Computational Methods for Reduced Order Modeling of Coupled Domain Simulations Fouad Bennini, Jan Mehner and Wolfram Dötzel Chemnitz University of Technology, D-09107 Chemnitz, Germany Department of Microsystems and Precision Engineering
[email protected] SUMMARY This paper deals with computer aided generation of reduced-order models for fast static and dynamic simulations of micro electromechanical systems. Following previous work, we present improved methods to reduce the computational effort of parameter extraction techniques, to consider multiple electrodes and the extension of analysing capabilities. A micromirror example will be used to demonstrate the practical suitability of reduced-order models for system simulations and feedback controller design. Keywords: Reduced-order modeling, Modal decomposition, Component and system simulation.
INTRODUCTION With the rapid development in the field of microtechnologies there is a growing need for fast and accurate design tools. Finite element and boundary element tools have been successfully applied for the design of micro electromechanical devices on the physical level of abstraction. Comprehensive algorithms enable one to get a precise behavioural description of each single domain (mechanical, thermal, electrostatic, fluidic etc.) including some of their most important interactions. Those methods are very accurate and helpful to study complicated problems but in practice they are difficult to handle and much too cumbersome for daily design tasks. Moreover, if one tries to perform system simulations or wants to consider electronic circuitry and continuous components at once, finite element or boundary element models are inappropriate. Therefore, novel simulation techniques and models with strongly reduced parameter sets are required. Those models are widely known as reduced order or macromodels. Reduced-order modeling has a long history in computational mechanics. Substructuring and modal superposition method became state of the art for linear systems. Bathe [1] surveyed different methodologies to extend both techniques to nonlinear mechanical systems
as well. Unfortunately, most approaches are confined to single domain simulations. A proper link to other physical disciplines is offered by energy equations known from multi-body simulators. Mechanisms driven by electromagnetic forces are described by energy terms established for rigid body motions. To extend their capabilities to compliant structures, deformation states can be described by shape functions which are superimposed to the rigid body motion. Since a few shape functions are sufficient to capture the deformation state of most mechanical systems this method appears convenient to model microsystems for circuit and system simulations. This methodology was successfully implemented in [2]. The ultimate goal of reduced order modeling is to obtain an accurate black-box model of the microsystem’s behaviour. Interface signals are limited to the voltagecurrent-relationship at each electrode, essential input quantities such as external loads (e.g. gravitation, pressure) and significant output quantities (e.g. a subset of displacements at characteristic model points). Since all model information are extracted from 3D finite element simulations we are able to capture flexible regions, mechanical non-linearities, electrostatic fringing fields and large signal behaviour with the same order of accuracy. Furthermore, due to the voltagecurrent interface those models are directly applicable in EDA environment.
THEORETICAL BACKGROUND Analysing continuous systems governed by partial differential equations has a long history. For complex problems there is in general no direct access to an analytical solution available. A common engineering approach is to approximate the unknown solution u by a series of weighted linearly independent shape functions (Galerkin method): n
u (t , x , y , z ) ≈
∑ q (t ) ⋅ ϕ ( x, y, z ) i
i
(1)
i =1
where qi are time dependent scale factors (the unknown parameters) and ϕi are spatial shape functions, in our case eigenmodes of the linear system. The deformation
state of the mechanical structure is now restricted to a linear combination of m modes but the number of DOF’s are reduced from several thousand parameters as occur in FEA to about ten parameters which are the mode amplitudes qi. An essential prerequisite to establish the equations of motion are proper energy terms, which cover the external (e.g. electrostatic energy) and internal (elastostatic and kinetic) energy state. Both are derived from a series of finite element runs at significant deflection states fitted to a polynomial function. Equilibrium of each mode occurs if d ∂E k ∂E p − = Qi i = 1...n , (2) dt ∂qi ∂qi where Ek is the kinetic energy, Ep is the potential energy and Qi are generalised forces containing damping and exterior loads. Ep includes nonlinear strain energy Emech(qi ) and electrostatic field energy Eelec(qi ) as well. For systems with several electrodes, the generalised electrostatic energy can be expressed as follows:
∑∑ c
kj ( q1 ,..., q n ) ⋅ Vk
⋅V j
(3)
j
where ckj is the mutual capacitance between electrodes k and j and Vk, Vj the applied electrode voltages respectively. Energy dissipation effects are considered via modal damping ratios, obtained either from analytical solutions (squeeze or slide film damping) or numerical fluid flow simulations. Finally the governing equation for each mode can be expressed by: mi qi + 2ξ iωi mi qi +
∑ϕ j
i
T
∂Emech (q1 , q2 ,..., qi ,..., qn ) = ∂qi
Fj +
1 2
∑∑ k
j
∂ckj (q1 ,..., qi ,..., qn ) ∂qi
(4) ⋅ VkV j
where ξi is the modal damping ratio of mode i, mi the modal mass and ωi the i-th eigenfrequency.
