC current flow:
>>> electrokinetic += integral(sur, -sigma * grad(dof(v) * grad(tf(v))))
>>>
>>> electrokinetic.solve()
>>> v.write(sur, "v.pos", 2)
>>> (-grad(v)*sigma).write(sur, "j.pos", 2)
>>>
>>> resistance = V.getvalue()/I.getvalue()
>>> print(f"Resistance is {resistance} Ohm")
```

`setupdateaccuracy`

`def setupdateaccuracy( self, extraintegrationorder: int ) ‑> None`

This method allows tuning the integration order in the projection used to update the field value after hp-adaptivity. A positive/negative argument increases/decreases the accuracy but slows down/speeds up the update.

###### Example

```
>>> ...
>>> v.setupdateaccuracy(2)
```

`setvalue`

`def setvalue( *args, **kwargs ) ‑> None`

This sets the field value on the region `physreg`

to `input`

expression. An extra int argument `extraintegrationdegree`

can be used to increase or decrease the default integration order when computing the projection of the expression on the field. Increasing it can give a more accurate computation of the expression but might take longer. The default integration order is equal to "*field order* ".

###### Examples

**Example 1:** `field.setvalue(physreg:int, input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> v.setvalue(vol, 12)
```

This sets the field value on region *vol* to .

**Example 2:** `field.setvalue(physreg:int, meshdeform:expression, input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> u = feild("h1xyz")
>>> v.setorder(vol, 1)
>>> u.setorder(vol, 1)
>>> v.setvalue(vol, u, expression(12))
```

This sets the field value on region *vol* to expression but on a mesh deformed by `meshdeform`

.

**Example 3:** `field.setvalue(physreg:int, numfftharms:int, input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v1 = field("h1", [2,3])
>>> v2 = field("h1", [1,4,5])
>>> v1.setorder(vol, 1)
>>> v2.setorder(vol, 1)
>>> v2.setvalue(vol, 5, v1*v1)
>>> v2.write(vol, "v2.vtk", 1)
```

This calls an FFT for the calculation required for nonlinear multi-harmonic expressions. The FFT is computed at `numfftharms`

timesteps.

**Example 4:** `field.setvalue(physreg:int, numfftharms:int, meshdeform:expression, input:expression, extraintegrationdegree:int=0)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v1 = field("h1", [2,3])
>>> v2 = field("h1", [1,4,5])
>>> u = field("h1xyz")
>>> v1.setorder(vol, 1)
>>> v2.setorder(vol, 1)
>>> u.setorder(vol, 1)
>>> v2.setvalue(vol, 5, u, v1*v1)
```

This calls an FFT for the calculation and the expression is evaluated on a mesh deformed by `meshdeform`

.

**Example 5:** `field.setvalue(physreg:int)`

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> v.setvalue(vol)
```

This sets the field value on region *vol* to .

`sin`

`def sin( self, freqindex: int ) ‑> quanscient.field`

This gets the "h1xyz" type field that is the harmonic at `freqindex`

times the fundamental frequency in field .

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> u = field("h1xyz", [1,2,3,4,5])
>>> us = u.sin(2) # gets the harmonic 4
```

`write`

`def write( *args, **kwargs ) ‑> None`

`writeraw`

`def writeraw( self, physreg: int, filename: str, isbinary: bool = False, extradata: List[float] = [] ) ‑> None`

This writes a (possibly multi-harmonic) field on a given region to disk in the **compact .slz sparselizard format**. If `isbinary=False`

the output format is in ASCII and with `isbinary=True`

the output is in binary format. In the latter case, the .slz.gz extension can also be used to write to gz compressed -slz format (the most compact version). While the binary file is more compact on disk it might be less portable across different platforms than the ASCII version. The last input argument allows storing extra data (timestep, parameter values, ..) that can be loaded back from the `loadraw`

output.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v=field("h1xyz"); x=field("x"); y=field("y"); z=field("z")
>>> v.setorder(vol, 2)
>>> v.setvalue(vol, array3x1(x*x, y*y, z*z))
>>> v.writeraw(vol, "v.slz.gz", True)
```

`formulation`

`class formulation`

The formulation object holds the port relations and the weak form terms of the problem to solve.

#### Example

The following creates an empty formulation object:

```
>>> mymesh = mesh("disk.msh")
>>> myformulation = formulation()
```

#### Methods

`A`

`def A( self, keepfragments: bool = False ) ‑> quanscient.mat`

This gives the matrix (of ) that was assembled during the `formulation.generate()`

call. By default the `keepfragments`

argument is False which means that the generated matrix is no longer kept in the formulation after returning it to a `mat`

object. However, if you select True for `keepfragments`

it means the generated matrix is kept in the formulation and will be added to the matrix assembled in any subsequent `formulation.generate()`

call.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generate()
>>> A = projection.A(); # equivalent to A = projection.A(keepfragments=False)
```

###### See Also

`formulation.rhs()`

, `formulation.b()`

`C`

`def C( self, keepfragments: bool = False ) ‑> quanscient.mat`

This gives the damping matrix that was assembled during the `formulation.generate()`

call. The damping matrix is a matrix that is assembled with only those terms in the formulation which have a dof and that dof has a first-order time derivative applied to it (i.e ). For multi-harmonic simulations damping matrix is empty.

By default, the `keepfragments`

argument is False which means that the generated matrix is no longer kept in the formulation after returning it to a `mat`

object. However, if you select True for `keepfragments`

it means the generated matrix is kept in the formulation and will be added to the matrix assembled in any subsequent `formulation.generate()`

call.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generate()
>>> C = projection.C(); # equivalent to C = projection.C(keepfragments=False)
```

###### See Also

`formulation.K()`

, `formulation.M()`

`K`

`def K( self, keepfragments: bool = False ) ‑> quanscient.mat`

This gives the stiffness matrix that was assembled during the `formulation.generate()`

call. The stiffness matrix is a matrix that is assembled with only those terms in the formulation which have a dof and that dof has no time derivative applied to it. For multi-harmonic formulations, the stiffness matrix holds the assembly of all the terms.

By default, the `keepfragments`

argument is False which means that the generated matrix is no longer kept in the formulation after returning it to a `mat`

object. However, if you select True for `keepfragments`

it means the generated matrix is kept in the formulation and will be added to the matrix assembled in any subsequent `formulation.generate()`

call.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generate()
>>> K = projection.K(); # equivalent to K = projection.K(keepfragments=False)
```

###### See Also

`formulation.C()`

, `formulation.M()`

`M`

`def M( self, keepfragments: bool = False ) ‑> quanscient.mat`

This gives the mass matrix that was assembled during the `formulation.generate()`

call. The mass matrix is a matrix that is assembled with only those terms in the formulation which have a dof and that dof has a second-order time derivative applied to it (i.e ). For multi-harmonic simulations mass matrix is empty.

By default, the `keepfragments`

argument is False which means that the generated matrix is no longer kept in the formulation after returning it to a `mat`

object. However, if you select True for `keepfragments`

it means the generated matrix is kept in the formulation and will be added to the matrix assembled in any subsequent `formulation.generate()`

call.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generate()
>>> M = projection.M(); # equivalent to M = projection.M(keepfragments=False)
```

###### See Also

`formulation.K()`

, `formulation.C()`

`allcountdofs`

`def allcountdofs( self ) ‑> int`

This is a collective MPI operation and hence must be called by all the ranks. It returns on every rank the global number of degrees of freedom defined in the scattered formulation. The count is exact if for each field the number of unknowns associated to each element matches across touching ranks. It is an estimation otherwise.

`allsolve`

`def allsolve( *args, **kwargs ) ‑> List[float]`

This is a collective MPI operation and hence must be called by all the ranks. This solves the formulation on all the ranks using DDM. The initial solution is taken from the fields' state. The relative residual history is returned. This method can be used for both linear and nonlinear problems.

###### Examples

**Example 1:** `formulation.allsolve(relrestol:double, maxnumit:int, soltype:str="lu", verbosity:int=1)`

This is used for linear problems. The `relrestol`

and `maxnumit`

arguments correspond to the stopping criteria for DDM solver. The DDM iterations stop if either relative residual tolerance is less than `relrestol`

or if the number of DDM iteration reaches `maxnumit`

.

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> projection.allsolve(1e-8, 500)
>>> v.write(vol, f"v_{getrank()}.vtu", 1)
```

**Example 2:** `formulation.allsolve(relrestol:double, maxnumit:int, nltol:double, maxnumnlit:int, relaxvalue:double=1, soltype:str="lu", verbosity:int=1)`

This is used for nonlinear problems. The `relrestol`

and `maxnumit`

arguments correspond to the stopping criteria for DDM solver. The DDM iteration stops if either relative residual tolerance is less than `relrestol`

or if the number of DDM iterations reaches `maxnumit`

. A nonlinear fixed-point iteration is performed for at most `maxnumnlit`

or until the relative error (norm of relative solution vector change) is smaller than the tolerance prescribed in `nltol`

. A relaxation value can be provided with `relaxvalue`

argument. Usually, a relaxation value less than (under-relaxation) is used to avoid divergence of a solution.

```
>>> ...
>>> projections.allsolve(1e-8, 500, 1e-6, 200, 0.75)
```

`b`

`def b( self, keepvector: bool = False, dirichletandportupdate: bool = True ) ‑> quanscient.vec`

This returns the rhs vector that was assembled during the `formulation.generate()`

call. By default the `keepvector`

argument is False which means that the generated rhs vector is no longer kept in the formulation after returning it to a `vec`

object. However, if you select True for `keepvector`

it means the generated rhs vector is kept in the formulation and will be added to the rhs vector assembled in any subsequent `formulation.generate()`

call.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generate()
>>> b = projection.b(); # equivalent to b = projection.b(keepvector=False)
```

###### See Also

`formulation.rhs()`

, `formulation.A()`

`countdofs`

`def countdofs( self ) ‑> int`

This returns the number of degrees of freedom defined in the formulation.

###### Example

```
>>> vol=1; sur=2
>>> mymesh = mesh("disk.msh")
>>>
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> numdofs = projection.countdofs()
82
```

`generate`

`def generate( *args, **kwargs ) ‑> None`

This assembles all the terms in the formulation.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2) # here 0 is the extra integration order, 2 is the block number assigned.
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v)) # default block number is 0
>>> projection += integral(vol, dtdt(dof(v))*tf(v)) # default block number is 0
>>>
>>> projection.generate()
```

A block number can be passed as an argument to generate only the necessary terms in the formulation. For example, the following generates only the block number 2. (i.e the first integral term)

```
>>> projection.generate(2)
```

A list of block numbers can also be passed as an argument. For example, the following generates all terms with block numbers 0 and 2. For this formulation, it means all terms are generated since these are the only block numbers existing. and 0 (default) block numbers.

```
>>> projection.generate([0,2])
```

`generatedampingmatrix`

`def generatedampingmatrix( self ) ‑> None`

This assembles only those terms in the formulation which have a dof and that dof has a first-order time derivative applied to it (i.e ). For multi-harmonic simulations, it generates nothing.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generatedampingmatrix() # Here it only generates 'dt(dof(v))*tf(v)'
```

###### See Also

`formulation.generate()`

, `formulation.generatestiffnessmatrix()`

, `formulation.generatemassmatrix()`

, `formulation.rhs()`

`generatemassmatrix`

`def generatemassmatrix( self ) ‑> None`

This assembles only those terms in the formulation which have a dof and that dof has a second-order time derivative applied to it (i.e ). For multi-harmonic formulations, it generates nothing.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generatemassmatrix() # Here it only generates 'dtdt(dof(v))*tf(v)'
```

###### See Also

`formulation.generate()`

, `formulation.generatestiffnessmatrix()`

, `formulation.generatedampingmatrix()`

, `formulation.rhs()`

`generaterhs`

`def generaterhs( self ) ‑> None`

This assembles only the terms in the formulation which have no dof.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generaterhs() # Here it only generates '-2*tf(v)'
```

###### See Also

`formulation.generate()`

, `formulation.generatestiffnessmatrix()`

, `formulation.generatedampingmatrix()`

, `formulation.generatemassmatrix()`

`generatestiffnessmatrix`

`def generatestiffnessmatrix( self ) ‑> None`

This assembles only those terms in the formulation which have a dof and that **dof has no time derivative applied to it. For multi-harmonic formulations it generates all the terms.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generatestiffnessmatrix() # Here it only generates dof(v)*tf(v)
```

###### See Also

`formulation.generate()`

, `formulation.generatedampingmatrix()`

, `formulation.generatemassmatrix()`

, `formulation.rhs()`

`getmatrix`

`def getmatrix( self, KCM: int, keepfragments: bool = False, additionalconstraints: List[indexmat] = [] ) ‑> quanscient.mat`

Depending on `KCM`

argument value, it returns the corresponding matrix as follows:

- , returns the stiffness matrix
- , returns the damping matrix
- , returns the mass matrix

`keepfragments`

argument is False which means that the generated matrix is no longer kept in the formulation after returning it to a `mat`

object. However, if you select True for `keepfragments`

it means the generated matrix is kept in the formulation and will be added to the matrix assembled in any subsequent `formulation.generate()`

call.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generate()
>>> K = projection.getmatrix(0); # equivalent to K = projection.getmatrix(KCM=0, keepfragments=False) or just K = projection.K()
>>> C = projection.getmatrix(1); # equivalent to C = projection.getmatrix(KCM=1, keepfragments=False) or just C = projection.C()
>>> M = projection.getmatrix(2); # equivalent to M = projection.getmatrix(KCM=2, keepfragments=False) or just M = projection.M()
```

###### See Also

`formulation.K()`

, `formulation.C()`

, `formulation.M()`

`islinear`

`def islinear( self ) ‑> bool`

This returns True if the formulation defined is linear, otherwise, it returns False.

###### Examples

```
>>> vol=1; sur=2
>>> mymesh = mesh("disk.msh")
>>>
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> projection.islinear()
True
```

`rhs`

`def rhs( self, keepvector: bool = False, dirichletandportupdate: bool = True ) ‑> quanscient.vec`

This returns the rhs vector that was assembled during the `formulation.generate()`

call. By default the `keepvector`

argument is False which means that the generated rhs vector is no longer kept in the formulation after returning it to a `vec`

object. However, if you select True for `keepvector`

it means the generated rhs vector is kept in the formulation and will be added to the rhs vector assembled in any subsequent `formulation.generate()`

call. This is the same as `formulation.b()`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v), 0, 2)
>>> projection += integral(vol, dt(dof(v))*tf(v) - 2*tf(v))
>>> projection += integral(vol, dtdt(dof(v))*tf(v))
>>>
>>> projection.generate()
>>> rhs = projection.rhs(); # equivalent to rhs = projection.rhs(keepvector=False)
```

###### See Also

`formulation.b()`

, `formulation.A()`

`solve`

`def solve( self, soltype: str = 'lu', diagscaling: bool = False, blockstoconsider: List[int] = [-1] ) ‑> None`

This generates the formulation, solves the algebraic problem with a direct solver then saves all the data in vector to the fields defined in the formulation.

###### Parameters

** soltype** :

`str`

: Direct solver type: "lu" or "cholesky". Default is "lu".** diagscaling** :

`bool`

: If True, diagonal scaling preconditioning is applied. Default is False.** blockstoconsider** :

`List[int]`

: List of blocks considered for solving. Default is meaning all the blocks are considered.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> projection.solve(); # equivalent to projection.solve("lu", False, -1);
>>> v.write(vol, "v.vtu", 1)
```

`genalpha`

`class genalpha( formul: formulation, dtxinit: vec, dtdtxinit: vec, verbosity: int = 3, isrhskcmconstant: List[bool] = [False, False, False, False] )`

This defines the genalpha object to solve in time the formulation `formul`

with the fields' state as an initial solution for , `dtxinit`

for , `dtdtxinit`

for . The `isrhskcmconstant`

list can be used to specify whether the RHS vector, K matrix, C matrix and M matrix are constant in time (default is False). If `isrhskcmconstant[i]=True`

, the corresponding vector/matrix is generated once and then reused. In `rhskcconstant[i]`

:

- corresponds to the rhs vector
- corresponds to the K matrix
- corresponds to the C matrix
- corresponds to the M matrix

If the K, C and M matrices are constant in time, the factorization of the algebraic problem is also reused.

The genalpha object allows performing a generalized alpha time resolution for a problem of the form , be it linear or nonlinear. The solutions for as well as and are made available. For nonlinear problems, a fixed-point iteration is performed at every timestep until the relative error (norm of relative solution vector change) is less than the prescribed tolerance.

The generalized alpha method comes with four parameters (, , and ) that can be tuned to adjust the properties of the time resolution method (convergence order, stability, high frequency, damping and so on). When both parameters are set to zero, a classical Newmark iteration is obtained. By default, the parameters are set to (, , and ) which corresponds to an unconditionally stable Newmark iteration.

A convenient way proposed to set the four parameters is to specify a high-frequency dissipation level and let the four parameters be deduced accordingly. This gives a set of parameters leading to an unconditionally stable, second-order accurate algorithm possessing an optimal combination of high-frequency and low-frequency dissipation. More information on the generalized alpha method can be found in the paper *A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-alpha method*.

#### Notes

Even if the rhs vector can be reused the Dirichlet constraints will nevertheless be recomputed at each timestep.

#### Methods

`allnext`

`def allnext( *args, **kwargs ) ‑> None`

This is a collective MPI operation and hence must be called by all ranks. It is similar to the `genalpha.next()`

function but the resolution is performed on all ranks using DDM. This method runs the generalized alpha algorithm for one timestep. After the call, the time and field values are updated on all the DDM ranks. This method can be used for both linear and nonlinear problems.

###### Examples

**Example 1:** `genalpha.allnext(relrestol: double, maxnumit: int, timestep: double)`

This is used for linear problems. The `relrestol`

and `maxnumit`

arguments correspond to the stopping criteria for DDM solver. The DDM iterations stop if either relative residual tolerance is less than `relrestol`

or if the number of DDM iterations reaches `maxnumit`

. Use `timestep=-1`

for automatic time adaptivity.

**Example 2:** `genalpha.next(relrestol: double, maxnumit: int, timestep: double, maxnumnlit:int)`

This is used for nonlinear problems. The `relrestol`

and `maxnumit`

arguments correspond to the stopping criteria for DDM solver. The DDM iteration stops if either relative residual tolerance is less than `relrestol`

or if the number of DDM iterations reaches `maxnumit`

. A nonlinear fixed-point iteration is performed for at most `maxnumnlit`

or until the relative error (norm of relative solution vector change) is smaller than the tolerance prescribed in `genalpha.settolerance()`

. Set `timestep=-1`

for automatic time-adaptivity and `maxnumnlit=-1`

for unlimited nonlinear iterations. This method returns the number of nonlinear iterations performed.

###### See Also

`count`

`def count( self ) ‑> int`

This counts the total number of steps computed.

`gettimederivative`

`def gettimederivative( self ) ‑> List[quanscient.vec]`

This returns a list containing current time derivative solutions. The first element in the list contains the solution of the first time derivative and the second element contains the solution of the second time derivative .

###### See Also

`gettimes`

`def gettimes( self ) ‑> List[float]`

This returns all the time values stepped through.

`gettimestep`

`def gettimestep( self ) ‑> float`

This returns the current timestep.

`next`

`def next( *args, **kwargs ) ‑> None`

This runs the generalized alpha algorithm for one timestep. After the call, the time and field values are updated. This method can be used for both linear and nonlinear problems.

###### Examples

**Example 1:** `genalpha.next(timestep: double)`

This is used for linear problems. \boldsymbol{Use for automatic time-adaptivity}.

**Example 2:** `genalpha.next(timestep: double, maxnumnlit:int)`

This is used for nonlinear problems. A nonlinear fixed-point iteration is performed for at most `maxnumnlit`

or until the relative error (norm of relative solution vector change) is smaller than the tolerance prescribed in `genalpha.settolerance()`

. Set `timestep=-1`

for automatic time-adaptivity and `maxnumnlit=-1`

for unlimited nonlinear iterations. This method returns the number of nonlinear iterations performed.

`postsolve`

`def postsolve( self, formuls: List[formulation] ) ‑> None`

This defines the set of formulations that must be solved every resolution of the formulation provided to the genalpha constructor. The formulations provided here must lead to a system of the form (no damping or mass matrix allowed).

###### See Also

`presolve`

`def presolve( self, formuls: List[formulation] ) ‑> None`

This defines the set of formulations that must be solved every resolution of the formulation provided to the genalpha constructor. The formulations provided here must lead to a system of the form (no damping or mass matrix allowed).

###### See Also

`setadaptivity`

`def setadaptivity( self, tol: float, mints: float, maxts: float, reffact: float = 0.5, coarfact: float = 2.0, coarthres: float = 0.5 ) ‑> None`

This sets the configuration for automatic time adaptivity. The timestep will be adjusted between the minimum timestep () and maximum timesteps () to reach the requested relative error tolerance (). The relative error is defined as

to measure the relative deviation from a constant time derivative.

Arguments and give the factor to use when the time step is refined or coarsened respectively. The timestep is refined when the relative error is above or when the maximum number of nonlinear iterations is reached. The timestep is coarsened when the relative error is below the product and the nonlinear loop has converged in less than the maximum number of iterations.

`setparameter`

`def setparameter( *args, **kwargs ) ‑> None`

This is used to set the parameters of the generalized alpha method.

To set the four parameters (, , and ), four arguments are passed, one for each parameter:

```
>>> genalpha.setparameter(b: double, g: double, ad: double, am: double)
```

To set the high-frequency dissipation (), only one argument is passed:

```
>>> genalpha.setparameter(rinf: double)
The range of high-frequency dissipation is in the range $0 \leq \rho_{\infty} \leq 1$. The four generalized alpha
parameters are optimally deduced from ($\rho_{\infty}$). The deduced parameters lead to an unconditionally stable,
second-order accurate algorithm possessing an optimal combination of high-frequency and low-frequency dissipation. Lower
($\rho_{\infty}$) values lead to more dissipation.
```

`setrelaxationfactor`

`def setrelaxationfactor( self, relaxfact: float ) ‑> None`

This sets the relaxation factor for the fixed-point nonlinear iteration performed at every timestep for nonlinear problems. If the relaxation factor is not set, the default value of is set. If is the solution obtained at a current iteration, is solution at previous iteration, then the new solution at the current iteration is updated as

where is the relaxation factor.

`settimederivative`

`def settimederivative( self, sol: List[vec] ) ‑> None`

This sets the current solution for the time derivatives and to `sol[0]`

and `sol[1]`

respectively.

###### See Also

`settimestep`

`def settimestep( self, timestep: float ) ‑> None`

This sets the current timestep.

`settolerance`

`def settolerance( self, nltol: float ) ‑> None`

This sets the tolerance for the fixed-point nonlinear iteration performed at every timestep for nonlinear problems. If the tolerance is not set, the default value of is considered.

`setverbosity`

`def setverbosity( self, verbosity: int ) ‑> None`

This sets the verbosity level. For debugging, higher verbosity is recommended. If the verbosity is not set, the default value of is considered.

`impliciteuler`

`class impliciteuler( formul: formulation, dtxinit: vec, verbosity: int = 3, isrhskcconstant: List[bool] = [False, False, False] )`

This defines the impliciteuler object to solve in time the formulation `formul`

with the fields' state as an initial solution for and `dtxinit`

for . The `isrhskcconstant`

list can be used to specify whether the RHS vector, K matrix and C matrix are constant in time (default is False). If `isrhskcconstant[i]=True`

, the corresponding vector/matrix is generated once and then reused. In `rhskcconstant[i]`

:

- corresponds to the rhs vector
- corresponds to the K matrix
- corresponds to the C matrix

If the K and C matrix are constant in time, the factorization of the algebraic problem is also reused.

The impliciteuler object allows performing an implicit (backward) Euler time resolution for a problem of the form , be it linear or nonlinear. The solutions for , as well as , are made available. For nonlinear problems, a fixed-point iteration is performed at every timestep until the relative error (norm of relative solution vector change) is less than the prescribed tolerance.

#### Notes

Even if the rhs vector can be reused the Dirichlet constraints will nevertheless be recomputed at each timestep.

#### Methods

`allnext`

`def allnext( *args, **kwargs ) ‑> None`

This is a collective MPI operation and hence must be called by all ranks. It is similar to the `impliciteuler.next()`

function but the resolution is performed on all ranks using DDM. This method runs the implicit Euler algorithm for one timestep. After the call, the time and field values are updated on all the DDM ranks. This method can be used for both linear and nonlinear problems.

###### Examples

**Example 1:** `impliciteuler.allnext(relrestol: double, maxnumit: int, timestep: double)`

This is used for linear problems. The `relrestol`

and `maxnumit`

arguments correspond to the stopping criteria for DDM solver. The DDM iterations stop if either relative residual tolerance is less than `relrestol`

or if the number of DDM iterations reaches `maxnumit`

. Use `timestep=-1`

for automatic time adaptivity.

**Example 2:** `impliciteuler.next(relrestol: double, maxnumit: int, timestep: double, maxnumnlit:int)`

This is used for nonlinear problems. The `relrestol`

and `maxnumit`

arguments correspond to the stopping criteria for DDM solver. The DDM iteration stops if either relative residual tolerance is less than `relrestol`

or if the number of DDM iterations reaches `maxnumit`

. A nonlinear fixed-point iteration is performed for at most `maxnumnlit`

or until the relative error (norm of relative solution vector change) is smaller than the tolerance prescribed in `impliciteuler.settolerance()`

. Set `timestep=-1`

for automatic time-adaptivity and `maxnumnlit=-1`

for unlimited nonlinear iterations. This method returns the number of nonlinear iterations performed.

###### See Also

`count`

`def count( self ) ‑> int`

This counts the total number of steps computed.

`gettimederivative`

`def gettimederivative( self ) ‑> quanscient.vec`

This returns the current solution for the first time derivative .

###### See Also

`impliciteuler.settimederivative()`

`gettimes`

`def gettimes( self ) ‑> List[float]`

This returns all the time values stepped through.

`gettimestep`

`def gettimestep( self ) ‑> float`

This returns the current timestep.

`next`

`def next( *args, **kwargs ) ‑> None`

This runs the implicit Euler algorithm for one timestep. After the call, the time and field values are updated. This method can be used for both linear and nonlinear problems depending on the number of arguments passed.

###### Examples

**Example 1:** `implicit.next(timestep: double)`

This is used for linear problems. \boldsymbol{Use for automatic time-adaptivity}.

**Example 2:** `implicit.next(timestep: double, maxnumnlit:int)`

This is used for nonlinear problems. A nonlinear fixed-point iteration is performed for at most `maxnumnlit`

or until the relative error (norm of relative solution vector change) is smaller than the tolerance prescribed in `impliciteuler.settolerance()`

. Set `timestep=-1`

for automatic time-adaptivity and `maxnumnlit=-1`

for unlimited nonlinear iterations. This method returns the number of nonlinear iterations performed.

`postsolve`

`def postsolve( self, formuls: List[formulation] ) ‑> None`

This defines the set of formulations that must be solved every resolution of the formulation provided to the impliciteuler constructor. The formulations provided here must lead to a system of the form (no damping or mass matrix allowed).

###### See Also

`presolve`

`def presolve( self, formuls: List[formulation] ) ‑> None`

This defines the set of formulations that must be solved every resolution of the formulation provided to the impliciteuler constructor. The formulations provided here must lead to a system of the form (no damping or mass matrix allowed).

###### See Also

`setadaptivity`

`def setadaptivity( self, tol: float, mints: float, maxts: float, reffact: float = 0.5, coarfact: float = 2.0, coarthres: float = 0.5 ) ‑> None`

This sets the configuration for automatic time adaptivity. The timestep will be adjusted between the minimum timestep () and maximum timesteps () to reach the requested relative error tolerance (). The relative error is defined as

to measure the relative deviation from a constant time derivative.

Arguments and give the factor to use when the time step is refined or coarsened respectively. The timestep is refined when the relative error is above or when the maximum number of nonlinear iterations is reached. The timestep is coarsened when the relative error is below the product and the nonlinear loop has converged in less than the maximum number of iterations.

`setrelaxationfactor`

`def setrelaxationfactor( self, relaxfact: float ) ‑> None`

This sets the relaxation factor for the fixed-point nonlinear iteration performed at every timestep for nonlinear problems. If the relaxation factor is not set, the default value of is set. If is the solution obtained at a current iteration, is solution at previous iteration, then the new solution at the current iteration is updated as

where is the relaxation factor.

`settimederivative`

`def settimederivative( self, sol: vec ) ‑> None`

This sets the current solution for the first time derivatives to `sol[0]`

.

