Simulate pull-in of an electrostatic MEMS actuator

In this tutorial you set up a coupled electrostatics, solid mechanics, and mesh-deformation model of a simple MEMS capacitor and find the pull-in voltage—the bias at which the structure becomes electromechanically unstable. Pull-in matters because electrostatic actuation in small gaps can no longer be balanced by elastic restoring force at that point.

What you’ll learn

• How to set up a parallel-plate model with a vacuum gap, variables for all key lengths, and shared regions for materials and physics.
• How to couple Electrostatics, Solid mechanics (including lumped spring), and Mesh deformation to capture gap closure.
• How to run a voltage sweep and plot displacement against applied voltage, then compare the result to the analytical estimate.

Prerequisites: you should be comfortable with the project layout in Introduction to Quanscient Allsolve (Geometry, Physics, Simulations, and the Common sidebar).

Model definition

The model consists of two parallel square plates with equal thickness, separated by a vacuum gap. A DC voltage is applied to the bottom plate while the top plate is grounded. The bottom plate is clamped and the top plate is attached to a spring with stiffness $K$. The displacement of the top plate is constrained to move only in the $Z$-direction. The interaction between the top plate and attached spring is determined using a lumped model:

$$F_z = -K \cdot U_z$$

Parameters

The model parameters, that should be defined as variables in Allsolve, are defined as follows:

• Length of the square plate, $L = 50$ μm
• Height of the square plate, $H = 3$ μm
• Gap between the two parallel square plates, $d = 1$ μm
• Stiffness of the spring, $K = 10000$ N/m
• Overlapping area between the two parallel plates, $A = L \times L ~$ m²

Geometric elements

Element	Size	Center point (X, Y, Z)
Bottom plate	L $\times$ L $\times$ H	(0, 0, 0)
Top plate	L $\times$ L $\times$ H	(0, 0, H + d)
Vacuum box	L $\times$ L $\times$ d	(0, 0, H/2 + d/2)

Output Results

• Plot of applied voltage vs resulting displacement.

Material Data

1. Vacuum for the gap between the parallel plates
   • Electric permittivity: $\epsilon$ = $\epsilon_0 ~$ [F/m]
2. Silicon dioxide for the top and bottom plate
   • Young’s modulus: $E$ = $70 ~$ [GPa]
   • Poisson’s ratio: $\nu$ = $0.17$
   • Electric permittivity: $\epsilon$ = $3.9 \cdot \epsilon_0 ~$ [F/m]

Note: Electric permittivity in a vacuum

$\epsilon_0$ = $\dfrac{1}{\mu_0 c^2}$ = $8.854 \cdot 10^{-12} ~$ F/m

Boundary conditions

• Bottom plate

  • Electrode: $V$ = $V_{\rm DC}$
  • Clamped: $[u_x, u_y, u_z]$ = $[0, 0, 0]$

• Top plate

  • Grounded: $V$ = $0$
  • in-plane clamp: $[u_x, u_y]$ = $[0, 0]$

Note

A variable called Vdc will be defined in Allsolve, which provides a temporary value for the voltage. Later, this variable is swept over by defining a sweep override which takes a vector of voltage values.

Analytical solution

For the model setup defined above, the analytical solution to the pull-in voltage $V_{\rm p}$ is given by the formula: ¹

$$V_{\rm p} = \sqrt{\frac{8}{27}\frac{Kd^3}{\epsilon A}}$$

where,

• $K$ is the spring stiffness [N/m].
• $d$ is the initial gap between the parallel plates [m].
• $\epsilon$ is the electric permittivity of the gap medium [F/m].
• $A$ is the overlapping area between the parallel plates [m²].

The corresponding displacement $u_{\rm p}$ of the top plate at pull-in voltage is one-third of the initial gap between the plates.

$$x = \frac{1}{3} d$$

The pull-in voltage and the corresponding displacement based on the values defined in Parameters:

$$V_{\rm p} = 365.9 ~ volts \\ u_{\rm p} = 0.3333 ~ μm$$

Step-by-step guide

The steps below use regions to select volumes and surfaces. After Fragment all (finalize), entity tags can differ from project to project, so the tutorial does not rely on fixed tag numbers. Name regions clearly and use them everywhere you need a target (materials, physics, mesh rules, and outputs).

Step 1 — Build the geometry

1. Start with a new project, and name it as Pull-in analysis.

2. In the project geometry options, start from a box (or add a Box from Geometry elements).

3. Open the Common sidebar, Definitions tab, and add the following variables.

   Name	Description	Expression
   L	Plate size [m]	50e-6
   H	Plate thickness [m]	3e-6
   d	Gap between plates [m]	1e-6
   Vdc	DC Voltage [V]	1
   K	Spring stiffness [N/m]	1e4
   A	Overlap area between plates [m²]	L*L

   You can import the variables as a csv file:

   name,description,expression
   L,Plate size [m],50e-6
   H,Plate thickness [m],3e-6
   d,Gap between plates [m],1e-6
   Vdc,DC Voltage [V],1
   K,Spring stiffness [N/m],1e4
   A,Overlap area between plates [m²],L*L

4. Return to Geometry and set the first Box so its Size (X, Y, Z) is L, L, H and the center is (0, 0, 0), matching the bottom plate row in Geometric elements.

   

   First box: set size, center, and then build.

5. Add a Translation to make a second box: set Target to the bottom-plate volume (the first box; pick the volume in the model view or from Entities in the Common sidebar). Set Copy to Yes and Translation in Z to H + d. Repeat count 1 adds the top plate as a second volume.

   

   Translate the first volume along Z by H + d, with copy enabled, to form the top plate.