THE MACROMODEL GENERATOR Following the top down design strategy, one can split the reduced order modeling procedure in three phases. The first phase is called macromodel generation pass and includes all necessary steps from finite element parameter extraction, energy function fit up to establishing the final component description. This process is computationally expensive but has to be done just once producing a black-box model with functional
Micromirror cell
X
k
Y
1 2
Z
Eelec (q1 ,..., qn ) =
access only at its interface pins. The second phase is called macromodel use pass. The black-box model is placed in the same or another design environment and enables very fast simulations. The third phase is called expansion pass, where the deformation state, stress distribution or electrostatic field quantities of the entire microstructure can be visualised at chosen time steps. Presently, the macromodel generator which performs the generation pass is implemented in MATLAB as "EMECH/MAC". It requires access to ANSYS/Multiphysics and to a detailed user-defined FE-model of the micromechanical structure. The practical suitability of the reduced order models for system simulation shall be demonstrated at a micromirror cell shown in Fig. 1. This example is computationally expensive not only due to plate warping and fringing field relevance but also due to the essential interactions of counter electrodes which are very close together. The mutual capacitance between both counter electrodes is likewise a function of the mirror deflection. Its change affects the total electrostatic energy and consequently the electrostatic forces, too.
Micromirror array for light deflection Fig. 1: Micromirror array as demonstration example for the macromodeling technique One of the most crucial issues of shape function methods is to determine which modes should be included and to estimate a significant range of amplitudes. Several criterions are offered in EMECH/MAC, for instance user specified modes, the lowest eigenmodes of the linear system, modes in a given direction and modes which contribute to a typical load case. The steps of the generation pass are shown in Fig. 2 for the micromirror array.
MN
Mirror plate Y
Z
C02
X
C01 C12
MX
Mode 1: Dominant
Y X
MX
MN
Y Z
X
MX
Mode shape relevance
q1 0.0 0.1 0.2 0.0 0.0 0.0 0.0
q2 0.0 0.0 0.0 0.1 0.2 0.0 0.0
q3 0.0 0.0 0.0 0.0 0.0 0.02 0.04
Em e c h 0.0 1 . 2 7e -1 2 . 5 9e -1 3 . 0 1e -2 6 . 0 5e -2 7 . 3 e-4 1 . 5 0e -3
q1 0.0 0.1 0.2 0.0 0.0 0.0 0.0
q2 0.0 0.0 0.0 0.1 0.2 0.0 0.0
q3 0.0 0.0 0.0 0.0 0.0 0.02 0.04
C01 1 . 2 0e -6 1 . 4 0e -6 1 . 6 0e -6 1 . 2 5e -6 1 . 3 0e -6 1 . 0 5e -6 1 . 1 0e -6
q1x10-6 q2x10-6
C01 C02 C12
No n lin ear s trai n en erg y f un c t io n
MN
Mode 3: Dominant
Mod e Freq . (Hz) Con. factor (% ) 1 34028 81,58 3 58550 18,27 7 420059 0,09 2 43377 0,03 5 238955 0,01 ... ... ...