###### See Also

`impliciteuler.gettimederivative()`

`settimestep`

`def settimestep( self, timestep: float ) ‑> None`

This sets the current timestep.

`settolerance`

`def settolerance( self, nltol: float ) ‑> None`

This sets the tolerance for the fixed-point nonlinear iteration performed at every timestep for nonlinear problems. If the tolerance is not set, the default value of is considered.

`setverbosity`

`def setverbosity( self, verbosity: int ) ‑> None`

This sets the verbosity level. For debugging, higher verbosity is recommended. If the verbosity is not set, the default value of is considered.

`indexmat`

`class indexmat( **kwargs )`

The `indexmat`

object stores a row-major array of integers that corresponds to a dense matrix. For storing an array of doubles, see `densemat`

object.

#### Examples

There are many ways of instantiating an `indexmat`

object. There are listed below:

**Example 1**: `indexmat(numberofrows:int, numberofcolumns:int)`

The following creates a matrix with 2 rows and 3 columns. The entries may be undefined.

```
>>> B = indexmat(2,3)
```

**Example 2**: `indexmat(numberofrows:int, numberofcolumns:int, initvalue:int)`

This creates a matrix with 2 rows and 3 columns. All entries are assigned the value `initvalue`

.

```
>>> B = indexmat(2,3, 12)
>>> B.print()
Matrix size is 2x3
12 12 12
12 12 12
```

**Example 3**: `indexmat(numberofrows:int, numberofcolumns:int, valvec:List[int])`

This creates a matrix with 2 rows and 3 columns. The entries are assigned the values of `valvec`

. The length of `valvec`

is expected to be equal to the total count of entries in the matrix. So for creating a matrix of size , length of `valvec`

must be 6.

```
>>> B = indexmat(2,3, [1,2,3,4,5,6])
>>> B.print()
Matrix size is 2x3
1 2 3
4 5 6
```

**Example 4**: `indexmat(numberofrows:int, numberofcolumns:int, init:int, step:int)`

This creates a matrix with 2 rows and 3 columns. The first entry is assigned the value `init`

and the consecutive entries are assigned values that increase by steps of `step`

.

```
>>> B = indexmat(2,3, 0, 1)
>>> B.print()
Matrix size is 2x3
0 1 2
3 4 5
```

**Example 5**: `indexmat(input:List[indexmat])`

This creates a matrix that is the vertical concatenation of `input`

matrices. Since the concatenation occurs vertically, the number of columns in all the input matrices must match.

```
>>> A = indexmat(2,3, 0)
>>> B = indexmat(1,3, 2)
>>> AB = indexmat([A,B])
>>> AB.print()
Matrix size is 3x3
0 0 0
0 0 0
2 2 2
```

#### Methods

`count`

`def count( self ) ‑> int`

This counts and returns the total number of entries in the dense matrix.

###### Example

```
>>> B = indexmat(2,3)
>>> B.count()
6
```

`countcolumns`

`def countcolumns( self ) ‑> int`

This counts and returns the number of columns in the dense matrix.

###### Example

```
>>> B = indexmat(2,3)
>>> B.countcolumns()
3
```

`countrows`

`def countrows( self ) ‑> int`

This counts and returns the number of rows in the dense matrix.

###### Example

```
>>> B = indexmat(2,3)
>>> B.countrows()
2
```

`print`

`def print( self ) ‑> None`

This prints the entries of the dense matrix.

###### Example

```
>>> B = indexmat(2,3, 0,1)
>>> B.print()
Matrix size is 2x3
0 1 2
3 4 5
```

`printsize`

`def printsize( self ) ‑> None`

This prints the size of the dense matrix.

###### Example

```
>>> B = indexmat(2,3)
>>> B.printsize()
Matrix size is 2x3
```

`integration`

`class integration`

`logformat`

`class logformat( value: int )`

Holds possible log formats

Members -----=

plain

json

#### Class variables

`json`

Type: `quanscient.logformat`

`plain`

Type: `quanscient.logformat`

`mat`

`class mat( **kwargs )`

The `mat`

object holds a sparse algebraic square matrix.

#### Raises

`RuntimeError`

: If a mesh object is not available before creating a `mat`

object.

#### Notes

Before creating a `mat`

object, ensure that a mesh object is available. If a mesh object is not already available, create an empty mesh object.

#### Examples

There are many ways of instantiating an `indexmat`

object. There are listed below:

**Example 1:** `mat(matsize:int, rowaddresses:indexmat, coladdresess:indexmat, vals:densemat)`

This creates a sparse matrix object of size `matsize`

`matsize`

. The `rowaddresses`

and `coladdresess`

provide the location (row, col) of non-zero values in the sparse matrix. The non-zero values are provided in the dense matrix `vals`

. Note that a mesh object must already be available before instantiating `mat`

object.

```
>>> rows = indexmat(7,1, [0,0,1,1,1,2,2])
>>> cols = indexmat(7,1, [0,1,0,1,2,1,2])
>>> vals = densemat(7,1, [11,12,13,14,15,16,17])
>>>
>>> mymesh = mesh()
>>> A = mat(3, rows, cols, vals)
>>> A.print()
A block 3x3:
```

Mat Object: 1 MPI processes type: seqaij row 0: (0, 11.) (1, 12.) row 1: (0, 13.) (1, 14.) (2, 15.) row 2: (1, 16.) (2, 17.)

D block 3x0:

Mat Object: 1 MPI processes type: seqaij row 0: row 1: row 2:

**Example 2:** `mat(myformulation:formulation, rowaddresses:indexmat, coladdresses:indexmat, vals:densemat)`

This creates a sparse matrix object whose `dof()`

structure is the one in the formulation `projection`

. The `rowaddresses`

and `coladdresess`

provide the location (row, col) of non-zero values in the sparse matrix. The non-zero values are provided in the dense matrix `vals`

.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>>
>>> addresses = indexmat(numberofrows=projection.countdofs(), numberofcolumns=1, init=0, step=1)
>>> vals = densemat(numberofrows=projection.countdofs(), numberofcolumns=1, init=12)
>>>
>>> A = mat(formulation=projection, rowaddresses=addresses, coladdresses=addresses, densemat=vals)
```

#### Methods

`copy`

`def copy( self ) ‑> quanscient.mat`

This creates a full copy of the matrix. Only the values are copied. (E.g: the `mat.reusefactorization()`

is set back to the default *no* reuse.)

###### Example

```
>>> A =
>>> copiedmat = A.copy()
```

`countcolumns`

`def countcolumns( self ) ‑> int`

This counts and returns the number of columns in the matrix.

###### Example

```
>>> numcols = A.countcolumns()
```

`countnnz`

`def countnnz( self ) ‑> int`

This counts and returns the number of non-zero entries in the matrix which is the sub-matrix of with eliminated Dirichlet constraints. Refer `mat.getainds()`

.

If the requested information is not available, then is returned.

###### Example

```
>>> numnnz = A.countnnz()
```

`countrows`

`def countrows( self ) ‑> int`

This counts and returns the number of rows in the matrix.

###### Example

```
>>> numrows = A.countrows()
```

`getainds`

`def getainds( self ) ‑> quanscient.indexmat`

Let us call *dinds* the set of unknowns that have a Dirichlet constraint and *ainds* the remaining unknowns. The `mat`

object holds sub-matrices and such that

where is a square matrix equal to with eliminated Dirichlet constraints. is an all zero matrix and is the square identity matrix of all Dirichlet constraints. Matrices and are stored with their local indexing. The methods `mat.getainds()`

and `mat.getdinds()`

gives the global indexing (i.e index in ) of each local index in and .

###### Example

```
>>> ainds = A.getainds()
```

###### See Also

`getdinds`

`def getdinds( self ) ‑> quanscient.indexmat`

This outputs *dinds*.

###### Example

```
>>> dinds = A.getdinds()
```

###### See Also

`print`

`def print( self ) ‑> None`

This prints the matrix size and values.

###### Example

```
>>> A.print()
```

`reusefactorization`

`def reusefactorization( self ) ‑> None`

The matrix factorization will be reused in `allsolve()`

.

`mesh`

`class mesh( **kwargs )`

The mesh object holds the finite element mesh of the geometry.

#### Examples

A mesh object based on a mesh file can be created through the native reader or via the GMSH API. To get more information on the physical regions of the mesh, the `verbosity`

argument can be set to .

```
>>> # Creating a mesh object with the native reader:
>>> mymesh = mesh("disk.msh")
>>>
>>> # Creating a mesh object with GMSH API:
>>> mymesh = mesh("gmsh:disk.msh")
```

In the domain decomposition framework, creating a mesh object requires two additional arguments: `globalgeometryskin`

and `numoverlaplayers`

. Furthermore, the mesh is treated as a part of a global mesh. Each MPI rank owns only a part of the global mesh and all ranks must perform the call collectively. The argument `globalgeometryskin`

is the part of the global mesh skin that belongs to the current rank. It can only hold elements of dimension one lower than the geometry dimension. The global mesh skin cannot intersect itself. The mesh parts are overlapped by the number of overlap layers requested. More than one overlap layer cannot be guaranteed everywhere as the overlapping is limited to the direct neighbouring domains. `mesh(filename:str, globalgeometryskin:int, numoverlaplayers:int, verbosity:int=1)`

In the above examples, the mesh objects were created based on a mesh file. Similarly, mesh objects can be created based on `shape`

objects.

```
>>> # define physical regions
>>> faceregionnumber=1; lineregionnumber=2
>>>
>>> # define a quadrangle shape object
>>> quadface = shape("quadrangle", faceregionnumber, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [10,6,10,6])
>>>
>>> # get the leftline from the contour of the quad shape object
>>> contourlines = quadface.getsons() # returns a list
>>> leftline = contourlines[3]
>>> leftline.setphysicalregion(lineregionnumber)
>>>
>>> # create mesh object based on the quadrangle shape and its left-side line
>>> mymesh = mesh([quadface, leftline])
>>> mymesh.write("quadmesh.msh")
```

Creating a mesh object from the shape object can be carried out also in the domain decomposition framework using the following syntax: `mesh(inputshapes:List[shape], globalgeometryskin:int, numoverlaplayers:int, verbosity:int=1)`

It is also possible to combine multiple meshes. Elements shared by the input meshes can either be merged or not by setting the bool value for argument `mergeduplicates`

. For every input mesh, a new physical region containing all elements is created. Set verbosity equal to 2 to get information on physical regions in the mesh.

```
>>> mymesh = mesh("disk.msh")
>>> mymesh.shift(2,1,0) # all entities are shifted by x,y,z amount
>>> mymesh.write("shifted.msh")
>>>
>>> mergedmesh = mesh(True, ["disk.msh", "shifted.msh"])
>>> mergedmesh.write("merged.msh")
>>>
>>> mergedmesh = mesh("merged.msh", 2)
```

#### Methods

`extrude`

`def extrude( self, newphysreg: int, newbndphysreg: int, bnd: int, extrudelens: List[float] ) ‑> None`

This extrudes the boundary region *bnd*. After the extrusion process, *newphysreg* will contain the extruded region and *newbndphysreg* will contain the extrusion end boundary.

###### Parameters

** newphysreg** :

`int`

: The physical region that will contain the extruded region.** newbndphysreg** :

`int`

: The physical region that will contain the end boundary of extrusion.** bnd** :

`int`

: The boundary region that is extruded.** extrudelens** :

`List[float]`

: A list specifying the size of each layer in the extrusion. The length of list determines the number of mesh layers in the extrusion. If is given as the extrusion length for each layer, an optimal value is automatically calculated.###### Example

We use the `disk.msh`

for the example here.

```
>>> vol=1; sur=2; top=3; circle=4 # physical regions defined in disk.msh
>>> volextruded=5; bndextruded=6; # new physical regions that will be utilized in extrusion
>>> mymesh = mesh()
>>>
>>> # predefine extrusion
>>> mymesh.extrude(newphysreg = volextruded, newbndphysreg = bndextruded,
... bnd = sur, extrudelens = [0.1,0.05])
>>>
>>> # extrusion is performed when the mesh is loaded.
>>> mymesh.load("disk.msh")
>>> mymesh.write("diskpml.msh")
```

###### See Also

`getdimension`

`def getdimension( self ) ‑> int`

This returns the dimension of the highest dimension element in the mesh 0D, 1D, 2D or 3D.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> dim = mymesh.getdimension()
>>> dim
3
```

`getdimensions`

`def getdimensions( self ) ‑> List[float]`

This returns the x, y and z mesh dimensions in meters.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> dims = mymesh.getdimensions()
>>> dims
[2.0, 2.0, 0.1]
```

`getphysicalregionnumbers`

`def getphysicalregionnumbers( self, dim: int = -1 ) ‑> List[int]`

This returns all physical region numbers of a given dimension. Use or no argument to get the regions of all dimensions.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> allphysregs = mymesh.getphysicalregionnumbers()
[4, 2, 3, 1]
```

`load`

`def load( *args, **kwargs ) ‑> None`

This method allows an empty mesh object to be populated with mesh data. It takes in the same corresponding arguments as required in instantiating a mesh object directly. The only difference with direct instantiation is that this method requires that an empty mesh object is already created. If this method is called by a non-empty mesh object any existing mesh data are lost.

###### Examples

```
>>> # Create an empty mesh object
>>> mymesh = mesh()
>>>
>>> # Load a mesh file with the native reader:
>>> mymesh.load("disk.msh", 2)
>>>
>>> # Load a mesh file with GMSH API:
>>> mymesh = mesh("gmsh:disk.msh", 2)
```

Loading a mesh from the shape objects:

```
>>> # define physical regions
>>> faceregionnumber=1; lineregionnumber=2
>>>
>>> # define a quadrangle shape object
>>> quadface = shape("quadrangle", faceregionnumber, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [10,6,10,6])
>>>
>>> # get the leftline from the contour of the quad shape object
>>> contourlines = quadface.getsons() # returns a list
>>> leftline = contourlines[3]
>>> leftline.setphysicalregion(lineregionnumber)
>>>
>>> # Load a mesh from the quadrangle shape and its left-side line
>>> mymesh = mesh() # creates an empty mesh object
>>> mymesh.load([quadface, leftline])
>>> mymesh.write("quadmesh.msh")
```

Combing multiple meshes with `load`

method

```
>>> mymesh = mesh("disk.msh")
>>> mymesh.shift(2,1,0) # all entities are shifted by x,y,z amount
>>> mymesh.write("shifted.msh")
>>>
>>> mergedmesh = mesh()
>>> mergedmesh.load(True, ["disk.msh", "shifted.msh"])
>>> mergedmesh.write("merged.msh")
>>>
>>> mergedmesh = mesh("merged.msh", 2)
```

`move`

`def move( *args, **kwargs ) ‑> None`

This moves the whole or part of the mesh object by the x, y and z components of expression u in the x, y and z direction.

