A Parametric Multilevel MEMS Simulation Methodology using Finite Element Method and Mesh Morphing

Vladimir A. Kolchuzhin and Jan E. Mehner Chemnitz University of Technology, Department of Micro systems and Precision Engineering

Reichenhainer Str. 70, 09126 Chemnitz, Germany vladimir.kolchuzhin@etit.tu-chemnitz.de

Abstract The MEMS designing is a very challenging and

interdisciplinary task. The paper deals with advanced computational methodology required to extend the considered effects within the ordinary FE-analyses taking into account parameter variation at the physic and system levels.

1. Introduction CAD systems for MEMS began in the early 1990s,

with the primary focus being on the 3D model generation, algorithms for fast electrostatic analysis of complicated structures and methods for solving coupled partial differential equations [1-4]. Like the mechanical CAD systems, MEMS CAD systems were integrated with FEM and BEM tools within the design environment, generating mesh from 3D models or 2D layout (layout-based design). Different design methodologies and a variety of software tools are utilized in order to analyze complex geometrical structures, to account for interactions among different physical domains. CAD for MEMS spans many levels of abstraction from physical, device to system level like the VLST design methodology. The structured MEMS design tools use the formation of standard data representations and component libraries [5-6]. Modern MEMS CAD systems tend to integrate with the EDA tools generating compact model from physics based models remains tedious and lacks automatic adaptation tools [6-9].

Modeling is an important step of the MEMS design. The simulation of MEMS components consist of several iteration steps. The physical behavior of 3D continuums is described by partial differential equations which are typically solved by the finite element method. The levels of difficulty of the physical simulations can be classified using three axes: the parameters, the physical domains and the response, as illustrated in Fig. I. An analysis of complex micromechanical systems involves multiple physical domains, including mechanical, electrostatic, magnetic, thermal, and fluidic domains or a mixture of them. The FE method allows for interactions among different physical domains to obtain the static, modal, frequency and transient responses. Different types of design parameters can be handled in modeling of MEMS component. The most obvious ones are continuous parameters such as geometrical dimensions, material properties, etc. The designer can also deal with discrete parameters such as boundary conditions or loads. Process issues involving dimensional variations can highly change the transfer function of the MEMS components as well as

have an influence on the effect of temperature and packaging.

Figure 1: OLAP cube for the physic level simulation

Parametric modeling has become the basis for most mechanical CAD systems. Designers spend a significant amount of time in geometry manipulation for meshing. In mechanical design, a lot of design tasks are variational. Therefore, the parameterized CAD models by mesh morphing can be reused in a design step. Currently, the parametric model is extracted by series of discrete FE solutions and subsequent interpolation procedures (multivariate polynomial and rational fitting, Gaussian process regression). Especially for a large set of design variables, data sampling and fit become time consuming and prone to errors. Parametric design is a revolutionary paradigm of CAE systems [10], which allows designers to take into account the parameters variation of a model in single step without redesign, e.g. ANSYS Variational Technology is a tool to provide accurate, high-order response surfaces based on a single finite element analysis [11]. The geometrical parameters can be automatically recognized from a CAD model. The application of Higher Order Derivatives (HOD) analysis to FE equations as a way to increase the efficiency and robustness was started in the middle of the 1990s [10]. The model response can be expanded in the vicinity of the initial position with regard to dimensional and physical parameters in a single FE run as multivariate Taylor series or its Pade equivalent.

From a large-scale 3D multi-physics simulation to compact modeling, one is interested to obtain the model response with respect to design parameters. Such simulations provide full information of the device behavior and lead to MEMS components with optimized performance parameters at system-level. Recently, we presented the application HOD approach for the

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parametric MEMS simulation at physic level [13] and parametric Reduced Order Modeling (ROM) at subsystem level [12,14]. Starting from a full order fmite element model with structural, electrostatic and thin film fluid elements, an automated generation step for parametric macromodel must be performed in order to create database information and export to VHDL-AMS. The results are compact behavioral descriptions taking into account the design parameters which can be transferred to the electronic and system simulators for virtual prototyping and device analyses.

The paper is focused on a parametric methodology to multilevel MEMS simulation, which support appropriate links available to the designer to switch between different levels. At each of these levels, a design can be viewed in physical, schematic or behavioral form. The capabilities and weaknesses of these methods are discussed. The simulation of a MEMS resonator fabricated by Bonding and Deep Reactive Ion Etching (BDRlE) technology illustrates the application of the approach.

