System Level Modeling of Microsystems using Order Reduction Methods

Transcription

1 System Level Modeling of Microsystems using Order Reduction Methods Sven Reitz, Jens Bastian, Joachim Haase, Peter Schneider, Peter Schwarz Fraunhofer Institute for Integrated Circuits, Branch Lab Design Automation EAS Zeunerstraße 38, D Dresden, Germany ABSTRACT In the development of microsystems, FEM simulators are used to investigate the behavior of system components with high accuracy. Generally, FEM simulations are time consuming. System-level models of all components are needed to allow a fast but sufficiently exact investigation of the system behavior to simulate entire microsystems. Typically, microsystems consist of nonelectrical components and electronic circuits. Providing models for electronic components and languages to describe the behavior of nonelectrical subsystems, simulators like Eldo, Saber, and VHDL-AMS simulators become more and more popular in the development of microsystems. For simple structures such as mechanical beams, models of microsystem components can be derived from analytical descriptions. Another possibility to consider more complex structures is to use FEM descriptions to generate models for system simulation. Some FEM simulators like ANSYS allow to access the numerical values of the system matrices. They are established based on the description of geometry and material data. Usually, these system matrices are very large ( up to system variables or more). For system simulation, models with about 10 up to 100 variables are often required. Therefore, methods for order reduction are applied to derive smaller system matrices. An improvement of an order reduction method based on a projection method is introduced in the paper. Using the reduced systems, behavioral models in languages like MAST, HDL-A or VHDL-AMS can be generated automatically. The described method was applied successfully to simulate mechanical microsystem components on system level. Keywords: system simulation, multi-domain, FEM, modeling tools, order reduction 1. INTRODUCTION Modeling for microsystem design 1 is a complicated task if complex structures and coupled field effects have to be considered. However, some parts of the modeling process may be supported by systematic methods, algorithms, and tools. In Fig. 1, a synopsis of some established modeling techniques is given. Two main approaches may be distinguished: Starting with the geometrical structure of the microsystem, a manual decomposition into subsystems or components is carried out. By further refinement of the partitioning in each physical domain often in a hierarchical manner the entire system may be decomposed into a lot of relatively simple basic elements. For simple structures like beams, plates etc. it is possible to describe the elements in an analytical form and with explicit geometrical dimensions and material parameters ( parameterizable models ) (Fig. 1 left). To consider more complex structures it is possible to generate models based on approximation of measured data or results from FEM simulations (Fig. 1 right). Another possibility is to use FEM descriptions from component design as an input for order reduction methods to generate system level models (Fig. 1 middle). Improvements and use of order reduction methods will be discussed in the following. Often, the behavior of the subsystems can be described by partial differential equations (PDE), e.g. electrical conductors, micromechanical sensors or

2 Microsystem Geometrical structure Mathematical description: PDE Modeling (manually): decomposition into multipoles Discretization (FEM, FDM,...) Generalized KIRCHHOFFian networks consisting of basic elements DAE, ODE, algebraic equations Mathematical description of basic elements (analytically) Order reduction Simulation in time or frequency domain Reduced system matrices Simulation results Black-box model generation Parameterizable analytical element models Numerically generated behavioral models Figure 1: Different ways to model a microsystem heat conduction. The semi-discretization of these PDEs leads to high dimensional systems of linear ordinary differential equations (ODE). For simulation of linearly modeled components with FEM simulators based on geometry and material data, the system of differential equations is established and solved numerically. The semi-discretized systems established by FEM simulators normally have the form M d2 x dt 2 + D----- dx + Kx = Bf dt (1) x a = B a T x (2) with x R N, f R m, M, D, K R N N, B R N m, B a R N p and typically N = 1000,, 10000, The interpretation of matrices, unknowns, and forcing stimuli depends on the application. For mechanical systems, M, D, and K are the mass, damping, and stiffness matrices. For thermal problems, M = 0, D is the heat capacitance matrix and K the conductivity matrix. B is the (generalized) incidence matrix of m points with applied forces f, thermal power, currents or the like. Generalized means that the entries of the incidence matrix do not have to be only 1 or 0 but also weighting factors may be used. x may be the vector of displacements in all discretization points for mechanical systems or the vector of temperatures in thermal applications. B a is the (generalized) incidence matrix of active (boundary) nodes and/or interesting observation points x a, respectively. If only active nodes are considered it is identical to B. In system simulation the behavior of a component only at certain points is relevant for the overall system knowledge. In the above terms these points are the active nodes and observation points. It is possible to reduce the computing time and the needed storage by assigning low dimensional systems to the large systems (1). The lower-dimensional system has to allow the determination of the behavior in the active nodes and observation points with a sufficient accuracy. The assignment of a low order system to the original one is called order reduction. The reduced order system is described by M d2 x D dx, (3) dt K x = B f dt x a = T B a x (4)

