HB 003 - MEMS Microspeaker

Electrostatically actuated silicon-based microspeakers are an emerging technology due to their obvious advantages, providing high sound quality over a wide range of frequencies ^{1 2}.

Multiharmonic analysis is conducted for a MEMS microspeaker in this example using a harmonic balance method.

Demo project: mems-microspeakers-webinar. The demo was presented in a Quanscient webinar ³.

Simulation setup guide

Below, you’ll find a simplified guide for setting up a multiharmonic analysis simulation for a MEMS microspeaker in Quanscient Allsolve.

Step 0 - Define shared expressions

Start out in the Properties section by defining the following shared expressions:

Name	Description	Expression
L	Length [m]	1000e-6
W	Width [m]	75e-6
T	Thickness [m]	3e-6
g0	Initial gap [m]	10e-6
freq	Frequency [Hz]	100
Voltage	Voltage [V]	30

Step 1 - Create the geometry

Geometric entity tags

Geometric entities (volumes, surfaces, curves and points) are referenced using tags throughout this tutorial.

Carefully replicate the geometry construction steps below in order to have correct tags. Incorrect entity tags can lead to errors or unexpected behavior during simulations.

In the Model section, create the model geometry by creating Box elements as follows.

1. Create the first box:

   Name	Element type	Center point (m)	Size (m)	Rotation (deg)
   box	Box	X: 0	X: L	X: 0
   		Y: 0	Y: T	Y: 0
   		Z: 0	Z: W	Z: 0

2. Create the second box by copying the first one, and editing Center point Y and Size Y parameters:

   Name	Element type	Center point (m)	Size (m)	Rotation (deg)
   box 2	Box	X: 0	X: L	X: 0
   		Y: T/2 + g0/2	Y: g0	Y: 0
   		Z: 0	Z: W	Z: 0

3. Create the third box by copying the second one, and editing Center point Y and Size Y parameters:

   Name	Element type	Center point (m)	Size (m)	Rotation (deg)
   box 2 2	Box	X: 0	X: L	X: 0
   		Y: -T/2 - g0*1.5	Y: 3*g0	Y: 0
   		Z: 0	Z: W	Z: 0

Finished geometry:

Step 2 - Define the materials

Go to the Properties section.

Material 1 - Air

1. Assign Air to the 2 outer box volumes (13, 25).
2. Share the target region.
3. Save the material settings.

Material 2 - Monocrystalline silicon

1. Assign Monocrystalline silicon to the intermediate box volume (1).
2. Share the target region.
3. Save the material settings.

Step 3 - Define the physics

Go to the Physics section.

In this example, Solid mechanics, Electrostatics, Laminar flow and Mesh deformation physics are required. Add all of them before moving on to define interactions.

Physics 1 - Solid mechanics

• As solid mechanics target, select the Monocrystalline silicon region.
• Add Clamp.
  • As Target, select the outer edge surfaces of the silicon volume in the positive-X and negative-X ends.
  • 
• Add the Solid mechanics - Mesh deformation coupling Large displacement.
• Add the Solid mechanics - Electrostatics coupling Electric force.
• Add Geometric nonlinearity.

Physics 2 - Electrostatics

• Let electrostatics target default to the whole geometry.
• Add Constraint.
  • As Target, select the Monocrystalline silicon region.
  • Set Constraint value to Voltage*sin(2*pi*0.5*freq*t).
  • 
• Add Constraint 2.
  • As Target, select the air box surface 9.
  • Set Constraint value to 0.
  • 
• Add the Electrostatics - Mesh deformation coupling Large displacement.

Physics 3 - Laminar flow

• As laminar flow target, select the Air region.
• Add Velocity constraint and name it as noslip.
  • As Target, select surfaces 7 - 10, 12, 13, 15, 16.
  • Set Constraint value to [1, 0; 1, 0; 1, 0].
  • 

Tip: Text-based target picking

To copy-paste a list of tags while picking targets, utilize text-based picking:

• Add Pressure constraint.
  • As Target, select the air box surfaces that were not included in noslip target (11, 14).
  • Set Constraint value to 0.
  • 
• Add the Laminar flow - Solid mechanics coupling Fluid structure.
  • As Target, select the silicon and air interface surfaces (3, 4).
• Add the Laminar flow - Mesh deformation coupling Large displacement.

