HB 003 - MEMS Microspeaker

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

In this example, harmonic balance is considered for a MEMS microspeaker. The example corresponds with the demo project mems-microspeakers-webinar, the results of which were covered in a Quanscient webinar ³.

Simulation setup guide

Below, you’ll find a simplified guide for setting this up 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 the 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 the Center point Y and Size Y parameters again:

   Name	Element type	Center point (m)	Size (m)	Rotation (deg)
   box 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

Now, your model geometry is finished, and should look like in the image below.

 

Step 2 - Define the materials

Proceed to the Properties section to define the model materials.

In this example, the Air and Monocrystalline silicon materials are used.

Air

First, pick the Air material from the material library and assign it to the 2 outer boxes (volumes 13, 25). Save the target as a shared region.

Monocrystalline silicon

Then, pick the Monocrystalline silicon material from the material library and assign it to the intermediary box (volume 1). Save the target as a shared region.

Now, your model materials are defined.

 

Step 3 - Define the physics

Proceed to the Physics section to define the physics.

In this example, the Solid mechanics, Electrostatics, Laminar flow and Mesh deformation physics are required.

Solid mechanics

• As solid mechanics target, select the Monocrystalline silicon region (volume 1).
• Add Clamp.
  • As Target, select the outer edge surfaces of the silicon volume along the X axis (surfaces 1, 2).
  • 
• Add the Solid mechanics - Mesh deformation coupling Large displacement.
• Add the Solid mechanics - Electrostatics coupling Electric force.

Electrostatics

• Let electrostatics target default to the whole geometry.
• Add Constraint.
  • As Target, select the silicon region (volume 1).
  • 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.

Laminar flow

• As laminar flow target, select the air region (volumes 13, 25).
• 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 directly from Docs while picking targets, utilize text-based picking:

1. Click the filter icon.
2. Click Text.
3. Paste your list of tags, such as 7 - 10, 12, 13, 15, 16.
4. Click Set tags.

• Add Pressure constraint.
  • As Target, select surfaces 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.

Mesh deformation

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

Tip: Expression editor

To copy-paste a matrix format constraint value directly from Docs, utilize the expression editor:

1. Click Switch to expression editor.
2. Paste your constraint value, such as [1, compx(u); 1, compy(u); 1, compz(u)].

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

Now, your physics, as well as their interactions and couplings are defined. Before moving on, check that your physics tree looks like in the image below.

 

Step 4 - Generate the mesh

Proceed to the Simulations section and create a new mesh using structured meshing:

1. Set Mesh quality to Expert settings.
2. Under Structured meshing, click Add structured entity and then Volume 3 times in total.
3. As structured entity targets, select volumes 1, 13 and 25, respectively.
4. Click Apply to save your work so far.
5. Set the segment counts A, B and C for the first structured entity as 5, 3 and 10, respectively. At this point, Allsolve will give you warnings about different segment length ratios and multiple definitions for curves, but you needn’t worry about them.
6. Set the segment counts A, B and C for the second structured entity as 5, 5 and 10, respectively.
7. Set the segment counts A, B and C for the third structured entity as 5, 5 and 10, respectively.
8. Click Apply & mesh.

Now, your mesh is finished, and should look like in the image below.

 

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.
  • Set Node type to 1 CPU, 16 GB.
  • Click Apply to save your work so far.
• 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(1, 20, 20).
• Add Outputs to your liking.

Adding a custom output with scripting (optional)

Once you have defined all the options, inputs and outputs for your simulation in the GUI, you can add an additional custom output with scripting:

1. Enable scripting mode.

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

   ##########################################
   # CUSTOM 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)
   
   ##########################################

Run your simulation by clicking Not 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 as animations.

 

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

 

Animation 2: Displacement of the complete microspeaker plate.

 

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

 

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 ↩