MX
Y Z
X
Emec h(q1,q 2,q3=0)
Z
Cap ac it an c e f u n c t io n
C01(q1,q 2,q3=0)
Static deflection at a test voltage
MN
Mode 7: Relevant
FE data table
q1x10-6
q2x10-6
Fig. 2: Generation pass flow The amplitude range must be treated carefully to prevent singularities (e.g. electrodes touching the ground) and to consider changed stiffness of modes in large deflection cases. Furthermore, the range of values should be larger than the operating range since most fit functions, especially polynomials, tend to oscillate at their interval boundaries. The dependence of electrostatic and elastostatic energy functions from the mode amplitudes must be captured correctly to guarantee accurate force and stiffness terms. To reduce the computational effort of function fit we distinguish between dominant and relevant modes. Dominant modes are characterised by large amplitude. Their interactions to all modes, dominant and relevant, are regarded throughout. Relevant modes contribute to the final solution but do not affect each other. Usually three modes are dominant and up to ten modes are relevant for most applications. Several types of fit functions are investigated. Global polynomials are easy to handle but tend to oscillations at strong slope changes. Nevertheless, choosing a
tailored polynomial coefficient set for the specific problem leads to satisfying results. EMECH/MAC enables the user to choose between full Lagrangian polynomial type, Pascal type and a reduced set where all products with more than two variables are eliminated. Noteworthy is that rational polynomials are not necessary for capacity functions since simple inverse polynomials satisfy such function characteristic as well.
MACROMODEL APPLICATION Static and harmonic analysis The functionality of the generated macromodel can be proven within a use pass implemented in EMECH/MAC (Fig. 3). A static analysis including a DC voltage sweep traces the operating point of the structure. A harmonic transfer function analysis of the system shows the resonance frequency shift due to the polarisation voltage. Such harmonic analyses are indispensable for sensor and actuator characterization. However, they can
ST A TIC A NA L YS IS
H A R MO NIC R ESP O N SE Polarisation volt.= 200 V Polarisation volt.= 100 V Exitation volt.=1 V
1 2 3
Node1 Node3 Node2 Coupled field (ANSYS)
P u ll- I n
Fig. 3: Simulation results of the component (static and harmonic case)
Z
X
Y
Actual a ng le
150 V
0V
+
Control unit
300 V
Tilt angle (deg)
a)
b)
Ac tual angle
Set angle
c)
Time (s)
Fig. 4: Example of a system simulation of the generated macromodel (micromirror with a control unit) only be done when the electrostatic stiffness terms are included in the macromodel description. Direct computation of electrostatic stiffness terms in FEA tools is impossible to date for general 3D meshes. Only for special geometries like narrow gaps a new class of reduced-order elements is recently available which calculates the electrostatic stiffness from given capacitance functions [3]. Using macromodels for MEMS simulations there is no restriction since we work with analytical energy functions the second derivatives of which lead to appropriate stiffness terms. System level simulation System level simulations are demonstrated at a voltagecontrolled micromirror cell for image projection and scanning applications (Fig. 4). A constant scan velocity in forward direction with fast return is equivalent to a saw-tooth like deflection function. This function is only achievable in high quality by feedback controller circuits. In practice the actual angle is detected optically with a second laser beam. The MEMS component is modeled in ANSYS with about 5000 solid elements. Energy data are extracted for three dominant modes and three electrodes in about 12 hours, while the function fit itself takes just a few minutes. Despite the computational effort in the generation pass, advantages become obvious at the use pass in SIMULINK. Transient simulations can be done in almost the same time as single degree of freedom models would need. Not only the tilt angle but also transverse motion and plate warping agrees well to fully coupled 3D finite element simulations.
CONCLUSIONS We have demonstrated an automated procedure for generating macromodels using modal basis functions. The generated models comprise nonlinear electromechanical coupling and are suited for all common analysis types in system simulation. Nonlinearities are stored by polynomial functions where a small number of polynomial coefficients has been found sufficient. A special interface to convert the models into VHDLAMS is under construction to be compatible to most commercial EDA tools. As a next step, reduced order models will be improved in terms of energy dissipation effects. Damping was considered by constant modal damping ratios. Actually, damping is strongly deflection dependent and stiffness correction terms caused by squeezed air films are not negligible. Both can be efficiently described by modal methods and will be implemented in future.
REFERENCES [1] K. J. Bathe and S. Gracewski, “On nonlinear dynamic analysis using substructuring and mode superposition”, Comp. & structures, No.13, 1981 [2] J. E. Mehner, L. D. Gabbay and S. D. Senturia, “Computer-Aided Generation of Nonlinear Reduced-Order Dynamic Macromodels: StressStiffened Case” J. Microelectromech. Syst., Vol.9, June 2000, pp. 270-277, [3] D. Ostergaard, M. Gyimesi: "Finite Element Based Reduced Order Modeling of Micro Electro Mechanical Systems (MEMS)." MSM 2000, San Diego, CA
