Efficient Multi-Physics Transient Analysis Incorporating Feedback-Dependent Boundary Conditions

Transcription

1 Efficient Multi-Physics Transient Analysis Incorporating Feedback-Dependent Boundary Conditions Balasaheb Kawade, D. H. S. Maithripala, and Jordan M. Berg {b.kawade, sanjeeva.maithripala, Department of Mechanical Engineering Texas Tech University Lubbock, TX , USA Abstract This paper describes the ANSYS simulation of an electrostatically-actuated MEMS device, in which the drive voltages are computed using feedback control laws. The device consists of a movable electrode suspended on flexible, elastic structures, and one or more fixed drive electrodes. The feedback control laws are used to determine the level of a variable voltage supply in a drive circuit for each drive electrode, and take inputs such as the total charge on the drive electrode, and the position of various points on the movable electrode. The simulation requires the solution of coupled structural and electrostatic field equations, and presents two challenges for a standard ANSYS multifield analysis. The first is that the boundary conditions for each load step are not known beforehand, but are generated by the controller logic based on the output of the previous simulation step results. The second is that the elements used to model the drive electrode control circuitry are incompatible with the electrostatic elements. We present several extensions that enable this analysis. We eliminate the circuit elements from the model, and instead propagate the associated states in an APDL macro. To allow efficient solution of the closed-loop model we incorporate an adaptive step size Runge-Kutta integration routine within this macro. Implementation of the adaptive step size routine speeds some transient simulations by a factor of more than 100. We present results for representative MEMS devices including a one-dof piston microactuator and a two-dof rotating/translating microactuator. Introduction Finite element analysis (FEA) is a powerful analysis tool, but it can be computationally demanding for complex physics and model geometries. The simulation of microelectromechanical systems (MEMS) often involves features such as coupling between multiple physics domains, large deflections, and contact, and presents many challenges to FEA. Thus, efficient FEA simulation techniques for MEMS analysis are an active research topic [9, 10, 11, 12]. The current paper considers electrostatically-actuated MEMS, and neglects effects such as squeeze-film damping. The ANSYS FEA software package contains semiautomated tools for simulation of such systems, including the multi-physics solver, the ESSOLV macro, and the ROM144, TRANS126, and TRANS109 elements. Each tool has specific strengths, but none is without problems. The multi-physics solver enables integration of multiple fields in a single analysis. However, neither non-structural analysis nor the multi-physics solver support multiframe restarts. Further, the multifield solver constrains the end time of a load step to be an integer multiple of the current step size. Automatic remeshing, which replaces the distorted mesh of a non-structural field with a quality mesh, is not available for the multiphysics and ROM analysis as of ANSYS release 10.0 [13]. Despite these problems, the multi-physics solver is the most powerful, flexible, and general approach to MEMS analysis available in ANSYS. Electrostatic actuation is a widely used actuation technique that exploits the attractive Coulomb forces that arise between capacitively coupled electrodes. A simple electrostatically-actuated MEMS is shown schematically in Fig.1. Such a system will exhibit the well-known nonlinear bifurcation phenomenon called pull-in. When the pull-in voltage is exceeded, no equilibrium points exist for the movable electrode, which will be drawn to the static electrode [16]. In its simplest form, pull-in restricts the stable operating range of parallel plate electrostatic devices to one-third of the zero voltage capacitive gap. Several control strategies have been proposed or implemented to extend this operating range [6, 7, 8, 15, 17]. The feedback control laws presented in [1, 3, 4, 5] have been developed for several microactuator configurations.

2 Implementation issues are discussed in [2]. The present paper describes the ANSYS validation of these control laws. ANSYS simulation can incorporate effects such as geometric nonlinearities, large deflections, fringing fields, electrode coupling, and non-uniform charge distribution that are usually neglected in an analytical treatment. Figure 1. Piston microactuator (parallel plate capacitor) with a drive circuit for feedback control While the MEMS device itself is modeled using structural and electrostatic elements, it is controlled through a drive circuit, as shown in Fig. 4. Circuit and electrostatic elements in ANSYS both have a voltage degree of freedom, but with different associated reaction forces current and charge, respectively. Therefore, these elements cannot be combined in a single analysis [13]. The TRANS126 transducer element is compatible with discrete circuit elements, but ignores fringing fields and is not well-suited to cases where those fields are significant. In order to handle more general situations, and to use electrostatic and structural elements with the multi-field solver in conjunction with a discrete driving circuit, we propagate the circuit states externally to the FEA analysis, using a macro written in the ANSYS Parametric Design Language (APDL). The closed-loop control algorithm varies electrode voltages to control the position of the movable electrode. The voltages are determined based on measured voltages, positions, and velocities. Thus the boundary conditions at a particular time are not known in advance, but must be calculated and applied at that time step. This means that the stepsize controls incorporated in the ANSYS package cannot be applied. However fixed stepsize integration is typically extremely inefficient. Thus, an adaptive stepsize Runge-Kutta integration scheme is incorporated into the APDL executive macro. Finally, the results of simulation of several benchmark control problems are described. The adaptive stepsize routine is seen to improve performance by a factor of more than 100. Simulation of both static and dynamic output feedback control laws are demonstrated on a piston microactuator and a breathing-mode micromirror. The dynamic output feedback case is computationally demanding, but due to the implementation features discussed it is tractable. Closed Loop Feedback Control Laws The feedback control laws are developed for 1-DOF and 2-DOF microactuators. Details about these control laws can be found in [1, 2, 3, 4, 5] 1-DOF Piston microactuator Figure 1 schematically shows a 1-DOF piston microactuator. Static and dynamic closed loop feedback control laws for piston microactuator are given by equations (1) and (2). Equation (1) is a static output feedback control law for the control electrode drive voltage u. V d is the control electrode voltage itself, Q is the charge on the drive electrode, and K > 0 is a gain. Equation (2) is a dynamic output feedback control