###### Examples

```
>>> vol = 1
>>> mymesh = mesh("disk.msh")
>>> x=field("x"); y=field("y")
>>>
>>> # Move the whole mesh object
>>> mymesh.move(array3x1(0,0, sin(x*y)))
>>> mymesh.write("moved.msh")
>>>
>>> # Move only the mesh part on the physical region 'vol'
>>> mymesh.move(vol, array3x1(0,0, sin(x*y)))
>>> mymesh.write("moved.msh")
```

`partition`

`def partition( *args, **kwargs ) ‑> None`

This requests a DDM partition of the mesh.

###### Examples

**Example 1**: `partition()`

```
>>> mymesh = mesh()
>>> mymesh.partition()
>>> mymesh.load("disk.msh")
```

**Example 2**: `partition(groupsphysregs: List[List[int]], groupsnumranks: List[int])`

Here the user has the flexibility to selective pick physical regions into certain groups. In the below example, all the entities of the physical region "sur" is grouped to the rank number .

```
>>> vol=1; sur=2; top=3
>>> mymesh = mesh()
>>> mymesh.partition([[sur]], [0])
>>> mymesh.load("disk.msh")
```

`printdimensions`

`def printdimensions( self ) ‑> List[float]`

This prints and returns the x, y and z mesh dimensions.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> mymesh.printdimensions()
Mesh dimensions:
x: 2 m
y: 2 m
z: 0.1 m
[2.0, 2.0, 0.1]
```

`rotate`

`def rotate( *args, **kwargs ) ‑> None`

This rotates the whole or part of the mesh object first by `ax`

degrees around x axis followed by `ay`

degrees around the y-axis and then by `az`

degrees around the z-axis.

###### Examples

```
>>> vol = 1
>>> mymesh = mesh("disk.msh")
>>>
>>> # rotate the whole mesh object
>>> mymesh.rotate(20,60,90)
>>> mymesh.write("rotated.msh")
>>>
>>> # rotate only the mesh part on the physical region 'vol'
>>> mymesh.rotate(vol, 20,60,90)
>>> mymesh.write("rotated.msh")
```

`scale`

`def scale( *args, **kwargs ) ‑> None`

This scales the whole or part of the mesh object first by a factor `x`

, `y`

and `z`

respectively in the x, y and z direction.

###### Examples

```
>>> vol = 1
>>> mymesh = mesh("disk.msh")
>>>
>>> # scale the whole mesh object
>>> mymesh.scale(0.1,0.2,1.0)
>>> mymesh.write("scaled.msh")
>>>
>>> # scale only the mesh part on the physical region 'vol'
>>> mymesh.scale(vol, 0.1,0.2,1.0)
>>> mymesh.write("scaled.msh")
```

`selectanynode`

`def selectanynode( *args, **kwargs ) ‑> None`

This tells the mesh object to create a new physical region `newphysreg`

that contains a single node arbitrarily chosen in the region `physregtoselectfrom`

. If no region is selected (i.e. if `physregtoexcludefrom`

is empty) or if the argument `physregtoexcludefrom`

is not provided, then the arbitrary node is chosen considering the whole domain. The new region `newphysreg`

is created when the `mesh.load()`

method is called on the mesh object.

###### Examples

**Example 1**: `mesh.selectanynode(newphysreg:int, physregtoselectfrom:int)`

```
>>> vol=1; anynode=12
>>> mymesh = mesh()
>>> mymesh.selectanynode(excluded, vol, [box]) # a node is chosen from 'vol' region
>>> mymesh.load("disk.msh")
>>> mymesh.write("out.msh")
```

**Example 2**: `mesh.selectanynode(newphysreg:int)`

```
>>> vol=1; anynode=12
>>> mymesh = mesh()
>>> mymesh.selectanynode(excluded, [box]) # a node is chosen from the whole domain
>>> mymesh.load("disk.msh")
>>> mymesh.write("out.msh")
```

`selectbox`

`def selectbox( *args, **kwargs ) ‑> None`

This tells the mesh object to create a new physical region `newphysreg`

that contains elements of the region `physregtoselectfrom`

that are in the box delimited by [,, ,, ,] given in `boxlimit`

. If no region is selected (i.e. if `physregtoselectfrom`

is empty) or if the argument `physregtoselectfrom`

is not provided, then the box region is created considering the whole domain.

The new region `newphysreg`

is created when the `mesh.load()`

method is called on the mesh object. The elements populated in the new region `newphysreg`

are of dimension `selecteddim`

.

###### Examples

**Example 1**: `mesh.selectbox(newphysreg:int, physregtobox:int, selecteddim:int, boxlimit:List[double])`

```
>>> vol=1; boxregion=12
>>> mymesh = mesh()
>>> mymesh.selectbox(boxregion, vol, 3, [0,1,0, 1,0,0.1]) # select box region from the 'vol' region
>>> mymesh.load("disk.msh")
>>>
>>> v=field("h1"); x=field("x"); y=field("y"); z=field("z")
>>> v.setorder(vol, 1)
>>> v.setvalue(vol, x*y*z)
>>> v.write(vol, "v.vtk", 1)
>>> v.write(boxregion, "vboxregion.vtk", 1)
```

**Example 2**: `mesh.selectbox(newphysreg:int, selecteddim:int, boxlimit:List[double])`

```
>>> vol=1; boxregion=12
>>> mymesh = mesh()
>>> mymesh.selectbox(boxregion, 3, [0,1,0, 1,0,0.1]) # select box region from the whole domain
>>> mymesh.load("disk.msh")
>>>
>>> v=field("h1"); x=field("x"); y=field("y"); z=field("z")
>>> v.setorder(vol, 1)
>>> v.setvalue(vol, x*y*z)
>>> v.write(vol, "v.vtk", 1)
>>> v.write(boxregion, "vboxregion.vtk", 1)
```

`selectexclusion`

`def selectexclusion( *args, **kwargs ) ‑> None`

This tells the mesh object to create a new physical region `newphysreg`

that contains the elements of the region `physregtoexcludefrom`

that are not in `physregtoexclude`

. If no region is selected (i.e. if `physregtoexcludefrom`

is empty) or if the argument `physregtoexcludefrom`

is not provided, then the new region is created considering the whole domain. The new region `newphysreg`

is created when the `mesh.load()`

method is called on the mesh object.

###### Examples

**Example 1**: mesh.selectexclusion(newphysreg:int, physregtoexlcudefrom:int, physregtoexclude:int)`

```
>>> vol=1; sur=2; top=3; box=11; excluded=12
>>> mymesh = mesh()
>>> mymesh.selectbox(box, vol, 3, [0,2, -2,2, -2,2])
>>> mymesh.selectexclusion(excluded, vol, [box]) # physregtoexcludefrom = 'vol'
>>> mymesh.load("disk.msh")
>>> mymesh.write("out.msh")
```

**Example 2**: mesh.selectexclusion(newphysreg:int, physregtoexclude:int)`

```
>>> vol=1; sur=2; top=3; box=11; excluded=12
>>> mymesh = mesh()
>>> mymesh.selectbox(box, vol, 3, [0,2, -2,2, -2,2])
>>> mymesh.selectexclusion(excluded, [box]) # physregtoexcludefrom = whole domain
>>> mymesh.load("disk.msh")
>>> mymesh.write("out.msh")
```

`selectlayer`

`def selectlayer( *args, **kwargs ) ‑> None`

This tells the mesh object to create a new physical region `newphysreg`

that contains the layer of elements of the region `physregtoselectfrom`

that touches the region `physregtostartgrowth`

. If no region is selected (i.e. if `physregtoselectfrom`

is empty) or if the argument `physregtoselectfrom`

is not provided, then the layer region is created considering the whole domain. When multiple layers are requested through the argument `numlayers`

, they are grown on top of each other. The new region `newphysreg`

is created when the `mesh.load()`

method is called on the mesh object.

###### Examples

**Example 1**: `mesh.selectlayer(newphysreg:int, physregtoselectfrom:int, physregtostartgrowth:int, numlayers:int)`

```
>>> vol=1; sur=2; top=3; layerregion=12
>>> mymesh = mesh()
>>> mymesh.selectlayer(layerregion, vol, sur, 1) # select layer region from the 'vol' region
>>> mymesh.load("disk.msh")
>>> mymesh.write("out.msh")
```

**Example 2**: `mesh.selectlayer(newphysreg:int, physregtostartgrowth:int, numlayers:int)`

```
>>> vol=1; sur=2; top=3; layerregion=12
>>> mymesh = mesh()
>>> mymesh.selectlayer(layerregion, sur, 1) # select layer region from the whole domain
>>> mymesh.load("disk.msh")
>>> mymesh.write("out.msh")
```

`selectskin`

`def selectskin( *args, **kwargs ) ‑> None`

This tells the mesh object to create a new physical region `newphysreg`

that contains elements that form the skin of the selected physical regions. If no region is selected (i.e. if `physregtoselectfrom`

is empty) or if the argument `physregtoselectfrom`

is not provided, then the skin region is created considering the whole domain.

The skin region `newphysreg`

is created when the `mesh.load()`

method is called on the mesh object. The dimension of the skin region is always one dimension less than that of the physical regions selected. Note that space derivatives or 'hcurl' field evaluations on a surface do not usually lead to the same values as a volume evaluation.

###### Examples

**Example 1**: `mesh.selectskin(newphysreg:int, physregtoskin)`

```
>>> vol=1; skin=12
>>> mymesh = mesh()
>>> mymesh.selectskin(skin, vol) # select skin region from the 'vol' region
>>>
>>> mymesh.load("disk.msh")
>>> v=field("h1"); x=field("x"); y=field("y"); z=field("z")
>>> v.setorder(vol, 1)
>>> v.setvalue(vol, x*y*z)
>>> v.write(vol, "v.vtk", 1)
>>> v.write(skin, "vskin.vtk", 1)
```

**Example 2**: `mesh.selectskin(newphysreg:int)`

```
>>> vol=1; skin=12
>>> mymesh = mesh()
>>> mymesh.selectskin(skin) # select skin region from the whole domain
>>>
>>> mymesh.load("disk.msh")
>>> v=field("h1"); x=field("x"); y=field("y"); z=field("z")
>>> v.setorder(vol, 1)
>>> v.setvalue(vol, x*y*z)
>>> v.write(vol, "v.vtk", 1)
>>> v.write(skin, "vskin.vtk", 1)s
```

`selectsphere`

`def selectsphere( *args, **kwargs ) ‑> None`

This tells the mesh object to create a new physical region `newphysreg`

that contains elements of the region `physregtoselectfrom`

that are in the sphere of prescribed radius and of center [, ,] as given in `centercoords`

. If no region is selected (i.e. if `physregtoselectfrom`

is empty) or if the argument `physregtoselectfrom`

is not provided, then the sphere region is created considering the whole domain.

The new region `newphysreg`

is created when the `mesh.load()`

method is called on the mesh object. The elements populated in the new region `newphysreg`

are of dimension `selecteddim`

.

###### Examples

**Example 1**: `mesh.selectsphere(newphysreg:int, physregtosphere:int, selecteddim:int, centercoords:List[double], radius:double)`

```
>>> vol=1; sphereregion=12
>>> mymesh = mesh()
>>> mymesh.selectsphere(sphereregion, vol, 3, [1,0,0], 1) # select sphere region from the 'vol' region
>>> mymesh.load("disk.msh")
>>>
>>> v=field("h1"); x=field("x"); y=field("y"); z=field("z")
>>> v.setorder(vol, 1)
>>> v.setvalue(vol, x*y*z)
>>> v.write(vol, "v.vtk", 1)
>>> v.write(sphereregion, "vsphereregion.vtk", 1)
```

**Example 2**: `mesh.selectsphere(newphysreg:int, selecteddim:int, centercoords:List[double], radius:double)`

```
>>> vol=1; sphereregion=12
>>> mymesh = mesh()
>>> ymesh.selectsphere(sphereregion, 3, [1,0,0], 1) # select sphere region from the whole domain
>>> mymesh.load("disk.msh")
>>>
>>> v=field("h1"); x=field("x"); y=field("y"); z=field("z")
>>> v.setorder(vol, 1)
>>> v.setvalue(vol, x*y*z)
>>> v.write(vol, "v.vtk", 1)
>>> v.write(sphereregion, "vsphereregion.vtk", 1)
```

`setadaptivity`

`def setadaptivity( self, criterion: expression, lownumsplits: int, highnumsplits: int ) ‑> None`

Each element in the mesh will be adapted (refined/coarsened) based on the value of a **positive** criterion (h-adaptivity). The max range of the criterion is split into a number of intervals equal to the number of refinement levels in the range `lownumsplits`

and `highnumsplits`

. All intervals have the same size. The barycenter value of the criterion on each element is considered to select the interval, and therefore the corresponding refinement of each mesh element. As an example, for a criterion with the highest value of 900 over the entire domain and a low/high refinement level requested of 1/3 the refinement on mesh elements with criterion value in the range 0 to 300, 300 to 600, 600 to 900 will be 1, 2, 3 levels respectively.

###### Example

```
>>> all = 1
>>> q = shape("quadrangle", all, [0,0,0, 1,0,0, 1.2,1,0, 0,1,0], [5,5,5,5])
>>> mymesh = mesh([q])
>>> x = field("x"); y = field("y")
>>> criterion = 1 + sin(10*x)*sin(10*y)
>>>
>>> mymesh.setadaptivity(criterion, 0, 5)
>>>
>>> for i in range(5):
... criterion.write(all, f"criterion_{100+i}.vtk", 1)
... adapt(1)
```

`setcohomologycuts`

`def setcohomologycuts( *args, **kwargs ) ‑> None`

This makes the mesh object aware of the cohomology cut regions.

###### Example

