Expressions

Expressions in Quanscient Allsolve

Quanscient Allsolve uses the power of expressions all around the application. You can define and use them for most inputs, wherever you see an icon. An expression can consist of numbers, matrices, simple arithmetics and predefined variables and functions among other things.

Some functions and variables are predefined as expressions for all projects but a project can have any amount of custom shared expressions.

Expression language

The expression language of Allsolve is a subset of Octave. In expressions, everything is a matrix of doubles. Even scalars are considered 1 x 1 matrices.

The supported features are listed below.

Numbers

Double precision number in any format, such as 41, 8.91, -0.91233, +4.655e-23.

Constants

Basic mathematical constants are predefined: pi, mu0 and epsilon0. Constants can be used in place of numbers, like 2 * pow(pi,2).

Expression API

Basic math constants

`pi`

The constant pi

`mu0`

Vacuum magnetic permeability

`epsilon0`

Vacuum electric permittivity

Quanscient variables

`t`

Time in a transient simulation

`x`

The x-coordinate of the geometry

`y`

The y-coordinate of the geometry

`z`

The z-coordinate of the geometry

Basic math functions

`sin`

The sine function

input: expression

`cos`

The cosine function

input: expression

`tan`

The tangent function

input: expression

`asin`

The inverse sine function

input: expression

`acos`

The inverse cosine function

input: expression

`atan`

The inverse tangent function

input: expression

`abs`

The absolute value function

input: expression

`sqrt`

The square root function

input: expression

`log`

The log function

input: expression

`log10`

The log10 function

input: expression

`exp`

The exponent function

input: expression

`pow`

The power function

base: expression
exponent: expression

Return an expression equal to base to the power of exponent

`mod`

The modulo function

input: expression
modval: float - divisor value

Returns an expression equal to the remainder resulting from the division of input by modval

Quanscient functions

`dx`

The dx function

input: expression

Returns the x space derivative expression.

`dy`

The dy function

input: expression

Returns the y space derivative expression.

`dz`

The dz function

input: expression

Returns the z space derivative expression.

`dt`

The dt function

input: expression

Returns the first-order time derivative expression.

`dtdt`

The dtdt function

input: expression

Returns the second-order time derivative expression.

`dtdtdt`

The dtdtdt function

input: expression

Returns the third-order time derivative expression.

`dtdtdtdt`

The dtdtdtdt function

input: expression

Returns the fourth-order time derivative expression.

`max`

The max function

a: expression
b: expression

Returns the maximum value of given values. Can be used with 2 expressions, fields or parameters.

`min`

The min function

a: expression
b: expression

Returns the minimum value of given values. Can be used with 2 expressions, fields or parameters.

`maxvalue`

The expression max function

region: Region
expression: expression
refinement: int

Returns the maximum value of a scalar expression on a region. For example maxvalue(reg.surface, compx(u), 5). Vector expressions needs to be converted to scalar with eg. norm or compx functions. The accuracy of the value of an expression can be improved by providing a higher refinement value.

`minvalue`

The expression min function

region: Region
expression: expression
refinement: int

Returns the minimum value of a scalar expression on a region. For example minvalue(reg.surface, compx(u), 5). Vector expressions need to be converted to scalar with eg. norm or compx functions. The accuracy of the value of an expression can be improved by providing a higher refinement value.

`probe`

Probe the value of an expression on a point region

region: Region - A point region
expression: expression - Expression to evaluate at point.

`interpolate`

Interpolate the value of an expression on a region in given coordinates.

region: Region
expression: expression
coords: [float, float, float]

Interpolate the value of an expression on a region in given coordinates [x, y, z].

The expression must be scalar. Vector expressions needs to be converted to scalar with eg. norm or compx functions.

`ifpositive`

Return separate values based on compared value.

condExpr: expression
trueExpr: expression
falseExpr: expression

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.

Example

At points, where x + y >= 0, value is 1, otherwise -1. ifpositive(x+y, 1, -1)

`comp`

The comp function

component: int

Get the $x$ , $y$ or $z$ component of a field with subfields.

`compx`

The compx function

Get the x component of a field. Equal to comp(0)

`compy`

The compy function

Get the y component of a field. Equal to comp(1)

`compz`

The compz function

Get the z component of a field. Equal to comp(2)

`transpose`

The transpose function

input: expression

Returns an expression that is the transpose of the input vector or matrix expression

`inverse`

The matrix inverse function

Returns the inverse of a square matrix.

`determinant`

The matrix determinant function

input: expression

Returns the determinant of a square matrix.

`grad`

The gradient function

input: expression

For a scalar input expression, this is mathematically treated as the gradient of a scalar ( $\nabla{v}$ ) 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 ( $\nabla{\boldsymbol{u}}$ ) and the output has one row per component of the input and one column per space derivative.

`div`

The divergence function

input: expression

Computes the divergence of a vector expression. The returned expression is a scalar. $\nabla \cdot \boldsymbol{u}$

`curl`

The curl function

input: expression

Computes the curl of a vector expression. The returned expression is a vector. $\boldsymbol{w} = \nabla \times \boldsymbol{u}$

`norm`

The norm function

input: expression

gives the $L2$ norm of an expression input.