3 law for the control electrode drive voltage u. Again, V d is the control electrode voltage, Q is the charge on the drive electrode, and K > 0 is a gain. In addition, R, ε, and A are model parameters the driving circuit resistance, the permittivity of the gap, and the area of the drive electrode, respectively. Variable v is the movable electrode velocity. Since this quantity is typically not measurable, except in the laboratory, it must be estimated using a dynamic observer. This is the reason we refer to (2) as the dynamic output feedback control. The control laws are discussed in further detail in [1, 3, 5] d ( ) u = V K Q Q (1) Q+ Q u = Vd R v K Q Q 2ε A ( ) (2) 2-DOF Breathing mode microactuator A breathing mode microactuator is shown in Figures 2 and 3. This microactuator translates in the Y- direction and rotates about the Z-axis, as shown schematically in Fig. 2, and in the ANSYS model in Fig. 3. Figure 2. Breathing mode microactuator: Electromechanical circuit Figure 3. Breathing mode microactuator and driving electrodes: ANSYS Model Static and dynamic closed loop feedback control laws for the breathing mode microactuator are given by following equations. Here equations (3) and (4) are the static output feedback control laws for each of the two control electrodes, where the quantities are the same as for the 1-DOF device, for the appropriate electrode as indicated by the subscripts. Equations (5) and (6) are the dynamic output feedback control laws, analogous to (2). Here v is the velocity of the movable electrode center of mass (c.m.), and Ω is the angular velocity about the c.m. ( ) u = V K Q Q (3) 1 d1 1 1 ( ) u = V K Q Q (4) 2 d Q + Q u = V R l Ω v K Q Q ( x 2 ) ( ) d ε A1 u = V Q R + Q l Ω v K Q Q ( x 2 ) ( ) d ε A2 (5) (6)

4 Procedure Integration of Circuit and Electrostatic Analyses This section details the solution procedure at each time step, including evaluation of the control law. Multiple drive electrodes are assumed, each associated with a circuit that, for the present study, is assumed to consist of a voltage source u i and a series resistance, r i. The current in the i th drive circuit is I i. Given a source voltage u i, the drive electrode voltage V di depends on I i. Because of element type incompatibilities, the V di and I i cannot be found within the ANSYS solver, and are instead computed between time steps in an APDL macro. This is done by solving state equations for the V di. The following general procedure is implemented at each time step: At the start of the n th time step, the movable electrode configuration q (n) and velocity v (n) are known from the previous time step, as are the charges Q (n) i and voltages V (n) di on the drive electrodes (all these quantities must be initialized for the first time step) and the lumped capacitances, C (n) ij = C ij (q (n) ). Based on these values the control voltages u (n) i, which may depend on any or all of these, are computed. The state equations for the V di include the lumped capacitances C ij, the V di, and the u i. These equations are used to obtain V (n+1) di. The V (n+1) di are applied as boundary conditions in an ANSYS analysis, which returns as output q (n+1), v (n+1), and Q (n+1) i. The q (n+1) are input to the CMATRIX macro to obtain C (n+1) ij = C ij (q (n+1) ). The process is repeated until the desired simulation time is reached. Figure 4. Micromirror model and driving circuit schematic The following is a general implementation for static output feedback control of the micromirror model shown in Fig. 4. At the n th load step the lumped capacitances and electrode voltages are related to the electrode charges by ( ) j n n n n n n i = ii di + im di dm m= 1 Q C V C V V (7) Differentiating equations (7) with respect to time, we get the following system of equations: To solve the system (8) for the n n ( V di dm ) n n n j n dq dc n dv dc d V i ii di im n n n = V + + di ( V V di dm ) + Cim (8) dt dt dt m= 1 dt dt n dv di dt, we need lumped capacitances C, C and currents I n. Lumped n capacitances are calculated by applying the ANSYS CMATRIX macro at each load step. The currents I i, the control electrode voltages V (n+1) dj and the drive voltages u (n+1) j for the (n+1) th load step are calculated by (9), (10), and (11): n im n ii i

