MUTs 001 - PMUT array

What you’ll learn

• How to build a PMUT array using the grid and translate operations, and define materials with geometry selector and computed regions.
• How to couple Elastic waves, Acoustic waves, and Electrostatics with piezoelectricity and AML perfectly matched layers for ultrasound simulation.
• How to run a harmonic frequency sweep and visualize acoustic pressure fields with slice filters.

Prerequisites: complete any one step-by-step tutorial first so you are comfortable with the Allsolve project layout (Geometry, Physics, Simulations, Results).

In this example, ultrasound emitted by a 6x6 array of PMUTs (Piezoelectric micromachined ultrasonic transducers) is modeled and simulated.

This example corresponds to the demo project 6x6 PMUT array

Simulation setup guide

Here you’ll find an example case level guide on modeling and setting up physics for a simple PMUT array in Quanscient Allsolve. See the last two steps for typical simulation options and visualizing results for harmonic ultrasound.

Variables

Allsolve supports inline variable creation, so variables can be created on the fly during modeling. Optionally, you can define all the variables first, and then move on to modeling.

Here is a list of all the variables used in this tutorial:

Name	Expression	Description
R	100e-6	Cavity radius
thmem	10e-6	Membrane thickness
thpiezo	2e-6	Piezo thickness
thcavity	20e-6	Cavity thickness
n	6	Array size (even numbers only)
Rair	5*R*n	Air bubble radius
freq	1e6	Frequency

Step 1 - Modeling

1. Build one of the PMUTs by creating 3 cylinders in a stack:

Name	Element type	Center point (m)	Size (m)	Rotation (deg)
Membrane	Cylinder	x: 0	Radius: R	x: 90
		y: 0	Height: thmem	y: 0
		z: 0		z: 0

Name	Element type	Center point (m)	Size (m)	Rotation (deg)
Piezo	Cylinder	x: 0	Radius: 0.67 * R	x: 90
		y: 0	Height: thpiezo	y: 0
		z: thmem/2+thpiezo/2		z: 0

Name	Element type	Center point (m)	Size (m)	Rotation (deg)
Cavity	Cylinder	x: 0	Radius: R	x: 90
		y: 0	Height: thcavity	y: 0
		z: -thcavity/2-thmem/2		z: 0

2. Use the Grid operation to make the 6x6 array of PMUTs:

Name	Element type	Target volumes	Translation	Repeat count
Grid	Grid operation	all 3 cylinder volumes of the single PMUT	x: 4*R	x: n
			y: 4*R	y: n
			z: 0	z: 0

3. Select all volumes in the grid and translate them so that your array is centered at the origin [0,0,0]:

Name	Element type	Target volumes	Translation
Translate	Translate operation	All volumes in the grid	x: -4*R*(n-1)/2
			y: -4*R*(n-1)/2
			z: 0

Tip

Use the Show axes tool to check the location of your array after translating.

4. Build a sphere and 2 mostly overlapping boxes for the air bubble and baseplate:

Name	Element type	Center point (m)	Size (m)
Sphere	Sphere	x: 0	Radius: Rair
		y: 0
		z: 0

Name	Element type	Center point (m)	Size (m)	Rotation (deg)
Box	Box	x: 0	x: 2*Rair	x: 90
		y: 0	x: 2*Rair	y: 0
		z: Rair+thmem/2	x: 2*Rair	z: 0

Name	Element type	Center point (m)	Size (m)	Rotation (deg)
Box 2	Box	x: 0	x: 2*Rair	x: 90
		y: 0	x: 2*Rair	y: 0
		z: Rair-thmem/2-thcavity	x: 2*Rair	z: 0

5. Apply the Fragment all operation.

6. Remove extra box and sphere sections (4 volumes in total, selected as in picture below).

7. Remove the cavity disc volume under each PMUT in the array.

Step 2 - Materials

Proceed to the Physics section to define the model materials.

1. Assign Air from the material library to the air bubble volume. Save the target as a region.
2. Create a Geometry selector type region called pzt. Include the small top disc of each PMUT in the region. This is easiest to do by setting both Minimum size and Maximum size with one of the small discs. Use the Pick similar tool to select the disc for min and max size.
3. Assign PZT from the material library to the pzt region.
4. Create a region called all, containing all volumes. It is easiest to do this by creating a computed region and picking the Complement of nothing. This way the all region is also updated if your geometry changes.
5. Create a Computed region type region. Call it mono-si, and include all the remaining volumes, i.e. the membrane disc of each PMUT as well as the large circular baseplate. To pick these volumes easily, use the Difference operation. Select regions all, pzt and Air target for the operation. The difference of these regions yields the volumes highlighted below (air bubble hidden):
6. Assign Monocrystalline silicon from the material library to the mono-si region.

Step 3 - Physics

In total, 3 physics are used in this example: Elastic waves, Acoustic waves and Electrostatics. Add all of them to your project, after which you can move on to defining their interactions.

Physics 1 - Elastic waves

• As elastic waves target, select all volumes except the air bubble.
• Add a Clamp interaction.
  • As clamp target, select the baseplate volume.
• Add the Elastic waves–Electrostatics coupling Piezoelectricity.
  • As piezoelectricity target, select the pzt region.

Physics 2 - Acoustic waves

• As acoustic waves target, select the air bubble.
• Add Perfectly matched layer.
  • As perfectly matched layer target, select the air bubble dome surface.
  • Set PML Type to AML.
    • AML is suitable for rounded surfaces such as an air bubble, whereas box PML is built for XYZ-aligned box surfaces.
• Add the Acoustic waves–Elastic waves coupling Acoustic structure.

Physics 3 - Electrostatics

• As electrostatics target, select the pzt region.
• Add a Constraint interaction.
  • As constraint target, select the bottom surface of each PZT disc on the array, 36 in total. To avoid clicking a large number of surfaces manually, you can define a geometry selector region with a bounding box for the surfaces. To avoid picking the larger PMUT membrane surfaces that exist on the same plane, use Pick similar for max size.

    Using geometry selector for planar surfaces

    To pick surfaces on a certain Z-plane with geometry selector, it’s recommended to use the exact same value for Z Min and Z Max. Surface localization in a very thin Z-slice can be inaccurate due to floating point errors. If Z min and max are set as the same value however, a buffer is set to both directions to pick planar surfaces reliably.

  • Set constraint value to 0.
• Add a Lump V/Q interaction.
  • As lump V/Q target, select the top surface of each PZT disc on the PMUTs. Again, define a geometry selector region with a bounding box to save some clicks.
  • Set Actuation mode to Voltage.
  • Set voltage to sin(2*pi*freq*t).

Step 4 - Meshing

• Add a mesh.
• Open the collapsible menu for Mesh element size.
• Set Max size to 1e-4.
• Save settings & mesh.

Step 5 - Simulation options

Create a simulation with following options:

• Set Analysis type to Harmonic.
• Set Fundamental frequency to freq.
• Set Number of FFT samples to 3.
• Set Node count to 10.
• As the Mesh, select the mesh you created.
• For Inputs, create a sweep:
  • Create a Sweep override over variable freq.
  • Set Override expression to linspace(0.7e6, 0.9e6, 41).
  • Add the sweep to simulation inputs.
• In Outputs, add:
  • u harmonic 2 field
    • Skin only
  • p harmonic 2 field
    • Skin only
  • See demo project for custom value output examples.

Step 6 - Results

In the Simulations section, add visualizations and plots as needed. The u field can be visualized in the air bubble using a slice filter for example: