ii COMPLEXITY REDUCTION TECHNIQUES FOR ADVANCED MEMS ACTUATORS SIMULATION iv COMPLEXITY REDUCTION TECHNIQUES FOR ADVANCED MEMS ACTUATORS SIMULATION Vorgelegt von Jan Lienemann Lehrstuhl für Simulation Institut für Mikrosystemtechnik (IMTEK) Albert-Ludwigs-Universität Freiburg Dissertation zur Erlangung des Doktorgrads der Fakultät für Angewandte Wissenschaften der Albert-Ludwigs-Universität Freiburg im Breisgau Adresse Lehrstuhl für Simulation Institut für Mikrosystemtechnik (IMTEK) Albert-Ludwigs-Universität Freiburg Georges-Köhler-Allee 103 79110 Freiburg Dekan Prof. Dr. Bernhard Nebel Autor Jan Lienemann Tag der Prüfung 21. 12. 2006 Gutachter Prof. Dr. Jan G. Korvink Prof. Dr. Ulrike Wallrabe Vorsitz Prof. Dr. Jürgen Wilde Beisitz Prof. Dr. Gerald Urban To my parents Erklärung nach S5(2) der Promotionsordnung Ich erkläre hiermit, dass ich die vorliegende Arbeit oh- ne unzulässige Hilfe Dritter und ohne Benutzung an- derer als der angegebenen Hilfsmittel angefertigt habe. Die aus anderen Quellen direkt oder indirekt übernom- menen Daten und Konzepte sind unter Angabe der Quelle gekennzeichnet. Insbesondere habe ich hierfür nicht die entgeltliche Hilfe von Vermittlungs- oder Be- ratungsdiensten (Promotionsberaterinnen oder Pro- motionsberater oder anderer Personen) in Anspruch genommen. Niemand hat von mir unmittelbar oder mittelbar geldwerte Leistungen für Arbeiten erhalten, die im Zusammenhang mit dem Inhalt der vorgelegten Dissertation stehen. Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt. Freiburg, den 17. Januar 2007 Jan Lienemann Contents I. Introduction 5 1. Overview 7 1.1. Electrowetting ........................... 7 1.1.1. Electrowetting Arrays ................... 10 1.1.2. Electrowetting Devices ................... 11 1.1.3. Device Design ........................ 12 1.1.4. Computer Simulation Aided Design ........... 14 1.2. Model Order Reduction ...................... 17 1.2.1. Model Order Reduction Versus Compact Modelling? . 18 1.3. Major Results ............................ 23 1.3.1. Modelling and Simulation of EWOD ........... 23 1.3.2. Model Order Reduction With Weak Nonlinearities . 24 1.4. Thesis Overview .......................... 24 2. MEMS Actuators 27 2.1. MST and Properties of MEMS .................. 28 2.1.1. MEMS as Transducers ................... 28 2.2. Scaling Effects ........................... 30 2.3. Computational Issues ....................... 32 2.3.1. Complexity in Numerical Modelling ........... 34 2.3.2. PDEs, Material and Geometry .............. 35 2.3.3. Coupling .......................... 36 2.4. Application Examples ....................... 37 2.4.1. IBM Scanning-Probe Data Storage Device ........ 38 2.4.2. Imego Butterfly Gyro ................... 42 2.4.3. IRST RF Switch ...................... 44 2.4.4. Bondwire Model ...................... 46 2.4.5. Heat Transfer Model .................... 46 ix Contents II. Theory 49 3. Modelling and Simulation of MEMS 51 3.1. Notation and Basic Theory .................... 51 3.1.1. Symbols and Notation ................... 51 3.1.2. Linear Algebra ....................... 53 3.2. Structure of Equations ....................... 54 3.2.1. Newton Raphson Procedure ................ 57 3.2.2. Homotopy Methods .................... 58 3.2.3. Path Following Schemes .................. 59 3.3. Numerical Time Integration .................... 60 3.3.1. First Order Systems .................... 60 3.3.2. Second Order Systems ................... 61 3.3.3. Conversion to First Order ................. 63 3.3.4. Nonlinear Systems ..................... 64 3.4. Harmonic and Modal Analysis .................. 64 3.5. Lagrangian Mechanics ....................... 65 3.6. Structural Continuum Mechanics ................. 67 3.6.1. Stress and Strain ...................... 67 3.6.2. Nonlinearities ........................ 70 3.6.3. Beams ............................ 71 3.6.4. Contact ........................... 73 3.7. Electrostatics ............................ 73 3.7.1. Transducer Elements .................... 76 3.7.2. Example: Capacitor with Movable Plate ......... 79 3.7.3. The ANSYS TRANS126 Element ............. 85 3.8. Spatial Discretisation Methods .................. 86 3.8.1. The Finite Element Method (FEM) ........... 87 3.8.2. FEM for a Coupled Analysis ............... 92 3.8.3. Geometric Nonlinearities .................. 101 3.9. Fluidics ...............................104 3.9.1. Fluid Dynamics ....................... 105 3.9.2. Wetting on Surfaces .................... 107 3.9.3. Free Surfaces in the Navier-Stokes Equations . 111 3.10. Electrowetting ...........................112 x Contents 4. Model Order Reduction 117 4.1. System Theory ...........................117 4.1.1. Transfer functions ..................... 