5 I n n n n dqi ui Vdi i dt Ri = = (9) n n 1 n dvdi di di n V + = V + t (10) dt ( ) u + = V K Q Q (11) n 1 n n i di i i Runge-Kutta Adaptive Step Size Integration: Because the drive electrode voltages are not known prior to each time step, the standard ANSYS load step utilities, including built-in stepsize control, cannot be used. Therefore a fourth-order adaptive stepsize Runge-Kutta method is incorporated into the APDL macro to improve computational efficiency. The implementation follows [14]. Let y 1 be an FEA solution found with a stepsize of 2 t, and y 2 be another FEA solution found with two steps of size t each. The stepsize for the next time step is then 0 tn+ 1 = S tn 1 (12) To implement this method, a target resolution is set for the norm of 1. Below this limit, a significant increase (four times the current time step) in the time step is permitted. In the ANSYS multi-physics solver, multiframe restart is not supported and the end time at given load step must be an integer multiple of the current time step. Further, multiframe restart is supported only by the nonlinear structural static or transient analysis. To work around these limitations, which complicate implementation of the adaptive stepsize Runge-Kutta logic, we proceed as follows: an ANSYS transient simulation is started with a small stepsize. After completion of the n th load step, the restart database and files are saved in a special directory. The upper limit on stepsize to be applied in the next time step, t n+1, is calculated from (12). The actual stepsize is selected so that (t n + 2 t) and (t n + t) are integer multiple of t. Figure 5 is a flowchart of the implementation. The implementation is done in four steps: 1 to 4. Figure 5. Flow chart of implementation of Runge-Kutta adaptive step size integration for feedback control transient simulation in ANSYS

6 Results & Discussion The procedures described above have been applied to several control test beds. Figures 6, 7, and 8 are simulation results, using the multi-physics solver, of static feedback control of the piston microactuator. Figures 9, 10, and 11 are simulation results, using the multi-physics solver, of dynamic feedback control of the piston microactuator. As seen from the plots, the static feedback control law benefits greatly from stepsize adaptation, while the dynamic output feedback law benefits much less. After an initial increase from 1E 10 to 1E-8, stepsize in the dynamic feedback case is seen to stabilize, oscillating about a mean value. For these cases, adaptation is turned off after 20 time steps to save on computational overhead. This difference can be justified by analyzing of the charge behavior, as seen in Figures 8 and 11. Charge settles very fast in the static feedback case, while it settles slowly in the dynamic feedback case. The slower stabilization of charge in the latter case makes step size adaptation inefficient after the initial transient. Figure 6. Static feedback control for a piston microactuator (Multi-Physics analysis) : Step Size Figure 7. Static feedback control for a piston microactuator (Multi-Physics analysis) : Gap

7 Figure 8. Static feedback control for a piston microactuator (Multi-Physics analysis) : Charge Figure 9. Dynamic feedback control for a piston microactuator (Multi-Physics analysis) : Step Size

8 Figure 10. Dynamic feedback control for a piston microactuator (Multi-Physics analysis) : Gap Figure 11. Dynamic feedback control for a piston microactuator (Multi-Physics analysis) : Charge Fig. 12 shows results of static feedback control simulation of piston microactuator (parameters such as feedback control constant K and damping are altered for the following results) using TRANS126 elements. Since TRANS126 is used in a direct coupled field analysis, limitations due to the multi-physics solver tool do not apply. Therefore the step size generated by the Runge-Kutta equation can be used directly. This makes the stepsize characteristics of TRANS126 different than that generated through the multi-physics solver.

9 Figure 12. Dynamic feedback control for a piston microactuator (TRANS126) : Gap and Step Size Figures 13, 14, and 15 show results of static and dynamic feedback control simulation of the 2-DOF microactuator using the multi-physics solver. The ANSYS model of this micromirror is shown in Fig. 3. As seen from the plots, this step size has a similar trend to that of the step size for the piston microactuator. Figure 13. Static and Dynamic feedback control for 2-DOF microactuator (Multi-Physics Solver): Time step

10 Figure 14. Static and Dynamic feedback control for 2-DOF microactuator (Multi-Physics Solver): C. M. Gap Figure 15. Static and Dynamic feedback control 2-DOF microactuator (using the multiphysics solver): Angle of Tilt (Radians) Conclusion ANSYS simulation of electrostatic MEMS for validation of control laws presents challenges due to element type incompatibilities and stepsize management. We have embedded the FEA simulation within an APDL macro that updates the drive circuit states and implements adaptive stepsize control. This simulation facility has been successfully applied to a number of device simulations. Further, Runge-Kutta adaptive step size integration method is observed to be very efficient for transient simulation of static feedback control systems. This methodology has improved computational efficiency of systems involving feedback dependent boundary conditions.