`crossproduct`

The cross product function

a: expression
b: expression

Computes the cross-product of two vector expressions. The returned expression is a vector. $\boldsymbol{c} = \boldsymbol{a} \times \boldsymbol{b}$

`doubledotproduct`

The double dot product function

a: expression
b: expression

Computes the double-dot product of two matrix expressions. The returned expression is a scalar. $\boldsymbol{A:B} = \sum_{i,j} A_{ij} B_{ij}$

Example

doubledotproduct([1, 2; 3, 4], [11, 12; 13, 14]) is resolved to 130

`cn`

cn(n) is a shortcut for cos(2 * pi * f * n * t)

n: float - fundamental frequency multiplier

`sn`

sn(n) is a shortcut for sin(2 * pi * f * n * t)

n: float - fundamental frequency multiplier

`getharmonic`

Returns the harmnum harmonic of the expression

harmnum: int
input: expression

`harm`

harm(n, expression) returns the nth harmonic of the expression

harmnum: int
input: expression

`makeharmonic`

makeharmonic creates a harmonic expression

harmonicNumbers: List[int]
expr: expression

Create harmonic expression from given harmonic indices and an expression.

`on`

The on function

physreg: Region
expr: 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.

Integrate to get the total current (flux) through a surface: integrate(reg.integration_surf, on(reg.integration_vol, transpose(j)) * normal(reg.integration_vol), 5)

`normal`

Returns the normal of a region

physreg: Region - region with project dimension

Given a region of project dimension, returns the normal pointing out of the region. For example in a 3D project, the region must be a volume. For 2D, a surface.

Integrate to get the total current (flux) through a surface: integrate(reg.integration_surf, on(reg.integration_vol, transpose(j)) * normal(reg.integration_vol), 5)

`linspace`

Produces a row vector of length numValues with linearly spaced values from start to end

start: float
end: float
numValues: int

linspace produces a row vector of length numValues with linearly spaced values from start to end. Can be used to create override vectors for sweeps.

linspace(1, 3, 5) is resolved to [1, 1.5, 2, 2.5, 3]

`logspace`

Produces a row vector of length numValues with logarithmically spaced values from 10^(start) to 10^(end), with default base 10

start: float
end: float
numValues: int
base: int

logspace produces a row vector of length numValues with logarithmically spaced values from 10^(start) to 10^(end), with default base 10. If end is pi, values go from 10^(start) to pi instead. Can be used to create override vectors for sweeps.

logspace(1, 3, 5) is resolved to [10, 31.623, 100, 316.228, 1000]

`repelem`

Repeats elements of a vector numRepeat times

vec: vector
numRepeat: int

Repeats elements of a vector numRepeat times. Can be used to create override vectors for multivariate sweeps so that numRepeat varies based on other vector lengths.

repelem([1, 2, 3], 3) is resolved to [1, 1, 1, 2, 2, 2, 3, 3, 3]

`repmat`

Repeats matrix rowRepeat times in row direction and colRepeat times in column direction

mat: matrix
rowRepeat: int
colRepeat: int

Repeats matrix rowRepeat times in row direction and colRepeat times in column direction. Can be used to create override vectors for multivariate sweeps so that rowRepeat is always 1 and colRepeat varies based on other vector lengths.

repmat([1, 2, 3], 1, 3) is resolved to [1, 2, 3, 1, 2, 3, 1, 2, 3]

`integrate`

Integrates an expression over a region

region: Region
expr: expression
order: int

Integrates an expression over the physical region region. Examples Calculate volume/area/length: integrate(region, 1, 3).

Integrate to get the total current (flux) through a surface: integrate(reg.integration_surf, on(reg.integration_vol, transpose(j)) * normal(reg.integration_vol), 5)

`wavelet`

Create a wavelet with given frequency and delay

frequency: expression
delay: expression

`ramp`

Create a ramp signal

rampuptime: expression - The time to transition linearly from value 0 to 1
holdtime: expression - The time the value stays at 1
rampdowntime: expression - The time to transition linearly from value 1 to 0
delay: expression - The time at which the rampup starts

It is a signal that starts initially at value 0. Then at 0 seconds + delay in seconds it transitions from value 0 to 1 in a linear increase. That transition happens in rampuptime seconds. This is followed by a flat value 1 that is held for holdtime seconds. This is followed by a linear decrease from value 1 to 0 that happens in rampdowntime seconds. The value is then forever 0 after that.

`vonmises`

The von Mises function

stress: 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 $(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{yz},\sigma_{xz},\sigma_{xy})$ .

For 2D plane stress problems all $z$ related components of the stress tensor are $0$ . For plane strain problems do not forget the term $(\sigma_{zz} = \nu \cdot (\sigma_{xx} + \sigma_{yy}))$ .

`lineinterpolate`

Interpolates the expression at a series of points along a line.

region: Region
expression: expression
firstCoords: [float, float, float]
lastCoords: [float, float, float]
numSamples: int

This method interpolates the expression at a series of points along a line inside a region. The line for interpolation is defined by a starting and an end point whose [x,y,z] coordinates are provided in the firstCoords and lastCoords arguments.

The numSamples argument determines the number of sample points considered along the line.

If a requested interpolation point along the line cannot be found (because it is outside of region or because the interpolation algorithm fails to converge, as can happen on curved 3D elements) then an error occurs.

To be able to plot the values of nicely, another output should be created that contains the coordinate points. For example outputs:

interpolatedValues = lineinterpolate(reg.box, [0, 0, 0], [0, 100, 0], 11)
yCoords = linspace(0, 100, 11)