118 4.1.2. Observability, Controllability & Minimal Representation 120 4.1.3. Passivity and Stability ................... 120 4.2. MOR for Linear Systems ...................... 122 4.2.1. Guyan Method ....................... 123 4.2.2. SVD Based Approximation Methods ........... 125 4.2.3. Krylov Subspace Methods and Padé Approximants . 128 4.2.4. Krylov Subspace Methods for Second Order Systems . 136 4.2.5. Other MOR Methods for Linear Systems ........ 141 4.3. MOR for Nonlinear Systems .................... 142 4.3.1. Proper Orthogonal Decomposition (POD) ........ 145 4.3.2. System Matrix Optimisation ............... 147 4.3.3. Balancing and Optimisation ................ 149 4.3.4. Polynomial Projection ................... 150 4.3.5. Other MOR Methods for Nonlinear Systems ....... 152 III. Implementation 155 5. EDEW–Tool for Simulation and Optimisation of Electrowetting 157 5.1. Surface Evolver Model ....................... 157 5.1.1. Numerical Representation ................. 158 5.1.2. Substrate-Liquid Interfaces ................ 158 5.1.3. Electrowetting Model ................... 159 5.2. EDEW ................................160 5.3. Limits ................................163 6. Polynomial Model Order Reduction Framework 169 6.1. Scanning-Probe Data Storage Device Model ........... 171 6.2. Representation of Polynomial Systems .............. 173 6.3. ANSYS Interface ..........................176 6.4. Series Expansion of Nonlinearities ................ 178 6.5. Model Order Reduction and Time Integration .......... 179 6.6. Verilog-A Export ..........................181 6.7. Limits ................................181 xi Contents IV. Results 183 7. EDEW 185 7.1. Droplet Motion ...........................185 7.2. Droplet Splitting ..........................187 7.3. Rising Fluid in Tube ........................188 7.4. Pinch-Off in Confined Setup .................... 188 7.5. Channels ..............................191 7.6. Optimisation of Electrode Fine Structure ............ 192 7.6.1. Influence of the Spike Shape ................ 194 7.6.2. Influence of the Spike Length ............... 195 7.6.3. Comparison with Geometric Model ............ 195 7.7. Results Summary ..........................197 8. Model Order Reduction 199 8.1. Arnoldi vs. Guyan – Second Order Case ............. 199 8.2. Polynomial Approximation .................... 205 8.3. Polynomial Reduction ....................... 206 8.4. Results with Other Reduction Approaches ............ 210 8.5. Combined Approaches ....................... 214 8.6. Results Summary ..........................215 9. Conclusions and Outlook 217 9.1. Conclusions .............................219 9.2. Outlook ...............................219 Appendices 221 A. Model Data of Numerical Examples for MEMS 223 B. Theory 225 B.1. Lagrangian Mechanics ....................... 225 B.1.1. Coordinates .........................225 B.1.2. Virtual Displacements ................... 226 B.1.3. Equations of Motion .................... 226 B.2. Structural Continuum Mechanics ................. 228 B.3. Arnoldi method ...........................230 xii Contents C. DSI Format 233 C.1. General ...............................233 C.2. File Header .............................235 C.3. System matrices and vectors .................... 236 C.4. Example ...............................237 D. ANSYS Interface 241 D.1. ANSYS Binary File Fields ..................... 241 D.2. Binary File Write-Out Script ................... 244 D.3. TRANS126 Export Script ..................... 244 D.4. Verilog-A Export Example ..................... 245 Bibliography 247 Acknowledgements 279 Curriculum Vitae 281 Nomenclature 283 Index 291 xiii xiv List of Figures 1.1. Typical setup of an electrowetting device. ............ 9 1.2. The main operations of a microfluidic electrowetting array. 10 1.3. Different actuation setups for electrowetting ........... 12 1.4. Different modelling approaches for a p-n-p transistor. ...... 20 2.1. The Y chart. ............................ 33 2.2. The three contributions to complexity. .............. 34 2.3. A MEMS RF switch (illustration). ................ 36 2.4. A clamped-clamped beam’s force-displacement curve. ...... 37 2.5. Setup of the storage device. .................... 39 2.6. Dimensions of the probe. ...................... 39 2.7. SEM image of a single probe of the storage device. ....... 40 2.8. 3D Visualisation of the Imego butterfly gyro. .......... 42 2.9. The Imego butterfly gyro. ..................... 42 2.10. Schematic layout of the butterfly design. ............. 43 2.11. The RF switch model. ....................... 45 2.12. First eigenmodes of the RF switch model. ............ 45 2.13. The bond wire model. ....................... 47 2.14. The modelled