11 References 1) D. H. S. Maithripala, B. D. Kawade, J. M. Berg, W. P. Dayawansa, A General Modelling and Control Framework for electrostatically actuated mechanical Systems, International Journal of Robust and Nonlinear Control, To Appear. 2) Robert C. Anderson, Balasaheb Kawade, Kandiah Ragulan, D. H. S. Maithripala, Jordan M. Berg, Richard O. Gale, W. P. Dayawansa, Integrated Charge and Position Sensing for Feedback Control of Electrostatic MEMS, SPIE Conference on Smart Structures and Materials 2005: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, San Diego, CA, March, ) D. H. S. Maithripala, Jordan M Berg, W. P. Dayawansa, Control of an Electrostatic MEMS using Static and Dynamic Output Feedback, ASME Journal of Dynamical Systems Measurement and Control, to appear, September ) D. H. S. Maithripala, Jordan M. Berg, W. P. Dayawansa, Capacitive Stabilization of an Electrostatic Actuator: An Output Feedback Viewpoint, Proceedings of the 2003 American Control Conference, Denver, CO, June 4 6, 2003, pp ) Sanjeeva Maithripala, Jordan M Berg, and W. P. Dayawansa, Nonlinear Dynamic Output Feedback Stabilization of Electrostatically Actuated MEMS, Proceedings of the CDC, Maui, HW, ) Joseph I. Seegar, and Bernhard E. Boser, Dynamics and Control of Parallel Plate Actuators Beyond the Electrostatic Instability, Proceedings of the Tenth International Conference on Solid- State Sensors and Actuators (Transducers 99), Sendai, Japan, 7 9 June 1999; ) Joseph I. Seegar, and Bernhard E. Boser, Charge Control of Parallel-Plate, Electrostatic Actuators and the Tip-In Instability, Journal of Microelectromechanical Systems 2003; 12(5): ) Jinghong Chen, Wendellin Weingartner, Alexi-Azarov, and Randy C. Giles, Tilt Angle Stabilization of Electrostatically Actuated Micromechanical Mirrors Beyond the Pull-In point, Journal of Microelectromechanical Systems 2003; 13(6) : ) Elmer S. Hung, and Stephan D. Senturia, Generating Efficient Dynamic Models for Microelectromechanical Systems from a Few Finite-Element Simulation Runs, IEEE Journal of Microelectromechanical Systems 1999; 8(3): ) Mohammad I. Younis, Eihab M. Abdel-Rahman, and Ali Nayfeh, A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS, Journal of Microelectromechanical Systems 2003; 12(5) : ) Gang Li, and N. R. Aluru, Efficient Mixed-Domain Analysis of Electrostatic MEMS, IEEE Transactions on Computer-Aided design of integrated circuits and systems, 22(9): ) Ofir Bochobza-degani, David Elata, and Yael Nemirovsky, An Efficient DIPIE Algorithm for CAD of Electrostatically Actuated MEMS Devices, Journal of Microelectromechanical Systems 2003; 12(5): ) ANSYS Release 10.0 Documentation, ANSYS Inc, Canonsburg, PA, ) William H. Press, Bian P. Flannery, Saul A. Teukolsky, and William T. Vetterling, Numerical Recipies, ) Yu Sun, D. Piyabongkarn, A. Sezen, B. J. Nelson, R. Rajamani, A high-aspect-ratio tow-axis electrostatic miroactuator with extended travel range, Sensors and Actuators A, 2002, 102(1, 2): ) Yael Nemirovsky, Ofir Bochobza-Degani, A Methodology and Model for the Pull-In Parameters of Electrostatic Actuators, Journal of Microelectromechanical Systems 2001; 10(4): ) Edward L. Chan, Robert W. Dutton, Electrostatic Micromechanical Actuators with Extended Range of Travel, Journal of Microelectromechanical Systems 2000; 9(3):

12 Nomenclature R, R 1, R 2 Resistances Q,Q 1,Q 2 Charges on respective drive electrodes u, u 1, u 2 Control voltages supplied to respective drive electrodes V d,v d1,v d2 l x Ω Voltages on respective drive electrodes Length of a drive electrode Angular velocity of a microactuator A, A 1, A 2 Areas of respective drive electrodes Ε K V θ S 0 1 t n Permittivity of a dielectric medium Feedback control constant (gain) Velocity of c.m. in Y direction Angle of tilt of a microactuator Safety Factor, few percent smaller than unity Desired accuracy Actual error Time Step of n th load step α 0.20 if 0 1 Q, Q 1, Q if 0 < 1 Equilibrium charges on respective drive electrodes