3 with x R n, M, D, K R n n, B R n m, B a R n p. The dimension n of the reduced system is determined by its vector of unknowns x R n with n «N, i.e. the dimension of the reduced system (3) is considerably smaller than the size of the original system (1). Typically is n = 10, The objective of order reduction is to generate automatically the matrices M, D, K using the matrices of the original system (1). There exist different algorithms that support this task. One of them is the Guyan algorithm. Its application to behavioral system modeling was discussed e.g. in 2. x = ( x r, x i ) T represents here the displacements x r at the boundary nodes and x i at the observation points of the component (see Fig. 2). An advantage of the method is that the static behavior is exactly described. But in the case of the investigation of the dynamic behavior problems may occur. That is why a modification of a reduced order method based on a projection method is introduced in section 2. The following section 3 describes the general procedure for the generation of behavioral models using different languages 3 like MAST, HDL-A, VHDL-AMS 4, and Modelica. Experiences are summarized in section MODIFIED PROJECTION METHOD 2.1 Formulation of Reduced Order Systems using Projection Methods As mentioned above, an often applied method for order reduction is the projection of (1) into a lower-dimensional subspace. This is done with an orthogonal projection matrix V R N n. The reduced order system (3) and (4) has the form V T MV d2 x , (5) dt 2 + V T DV dx V T KVx = V T Bf dt x = Vx. (6) In relation to (3), M = V T MV, D = V T DV, K = V T KV, B = V T B, B a = V T B a can be determined. In modal reduction methods, V is constructed as an orthonormal base of the most important eigenvectors of (1) for the undamped case ( D = 0 ) 5. For first order systems (i.e. M = 0 ) Krylov subspace methods have become popular during the past years. After matching the moments of the original system (1) and the reduced system (3), V is constructed from an orthonormal base of a Krylov subspace of R N, e.g. with a Lanczos algorithm 6 or with an Arnoldi algorithm 7. Recently the ENOR algorithm 8 has been developed for the second order system (1). A modification of this algorithm was developed and investigated by the authors and is presented in this section. Two examples in section 4 demonstrate the efficiency of the new algorithm. The considerations below are based on the ENOR (Efficient Nodal Order Reduction) algorithm originally developed for RLC circuits. Both algorithms ENOR and the modified algorithm have the advantage that they can be used for second order ODEs without transforming them into first order systems. That conversion would double the dimension of the system, and symmetry and positive definiteness of the original matrices could be lost. The first enhancement of the new algorithm is a modification for the static case, i.e. an expansion around the Laplace variable s 0 = 0. The approach will be discussed in section 2.1 and the resulting algorithm will be presented in section 2.2. The resulting projection matrix V from the expansion around s 0 = 0 provides the exact solution for the static case ( M = D = 0, i.e. all derivatives are zero) of the original system in the selected nodes. The second enhancement is the merging of projection matrices resulting from expansions around different frequencies s 0 to a new projection matrix. This approach is described in section 2.4. Similar considerations were performed in 9 but they were not used in this context since then as far as the authors know. With this enhancement it is possible to reach an exact stationary approximation as well as a good approximation for the time domain based on the reduced system.

4 2.2 Derivation of the Projection Matrix using Moment Matching The Laplace transform of (1) is The projection matrix will be determined from the pulse response of (1), i.e. matrix I R m m is equal to the number of columns of B. This leads to ( s 2 M + sd + K)X( s) = B F( s). (7) Fs ( ) = I. Here the dimension of the identity ( s 2 M + sd+ K)X( s) = B. (8) Now X is expanded about the point (angular frequency) s 0, i.e. in powers of z = s s 0. It follows and finally After equating powers of z k we get the block moments X k by solving the resulting system of linear equations. Then the projection matrix is composed by V = X 0 X 1. Since the linear hull of the columns of X k, colspan{ X 0, X 1, }, is important and not the X k themselves, we will orthonormalize the X k on the fly to insure that V is well conditioned. The above procedure has the advantage over the ENOR algorithm 8 that an expansion around s 0 = 0 is possible. This d comes in very handy because it ensures that the stationary case ( 2 x ) will be represented exactly. This dt 2 = 0, dx = 0 dt means that (5), (6) provides the exact solution of (1), (2). The reduced system is passive if M, D, and K are positive semidefinite 8, Modified ENOR Algorithm Equating powers of z k leads to the following algorithm. Here s 0 is the frequency around which is expanded and q is the number of iterations. Without deflation the reduced system has the dimension n = qm. The X k can be orthogonalized by the modified Gram-Schmidt method or QR decomposition (Householder method). If M, D, and K are symmetric and positive definite, Kˆ can be Cholesky factorized in step 2. Otherwise an LU decomposition will be performed. Input: M, D, K, B, s 0, q Output: V 1. Compute Kˆ = 2 s 0M + s0 D+ K and Dˆ = 2s 0 M + D. Set X 1 = Cholesky factorize Kˆ. 3. Compute X 0 by solving Kˆ X 0 = B. Orthonormalize X 0. This yields V 0. Set V = V 0 and X 0 = V 0 Set k = 1. ( s 2 M + sd + K) ( X 0 + ( s s 0 )X 1 + ( s s 0 ) 2 X 2 + ) = B (( s 0 + z) 2 M + ( s 0 + z)d + K) ( X 0 + zx 1 + z 2 X 2 + ) = B 4. Calculate X k by solving Kˆ X k = Dˆ X k 1 MX k 2. Orthonormalize V X k. This yields VV k. Set X k = V k. Set k = k + 1. If k < q go to step 4.