Physics 4 - Mesh deformation

• Let mesh deformation target default to the whole geometry.
• Add Constraint.
  • As Target, select the Monocrystalline silicon region.
  • Set Constraint value to [1, compx(u); 1, compy(u); 1, compz(u)].
  • 

Tip: Expression editor

To copy-paste matrix values while inputting constraint values, utilize the expression editor:

• Add Constraint 2.
  • As Target, select surfaces 7 - 9, 12 - 14.
  • Set Constraint value to [1, 0; 1, 0; 1, 0].
  • 

Finished physics tree:

Step 4 - Mesh the geometry

Go to the Simulations section.

Create a new mesh using structured meshing:

1. Set Mesh quality to Expert settings.
2. Add 3 Volume structured mesh entities, each targeting a different volume in the geometry.
3. For the structured entity targeting volume 1, set the segment counts A, B and C as 5, 3 and 10, respectively.
4. For the structured entity targeting volume 13, set the segment counts A, B and C as 5, 5 and 10, respectively.
5. For the structured entity targeting volume 25, set the segment counts A, B and C as 5, 5 and 10, respectively.
6. Save the settings & mesh.

Finished mesh:

Step 5 - Simulate

In the Simulations section, create a new simulation:

• In Simulation settings:
  • Set Analysis type to Multiharmonic.
  • Set Fundamental frequency to freq.
  • Set Harmonics to 1 2 3 4 5 6 7 8 9.
  • Set Number of FFT samples to 20.
  • Set Node count to 10.
  • Use the default Node type (1 CPU, 16 GB).
• In Mesh, select the mesh you created.
• In Inputs:
  • Add freq sweep.
    • Set Override expression to linspace(20, 20000, 20).
  • Add Voltage sweep.
    • Set Override expression to linspace(5, 25, 20).
• Add Outputs to your liking.

2-variable sweeps

The surface plot in Results contains all pairings of frequencies and voltages within the given limits. Here, only the diagonal is included.

To get the full surface plot, some larger override vectors are required. For defining the override vectors, see 2-variable sweeps.

Adding a THD output via scripting (optional)

Once you have defined all the options, inputs and outputs for your simulation through the GUI, you can add a Total Harmonic Distortion (THD) custom output with scripting:

1. Enable scripting mode.

2. Add this snippet of code at the end of your simulation.py file.

   ##########################################
   # THD_F CUSTOM VALUE OUTPUT
   
   probe = [0,0,0]
   ui = []
   THD = 0
   for i in range(9):
       ui.append(qs.allinterpolate(reg.solid_mechanics, qs.harm(i+1, qs.compy(fld.u), 20), probe))
       if(i>2):
           THD = THD + ui[i]*ui[i]
   
   THD = qs.evaluate(qs.sqrt(THD/(ui[0]*ui[0]+ui[1]*ui[1]+1e-50)))
   qs.setoutputvalue("THD_F", THD)
   
   ##########################################

3. Save the script.

Your simulation is now ready to run.

Step 6 - Plot & visualize

In the Simulations section, add plots to see value output results, or visualizations to see field output results.

One visualization example is given below, generated with the following options:

• Field output:
  • Displacement field u harmonic 1
  • Out of sweep subsimulations 0 - 19, Simulation 0 was used
• Color preset:
  • Cool to warm
• Warp:
  • Scale factor 500000

Results

Below are a few examples of simulation results post-processed with external software.

Plot 1: The custom value output THD_F plotted as a function of frequency and voltage

Animation 1: Displacement, microspeaker plate cut in half at the X-axis symmetry plane

Animation 2: Displacement of the complete microspeaker plate

References

Footnotes

1. Kaiser, B. et al. Concept and proof for an all-silicon MEMS micro speaker utilizing air chambers. Microsyst Nanoeng 5, 43 (2019). https://doi.org/10.1038/s41378-019-0095-9. ↩

2. Melnikov, A., Schenk, H.A.G., Monsalve, J.M. et al. Coulomb-actuated microbeams revisited: experimental and numerical modal decomposition of the saddle-node bifurcation. Microsyst Nanoeng 7, 41 (2021). https://doi.org/10.1038/s41378-021-00265-y ↩

3. Dr. Andrew Tweedie, Dr. Abhishek Deshmukh, Jukka Knuutinen. Faster and more reliable MEMS design with cloud-based multiphysics simulations. Quanscient webinars (2024). https://quanscient.com/events/mems-webinar-06-24/register ↩