Electrostatic voltage isosurfaces around comb drive
Electrostatic field analysis is important in MEMS to determine both capacitance and electrostatic forces which are typically used to actuate devices such as comb drives. Electrostatic field analysis determines the electric field and electric scalar potential (voltage) distribution caused by charge distributions or applied potential. You can apply two types of loads voltage and charge densities. An electrostatic analysis is assumed to be linear, and hence the electric field is proportional to the applied voltage.
ANSYS offer both H and adaptive P-element electrostatic elements and the open domain can be modeled either using infinite boundary elements (INFIN110, and INFIN111) or Trefftz domain technology. We cover the Trefftz approach in a separate section on this page, click here for details.
Iinterior view of an electrostatic octo-pole benchmark used to validate ANSYS solution accuracy at the nano-scale level (Courtesy Andreas Heike)
Traditional H-finite elements rely on mesh refinement to achieve a desired accuracy. P-Adaptive elements rely on increasing the polynomial order of the element to achieve a desired accuracy. Brick / wedge (SOLID128) and tetrahedral (SOLID127) P-element shapes are available. The polynomial order of each element is increased automatically (from level 2 - 8) to satisfy convergence to a prescribed degree of accuracy. The P-elements supports the same electrostatic boundary conditions as H elements, loads, ODE coupling and constraint equations. P elements also work with Trefftz Domain & Capacitance extraction functions.
Top view of a section of "teeth" from a lateral MEMS comb drive. Its shows the P-element mesh and resulting electrostatic field contour plot. The P-element polynomial orders are color coded.
ANSYS offers both H and adaptive P-element electrostatic elements and the open domain can be modeled either using infinite boundary elements (INFIN110, and INFIN111) or the Trefftz method. The Trefftz method is a hybrid finite element--Trefftz method (hereafter referred to as the Trefftz method). The Trefftz method bears the name of the founder of boundary element techniques.
The Trefftz method combines the efficiency of boundary techniques in open domain treatment with a finite element-like positive definite stiffness matrix. i.e. The open domain is not meshed, thus reducing the problem size. It allows treatment of complex surface geometry, even with high aspect ratios common in MEMS. The Trefftz method implemented in ANSYS is an easy to use, accurate method for handling open boundary domains in electrostatics.
The Trefftz Method may be used to connect multiple finite element electrostatic field domains, eliminating the need to mesh the field regions between component regions. Advantages include a substantial reduction in the size of the Finite Element model size since you can have any number of unconnected finite element domains connected by the Trefftz regions.
Electrostatic field around two charged spheres:
The Trefftz approach has proven to be an extremely accurate and efficient method for electrostatic analysis. It combines the advantages of finite element with boundary element technology. For example, a Trefftz analysis of a charged, isolated sphere produces very accurate results with a relatively small meshed domain. About 200 degrees of freedom (DOF's) yielded an accuracy of within 3% of the closed form solution. i.e. 110.2 pF versus 113 pF for the closed form solution.
Electrostatic field contours for single charged sphere benchmark.
Further refinement of the mesh yields rapid accuracy convergence to the closed form solution. The following graph illustrates the accuracy improvements by subsequently doubling the mesh density. After quadrupling the mesh density (800 DOF's) the Trefftz analysis is within 1E-3 of the closed form solution. By comparison, traditional finite element (infinite) element approach would require in access 25,000 DOF's to achieve that level of accuracy.
Mesh refinement accuracy improvements