5 2.4 Further Improvements The above algorithm can be started for n s different values s 0, i, i = 1,, n s. The resulting matrices V i are merged, i.e. V is composed by the orthonormalized column vectors of V i, i = 1,, n s, V = orth V 1 V ns. If one of the matrices V i is calculated by an expansion around s 0 = 0 it is assured that by using V the reduced system is exact for the static case, too. Using expansions around higher s 0 provides better approximations at higher frequencies and in the time domain 9. This means that merging projection matrices in this way leads to models with an exact static solution and improved approximation accuracy in time and frequency domain. 3. GENERATION OF BEHAVIORAL MODELS f 1 d 1 f 2 d 2 boundary nodes observation points a) c) f d x i b) Figure 2: Approach for model generation d) behavioral model The general way to use order reduction methods in behavioral modeling is shown in Fig. 2. It consists of the following steps Starting point is the FEM description of the structure (Fig. 2a). Simulation engines like ANSYS can be used for it. After semi-discretization (meshing in terms of FEM, Fig. 2b) the nodes which have to be considered (at the boundary and if necessary some internal observation points) are selected (Fig. 2c). Thus, flow and difference quantities of the behavioral model are defined. Then order reduction can be applied. If the Guyan substructuring algorithm 11 of ANSYS is used, then the reduced matrices can be directly written out. Another way is to write out the matrices of the original system (1) and to apply the algorithm discussed above. Based on the matrices behavioral models can be generated (Fig. 2d). This prodedure is supported by an add-on tool for FEM simulators 2 which supports the behavioral languages MAST, HDL- A, VHDL-AMS, and Modelica. Experiments using the internal order reduction of ANSYS provide good results. Beside the Guyan algorithm the method introduced in section 2 was successfully applied to model mechanical components and thermal effects. Results will be presented in the next section. The application of the methods in other physical domains is currently carried out.

6 4. RESULTS USING THE MODIFIED PROJECTION METHOD 4.1 Micromechanical Sensor In Fig. 3 left a micromechanical acceleration sensor by courtesy of Robert Bosch GmbH is shown 12. The moveable part, the seismic mass, is connected to the housing by four beams and will be deflected if the sensor is accelerated. fixation beams electrodes behavioral model seismic mass Figure 3: Micromechanical acceleration sensor The sensor was modeled with ANSYS. The exported model from ANSYS had 3019 nodes with 6 degrees of freedom (3 translational and 3 rotational ones). Having regard to the Dirichlet boundary conditions in the fixation points, the system (1) has a dimension of N = The results of the small signal analysis, the displacement of the seismic mass depending on the frequency of the acceleration force of the original and reduced systems are shown in Fig Reduced systems dim=18102 dim=4 dim=10 dim= db Frequency (Hz) Figure 4: Frequency response for the original system and systems reduced with the new algorithm. The plot of the curve for dim=30 is identical to the original system (dim=18102).

7 The modified ENOR algorithm was used to compute different projection matrices. The Laplace transformed system was expanded around s =, i.e Hz Hz with the selected units. A combination of different projection matrices was not neccessary to improve the accuracy. Yet the reduced system of the dimension n = 10 results in a very good 2π approximation of the original system. The graph achieved by a reduced system of the dimension n = 30 is nearly identical to the graph of the original system. 451 frequency samples were used for the simulation. While the time needed for the AC simulation of the full system was 3.5 hours, the AC simulations for the reduced systems including the reduction was performed in less than 15 seconds. 4.2 Thermal model This example shows the results of merging projection matrices. A beam is considered with the constant temperature K at one end and an input of 10 mw heat power at the other, Fig. 5. The exported ANSYS model has the dimension N = 725 after considering the Dirichlet boundary condition at one end. 10 mw K T a Figure 5: Heat propagation in a beam The temperature T a in the middle of the beam will be observed. Fig. 6 shows results of the transient simulation for reduced systems for which the expansion was performed around s 0 = 0 (dimension n = 2 ) and s 0 = 10 4 (dimension n = 4 ). As it can be seen, the expansion around s 0 = 0 is exact for the static case. The transients for original and reduced system differ. 295 Reduced system, s 0 =0, dim=2 295 Reduced system, s 0 =10 4, dim= Temperature in K Temperature in K original s 0 = Time in s x 10 3 original s 0 =1e Time in s x 10 3 Figure 6: Time domain response for the original system and systems reduced with the new algorithm