```
>>> mymesh = mesh()
>>> mymesh.setcohomologycuts([chreg1, chreg2])
>>> mymesh.load("disk.msh")
```

`setphysicalregions`

`def setphysicalregions( self, dims: List[int], nums: List[int], geometryentities: List[List[int]] ) ‑> None`

`shift`

`def shift( *args, **kwargs ) ‑> None`

This translates the whole or part of the mesh object by x, y and z amount in the x, y and z direction.

###### Examples

```
>>> vol = 1
>>> mymesh = mesh("disk.msh")
>>> x=field("x"); y=field("y")
>>>
>>> # shift/translate the whole mesh object
>>> mymesh.shift(1.0, 2.0, 3.0)
>>> mymesh.write("shifted.msh")
>>>
>>> # shift/translate only the mesh part on physical region 'vol'
>>> mymesh.shift(vol, 1.0, 2.0, 3.0)
>>> mymesh.write("shifted.msh")
```

`split`

`def split( self, n: int = 1 ) ‑> None`

This splits each element in the mesh `n`

times. Element quality is maximized and element curvature is taken into account. Each element is split recursively `n`

times as follows:

- point \leftarrow 1 point
- line \leftarrow 2 lines
- triangle \leftarrow 4 triangles
- quadrangle \leftarrow 4 quadrangles
- tetrahedron \leftarrow 8 tetrahedra
- hexahedron \leftarrow 8 hexahedra
- prism \leftarrow 8 prisms
- pyramid \leftarrow 6 pyramids + 4 tetrahedra

###### Example

```
>>> mymesh = mesh()
>>> mymesh.split()
>>> mymesh.load("disk.msh")
>>> mymesh.write("splitdisk.msh")
```

`use`

`def use( self ) ‑> None`

This allows one to select which mesh to use in case multiple meshes are available. This call invalidates all objects that are based on the previously selected mesh for as long as the latter is not selected again.

###### Example

```
>>> finemesh = mesh()
>>> finemesh.split(2)
>>> finemesh.load("disk.msh")
>>> coarsemesh = mesh("disk.msh")
>>> finemesh.use()
```

`write`

`def write( *args, **kwargs ) ‑> None`

This writes the mesh object to a given input filename.

###### Examples

```
>>> # mesh data
>>> mymesh = mesh("disk.msh")
>>> vol=1; top=2; sur=3
```

If a physical region is passed in the first argument, then only part of the mesh object included in that physical region is written:

```
>>> mymesh.write(vol, "out.msh")
```

If only the file `name`

is provided as an argument, then all the physical regions of the mesh are written.

```
>>> mymesh.write("out.msh)
>>> # or equivalently:
>>> mymesh.write("out1.msh", physregs=[-1], option=1)
```

The argument `physregs`

is the list of physical regions that will be written if the argument `option=1`

. If `option=-1`

then all the physical regions except the ones in the list `physregs`

will be written. The default value for `physregs=-1`

which is equivalent to considering all the physical regions (the option argument is ignored when `physregs=-1`

).

```
>>> mymesh.write("out2.msh", [-1], 1) # all physical regions are written
>>> mymesh.write("out3.msh", [-1], -1) # all physical region will be written, ignores 'option' argument
>>> mymesh.write("out4.msh", [1,2], 1) # physical regions 1 and 2 will be written
>>> mymesh.write("out5.msh", [1,2], -1) # all physical regions except 1 and 2 will be written
```

`parameter`

`class parameter( **kwargs )`

The parameter object can hold different expression objects on different geometric regions.

#### Examples

A parameter object can be a scalar. The following creates an empty object.

```
>>> mymesh = mesh("disk.msh")
>>> E = parameter()
```

A parameter object can also be a 2D array. The following creates an empty object.

```
>>> E = parameter(3,3) # a 3x3 parameter matrix
```

#### Methods

`addvalue`

`def addvalue( self, physreg: int, input: expression ) ‑> None`

`allintegrate`

`def allintegrate( *args, **kwargs ) ‑> float`

This is a collective MPI operation and hence must be called by all ranks. This integrates a parameter across all the DDM ranks.

###### Examples

...

```
>>> myparameter = parameter()
>>> myparameter.setvalue(vol, 12.0)
>>> integralvalue = myparameter.allintegrate(vol, 4)
>>>
>>> # allintegrate on a mesh deformed by field 'u'
>>> integralvalueondeformedmesh = myparameter.allintegrate(vol, u, 4)
```

`allinterpolate`

`def allinterpolate( *args, **kwargs ) ‑> List[float]`

This is a collective MPI operation and hence must be called by all ranks. Its functionality is as described in `parameter.interpolate()`

but considers the physical region partitioned across the DDM ranks. The *xyz* coordinate argument must be the same for all ranks.

###### Examples

```
>>> ...
>>> p = parameter(3,1)
>>> p.setvalue(vol, array3x1(x,y,z))
>>> interpolated = p.allinterpolate (vol, {0.5,0.6,0.05});
>>>
>>> # allinterpolate on a mesh deformed by field 'u'
>>> interpolated = p.allinterpolate (vol, u, {0.5,0.6,0.05});
```

`allmax`

`def allmax( *args, **kwargs ) ‑> List[float]`

This is a collective MPI operation and hence must be called by all ranks. It computes the max value across all the DDM ranks. The evaluation of the max value can be restricted to a box or evaluated on a deformed mesh similar to as described in `parameter.max()`

.

###### See Also

`parameter.max()`

, `parameter.allmin()`

`allmin`

`def allmin( *args, **kwargs ) ‑> List[float]`

This is a collective MPI operation and hence must be called by all ranks. It computes the min value across all the DDM ranks. The evaluation of the min value can be restricted to a box or evaluated on a deformed mesh similar to as described in `parameter.min()`

.

###### See Also

`parameter.min()`

, `parameter.allmax()`

`atbarycenter`

`def atbarycenter( self, physreg: int, onefield: field ) ‑> quanscient.vec`

This outputs a `vec`

object whose structure is based on the field argument `onefield`

and which contains the parameter evaluated at the barycenter of each **reference** element of physical region `physreg`

. The barycenter of the reference element might not be identical to the barycenter of the actual element in the mesh (for curved elements, for general quadrangles, hexahedra and prisms). The evaluation at barycenter is constant on each mesh element.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x = field("x"); f = field("one")
>>>
>>> # Evaluating the parameter
>>> p = parameter()
>>> p.setvalue(vol, 12*x)
>>> p.write(vol, "parameter.vtk", 1)
>>>
>>>> # Evaluating the same parameter at barycenter
>>> myvec = p.atbarycenter(vol, f)
>>> f.setdata(vol, myvec)
>>> f.write(vol, "barycentervalues.vtk", 1)
```

`countcolumns`

`def countcolumns( self ) ‑> int`

This returns the number of columns in the parameter.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> E = parameter(2,3)
>>> E.countcolumns()
3
```

`countrows`

`def countrows( self ) ‑> int`

This returns the number of rows in the parameter.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> E = parameter(2,3)
>>> E.countrows()
2
```

`integrate`

`def integrate( *args, **kwargs ) ‑> float`

This integrates a parameter over the physical region `physreg`

. The integration is exact up to the order of polynomials specified in the argument `integrationorder`

.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> myparameter = parameter()
>>> myparameter.setvalue(vol, 12.0)
>>> integralvalue = myparameter.integrate(vol, 4)
>>>
>>> # integrate on a mesh deformed by field 'u'
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> integralvalueondeformedmesh = myparameter.integrate(vol, u, 4)
```

`interpolate`

`def interpolate( *args, **kwargs ) ‑> None`

This interpolates the parameter at a single point whose [x,y,z] coordinate is provided as an argument. The flattened interpolated parameter values are returned if the point was found in the elements of the physical region `physreg`

. If not found an empty list is returned. A parameter can also be interpolated on a deformed mesh by passing its corresponding field.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x=field("x"); y=field("y"); z=field("z")
>>> xyzcoord = [0.5,0.6,0.05]
>>>
>>> p = parameter(3,1)
>>> p.setvalue(vol, array3x1(x,y,z))
>>> interpolated = p.interpolate(vol, xyzcoord)
>>> interpolated
[0.5, 0.6, 0.05]
>>>
>>> # interpolation on the mesh deformed by field 'u'
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> interpolated = p.interpolate(vol, u, xyzcoord)
```

`max`

`def max( *args, **kwargs ) ‑> List[float]`

This gives the max value of the parameter over the geometric region `physreg`

. The max value is obtained by splitting all elements `refinement`

times in each direction. Increasing the refinement will thus lead to a more accurate max value, but at an increased computational cost. The max value is exact when the refinement nodes added to the elements correspond to the position of max. For a first-order nodal shape function interpolation, on a mesh that is not curved, the max is always exact to machine precision.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x = field("x")
>>> p = 2*x
>>> maxdata = p.max(vol, 1)
>>> maxdata[0]
2.0
```

The search of the max value can be restricted to a box delimited by the last argument whose form is [xboxmin,xboxmax, yboxmin, yboxmin, zboxmax, zboxmin]. The output returned is a list of form [maxvalue, xcoordmax, ycoordmax, zcoordmax] or an empty list if the physical region argument is empty or is not in the box provided. If the argument defining the box is not provided, then the whole geometric region is considered for evaluating the max value.

```
>>> maxdatainbox = p.max(vol, 5, [-2,0, -2,2, -2,2])
```

The max value can also be evaluated on the geometry deformed by a field (possibly a curved mesh). The max location and the delimiting box are on the undeformed mesh.

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> maxdataondeformedmesh = p.max(vol, u, 1)
```

###### See Also

`min`

`def min( *args, **kwargs ) ‑> List[float]`

This gives the min value of the parameter over the geometric region `physreg`

. The min value is obtained by splitting all elements `refinement`

times in each direction. Increasing the refinement will thus lead to a more accurate min value, but at an increased computational cost. The min value is exact when the refinement nodes added to the elements correspond to the position of min. For a first-order nodal shape function interpolation, on a mesh that is not curved, the min is always exact to machine precision.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> x = field("x")
>>> p = 2*x
>>> mindata = p.min(vol, 1)
>>> mindata[0]
-2.0
```

The search of the min value can be restricted to a box delimited by the last argument whose form is [xboxmin,xboxmax, yboxmin, yboxmin, zboxmax, zboxmin]. The output returned is a list of the form [maxvalue, xcoordmax, ycoordmax, zcoordmax] or an empty list if the physical region argument is empty or is not in the box provided. If the argument defining the box is not provided, then the whole geometric region is considered for evaluating the min value.

```
>>> mindatainbox = p.min(vol, 5, [-2,0, -2,2, -2,2])
```

The min value can also be evaluated on the geometry deformed by a field (possibly a curved mesh). The min location and the delimiting box are on the undeformed mesh.

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> u.setorder(vol, 1)
>>> mindataondeformedmesh = p.min(vol, u, 1)
```

###### See Also

`print`

`def print( self ) ‑> None`

This prints the information on the parameter to the console.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> E = parameter()
>>> E.setvalue(vol, 150e9)
>>> E.print()
```

`setvalue`

`def setvalue( self, physreg: int, input: expression, op: str = 'set' ) ‑> None`

This sets the parameter `ìnput`

on the physical region `physreg`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> E = parameter()
>>> E.setvalue(vol, 150e9)
```

`write`

`def write( *args, **kwargs ) ‑> None`

This evaluates a parameter in the physical region `physreg`

and writes it to the file `filename`

. The `lagrangeorder`

is the order of interpolation for the evaluation of the parameter.

###### Examples

```
>>> # setup
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> u = field("h1xyz")
>>> v = field("h1", [1,2,3])
>>> u.setorder(vol, 1)
>>> v.setorder(vol, 1)
>>>
>>> # interpolation order for writing a parameter
>>> p = parameter()
>>> p.setvalue(1e8*u)
>>> p.write(vol, "uorder1.vtk", 1) # interpolation order is 1
>>> p.write(vol, "uorder3.vtk", 3) # interpolation order is 3
```

`port`

`class port( **kwargs )`

The port object represents a scalar lumped quantity.

#### Examples

A port object with an initial zero value is created as:

```
>>> V = port()
```

A multi-harmonic port object with an initial zero value can be created by passing a list of harmonic numbers. Refer to the multi-harmonic field constructor for the meaning of the harmonic numbers.

```
>>> V = port([2,3])
```

#### Methods

`cos`

`def cos( self, freqindex: int ) ‑> quanscient.port`

This gets a port that is the harmonic at `freqindex`

times the fundamental frequency in port V.

###### Example

```
>>> V = port([1,2,3,4,5])
>>> Vs = V.cos(0)
>>> Vs.getharmonics()
1
```

###### See Also

`getharmonics`

`def getharmonics( self ) ‑> List[int]`

This returns the list of harmonics of the port object.

###### Example

```
>>> V = port([1,2,3])
>>> harms = V.getharmonics()
>>> harms
[1, 2, 3]
```

`getname`

`def getname( self ) ‑> str`

This gets the name of the port object.

###### Example

```
>>> V = port()
>>> V.setname("LumpedMass")
>>> V.getname()
'LumpedMass'
```

###### See Also

`getvalue`

`def getvalue( self ) ‑> float`

This returns the value of the port object.

###### Example

```
>>> V = port()
>>> V.setvalue(-1.2)
>>> val = V.getvalue()
>>> val
-1.2
```

###### See Also

`harmonic`

`def harmonic( *args, **kwargs ) ‑> quanscient.port`

This returns a port that is the harmonic/list of harmonics of the port object.

###### Example

```
>>> V = port([1,2,3])
>>> V23 = V.harmonic([2,3])
>>> V23.getharmonics()
[2, 3]
```

`print`

`def print( self ) ‑> None`

This prints the information of the port object.

###### Example

```
>>> V = port([2,3]) # create a multi-harmonic port object
>>> V.harmonic(2).setvalue(1) # set the value of 2nd harmonic
>>> V.harmonic(3).setvalue(0.5) # set the value of 3rd harmonic
>>> V.print()
Port harmonic 2 has value 1
Port harmonic 3 has value 0.5
```

`setname`

`def setname( self, name: str ) ‑> None`

This sets the name for the port object.

###### Example

```
>>> V = port()
>>> V.setname("LumpedMass")
>>> V.print()
Port LumpedMass has value 0
```

###### See Also

`setvalue`

`def setvalue( self, portval: float ) ‑> None`

This sets the value of the port object.