2. Methodology description Taylor series expansion is a common engineering

approach to estimate the structural response versus parameter variation. In the standard matrix methods of analysis, based on lumped element idealization, as well as mesh-methods, the structure being analyzed is approximated as an assembly of discrete elements connected at nodes. The idea of the HOD approach is to compute not only the governing system matrices but also high order derivatives with regard to design parameters. An Automatic Differentiation (AD) algorithm is applied to a FE code (element stiffness matrices, finite element equations, and post-processing data) to compute exact high order derivatives with respect to design parameters. The AD utilizes the generalized differentiation rules and gives an exact (up to the machine precision) numerically representation of high order derivatives. The derivatives of a global FE matrix K can be obtained from a superposition of derivatives of the elementary matrices k. The Taylor expansion of the FE solution can be obtained by differentiation of the corresponding FE problem (static, modal or harmonic). As a result, the Taylor vectors of the model response can be expanded in the vicinity of the initial position capturing parameters. More details on HOD algorithm can be found in [10,11,13].

2. 1. Mesh morphing In contrast to ordinary FEM, one need a special

parametric FE-model what captures the variation of geometrical design variables without a remesh procedure. Mesh-morphing affects only the nodes. The associativity of the nodes and elements is retained with the solid modeling entities. Unlike sensitivity analysis of a shape design variable, where only finite elements situated at the surface are perturbed, the HOD approach requires the perturbation of the internal nodes. Parametric mesh­morphing means that the node table contains not only a single numerical value for each spatial direction as it is

supported by most mesh tools, rather each node i will be described by analytical functions {x/p), Yi(P), Zi(P)} with respect to geometrical parameters p.

The description of the shape change at the outer boundaries is necessary at the first and can be described as a continuous function of the parameters. Any 3D linear transformation such as translation, rotation, scaling and combinations above-mentioned can be represented by a coordinate transformation matrix T. A wide variety of configurations can be created with these linear transformations:

T(p,xJ = Xi +v(xJp. The linear approximation means that the design velocity vex) can be easily computed if the initial and fmal meshes are known. The crucial aspect of using mesh-morphing is the perturbation of the internal nodes computation after the boundary perturbation with regard to geometrical parameter p. A convenient way to transform global parameters to internal nodal coordinates is a Laplacian smoothing used for mesh-morphing in coupled domain analyses. Generally, internal FE nodes must move smoothly with respect to dimensional modifications, especially in case of large displacements or complicated shape (e.g. perforation holes, sharp notches). Especially geometrical parameter must carefully be mapped to the node table since local distortions affect the accuracy of the Taylor series solution. The accuracy of parametric solutions depends mainly on the quality of mesh perturbations caused by mapping of global parameters p to the nodal table [10].

The simple automated algorithm of the parametric mesh-morphing is proposed here. Tn contract to an iterative Laplacian smoothing, the electrical analogy can be effective used to compute design velocities for nodal transformations with regard to the geometrical parameter. Using the outer boundary as boundary condition for Laplace problem, the design velocity at interior points can be computed. After all design boundary velocity is calculated, the perturbation is extended to the interior domain. In case of the n design variables, the method requires n additional linear analyses in order to compute design velocity v.

2.2. Parametric substructuring technique Parametric technique requires large increases in

system memory to store the derivatives of system matrix. Tn order to save system memory and to build parametric component libraries, the combine of substructure technique and parametric analysis is proposed. The FE­model can be divided into superelements, which are independent or variable with regard to parameter. It represents a set of elements that are reduced to act as one superelement. The substructure technique may be used in any linear analysis type: static, modal (Component Mode Synthesis), harmonic and transient analyses. This technique belongs to the standard procedures of the FE software. The objective here is to illustrate that the CMS analysis can be carried out in an accurate way based on

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the results of parameterized superelements. The resonator is divided into four super-elements as depicted in Fig. 2a. The Fig. 2b. plots the sparse adjacency matrix of the connectivity graph of the resonator.

Superelement: bond pad

. Interface: � spnng

master nodes

seismic mass Superelement: finger

FE-mesh of resonator \. : ... J

Kmass

b) sparsity pattern of stijJness matrix

Figure 2: Substructure of MEMS resonator

The width w, of the spring is varied as shown in Fig. 3.

