## Fast Simulation of Electro-Thermal MEMS: Efficient Dynamic Compact ModelsFast Simulation of Electro-Thermal MEMS provides the reader with a complete methodology and software environment for creating efficient dynamic compact models for electro-thermal MEMS devices. It supplies the basic knowledge and understanding for using model order reduction at the engineering level. Emphasis is placed on the application of the Arnoldi method for effective order reduction of thermal systems. This tutorial is written for MEMS engineers and is enriched with many case studies which equip readers with the know-how to facilitate the simulation of a specific problem. |

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### Contents

1 | |

2 | |

12 MEMS and Electrothermal Simulation | 4 |

13 Thematic Outline | 5 |

Dynamic Electrothermal Simulation of Microsystems | 7 |

21 Overview of Coupled Electrothermal and Thermoelectric Effects | 8 |

22 Joule Heating in Microsystems | 10 |

23 Physical Model Heat Transfer Equation | 13 |

532 Reduction of Systems with Nonlinear Input Functions | 82 |

533 SystemLevel Simulation | 84 |

534 Computational Efficiency | 86 |

54 Arnoldi versus Guyan versus Control Theory Methods | 88 |

55 Inverse Thermal Problem via Model Order Reduction | 91 |

56 Summary | 96 |

Advanced Development | 97 |

611 Convergence of Relative Error | 99 |

24 Decoupling the Heat Transfer Equation from Other Physical Domains | 15 |

25 Solving the Heat Transfer Partial Differential Equation | 17 |

251 Linearization | 18 |

253 Numerical Methods | 19 |

26 Dynamic Compact Thermal Modeling | 21 |

262 Modal Approaches | 23 |

263 Model Order Reduction | 24 |

27 Summary | 28 |

Overview of Linear Model Order Reduction | 31 |

311 Balanced Truncation Approximation | 32 |

312 Singular Perturbation Approximation | 35 |

314 Comparison of Methods | 36 |

321 Lanczos Algorithm | 39 |

322 Arnoldi Process | 41 |

323 Arnoldi versus Lanczos | 43 |

33 Guyan Reduction | 45 |

332 Dynamic Matrix Condensation | 46 |

34 Summary | 48 |

Selected Model Reduction Software | 50 |

41 SLICOT Library | 52 |

43 Using Package Post4MOR and Mathlink Interface to SLICOT in Mathematica | 57 |

44 Summary | 59 |

Application of Model Reduction to Electrothermal MEMS | 61 |

51 MEMS Case Studies | 62 |

512 Tunable Optical Filter | 65 |

513 Microhotplate Gas Sensor | 67 |

52 Preparation for Automatic Model Order Reduction | 70 |

53 Model Reduction of Thermal MEMS via the Arnoldi Algorithm | 74 |

531 Approximation of the Complete Output | 81 |

612 Convergence of Hankel Singular Values | 107 |

613 Sequential Model Order Reduction | 112 |

614 Advantages and Disadvantages of the Proposed Strategies | 115 |

62 Order Reduction of Interconnected Thermal MEMS Models | 116 |

621 Microhotplate Array | 118 |

622 Block Arnoldi | 120 |

623 Coupling of Reduced Order Models via Substructuring | 127 |

624 Coupling of Reduced Order Models in the General Case | 131 |

625 Overview of Proposed Strategies | 137 |

631 Parametric Model Reduction | 138 |

632 Model Reduction for SecondOrder Systems | 140 |

64 Summary | 141 |

References | 142 |

Appendix | 157 |

811 Basic Theory | 158 |

812 Preparing ANSYS files | 159 |

813 Performing Model Reduction | 162 |

814 Advanced Options | 163 |

Summary | 164 |

816 References | 165 |

82 Post4MOR | 166 |

822 SimulationResult | 169 |

823 Functions for Transient and Harmonic Simulation | 172 |

83 Mathlink interface to SLICOT | 173 |

831 How to Compile the Interface | 174 |

833 How to Call SLICOT | 175 |

176 | |

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### Common terms and phrases

applied approximation Arnoldi algorithm Arnoldi process Arnoldi-based block Arnoldi boundary conditions circuit compact model compact thermal modeling computational control theory control theory methods convergence coupling decoupling dimension E. B. Rudnyi electrical electro-thermal simulation Error indicator expansion point full-scale model Hankel singular values heat conductivity heat sources heat transfer heat transfer equation heater Hence IEEE Transactions integration interface iteration J. G. Korvink Joule heating Krylov subspace Lanczos algorithm load vector Mathematica mathematical Mathlink membrane MEMS MEMS devices MEMS models microsystems microthruster microthruster model model order reduction model reduction MOR for ANSYS nodes nonlinear ODE system optical filter optimization option order 50 ordinary differential equations Packaging Padé approximants parameters projection Reduced model order reduced order models reduced system relative error resistor Section silicon SLICOT solution specified step response structure system matrices System order temperature distribution tion transfer function transient True error