###### Examples

```
>>> V = port()
>>> V.setvalue(10.0)
>>> V.print()
Port has value 10
```

To set the value of a multi-harmonic port:

```
>>> V = port([2,3]) # create a multi-harmonic port object
>>> V.harmonic(2).setvalue(1) # set the value of 2nd harmonic
>>> V.harmonic(3).setvalue(0.5) # set the value of 3rd harmonic
>>> V.print()
Port harmonic 2 has value 1
Port harmonic 3 has value 0.5
```

###### See Also

`port.harmonic()`

, `port.getvalue()`

`sin`

`def sin( self, freqindex: int ) ‑> quanscient.port`

This gets a port that is the harmonic at `freqindex`

times the fundamental frequency in port V.

###### Example

```
>>> V = port([1,2,3,4,5])
>>> Vs = V.sin(2)
>>> Vs.getharmonics()
4
```

###### See Also

`preconditioner`

`class preconditioner`

`shape`

`class shape( **kwargs )`

The shape objects are meshed geometric entities. The mesh created based on shapes can be written in `.msh`

format at any time for visualization in GMSH. It might be needed to change the `color`

and `visibility`

options in the menu `Tools > Options > Mesh`

of GMSH.

#### Examples

Depending on the number and type of arguments, different shape objects can be created for different purposes.

- Creates a shape with the coordinates of all nodes provided as input:
- Example 2: for points and lines:
`myshape = shape(shapename:str, physreg:int, coords:List[double])`

- Example 2: for points and lines:
- Creates a shape based on the coordinates of the corner nodes in the shape
- Example 3: for lines and arcs:
`myshape = shape(shapename:str, physreg:int, coords:List[double], nummeshpts:int)`

- Example 4: for triangles and quadrangles:
`myshape = shape(shapename:str, physreg:int, coords:List[double], nummeshpts:List[int])`

- Example 3: for lines and arcs:
- Creates a shape based on sub-shapes provided.
- Example 5: for lines and arcs:
`myshape = shape(shapename:str, physreg:int, subshapes:List[shape], nummeshpts:int)`

- Example 6: for straight-edged triangles and quadrangles:
`myshape = shape(shapename:str, physreg:int, subshapes:List[shape], nummeshpts:List[int])`

- Example 7: for curved triangles and quadrangles. Also, union of several shapes:
`myshape = shape(shapename:str, physreg:int, subshapes:List[shape])`

- Example 5: for lines and arcs:
- Creates a disk shape.
- Example 8:
`myshape = shape(shapename:str, physreg:int, centercoords:List[double], radius:double, nummeshpts:int)`

- Example 9:
`myshape = shape(shapename:str, physreg:int, centerpoint:shape, radius:double, nummeshpts:int)`

- Example 8:

**Example 1:** Creating an empty shape object

```
>>> myshape = shape()
```

**Example 2:** `myshape = shape(shapename:str, physreg:int, coords:List[double])`

. This can be used to create a line going through a list of nodes whose x,y,z coordinates are provided. A physical region number is also provided to have access to the geometric regions of interest in the finite element simulation.

```
>>> linephysicalregion = 1
>>> myline = shape("line", linephysicalregion, [0,0,0, 0.5,0.5,0, 1,1,0, 1.5,1,0, 2,1,0])
>>> mymesh = mesh([myline])
>>> mymesh.write("meshed.msh")
```

If the nodes in the mesh need to be accessed, a point shape object can be created with corresponding nodal coordinates. The nodes can then be accessed through the physical region provided.

```
>>> pointphysicalregion = 2
>>> point1_coords = [0,0,0]
>>> point2_coords = [2,1,0]
>>> mypoint1 = shape("point", pointphysicalregion, point1_coords)
>>> mypoint2 = shape("point", pointphysicalregion, point2_coords)
>>> # Points 1 and 2 are now available in the `pointphysicalregion=2`.
>>>
>>> p = field("h1")
>>> p.setorder(linephysicalregion, 1)
>>> p.setconstraint(pointphysicalregion, 2) # Dirichlet boundary constraint will be applied on points 1 and 2
```

**Example 3:** `myshape = shape(shapename:str, physreg:int, coords:List[double], nummeshpts:int)`

This can be used to create: a straight line between the first (x1,y1,z1) and last point (x2,y2,z2) provided. a circular arc between the first (x1,y1,z1) and second point (x2,y2,z2) whose center is the third point (x3,y3,z3). The `nummeshpts`

argument corresponds to the number of nodes in the meshed shape. At least two nodes are expected.

```
>>> linephysicalregion=1; arcphysicalregion=1
>>> myline = shape("line", linephysicalregion, [0,0,0, 1,-1,1], 10) # creates a line mesh with 10 nodes
>>> myarc = shape("arc", arcphysicalregion, [1,0,0, 0,1,0, 0,0,0], 8) # creates an arc mesh with 8 nodes
>>> mymesh = mesh([myline, myarc])
>>> mymesh.write("meshed.msh")
```

**Example 4:** `myshape = shape(shapename:str, physreg:int, coords:List[double], nummeshpts:List[int])`

This can be used to create: a straight-edge quadrangle with a full quadrangle structured mesh. a straight-edge triangle with s structured mesh made of triangles along the edge linking the second and third node and quadrangles everywhere else.

```
>>> quadranglephysicalregion=1; trianglephysicalregion=2
>>> myquadrangle = shape("quadrangle", quadranglephysicalregion, [0,0,0, 1,0,0, 1,1,0, -0.5,1,0], [12,10,12,10])
>>> mytriangle = shape("triangle", trianglephysicalregion, [1,0,0, 2,0,0, 1,1,0], [10,10,10])
>>> mymesh = mesh([myquadrangle, mytriangle])
>>> mymesh.write("meshed.msh")
The <code>coords</code> argument provides the <code>x,y,z</code>coordinates of the corner nodes. E.g. (0,0,0), (1,0,0), (1,1,0) and (-0.5,1,0) for the quadrangle.
The <code>nummeshpts</code> argument specifies the number of nodes to mesh each of the contour lines. At least two nodes are expected for each contour line.
All contour lines must have the same number of nodes for the triangle shape while for the quadrangle shape the contour lines facing each other
must have the same number of nodes.
```

**Example 5:** `myshape = shape(shapename:str, physreg:int, subshapes:List[shape], nummeshpts:int)`

This can be used to create the following shapes from the list of subshapes provided: a straight line between the first (x1,y1,z1) and last point (x2,y2,z2) provided. a circle arc between the first (x1,y1,z1) and second point (x2,y2,z2) whose center is the third point. The `nummeshpts`

argument corresponds to the number of nodes in the meshed shape.

```
>>> # Point subshapes
>>> point1 = shape("point", -1, [0,0,0])
>>> point2 = shape("point", -1, [1,0,0])
>>> point3 = shape("point", -1, [0,1,0])
>>> point4 = shape("point", -1, [1,-1,1])
>>>>
>>> # Creating line and arc shapes from point subshapes
>>> linephysicalregion=1; arcphysicalregion=2
>>> myline = shape("line", linephysicalregion, [point1, point4], 10)
>>> myarc = shape("arc", arcphysicalregion, [point2, point3, point1], 8)
>>> mymesh = mesh([myline, myarc])
>>> mymesh.write("meshed.msh")
```

**Example 6:** `myshape = shape(shapename:str, physreg:int, subshapes:List[shape], nummeshpts:List[int])`

This can be used to create the following shapes from the list of subshpaes provided: a straight-edge quadrangle with a full quadrangle structured mesh a straight-edge triangle with a structured mesh made of triangles along the edge linking the second and third nodes and quadrangles everywhere. The `subshapes`

argument provides the list of corner point shapes. The `nummeshpts`

argument gives the number of nodes to mesh each of the contour lines. At least two nodes are expected for each contour line. All contour lines must have the same number of nodes for the triangle shape while for the quadrangle shape the contour lines facing each other must have the same number of nodes.

```
>>> # Point subshapes
>>> point1 = shape("point", -1, [0,0,0])
>>> point2 = shape("point", -1, [1,0,0])
>>> point3 = shape("point", -1, [1,1,0])
>>> point4 = shape("point", -1, [0,1,0])
>>> point5 = shape("point", -1, [2,0,0])
>>>
>>> # Creating triangle and quadrangle shape from subshapes of corner points
>>> quadranglephysicalregion=1; trianglephysicalregion=2
>>> myquadrangle = shape("quadrangle", quadranglephysicalregion, [point1, point2, point3, point4], [6,8,6,8])
>>> mytriangle = shape("triangle", trianglephysicalregion, [point2, point5, point3], [8,8,8])
>>> mymesh = mesh([myquadrangle, mytriangle])
>>> mymesh.write("meshed.msh")
```

**Example 7:** `myshape = shape(shapename:str, physreg:int, subshapes:List[shape])`

This can be used to create: a curved quadrangle with full quadrangle structured mesh. a curved triangle with structured mesh made of triangles along the edge linking the second and third nodes and quadrangles everywhere. a shape that is the union of several shapes of the same dimension. The `subshapes`

argument provides the contour shapes (clockwise or anti-clockwise). All contour lines must have the same number of nodes for the triangle shape while for quadrangle shape the contour lines facing each other must have the same number of nodes.

```
>>> # Creating subshapes
>>> line1 = shape("line", -1, [-1,-1,0, 1,-1,0], 10)
>>> arc2 = shape("arc", -1, [1,-1,0, 1,1,0, 0,0,0], 12)
>>> line3 = shape("line", -1, [1,1,0, -1,1,0], 10)
>>> line4 = shape("line", -1, [-1,1,0, -1,-1,0], 12)
>>> line5 = shape("line", -1, [1,-1,0, 3,-1,0], 12)
>>> arc6 = shape("arc", -1, [3,-1,0, 1,1,0, 1.6,-0.4,0], 12)
>>>
>>> quadranglephysicalregion=1; trianglephysicalregion=2; unionphysicalregion=3
>>> myquadrangle = shape("quadrangle", quadranglephysicalregion, [line1, arc2, line3, line4])
>>> mytriangle = shape("triangle", trianglephysicalregion,[line5, arc6, arc2])
>>> myunion = shape("union", unionphysicalregion, [line1, arc2, line3, line4])
>>>
>>> mymesh = mesh([myquadrangle, mytriangle, myunion])
>>> mymesh.write("meshed.msh")
```

**Example 8:** `myshape = shape(shapename:str, physreg:int, centercoords:List[double], radius:double, nummeshpts:int)`

This is used to create a 2D disk with structured mesh centered around `centercoords`

. The `nummeshpts`

argument corresponds to the number of nodes in the contour circle of the disk. Since the disk has a structured mesh, the number of mesh nodes must be a multiple of 4. The `radius`

argument provides the radius of the disk.

```
>>> diskphysicalregion=1
>>> mydisk = shape("disk", diskphysicalregion, [1,0,0], 2, 40)
>>> mymesh = mesh([mydisk])
>>> mymesh.write("meshed.msh")
```

**Example 9:** `myshape = shape(shapename:str, physreg:int, centerpoint:shape, radius:double, nummeshpts:int)`

This is used to create a 2D disk with structured mesh centered around point shape `centerpoint`

. The `nummeshpts`

argument corresponds to the number of nodes in the contour circle of the disk. Since the disk has a structured mesh, the number of mesh nodes must be a multiple of 4. The `radius`

argument provides the radius of the disk.

```
>>> diskphysicalregion=1
>>> centerpoint = shape("point", -1, [1,0,0])
>>> mydisk = shape("disk", diskphysicalregion, centerpoint, 2, 40)
>>> mymesh = mesh([mydisk])
>>> mymesh.write("meshed.msh")
```

#### See Also

#### Methods

`duplicate`

`def duplicate( self ) ‑> quanscient.shape`

This outputs a shape that is a duplicate of the initial shape. All the subshapes are duplicated recursively as well but the object equality relations between subshapes are identical between a shape and its duplicate.

###### Example

```
>>> myquadrangle = shape("quadrangle", 1, [-1,-1,0, 1,-1,0, 1,1,0, -1,1,0], [6,8,6,8])
>>> otherquadrangle = myquadrangle.duplicate()
```

###### See Also

`shape.move()`

, `shape.shift()`

, `shape.scale()`

, `shape.rotate()`

`extrude`

`def extrude( *args, **kwargs ) ‑> quanscient.shape`

A given shape is extruded in the requested direction (z by default) to form a higher dimensional shape. The extrude function works for 0D, 1D and 2D shapes.

###### Parameters

** physreg** :

`int`

: Physical region to which the extruded shape is set.** height** :

`double`

: Height of extrusion in the requested direction.** numlayers** :

`int`

: Value specifying the number of node layers the extruded mesh should contain.** extrudedirection** :

`List[int]`

: Unit vector specifying the direction of extrusion.###### Examples

**Example 1:** `myshape = shape.extrude(physreg:int, height:double, numlayers:int, extrudedirection:List[double])`

```
>>> myquadrangle = shape("quadrangle", 1, [-1,-1,0, 1,-1,0, 1,1,0, -1,1,0], [2,2,2,2])
>>> volumephysicalregion = 100
>>> myvolume = myquadrangle.extrude(volumephysicalregion, 1.4, 6, [0,0,1])
>>> mymesh = mesh([myvolume])
>>> mymesh.write("meshed.msh")
```

**Example 2:** `myshape = shape.extrude(physreg:List[int], height:List[double], numlayers:[int], extrudedirection:List[double])`

. This extends the `extrude`

function to multiblock extrusion.

```
>>> mytriangle = shape("triangle", 1, [0,0,0, 1,0,0, 0,1,0], [6,6,6])
>>>
>>> '''
>>> Creating multiblock extrusion:
>>> ---------|----------|----------|-----------
>>> block | physreg | height | numlayers
>>> ---------|----------|----------|-----------
>>> Block 1: | 11 | 0.5 | 3
>>> Block 2: | 12 | 0.3 | 5
>>> ---------|----------|----------|-----------
>>> In block 1, the initial shape is extruded to a height of 0.5 and contains 3 node layers and the extruded shape is set to physical region 11.
>>> In block 2, the initial shape is extruded to a height of 0.3 (starting from height 0.5) and contains 5 node layers and the extruded shape is set to physical region 12.
>>> '''
>>> myvolumes = mytriangle.extrude([11,12], [0.5,0.3], [3,5], [0,0,1]) # creates two extruded shapes
>>> mymesh = mesh(myvolumes)
>>> mymesh.write("meshed.msh")
```

`getcoords`

`def getcoords( self ) ‑> List[float]`

This returns the coordinates of all nodes in the shape mesh.