8 The reduced system after expansion around s 0 = 10 4 shows little difference to the original system in the beginning but is not exact for the static case. The response in Fig. 7 was calculated with a projection matrix which was merged from the first two columns of both previous matrices (dimension n = 4 ). This model is exact for the static case and provides a very good approximation of the time response. 295 Reduced system, s 0 =0 and s 0 =10 4, dim=4 290 Temperature in K original merged Time in s x 10 3 Figure 7: Time domain response for the original system and a system reduced by a merged projection matrix 5. CONCLUSIONS We have discussed an approach that allows the usage of analytical FEM formulas for the construction of behavioral models, to derive behavioral models with fixed numerical values for components from FEM descriptions, the implementation of models in different languages (MAST, HDL-A, VHDL-AMS, Modelica). A new order reduction method is applied to the large systems of linear ODEs from the FEM description. Similar to the ENOR algorithm 8 it is based on moment matching. By merging projection matrices resulting from expansions around different frequencies, the reduced models are exact for the static case and provide good approximations for time and frequency domain. The reduced models can be used in a Spice-compatible circuit and system simulator. The approach is demonstrated with examples. The new algorithm was successfully applied to model a acceleration sensor and a thermal problem. The generated model descriptions may be easily connected to each other or to behavioral models covering other physical domains inside one circuit and system simulator. The described approach is especially suitable to derive linear models of micromechanical structures. It is planned to support the generation of behavioral models of further physical domains, to verify the method and the model generation using further examples, and to extend the amount of supported description languages if new ones are in common.

9 ACKNOWLEDGMENTS This work has been funded within the project EKOSAS under label 16SV1161/7 by the german ministry for education and research (BMBF = Bundesministerium für Bildung und Forschung) and within the project SFB (Sonderforschungsbereich Automatisierter Systementwurf). The authors of this paper are responsible for its content by their own. REFERENCES 1. Senturia S. D.: CAD Challenges for Microsensors, Microactuators, and Microsystems. Proc. of the IEEE 86(1998)8, S. Reitz, J. Becker, J. Haase, P. Schwarz: Generierung von Verhaltensmodellen aus ANSYS-Beschreibungen. 18. CAD-FEM Users Meeting, Friedrichshafen, , 1.3.5, Mukherjee, T.; Fedder, G. K.; Ramaswamy, D.; White, J.: Emerging Simulation Approaches for Micromachined Devices. IEEE Transactions on CAD 19(2000)12, Christen, E.; Bakalar, K.: VHDL-AMS A Hardware description Language for Analog and Mixed-Signal Applications. IEEE Transactions on CAD-II 46(1999)10, Gabbay, L. D.; Mehner, J. E.; Senturia, S. D.: Computer-Aided Generation of Nonlinear Reduced-Order Dynamic Macromodels (Part I and II). Journ. of Microelectromechanical Systems 9(2000)2, Feldmann, P.; Freund, R. W.: Efficient Linear Circuit Analysis by Padé Approximation via the Lanczos Process. IEEE Transactions on CAD 14(1995)5, Elfadel, I. M.; Ling, D. D.: A Block Rational Arnoldi Algorithm for Multipoint Passive Model-Order Reduction of Multiport RLC Networks. Proc. ICCAD, Nov. 1997, Sheehan, B. N.: ENOR: Model Order Reduction of RLC Circuits Using Nodal Equations for Efficient Factorization. Proc. 36th DAC 1999, de Villemagne, C.; Skelton, R. L.: Model reductions using projection formulation. Int. J. Control 46(1987)6, Freund, R. W.: Passive reduced-order modeling via Krylov-subspace methods. Numerical Analysis Manuscript No , Bell Laboratories, Guyan, R. J.: Reduction of Stiffness and Mass Matrices. AIAA Journal 3(1965)2, Haase, J.; Reitz, S.; Schwarz, P.: Behavioural modeling for heterogeneous systems based on FEM descriptions. Proc. BMAS 99, Orlando,