I�terface: master nodes

Figure 3: Mesh-morphing of the spring superelement

This IS a linear transformation; therefore the parameterized stiffness matrix of spring can be represented by

K .I1'r;nl( (p) = K .I1'r;111( (Po) + K ;�;";nl( (Po )(p - Po) :

n=l �

\\� n=O

Figure 4. Parameterized superelement

2.3 Parametric macromodeling The ROM modeling of coupled electrostatic-structural

domains based on the Mode Superposition Method (MSM) is a very effective technique for fast transient simulation of MEMS components. The strain energy, the mutual capacitances, the damping coefficients and the modal load forces are the parameters characterizing the coupled electromechanical system. The generation process of macromodel involves a set of parametric analysis with regard to the geometrical parameters and modal amplitudes: at each point the microstructure is displaced to a linear combination of selected mode shapes in order to calculate the strain energy in structural domain and the capacitances in electrostatic domain More details can be found in [12,14].

3. Example The comb-drive consisting of interdigitated fmgers is

one of the main component of MEMS resonators. Optimized comb drives with linear, quadratic and cubic force profiles have already been designed [15].

The parametric model extraction of the capacitive cell using one FE solution is presented as an application of the general HOD methodology. Three geometrical parameters

(side-wall angle a and shifts in x and y-direction) are used to parameterize the model.

The perturbation of internal nodes with respect to shifts in x and y-direction is obtained by solving Laplace's equation with Dirichlet boundary conditions. The initial

mesh (a=O and shifts ux=O and uy=O) and countor plot of design velocity for perturbation of internal nodes are shown in Fig. Sa. The perturbated meshes with regard to

the a and uy are visualized in Fig. Sb-c respectively.

Figs. 6-7 illustrate the capacitance relationship depending on design parameters. It is essential that the response functions not only provide the capacitance data but also the first (Fig. 7c) and second (Fig. 7d) derivatives needed for Maxwell force and electrostatic softening computations. The obvious disadvantage is that applicable range limited by mesh-morphing procedures. The large

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mesh perturbation of electrostatic FE mesh is a bottleneck and cannot be applied in parametric analysis.

� � 4 Figure 5: Mesh-morphing for parametric modelling

a) initial mesh and contour plot of design velocity

v(x,y,z) 6[0 . . . 1J for mesh perturbations; b) mesh

perturbation with a= 1.00,- c) mesh perturbation with

a = 1. 0 0 and uy= 1. Oflm;

4. Conclusions The use of the derivative-based technology is

demonstrated as a good alternative to existing classical techniques utilized for component and system design of MEMS.

The method is very efficient for large matrices, because the factorization of the system increases with the matrix dimensions. By using the HOD method, the simulation results become directly polynomial functions in terms of design parameters. It is necessary to point out the need for additional system memory, for parametric mesh-morphing procedures.

Parameterization has both advantage and disadvantage. Parameterization increases complexity of the problem as designer must model not only the initial concept, but a structure that guides variation. Changes can cause invalid model. It has been found that this technique is limited to moderate parameters variations ±15% due to the mesh-morphing procedures. However, this relatively small variation is enough for modeling of MEMS fabrication tolerances. Positively, parameterization can reduce the time and effort required for change and reuse, and can yield better understandings of the conceptual structure. The number of parameters is a criterion: a model with the one or few independent parameters can be deduced from curve fitting or numerical and experimental observations.

tI:,7.0

l g C ,:s .� §-

u 6.0 -

5.0 -

0.0 0.2 0.4 0.6 0.8 1.0 a, deg.

Figure 6: Extracted at the one evaluation point

capacitance of the cell with regard to side-wall angle

tI: 8.0 -�-----�� oJ u § 'G 7

.0 '" §-

u

6.0

1.2

4.0 '----"---�--�� 4.0 �' -�--�---� -2.0 a)

E X 10.10 i:;:: 5.0

4.0

0.0 2.0 uX.l1m

o S _

aC I l-ax X�O

-1.0 b)

Vl 3.0

2.5

2.0

1.5

-0.5 0.0 0.5 1.0 uY.l1m

3.0 1.0 '-----�-�--��� 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 c) a, deg. d) a, deg. Figure 7: Extracted at the one evaluation point capacitance relationship depending on design

parameters

When the number of parameters grows, it becomes more difficult to span the complete parameter space, since each parameter lets the number of possible variations grow in an exponential way. For parametric problem with two coupled parameters, the presented method is very effective than ordinary FE runs. The parametric database allows interactive update of model response for any parameter values. Consequently, HODM is particularly attractive where the number of evaluation point is large (e.g. Monte Carlo simulation to proof the interplay of packages and sensor with the electronic circuit). Behavior model extraction and design optimization is a main field of applications of the presented simulation methodology.

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Acknowledgments This work has been done within the Research Unit

1713 which is funded by the German Research Association (DFG).

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