###### Examples

```
>>> myquadrangle = shape("quadrangle", 555, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [2,2,2,2])
>>> myquadrangle.getcoords()
[0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0]
```

`getcurvatureorder`

`def getcurvatureorder( self ) ‑> int`

This returns the curvature order of a given shape.

###### Example

```
>>> q = shape("quadrangle", 1, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [2,2,2,2])
>>> q.getcurvatureorder()
1
```

`getdimension`

`def getdimension( self ) ‑> int`

This gives the shape dimension (0D, 1D, 2D or 3D).

###### Examples

```
>>> mypoint = shape("point", 111, [0,0,0])
>>> mypoint.getdimension()
0
>>>
>>> myline = shape("line", 222, [0,0,0, 0.5,0.5,0, 1,1,0, 1.5,1,0, 2,1,0])
>>> myline.getdimension()
1
>>>
>>> myarc = shape("arc", 333, [1,0,0, 0,1,0, 0,0,0], 8)
>>> myarc.getdimension()
1
>>>
>>> mytriangle = shape("triangle", 444, [1,0,0, 2,0,0, 1,1,0], [10,10,10])
>>> mytriangle.getdimension()
2
>>>
>>> myquadrangle = shape("quadrangle", 555, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [6,8,6,8])
>>> myquadrangle.getdimension()
2
>>>
>>> mylines = myquadrangle.getsons()
>>> mylines[0].getdimension()
1
```

`getname`

`def getname( self ) ‑> str`

This returns the name of the shape.

###### Examples

```
>>> mypoint = shape("point", 111, [0,0,0])
>>> mypoint.getname()
'point'
>>>
>>> myline = shape("line", 222, [0,0,0, 0.5,0.5,0, 1,1,0, 1.5,1,0, 2,1,0])
>>> myline.getname()
'line'
>>>
>>> myarc = shape("arc", 333, [1,0,0, 0,1,0, 0,0,0], 8)
>>> myarc.getname()
'arc'
>>>
>>> mytriangle = shape("triangle", 444, [1,0,0, 2,0,0, 1,1,0], [10,10,10])
>>> mytriangle.getname()
'triangle'
>>>
>>> myquadrangle = shape("quadrangle", 555, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [6,8,6,8])
>>> myquadrangle.getname()
'quadrangle'
>>>
>>> mylines = myquadrangle.getsons()
>>> mylines[0].getname()
'line'
```

`getphysicalregion`

`def getphysicalregion( self ) ‑> int`

This gives the physical region number for a given shape. The physical region is used in the finite element simulation to identify a region.

###### Returns

`int:`

: Returns -1 if the physical region was not set, else a corresponding positive integer.

###### Examples

```
>>> myquadrangle = shape("quadrangle", 111, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [6,8,6,8])
>>> mylines = myquadrangle.getsons()
>>>
>>> myline1 = mylines[0]
>>> myline1.setphysicalregion(2)
>>> myline1.getphysicalregion()
2
>>>
>>> myline2 = mylines[1]
>>> myline1.getphysicalregion()
-1
```

###### See Also

`shape.setphysicalregion()`

, `shape.getsons()`

`getsons`

`def getsons( self ) ‑> List[quanscient.shape]`

This returns a list containing the direct subshapes of the shape object. For a quadrangle, its 4 contour lines are returned. For a triangle, its 3 contour lines are returned.

###### Example

```
>>> myquadrangle = shape("quadrangle", 111, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [6,8,6,8])
>>> mylines = myquadrangle.getsons()
>>> for myline in mylines:
... myline.setphysicalregion(2)
...
>>> mymesh = mesh(mylines)
>>> mymesh.write("meshed.msh")
```

`move`

`def move( self, u: expression ) ‑> None`

This moves the shape (and all its subshapes recursively) in the x,y and z direction by a value provided in the 3x1 expression array. When moving multiple shapes that share common subshapes, ensure that subshapes are not moved multiple times.

###### Parameter

u: `expression`

3x1 array expression that specifies the values by which shape is moved in x,y,z direction

###### Example

```
>>> x=field("x"); y=field("y"); z=field("z")
>>> myquadrangle = shape("quadrangle", 1, [-1,-1,0, 1,-1,0, 1,1,0, -1,1,0], [12,16,12,16])
>>> myquadrangle.move(array3x1(0,x,sin(x*y)))
>>> mymesh = mesh([myquadrangle])
>>> mymesh.write("meshed.msh")
```

###### See Also

`shape.shift()`

, `shape.scale()`

, `shape.rotate()`

, `shape.duplicate()`

`rotate`

`def rotate( self, alphax: float, alphay: float, alphaz: float ) ‑> None`

This rotates the shape (and all its subshapes recursively) first by `alphax`

degrees around the x-axis, then `alphay`

degrees around the y-axis and finally `alphaz`

degrees around the z-axis. When rotating multiple shapes that share common subshapes make sure that subshapes are not rotated multiple times.

###### Parameters

** alphax** :

`double`

: Degrees by which the shape is rotated around the x-axis.** alphay** :

`double`

: Degrees by which the shape is rotated around the y-axis.** alphaz** :

`double`

: Degrees by which the shape is rotated around the z-axis.###### Example

```
>>> myquadrangle = shape("quadrangle", 1, [-1,-1,0, 1,-1,0, 1,1,0, -1,1,0], [6,8,6,8])
>>> myquadrangle.rotate(0,0,45)
>>> mymesh = mesh([myquadrangle])
>>> mymesh.write("meshed.msh")
```

###### See Also

`shape.move()`

, `shape.shift()`

, `shape.scale()`

, `shape.duplicate()`

`scale`

`def scale( self, scalex: float, scaley: float, scalez: float ) ‑> None`

This scales the shape (and all its subspaces recursively) in the x,y and z directions by a given factor. A factor of 1 keeps the shape unchanged. When scaling multiple shapes that share common subshapes make sure the subshapes are not scaled multiple times.

###### Parameters

** scalex** :

`double`

: Value by which the shape is scaled in x-direction.** scaley** :

`double`

: Value by which the shape is scaled in y-direction.** scalez** :

`double`

: Value by which the shape is scaled in z-direction.###### Example

```
>>> myquadrangle = shape("quadrangle", 1, [-1,-1,0, 1,-1,0, 1,1,0, -1,1,0], [6,8,6,8])
>>> myquadrangle.scale(2,0.5,2)
>>> mymesh = mesh([myquadrangle])
>>> mymesh.write("meshed.msh")
```

###### See Also

`shape.move()`

, `shape.shift()`

, `shape.rotate()`

, `shape.duplicate()`

`setphysicalregion`

`def setphysicalregion( self, physreg: int ) ‑> None`

This sets the physical region number for a given shape. Subshapes are not affected. The physical region is used in the finite element simulation to identify a region.

###### Parameters

** physreg** :

`int`

: An integer identifying the physical region.###### Example

```
>>> quadphysicalregion=1; linephysicalregion=2
>>> myquadrangle = shape("quadrangle", quadphysicalregion, [0,0,0, 1,0,0, 1,1,0, 0,1,0], [6,8,6,8])
>>> myline = myquadrangle.getsons()[0] # the shape 'myline' is not associated with any physical region yet.
>>> myline.setphysicalregion(linephysicalregion)
>>> mymesh = mesh([myquadrangle, myline])
>>> mymesh.write("meshed.msh")
```

###### See Also

`shape.getphysicalregion()`

, `shape.getsons()`

`shift`

`def shift( self, shiftx: float, shifty: float, shiftz: float ) ‑> None`

This shifts the shape (and all its subshapes recursively) in the x,y, and z directions by a double value. When shifting multiple shapes that share common subshapes make sure the subspaces are not shifted multiple times.

###### Parameters

** shiftx** :

`double`

: Value by which the shape is moved in the x-direction.** shifty** :

`double`

: Value by which the shape is moved in the y-direction.** shiftz** :

`double`

: Value by which the shape is moved in the z-direction.###### Example

```
>>> myquadrangle = shape("quadrangle", 1, [-1,-1,0, 1,-1,0, 1,1,0, -1,1,0], [6,8,6,8])
>>> myquadrangle.shift(1,1,2)
>>> mymesh = mesh([myquadrangle])
>>> mymesh.write("meshed.msh")
```

###### See Also

`shape.move()`

, `shape.scale()`

, `shape.rotate()`

, `shape.duplicate()`

`spanningtree`

`class spanningtree( physregs: List[int] )`

The spanningtree object holds a spanning tree whose edges go through all nodes in the mesh without forming a loop. The tree growth starts on the physical regions provided as an argument.

#### Parameters

** physregs** :

`List[int]`

: List of physical regions where the spanning tree is first fully grown before being extended everywhere.#### Example

A spanning tree object is created by passing the physical regions 'sur' and 'top'. Hence, here the tree is first fully grown on face regions 'sur' and 'top' before extending everywhere.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3; # physical regions
>>> spantree = spannintree([sur, top])
```

#### Methods

`countedgesintree`

`def countedgesintree( self ) ‑> int`

This returns the number of edges in the spanning tree.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3; # physical regions
>>> spantree = spannintree([sur, top])
>>> spantree.countedgesintree()
1859
```

`write`

`def write( self, filename: str ) ‑> None`

This writes the tree into a file for visualization.

###### Parameters

** filename** :

`str`

: Name of the file to which the data samples are written.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2; top=3; # physical regions
>>> spantree = spannintree([sur, top])
>>> spantree.write("spantree.vtk")
```

`spline`

`class spline( **kwargs )`

The spline object allows interpolation in a discrete data set using cubic (natural) splines.

#### Raises

`RuntimeError`

: If a mesh object is not available before creating a `spline`

object.

#### Notes

Before creating a `spline`

object, ensure that a mesh object is available. If a mesh object is not already available, create an empty mesh object.

#### Examples

Say the data samples are in a text file, with each data separated by "," as shown below:

```
>>> 273,5e9,300,4e9,320,2.5e9,340,1e9
```

A spline object can then be created by reading the x-y data contained in the text file.

```
>>> mymesh = mesh() # if a mesh object is not already available.
>>> spl1 = spline(filename="measured.txt", delimiter="\n")
```

The x-y data samples can also be provided in two separate lists or tuples. In that case, a spline object is created as follows:

```
>>> mymesh = mesh() # if a mesh object is not already available.
>>> temperature = (273, 300, 320, 340)
>>> youngsmodulus = (5e9, 4e9, 2.5e9, 1e9)
>>> spl2 = spline(xin=temperature, yin=youngsmodulus)
```

#### Notes

The ordering of the samples provided does not matter. Internally they are always sorted in the ascending order of data.

#### Methods

`evalat`

`def evalat( *args, **kwargs ) ‑> float`

Interpolates at given point(s) that falls within the original data range provided.

###### Parameters

** input** :

`double`

or `List[double]`

or `Tuple(double)`

: An point or a list/tuple of points for interpolation.###### Returns

`double`

: an interpolated scalar value if the input was a scalar.

`List[double]`

: a list of interpolated values if the input was a list/tuple.

###### Raises

`RuntimeError`

: If at least one of the inputs for interpolation is outside the range of the original data samples provided.

###### Examples

```
>>> mymesh = mesh() # if a mesh object is not already available.
>>>
>>> temperature = (320, 273, 340, 300) # original x input
>>> youngsmodulus = (2.5e9, 5e9, 1e9, 4e9) # original y input
>>> spl = spline(temperature, youngsmodulus)
```

**Example 1:**

```
>>> spl.evalat(303)
4199390995.630462
```

**Example 2:**

```
>>> spl.evalat([298,304,275])
[3912277862.548358, 4278701622.971286, 4835693423.344276]
```

**Example 3:**

```
>>> spl.evalat(250)
RuntimeError: Error in 'spline' object: data requested in the interval (250,250) is out of the provided data range (273,340)
```

**Example 4:**

```
>>> spl.evalat([290, 310, 400])
RuntimeError: Error in 'spline' object: data requested in the interval (290,400) is out of the provided data range (273,340)
```

`getderivative`

`def getderivative( self ) ‑> quanscient.spline`

This returns the derivative of the spline.

The spline polynomial and its derivative :

###### Returns

`spline`

: spline object that holds the data samples corresponding to the derivative of the spline.

###### Example

```
>>> mymesh = mesh() # if a mesh object is not already available.
>>>
>>> temperature = (273, 300, 320, 340)
>>> youngsmodulus = (5e9, 4e9, 2.5e9, 1e9)
>>>
>>> spl = spline(temperature, youngsmodulus)
>>> spl.write("spline_data.txt", 0, ",")
273,5000000000,300,4000000000,320,5000000000,340,1000000000
>>>
>>> dspl = spl.getderivative()
>>> dspl.write("splinederivative_data.txt", 0, ",")
273,-82402205.576362908,300,53693300.041614674,320,-58198085.726175644,340,-270900957.13691229
```

`getxmax`

`def getxmax( self ) ‑> float`

This returns the maximum value that input takes in the original data provided.

###### Example

```
>>> mymesh = mesh() # if a mesh object is not already available.
>>>
>>> temperature = (320, 273, 340, 300) # original x input
>>> youngsmodulus = (2.5e9, 5e9, 1e9, 4e9) # original y input
>>> spl = spline(temperature, youngsmodulus)
>>> spl.getxmax()
340
```

###### See Also

`getxmin`

`def getxmin( self ) ‑> float`

This returns the minimum value that input takes in the original data provided.

###### Example

```
>>> mymesh = mesh() # if a mesh object is not already available.
>>>
>>> temperature = (320, 273, 340, 300) # original x input
>>> youngsmodulus = (2.5e9, 5e9, 1e9, 4e9) # original y input
>>> spl = spline(temperature, youngsmodulus)
>>> spl.getxmin()
273
```

###### See Also

`set`

`def set( self, xin: List[float], yin: List[float] ) ‑> None`

`write`

`def write( self, filename: str, numsplits: int, delimiter: str = '\n' ) ‑> None`

This writes to file a refined version of the original data samples with data sorted in ascending order. It can be used to visualize the interpolation obtained with cubic splines.

###### Parameters

** filename** :

`str`

: Name of the file to which the data samples are written.** numsplits** :

`int`

: The number of additional points between two successive input considered for evaluation and subsequent writing. Minimum value required is .** delimiter** :

`str`

, optional : String specifying the delimiter. The default value is "\n".###### Raises

`RuntimeError`

: If `numsplits < 0`

. Minimum value required is .

###### Examples

Create a spline object:

```
>>> mymesh = mesh() # if a mesh object is not already available.
>>>
>>> temperature = (320, 273, 340, 300) # original x input
>>> youngsmodulus = (2.5e9, 5e9, 1e9, 4e9) # original y input
>>> spl = spline(temperature, youngsmodulus)
```

**Example 1:** If `numsplits = 0`

, no additional points are considered and the original data samples are written.

```
>>> numsplits = 0
>>> spl.write("spline_data.txt", numsplits, ",")
273,5000000000,300,4000000000,320,5000000000,340,1000000000
```

**Example 2:** If `numsplits = 1`

, between two successive inputs, one additional point is considered for evaluation and subsequent writing.

```
>>> numsplits = 1
>>> spl.write("spline_data.txt", numsplits, "\n")
273
5000000000
286.5
4040677668.5393257
300
4000000000
310
4779728464.4194756
320
5000000000
330
3531757178.5268416
340
1000000000
```

Similarly, if `numsplits = 2`

, between two successive inputs, two additional points are considered for evaluation and subsequent writing.

`universe`

`class universe`

#### Class variables

`cohomologycuts`

`ddmcoefs`

Type: `list`

`extraintegrationorder`

`gmresmodifiedgramschmidt`

`maxnumthreads`

`mortarsearchmiss`

`mortarsearchradius`

`roundoffnoiselevel`

`writeownedonly`

`writetobinary`

#### Static methods

`getmaxnumthreads`

`def getmaxnumthreads() ‑> int`

`setmaxnumthreads`

`def setmaxnumthreads( mnt: int ) ‑> None`

`vec`

`class vec( **kwargs )`

The `vec`

object holds a vector, be it the solution vector of an algebraic problem or its right-hand side.

#### Examples

**Example 1:** `vec(formul:formulation)`

This creates an all-zero vector whose structure and size is the one of formulation *projection*.

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> b = vec(projection)
```

**Example 2:** `vec(vecsize:int, addresses:indexmat, vals:densemat)`

This creates a vector with given values at given addresses.

```
>>> allinitialize()
>>> addresses = indexmat(3,1, [0,1,2])
>>> vals = densemat(3,1, [5,10,20])
>>>
>>> b = vec(3, addresses, vals)
>>> b.print()
```

Vec Object: 1 MPI processes type: seq 5. 10. 20.

```
>>> allfinalize()
```

#### Methods

`copy`

`def copy( self ) ‑> quanscient.vec`

This creates a full copy of the vector object.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> sol = vec(projection)
>>>
>>> copiedvec = sol.copy()
```

`getallvalues`

`def getallvalues( self ) ‑> quanscient.densemat`

This gets the values of all the entries of the vector in sequential order.

###### Returns

`densemat`

: A column matrix with the number of rows equal to the vector object size.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2 >>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> myvec = vec(projection) # creates a zero vector
>>>
>>> vals = densemat(myvec.size(),1, 12)
>>> myvec.setallvalues(vals) # all the entries now contain a value of 12
>>>
>>> vecvals = myvec.getallvalues()
```

###### See Also

`vec.setvalues()`

, `vec.setallvalues()`

, `vec.setvalue()`

, `vec.getvalues()`

, `vec.getvalue()`

`getvalue`

`def getvalue( *args, **kwargs ) ‑> float`

This gets the value of the vector object at the given `address`

. The `address`

provides the index at which the entry is requested.

###### Parameters

** address** :

`int`

: Index at which the entry of a vector is requested.###### Returns

`densemat`

: A matrix of size .

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2 >>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> myvec = vec(projection) # creates a zero vector
>>>
>>> vals = densemat(myvec.size(),1, 12)
>>> myvec.setallvalues(vals) # all the entries now contain a value of 12
>>>
>>> vecvals = myvec.getvalue(2)
```

###### See Also

`vec.setvalues()`

, `vec.setallvalues()`

, `vec.setvalue()`

, `vec.getvalues()`

, `vec.getallvalues()`

`getvalues`

`def getvalues( self, addresses: indexmat ) ‑> quanscient.densemat`

This gets the values in the vector that are at the indices given in `addresses`

.

###### Parameters

** addresses** :

`indexmat`

: A column matrix storing the indices at which the entries of a vector are requested.###### Returns

`densemat`

: A column matrix with number of rows equal to the length of the `addresses`

.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2 >>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> myvec = vec(projection) # creates a zero vector
>>>
>>> vals = densemat(myvec.size(),1, 12)
>>> myvec.setallvalues(vals) # all the entries now contain a value of 12
>>>
>>> addresses = indexmat(myvec.size(),1, 0,1)
>>> vecvals = myvec.getvalues(addresses)
```

###### See Also

`vec.setvalues()`

, `vec.setallvalues()`

, `vec.setvalue()`

, `vec.getallvalues()`

, `vec.getvalue()`

`load`

`def load( self, filename: str ) ‑> None`

This loads the data of a vector object from a file (either `.bin`

or `.txt`

ASCII format). This only works correctly if the dof structure of the calling vector object is the same as that of the vector object from which the file was written to the disk. In other words, the same set of formulation contributions must be defined in the same order.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> sol = vec(projection)
>>> sol.write("vecdata.bin") # writes the vector data to a file
>>>
>>> loadedvec = vec(projection)
>>> loadedvec.load("vecdata.bin") # loads the data to the vector object
```

###### See Also

`noautomaticupdate`

`def noautomaticupdate( self ) ‑> None`

After this call, the vector object will not have its value automatically updated after hp-adaptivity. If the automatic update is not needed then this call is recommended to avoid a possibly costly update to the vector values.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> sol = vec(projection)
>>>
>>> sol.noautomaticupdate()
```

`norm`

`def norm( self, type: str = '2' ) ‑> float`

This returns the , or norm of the vector. The default is the norm.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> sol = vec(projection)
>>>
>>> vals = densemat(sol.size(),1, 12)
>>> sol.setallvalues(vals) # all the entries now contain a value of 12
>>>
>>> sol.norm()
108.6646
```

###### See Also

`permute`

`def permute( self, rowpermute: indexmat, invertit: bool = False ) ‑> None`

This rearranges the vector in the order of indices prescribed in `rowpermute`

. The inverse permutation is performed if the boolean flag `invertit`

is set to True. The `rowpermute`

describes the mapping or inverse mapping function.

###### Parameters

** rowpermute** :

`indexmat`

: A column matrix storing the order of vector indices for the permutation.** invertit** :

`bool`

, default`: False`

: If set to True, inverse permutation is performed.###### Example

```
>>> rows = indexmat(6,1, [0,1,2,3,4,5])
>>> vals = densemat(6,1, [00,10,20,30,40,50])
>>>
>>> v = vec(6, rows, vals)
>>> permuterows = indexmat(6,1, [3,1,4,5,0,2])
>>> v.permute(permuterows, invertit=False)
>>> v.print()
Vec Object: 1 MPI processes
type: seq
30.
10.
40.
50.
0.
20.
>>>
>>> # Inverting the permutation on the above will give back the original order of the vector.
>>> v = vec(6, rows, vals)
>>> permuterows = indexmat(6,1, [3,1,4,5,0,2])
>>> v.permute(permuterows, invertit=True)
>>> v.print()
Vec Object: 1 MPI processes
type: seq
0.
10.
20.
30.
40.
50.
```

`print`

`def print( self ) ‑> None`

This prints the values of the vector object.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> sol = vec(projection)
>>>
>>> sol.print()
```

`setallvalues`

`def setallvalues( self, valsmat: densemat, op: str = 'set' ) ‑> None`

This replaces all the entries in the vector object by the values in `valsmat`

. The addresses of `valsmat`

are assumed to be in sequential order. If `op='set'`

, the values are replaced and if `op='add'`

the values are instead added to existing ones. This method works on all the entries.

###### Parameters

** valsmat** :

`densemat`

: A column matrix storing the values that are replaced in or added to the vector object.** op** :

`str`

, default`: 'set'`

: Equal to 'set' if the values must be replaced. For adding values to existing ones use 'add' instead.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2 >>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> myvec = vec(projection) # creates a zero vector
>>>
>>> vals = densemat(myvec.size(),1, 12)
>>> myvec.setallvalues(vals)
```

###### See Also

`vec.setvalues()`

, `vec.setvalue()`

, `vec.getvalues()`

, `vec.getallvalues()`

, `vec.getvalue()`

`setdata`

`def setdata( *args, **kwargs ) ‑> None`

This sets to the vector the data from the fields and ports defined in the formulation.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> sol = vec(projection) # creates a zero vector
This transfers the data of a field <code>myfield</code> to the addresses in the vector object corresponding to the physical region <code>physreg</code>.
>>> sol.setdata(vol, v); # populates the vector with the data from the field v
```

**Example 2:** `vec.setdata()`

This transfers to the vector the data from all fields and ports defined in the associated formulation.

```
>>> ...
>>> sol.setdata();
```

`setvalue`

`def setvalue( *args, **kwargs ) ‑> None`

This replaces the value in the vector object at the given `address`

with the values in `value`

. The 'address' provides the index at which the entry is replaced by `value`

. If `op='set'`

, the value is replaced and if `op='add'`

the value is added to the existing one. This method works only on a given single entry.

###### Parameters

** address** :

`int`

: Index at which the entry of a vector is replaced/added.** value** :

`double`

: Value that is set/added in the vector object.** op** :

`str`

, default`: 'set'`

: Equal to 'set' if the value must be replaced. For adding values to existing ones use 'add' instead.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2 >>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> myvec = vec(projection) # creates a zero vector
>>>
>>> myvec.setvalue(2, 2.32)
```

###### See Also

`vec.setallvalues()`

, `vec.setallvalues()`

, `vec.getvalues()`

, `vec.getallvalues()`

, `vec.getvalue()`

`setvalues`

`def setvalues( self, addresses: indexmat, valsmat: densemat, op: str = 'set' ) ‑> None`

This replaces the values in the vector object at the given `addresses`

with the values in `valsmat`

. If `op='set'`

, the values are replaced and if `op='add'`

the values are instead added to existing ones. This method works only on entries given in the `addresses`

.

###### Parameters

** addresses** :

`indexmat`

: A column matrix storing the indices at which the entries of a vector are replaced/added.** valsmat** :

`densemat`

: A column matrix storing the values that are replaced in or added to the vector object.** op** :

`str`

, default`: 'set'`

: Equal to 'set' if the values must be replaced. For adding values to existing ones use 'add' instead.###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2 >>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> myvec = vec(projection) # creates a zero vector
>>>
>>> addresses = indexmat(myvec.size(),1, 0,1)
>>> vals = densemat(myvec.size(),1, 12)
>>>
>>> myvec.setvalues(addresses, vals) # op is the default 'set'. All entries are replaced by value 12.
>>> myvec.setvalues(addresses, vals, 'set') # All entries are replaced by value 12.
>>> myvec.setvalues(addresses, vals, 'add') # Value 12 is added to all entries.
```

###### See Also

`vec.setallvalues()`

, `vec.setvalue()`

, `vec.getvalues()`

, `vec.getallvalues()`

, `vec.getvalue()`

`size`

`def size( self ) ‑> int`

This returns the size of the vector object. If the vector was instantiated from a formulation, then the vector size is equal to the number of dofs in that formulation.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> b = vec(projection)
>>> b.size()
82
```

`sum`

`def sum( self ) ‑> float`

This returns the sum of all the values in the vector.

###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> sol = vec(projection)
>>>
>>> vals = densemat(sol.size(),1, 12)
>>> sol.setallvalues(vals) # all the entries now contain a value of 12
>>>
>>> sol.sum()
984.0
```

###### See Also

`updateconstraints`

`def updateconstraints( self ) ‑> None`

This updates the values of all Dirichlet constraint entries in the vector.

###### Example

```
>>> mymesh = mesh("disk.msh")
>>> vol=1; sur=2
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>>
>>> b = vec(projection)
>>>
>>> v.setconstraint(sur, 1)
>>> b.updateconstraints()
```

`write`

`def write( self, filename: str ) ‑> None`

This writes all the data in the vector object to disk in a lossless and compact form. The file can be written in binary `.bin`

format (extremely compact but less portable) or in ASCII `.txt`

format (portable).

###### Parameters

** filename** :

`str`

: The name of the file to which the data from the vector object is written.###### Examples

```
>>> mymesh = mesh("disk.msh")
>>> vol = 1
>>> v = field("h1")
>>> v.setorder(vol, 1)
>>> projection = formulation()
>>> projection += integral(vol, dof(v)*tf(v) - 2*tf(v))
>>> sol = vec(projection)
>>>
>>> sol.write("vecdata.txt") # writes the vector data to a file
```

###### See Also

`vectorfieldselect`

`class vectorfieldselect`

#### Methods

`setdata`

`def setdata( self, physreg: int, myfield: field, op: str = 'set' ) ‑> None`

`wallclock`

`class wallclock`

This initializes the wall clock object.

#### Methods

`pause`

`def pause( self ) ‑> None`

This pauses the clock. The `wallclock.pause()`

and `wallclock.resume()`

functions allow to time selected operations in loop.

###### Example

```
>>> myclock = wallclock()
>>> myclock.pause()
>>> # Do something
>>> myclock.resume()
>>> myclock.print()
```

###### See Also

`print`

`def print( self, toprint: str = '' ) ‑> None`

This prints the time elapsed in the most appropriate format (, , or ). It also prints the message passed in the argument `toprint`

(if any).

###### Parameters

** toprint** :

`str`

, optional : message to print. The default value is an empty string "".###### Example

```
>>> myclock = wallclock()
>>> myclock.print("Time elapsed") # or myclock.print()
```

`resume`

`def resume( self ) ‑> None`

This resumes the clock. The `wallclock.pause()`

and `wallclock.resume()`

functions allow to time selected operations in loop.

###### Example

```
>>> myclock = wallclock()
>>> myclock.pause()
>>>
>>> for i in range (0, 10):
>>> myclock.resume()
>>> # Do something and time it
>>> myclock.pause()
>>> # Do something else
>>> myclock.print()
```

###### See Also

`tic`

`def tic( self ) ‑> None`

This resets the clock.

###### Example

```
>>> myclock = wallclock()
>>> myclock.tic()
```

###### See Also

`toc`

`def toc( self ) ‑> float`

This returns the time elapsed (in ).

###### Example

```
>>> myclock = wallclock()
>>> timeelapsed = myclock.toc()
```

###### See Also

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