6. Add a third Box that fills the gap between the plates. Use Size (X, Y, Z) = (L, L, d) and center (0, 0, H/2 + d/2). Then Confirm model changes so the vacuum cell between the plates exists as a volume in the final built model.

   

   Final geometry: two plates and a box for the gap region.

Step 2 — Create regions

In the Common sidebar, open the Definitions tab. Under Regions, add regions (see Regions) so you can pick stable names for materials and physics. Use the model view or Entities to choose which tags belong in each region. Suggested names:

Region (suggested name)	What to include
plates	Both plate volumes (silicon dioxide)
vacuum_gap	The gap volume (vacuum)
bottom_plate	The bottom plate volume only
top_plate	The top plate volume only
top_plate_top	The top exterior surface of the top plate (a surface region, for Lump U/F)

Step 3 — Define the materials

Open the Physics section and add Materials:

1. Add a Vacuum material. Set its target to the region you defined for the gap (for example vacuum_gap).

2. Add a Silicon dioxide material. Set its target to the region that contains both plate volumes (for example plates).

Vacuum is assigned to the gap region and silicon dioxide to the plates region.

Step 4 — Define physics and interactions

In the Physics section:

1. Add Solid mechanics, Electrostatics, and Mesh deformation.

   

   Add the three physics before wiring interactions.

   Tip: Physics

   Add all physics first, then define their interactions. Some couplings require every relevant physics to exist before you can add the interaction.

Solid mechanics

1. Set the Solid mechanics target to the same silicon dioxide volume region you used for the material (for example plates).

   Physics	Target
   Solid mechanics	Region for both plates (e.g. plates)

2. Add Clamp on the bottom plate.

   Interaction name	Interaction type	Target
   Clamp	Clamp	bottom_plate

3. Add Constraint for in-plane fixation of the top plate:

   Interaction name	Interaction type	Target	Value
   In-plane clamp	Constraint	top_plate	[1, 0; 1, 0; 0, 0]

   The in-plane clamp constrains translation in X and Y directions on the top plate. Z remains free for pull-in motion.

   

   Constraint pattern for the top plate.

4. Add Lump U/F to attach the lumped spring on the top face:

   Interaction name	Interaction type	Target	Actuation mode	Value
   Lump U/F	Lump U/F	top_plate_top	Circuit coupling	[0, 0; 0, 0; 1, lump.Fz + K*lump.Uz]

   

   Lumped spring via circuit coupling on the top face.

5. Add Electric force (Solid mechanics ↔ Electrostatics).

6. Add Large displacement (Solid mechanics ↔ Mesh deformation).

Electrostatics

1. Let the electrostatics target be the whole model (all volumes: plates and gap), unless you have a reason to restrict it.

2. Add a Constraint for the ground (top plate at 0 V):

   Interaction name	Interaction type	Target	Value
   ground	Constraint	top_plate	0

3. Add a Constraint for the electrode (bottom plate at Vdc):

   Interaction name	Interaction type	Target	Value
   electrode	Constraint	bottom_plate	Vdc

4. Add Large displacement (Electrostatics ↔ Mesh deformation).

Mesh deformation

1. Let the mesh deformation target be the whole model (all volumes).

2. Add a second Constraint for the vacuum region so the mesh does not shear in-plane in the gap (same pattern as the old in-plane lock on the gap volume):

   Interaction name	Interaction type	Target	Value
   In-plane constraint	Constraint	vacuum_gap	[1, 0; 1, 0; 0, 0]

   

   Gap region: in-plane mesh constraint.

All required physics and interactions are in place.

Completed physics and interaction list.

Step 5 — Generate the mesh

1. Go to the Simulations section.

2. Add a new mesh.

3. Set Autorefine to Disabled.

4. Open the Mesh element size options.

5. Set Scale factor to 0.75.

6. Under Customizations, add Mesh extrusion —> Simple extrusion:

   Customization	Overlap mode	Target	Sublayers
   Simple extrusion	Prevent	All three box volumes	10, 8, 10

7. Run meshing and check the preview.

Preview the mesh before simulating.

Step 6 — Apply settings and run the simulation

1. Add a new simulation.

2. Set Analysis type to Static.

3. Select the mesh you created.

4. In the Common sidebar —> Definitions, add a Sweep Variable override. Override the Vdc variable with expression linspace(5, 365, 73). Add the sweep in your simulation inputs.

5. Add a Custom Value output called max displacement in um. Set the Output expression to maxvalue(reg.plates, -compz(u), 5) * 1e6. Here reg.plates is a region handle that matches the Solid mechanics target.

6. Run the simulation.

Run the static sweep over `Vdc`.

Step 7 — Plot results

1. Add a plot with Vdc on the X axis and max displacement in um on the Y axis.

Tip: Plot results in real time

You do not need to wait for the full sweep to finish before plotting. After the first sweep job completes, add the plot; it will update as more points finish. Use Logs to see when jobs are done.

Pull-in: displacement grows with voltage until the system becomes unstable in this idealized model.

Summary

You defined variables and regions, applied materials to the gap and plates, coupled electrostatics, solid mechanics (with a lumped spring on the top face), and mesh deformation, meshed the model, ran a static sweep over Vdc, and plotted displacement. Compare the pull-in point on the graph to the analytical values for the same $K$, $d$, $\epsilon$, and $A$.

References

Footnotes

1. Kaajakari, V. MEMS Tutorial: Pull-in voltage in electrostatic microactuators, 1–2. https://www.kaajakari.net/~ville/research/tutorials/pull_in_tutorial.pdf ↩