Model Order Reduction of Electro-Thermal MEMS

Dissertation zur Erlangung des Doktorgrades

der Fakultät für Angewandte Wissenschaften der Albert-Ludwigs Universität Freiburg im Breisgau

Tamara Bechtold

2005 Dekan: Prof. Dr. J. G. Korvink

Referenten: Prof. Dr. J. G. Korvink, Prof. Dr. Ch. Ament

Datum: 25. Juli 2005 CONTENTS

Abstract 1 Zusammenfassung 3

1 Introduction 5 1.1 MEMS and Model Order Reduction 5 1.2 MEMS and Electro-Thermal Simulation 6 1.3 Thesis Overview 7 1.4 Major Results 9

2 Dynamic Electro-Thermal Simulation of Microsystems 11 2.1 Overview of Coupled Electro-Thermal and Thermo-Electric Effects 13 2.2 Joule Heating in Microsystems 15 2.3 Physical Model - Heat Transfer Equation 19 2.4 Coupling of Heat Transfer Equation to Other Physical Domains 21 2.5 Solving the Heat Transfer Partial Differential Equation 24 2.5.1 Linearization 25 2.5.2 Analytical Solutions 27 2.5.3 Numerical Methods 27 2.6 Dynamic Compact Thermal Modeling 30 2.6.1 RC Ladder Approach 30 2.6.2 Modal Approaches 33 2.6.3 Model Order Reduction 34 2.7 Conclusion 39 3 Linear Model Order Reduction Methods 43 3.1 Control Theory Methods 43 3.1.1 Balanced Truncation Approximation 44 3.1.2 Singular Perturbation Approximation 47 3.1.3 Hankel Norm Approximation 48 3.1.4 Comparison of Methods 49 3.2 Krylov Subspace Methods 50 3.2.1 Lanczos Algorithm 53 3.2.2 Arnoldi Process 56 3.2.3 Arnoldi versus Lanczos 59 3.3 Guyan Reduction 62 3.3.1 Static Matrix Condensation 62 3.3.2 Dynamic Matrix Condensation 63 3.4 Conclusion 67

4 Applications 69 4.1 Pyrotechnical Microthruster 69 4.2 Tunable Optical Filter 73 4.3 Microhotplate Gas Sensor 77 4.4 Preparation for Automatic Model Order Reduction 82

5 Numerical Results 87 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm 87 5.1.1 Approximation of the Complete Output 97 5.1.2 Reduction of Weakly Nonlinear Systems 99 5.1.3 System-Level Simulation 103 5.1.4 Calculation Efficiency 105 5.2 Comparison Between the Arnoldi Algorithm, the Guyan Algorithm and the Control Theory Methods 107 5.3 Conclusion 111 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algo- rithm 113 6.1 Convergence of Relative Error 115 6.2 Convergence of Hankel Singular Values 125 6.3 Sequential Model Order Reduction 134 6.4 Conclusion 137

7 Model Reduction of Interconnected Systems 141 7.1 Microhotplate Array 142 7.2 Block Arnoldi 146 7.2.1 Interconnected System Behavioral Simulation 152 7.3 Coupling of Reduced Order Models via Substructuring 154 7.4 Coupling of Reduced Order Models in the General Case 160 7.4.1 Coupling by Fluxes 160 7.4.2 Structure Preserving Model Order Reduction 165 7.5 Conclusion 168

8 Conclusion and Outlook 171 Appendix 175 A.1 Material Properties 175 A.2 MAST Templates 177 References 181 Publications List 199 Acknowledgements 201

ABSTRACT

The modeling of electro-thermal processes, for example Joule heating, is becoming increasingly important in microsystems (in the following MEMS) development. In microelectronic systems, for example, high temperatures may cause the malfunction or even destruction of the device. Other devices, such as microsensors and microactuators need high temperatures to improve transduction efficiency. In both cases the designer should be able to predict the temperature distribution for the given electrical input and the impact of the temperature on the device‘s electronics in turn. In other words, one must run a joint electro-thermal simulation. Conventionally, in each sequence of electro-thermal simulation the temperature field is computed on a discrete grid whose size, due to increasingly complex microstructures, easily exceeds 100,000 degrees of freedom (DOF), i. e. ordinary differential equations. Although modern computers are able to handle engineering problems of this size, the system-level simulation would become prohibitive if full models were directly used. Hence, the reduction of the problem’s size is the first milestone of efficient MEMS modeling and simulation. This thesis presents the application of mathematical model order reduction (MOR) methods (preferably Arnoldi algorithm) to the automatic generation of accurate dynamic compact thermal models (DCTM) of electro-thermal microsystems. Unlike conventional approaches to DCTM, which are based on the lumped- element decomposition of the model followed by parameter fitting, mathematical MOR is formal, robust and can be made fully automated. The reduced order models can be formally converted into Hardware Description Language (HDL) form and directly used in system-level simulation. The results obtained in this thesis led to the creation of the software tool mor4ansys at the chair for simulation of the university of Freiburg. Presently it is possible to use mor4ansys to automatically create reduced order thermal models (with approximately 50 DOF) directly from ANSYS models (with more than 100,000 DOF).

1 In this work we apply model order reduction to successfully create dynamic compact thermal models of several novel MEMS devices. We consider the different aspects of the Arnoldi algorithm which make it stand out against other linear MOR methods, such as the approximation of the complete output and the reduction of weekly nonlinear systems. We propose three heuristic methods to estimate the error of Arnoldi-based reduction. We present and apply two methods for model order reduction of thermal MEMS-array structures, namely Block Arnoldi and Guyan-based substructuring and describe a general technique of coupling two reduced thermal models via surface fluxes.

2 ZUSAMMENFASSUNG

Die Modellierung elektro-thermischer Prozesse, wie z.B. Joule’sche Wärmegene- rierung, gewinnt zunehmend an Bedeutung in der Entwicklung von Mikrosyste- men (im Folgenden MEMS genannt). In mikroelektronischen Systemen können hohe Temperaturen Fehlfunktionen verursachen oder sogar zur Zerstörung des Bauelements führen. Andere Komponenten, wie Mikrosensoren oder -aktoren, benötigen hohe Temperaturen zur Verbesserung ihrer Energiewandlungseigen- schaften. In beiden Fällen sollte der Designer in der Lage sein, für vorgegebene elektrische Eingangsleistung die Temperaturverteilung sowie den Einfluss der Temperatur auf die Elektronik zu bestimmen. In anderen Worten, er muss eine elektro-thermische Simulation durchführen. In jedem Iterationsschritt einer konventionellen elektro-thermischen Simulation wird die Temperaturverteilung auf einem diskreten Gitter berechnet. Aufgrund der zunehmenden Komplexität von Mikrostrukturen übersteigt die Größe solcher Gitter (bzw. Anzahl der Differentialgleichungen) leicht 100.000. Obwohl moderne Computer in der Lage sind Probleme dieser Größe zu lösen, ist eine Simulation von solchen Modellen auf Systemebene kaum möglich. Die Reduktion der Modellgröße stellt demnach einen Meilenstein zur effizienten Modellierung und Simulation von (nicht nur elektro-thermischen) MEMS dar. Die vorliegende Arbeit stellt die Anwendung mathematischer Methoden zur Modellordnungsreduktion (MOR) (vorzugsweise den Arnoldi Algorithmus) zur automatischen Generierung von präzisen kompakten dynamisch-thermischen Modellen (DCTM) von elektro-thermischen Mikrosystemen vor. Alternativ zu konventionellen Verfahren zur Erzeugung von DCTM, welche auf dem Erstellen eines Ersatzschaltbildes mit anschließender Parameteroptimierung beruhen, stellt die mathematische MOR einen formalen, robusten und vollständig automatisier- baren Weg dar. Die reduzierten Modelle lassen sich in eine Hardware-Beschrei- bungssprache (HDL) konvertieren und sind somit in einer Simulation auf Syste- mebene nutzbar.

3 Die in dieser Arbeit erzielten Ergebnisse führten am Lehrstuhl für Simulation der Universität Freiburg zur Erstellung des Software-Pakets mor4ansys. Gegenwärtig ist es möglich, mor4ansys zur automatischen Erstellung von kompakten thermischen Modellen (mit ungefähr 50 Freiheitsgraden) direkt aus ANSYS Modellen (mit mehr als 100.000 Freiheitsgraden) zu verwenden. Wir haben die Modellordnungsreduktion erfolgreich auf die Erstellung von kompakten dynamisch-thermischen Modellen von mehreren neuen MEMS Kompo- nenten angewandt. Weiterhin haben wir verschiedene Aspekte des Arnoldi-Algo- rithmus, wie die Approximation des gesamten Temperaturfeldes und die Reduk- tion von schwach nichtlinearen Systemen, welche ihn von anderen linearen MOR- Methoden abheben, untersucht. Wir haben drei heuristische Methoden zur Abschätzung des Fehlers der Arnoldi-Reduktion vorgeschlagen. Letztendlich haben wir zwei Methoden zur Modellordnungsreduktion von thermischen MEMS-Arrays präsentiert and angewandt. Diese sind Block-Arnoldi und Guyan- basierte Unterteilung (implementiert unter dem Namen substructuring in ANSYS Simulator). Wir haben weiterhin, einen generellen Weg zur Kopplung von reduzierten thermischen Modellen über die Oberflächenflüsse beschrieben.

4 1.1 MEMS and Model Order Reduction

1INTRODUCTION

This thesis presents the application of mathematical model order reduction for the automatic generation of accurate dynamic compact thermal models of electro- thermal microsystems. The reduced order models are convertible into Hardware Description Language (HDL) form and can be directly used in system-level simulation.

1.1 MEMS and Model Order Reduction

The development of increasingly complex microstructures (in the following we will call all microsystems MEMS, even if non-micro-electromechanical functions are employed) demands sophisticated simulation techniques for design, control and optimization. Often, a system-level simulation which additionally to device includes the driving circuitry is indispensable. Due to the complex nature of MEMS, the discrete models resulting from, for example, the finite element method (FEM) are usually very large (100,000 equations are the engineering standard). Even though modern computers are able to handle this size of engineering problems, the system-level simulation would become prohibitive if the full models were directly used. Hence, the need for efficient computational techniques is greater than ever. Although no universal simulation strategy currently exist which could cover all the MEMS design situations [1], the reduction of the problem size could in any case drastically reduce the computational work (see Figure 1.1). Note that compact modeling itself had been developed in electrical engineering long before MEMS existed. It aims at creating a small size equivalent network of resistors, capacitors, inductors, etc. which accurately describes the dynamics of the device and can be directly inserted into SPICE-like simulators. Naturally, MEMS engineers try to use the same methodology. However, this approach requires the designer to choose the correct network structure intuitively, i. e. without strict guidelines, and to perform a time-consuming parameter extraction based

5 1 Introduction on fitting the model to measured or simulated curves. An alternative to “classical compact modeling” is mathematical model order reduction (MOR), which is formal, robust and can be made fully automated. It is based on the formal conversion of governing partial differential equation (PDE) systems to low-dimensional ordinary differential equation (ODE) systems without representing the latter as an electrical circuit. Hence, it can be considered as “compact modeling on demand” [2]. The development of efficient model order reduction methods for automatically creating accurate low-order dynamic models is about to become a major subject of MEMS simulation and modeling research.

Physics & Reduced system FEM System of MOR Geometry n ODEs of r << n ODEs

= + x˙ r []Ar xr br u = T yr cr xr

physical level device level system level

Figure 1.1 Motivation for model order reduction.

1.2 MEMS and Electro-Thermal Simulation

The modeling of electro-thermal processes, e. g., Joule heating, becomes increasingly important during microsystems development [3]. For example, with the decreasing size and growing complexity of micro-electronic systems, the power dissipation of integrated circuits has become a critical concern. The thermal inﬂu- ence upon the device caused by each transistor’s self-heating and the thermal interaction with tightly placed neighboring devices cannot be neglected because excessive temperatures may cause the malfunction or even destruction of the device. Whereas Joule heating in microelectronics is a “parasitic” effect, some other devices like microsensors and microactuators use it (directly or indirectly) as a functioning principle. For example, microhotplate-based devices can use

6 1.3 Thesis Overview

Joule heating to achieve the operating temperature. Micromechanical devices with electro-thermal actuation can use Joule heating for electro-thermal expansion. Microfluidic devices employ Joule heating to expel micro droplets out of micro fabricated reservoirs, etc. In all mentioned design situations, the engineer’s task is to predict the temperature distribution for the given electrical input and the impact of the temperature on the device electronics in turn, i. e. to run an electro-thermal simulation. To go a step further, in each sequence of joint electro-thermal simulation the temperature field is computed on a discrete grid whose size, as already mentioned, easily exceeds 100,000 degrees of freedom (DOF), i. e. ordinary differential equations. Nevertheless, as we have started this research, no efforts to apply model order reduction to electro-thermal MEMS models have been made so far. Previously, order reduction approaches were only used for electric circuits and structural mechanic problems, although thermal problems which result in the ﬁrst order (linear) ODE systems after spatial discretization are the “simplest” case from the MOR point of view. In our opinion, model order reduction of electro-thermal models ranks among the most important MEMS modeling strategies.

1.3 Thesis Overview

In chapter 2 the solution of the heat transfer partial differential equation, which constitutes the central part of electro-thermal microsystem simulation, is discussed. Different levels of solution approaches are presented, starting from the most rigorous, when the heat transfer can only be modeled by a set of coupled partial differential equations, and finishing with the final level of approximation, which is compact thermal modeling. In chapter 3 we present the most important algorithms for model order reduction of linear first order ODE systems. All model order reduction methods are based on the projection of the large-scale system to some subspace. Different projections characterize different methods. The central focus of this thesis is the Arnoldi algorithm, which is based on the projection to so-called Krylov subspace. We have also researched the mathematically optimal control theory methods, which already have a long tradition in model order reduction of moderate size linear sys-

7 1 Introduction tems, and the commercially available Guyan algorithm (implemented in ANSYS), which has been mostly used for mechanical engineering problems. In chapter 4 we present three novel MEMS devices used as case studies for model order reduction. These are the pyrotechnical microthruster, the thermally tunable optical filter and the microhotplate gas sensor. The formal switch from electro- thermal to “pure” thermal modeling, under the assumption of homogenous heat generation within the lumped resistor, is made in this chapter as well. Chapter 5 presents the numerical results of model reduction for the three case studies. It further considers different aspects of the Arnoldi algorithm, such as the approximation of the complete output and the reduction of weakly nonlinear systems. Results of Arnoldi- based reduction are compared with the results of control theory methods and those of the Guyan algorithm. The results of system-level simulation with SABER are presented in this chapter as well. In chapter 6 we suggest three heuristic strategies for error estimation of Arnoldi- based reduction. The first strategy is based on the convergence of relative error between two successive reduced models of the order r and r+1, the second is based on the convergence of the so-called Hankel Singular Values (characteristic of the system defined in control theory) belonging to reduced order models and the third strategy is based on the sequential application of Arnoldi and mathematically superior control theory methods. In chapter 7 we discuss the difficult topic of coupling reduced order models, which is an indispensable procedure for effective compact modeling of MEMS arrays. We have applied two methods for model order reduction of a novel gas sensor array device, namely Block Arnoldi and Guyan-based substructuring. Additionally, we describe a general technique of coupling two reduced thermal models via the surface fluxes. The results of SABER simulation for the reduced gas sensor array device are also given in this chapter. During the MOR methodology development the Arnoldi algorithm was proto- typed in MATHEMATICA and finally was implemented in C++ as a part of the order reduction tool mor4ansys [4], which automatically creates reduced order thermal models directly from ANSYS models.

8 1.4 Major Results

1.4 Major Results

METHODOLOGY FOR APPLYING MODEL ORDER REDUCTION TO ELECTRO- THERMAL MEMS MODELS We have applied model order reduction to successfully create dynamic compact thermal full-scale FE model ≤ ηM 2 ηP 2 models of several novel MEMS UU– h c ∑ T + T ∈ devices. We have tested and TS compared the advantages and disadvantages of different linear reduced order model model order reduction methods. It is currently possible to use software tool mor4ansys [4] to automatically create reduced order thermal models (with around 50 DOF), directly from ANSYS models (with more than 100,000 DOF). Model order reduction is automatic and based on Pade approximation of transfer function via the Arnoldi algorithm. Reduced models are easily convertible into HDL form, and can be directly used for system-level simulation. We have tested them using SABER simulator.

APPROXIMATION OF COMPLETE OUTPUT AND WEAKLY NONLINEAR 1ST ORDER ODE SYSTEMS USING THE ARNOLDI ALGORITHM Due to the fact that the Arnoldi reduction algorithm does not CT˙ + KT = FQ() t, T take into account either input or ≤ … ηM 2 ηP 2 UU– h10c ∑0 T + T output vector (matrix), it is pos- ∈ 01…TS0 sible to approximate not only a y = ⋅ T . . . single output response but also . . . the transient thermal response in 00… 1 all finite element nodes of the device. It is further possible to transfer the nonlinearities of the input function into the reduced system, i. e. weakly nonlinear systems can be reduced as well. This is numerically demonstrated in chapter 5.

9 1 Introduction

STOP CRITERIA FOR FULLY AUTOMATIC MOR VIA THE ARNOLDI ALGORITHM We have suggested three heuristic methods to estimate the 0 True error E (s) 10 r Error indicator Ê (s) error of Arnoldi-based reduc- -3 r 10 M 2 P 2 UU– ≤ c ∑ η T + η T tion. The first method is based -6 h 10 TS∈ -9 on the convergence of relative 10 -12 error between two successive 10 -15

reduced order models of the Error magnitude for 1000 rad/s 10 01020304050 order r and r+1, the second is System order based on the convergence of Hankel Singular Values of the reduced order models and the third method is based on the sequential application of Arnoldi and mathematically superior control theory methods. This is done in chapter 6.

COUPLING OF REDUCED MODELS We present and apply two methods for model order reduction of thermal MEMS-array structures, ≤ ηM 2 ηP 2 Block Arnoldi and Guyan-based UU– h c ∑ T + T substructuring. We further TS∈ describe a general technique of coupling two reduced thermal models gained by projection (e. g. when the Arnoldi algorithm is used). The coupling is done via surface fluxes. In chapter 7 we discuss the possibilities of how to find appropriate projection subspaces in this case.

10 2DYNAMIC ELECTRO- THERMAL SIMULATION OF MICROSYSTEMS

Thermal management has become a crucial part of designing modern microelectronic and micro-electro-mechanical components and systems. Only after a careful study of thermal effects in microsystems can we predict their performance, reliability and yield. In integrated circuits, thermal management is important on all levels, starting from the transistor level [5], [6], over the chip level [7]-[9], the package level [10] - [12] and the printed circuit board (PCB) level [13], to the system level [14]. Self- heating of transistors changes their electric properties and this can affect the circuitry. High current during electrostatic discharge may increase the local temperature too much and destroy the transistor. Self-heating of interconnect may reduce its average time to failure. It is important to position elements on a chip in such a way that the effect of heating is minimal in respect to the chip functioning. A package should be designed in such a way as to allow effective heat removal. Finally, the placement of semiconductor devices on PCB should also be optimal from the thermal management viewpoint. It is worthy of note that the reliability of the whole board is tied to its temperature regime because there are many mechanisms that lead to enhanced board degradation at elevated temperatures. Numerous MEMS devices, such as thermal actuators [15], thermal flow sensors [16], microhotplate gas sensors [17], [18], tunable optical filters [19], [21] and many others are based on thermal effects. The heating changes optical properties in optical filters, reduces activation energies to allow metal oxides to detect gases and causes mechanical stress and thus movement in actuators. The heat transfer through the moving fluid allows us to measure the flow rate in heat-flow sensors.

11 2 Dynamic Electro-Thermal Simulation of Microsystems

All diverse engineering problems mentioned above have in common that an engineer should be able to predict the temperature distribution for the given electrical input and to estimate how the temperature in turn affects the electric part. In other words, he must run a joint electro-thermal simulation. As for the simulation of the electrical domain, there are usually already available tools, so the real task becomes to enhance them in order to take into account a variety of electro-thermal and thermo-electric effects. Hence, the solution of the heat transfer equation constitutes the central part of electro-thermal simulation. The main goal of this chapter is to present a unified description of different levels of solution approaches for the heat transfer partial differential equation. Of course, in many cases the coupling of the heat transfer equation to other physical domains must be taken into account as well. Throughout this chapter, we will briefly review different couplings and discuss what is necessary to decouple the heat transfer equation in order to be able to solve it separately. Section 2.1 gives a short overview of different electro-thermal and thermo-electric effects in microsystems. Throughout this thesis we will take into account only the Joule heating effect, although the model order reduction approach can be used for other effects as well. The importance of the Joule heating effect for MEMS and microelectronic systems will be highlighted through examples in section 2.2. Sec- tions 2.3 and 2.4 describe physical modeling with heat transfer equation and its couplings to other physical domains at the device level. Throughout section 2.5 we discuss how to solve the parabolic heat transfer equation on its own. The first topic is the linearization of the heat transfer equation, because the solution of the linear equation is much faster than that of the non-linear one. The analytical solutions are briefly reviewed. We consider further the most frequent approach, which is based on spatial discretization of the thermal domain (often called the brute force method). The final level of approximation, dynamic compact thermal modeling (DCTM) is described in section 2.6. It allows us to effectively perform transient electro-thermal simulations on the system level. We include here the conventional, non automatic approaches, such as RC-ladder network, semi automatic approaches such as modal approximation, as well as the increasingly popular model order reduction (MOR) approach, which can be made fully automatic.

12 2.1 Overview of Coupled Electro-Thermal and Thermo-Electric Effects

Finally, section 2.7 concludes the chapter by displaying a schematics of discussed modeling approaches (see Figure 2.12).

2.1 Overview of Coupled Electro-Thermal and Thermo- Electric Effects

A classification of the most important electro-thermal effects in semiconductor and IC-related materials utilized in microsystems is given in Table 2.1. A brief explanation of each effect is presented below(SI units are used). Table 2.1 Electro-thermal signal conversion effects in microsystems (see also [3]).

Thermal Electrical thermoresistance Thermal heat conduction Seebeck effect pyroelectricity Joule heating Electrical Peltier effect electrical conduction Thomson effect

Heat conduction at the macroscopic level means that when a temperature gradient exists within a solid body, heat energy will flow from the region of high temperature to the region of low temperature. This phenomenon is described by Fou- rier's law:

q = –κ∇T (2.1)

This equation determines the heat flux vectorq (the amount of heat penetrating a unit of area per unit of time) in W/m2 for a given temperature profileT and thermal conductivityκ of the body (a material property that describes the rate at which heat flows within a body for a given temperature difference) in W/(mK). All quantities relate to a specific point. The minus sign ensures that heat flows down the temperature gradient. Electrical conduction describes the electrical current flow in the presence of the electrical potential gradient. If, in (2.1), the temperature is replaced by the

13 2 Dynamic Electro-Thermal Simulation of Microsystems electric potentialϕ and the heat flux vector by the electric current density vector j , the solution of a corresponding problem of electric conduction is given by:

j = –σ∇ϕ (2.2) whereσ is the specific electric conductivity in S/m. Joule heating is the dominant mechanism for heat generation due to the flow of the electrical current through the material. It is defined by Joule’s law, which in an ohmic conductor has the form:

2 Qj= ρ (2.3) where j is the current density vector in A/m2,ρ is the specific electric resistivity inΩm and Q is the generated heat per unit volume in W/m3. This is usually the main effect responsible for heat generation in resistively heated microsystems. A number of applications will be presented in section 2.2. The thermoresistance effect states that the specific electric resistivityρ of the material can be expressed as a function of temperatureT :

(2.4) ρ() ρ()α β 2 … T = 0 1 ++T T + whereαβ and are the temperature coefficients of the material in K-1 and K-2 ρ respectively, and0 is the specific electric resistivity at the zero temperature. Pos- itive temperature coefficients are peculiar to pure metals and some alloys. In a number of microsystem applications, metal resistors out of platinum, nickel, copper etc., are used as heating and sensing elements [17], [19]. In such cases (2.4) is on one side of paramount importance for temperature control, and on the other side it contributes to the nonlinearity of the governing heat transfer equation via (2.3). The Seebeck and Peltier effects are associated with a junction of two different conductors or semiconductors, a so-called thermocouple. If there is a maintained temperature difference between the junction and the free ends, an open-circuit Seebeck voltage is obtained at the non-connected end. The Seebeck effect is

14 2.2 Joule Heating in Microsystems investigated for use in miniaturized voltage sources [22]. Those devices utilize heat generated by the human body and supply electronic devices (e.g. watches) with a minimum amount of electrical power. It is further used in thermoelectric infrared gas sensors [23]. The Peltier effect describes the generation or absorption (depending on the direction of the current) of heat in a thermocouple when the current flows through the junction in the absence of any temperature gradient. Con- trary to Joule heating, the Peltier effect is reversible. It can be effectively used for heat transport opposite the temperature gradient and hence for cooling down the environment. It is mainly used for controlling the temperature of chips while measurements are being made. On-chip integrated Peltier elements are ideally suited for highly localized on-chip thermal stabilization [24]. The Thomson effect is complementary to the Peltier effect. In the Thomson effect two dissimilar materials are not needed but a current passed along a conductor, when a temperature gradient is maintained, results in either absorbance or generation of additional heat (in addition to Joule heat). The pyroelectric effect describes the change of the electric polarization induced by a change of temperature in some nonlinear dielectric materials, such as PZT-ceramics, PbTiO3, PVC etc. Pyroelectric materials are used within the thermal sensors and infrared detectors. Pyroelectric motion (infrared) sensors are used in a wide variety of applications. They are commonly found in security products, such as burglar alarms, motion detectors and intrusion detection systems [25]. These sensors are also useful in environmental systems, lighting controls, visitor announcers, robotics, and artiﬁcial intelligence. They work by detecting the infrared heat emitted by the human body.

2.2 Joule Heating in Microsystems

Joule heating happens to be the dominant mechanism for heat generation in microsystems. Some devices like microsensors and microactuators are designed to optimize the Joule heating effect in a controlled manner in order to improve transduction efficiency. For the others, like integrated circuits (ICs), this effect is “parasitic“, and hence their design aims at the suppression of Joule heating. Below we review several MEMS devices whose working principle is directly or indi-

15 2 Dynamic Electro-Thermal Simulation of Microsystems rectly based on Joule heating and mention the most common “parasitic” effects based on Joule heating in microelectronic devices and their packaging. Micro-hotplate-based devices, such as gas sensors [27]-[32], optical filters [19], thermal flow sensors [16], solid fuel microthrusters [33], thermal infra-red emitters [34] etc. use Joule heating to realize a desired functionality. Maximum temperatures for the operation of these devices reach several hundred°C .In order to significantly reduce the required electrical heating power, the resistive heater structures are placed on a thermally isolated micromachined platform called a membrane (see Figure 2.1). Elevated temperatures may be essential for the onset of operation of the device (e.g. by gas sensors, microthrusters or infra-red emitters) or controlled temperate changes are used for tuning the structure‘s characteristics (e. g. in optical filters).

Figure 2.1 Photomicrograph of a single microhotplate element from [35].

Micromechanical devices with electro-thermal actuation areusedtoper- form direct mechanical actuation through thermal expansion resulting from Joule heating of selected microstructures. Both in-plane actuators, based on thermally induced extension [37], and out-of-plane actuators [38], based on different thermal expansion coefficients that rely on the thermal expansion of silicon, polysilicon, nickel, or related materials have been extensively studied and used in MEMS structures. Thermal microactuators based on polymers such as polyimide, thermal bimorphs, pseudo-bimorphs, buckling beams, compliant structures and high- aspect-ratio structures have also been investigated (see [35] and the references there). The working principle of embedded electro-thermal-compliant actuator is

16 2.2 Joule Heating in Microsystems shown in Figure 2.2. The wide arms, which have lower electrical resistance than the narrow arms, draw more current and get hotter. As a result, the structure bends.

a) b) Figure 2.2 An example of electro-thermal embedded actuation: (a) original configuration; (b) deformed configuration superimposed on the original. From [36].

Microfluidic devices employ Joule heating to expel micro droplets out of micro fabricated reservoirs. In the reservoirs air-bubbles are generated within the liquid phase at the bottom of the reservoir that is above the heater, as schematically shown in Figure 2.3. As the properties of micro-scale bubbles are dominated by surface tension effects, they tend to be very stable and hence useful for generating mechanical work. The dramatic increase in volume, which is due to bubble generation, forces small amounts of liquid to leave the chamber through a micro-nozzle. This technology is used in a broad spectrum of MEMS devices starting from ink jet print heads [39]. The thermopneumatic effect is further used to vaporize a working fluid through Joule heating. The increased gas pressure actuates a dia- phragm. This deflection can be used in micropumps and microvalves [40] - [45] as components of micro-fluidic systems. Microelectronic devices and their packaging are significantly influenced by temperature. In contrast to the previously described devices, Joule heating is not a desirable effect in microelectronic systems, but rather a “parasitic” one. Joule heating activates damaging mechanisms such as corrosion of metallization and bond-pads, fatigue of wires or bonds, contact spiking, electrostatic discharge,

17 2 Dynamic Electro-Thermal Simulation of Microsystems

trapped fluid

glass silicon

closed (cold)

open (hot) flow Figure 2.3 Functioning principle of a thermopneumatic microvalve, from [46]. electrical overstress (causing alone over 40% of all failure), electro-migration, die thermal breakdown, damage of wire bonded interconnections, flip-chip joints damage, cracking in plastic packaging etc. In addition to all of this there is almost no room for heat dissipation due to the huge number of transistors per unit area (modern VLSI contain 104 to 107 transistors per chip) and also no longer enough time due to higher frequency. Today‘s heat flux densities on the chip easily exceed 100W/cm2 [47]. Not only the operating temperature itself, but also fluctuations are dangerous for the device’s reliability. It was already known in 1989 that a temperature oscillation of only 15 K increases the failure rate by eight times [47]. Due to the fact that temperature gradients are steeper in transient than in steady-state phase [49], transient (dynamic) thermal stresses are more dangerous. Violent operating conditions include the following dynamic effects: avalanche and electro-static discharge. Further effects such as temperature oscillations, current and voltage thermal drift or temperature gradient over the device [47], may degrade the device’s performance. It should be noted that heat transfer modeling on its own varies in its importance for the above applications. In microhotplate-based devices it plays a major role and can be easily decoupled from other physical domains. In microfluidic devices, on the other hand, it plays a relatively small role, because it is strongly coupled to Navier-Stockes equations. In Figure 2.4 the qualitative importance of heat transfer

18 2.3 Physical Model - Heat Transfer Equation modeling for the different types of MEMS devices is schematically presented. However, it is difficult to set clear borders, because each engineering problem has to be considered on its own.

microfluidic thermal actuator micro hotplate

thermal domain other domains

Figure 2.4 Qualitative importance of heat transfer modeling for different types of MEMS devices.

2.3 Physical Model - Heat Transfer Equation

The most general way to model heat transfer on the device level in crystalline semiconductors is the phonon Boltzmann transport equation. It describes the heat exchange between heat carriers (phonons), which are energy quanta of lattice vibrations [48] and the lattice. It should be used when the heat carrier mean free path is comparable with the characteristic dimensions at microscale. To give an example, the phonon’s mean free path in Silicon at 300K is about 300nm. How- ever, if one can assume that the Boltzmann-Maxwell distribution is valid for any small volume, or in other words that the temperature is defined at any point within the domain, a hyperbolic heat equation is used. It predicts a finite wave speed of heat propagation and is valued at a very small time scale of femtosecond, for example during laser heating of thin metal films. If we further, assume that the speed of the thermal waves is infinite, the parabolic heat transfer equation can be used instead. Its solution is our main topic, since the assumptions mentioned above are valid for the devices in question.

19 2 Dynamic Electro-Thermal Simulation of Microsystems

The parabolic heat transfer equation specifies the complete spatial and time profile of a temperature distribution within a computational domainΩ , limited by the boundary∂Ω . In solid it has a form:

∂T ∇•()κ∇T + Q – ρC = 0 (2.5) p ∂t

κ() () wherer is the thermal conductivity in W/mK at the position r,C p r is the specific heat capacity (a material property that indicates the amount of energy a body stores for each degree increase in temperature, on a per unit mass basis) in J/kgK,ρ()r is the mass density in kg/m3,Trt(), is the temperature distribution andQrt(), is the heat generation rate per unit volume in W/m3. (2.5) states that the temperature profile within a body depends upon the rate of its internally-generated heat, its capacity to store some of this heat by raising its temperature and its rate of thermal conduction to its boundaries (where the heat is transferred to the surrounding environment). The solution of (2.5) requires the determination of initial and boundary conditions (BC) for the computational domainΩ . The temperature distributionT 0 att = 0 serves as an initial condition: ∀ ∈ Ω (), () r ,t = 0 ,Trt = T 0 r (2.6)

Prescribed temperature distribution on the boundary in time can be modeled by Dirichlet boundary condition as follows:

∀ ∈ ∂Ω () () r ,Tr = T prescribed t (2.7)

Quite often, the boundary temperature is assumed to be constant and equal to the temperature of the surroundings and can be set to zero without a loss of generality.

Prescribed normal heat flux through the body boundariesq⊥ in time can be modeled by Neumann boundary condition as follows:

∀ ∈ ∂Ω () () r ,q⊥ r = q prescribed t (2.8)

Convective heat transfer takes place when the whole subvolumes move from one place at a certain temperature to another at a different temperature. Hence, the

20 2.4 Coupling of Heat Transfer Equation to Other Physical Domains convection is created by fluid flow. In many cases however, it is possible to eliminate the fluid flow from the computational domain by replacing it with a so-called convection boundary condition. It assumes that the normal heat flux through the boundary is proportional to the temperature difference between the boundary and the ambient temperature of the adjacent fluid:

∀ ∈ ∂Ω () () r ,q⊥ r = hT– Tambient (2.9) whereh is the heat transfer coefficient, which characterizes the thermal flow as well as the thermal contact between the conducting solid and adjacent fluid with temperatureT ambient . Again, quite often it isT ambient = 0 . It should be noted that the convection boundary condition turns into Neumann forh = 0 , or into Dirichlet forh → ∞ .

Heatradiationmechanismcanbemodeledbysocalledradiation boundary condition as follows:

∀ ∈ ∂Ω () ε⋅⋅ σ ()4 4 r ,qemitted r = AT surface – T ambient (2.10) whereq is the emitted heat transfer rate (power),ε is the surface incivility, emitted –8 2 4 σ is the Stefan-Boltzmann constant (5.669⋅ 10 m K ) andA is the radiation surface. It is worthy of note that in practice it is important to be able to solve (2.5) with different boundary conditions, that is, to create a boundary conditions independent (BCI) model [63]. This is possible at the level of detailed solution of (2.5) (see section 2.5.3) but is difficult at the level of compact models (see section 2.6.1).

2.4 Coupling of Heat Transfer Equation to Other Physical Domains

In the general case the heat transfer equation (2.5) is coupled with other partial differential equations already at the device level. Let us consider the most important coupling and discuss what approximations are necessary to decouple the

21 2 Dynamic Electro-Thermal Simulation of Microsystems equations in order to be able to efficiently solve (2.5) on its own, preferably via spatial discretization and MOR. The first coupling comes through the environment, since the device is usually surrounded by moving fluid (either gas or liquid), which serves to remove heat by natural or forced convection. The heat transfer in the moving fluid is generally a part of the solution of Navier-Stokes equations. They can be coupled with (2.5) in order to describe the overall heat transfer, which makes the resulting set of equations quite complicated. In some cases it is possible to decouple them by applying the convection boundary condition (2.9). If there is a convective flow with specified flow velocityv within a computational domain, the heat transfer equation (2.5) gains additional, so-called “thermal ρ ∇ flow term”C pvT . This term complicates the analytical solutions. However, numerical methods for solving (2.5) can treat the thermal flow term with small additional effort and moreover MOR-based solutions can be used as well [50]. If the flow velocity within a computational domain cannot be specified, as in the extreme case of bubbles formation by microfluidic devices for example, there is no way of decoupling the (2.5) from Navier-Stokes equations. Hence, the solution of spatially discretized coupled system becomes time consuming. In the case of thermo-mechanical coupling, a good approximation is the solution of the thermal problem on the Lagrangian grid [51]. Then, it is possible to solve the thermal problem on its own and to use the temperature distribution to estimate additional stress, which is due to thermal expansion. The electro-thermal coupling is present through the heat generation rate Q, emerging mostly from the Joule effect (2.3). In general, in order to solve for the current in (2.3), it is necessary to solve the Poisson equation, which (in the case of an isotropic resistive heater which is free from electrical charges) has a form:

∇• j= ∇•()σϕ∇ = 0 (2.11)

It is possible to solve the Poisson equation for the unknown electric potential field, provided that the geometry of the heater, its specific conductivity and the boundary conditions are specified [26]. (2.11) depends only implicitly on time, due to possible changes in boundary values or changes in conductivity. This is because

22 2.4 Coupling of Heat Transfer Equation to Other Physical Domains the speed of the electron propagation is very high. We assume that (2.11) holds instantly at any given time. It should be noted that if the electrical current frequency reaches the kHz range, i.e. when capacitance and inductive effects have to be accounted for, a higher level model based on the solution of the electromag- netic Maxwell equations is required. By inserting (2.11) into (2.3) the heat generation rate within an ohmic conductor changes into:

2 j σϕ⋅ ()∇ 2 Q ==-----σ (2.12)

By assuming homogeneous heat generation over a lumped resistor (2.12) simpli- fies to the well known:

2 2 U QI==R ------(2.13) R where I is the current passing the lumped resistor, due to potential difference U over it, R is its resistivity and Q is the total heat generated within the resistor’s volume. By replacing Q in (2.5) with (2.13) the coupling to the Poisson equation is avoided. However, (2.13) may not offer a good approximation in all cases. For example if resistors with complex geometries are present, homogeneous heat generation can be hardly assumed and in this case the consistent electro-thermal simulation is needed (that is a simultaneous solution of (2.12) and (2.5)), as shown in Figure 2.5.

The electro-thermal coupling at the system level (with assuming a homogeneous heat generation within a lumped heater) for the special case of resistively heated microdevices, is schematically presented in Figure 2.6. Note that the coupling becomes more complicated in case of ICs, i. e. when the temperature impact on semiconductor devices is taken into account [54]. In order to obtain a thermal system

CT˙ + KT = FQ⋅ (2.14)

23 2 Dynamic Electro-Thermal Simulation of Microsystems

T potential mesh for (2.12) thermal mesh for (2.5)

electrical simulation

U R

Figure 2.5 Sequence of the coupled electro-thermal simulation when coupling is on the device and on the system level. shown in Figure 2.6 a spatial discretisation of (2.5) over the complete simulation domain (e.g., the whole chip) is necessary. Obviously, in order to effectively perform a system level simulation, it is necessary to keep the dimension of (2.14) as moderate as possible. This is exactly the goal of compact modeling described in section 2.6.

2.5 Solving the Heat Transfer Partial Differential Equation

In the following as a border between analytical and numerical solutions of (2.5) we will use the discretization of the computational domain in space, which necessarily leads to an ordinary differential equation system. One can start with a set of (),, functions in the computational domainf i xyz for the infinite series expansion () and then proceed systematically to determine the unknown coefficients ai t

24 2.5 Solving the Heat Transfer Partial Differential Equation

electrical domain

∆UR(∆T) I

R I 2

thermal domain

∆T C T + K T = F I 2R

Figure 2.6 Schematics of coupled electro-thermal simulation at system level for the resistively heated microsystems. Homogeneous heat generation is assumed. within the series, in order to solve the heat transfer equation. The separation of variables method gives:

∞ (),,, () (),, (2.15) Txyzt = ∑ ai t f i xyz i = 0

In general if the expansion functionsai andf i exactly satisfy the BCs for the problem, we refer to a solution as analytical. An alternative is to split the domain into many small pieces, that is to discretize it in the space. Then one can approximate the temperature field within an element with local shape functions and express the whole temperature field in a piece-wise fashion. In this case, we refer to a solution as numerical.

2.5.1 Linearization When material properties in the heat transfer equation are constant, it is called linear. In this case, there are many mathematical benefits, for example one can use the superposition principle. Computationally it is definitely much easier to solve a linear partial differential equation. Almost all analytical solutions require the

25 2 Dynamic Electro-Thermal Simulation of Microsystems heat transfer equation to be linear. For numerical methods this is not required but it is no doubt advantageous to do so if possible. In general, however, the material properties are temperature dependent and (2.5) has a form:

∂ ∇•()κ()T ∇T + Q – ρC ()T ()T = 0 (2.16) p ∂t

It is always possible to perform linearization around the operation point (temperature) in order to convert a non-linear heat transfer equation to a linear one. After obtaining a set of linear models around a chosen set of temperatures in this way, one can use a sort of weighting function, as done in [52] to e. g. extract a nonlinear compact thermal model.

In some special cases it is further possible to use certain transformations that lin- earize (2.16). Under the assumptions that only the thermal conductivityκ is temperature dependent and thatκ is not a function of space, the authors in [53] suggest using the Kirchoff transformation of the form:

T θ 1 κ() = T s + κ----- ∫ T dT (2.17) s T s

κ κ() withs = T s andT s is the heat sink temperature, which fully linearizes a static part of (2.16). By additionally defining a new time variable,τ :

t τ ()θ ks = ∫k dt (2.18) 0

κ ⁄ ρ where diffusitivityks = s C p , the time-dependent non-linear heat transfer equation (2.16) becomes fully linearized:

∂θ ∇2θ Q 1 + κ----- – -----∂τ = 0 (2.19) s ks

26 2.5 Solving the Heat Transfer Partial Differential Equation

2.5.2 Analytical Solutions Most published analytical solutions assume a simple geometry of the computational domain. Examples are the thin rectangular heat source or a volume heat source on the top of an infinite medium. The latter is often used for modeling a bipolar transistor (see [54] and the references there). Another example is a stacked layer structure that can be infinitely extended in length and with directions, used for modeling a power FET transistor [55]. The most general analytical solution so far is based on [53], in which a linearized heat conduction equation (2.19) is ana- lytically solved in the rectangular (multilayer) thermal domain with arbitrarily distributed volume heat sources. It is in principle possible to divide a region with arbitrary geometry into rectangular thermal subvolumes and to apply the solution from [53]. However, in this case the problem of coupling these subvolumes must be solved as well. Different analytical approaches use Green functions [56], [54], Fourier series [53], [57] or Fourier transform [55]. As a conclusion to this subsection we want to state that although the analytical solutions require lower computational effort and are preferred by microsystem designers in the above-mentioned special cases, it is unfortunately difficult to use them in general.

2.5.3 Numerical Methods There are several methods associated with a mesh, which partitions the arbitrary computational domain into smaller units. These are the finite difference method (FDM), the finite volume method (FVM), the finite element method (FEM) and the boundary element method (BEM). They semi-discretize the heat transfer partial differential equation (2.5) and transform it into a system of ordinary differential equations (2.14). An overview of these methods and a discussion of similarities and differences between them can be found in [1]. For the heat transfer equation (2.5), a transmission line matrix (TLM) approach can be used as well [58], [59]. This approach represents a physical model of heat flow as a sequence of voltage (temperature) pulses traveling through a matrix network of transmission lines. This method requires a rectangular mesh and a creation of an RC network.

27 2 Dynamic Electro-Thermal Simulation of Microsystems

Lastly, each spatial discretization can be transformed into an equivalent thermal RC network. In this subsection we will briefly discuss the resemblance of thermal and electrical circuits. We have mentioned the importance of creating a boundary conditions independent model. Let us briefly explain, how a typical commercial solver generates such a model. The first step is the “pure” discretization of the computational domain without having specified the heat generation rate or boundary conditions. It results in two system matrices, CBCI and KBCI, which are stored in the software database. In the general case, these matrices depend on temperature and are not formed explicitly but rather are stored as a list of element matrices. Both are sparse and CBCI is quite often lumped, which means that it is converted into a diagonal matrix. The dimension of CBCI and KBCI equals the number of the introduced finite element nodes. We refer to this matrices as BCI, because it is possible to apply boundary conditions to a model described with CBCI and KBCI without repeated discretization. After the Neumann and convection boundary conditions have been introduced, a system of ordinary differential equations is written as:

˙ CBCIT ++K BCITK∑ conT = ∑ f source + ∑ f Neumann + ∑ f con (2.20) where the sum contains all the heat sources and boundary conditions. The volume heat sources and the Neumann boundary conditions contribute only to the load vector, while the convection boundary conditions contribute to both the load vector and the heat conductivity matrix. It is possible to add the scaling factors to each heat source and boundary condition. Hence, the equation (2.20) can describe various external conditions without further need for any changes. The use of Dirichlet boundary conditions, however, changes (2.20) as follows:

˙ CT + KT = ∑ f source + ∑ f Neumann + ∑ f con + ∑ f Dirichlet (2.21) where the dimension of all vectors and matrices is reduced by the number of nodes to which the constant temperature is applied (Dirichlet nodes). (2.21) is obtained from (2.20) by crossing out columns and rows belonging to the Dirichlet nodes. C

28 2.5 Solving the Heat Transfer Partial Differential Equation

and K are obtained from CBCI and KBCI by crossing out those columns and rows which belong to the Dirichlet nodes. However, the use of Dirichlet boundary conditions makes (2.21) less flexible, as it is impossible to replace them with different boundary conditions when once applied. In (2.20), one can change BCs by simply using zero scaling factors. Luckily, one can always replace the Dirichlet boundary conditions with the convection BCs if a very high value for the film coefficient is used. Equations (2.20) and (2.21) are equivalent to (2.14) and can be directly plugged into the electric simulator as an equivalent RC-network. In order to highlight a resemblance of thermal and electrical circuits, let us observe a parallel RC circuit in Figure 2.7.

I1 I2 V1 V2

Figure 2.7 A parallel RC circuit.

It is described by:

11V˙ 1 11V I C – 1 + --- – 1 = 1 (2.22) 1 1 R 1 1 V I – V˙2 – 2 2

When comparing (2.22) with (2.14) it is easy to see that the heat capacity matrix can be released by capacitor elements, the heat conductivity matrix by resistor elements and the heat source vector (matrix) by current sources. The equivalent thermal networks were derived for different types of finite elements [60]. Figure 2.8 shows a conductive thermal network for a tetrahedral element with a convective boundary.

29 2 Dynamic Electro-Thermal Simulation of Microsystems

node 4

node 1 node 2 node 2

node 1 node 3 node 3

Figure 2.8 Thermal impendance network for a tetrahedral element for use away from the boundaries, from [60].

It should be noted that the transformation of (2.14) into an equivalent thermal RC network is exact, i. e. no approximation (in the sense of compacting the circuit) is made so far.

2.6 Dynamic Compact Thermal Modeling

As already mentioned, it is the dimension of (2.14) which makes the simulation time-consuming. For a variety of MEMS devices the order of the resulting thermal ODE system exceeds 100,000. Hence, it is prohibitive to use these models during system-level simulation. Instead, accurate dynamic compact thermal models (DCTM) are required. The methods for constructing DCTM can be divided into three general groups: non-automatic methods such as different RC ladder network approaches, semi-automatic methods such as modal approaches and model order reduction methods which can be made fully automatic. Through this section we will describe the main properties of the first two groups. Model order reduction methods will be the topic of the chapter 3.

2.6.1 RC Ladder Approach A large number of DCTM approaches are based on fitting an RC ladder network on the observed (measured or numerically computed) system response,

30 2.6 Dynamic Compact Thermal Modeling using a suitable optimization technique. In this case, however, the RC ladder network is based on an attempt to lump a distributed thermal domain. For 1D heat conduction, if we subdivide the domain into a large number of small slabs whose thicknesses L go to zero, we get a so-called Cauer ladder network (see Figure 2.9 left) with grounded capacitors and floating resistors. This can be transformed into a Foster network (Figure 2.9 right) via standard circuit transformation algorithms. While the Cauer network appears to be a small version of the large equivalent circuit described in the previous section (that is a fine mesh was replaced by a coarse one), the Foster network has no physical meaning. In both cases, the question is how to set a proper number of RC pairs or, in the case of 2D and 3D thermal conduction, how to choose a proper network structure at all.

R i Rj

C Ci j Cauer Foster Figure 2.9 Cauer RC ladder network (left) and Foster network (right).

In [62] the authors suggest a robust method based on computing a time-constant spectrum function connected to Foster network. A Foster network representation brings along an infinite number of time-constants (each RiCi pair represents a time τ ⋅ τ constanti = Ci Ri ) and so defines a continuous spectrum. The step response ()()⁄ τ of a single RiCi stage isRi 1 – exp –t i . Hence, the unit-step response of the Forster network can be constructed as a sum of these exponential terms:

() ()()⁄ τ Tt = ∑Ri 1 – exp –t i (2.23) i

31 2 Dynamic Electro-Thermal Simulation of Microsystems

After replacing Ri with a continuous spectrum, the above sum can be replaced by the integral over the wholeτ range:

∞ Tt()= ∫ R()ξ ()ξ1 – exp()–t ⁄ exp()ξ d (2.24) –∞ whereR()ξ is the time-constant spectrum function defined on thezt= ln() and ξτ= ln()logarithmic time axes. Lastly, the relationship betweenTt() and spectrum can be expressed by convolution as:

d Tz()= Rz()⊗ wz() (2.25) dz wherewz()= exp()zz– exp() . Hence, from the measured step response Tt() we can discretize to determine a suitable number of RC ladder elements in the Foster circuit required to represent a (preferably multilayer) system prior to ﬁtting. As the Foster network has no physical meaning, it has become an engineering practice to convert it back to Cauer form. This method also works for the 2D and 3D heat conduction, but is unfortunately applicable only to single conduction path, i. e., to single-input-single-output (SISO) problems.

For the multiple-input-multiple-output (MIMO) system, not only the number but also the structure of a compact RC ladder network must be specified. Recently, the authors in [45] have suggested the application of evolutionary algorithms for setting the correct topology of the compact model. Special challenge in dynamic compact thermal modeling, is to construct a boundary condition independent compact model, which would be reusable for different surroundings. This means that if e. g., a chip producer does not know the conditions under which the chip will be used, the compact thermal model must allow an engineer to research how the environmental changes inﬂuence the chip temperature. The two European projects, DELHPI and PROFIT have addressed the need to produce an accurate and boundary condition independent compact thermal models of a chip [63]-[67], which would simplify the chip cooling simulation of coupled thermal and ﬂuidic domain. The chip benchmarks representing

32 2.6 Dynamic Compact Thermal Modeling boundary condition independent requirements are described in [68]. Related discussions can be also found in references [69] and [70]. The goal of the PROFIT project was to extend the methodology to transient compact thermal models by using methods from [62] and [71]. The current solutions from both projects are mainly based on data ﬁtting for the apriori chosen resistor network.

In spite of the large number of related methods suggested in the last years (an extensive review can be found in [72]), the RC network extraction remains a non- automatic approach, which requires a designer to choose the correct number and position of the RC ladders without strict guidelines, and to perform a time-consuming parametrization.

2.6.2 Modal Approaches

It is well-known from structural mechanics that an elastic string vibration under the action of arbitrary external force, can be viewed as a linear combination of different vibrating modes, each one corresponding to a resonance of the string. Although their number is infinite, the actual response can be approximated quite well by considering only the few first harmonics. This is how the elastic string, which is a distributed system can be compacted into a lumped system with only a few degrees of freedom. In terms of the partial differential equation governing a string vibration this means that one has to compute the eigenvalues, which are the vibration frequencies, and the eigenfunctions, which are the spatial forms of vibrations for each mode. Out of the first few eigenfunctions a compact model can be constructed In an analog manner, the temperature can be expressed as series expansion around a set of “dominant” eigenfunctionsU j :

k (), ≅ ()⋅ () Trt ∑ U j r V j t (2.26) j = 1 whereV j are jet unknown expansion coefficients. By substituting (2.26) into (2.5), multiplying it byV i and integrating, it is possible to get a reduced model [72] of the form:

33 2 Dynamic Electro-Thermal Simulation of Microsystems

* * C dV() t ⁄ ()dt + K Vt()= st() (2.27)

* ρ * ∇()κ∇ whereCij = ∫ C pU iU jdr ,Kij = ∫U i U j dr ,si = ∫U iQrd , ij, ∈ []1, k . Unfortunately, the analytical solution of the eigenvalues problem for heat conduction partial differential equation (2.5) is available only for some simple geometries [73]. Hence, a first approximation is to perform spatial discretization and to compute the eigenvalues and corresponding eigenvectors of the system matrix of (2.14). To do this, we need to reduce (2.14) to a single matrix representation as:

AT˙ = TBut+ () (2.28)

–1 –1 whereAK= – C andBK= – FQ . The number of eigenvectors of A equals the number n of finite element nodes. It is possible to reduce the system (2.28) by × ∈ nr projection, where a projection matrixV modal C is composed of r eigenvectors of A.Nowthen-dimensional equation (2.28) can be projected onto r- dimensional subspace as follows:

T ⋅⋅ ⋅ T ⋅⋅T ⋅ () V modal AVmodal z˙ = V modal V modal zV+ modal Bu t (2.29) where z is a generalized variable vector. The question which remains is how to choose the “dominant” modes (eigenvectors). For the defined thermal output T yt()= E ⋅ T (E is either a vector or matrix), one can observe the transfer func- T –1 tionGs()= E ()sI– A B , and choose those poles (and associated eigenvalues) which are around the region of frequencies of interest. Nevertheless, the choice of important modes still requires the designers action, which makes a modal approach manual.

2.6.3 Model Order Reduction The only group of DCTM methods which can be made fully automatic, i. e. only with the minimal intervention by the designer, are the mathematical model order reduction (MOR) methods. Hence, among microelectronic and MEMS designers they are becoming increasingly popular [74]-[80].

34 2.6 Dynamic Compact Thermal Modeling

The ideas of mathematical model order reduction have been developed in the control theory and are applicable to first order linear ODE systems, such as (2.14). The development of model reduction of nonlinear systems is still in its early stages [81], [37]. According to the control theory, (2.14) should be written in the right- hand side state-space formulation as:

T˙ = AT+ Bu() t (2.30) T yt()= E ⋅ T where the system is treated as a black box, i. e. the internal state vector of temper- n aturesTR∈ is not directly accessible to an external observer. The controller can influence the system state through the input functions specified by the vector m ut()∈ R and distributed to the internal nodes in accordance to the input matrix nm× BR∈ . As the number of inputs is typically smallmn« , the matrixB has a small number of columns. Furthermore, the observer is interested in only a few p outputs, specified by the vectoryR∈ . The required outputs are selected from np× the complete state vector via output matrixER∈ . Hence, a high-dimensional ODE system, governed by a small number of external inputs, has to be solved in order to determine a small number of relevant outputs. Let us also mention at this place the all-important transfer function of the system (2.30), defined as:

T –1 Gs()= E ()sI– A B (2.31) where s is the laplace variable and I is a unity matrix of the dimension n. Although Gs()is a relatively small matrix with p rows and m columns, its computation requires the inverse of a large-scale system matrix A. While searching for a possibility to reduce the number of state variables, i. e., equations in (2.30), let us transform the state vector T using a transformation × ∈ nn matrixV n R as follows: ⋅ TV= n z (2.32)

35 2 Dynamic Electro-Thermal Simulation of Microsystems

Hereby z is the new state vector in terms of generalized coordinates expressed by the transformation matrix. It is important to understand that if T has the spatial and physical meaning as a vector of temperatures belonging to FE nodes, z has none of either. Rather, it is a vector of multiplication factors for the “global shape functions” given by the columns of matrix Vn. To make this more clear, let us take a closer look at discretization methods where the temperature over the whole domain is approximated as a piece-wise linear combination over the elements:

i = 1, n i = 1, n () 1 1() … e e() Tr = ∑ ci N i r + + ∑ ci N i r (2.33) Ω Ω 1 e where n is the number of free coefficientsci in each element (these can be the nodes temperatures for example) and e is the total number of elements. A property of each local shape function Ni is that it is one at the ith node and zero outside the finite element. Within the element it can linearly decay between one and zero. The only way to directly reduce a system based on (2.33) is to coarsen the mesh, i. e. to chose the shape functions which cover several elements. This, however, results in a significant lost of precision (see Figure 2.10 top).

On the other hand, each column of matrixV n in (2.32) can be seen as a linear combination of the local shape functions, i.e, as a global shape function over the whole heat transfer domain. This gives hope that it may be possible to truncate some of the generalized coordinates z and therefore reduce a dimension of the system (2.32) without losing much accuracy (see Figure 2.10 bottom). Both ways of compressing information, which are shown in Figure 2.10 have their advantages and disadvantages. By coarsening the mesh, one preserves the physical nodes, but looses the precision. By performing a mathematical model order reduction, one looses the physical nodes, but preserves high accuracy (as will be explained below). Note that those two methods do not exclude each other, but can rather be combined in order to maximally increase the efﬁciency.

Let us emphasize that after the transformation, equation (2.30) changes to:

36 2.6 Dynamic Compact Thermal Modeling

Original picture Order 6 approximation

Order 12 approximation Order 20 approximation

Figure 2.10 Order reduction via coarsening the mesh (top, from [82]) and via truncating the generalized coordinates (bottom, from [83]).

T T z˙ = W AV ⋅ z + W Bu() t n n n (2.34) () T ⋅ yt = E V n z

× , ∈ nn T ⋅ whereW n V n R are biorthigonal, which means thatW n V n = I . For the external observer (2.34) behaves exactly the same way as (2.30), i. e. one can easily prove that the transfer function (2.31) does not change. As already mentioned, model reduction is based on the idea that one can find such a transformation when one can accurately represent the state vector with just a few generalized coordinates. In other words, a transformation exists when one can truncate most of the generalized coordinates, that is to approximate:

37 2 Dynamic Electro-Thermal Simulation of Microsystems

z T ⋅ 1 = V r V nr– (2.35) z2 through

⋅ ε TV= r z1 + (2.36) with the error vectorε being small. This, on the other hand, changes (2.34) to a low dimensional system:

˙ () z1 = Arz1 + Brut (2.37) () T ⋅ yr t = Er z1

T T T T ⋅ withAr = W r AVr ,Br = W r B ,Er = V r E andW r V r = I . The number of inputs and outputs in (2.37) is the same as in (2.30), whereas the number of equations (dimension of the state vector) is smaller. This transformation is schematically shown in Figure 2.11.

Before: = T . . E

=+AB.

After: = T . =+.. Er Ar Br

User Input System Output Figure 2.11 Schematics of the system before and after model reduction step, from [85].

38 2.7 Conclusion

The equation (2.36) can be also seen as a projection of a n-dimensional state vector to a r-dimensional subspacern« , defined by the columns of matrix Vr (global shape functions). (2.37) is a projection of the whole system (2.30). The output of the reduced system is, however, not the same as that of (2.30) or (2.34), since we have introduced the truncation errorε . The goal of model order reduc- () () tion is to minimize this error either in the time domainmin yt – yr t or in the () () Laplace domainmin Gs – Gr s , where the transfer function of the reduced system is defined as:

() T ()–1 Gr s = Er sI– Ar Br (2.38)

There are several model reduction methods for linear ODE systems which produce smallε . They take the system matrices A and B of the linear system as input and perform linear algebra manipulations with them in a different manner to con- structW r andV r . Let us emphasize that, unlike the non-automatic RC-ladder approach, there is no explicit minimization procedure forε and in a way, one obtains the best topology of a reduced system simultaneously with its system matrices. These methods and their most important properties are discussed in chapter 3. Finally, let us get back to the physical sense of the internal nodes in the reduced models. The goal is to approximate the original system, which means, the smaller error, the more physical sense. If we think of RC parameters as the amplitudes for global functions, then any network, including Foster RC-network, has a valid physical sense for thermal modeling.

2.7 Conclusion

In this chapter we have highlighted the importance of heat transfer modeling, which is presently a central part of the electro-thermal simulation of microsystems. We have discussed in which cases it is possible to decouple the heat transfer PDE from other physical domains, in order to be able to solve it separately. Unfor- tunately, analytical solutions are only available for simple geometries, whereas either approximations or numerical methods must be used for complex geometries.

39 2 Dynamic Electro-Thermal Simulation of Microsystems

However, the numerical solution of the heat transfer PDE via e. g. finite elements is often impractical or even prohibitive if we want to simulate the whole system with a large number of interconnected devices. Again, the number of resulting ordinary differential equations (ODEs) for a single device easily exceeds 100,000. Even by using a domain decomposition technique on parallel computers, this huge number of unknowns demands large resources of CPU-time and memory. Hence, a reduction of the number of unknowns to a lower-dimensional system, known as dynamic compact thermal modeling has become a standard for microsystem simulation.

electro-thermal simulation of microsystems central part heat transfer PDE (Boltzmann, hyperbolic, parabolic) solution

analytical solution numerical solution (only simple geometries) (ODE via discretization)

mea sureme nts simulation results DCTM

RC parameter coarsening modal mathematical optimization the mesh approach MOR

Figure 2.12 Schematics of discussed and recommended (bold) research paths in electro-thermal modeling of microsystems.

We will emphasize compact thermal modeling via model order reduction, which follows the spatial discretization, because it presently offers an accurate and effective solution for electro-thermal modeling of microsystems and enables

40 2.7 Conclusion an automatic system-level modeling. Available model order reduction methods and their application to compact electro-thermal modeling of MEMS devices will be the topic of chapters 3-7. A schematics of the discussed and recommended researched paths described in this chapter is shown in Figure 2.12. To this end, we have presented the conventional, non automatic approaches (RC-ladder network), semi automatic approaches (modal approximation), and the increasingly popular model order reduction approaches, which can be made fully automatic. However, they all consider the DCTM of a single device only. Since microelectronic and MEMS are usually composed of subsystems that are interconnect to array structures for example, it is desirable, especially for a large number of subsystems, to extract a heat-transfer macromodel of each subsystem on its own and then to couple them back together. This is going to be the topic of chapter 7.

41 2 Dynamic Electro-Thermal Simulation of Microsystems

42 3.1 Control Theory Methods

3LINEAR MODEL ORDER REDUCTION METHODS

In the previous chapter we highlighted the importance of model order reduction techniques for dynamic compact thermal modeling of microsystems. The goal of this chapter is to describe a number of relevant MOR methods for linear systems. Let us emphasize that for MEMS case studies in this thesis, we assume material κ thermal properties andC p to be temperature independent, which means that spartial discretization results in a linear ODE system of the form (2.14).

3.1 Control Theory Methods

We start from the stable linear dynamic state space system of the form:

x˙()t = Ax() t + Bu() t (3.1) T yt()= E ⋅ xt()

n m Heretxt is the time variable,()∈ R is a state vector,ut()∈ R the input exci- p nn× tation vector, andyt()∈ R the output measurement vector.AR∈ is the nm× np× system matrix,BR∈ andER∈ are input and output distribution arrays, respectively.nm is the state space dimension and andp are the number of inputs and outputs, respectively. As already mentioned in the previous chapter, in most practical cases we can assume thatm andp are much smaller thann . Control theory offers a number of well-established tools for the automatic model reduction of stable linear systems [86]. Before describing different methods, let us explain two important characteristics of the system (3.1) called controllability and observability. A system is said to be controllable if for any initial state () () xt0 = x0 and any final statex1 , there exist an inputut that transfersx0 to

43 3 Linear Model Order Reduction Methods

x1 in a finite time. It is observable if it is possible to calculate any unknown initial () () statext0 = x0 uniquely from the given inputut and the measured output yt()over a finite time interval. In the following we will assume the system (3.1) to be a so called minimal realization, i. e. there are no states which are neither controllable nor observable at all. Note that the system resulting from spatial discretization of computational domain (like the MEMS case studies in this thesis) are always minimal realizations. In terms of control theory, controllability and observability are described through the grammians,PQ and , defined as:

∞ At T AT t Pe= ∫ BB e dt 0 (3.2) ∞ AT t T At Qe= ∫ E Ee dt 0

It can be shown [89] that the criteria for controllability and observability are that P and Q are both regular (i. e., have full rank). If, in addition, all eigenvalues of A have negative real parts (i. e., the system is stable), than the grammians can be computed as the unique and positive definite (symmetric reel matrixA is positive T definite ifx Ax > 0, ∀x ≠ 0 ) solutions of two Ljapunov equations belonging to the system (3.1):

T T AP+ PA = –BB (3.3) T T A QQA+ = –E E

Numerically stable and direct methods to solve Ljapunov equations and find the controllability and observability grammians can be found in [87].

3.1.1 Balanced Truncation Approximation The idea behind model reduction via balanced truncation is to obtain a reduced- order model by finding and deleting those states that are simultaneously least controllable and observable. B. C. Moore [88] was able to suggest such a similarity

44 3.1 Control Theory Methods transformation, which balances the system, i. e. sets both grammians equal and diagonal:

ˆ ˆ ()σ ,,…σ P ==Q diag 1 n (3.4)

σ ≥≥≥σ …σ with the diagonal entries in descending order1 2 n . The parameters σ i are called Hankel singular values (HSV) and are transformation invariant. They are property of the system and depend only on input-output behavior. λ Hankel singular values can be computed as the square roots of eigenvaluesi of the product ofPQ and :

σ λ ()⋅ , ,,… i ==i PQ i 1 n (3.5)

It can be shown that the Hankel singular values reflect the contributions of different entries of the state vector to system responses [89], [84]. Hence, to reduce the order of the balanced system (A˜ ,,B˜ E˜ ), it is sufficient to remove (truncate) those state variables (and corresponding blocks inA˜ , B˜ andE˜ ) related to the smallest Hankel singular values. Balancing and truncation can be done simultaneously by applying a projection to the original non-balanced system, using e. g. square root algorithm (Algorithm 3.1) [87], [90].

45 3 Linear Model Order Reduction Methods

Algorithm 3.1 The square root algorithm for calculating BTA of order r

Inputs: ABE,,

˜ ,,˜ ˜ Outputs: A B E' the matrices of the reduced, internally balanced system of order r, Wr r r r ˜ T T and V - mutually orthogonal projection matrices such thatA = W AV ,B˜ = W B and r T r r r r r ˜ V E Er = r 1. Compute the controllability grammian P and the observability grammian Q by solving the Ljapunov equations. 2. Find the Cholesky factorization of the solutions P and Q:

T T QLL P = LcLc , = o o T 3. Calculate the singular value decomposition of the matrixLo Lc : T L U ΣU Lo c = 1 2 nn× whereΣ ∈ R is a diagonal matrix of Hankel singular values in descending order. × × ∈ nr ∈ nr 4. Form the projection matricesW r R andV r R as: 1 1 –--- –--- 2 2 Σ W L U Σ V r = LcU 2r r and r = o 1r r whereU 1r andU 2r are the first r columns of matricesU 1 andU 2 , and 1 –--- 2 ⎛⎞1 1 Σ = diag⎜⎟------,,… ------r ⎝⎠σ σ 1 r

5. Apply the projection to the system (3.1) to find the order r truncated balanced realization as:

T T T ˜ W AV ˜ W B ˜ V E Ar = r r ,,Br = r Er = r

The controllability and observability grammians of the orderr reduced system ˜ ,,˜ ˜ ˜ ˜ ()σ ,,…σ (Ar Br Er ) are diagonal and equal,Pr ==Qr diag 1 r . Some alternatives to algorithm 3.1 can be found in [87].

46 3.1 Control Theory Methods

The most important characteristic of algorithm 3.1 is that it provides a global error bound between the transfer functions of the original and the reduced systems BTA() G(s) andGr s :

() BTA() ≤ ()σ …σ Gs – Gr s ∞ 2 r + 1 ++n (3.6) where the infinity norm. ∞ denotes the largest magnitude of the difference of BTA() transfer functions and G(s) andGr s are defined as in (2.31) and (2.38), respectively. The proof of (3.6) can be found in [84]. The discussion of different error norms is given in [86].

3.1.2 Singular Perturbation Approximation

The fact that balanced truncation approximation generally incurs an approximation error in the low-frequency region is undesirable in some practical applications. Hence, an algorithm which produces zero error at zero frequency, called singular perturbation technique, is obtained as follows. After decomposing the system matrices and the state vector of (3.1) as:

A11 A12 B1 x1 A = ,B = ,E = E1 E2 ,x = (3.7) A21 A22 B2 x2

× ∈ rr withA11 R , and assuming that A22 is invertible, the reduced system is defined by following truncation as [87]:

–1 –1 z˙ = ()A – A A A zB+ ()– A A B u 11 12 22 21 1 12 22 2 (3.8) ()–1 –1 yr = C1 – C2A22 A21 zC– 2A22 B2u

An important property of (3.8) is that the steady state gain matches that of the original system, that is:

SPA() () Gr 0 = G 0 (3.9)

47 3 Linear Model Order Reduction Methods

It turns out that if we first balance the system, and then truncate it as in (3.8), the reduced system fulfills the error bound (3.6) (see [87] for details):

() SPA() ≤ ()σ …σ Gs – Gr s ∞ 2 r + 1 ++n (3.10)

3.1.3 Hankel Norm Approximation () The Hankel norm. H ofGs is defined as the maximal Hankel singular value of the system (3.1):

() λ ()σ⋅ Gs H ==max PQ max (3.11)

The optimal Hankel norm approximation problem is the problem of finding an HNA() < approximationGr s of degreern such that the Hankel norm of the error () HNA() Gs – Gr s H is minimized. The lower bound for the above norm is given through [87]:

σ ≤ () HNA() r + 1 Gs – Gr s H (3.12)

() (3.12) holds for anyGr s with exactlyr stable poles. Since the infinity norm is never smaller than the Hankel norm [84], (3.12) also means:

σ ≤ () HNA() r + 1 Gs – Gr s ∞ (3.13)

An algorithm for the construction of an optimal Hankel norm approximation can HNA() be found in [87]. An optimal Hankel norm approximationGr s of order r fulfills the following upper bound:

HNA () ()∞ ≤ ()σ …σ Gs – G r s r + 1 ++n (3.14) which is half the bound for the balanced truncation case.

48 3.1 Control Theory Methods

3.1.4 Comparison of Methods Hankel norm approximation is considered by mathematicians to be the optimal method in terms of additionally specifying the lower bound (3.13) for the infinity error norm [83]. Nevertheless, the most frequently used method is balanced truncation approximation, due to the simplicity of implementation and the short computational time. The comparison of computational times for all three methods is given in [92]. It can be shown that all presented methods preserve the stability and passivity of the original system within a reduced system. In Table 3.1 we summa- rize their advantages and disadvantages. Table 3.1 Advantages and disadvantages of control theory methods

Advantages Disadvantages simple implementation, short does not approximate steady BTA computational times. state

SPA approximates steady state extra computational time

speciﬁes additionally a lower complex implementation and HNA error bound for the transfer no approximation of steady function. state

Apart from the three described methods, the control theory deduces a number of other reduction algorithms which are suitable for linear systems. However, none of them offers an error estimate. A good review of available methods can be found in [89] or [93].

49 3 Linear Model Order Reduction Methods

3.2 Krylov Subspace Methods

Similar to the control theory algorithms, the Krylov subspace techniques require a single matrix representation of a first order ODE system. In this case, however, it is more convenient to choose a left-hand side representation of the form:

Ax˙()t = xt()+ bu() t (3.15) T yt()= e ⋅ xt() with the same syntax as in equation (3.1). Note that in this section we consider only the Single-Input-Single-Output (SISO) case, i. e.mp==1 (for SISO we use the lower case letters b and e for the input and output arrays), which does not effect the generality of the methods. The SISO transfer functionGs() is a scalar valued rational function defined as:

T –1 Gs()= –e ()IsA– b (3.16)

() The Taylor series expansion ofGs abouts0 = 0 is given by:

∞ () T ()2 2 … i Gs ==–e IsAs++A + bm∑ is (3.17) i = 0

T i ,,,… wheremi = –e A b fori = 012 , are called moments abouts0 . Krylov subspace methods for model order reduction correspond to choosing () () Gr s as a Padé or Padé-type approximant ofGs . r-th Padé approximant of the () transfer functionGs about the expansion points0 , is a rational function:

() r – 1 … Pr 1 s ar 1s ++a1sa +0 G ()s ==------– ------– - (3.18) r () r r – 1 Qr s … brs ++++br – 1s b1s 1

50 3.2 Krylov Subspace Methods

() whose Taylor series arounds0 agrees with the Taylor series ofGs in the first 2r terms (moments), i. e. fors0 = 0 :

() () ()2r Gs = Gr s + Os (3.19) where

2r limOs()= 0 (3.20) → ss0

() We will also consider reduced order modeling based on functionsGr s for () which less than 2r moments match. In this caseGr s is called a Padé-type approximant. We refer the reader to [94] for an overview of Padé and Padé-type approximants. The equation (3.19) presents 2r conditions for the 2r degrees of freedom (coef- () ficients) ofGr s . Note that coefficient b0 in (3.18) is set to one, which eliminates () an arbitrary multiplicative factor inGr s . The coefficientsai andbi of polyno- () () () mialsPr – 1 s andQr s can be computed as follows. MultiplyingQr s on both sides of (3.19) yields:

() () () ()2r () (3.21) GsQr s = Pr – 1 s + Os Qr s

() If we developGs until the momentm2r , we get:

()… r … 2r ()r r – 1 … m0 +++++m1s mrs m2rs brs ++++br – 1s b1s 1 =

… r – 1 ()2r ()r r – 1 … = a0 +++a1s ar – 1s +Os brs ++++br – 1s b1s 1 which further yields:

51 3 Linear Model Order Reduction Methods

()r r – 1 … m0 brs ++++br – 1s b1s 1 + ()…r r – 1 … + m1sbrs ++++br – 1s b1s 1 + r – 1()r r – 1 … (3.22) + mr – 1s brs ++++br – 1s b1s 1 + r()…r r – 1 … + mrs brs ++++br – 1s b1s 1 + 2r()r r – 1 … + m2rs brs ++++br – 1s b1s 1 =

r – 1 … ()2r ()r r – 1 … = ar – 1s ++a1sa ++0 Os brs ++++br – 1s b1s 1 i By matching the coefficients ofs terms of (3.22) forir= , …, 2r – 1 , we get the following equation system for the coefficientsbi :

… m0 m1 mr – 1 br mr m m … m b m 1 2 r r – 1 = – r + 1 (3.23) ......

mr – 1 mr m2r – 2 b1 m2r – 1

i The coefficientsai can be determined by matching the coefficients ofs terms of (3.22) fori = 0, …, r – 1 as:

a0 = m0 a = m b + m 1 0 1 1 (3.24) . . … ar – 1 = m0br – 1 ++++m1br – 2 mr – 2b1 mr – 1

However, computing Padé approximants using explicit moment computations, as is done in Asymptotic Waveform Evaluation algorithm [95], is numerically unstable [91]. It turns out that the numerically stable way for moments evaluation

52 3.2 Krylov Subspace Methods leads over the so-called Krylov subspaces, defined as follows. Forr = 12,,… , the subspace:

R{}, {},,,… r – 1 K r Ab= span b Ab A b (3.25)

n ofR is called the r-th right Krylov subspace, induced by A and b, and the subspace:

L T ⎧⎫T T r – 1 K {}A , e = span⎨⎬ e,,, A e … ()A e (3.26) r ⎩⎭

n ofR is called the r-th left Krylov subspace, induced by A and b. Note that moments can be computed from (3.25) and (3.26) by computing the inner products:

i T i T ⎛⎞()T ⋅ ()i ⎛⎞()T ⋅ ()i + 1 m2i = –⎝⎠A e A b andm2i + 1 = –⎝⎠A e A b (3.27) fori = 01,,… ,r – 1 . Unfortunately, the vectors from (3.25) and (3.26) quickly become almost linearly dependant [91] and there is a rapid accumulation of rounding errors. The solution is to construct more stable basis vectors v1, v2, ..., vr and w 1 , w 2 , ..., w r saved in columns of matrices W and V,suchthat R{}, {},,… L{}T, {},,… K r Ab= span v1 vr andK r A e = span w1 wr .

There are two main approaches for constructing basis matrices Wr and Vr for Krylov subspaces the Lanczos algorithm and the Arnoldi process.

3.2.1 Lanczos Algorithm nn× n Given a matrixAR∈ , a right starting vectorbR∈ and a left starting n vectoreR∈ , the classical Lanczos reduction algorithm (process) generates a × , ∈ nr pair of biorthogonal basisW r V r R for subspaces (3.25) and (3.26), where

T ≠ wi v j = 0 , for allij (3.28)

53 3 Linear Model Order Reduction Methods andv = bb⁄ andw = ee⁄ and. is the 2 - norm. It generates further a 1 1 × ∈ rr tridiagonal matrixT r R , which is related to the original system matrix A as:

T W r AVr = T r (3.29) and can be considered as an oblique projection of A onto the subspace (3.25) while remaining perpendicular to subspace (3.26). Figure 3.1 illustrates the orthogonal and oblique projection of a vector.

L K r x x

x x R 0 x K r R K r

Figure 3.1 Example of an orthogonal and an oblique projection of a vector, from [85].

Note thatT r equals the system matrix of the reduced systemAr . An algorithm for the basic Lanczos process can be found in [91] or [96]. If the Lanczos algorithm is carried to the end with n being the last step, it can be viewed as a means of trid- iagonalizing A by a similarity transformation:

–1 V n AVn = T n (3.30)

This allows one to write a transfer function (3.16) as:

54 3.2 Krylov Subspace Methods

–1 () T ()–1 Gs ==–e IsV– nT nV n b T ()–1 T (3.31) ==–e V n IsT– n W n b –1 T T ⋅ ()⋅ = –e bi1 IsT– n i1

n where i1 is a first standard unit vector inR . Note that (3.31) is due to (3.28) and ⁄ ⁄ () the fact thatv1 = bb andw1 = ee . The moments ofGs are given by:

T i T i T ⋅ ⋅ mi ==–e A be– bi1 T n i1 (3.32)

() The reduced order transfer functionGr s is defined by:

() T ()–1 Gr s ==–er IsA– r br T ()–1 T (3.33) ==–e V r IsT– r W r b –1 T T ⋅ ()⋅ = –e bi1 IsT– r i1

× whereT r is therr leading principal submatrix ofT n andi1 is a first standard r () unit vector inR . The moments ofGr s are given by:

T i T T i m ==–e A b –e bi ⋅ T ⋅ i (3.34) ri r r r 1 r 1 i i T ⋅ ⋅ T ⋅ ⋅ ≤≤ Due toi1 T n i1 = i1 T r i1 for0 i 2r – 1 (this can be proved by mathematical induction [96]), we immediately see that the first 2r moments ofGs() and () () () Gr s match, i. e.Gr s is a Padé approximant ofGs . () The main advantage of the Lanczos algorithm is thatGr s is uniquely specified due to matching a maximal number of moments, 2r. This means that we can apply the projection (usingV r andW r as the basis for (3.25) and (3.26)) directly to the linear system with two matrices (2.14) or to the right-hand side representation (3.1) and get the same reduced system. Furthermore, as the basic Lanczos algo-

55 3 Linear Model Order Reduction Methods rithm is based on simple three-term recurrences (see [91] or [96]), it is reasonably fast for large r. However, it should be noted that in practical implementations, the T ≈ basic Lanczos algorithm stops prematurely, due towi vi 0 . This problem can be solved by using a much more complicated version of the Lanczos process, as suggested in [97]. Another disadvantage of the Padé approximants is that the generated reduced models may not be passive and stable in general, even though the original linear dynamic system was passive and stable [98]. Suggestions for improving the preservation of stability and passivity can be found in [99].

3.2.2 Arnoldi Process ,,,… The Arnoldi process produces only one sequence of the vectorsv1 v2 vr , which span the right Krylov subspace (3.25) and are othonormal:

⎧ T 1if, ij= ≤ , ≤ vi v j = ⎨ for all1 ij r (3.35) ⎩ 0if, ij≠

∈ rr× It further generates an upper Hessenberg matrixH r R (with , ∀(), , > hij = 0 ij ij– 1 ), which is related to the system matrix A as follows:

T ⋅⋅ V AV= H r (3.36)

The matrixH r can be considered as an orthogonal projection of the matrix A onto the Krylov-subspace (3.25) (see Figure 3.1 left) and it is equal to the system matrix of the reduced systemAr . Algorithm 3.2 (see [91]) presents a complete statement of the Arnoldi process for the SISO setup.

56 3.2 Krylov Subspace Methods

Algorithm 3.2 Arnoldi process nn× n Inputs: MatrixAR∈ and the starting vectorbR∈ and order r of the reduced system nr× Outputs: Projection matrixV ∈ R and the (upper Hassenberg) system matrix of the T r reduced system A V AV r = r r

v 1. Setˆ1 = b . ,,… Fori = 1 r do (build j-th Arnoldi vector vj): ˆ 2. Computehii, – 1 = vi .

Ifhii, – 1 = 0 , then stop. R{}, (The Krylov subspaceK r Ab is exhausted). ˆ ⁄ 3. Setvi = vi hii, – 1 .

4. Setvi + 1 = Avi . 5. Forj = 1,,… i do: T Seth = v vˆ andvˆ = vˆ – v ⋅ h . ji, j i + 1 i + 1 i + 1 j ji, End_For. End_For. … (),, , … V r = v1 vr ,.Ar ==H r hij, ij =1 r

System (3.15) is now reduced by projection:

T T T Ar = V r AVr , br = V r b ,er = V r e (3.37)

In order to demonstrate the moment matching property of Arnoldi process, we first emphasize its following properties:

T AVr = V rH r + hr + 1, rvr + 1ir and (3.38)

k k ,,… A bbV= rH r i1 , fork = 0 r – 1 (3.39)

57 3 Linear Model Order Reduction Methods

r wherei1 andir are the first and the r-th standard unit vectors inR . (3.38) is obvious from algorithm 3.2.

For property (3.39) we note that since the entries ofH r below its lower 2nd diagonal are all zeros, it can be proven by mathematical induction (we will not supply k () the evidence here) that the entries ofH r below its lowerk + 1 th diagonal are also all zeros. Hence,

k T ⋅ ⋅ ,,… ir H r i1 = 0 fork = 0 r – 1 (3.40)

⁄ Due to (3.38) and the fact thatv1 = bb , which impliesbbV= ri1 , we further get:

⋅ T Ab== A b Vri1 bVrH ri1 + bhr + 1, rvr + 1ir i1 =bVrH ri1 (3.41)

From (3.38) and (3.40) (evidence again by mathematical induction), it follows (3.39). Now, the moments ofGs() are defined as:

T k T k mk ==–e A be– bVrH r i1 =

T k fork = 0,,… r – 1 (3.42) –e V r H r bi1 ==⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ mrk T k –er Ar br

Hence, in this case only r moments are matched and

() T ()–1 T ()–1 Gr s ==–er IsA– r br –e V r IsH– r bi1 (3.43) is called a Padé-type approximant ofGs() . Consequently, the Arnoldi process is not invariant to system representation, i. e. if we apply the projection usingV r as the basis for (3.25)) to (3.1) or (2.14) instead of (3.15), we will get different reduced order systems. Another disadvantage of the Arnoldi method is that each new Arnoldi vector should be orthogonal to all previously generated vectors. This

58 3.2 Krylov Subspace Methods means that the computational effort grows disproportionately to the dimension of the subspace. However, it can be shown (see [98]) that if we apply the Arnoldi- based projection to the first order stable and passive ODE system (2.14), with symmetric and positive definite system matrices C and K, the reduced order model will also be stable and passive. So, in simple terms, the Arnoldi process trades some optimality (in the sense of matching as many moments as possible) to gain guaranteed stability and passivity.

3.2.3 Arnoldi versus Lanczos In Table 3.2 the most important properties of both reduction algorithms are summarized. Table 3.2 Comparison of Arnoldi and Lanczos.

Arnoldi Lanczos Accuracy of approximation r moments match 2r moments match ()2 () ()() Computational complexity O 2r n + 2rNz A O 16rn + 4rNz A Invariance properties no yes Numerical stability yes no Preservation of stability and pas- yes no sivity Complete output approximation yes no

Accuracy of approximation: We have seen that the Lanczos algorithm produces a reduced system closer to the original one, because the number of moments matched here is twice that of the Arnoldi process. This has a simple explanation. Model reduction by the Arnoldi process does not take into account output vector e at all, while model reduction by means of the Lanczos algorithm is made by an oblique projection on the right Krylov subspace (3.25) that takes into account the left Krylov subspace (3.26). It is possible to increase the number of matched moments by Arnoldi to 2r by modifying it to so-called 2-sided-Arnoldi [100], [101], which takes into account the output vector. However, the algorithm becomes more complex and loses its ability to preserve the stability and passivity of the original system.

59 3 Linear Model Order Reduction Methods

Computational complexity: One disadvantage of the Arnoldi method is that each new Arnoldi vector should be orthogonal to all previously generated vectors. This means that the computational efforts for orthogonalisation over the r steps of algorithm grow as O(2r2n) with the dimensions n of the full space and r of the Krylov-subspace. Additionally r steps of the Arnoldi procedure require r matrix- vector products at the cost of 2rNz(A), where Nz(A) is a number of non-zero elements of A (Nz(A)=n2 for a dense matrix). Thus, on average the computational efforts for the Arnoldi algorithm grow as O(2r2n + 2rNz(A)). The Lanczos algorithm requires less computational efforts for orthogonalisation. In each step it is necessary to deal with just two previously generated vectors (matrixT r is tridiagonal), which makes the orthogonalisation costs over the r steps of the algorithm grow only as O(16rn). Together with matrix-vector products, the computational efforts of the Lanczos algorithm are of O(16rn+4rNz(A)). Hence, for large r it is faster than Arnoldi. Invariance properties: Changing the representation of the original system does not change the input-output behavior of the reduced order models generated by the Lanczos algorithm. The Arnoldi process is not invariant to system representation due to matching only r moments, which is half of the number of ()} unknowns inGr s . Numerical stability: Because of rounding errors in both algorithms, the column vectors of basis V and W for the Krylov-subspaces (3.25) and (3.26) may become non-orthogonal. How quickly this happens depends on the chosen dimension r of the Krylov-subspaces. Since in the Arnoldi process each new vector should be orthogonal to all previously generated vectors, the rounding errors accu- mulate slower than by the Lanczos algorithm, where each new vector is orthogo- nalized only with respect to the last two generated vectors. Preservation of the stability and passivity of the original system: The original dynamic system can be stable, that is, when time goes to infinity the values of x remain finite (bounded) and passive. Such system does not generate energy. If so, then it is important that the reduced system also possesses these properties. Unfortunately, neither the Arnoldi nor the Lanczos algorithms in their basic form guarantee this and an extra effort is needed for preserving the properties of the original dynamic system. As the Arnoldi process is mathematically simpler than

60 3.2 Krylov Subspace Methods the Lanczos algorithm (this is emphasized by their names: process and algorithm), in engineering applications it is more frequently used. The coordinate transformed Arnoldi [102] guarantees stable model reduction and the “block Arnoldi plus con- gruent transform” (PRIMA) from [103] guarantees passivity by transforming the system matrices so that they are positive semi-definite. Approximation of the complete output: In general the Lanczos algorithm produces reduced-order models which are “optimized“ for particular output(s). This is because it takes into account the output vector c by using a basis W of the output Krylov-subspace (3.26). The Arnoldi process, on the other hand, does not take into account the output vector at all, which enables the approximation of the all outputs. In other words, it works even if the output matrix is a unity matrix, i. e.EI= nn× in (3.1) or (3.15). From a control theory viewpoint, we term such system setup a Single-Input-Complete-Output (SICO). This aspect will be numerically demonstrated in chapter 5. Nevertheless, both approaches are based on moment matching and are by nature local because they accurately approximate the transfer function (3.16) only near the expansion points0 . This can be improved by multi-point expansion, i. e. by expanding the transfer function about several points si and requiring the reduced transfer function to match the ﬁrst moments at all expansion points. This idea has been implemented in the so-called Rational Krylov methods [104] - [106]. The main methodological challenges in multi-point expansion methods are how to choose the expansion points and to determine how many of them are needed. Lastly, let us note that in both approaches, instead of just one starting vector v, one can take a number of starting vectors expressed by the matrix B. This leads to a generalization of the Arnoldi and Lanczos algorithms to the so-called block- Arnoldi and block-Lanczos algorithms [91], [98], which are suitable for the reduction of Multiple-Input-Multiple-Output (MIMO) systems. Block Arnoldi will be the topic of section 7.2.

61 3 Linear Model Order Reduction Methods

3.3 Guyan Reduction

In the last section of this chapter we will describe a commercially available [107] Guyan reduction, which has been commonly used in mechanical engineering for many years and can easily be applied to the thermal domain as well. The Guyan algorithm [108] is another model order reduction method which projects (similar to equations (2.36) and (2.37)) a high dimensional ODE system to a lower-dimensional one. A projection subspace is chosen, however, based on engineering intuition rather than on mathematical rigor. Practically, this means that the designer can choose the “important” FE nodes which are to be physically preserved within a reduced model.

3.3.1 Static Matrix Condensation

The system matrices obtained by spatial discretization of heat transfer PDE contain terminal nodes which connect to the external circuitry as well as to internal nodes. The large dimension of (2.14) could be effectively reduced by the elimination of internal nodes. For steady-state problems of the form:

KT⋅ = Fut⋅ () (3.44) it is possible to decompose the linear system (3.44) into terminal and internal equations by splitting the matrixK into four blocks:

F Kee Kei T e e ⋅ = (3.45) F Kie Kii T i i with the index sets e and i ranging over all external and internal nodes respectively. For the simplicity the input termut() has been incorporated in the load vector. It is further possible to eliminate the equations for the non-terminal nodes by means of linear algebra operations (e. g. the Schur complement) [109] as follows. By partitioning the temperature and load vectors consistently with (3.45), the system (3.44) can be decomposed as:

62 3.3 Guyan Reduction

⋅ ⋅ Fe = Kee T e + Kei T i (3.46)

⋅ ⋅ Fi = Kie T e + Kii T i (3.47)

In order to eliminate the equations for the internal nodes, one can expressT i from (3.47) as:

–1 ⋅ ()⋅ T i = Kii Fi – Kie T e (3.48) and insert it into (3.46) to get:

()⋅⋅–1 ⋅ ⋅ –1 ⋅ Kee – Kei Kii Kie T e = Fe – Kei Kii Fi (3.49)

The dimension of the reduced system (3.49) is determined by the number of –1 chosen terminal nodes. It is not necessary to computeKii explicitly, but rather the LU decomposition ofKii and back substitutions corresponding to each matrix-vector product can be made. Additional speed up is due to the sparse structure ofKii . The reduced heat conductivity matrix and load vector are then defined as:

–1 K = K – K ⋅ K ⋅ K r ee ei ii ie (3.50) ⋅ –1 ⋅ Fr = Fe – Kei Kii Fi

Please note that until now no approximation has been used and hence the condensation of the heat conductivity matrix is exact.

3.3.2 Dynamic Matrix Condensation Generalizing the elimination of internal nodes via Shur complement to the dynamic problems was first proposed by Guyan [110]. It was originally applied only to undamped structural analysis. Starting from the static structural equation

63 3 Linear Model Order Reduction Methods of the form (3.45), Guyan assumed that no loads were applied on the internal nodes (Fi = 0 ). In this case, after replacing the temperature vector with the dis- placement vector, the equation (3.48) changes into:

–1 ⋅ ⋅ xi = –Kii Kie xe (3.51) and the Shur complement amounts to a coordinate transformation of the form ⋅ xV= G xe with:

I V G = –1 (3.52) –Kii Kie

Here,I is the unity matrix, whose dimension corresponds to the number of terminal nodes. From (3.50) the reduced heat conductivity (stiffness) matrix and the load vector are seen to be built by projection:

T K = V ⋅⋅KV r G G (3.53) T ⋅ Fr = V G F

Guyan further employed the transformation (3.52) to the kinetic and potential energies of the structure, defined as:

1 T E = ---x˙ ⋅⋅Mx˙ k 2 (3.54) 1 T E = ---x ⋅⋅Kx p 2 to get:

64 3.3 Guyan Reduction

1 ˙ T ⋅⋅⋅T ⋅ ˙ Ek = ---xe V G MVG xe 2 (3.55) 1 T T E = ---x ⋅⋅⋅V KV ⋅ x p 2 e G G e

From (3.55) the reduced mass matrix is also seen to be built by projection:

T ⋅⋅ M r = V G MVG (3.56) or in terms of partitioning it consistently with (3.45):

T –1 ()–1 ()–1 (3.57) M r = M ee–M eiKii Kie – Kii Kie M ie – M iiKii Kie

There is no mathematical proof for (3.57) but rather the analogy to (3.53) is used. There has been a number of attempts to modify the Guyan algorithm for the damped structural problems, e.g. [111] and [112]. The commercial FE solver ANSYS [107] also offers the possibility of Guyan-based reduced order modeling for transient heat transfer problems (2.14). The computation ofKr andFr is done as in (3.50) and the reduced heat capacity matrix is given through:

–1 –1 –1 –1 (3.58) Cr = Cee – CeiKii Kie – KeiKii Cie +KeiKii CiiKii Kie which is again similar to (3.57), assuming a symmetricK matrix and therefore T ()–1 T –1 Kie = Kei andKii = Kii . Now a reduced thermal system has a form:

⋅ ˙ ⋅ Cr T e + Kr T e = Fr (3.59)

T ⋅⋅ T ⋅⋅ T ⋅ withKr = V G KVG ,Cr = V G CVG andFr = V G F . Even for ≠ the case whereFi 0 the projection matrixV G is used as deﬁned in (3.52). After the reduction, it is possible to expand the terminal degree of freedom values to

65 3 Linear Model Order Reduction Methods gain the complete temperature distribution of the device using the equation (3.48) for the steady-state. The main advantage of Guyan-based MOR is that it physically preserves the terminal nodes, as schematically shown in Figure 3.2. This is advantageous for the coupling of several reduced order models (as will be demonstrated in chapter 7). However, the modified Guyan method will provide sufficient accuracy for the reduction of thermal systems only if the chosen number of terminal nodes is large enough (see numerical results in chapter 5). Besides, there are no strict guidelines on how to chose the terminal nodes.

FE model Reduced model Guyan reduction

Terminal nodes Figure 3.2 Schematics of Guyan-based model order reduction with physical preservation of chosen terminal nodes.

66 3.4 Conclusion

3.4 Conclusion

In this chapter we have presented the most important methods for model order reduction of linear dynamic systems. These methods and their most important properties are summarized in Table 3.3. Table 3.3 Methods for model order reduction of linear dynamic systems.

Advantages Disadvantages

Control theory methods have a global error estimate, computational3 complexity is (Balanced Truncation can be used in a fully auto- On(), hence can be used Approximation, Singular matic manner only for systems with less Perturbation Approximation, than a few thousand Hankel Norm Approxima- unknowns tion) SVD-Krylov (low-rank have a global error estimate currently under development Grammian approximants) and the computational com-2 and matrix sign function plexity is less than On() methods Padé approximants computationaly very advan- do not have a global error (moment matching) via Kry- tageous, can be applied to estimate. Hence, it is neces- lov subspaces by means of very high-dimensional 1st sary to select the order of the either the Arnoldi or Lancsoz order linear systems reduced system manually algorithm Guyan-based methods preserve the physical nodes result in unnecessary large reduced order models

Control theory methods are mathematically optimal, i. e. offer a global error estimate for the difference between the transfer function of the original high- dimensional and reduced low-dimensional system. They show that model reduction of a linear dynamic systems is solved in principle, but require a computational 3 complexity ofOn() , and therefore are only applicable to small linear systems. The SVD-Krylov methods (SVD stays for singular value decomposition), which are based on low-rank Grammian approximants [113]-[116], or the matrix sign function methods [117] have resulted from the efforts to find computationaly effective strategies in order to apply control theory methods to large-scale systems. However, they are currently still under development. Up to now, the most of the practical work in model reduction of large linear dynamic systems has been tied to Padé approximants of the transfer function via Krylov subspaces by means

67 3 Linear Model Order Reduction Methods of either the Arnoldi or the Lanczos process. This methods have no global error estimate, but can easily be applied to high-dimensional ODE systems. Commer- cially available algorithms based on modification of Guyan’s method preserve the physical nodes but result in unnecessarily large reduced order models. Due to its mathematical simplicity and numerical stability we think that presently Arnoldi process is the most suitable tool for model order reduction of electro-thermal engineering problems. The numerical results from chapters 5 -7 proof this.

68 4.1 Pyrotechnical Microthruster

4APPLICATIONS

In this chapter we present three novel MEMS devices used as case studies for model order reduction. These are the pyrotechnical microthruster, the thermally tunable optical filter and the microhotplate gas sensor. In sections 4.1 - 4.3 the device description and the numerical simulation results computed in ANSYS are presented. Section 4.4 shows that for the explored case studies it is possible to decouple electrical and thermal simulations, which enables the application of the Arnoldi-based model order reduction.

4.1 Pyrotechnical Microthruster

Pyrotechnics is a technology that is several centuries old and was mainly developed for military needs. The trend towards system miniaturization, however, has resulted in a completely new technology called micropyrotechnics. Promising applications of this technology are the medical field, security systems and especially the space field. In recent years a large interest in MEMS devices has arisen within the space community [118]. Building a cluster of microsatellites (20 - 100kg) or nanosatel- lites (< 20 kg) should be cheaper and more robust than building a single huge satellite. These microspacecrafts could be used in various space research fields, such as asteroid mission, disaster and magnetospheric monitoring, inspection of other spacecrafts [119], [120] etc. A key point in the miniaturization of spacecrafts are the micropropulsion subsystems, since in all mentioned applications maneuvering plays a central role. The micropropulsion subsystems should be able to deliver very small (a fewµN to a few 100mN ) and accurate forces needed for the stabilization, the pointing and the station keeping of small satellites. In the last years, a number of micropropulsion systems using MEMS technology, such as micro cold gas microthrusters [121], subliming solids microthrusters [122], vaporizing liquid microthrusters [123], micro pulsed plasma thrusters [124], micro ion thrust-

69 4 Applications ers, bi-propellant [126], and solid propellant microthrusters [127], have been investigated. The short duration of thrust impulse makes pyrotechnical solid propellant microthrusters most suitable for achieving a low velocity increment needed, for example, for the station keeping of small satellites (short duration mission). In the last few years, a number of MEMS research groups from Europe, USA and Asia have been working on the development of these devices [128]-[134]. Their concept is based on the high rate combustion of one single propellant stored in a combustion chamber. The gas generated by the combustion of the propellant is accel- erated in a nozzle, thus delivering a thrust. The concept of solid propellant microthrusters offers several advantages: there is no liquid fuel and hence no leakage, there are no moving parts and the solid propellant ignition consumes little energy.

The solid propellant microthruster presented here was developed within a Euro- pean project Micropyros (founded under IST-99047). It integrates solid fuel with four bonded silicon micromachined wafers [135] and delivers an impulse-bit thrust within a sub millimeter volume of silicon by producing a large amount of energy from the ignitable substance contained within the microsystem. Micro- thruster fuel is ignited by passing an electric current through a polysilicon resistor embedded in a thin dielectric membrane, as shown in Figure 4.1. After the ignition phase, sustained combustion takes place and forms a high-pressure, high-temperature gas mixture. Under the pressure of the gas the membrane ruptures and an impulse is imparted to the carrier frame as the gas escapes from the tank. The lack of the restart ability is compensated for by the fabrication of an array structure. The dimensions of individual rockets are in the millimeter scale for the chamber and in the micrometer scale for the nozzle. Igniters, nozzles and intermediary wafers are fabricated using silicon micromachining and combustion chambers are fabricated using Foturan glass or silicon. A fabrication of the igniter wafer is done by thermally oxidizing a silicon wafer and coating it with silicon rich LPCVD (Low Pressure Chemical Vapor Deposition) nitride. In a third step, a polysilicon layer is deposited by LPCVD at 605˚C and doped by diffusion. The heater filament is patterned using a reactive ion etching (RIE). The electrical pads and electrical supply lines are fabricated in gold. To fabricate the membrane, the silicon is etched away by Deep Reactive Ion Etching

70 4.1 Pyrotechnical Microthruster

nozzle igniter intermediary chamber

chamber

Solid propellant Figure 4.1 A structure of pyrotechnical microthruster array. Picture courtesy of C. Rossi (LAAS-CNRS, France).

(DRIE). The nozzles are fabricated by DRIE. Two chamber materials are used: silicon and Foturan, which is a photostructurable glass that can be anisotropically wet etched. The main advantages of using silicon lie in its well-known technology fabrication and its high melting point. The silicon chambers were realized by DRIE. On the other hand, Foturan offers a lower heat conductivity that can result in better thermal insulation between the chambers, which could be of importance when considering the thermal cross-talk between a single rocket and its closest neighbors. A complete modeling strategy of microthruster array combines an electrical circuit, heat transfer, combustion, membrane rapture and gas dynamics. It happens however, that the main modeling aspects must include electro-thermal simulation and a system-level simulation of the array and the driving circuitry. All other aspects of microthruster operation can be modeled on the basic level, as suggested in [136]. In this thesis we consider the initial heating phase of the fuel, right up to the onset of ignition. Electro-thermal simulation helps to design a device in such a way as to reach the ignition temperature within the fuel and at the same time not to reach it in the neighboring microthrusters, that is, at the border of the computa-

71 4 Applications tional domain. Additionally, the resistor‘s temperature during the heating up phase must not become too high, as this could cause the destruction of the membrane. Subsequent model order reduction supplies a system-level model which is used for the joint simualation of the of the array and the driving circuitry. For evaluation purposes of MOR algorithms we use a simplified axi-symmetrical model of a single igniter chip (see Figure 4.2), which after finite element based spatial discretization of the governing heat transfer equation (2.5) results in a linear thermal system of 1,071 ODEs. The device solid model has been made and

observed output poly-Si

observedSOG output poly-Si

SOG

SiNx

SiN SiO2x

SiO2

silicon substrate igniting fuel Figure 4.2 Model structure of 2D axi-symmetrical model for a pyrotechnical silicon substrate igniting fuel microthruster (single igniter wafer). All dimensions are given in µm . meshed in ANSYS (with PLANE55 elements). All material properties are considered to be temperature independent. A table with values used for the material properties can be found in Appendix A.1. A heat generation rate Q was assumed uniform and applied to the heater’s surface. The initial temperature distribution was set to zero and Dirichlet boundary conditions were set toT = 0°C at the bottom of the chip. Numerical simulation results for the full ﬁnite element model are shown in Figure 4.3.

72 4.2 Tunable Optical Filter

Figure 4.3 Temperature distribution within the igniter wafer after 0.3s of heating ° with 80mW power.T ref = 0 C .

4.2 Tunable Optical Filter

Most of the data communications nowadays are performed using fiber-optic cables. The information is carried through the fiber-optics by modulated laser light signals which have a fixed wavelength. The main application field of such communication systems is to connect computer networks to the data ‘highway’. This connectivity is realized on various levels, ranging from local area networks to the ultra long-haul installations of fiber-optic cables (more than 10,000km) which connect networks from continent to continent. With ever-increasing amounts of data to be transferred, individually modulated optical channels (with different wavelengths), are being added to the established transmission systems. A major advantage over the electrical data transmission is that there is no crosstalk between the light signals. Modern communication systems carry up to 128 independent channels, and systems with 1,000 channels on a single fiber are being developed. With so many channels operating in parallel, network management, for example monitoring of the channel’s power level and data rate, is becoming very important. Furthermore, individual channels have to be dropped from one fiber and fed into another one. This is usually performed by converting the optical signals into the electrical domain and after switching, adding or dropping them, con-

73 4 Applications verting them back into the optical domain. This opto-electro-opto conversion is a major limitation for increasing the data rate in fiber-optic cables. In order to circumvent this ‘bottleneck’, so-called all-optical cross connects are being developed, where the switching is realized completely in the optical domain. Add-drop functionality is provided by tunable optical filters, which are able to separate the individual channels. Additionally, they serve as key components for optical channel monitoring and tunable lasers. Several state-of-the-art concepts exist for the realization of tunable optical filters which make use of e. g. acousto-optic effects [137], [138] or liquid crystals [139]. As the miniaturization of these systems improves their performance consid- erably (much lower power consumption), a new class of optical MEMS devices is arising. Micromechanically tunable optical filters [140], [141] feature dielectric mirrors in connection with a single air-cavity of variable thickness and achieve tuning by varying the distance between the mirrors. Thermo-optical tunable microfilters [142], [143] on the other hand, are based on “pure” thermal modula- tion of resonator’s optical thickness. The device described here [19], was developed within a DFG project AFON (funded under grant ZA 276/2-1) and represents a new class of tunable optical filters which are fabricated using silicon MEMS technology and based on the thermo-optic effect. It consists of an optical resonator which is formed out of two dielectric Bragg mirrors which are separated by a solid-state material, as shown in Figure 4.4. An important optical characteristics of such a resonator is its optical thickness, defined asnd⋅ , where n is the refractive index of the material and d is the distance between the mirrors. The light, which propagates inside the cavity, is reflected repeatedly between both mirrors so that interference occurs. Construc- λ ⁄ , ,,… tively interfering waves (those with wavelengths0 ==2nd m m 12 ) may pass the filter, while others are reflected. The transmission wavelengths are determined only by varying the temperature and thus the refractive index n of the material, i. e. without physically changing the separation of the mirrors. As the thickness d of the central cavity region (the region between the mirrors) is of the order of the wavelength of light (ca.1µm ), the power consumption (for the heating of the microcavity) decreases drastically compared to similar devices in [142] and [143].

74 4.2 Tunable Optical Filter

λ 0 mirror

n cavity

Figure 4.4 Schema of a solid-state resonator cavity (see also [20]).

The device is fabricated by thin film deposition of the optical layers on a silicon substrate and further processing is done using MEMS technology. Amorphous silicon was used for the cavity material because of its suitably high thermo-optic coefficient (change of n with temperature). It is important to mention that the changes in the physical dimensions of the cavity due to thermal expansion are negligible and hence have no influence on the shift in transmission wavelength. The Bragg mirrors consist of dielectric materials such as silicon dioxide, silicon nitride or amorphous silicon, which are all deposited by plasma enhanced chemical vapor deposition. Thin film metal resistors are structured on the filter layers for the fabrication of heating and temperature sensing resistors. The microcavity is heated by conducting the electrical current through the heating resistor. In order to reduce the power consumption, the silicon substrate material was removed from the back- side of the wafer under the filter by anisotropic etching. The removal of substrate material also prevents additional optical resonances. Thermal isolation is further improved by structuring the membrane into a plate and fixing it to the substrate through micromachined suspension arms, as shown in Figure 4.5. A comprehensive modeling approach for the tunable filter includes the simulation of electro-thermal and thermo-structural domains and the impact of these domains on the optical properties. Although the thermally induced structural deformation of the filter’s membrane shows an impact on the optical quality, it is not of primary concern because the overall deformation is limited to buckling of the membrane. The resulting spherical shape has a radius of curvature bigger than

75 4 Applications

lensed fiber suspension arm

collimated beam resistor

filter membrane Figure 4.5 A structure of a tunable optical filter. Picture courtesy of D. Hohlfeld (IMTEK, Germany).

1 mm. As the illuminated area at the center of the membrane is restricted to a diameter of50µm , the light still passes the filter almost vertically in this section. Hence, the tuning behavior is mainly determined by the membrane’s temperature distribution. An inhomogenous temperature distribution across the illuminated section of the filter membrane causes the filter shape to degrade significantly. Thus, simulation of heat transfer is of paramount concern in the design process of the optical filter. The transient thermal behavior must be carefully analyzed as well because the tuning and switching speed are important parameters for future operation of the filter in dynamically configurable optical networks. Beyond the electro-thermal device simulation the connected circuitry can be considered as well. Feedback control can be utilized for stabilization of the membrane’s temperature. Hence, a system-level simulation becomes necessary and requires a compact thermal model.

For evaluation purposes of MOR algorithms we use a two-dimensional device model, which neglects the heat transfer in z direction. Therefore, a weighted sum of material properties of different layers was assigned to corresponding surfaces, as shown in Figure 4.6. All material properties are considered to be temperature independent. A table with values used for the material properties can be found in Appendix A.1. A finite element-based spatial discretization of the governing heat transfer equation (2.5), performed in ANSYS (with PLANE55 elements), results in a linear thermal system of 1,668 ODEs. A heat generation rate Q was assumed

76 4.3 Microhotplate Gas Sensor 5

100 50 250

platinum heater & membrane layers membrane layers Figure 4.6 Model structure of 2D model for tunable optical filter. All dimensions are given inµm . uniform and applied to the heater’s surface. The initial temperature distribution is set to zero and Dirichlet boundary conditionsT = 0°C were set at all four edges of the model. The model contains a constant load vector, corresponding to the constant input power of 1mW. Throughout a single output, placed in the middle of the membrane will be observed. Numerical simulation results for the full finite element model are shown in Figure 4.7.

4.3 Microhotplate Gas Sensor

There is a large demand for gas sensing devices in various domains. They are desired in e. g. safety applications where combustible or toxic gases are present or in comfort applications, such as climate controls of buildings and vehicles where good air quality is required. Additionally, gas monitoring is needed in process control and laboratory analytics. All of these applications demand cheap, small and user-friendly gas sensing devices which show high sensitivity, selectivity and stability with respect to a given application. A large variety of gas sensors exist based on different sensing principles, e.g. semiconductor gas sensors, optical sensors, thermal conductivity sensors, different mass sensitive devices, etc. The main advantages of semiconductor gas sensors based on metal oxides are sensitivity to

77 4 Applications

Figure 4.7 Temperature distribution of optical filter after 0.25s of heating with ° 1 mW.T ref = 0 C . some relevant gases like CO, H2,NOx and hydrocarbons, simple signal processing, low production cost and small size. Already in the sixties it was demonstrated [144] that semiconducting metal oxides heated to ~300°C in air vary their conductivity according to the presence of reactive gases in the air. This initiated the commercial development of thick film sensors based mostly on SnO2. However, SnO2 gas sensors have low selectivity and high power consumption due to the use of bulky ceramic substrates. Hence, in the last few years a large amount of research has been devoted to the development of low power Si-micromashined gas sensors based on microhotplates [17], [145]-[148]. Usually, the main drawback of such implementations has been the fragility of the thin dielectric membrane used to achieve thermal isolation of the microhotplate from the surrounding wafer. The goal of the European project Glassgas (founded under IST-99-19003) was to develop a novel microhotplate gas sensor that uses a metal oxide as the sensing material and features low power operation and high mechanical stability. The fabrication approach enables the integration of various sensing areas in an array configuration [149]. The gas-dependent change in electrical conductivity relies upon diffusion of oxygen from an ambient gas (e. g. air) into the sensitive layer at room temperature. In the presence of a reducing or oxidizing gas which is able to react with the absorbed oxygen at temperatures between200°C and 400°C , the

78 4.3 Microhotplate Gas Sensor oxygen surface concentration and thus the electrical conductivity of metal oxide is altered. This device requires a homogeneous temperature distribution over its gas sensitive regions, good thermal decoupling between the hotplate and the silicon rim and good mechanical stability at high temperatures. This is achieved by supporting the silicon micromachined platform with glass pillars emanating from a glass cap above the silicon wafer, as shown in Figure 4.8. Glass (Pyrex #7740) was chosen due to its low thermal conductivity and anodic bondability to silicon.

glass structure

hotplate silicon rim

Figure 4.8 Gas sensor array structure. Picture courtesy of J. Wöllenstein (Fraun- hofer IPM, Germany).

Double-side polished Si-wafers are used for the fabrication of the sensor element. A thermal oxide is grown and afterwards photolithography and structuring for defining the diffusion area of the later silicon island is carried out. Boron diffusion is performed and subsequently annealed in an oxygen atmosphere. Then the oxide is removed from the front side and a new thin oxide layer is grown prior to the deposition of the silicon nitride membrane material. The gas sensor layout features four interdigital structures with symmetrical electrodes. The heater is located on the outer edge of the hotplate areas. The metal electrodes and the heating resistor are fabricated from platinum. The four different gas sensitive layers are deposited and structured in separate steps. After completing the silicon sensor chip, the microstructured glass cap is bonded to the front side of the wafer. The columns of the supporting glass structure are fabricated by mechanically sawing a 1 mm thick glass substrate. The front-side of the bonded wafer assembly is pro-

79 4 Applications tected during the following KOH etching step of the silicon’s back side. Etching stops at the membrane’s bottom side and at the highly p-doped silicon platform. Subsequent removal of the dielectric membrane material can be performed by dry etching from the back side, so that the silicon island is solely supported by the glass pillars. In this way the hotplate is almost completely themally isolated from the surrounding silicon substrate. Micromachined gas sensor is not only a challenge with respect to thermal design but also with respect to mechanical design. Only by choosing the right mechanical design can a large intrinsic or thermal-induced membrane stress leading to membrane deformation/breaking of the membrane be avoided. It is further necessary to build a chemometrics calibration model which correlates the set of sensors resistance measurements to the sensed gas concentration. Prior to fabrication, a thermal simulation is performed to determine the heating efficiency and temperature homogeneity of the gas sensitive regions. Another important thermal issue to be considered by the simulation is the thermal decoupling between hotplate and silicon rim. As the device is connected to circuitry for heating power control and sensing resistor readout, a system-level simulation is also needed. Hence, a compact thermal model must be generated. We have tested the applicability of Krylov-subspace based MOR to the large- scale linear systems using a 3D sensor model shown in Figure 4.9. All material properties are considered to be temperature independent. A table with values used for the material properties can be found in Appendix A.1. Temperature is assumed to be in degreesCelsius with an initial state of0°C and Dirichlet boundary con- ditionsT = 0°C are applied at the top and bottom of the chip. A finite element- based spatial discretization of the governing heat transfer equation (2.5) results in a linear thermal system of the 73,955 ODEs. The device solid model was made and meshed in ANSYS (with SOLID70 elements). The model contains a constant load vector corresponding to the constant input power of 340mW. Throughout this thesis two outputs for this model for linear and weak nonlinear simulation will be observed. Both are marked in Figure 4.9. Numerical simulation results of the temperature distribution over the chip are shown in Figure 4.10.

80 4.3 Microhotplate Gas Sensor

observed output for nonlinearities

observed output

glass silicon platinum Figure 4.9 Model structure of a 3D model for microhotplate gas sensor. All dimensions are given inµm .

Figure 4.10 Temperature distribution over the gas sensor chip after 5s of heating ° with constant heating power of 340mW.T ref = 0 C .

81 4 Applications

4.4 Preparation for Automatic Model Order Reduction

It is presently possible to apply the Arnoldi process only to solely linear and weakly non-linear (with non-linear input function) systems. For electro-thermal models this implies not only temperature-independent material properties, but also the decoupling of the governing equations (2.11), (2.12) and (2.14). After spatial discretization in ANSYS, the resulting system of coupled domain electro- thermal simulation (assuming constant material properties) has a form:

C 0 ˙ K 0 T Q()ϕ, t th ⋅ T + th ⋅ = (4.1) ϕ ϕ (), 00 ˙ 0 Kel ITt whereCth is a thermal specific heat matrix,Kth is a thermal conductivity matrix, Kel is electric conductivity matrix,Q is a heat generation vector (in our case, due to Joul heating only),IT is a nodal electric current vector and andϕ are the temperature and electric potential vectors. A simplification to the “pure” thermal model can be done by assuming a uniform heat generation rateQ over a lumped heater with electrical resistivityR and assuming that the electric power is completely transformed into the heating power. This changes (2.12) into:

2 2 U ()t Qt()==I ()t ⋅ R ------(4.2) R where I(t) is the electric current andUt() is the applied voltage over a lumped heater. By inserting (4.2) in (4.1) we get a linear thermal ODE system, which in the single-input-single-output representation has a form:

CT˙ + KT = FQt⋅ () (4.3) T ye= ⋅ T n whereFe, ∈ R are the load and the output vectors and n is the dimension of the system.

82 4.4 Preparation for Automatic Model Order Reduction

It turns out that for our case studies it is possible to approximate the temperature distribution of the electro-thermal simulation through the temperature distribution of the thermal simulation. It is done by adding all the Joule heat terms over the heating resistors volume (surface) in ANSYS and applying this value as homogeneous heat power over the same heater’s volume in pure thermal simulation. Figure 4.11 shows the Joule heat distribution over the heater of the gas sensor model, which is proportional to the squared current density distribution. It is approximately uniform. Abrupt changes in the geometry, such as right angles, lead to extreme numerical values (dark blue and red in Figure 4.11). They are physically not justiﬁed and should be evaluated with caution, as in the fabricated structure the corners are slightly rounded by the photolitography. Hence, in this case the total amount of heat determined by an electro-thermal simulation can be applied as a homogenous heat generation to the heater’s volume. Indeed, Figure 4.12 and Figure 4.13, which compare the temperature distribution over the heater for the coupled-domain simulation and for the thermal simulation, show almost identical results. The asymmetric temperature distribution over the heater is due to the asymmetrical model structure, displayed in Figure 4.9.

Figure 4.11 Joule heat distribution in W/m3 over the heater of the microhotplate gas sensor.

83 4 Applications

Figure 4.12 Temperature distribution in °C over the gas sensor’s heater for the coupled-domain simulation. Electrical voltageU = 1.77V was applied to the heater’s ends.

Figure 4.13 Temperature distribution in °C over the gas sensor’s heater for the thermal simulation. The sum of Joule heat terms from Figure 4.11, which is340mW , was applied as the homogeneous heat power.

84 4.4 Preparation for Automatic Model Order Reduction

We have obtained similar results for microthruster and optical filter models as well, which shows that the simplification from electro-thermal to pure thermal FE modeling is acceptable for these devices. In general, it is always acceptable when the current density distribution over the heated volume is approximately uniform. This happens when the heater’s material has a high thermal conductance (such as metal or silicon), and the heater itself is thermally isolated (by means of a membrane for example). In such a case, the high thermal conductance causes an instan- tenous distribution of generated heat, which results in a homogenous temperature profile.

85 4 Applications

86 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

5NUMERICAL RESULTS

In this chapter we present the model order reduction results for the electro-thermal MEMS case studies described in chapter 4. The accent is put on Arnoldi- based reduction. In section 5.1 the different aspects of Arnoldi-based model order reduction, such as the approximation of the complete output, the reduction of weekly nonlinear systems, the system-level simulation of the reduced model and the increase in the computational efficiency, are numerically demonstrated. Sec- tion 5.2 compares the results of Arnoldi-based model order reduction with reduction using Guyan’s algorithm and reduction using the control theory methods. Finally, section 5.3 concludes the chapter.

5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

Let us emphasize once more that the goal of Arnoldi-based model order reduction is to transform the equation system (4.3) into a system with the same form:

() Crz˙ + Krz = FrQt (5.1) T ⋅ yr = er z but with a much smaller dimensionrn« . Here, a generalized variablez can be seen as a projection of thenr -dimensional temperature vector to -dimensional subspace, subjected to some errorε :

r TVz= ⋅ + ε, zR∈ , rn« (5.2)

T ⋅ andyr = er z is the linear combination of the reduced states which corresponds to the required states y in equation (4.3).

87 5 Numerical Results

The matrixV (in the previous chapters we also called it Vr) in (5.2) is composed fromrn -dimensional vectors that form a basis for the right Krylov subspace of the dimension r:

R{}, {},,,… r – 1 K r Ab= span b Ab A b (5.3)

–1 –1 withAK= – Cb, = – K F . In fact, the whole equation (4.3) can be directly projected onto the reduced subspace by applying (5.2) (ε is neglected) and then T multiplying (4.3) from the left side byV . As already described in section 3.2, this projection process produces a reduced order system (5.1) according to the Padé-type approximation:

T Cr = V CV T K = V KV r (5.4) T Fr = V F T er = V e

It can be shown that the transfer functions of the systems (4.3) and (5.1) defined as:

T –1 Gs()= e ()sC+ K F and (5.5)

() T ()–1 Gr s = er sCr + Kr Fr (5.6) when developed into Taylor expansions around s0 = 0 :

88 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

∞ i () i T ()–1 –1 Gs = ∑ mis with mi = e –K C K F i = 0 (5.7) ∞ i T –1 i –1 G ()s = m s with m = e ()–K C K F r ∑ ri ri r r r r r i = 0 wherem andm are called the ith moments, match in the ﬁrst r moments: i ri

m ==m , i 0,,… r – 1 (5.8) i ri

Please note once more that the transformation matrixV is a direct output of the Arnoldi algorithm, and that neither input termQt() nor output vectore take part in order reduction. This brings along two additional important properties of Arnoldi algorithm: 1. The Single-Input-Single-Output (SISO) algorithm setup is sufficient to approximate not only a single output response but also the transient thermal response in all finite element nodes of the device. This will be numerically demonstrated in section 5.1.1. 2. The nonlinearities of the input function can be transferred into the reduced system, i. e. weakly non-linear systems can be reduced as well. This will be numerically demonstrated in section 5.1.2. Let us show using the case studies from chapter 4 how Arnoldi-based model order reduction works for the transient heat transfer problems in 2D and 3D. Microthruster: This 2D model (described in section 4.1) contains 1,071 ordinary differential equations. The Arnoldi process was applied to iteratively generate several reduced models with different orders. For each reduced model a trans- () 6 fer functionGr s over a relevant range of frequencies from1rad/s to 10 rad/s was computed. The magnitude of the frequency responses of the full-scale model and reduced models with orders 10, 20 and 50 are shown in Figure 5.1. As expected, we can observe a good match in the frequency domain around the

89 5 Numerical Results

expansion points0 , which we have chosen to be zero. Figure 5.2 shows a step response of the full-scale model in the observed output node (see Figure 4.2), for the constant input power of 80mW and vanishing step response errors (within the initial 0.05s) for the reduced models with orders 7, 10 and 20. The errors for all three reduced models are less than 0.02% after 0.02s. The results show that in the case of a microthruster model, it is possible to approximate an ODE system of 1,071 equations with only 7 equations with a maximal relative error less than 4%.

1 10

-2 10

-5 10 G(s) -8 10 Full-scale model (1,071 DOF) -11 Reduced model order 10 10 Reduced model order 20 -14 Reduced model order 50 10 0 2 4 6 10 10 10 10 Frequency (rad/s) Figure 5.1 Frequency response of the microthruster.

Optical filter: This 2D model (described in section 4.2) contains 1,668 ODEs. The Arnoldi process was applied to this model to iteratively generate several reduced models with different orders. For each reduced model a transfer function () 9 Gr s over a relevant range of frequencies from1rad/s to 10 rad/s was computed. The magnitude of the frequency responses of the full-scale model and reduced models with orders 1, 5 and 10 are shown in Figure 5.3. Again we can observe an excellent match in the frequency domain around the expansion point s0 = 0 . Figure 5.4 shows a step response of the full-scale model in the observed central membrane node (see Figure 4.6) for the constant input power of 1mW and vanishing step response errors (within the initial 0.001s ) for the reduced models with orders 1, 5 and 10. The errors for all three reduced models are less than 0.001% after 0.001s . Hence, in the case of optical filter model it is possible to

90 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

400 Full-scale model (1,071 DOF)

300 1

200 -1 -2 Reduced model order 7 -3 Relative error (%) Reduced model order 10 Temperature (°C) Temperature 100 Reduced model order 20 -4 0 0.01 0.02 0.03 0.04 0.05 Time (s) 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s)

Figure 5.2 Step response (outer plot) and step response errors (inner plot) of the microthruster for the constant input power of 80mW.

1 10

-1 10 G(s) Full-scale model (1,668 DOF) -3 10 Reduced model order 1 Reduced model order 5 Reduced model order 10 0 2 4 6 8 10 10 10 10 10 Frequency (rad/s)

Figure 5.3 Frequency response of the optical filter.

91 5 Numerical Results

Full-scale model (1,668 DOF) 200 0.02 150 Reduced model order 1 0.01 Reduced model order 5 100 Reduced model order 10

0 Rrelative error (%) Temperature (°C) Temperature 50

0 0.0002 0.0006 0.001 Time (s) 0 0 0.05 0.1 0.15 0.2 0.25 Time (s) Figure 5.4 Step response (outer plot) and step response errors (inner plot) of the optical filter for the constant input power of 1mW. approximate an ODE system of 1,068 equations with only 5 equations with a maximal relative error less than0.01% . Gas sensor: This 3D model (described in section 4.3) contains 73,955 ODEs. The Arnoldi process was applied to this model to iteratively generate several reduced models with different orders. For each reduced model a transfer function over a relevant range of frequencies from 0.01Hz to 10Hz was computed. The magnitude of the frequency responses of the full-scale model and the reduced models with orders 2, 5, and 10 which show a good match in the frequency domain around the expansion points0 = 0 are displayed in Figure 5.5. Figure 5.6 shows a step response of the full-scale model in the observed central membrane node (see Figure 4.9) for the constant input power of340mW and vanishing step response errors for the reduced models with orders 2, 5 and 10. The errors for all three reduced models are less than 0.5%after 2s . Hence, in the case of gas sensor model it is possible to approximate an ODE system of 73,955 equations with only 10 equations with a maximal relative error less than0.5% . As expected, the three presented thermal models show low-pass filter characteristics. This can be proved in general by observing the definition of the thermal –1 n × 1 system’s transfer function (5.5). A vector term()sC+ K FR∈ consists of

92 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

300

200 150 G(s) 100 Full-scale model (73,955 DOF) Reduced model order 2 Reduced model order 5 Reduced model order 10 0.01 0.05 0.1 0.5 1 5 10 Frequency (Hz) Figure 5.5 Frequency response of the gas sensor.

300 Full-scale model (73,955 DOF) 250 8 200 Reduced model order 2 6 Reduced model order 5 Reduced model order 10 150 4

2 100 Temperature (°C) Temperature 0 Rrelative error (%)

50 -2 012345 0 Time (s) 012345 Time (s) Figure 5.6 Step response (outer plot) and step response errors (inner plot) of a gas sensor in a central hotplate node, for the constant input power of 340 mW.

93 5 Numerical Results

()⁄ () () () n rational functions of the formPn – 1 s Pn s , wherePn – 1 s andPn s are polynomials in s of the ordern – 1 and n respectively. This follows from applying the Cramer’s rule as follows:

… … sC1 + K11 F1 K1n

det Fk

n K … F … sC + K () ⋅ n1 n n nn Gs = ∑ ei ------(5.9) … i = 1 sC1 + K11 K1n det … Kn1 sCn + Knn

From equation (5.9) is noticeable that the denominator is a polynomial sum of the ordern – 1 since the i-th column of()sC+ K is replaced by the constant vector F. The nominator is of the order n. Due to the positive diagonal entries of the heat conductivity matrix K, no zeros of G(s) ats = 0 are present. In order to exclude the vibratory systems with complex conjugated poles, we additionally note that, because C and K are both positive definite (with positive real eigenvalues [171]), system (4.3) has only real negative eigenvalues [150], i.e.Gs() is stable with real poles. Hence, all thermal models (4.3) can be represented by the equivalent thermal circuit shown in Figure 5.7, which is also a low pass filter. The approximation in Figure 5.7 is a first order approximation, whereas higher order approximations lead to the equivalent thermal networks discussed in chapter 2.5.3. In this simplified thermal circuit in Figure 5.7, a thermal resistorRth models a heat transfer between the hot areas (e. g. membrane) and the heat sink (e. g. substrate), whereas a thermal capacitorCth models the impossibility of the system to react instanta- neously to the heat sourceQRC . This thermal element, equivalent to its electrical τ ⁄ counterpart, has a time constant= 1 RthCth and therefore the transient thermal response (rise-time) of the device is determined by its thermal mass and its thermal resistance, as can be expected.

94 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

U2 Q = Rheater electrical domain Joule thermal domain heating :

∆T ∆ URheater( T) Q Rth Cth

thermoresistance:

∆ α ∆ Rheater( T) = Rheater,0 (1 + T)

Figure 5.7 Equivalent thermal circuit corresponding to the low-pass filter characteristic.

The next important practical issue is that the connection between the rise-time te and the cutoff frequencyf g for thermal models can be approximated through the ideal low-pass filter equation:

1 te = ------⋅ - (5.10) 2 f g

This gives the designer a guideline on how to choose the necessary frequency range of interest for error estimation (see chapter 6). For the presented electro- thermal case studies the frequency range of interest is between 0 and fg. So far we have represented the numerical results either in the frequency domain (Bode diagram - magnitudes of frequency responses) or in the time domain (step responses, errors), which is the common practice in electro-thermal engineering [76]. In principle, there are different representations of the dynamic systems, such as pole-zero diagrams, phase plots or Nyquist plots, which originate from control system theory [172]. In Figure 5.8 and Figure 5.9 we present the phase plot and the Nyquist diagram of the microthruster model. The phase angle ranges between –π andπ , but the plot could actually be unfolded since the phase increases with frequency. Which representation should be used depends on the frequency range

95 5 Numerical Results

3 2 Full-scale model (1,071 DOF) Reduced model order 20 1 0 -1 Phase angle (rad) -2 -3 10-2 100 102 104 106 Frequency (rad/s)

Figure 5.8 Phase plot of microthruster. )) ω Im(G(j 0 5 10 15 20 25 Re(G(jω))

-2 ω = 8 ω = 0 -4 Full-scale model (1,071 DOF) -6 Reduced model order 20 -8

-10

Figure 5.9 Nyquist diagram of microthruster’s frequency response.

96 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm of interest, knowledge of the system‘s stability etc. A Bode diagram offers a logarithmic representation of the frequency axes, i.e a huge frequency range can be visualized, as well as a direct correspondence between frequencies and magnitudes. A Niquist plot represents a frequency response in the complex plane and can be used to judge whether the system is stable (as well as the poles-zero plot). However, since we deal with stable (first order) thermal systems with real poles and use the Arnoldi reduction algorithm, which preserves stability (see section 3.2.3), the chosen representations suffice for observation of the approximation quality.

5.1.1 Approximation of the Complete Output The results presented so far refer to the Single-Input-Single-Output (SISO) rep- n × 1 resentation of (4.3), which implies that e and f are both vectors fromR . In more general cases however, the temperature response in several or even in all finite element nodes may be required. We will refer to the case when the whole temperature field is required as a Single-Input-Complete-Output (SICO) setup. In nn× such a caseeEI==nn× is an identity matrix fromR . An important property of the Arnoldi algorithm is that due to the fact that the output vector (matrix) e does not explicitly participate in model order reduction, it is possible to recover the complete transient thermal output from the reduced system. In other words, the Padé-type approximation of (5.5) for the arbitrary vector e, is warranted by only defining a transformation matrix V. Let us explain this. By applying a projection T (5.2) while neglectingε , the output row of (4.3), which isye= ⋅ T , changes T intoyr = e Vz . Hence, each row of the transformation matrix V “corresponds” to a single node temperature of the FE model. Which output (row) is chosen is defined by e. In SICO case, all the rows of V are required and

T ⋅⋅ ⋅ ≈ yr= E Vz= Vz T (5.11) stays for the complete temperature vector T. In order to demonstrate based on examples from chapter 4 that the projection error ε for thermal models indeed vanishes, let us define a mean square relative difference (MSRD) over the time for all the FE nodes as:

97 5 Numerical Results

2 n ⎛⎞T ()t – Tˆ ()t ⎜⎟i i ∑ ------() - ⎝⎠T i t MSRD() t = i------= 1 (5.12) n

() ˆ () whereT i t is the temperature of the i-th FE node in time andT i t is the i-th component of the vectorVz⋅ in time. Figure 5.10 shows a mean square relative difference (5.12) for the microthruster model computed over the initial 5ms. We can observe that already for the reduced system of order 20, a maximal MSRD for all 1,071 nodes amounts to only 0.14%. Hence, after the simulation of the reduced model it is possible to recover the transient solution for all the 1071 nodes by using equation (5.11).

MSRD (%) 3

20 1 15 system 10 7 order 0 5 0.02 Time (s) 0.04 Figure 5.10 Mean square relative difference (5.12) for all of the 1,071 finite element nodes of the microthruster model during the initial 0.05s.

Figure 5.11 additionally shows the MSRD (5.12) for the optical filter over the initial 5ms. In this case, a reduced model of order 5 shows a maximal MSRD for all 1,668 nodes less than 0.06%. It decreases even further for higher system orders. As in the previous example, the transient thermal response in all 1,668 FE nodes can be extremely well approximated by the reduced models.

98 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

MSRD (%)

0.008

0.006

0.004

0.002 15 10 0 7 system 0 5 order 0.02 Time (s) 0.04 Figure 5.11 Mean square relative difference (5.12) for all of the 1,668 finite element nodes of the optical filter during the initial 0.05s.

As already explained, in both cases, the Arnoldi reduction algorithm can be viewed as a projection from the full space to the reduced Krylov space (5.3) with an identity output matrix corresponding to the SICO system description. We forego displaying the results of the complete output approximation for the large-scale gas sensor model here, because this is usually not required in the engineering applications. Nevertheless, a possibility of approximating the complete transient field with a simple SISO algorithm setup remains an exclusive feature of the Arnoldi reduction algorithm.

5.1.2 Reduction of Weakly Nonlinear Systems When modeling the devices from chapter 4 it is necessary to take into account the dependence of the heater’s resistivity on temperature, which is given through:

() ⋅ ()α β 2 … RRT==R0 1 ++T T + (5.13)

° αβ whereR0 is resistivity at0 C and and are the temperature coefficients. (5.13) changes (4.2) into:

99 5 Numerical Results

2 U ()t QtT(), = ------(5.14) RT()

Thus, a linear equation system (4.3) changes into the weak non-linear system:

2 U ()t CT˙ + KT ==FQ() t, T F ⋅ ------RT() (5.15) T ye= ⋅ T

In terms of the finite element model, the dependence temperatureTRT in() , can be chosen as an arbitrary linear combination of statesT i :

⋅ Tc= ∑ i T i (5.16) i

It turns out that it is possible to reduce the equation system (5.15) by using linear Arnoldi process [153]. As the non-linear input termQtT(), , similar to the output vector (matrix) E, is not explicitly affected by the model order reduction, it is possible to transfer it into the reduced system, which will then be of the form:

˙ (), CrT r + KrT r = FrQtz (5.17) T ⋅ yr = er z or, using a projection (5.2):

T T T V CV z˙ + V KVz = V FQ() t, V∗ ⋅ z (5.18) T yr = e Vz

The reason why we have writtenV∗ instead of V in (5.18) lies in (5.16). In the cases wheni = 1 , heating power is a function of a single node temperature and a productV∗ ⋅ z defines a single additional output.V∗ is then a single row of a pro-

100 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm jection matrixVi . In case when> 1 ,V∗ is a linear combination of different rows of a projection matrixVV . In any case,∗ ⋅ z must be equal to (5.16). We will demonstrate (5.18) numerically for the microhotplate gas sensor example. In order to be able to control a heater’s resistivity, and therefore the temperature of the hotplate, equation (5.13) was taken into account by the ANSYS simulation of the full-scale model as well as by model order reduction as well. For a platinum sensor’s heater it is sufficient to assume a linear temperature dependence in the range of0°C to 500°C , i. e.β can be neglected. Measured dependence of the heater’s resistivity on temperature and consequently the dependence of the heating power on temperature for the constant input voltage of 14V are shown in Figure 5.12. A full-scale model was implemented using the non-linear heat generation rate, which is available as a table load in ANSYS7. For this procedure, the heater’s temperature was approximated through a single node temperature T N (see the observed output for nonlinearities in Figure 4.9), so that the heating power was expressed as:

2 (), ()() U QtT ==QTN t ------⋅ ()α (5.19) R0 1 + T N

Ω α × –3 ⁄ where U is a desired constant voltage, R0 = 274.94 and= 1.469 10 1K . Reduction from order 73,955 down to order 10 was done by linear Arnoldi. As the reduced model contains a constant load vector which corresponds to the input power of 340mW, the right hand side of (5.17) must be divided by 0.34 before multiplying it with (5.19). For integration,T N must be, of course, replaced with the corresponding linear combination of the reduced statesV∗ ⋅ z . Figure 5.13 shows the step response and the step response error between the full- scale and the reduced order 10 model in the observed central hotplate node (see Figure 4.9). The temperature dependent heating power according to the Figure 5.12 (right) has been taken into account. These results show that it is possible to approximate the full-scale model of order 73,955 through the reduced model of order 10, with maximal relative error of 1.2%. As expected, the obtained steady-state of the non-linear sensor model differs from the one for the linear thermal model (Figure 5.6), but the reduced models describe these steady-states cor- rectly in both cases.

101 5 Numerical Results

290

) 450 Ω 280 400 270 350 260

Power (mW) 250 Resistance ( 300 240 0 100 200 300 400 500 0 100 200 300 400 500 Temperature (°C) Temperature (°C)

Figure 5.12 Heater’s resistivity as a function of temperature (left). Heating power as a function of temperature, measured for the constant input voltage of 14V (right).

250 Full-scale model (73,955 DOF) Reduced model order 10 200 0.2 0 150 -0.2 -0.4 100 -0.6 -0.8 Reduced model order 10 Relative error (%) Temperature (°C) Temperature 50 -1 -1.2 012345 0 Time (s) 012345 Time (s)

Figure 5.13 Step response (outer plot) and step response error (inner plot) of the gas sensor in a central hotplate node for temperature dependent heating power according to Figure 5.12 (right).

102 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

5.1.3 System-Level Simulation In the work up to now, equation (5.1) was solved by numerical time integration using the NDSolve command of Mathematica. For industry level applications, the reduced models have to be expressed in an appropriate hardware description language (HDL). MAST, a language of the behavioral simulator SABER, for example, allows the direct implementation of the ODE systems into its templates. Figure 5.14 shows the structure of the implemented MAST model (see Appen- dix A.2) for a microhotplate gas sensor with a temperature dependant heater as described in section 5.1.2. It contains a reduced order thermal model with temper-

V 2 Q = R heater T B N

Cr z + Kr z = Fr Q Tout R heater

Ω Ω V A Q 1 2 Ω Ω T 3 4 ground R lead

R R α T heater = heater,0 (1 + N)

Figure 5.14 Model structure containing a linear reduced model of the gas sensor with back coupled temperature dependent heater. ature dependent input power (equation (5.17)). A template containing a (linear) reduced ODE system is back coupled to the heaters resistivity, which is dependent on a single node temperatureT N (see output node for nonlinearities in Figure 4.9). This temperature is computed as a particular linear combination of the reduced states:

∗ ⋅ T N = V z (5.20)

103 5 Numerical Results

The output node temperatureT out (see the observed output node in Figure 4.9) is also a linear combination of reduced states, marked withyr in (5.17). Basically, there is no difference between this implementation and the implementation in section 5.1.2. Figure 5.15 compares the numerical simulation results of the behavioral MAST model, which was integrated in SABER, the non-linear full-scale model, which was integrated in ANSYS and the reduced order model (5.17), which was integrated in Mathematica.

250

200

150

100 Full-scale model (73,955 DOF) Reduced model order 10 Temperature (°C) Temperature 50 MAST HDL model

0 012345 Time (s) Figure 5.15 Solution of the full-scale system (73,955 DOF) and of the reduced order 10 system in a central hotplate node of a gas sensor device. The reduced systems were integrated in Mathematica and SABER (MAST HDL model). The temperature dependence of the heating power is based on Figure 5.12.

As expected, the difference between the Mathematica and SABER solutions totals to the numerical integration errors between the different solvers. An important advantage of the behavioral model over the equation system (5.17), however, is the back coupling of the heater’s resistor. It allows the monitoring of the temperature through the change of resistance and furthermore, certain design changes (such as the change of the heater’s resistivity) are still possible after the model order reduction.

104 5.1 Model Reduction of Thermal MEMS via the Arnoldi Algorithm

5.1.4 Calculation Efficiency The ultimate goal of model order reduction is, of course, to increase computational efficiency with minimal loss of accuracy. The high precision of Arnoldi- based reduced order thermal models was numerically presented in the previous sections and here we discuss the decrease in computational time. In section 3.2.3 the computational complexity of the Arnoldi algorithm was estimated as:

2 () 2r n + 2rNz A (5.21) where Nz(A) is a number of non-zero elements of a single (preferably sparse) system matrix A [168]. In a case of thermal systems matrix A is not given explic- itely. Instead, the system is described by two large-scale symmetric (sparse) matri- –1 ces C and K. However, the direct computation ofAK= – C via matrix inverse would lead to the lost of both, symmetry and sparsity and would require high computational effort. Luckily, the explicit inverse of K can be avoided. If a direct solver is available, either Cholesky or LU sparse factorization ofK can be made once:

KLU= ⋅ (5.22) and then in each iteration of the Arnoldi algorithm, which is:

–1 v = bK= – F 1 (5.23) –1 ⋅ vi + 1 = K Cvi a fast back substitution (which only takes about 3% of factorization time) for solving (5.23) is performed using the following three steps: ⋅ 1)C is multiplied byvi ,aCv= i . This is a fast operation, because C is sparse. ⋅ –1 ⋅ ⋅ 2)The linear equationsLb= a are forward solved, so thatbL= Cvi . This is a fast operation, because L is a lower triangular matrix.

105 5 Numerical Results

3)The linear equationsUc⋅ = b are forward solved, so that ()–1 ⋅ –1 ⋅ ⋅ cU= L Cvi , which is required. Since, U is upper triangular, this is a fast operation as well. It is worth of noting that although the number of nonzero elements of the matrix K-Nz(K) is known, it is not possible to predict Nz(L) and Nz(U) because they depend on the connectivity of K (the distribution of zero entries). In order to speed up the computation and reduce memory storage, K can be reordered (in sense of moving the zeros towards main diagonal) [150]. Still, Nz(L) is about an order of ()≥ () magnitude larger than Nz(K) andN z K N z C because C is often diagonal. As a result, the steps 2) and 3) of a back substitution require more time than step 1). Hence, our observation is that the model order reduction of thermal systems via Arnoldi algorithm can be performed in approximately the same time as a single stationary system solution, that is the time needed for solvingKTFQt= () . Numerical experiments confirm this estimate. The required time totals to the time for factorization and the time for construction of a single Arnoldi vector (via steps 2) and 3)). Unfortunately, it is not possible to estimate the computational effort with more precision since it is impossible to predict Nz(L) and Nz(U) based on Nz(K) only. Table 5.1 Computational times in seconds on Sun Ultra-80 with 4Gb RAM and 450MHz.*Times for ANSYS stationary solution includes input and output operation times.

Dimension Nz(K) Stationary* Factoriza- Generation of the solution in tion in ﬁrst 50 Arnoldi ANSYS7.1 TAUCS vectors Microthruster 1071 5141 3 0.04 0.51 Optical ﬁlter 1668 6209 4 0.06 0.69 Gas sensor 73955 885141 200 93 135

Table 5.1 compares the computational times for the stationary solution of the full-scale models and for the construction of the reduced order models for case studies from chapter 4. In the second and third column the model’s dimension and the number of non-zero elements of the matrix K are displayed. The time needed for Arnoldi-based model order reduction of each model is the sum of the times needed for the factorization and for the generation of the Arnoldi vectors (the last

106 5.2 Comparison Between the Arnoldi Algorithm, the Guyan Algorithm and the two columns in Table 5.1). A time integration of the reduced, order 50 system in Mathematica lasts about 1s. The computational times for the transient solutions of the full-scale models in ANSYS with 30 time-steps are 120s for the microthruster model, 150s for the optical filter model and 6,840s for the gas sensor model. As the time for the extraction of the system matrices from ANSYS is 317s for the gas sensor model, the total computational time is reduced by a factor of 12.5 due to MOR. At the end of this subsection let us note that when the dimension of A grows large enough (e. g. 106 DOF), the LU factorization may take too much time. In such a case, (5.23) must be computed by an iterative method, which can also be based on Krylov-subspace (see [85] and the references in it).

5.2 Comparison Between the Arnoldi Algorithm, the Guyan Algorithm and the Control Theory Methods

All the algorithms described in chapter 3 offer the possibility for automatic model order reduction of the system (4.3). The mathematically superior control theory methods offer a global error bound as well. Unfortunately, due to the com- 3 putational effort ofOn() for these methods, they are impractical for large-scale systems. The Guyan algorithm can also be used for MOR of the large-scale thermal systems, but it requires a larger order of the reduced model than Arnoldi to achieve the same accuracy. In this section, the Arnoldi-based MOR is numerically compared to the MOR using the Guyan algorithm, the Singular Perturbation Approximation, the Hankel Norm Approximation and the Balanced Truncation Approximation. Figure 5.16 shows a comparison between the two reduced, order 20 models of the microthruster. One of them was computed by the Arnoldi process and the other by the Guyan algorithm (using ANSYS substructuring). Master degrees of freedom (terminal nodes) needed for the Guyan algorithm were chosen automatically by ANSYS. The maximal relative error by the Arnoldi-based reduction is less than 0.5% (see Figure 5.2), whereas this error by Guyan-based reduced order modeling ascends to over 64% (not shown). This large error by the Guyan algorithm-based reduction mostly occurs during the transient heating phase and vanishes within the

107 5 Numerical Results steady-state response, according to equations (3.45) and (3.50). The approximation error for the reduced heat capacity matrix (3.58) decreases as the orderr of the reduced system grows. This is demonstrated in Figure 5.17. However, the maximal relative error between the full-scale solution and the reduced solution of the order 200 still amounts to 6% (not shown), which is significantly worse than in the case of Arnoldi-based reduction to order 20.

400

300

200

100 Full-scale model (1,071 DOF) Temperature (°C) Temperature Reduced model order 20, via Arnoldi Reduced model order 20, via Guyan 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s)

Figure 5.16 Step response of the microthruster when using the Arnoldi and Guyan algorithms.

From the microthruster example we can see that the Guyan method offers less accuracy for the reduction of thermal models than Arnoldi. This is because the reduced order modeling based on Guyan makes an attempt to generalize equation (3.45) for a steady-state thermal response to the transient thermal problem (4.3) using a coordinate transformation of the form:

T e I = ⋅ T (5.24) –1 ⋅ e T i –Kii Kie

This leads to exact matrix condensation for the heat conductivity matrix K, but an approximated condensation for the heat capacity matrix C.

108 5.2 Comparison Between the Arnoldi Algorithm, the Guyan Algorithm and the

400

300

200 Full-scale model (1,071 DOF) 100 Reduced model order 200, via Guyan Temperature (°C) Temperature Reduced model order 100, via Guyan 0 Reduced model order 20, via Guyan 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s) Figure 5.17 Step responses of the microthruster when using Guyan reduction to different orders.

Arnoldi-based reduction, on the other hand, starts with moment matching for the transient thermal problem (4.3) as it is, and amounts to a coordinate transfor- nr× mation of the form (5.2), whereVR∈ is gained directly as an output of the Arnoldi algorithm andε vanishes for the described case studies. Another advantage of the Arnoldi algorithm over Guyan method is its iterative nature. Whereas with the Guyan algorithm the equations (3.50) and (3.58) must be recomputed each time the number of master degrees of freedom is changed, only an additional vector matrix multiplication is needed in each iterative step of the Arnoldi orthogonalisation procedure. Finally, the fact that no master degrees of freedom need to be chosen by the Arnoldi algorithm contributes to its convenience. Due to their high computational requirements, the control theory methods from chapter 3.1 can only be applied to small-scale models with no more than a few thousand degrees of freedom. Hence, we present here the numerical results of MOR of the microthruster model (with 1,071 DOF), using the three methods available in the SLICOT library [92]: the balanced truncation approximation (BTA), the singular perturbation approximation (SPA) and the Hankel norm approximation (HNA). For the specified error bound:

109 5 Numerical Results

() ()≤ Gs – Gr s 0.1 (5.25) a BTA algorithm within SLICOT estimated the size of the reduced system to be r = 7 . Figure 5.18 shows the relative error of the Arnoldi-based reduction compared to the relative error caused by the BTA-based reduction to the order 7. A maximal relative error by Arnoldi-based reduction approaches 4% during the transient heating phase, whereas in an optimal BTA-based reduction, this error amounts to less than 0.01%. However, the steady-state is better approximated by Arnoldi than by BTA. Figure 5.19 offers a closer look to the relative errors for BTA, SPA and HNA reduction. Out of the three control theory methods, only SPA preserves stationary state. HNA and BTA do not preserve the stationary state, but yield smaller errors within the transient phase than the SPA.

1 Arnoldi-based reduction to order 7 BTA-based reduction to order 7 0.1

10-3 Relative error (%) 10-5

0 0.2 0.4 0.6 0.8 1 Time (s) Figure 5.18 Relative error of the BTA and Arnoldi-based reduction for the microthruster model (1,071 DOF).

As we have already mentioned, due to their computational complexity the mathematically superior control theory methods are not directly applicable for large-scale FE models. Nevertheless, it is possible to make use of them for such models as well within the framework of sequential model order reduction. It is based on consecutively applying the Arnoldi algorithm and control-theory methods, and will be numerically demonstrated in chapter 6.

110 5.3 Conclusion

0.01

-0.01

-0.02 Relative error (%) BTA-based reduction to order 7 SPA-based reduction to order 7 -0.03 HNA-based reduction to order 7

0 0.2 0.4 0.6 0.8 1 Time (s)

Figure 5.19 Relative error of the BTA, SPA and HNA-based reduction for the microthruster model (1,071 DOF).

5.3 Conclusion

In this chapter we have numerically demonstrated that, although the Arnoldi algorithm is presently limited to linear systems and has no global error estimate, it can already be used for highly effective modeling and simulation of electro-thermal MEMS devices. It appears to work extremely well for heat transfer in 2D and 3D, which is consistent with the recent observations of other groups [150], [151] which have been published during our work.

Table 5.2 Main characteristics of Arnoldi algorithm when applied to electro-ther mal MEMS models Arnoldi for electro-thermal models approximation of the complete output reduction of weakly nonlinear systems increase in calculation efﬁciency formal conversion into HDL more accurate than Guyan more efﬁcient than control theory

111 5 Numerical Results

The Arnoldi algorithm allows the restoration of the whole temperature domain, and can thus be viewed as a fast time integration procedure. It allows the reduction of weakly non-linear systems, for the case when the nonlinearity appears in the input function. The computational complexity of the reduction of thermal models via Arnoldi is comparable to a single stationary system solution. For the same accuracy the Arnoldi algorithm requires a smaller system order than the commercially available Guyan algorithm does for the thermal systems. It is much more efficient than the mathematically superior control theory methods. The major characteristics of the Arnoldi-based reduction of electro-thermal models are summarized in Table 5.2.

112 6STOP CRITERIA FOR FULLY AUTOMATIC MODEL REDUCTION VIA THE ARNOLDI ALGORITHM

In order to apply Arnoldi-based model order reduction, the MEMS designer has to provide a discretized model (e. g. a finite element (FE) model) of the device and to specify which frequency band should be well approximated by the compact model. This is done by choosing one or more expansion points in the frequency domain. The next important step is to specify the desired order of the target reduced system. A key question is: which order of the reduced system do we need to select in order to achieve a desired accuracy. A reduced model is an approximation of the original large-scale model. Hence, the difference between the two can be characterized by some error norm. In order to automate the MOR process completely, one should be able to estimate this error as a function of the reduced model’s dimension. The automatic procedure from device-level to system-level modeling is schematically shown in Figure 6.1. Unfortunately, an effective error estimate for the Krylov-subspace methods is still an open research question. To our knowledge, only local (single-frequency) error estimates have been suggested so far [154]-[156]. Recent suggestions for the Arnoldi reduction algorithm can be found in [157] and [158]. The estimate [154] has been developed for the Padé approximation of transfer function via the Lanc- zos process and is based on the properties of the tridiagonal transformation matrix created by this process. Hence, it can not be applied directly to Arnoldi-based reduction. Estimates in [155] and [156] are based on residuals of the associated linear system involved in the transfer function. Evaluation of these general expressions, as well as the choice of the right frequency domain, remains a difficult task.

113 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

Figure 6.1 Compact model extraction. Eliminating the need for user iteration makes the process fully automatic.

A suggestion in [157] involves the numerical error accumulation during the Gramm-Schmidt orthogonalisation procedure, which leads to linear dependence of the r-th basic vector from the previous r-1 and can be checked by either check- ing the vector norm or the angle between the vectors. However, this stop criteria offers no relation to the actual approximation error between the original and the reduced order model. Furthermore, numerical error accumulation by computing the projection matrix V may become significant only after a large number of iterations (several hundred). For engineering applications, an error estimate should be implemented in the design ﬂow in such a way as to ﬁt in the iterative framework of Krylov-based MOR.

In this chapter, we propose three “heuristic” approaches for estimating the error of the reduced-order model computed via the Arnoldi algorithm. The idea is either to compute the relative error between the successive reduced order models (similar to [156]) or alternatively, to compute the Hankel singular values of the reduced model in each iteration of the Arnoldi algorithm. The third approach is based on sequential strategies [159], [160]. Since we can not yet prove rigorous error bounds, we refer to our results as error indicators. The proposed strategies can be used by MEMS designers to automatically create heat-transfer macromodels using the Arnoldi algorithm.

114 6.1 Convergence of Relative Error

6.1 Convergence of Relative Error

Probably the simplest approach to estimate the model error in either the time- or frequency-domain is to compute the difference between two “neighboring” reduced models with orderrr and+ 1 . Let us define a relative frequency- response error as:

() () Gs – Gr s E ()s = ------(6.1) r Gs()

() () whereGs andGr s are the transfer functions of the original and of the reduced order model (equations (5.5) and (6.6)), respectively. Let us further define a relative frequency-response error between two successive reduced order models as:

G ()s – G ()s ˆ () r r + 1 Er s = ------() (6.2) Gr s

We have observed that for all the case studies described in chapter 4:

()≈ ˆ () Er s Er s (6.3) holds for a wide range of frequencies around the expansion points0 = 0 . As already mentioned, this error indicator can be applied in the time-domain as well. Let us define a quadratic relative step-response error as:

⋅ ∆ N t ⎛⎞yt()– y ()t 2 ε() 1 ⋅ ⎜⎟i r i (6.4) r = ---- ∑ ------() N ⎝⎠yti ti = 0

() () whereyti andyr ti are the system outputs of the full and orderr reduced system inN discrete time-points spaced ∆t apart. Let us further deﬁne a quadratic relative step-response error between two successive reduced order models as:

115 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

⋅ ∆ N t ⎛⎞y ()t – y ()t 2 ε() 1 ⋅ ⎜⎟r i r + 1 i (6.5) ˆ r = ---- ∑ ------() - N ⎝⎠yr ti ti = 0

Again, it can be shown that for the electrothermal MEMS models from chapter 4:

ε()r ≈ εˆ()r (6.6)

Note that (6.4) and (6.5) are not functions of time, but rather of the system order, and that they require a time integration of system outputs for a chosen output node. Hence, they are slightly more expensive to compute than the frequency-response errors (6.1) and (6.2) and are susceptible to error accumulation. The equations (6.3) and (6.6) are the main result of this section, and will be numerically demonstrated using examples from chapter 4. () Figure 6.2 through Figure 6.6 compare the true errorEr s to the error indica- ˆ () torEr s for the microthruster model (1,071 DOF) at different frequencies.

0 10 True error Er(s) Error indicator Ê (s) -3 r 10 -6 10 -9 convergence 10 -12 10 -15 10 Error magnitude for 10 rad/s 01020304050 System order Figure 6.2 Error indicator in the frequency domain for the microthruster model (1,071DOF) at ω = 10rad/s .

ˆ Both curvesEr andEr match well in the frequency domain around the expansion point (which is zero in our case). Additionally we can observe two effects:

116 6.1 Convergence of Relative Error

0 10

-3 10

-6 10 convergence -9 10

-12 True error E (s) 10 r Ê (s) -15 Error indicator r Error magnitude for 100 rad/s 10 01020304050 System order Figure 6.3 Error indicator in the frequency domain for the microthruster model (1,071DOF) at ω = 100rad/s .

0 True error E (s) 10 r Error indicator Ê (s) -3 r 10 -6 10 -9 10 -12 10 -15

Error magnitude for 1000 rad/s 10 01020304050 System order Figure 6.4 Error indicator in the frequency domain for the microthruster model (1,071DOF) at ω = 1000rad/s .

3 At frequencies up to ω = 10 rad/s convergence occurs when a treshold reduced system is reached (Figure 6.2 and Figure 6.3). This means that for e. g. ω = 10rad/s it is not possible to approximate the system better with more than 10 Arnoldi vectors. The system order necessary to reach convergence increases towards higher frequencies (see 3D representation in Figure 6.5), since always

117 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

Error magnitude 100

10-5 True error Er(s) 10-10 Error indicator Êr(s) 10

100 20 40 1000 System order Frequency (rad/s)

points of convergence Figure 6.5 Error indicator for the microthruster model at different frequencies.

101 rad/s 5 -2 10

-5 10 oscillations

-8 10 True error Er(s)

Error indicator Êr(s) Error magnitude for 10 01020304050 System order Figure 6.6 Error indicator in the frequency domain for the microthruster model 5 (1071DOF) at ω = 10 rad/s . more terms are needed for accurate Taylor series expansion. The convergence occurs presumably because the machine’s numerical precision has been reached. The minimal errors of the approximation for the given machine precision, for the microthruster model, areEˆ 10 andEˆ 23 for the frequencies of 10rad/s and 100rad/s.

118 6.1 Convergence of Relative Error

At high frequencies, however, convergence disappears. Instead, we observe fluctuations (Figure 6.6) due to being too far away from the expansion point s = 0 . For an expansion around a higher frequency than zero, we expect to 0 5 achieve convergence atω = 10 rad/s as well. () Figure 6.7 through Figure 6.9 show a good match between the true error Er s ˆ () and the error indicatorEr s for the optical filter model (1,668DOF) at different frequencies.

0 10 True error Er(s) -4 Error indicator Ê (s) 10 r

-8 10 convergence

-12 10

-16 10 Error magnitude for 10 rad/s 01020304050 System order Figure 6.7 Error indicator in the frequency domain for the optical filter model (1,668 DOF) atω = 10rad⁄ s .

The minimal errors of the approximation for the given machine’s precision, for 3 the optical filter model, are Eˆ ,Eˆ andEˆ for the frequencies of10 , 10 and 4 6 16 80 10 rad/s respectively. Due to the slow change of magnitude ofGs() (only five 6 orders of magnitude between 1 rad/s and10 rad/s) in the case of optical filter, convergence is still reached at relatively high frequencies. If computing for still higher frequencies, fluctuations would probably appear again and an expansion > arounds0 0 would become necessary.

119 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

0 True error E (s) rad/s 10 r 3 -4 Error indicator Ê (s) 10 r

-8 convergence 10 -12 10

-16 10

Error magnitude for 10 01020304050 System order Figure 6.8 Error indicator in the frequency domain for the optical filter model 3 (1,668 DOF) atω = 10 rad/s .

0 True error E (s) 10 r rad/s

5 Error indicator Ê (s) -4 r 10 convergence -8 10

-12 10

-16 10 Error magnitude for 10 1 50 100 150 200 System order Figure 6.9 Error indicator in the frequency domain for the optical filter model 5 (1,668 DOF) atω = 10 rad/s .

() ˆ () Figure 6.10 and Figure 6.11 compare the true errorEr s to the estimate Er s for the large-scale model of gas sensor (73,955 DOF) at different frequencies. The results are of the same quality as for the 2D models in spite of the tremendous difference in the model’s dimension. Due to the relatively small range of observed frequencies (not far away from the expansion point), no fluctuations occur.

120 6.1 Convergence of Relative Error

102 True error E (s) -1 r 10 Error indicator Êr(s) -4 10 convergence -7 10 -10 10 -13 10 Error magnitude for 0.01 Hz 01020304050 System order Figure 6.10 Error indicator in the frequency domain for the gas sensor model (73,955 DOF) atf = 0.01Hz .

102

-1 10 -4 10 -7 10 -10 True error E (s) 10 r -13 Error indicator Ê (s) 10 r Error magnitude for 10 Hz

11020304050 System order Figure 6.11 Error indicator in the frequency domain for the gas sensor model (73,955 DOF) atf = 10Hz .

Let us now show the numerical results of the time domain error indicators according to the equations (6.5) and (6.6). Figure 6.12 shows a good match between the true error in time-domainε()r and the estimate εˆ()r for the microthruster model, computed for discrete times between 0s and 0.3s with ∆t = 0.01s .

121 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

-3 10 True error ε(r) Error indicator ε(r) -5 10

-7 10 Error magnitude

-9 10 11020304050 System order Figure 6.12 Time domain estimate of the microthruster model (1,071DOF).

The time interval for error estimate was chosen based on the rise time of the microthruster device, which is approximately 0.3s (see Figure 5.2). Due to the fact that the Arnoldi algorithm preserves a stationary-state for the expansion around zero frequency, the main error accumulation can be expected during the rise time of the device and therefore the chosen time interval must be long enough. The convergence of time-domain error (after the order 25 of the reduced system has been reached) is in this case also due to the limited numerical precision of the time integrator in Mathematica. NDSolve function in Mathematica4.1 comprises six significant digits. It is possible to increase the numerical precision of time-integration, however, this leads to a common trade-off between efficiency and accuracy. Figure 6.13 compares the true error in time-domainε()r to the estimate εˆ()r for the optical filter model, computed for discrete times between 0s and 0.25s with ∆t = 0.01s . The time interval for error estimate was again chosen based on the rise time of the device (see Figure 6.4). The time-domain estimate of the gas sensor model (Figure 6.13) shows that the integration in ANSYS6.1 with 50 time steps already brings along an error of about 1%. This can be corrected by increasing the number of integration time steps within ANSYS. With 200 time steps an improvement of about 8% (the difference between both true errors are computed as in (6.4)) appears. Eventually, the true error and error estimate fit together. Note that in this case the reduced solution is

122 6.1 Convergence of Relative Error

-4 10 True error ε(r) -5 10 Error indicator ε(r) -6 10 -7 10 -8 Error magnitude 10 -9 10 1 5 10 15 20 System order Figure 6.13 Time domain estimate of the optical filter model (1,668 DOF).

-1 10

-2 10

-3 10

-4 10 ε Error magnitude -5 True error (r) (50 time-steps) 10 True error ε(r) (200 time-steps) -6 10 Error indicator ε(r) 01020304050 System order Figure 6.14 Time-domain estimate of the gas sensor model (73,955 DOF). The arrow indicates that the true error decreases as the integration precision increases. in a way “better” than the full-scale solution of the ODE system computed by ANSYS6.1 due to switching to a more accurate integrator in Mathematica4.1. Hence, in order to circumvent numerical errors caused by additional time integration, it is simpler to use a frequency-domain estimate for which only the linear equation systems (in the Laplace-domain) have to be solved. The presented strat-

123 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm egy in frequency-domain is schematically summed up as an algorithm in Figure 6.15.

s = 0 set 0 calculate vector 1, set i =2 , set tol, choose jω

calculate vector i

G ()sG– ()s Eˆ ()s = r r+1 r () Gr s

< + Er tol exit

ii1= +

+ s oscillations vary 0

- + convergence exit

Figure 6.15 Error indicators algorithm, based on the convergence of relative error in the frequency domain (equation (6.3)).

The algorithm in Figure 6.15 is similar to [154] with the advantages that it is computationally simpler and that it includes the oscillations and convergence checks. Concerning the choice of the frequency range of interest, the low-pass filter characteristics of electro-thermal models can be used to connect the rise-time and the critical frequency according to equation (5.10) for example. Another challenge is a choice of the error norm itself. The expressions (6.2) and (6.5) must be

124 6.2 Convergence of Hankel Singular Values

()≈ ()≈ eventually modified ifGr s 0 oryr ti 0 in order not to end up with huge relative errors. In Examples 5.1 through 5.3 we have made computations with temperature values in Kelvin in order to avoid such numerical problems. It would have been further possible to replace the discrete time pointsti with a continuous time variable and therefore the sum through an integral:

t end () () yt – yr t ------dt , (6.7) ∫ yt() t = 0 but as in engineering applications the results are frequently given in some few discreet points, we have decided for simplicity to deﬁne error as in (6.4).

6.2 Convergence of Hankel Singular Values

The second approach proposed is based on the computation of Hankel Singular Values (HSV) of the reduced system. Let us emphasize once more that the Hankel Singular Values of the linear dynamic system (3.1) of the order n can be computed by solving two Lyapunov equations (3.3) after the controllability grammian P and the observability grammian Q. The Hankel Singular Values are defined as square roots of the eigenvalues of the product of both grammians. Well-established model order reduction methods from chapter 3.1 offer a global error bound for an approximante of order r, based on the sum of the tail of the ordered set of Hankel Σσ{}σ,,,σ …σ , ≥ σ singular values= 1 2 n i + 1 i beginning with entry r+1: ≤ ()σ …σ GG– r ∞ 2 r + 1 ++n (6.8)

HerbyGs() is the transfer function of the original state-space model (5.5) and () Gr s is the transfer function of it’s reduced order-r model, which is obtained by using projectors originating from the solutions to (3.3). As already mentioned, it is unfortunately not efficient to solve Ljapunov equations and compute HSV for large-scale systems. Luckily we have shown that, for our case studies from chapter 4, the frequency-response error of the Arnoldi reduction σˆ can be approximated by (6.8) by only computing the HSVij of the reduced

125 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm system in each iteration. In this way, after i iterations we have a matrix-like structure:

σˆ … 11 0 0 σˆ σˆ … 21 22 0 H i = (6.9) … σˆ σˆ …σˆ i1 i2 ii

σˆ whereij is the j-th HSV of the i-th order reduced model. We have observed for the electro-thermal MEMS models from chapter 4 that after a number of Arnoldi σˆ iterations the largestij of the created reduced order models converge towards the HSV of the original model. Figure 6.16 shows that for the large-scale gas sensor model the reduced system of order 50 already reproduces the original 8 largest HSV. Furthermore, in each iteration one new value is added towards the

2 0 -2

(HSV) -4 10 σ^ σ^ convergence i1 i5

log -6 σ^ σ^ i2 i6 σ^ σ^ -8 i3 i7 σ^ σ^ -10 i4 i8

01020304050 System order Figure 6.16 Largest 8 HSV of the Arnoldi reduced gas sensor models (order 1 to 50). end of the set, while the beginning values slowly converge. This means that after a number of iterations we can consider the largest original HSV (those which do not change any more when increasing the reduced system order) as known and use

126 6.2 Convergence of Hankel Singular Values

(6.8) to approximate the frequency response error. To demonstrate this, let us set an error bound of 10% for the transfer function of optical filter model and query the order of the reduced system needed to fulfill this error. This can be expressed as:

() ()≤ , Gs – Gr s 0.1 r = ? (6.10)

Figure 6.17 shows that already after the second iteration, the first two Hankel singular Values of optical filter seem to have converged, which means that the difference between two consecutive values is negligible. Hence, we can consider both as known.

1 acceptable Convergence convergence 0 threshold

-1

(HSV) -2 10

^ ^ σ 1 σ log i i5 -3 σ^ σ^ i2 i6 σ^ σ^ i3 i7 -4 σ^ σ^ i4 i8

0 5 10 15 20 System order Figure 6.17 Largest 8 HSV of the Arnoldi reduced filter models (order 1 to 20) normalized to the corresponding HSV of the original model.

σ Observing the matrixH 6 we further see that the third reduced HSVˆ i3 con- σ –2 verges towards3 = 0.04 , i. e. has an order of magnitude of10 .

127 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

118.6 0 0 0 0 0 118 0.62 0 0 0 0 118 0.6 0.02 0 0 0 H = –5 (6.11) 6 118 0.6 0.02 1.31⋅ 10 00 –3 –4 118 0.58 0.04 2.4⋅ 10 1.1⋅ 10 0 –3 –4 –6 118 0.58 0.04 1.5⋅ 10 1.5⋅ 10 3.8⋅ 10

σ ,,σ … If we assume the worst case, that is that4 5 are of the same order of mag- σ nitude as3 , and apply the estimate (6.8) we get:

() ()≤ ⋅⋅–2 ≈ Gs – G2 s 2 1666 10 33 (6.12)

As we assume a rapid decay of the HSV for our MEMS models [85], we need some HSV decay estimate in order to correct the right side of (6.12). In [161] an λ () effective decay estimate for the eigenvaluesi P of one grammian of the symmetric system has been proposed as:

λ () ⎛⎞k – 1 ()2 j + 1 ⁄ ()2k 2 k P κ()A – 1 ------≤ ⎜⎟∏ ------(6.13) λ () ()2 j + 1 ⁄ ()2k 1 P ⎝⎠κ() j = 0 A + 1

It is based only on the knowledge of the condition numberκ of A, which can be computed by iterative methods [162]. Since we have observed essentially the λ () same quality of results for the decay estimate of HSV andi P in our case stud- σ ies, we have used a formula (6.13) to estimate an upper bound fori :

σ˜ ⎛⎞i – 1 ()2 j + 1 ⁄ ()2i 2 i κ()A – 1 ------≤ ⎜⎟∏ ------(6.14) σ ()2 j + 1 ⁄ ()2i 1 ⎝⎠κ() j = 0 A + 1

σ˜ and so correct the right side of the inequality (6.12). In (6.14),i should be under- σ stood as an estimate ofi .

128 6.2 Convergence of Hankel Singular Values

Another recently proposed estimate [163] given by:

λ ()P k – 1 λ λ k –1 k – j ------λ ()- = ------⋅ ()λ ∏ ------(6.15) 1 P 2 Re k λ + λ j = 1 k j does not require symmetry of A, but depends on the complete knowledge of the spectrum of the system matrix, and is not practical for large-scale systems. The estimate (6.13) as well as a decay curve of the true Hankel Singular Values of the tunable optical ﬁlter are shown in Figure 6.18.

True HSV σi 0 Decay estimate σi -2.5 -5 (HSV)

10 -7.5

log -10 -12.5

01020304050 Hankel singular values index i Figure 6.18 Decay estimate of HSV for the tunable optical filter model (1,668 DOF).

σ˜ By adding up all thei estimates with an order of magnitude smaller or equal σ to3 we get:

1668 σ˜ ≈ , σ˜ ≤ σ ∑ i 0.085 j 3 (6.16) ij=

This sum already indicates that we could possibly fulfill our specified error bound with only two iterations. Indeed, for the original filter model it holds:

129 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

() ()≤ ⋅ ()σ …σ Gs – G2 s 2 3 ++1668 = 0.049 (6.17)

We can repeat the whole procedure for the microthruster model. We state the task as in (6.10), i. e. to search after the order of reduced model which would fulfill the 10% error bound. Figure 6.19 shows that for the microthruster model, the

(HSV) -1

10 σ1 σ5 convergence σ2 σ6 log -2 σ3 σ7 σ4 σ8 -3

01020304050 System order Figure 6.19 Largest 8 HSV of the Arnoldi reduced microthruster models (order 1 to 50) normalized to the corresponding HSV of the original model. reduced system of order 40 already reproduces the original 8 largest HSV. The matrix structureH 40 (not displayed) shows further that the eight reduced HSV σ σ –2 ˆ i8 converges towards8 = 0.05 , i. e. has an order of magnitude of10 . σ ,,,σ …σ Assuming the worst case, that is that8 9 1071 are of the same order of σ magnitude as8 , and applying the estimate (6.8) we get:

() ()≤ ⋅⋅()–2 ≈ Gs – G7 s 2 1071– 7 10 20 (6.18)

The right side of (6.18) can again be corrected through the decay estimate (6.14) σ˜ σ σ by adding all the estimatesi ofi which are equal or smaller than8 . In this way we get:

130 6.2 Convergence of Hankel Singular Values

1071 σ˜ ≈ , σ˜ ≤ σ ∑ i 0.14 j 8 (6.19) ij=

The estimate (6.14) as well as a decay curve of the true Hankel Singular Values of microthruster are shown in Figure 6.20.

2.5

-2.5 (HSV)

10 -5

log -7.5

-10 True HSV Decay estimate -12.5

01020304050 Hankel singular values Figure 6.20 Decay estimate of HSV for microthruster model (1,071 DOF).

As in the previous example, (6.19) also indicates that the error bound of 0.1 could be fulfilled with seven iterations. Indeed, for the original system it holds:

() ()≤ ⋅ ()σ …σ Gs – G7 s 2 8 ++1071 = 0.094 (6.20)

In the case of the large-scale model of a gas sensor it was not possible to compute true Hankel singular values, hence the estimate (6.14) was computed (Figure 6.21 shows the first 1,000 estimates). Let us follow the same logic as in the 2D examples. Once again, Figure 6.16 shows that after less than 50 iterations, the first eight Hankel singular values of the gas sensor model have converged. σ From the matrixH 50 (not displayed) we find out that the ninth reduced HSV ˆ i9 σ –2 converges towards9 = 0.02 , i. e. has an order of magnitude of10 . The worst case gives:

131 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

10 0 Decay estimateσ -10 i -20 (HSV) -30 10

log -40 -50 -60

0 200 400 600 800 1000 Hankel singular values index i Figure 6.21 Decay estimate of HSV for gas sensor model (73,955 DOF).

() ()≤ ⋅⋅–2 ≈ Gs – G8 s 2 73947 10 147894 (6.21)

σ˜ Luckily, by adding up all thei estimates with an order of magnitude smaller or σ equal to9 we get:

73955 σ˜ ≈ σ˜ ≤ σ ∑ i 0.2 ,j 9 (6.22) ij= which indicates that the reduced model of order eight could have an approximate error given by:

() ()≤ ⋅ ≈ Gs – G8 s 2 0.2 0.4 (6.23)

Let us mention that the sum in equation (6.22) must not necessarily be made over the complete model size, because already afteri = 1000 has been reached, σ˜ –64 the order of magnitude ofi is only10 . Hence, the remaining terms can be neglected. () () () Note once more thatG2 s in (6.17),G7 s in (6.20) andG8 s in (6.23) should be computed by e. g. balanced truncations and hence the method suggested

132 6.2 Convergence of Hankel Singular Values here only indicates where to stop in an iterative model order reduction based on the Arnoldi process. The presented strategy is schematically summarized in Figure 6.22.

calculate vector 1, set i =2 , set ε provide decay estimate σ˜ i , choose m

calculate vector i

compute σˆ and H ij

ii1= +

- + σ ≈ ε + convergence of first k k values - sum all the estimates - km> σ˜ ≤ σ i k +

increase ε + σ˜ ε increase m ∑ i » - exit

Figure 6.22 Error indicator algorithm based on the computation of HSV of the reduced system.

The challenge in the above algorithm is to choose the right number of the largest HSV m, which must have converged if the error boundε is to be fulfilled. In the case of the optical filter and microthruster, the convergence of 2 and 7 largest HSV respectively was enough to fulfill an error bound ofε = 0.1 . If one, how-

133 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm ever, needs too many iterations to meet the order of magnitude of ε among the converged HSV, the solution of the Lyapunov equations may become too time- consuming, and hence the error bound must be increased. Presently there are no guidelines on how to coordinate these numbers.

6.3 Sequential Model Order Reduction

The basic idea behind the sequential MOR is to first use the Arnoldi algorithm to reduce a large-scale ordinary differential equation system of order n to some order r1, and then to switch to one of the mathematically superior control theory methods for further reduction from order r1 to chosen order r2. The computational effort of such an approach totals to:

()2 () ()3 O 2r1n + 2r1N z A + Or1 (6.24) where Nz(A) is a number of non-zero elements of a single system matrix A. In this way we have an exact error estimate between the reduced system of order r1 and that of order r2. The problem of choosing the proper r1 remains. We suggest to choose r1 according to one of the stop criteria from sections 6.1 and 6.2. In fact, for the tested electro-thermal MEMS models,r1 = 50 turns out to be a reason- able choice, as shown in Figure 6.23, for the gas sensor model when first using Arnoldi for the reduction from 73,955 to 200, 150, 100 and 50 respectively and then singular perturbation approximation. Figure 6.24 to Figure 6.26 compare the results of sequential MOR for the gas sensor model withr1 = 50 and r2 = 5 using singular perturbation approximation (SPA), Hankel norm approximation (HNA) and balanced truncation approximation (BTA) to the “pure” Arnoldi reductions withr = 50 andr = 5 . We observe that the target reduced order 5 can be reached with smaller error if sequential MOR is used rather than the Arnoldi algorithm alone. To combine the error indicators from sections 6.1 and 6.2 with sequential MOR, we should first compute the transfer function of the reduced system or the ε HSV in each Arnoldi iteration until some prescribed error1 has been approximately fulfilled:

134 6.3 Sequential Model Order Reduction

-0.1

-0.2

-0.3 Arnoldi to 200 + SPA to 20

Relative error (%) -0.4 Arnoldi to 150 + SPA to 20 Arnoldi to 100 + SPA to 20 -0.5 Arnoldi to 50 + SPA to 20

012345 Time (s)

Figure 6.23 Sequential reduction of a gas sensor using Arnoldi with r1=200, 150, 100 and 50 and SPA withr2 = 20 .

2.5 2 Arnoldi to 50 + SPA to 5 1.5 Arnoldi order 50 Arnoldi order 5 1 0.5 0

Relative error (%) -0.5 -1

012345 Time (s) Figure 6.24 Sequential reduction of a gas sensor using Arnoldi and SPA with r1 = 50 and r2 = 5 .

135 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

2.5

2 Arnoldi to 50 + HNA to 5 1.5 Arnoldi order 50 Arnoldi order 5 1 0.5 0

Relative error (%) -0.5 -1

012345 Time (s) Figure 6.25 Sequential reduction of a gas sensor using Arnoldi and HNA with r1 = 50 and r2 = 5 .

2.5

2 Arnoldi to 50 + BTA to 5 1.5 Arnoldi order 50 Arnoldi order 5 1 0.5 0

Relative error (%) -0.5 -1

01234 5 Time (s) Figure 6.26 Sequential reduction of a gas sensor using Arnoldi and BTA with r1 = 50 and r2 = 5 .

136 6.4 Conclusion

() ()≤ ε Gs – Gr1 s 1 (6.25) and then use one of the three presented control theory methods to reduce the model further toward the target order r2 having:

() ()≤ ε Gr1 s – Gr2 s 2 (6.26)

Using a simple triangle rulexy– ≥ xy– , the error between the target reduced system with order r2 and the full-scale system can be now expressed as:

() ()≤ ε ε Gs – Gr2 s 1 + 2 (6.27)

It should be noted that inequality (6.27) is only a guideline on how to approximately choose a reduced order model.

6.4 Conclusion

The computation of the modeling error involves a common trade-off between computational efficiency and the accuracy of approximation. We have described three strategies and, as usual, each has its advantages and disadvantages (see Table 6.1). At the present stage, the convergence of relative error and sequential MOR can be recommended for practical use. They are both straightforward to implement. The extra computational time required for the convergence of the relative error is very small, provided it is estimated in the frequency domain. In this case, the strategy can be summarized in two main steps: 1. Choose an error required at the highest relevant (cutoff) frequency. 2. Solve a linear equation system for this frequency at each iteration.

137 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

This is the only method of the three for which it is possible to detect (through the oscillations) whether the required frequency is too far from the expansion point or not. Table 6.1 Advantages and disadvantages of the proposed strategies.

Convergence of Convergence of Sequential relative error HSV MOR

Simple implementation + - +

low high medium Computational (add. linear solve in (sol. of Lyapunov (single sol. of complexity each iteration) eq. in each iteration) Lyapunov eq.) Detection wether expansion point is too +-- far away

Lowest system order - needs further research + for the speciﬁed error

The computational time for sequential model reduction is higher than that for the convergence of the relative error, because first a maximum number of Arnoldi vectors (r1 ) has to be estimated. Our tests show that for electro-thermal MEMS modelsr1 = 50 might be sufficient. Now, the further reduction fromr1 to r2 (which also has to be chosen) by BTA, SPA or HNA takes additional time. On the other hand, the order of the final reduced model might be lower than if it was determined by the convergence of the relative error for the same prescribed error bound. This is because the model of orderr2 includes information from r1 > Arnoldi vectors, althoughr1 r2 . The convergence of the Hankel singular values requires the largest computational effort, because it is necessary to solve Lyapunov equations for the reduced model in each iteration. It also requires a decay estimate for the HSV. However, it is better suited for theoretical analysis than the other two methods. The advantages and disadvantages of each method are summarized in Table 6.1.

138 6.4 Conclusion

At this time we offer no theoretical justification for the proposed strategies. Whether they will function in general remains a question which requires further research, particularly with more complex transfer functions than those of the low- pass filters. We have shown, however, that they currently offer sufficient accuracy for the presented engineering problems. Due to the simplicity of implementation, the high computational efficiency and a low susceptibility to numerical error accumulation, convergence of the relative error in the frequency domain followed by one of the control theory methods takes priority for the automatic generation of dynamic compact thermal models. Nevertheless, the convergence of the HSV and its application for error estimation certainly deserves further research.

139 6 Stop Criteria for Fully Automatic Model Reduction via the Arnoldi Algorithm

140 7MODEL REDUCTION OF INTERCONNECTED SYSTEMS

In the previous two chapters the large-scale linear systems originating from the spatially discretized heat transfer equation (4.3) were reduced by the Arnoldi algorithm without taking into account their structure. As such systems are often composed of subsystems that are interconnected, array structures for example, it is desirable, especially with a large number of subsystems, to reduce each subsystem on its own and then to couple them back together. Hence, we seek a kind of compact thermal multiport representation which allows thermal fluxes to cross the boundaries and enables straightforward coupling to the next thermal multiport. The main problem thereby is that the thermal flow is not lumped by nature as, for example, the electrical flow is along metallic wire interconnects. The ratio of elec- 8 trical conductivity of metals and that of insulators is of the order of10 . Hence, the electrical current flow takes place almost solely in metal paths. This is not the case with heat flow because the ratio of thermal conductivities in microtechnology 2 is only of the order of10 (see Figure 7.1). Therefore, it is unclear how to lump the thermal fluxes at shared surfaces between two finite element (FE) models in order to form the thermal ports (Figure 7.2) which would serve to couple together several compact models. As a matter of fact, there appear to be very few works [164], [165] on how to couple (dynamic) compact thermal models. The goal of this chapter is to present and discuss the currently available solutions for model order reduction of thermal micro-array structures. In section 7.1 the MEMS case study, a MOS-transistor-based microhotplate array model, is presented. It is a high dimensional model, containing several hundreds of FE nodes at each shared interface. Section 7.2 describes the possibility of reducing the entire array, i. e., without decoupling the parts, via the block Arnoldi procedure, which

141 7 Model Reduction of Interconnected Systems is suitable for multiple-input-multiple-output (MIMO) linear systems. It offers a simple and effective solution for the reduction of array structures, but “hides” the problem of coupling the dynamic compact thermal models. Section 7.3 discusses the possibility of coupling reduced order models via substructuring based on a modified Guyan algorithm. It is usually used in structural mechanics, but is available for the thermal domain as well. As it demands the preservation of all shared (coupling) nodes, the resulting orders of compact models are much larger than by block Arnoldi. Section 7.4 addresses some possibilities of coupling reduced models in general case via additional flux inputs and introduces a structure preserving MOR. Finally, section 7.5 concludes the chapter with a brief comparison of the available methods.

Electric flow Heat flow κ κ 8 κ κ 2 cond / ins = 10 cond / ins = 10

Conductor

V1 V2 T1 T2

Insulator

Figure 7.1 Comparison of Distributive effects: electric flow is lumped and heat flow is not.

7.1 Microhotplate Array

A microhotplate array model is built upon a real gas sensor device fabricated in industrial CMOS technology [18], [148]. It is composed of 500µm× 500µm CMOS multilayer fully suspended membrane and a ring PMOS transistor heater of5µm gate-length and 720µm overall gate width buried in a silicon island under the membrane, as shown in Figure 7.3. Figure 7.4 shows an example of a fabricated chip with digital circuitry in the lower part, analog circuitry in the middle and an array of three microhotplates in the upper section. The microhotplates are covered with drop-deposited noncrystalline SnO2 as sensitive layers. Full advan-

142 7.1 Microhotplate Array

thermal multiports

Figure 7.2 Continuous thermal flux through the shared interface of two FE models. The goal is to model the “FE cubes” as thermal multiports i. e. to lump a flux. tage is taken of the features offered by applying CMOS-technology. All sensor values can be set and read out via the digital interface, which drastically reduces the packaging complexity since the number of bond wires is the same as for a single microhotplate.

thick-film SnO resistor 2 membrane (dielectric layers)

Si-island bulk silicon PMOS-ring-transistor heater Figure 7.3 Cross-section of a single microhotplate.

Before the microhotplate devices are fabricated, the designs undergo an extensive simulation process, such as thermal modelling, using finite-element simulations. Modeling the transducer and hotplate behavior is an extremely important

143 7 Model Reduction of Interconnected Systems

analog circuitry & converters microhotplates with MOS-transistor heater digital circuitry (controller & interface)

packaging 4.4 x 5.4 mm2 bondpads Figure 7.4 Micrograph of the chip with microhotplate array and circuitry. Pic- ture courtesy of M. Graf (ETH Zürich, Switzerland). step towards the formulation of a compact sensor model for monolithic system realizations. The parameters of interest include the hotplate thermal resistance and thermal time constant, a prediction of which facilitates and accelerates the design of the monolithic array systems for a given microhotplate design. For test purposes, we built a finite element model of22× microhotplate array structure which contains 100,934 DOF (meshed and discretized with ANSYS using SOLID90 elements) and several hundreds FE nodes at each shared surface. We modeled the membrane as a single layer with material properties as proposed in [166]. The transistor heater is modeled as a lumped element circular poly silicon heater buried in Si island under the membrane, which allows for a pure thermal simulation. Our model captures the thermal efficiency of the real device, which is6°C/mW [167], to within 70%. For the reduction of interconnected thermal systems it is necessary to describe the thermal crosstalk within a reduced order model. To accomplish this, we mod- elled a22× array as a part of still larger array. At all four side walls of the chip, the convection boundary conditions (2.9) were applied. The heat transfer coeffi- 4 ° cient was set to h =10 and the ambient temperature was set toT ambient = 0 C . The initial temperature was also set to zero. The heat source power for each circular heater was set to40mW . In sections 7.2 and 7.3 it will be shown that it is possible to turn the heat sources on and off also after the model order reduction step.

144 7.1 Microhotplate Array

Temperature contour plots in Figure 7.5 through 7.7 demonstrate that the chosen BCs result in a temperature increase of about10°C per added heat source every- where on the chip. This effect must be taken into account if, e. g. no automatic temperature control is employed. Note, however, that the model only resembles the real device and hence the true crosstalk level may be different.

Figure 7.5 Steady-state temperature distribution in°C of a22× microhotplate array with a single heat source.

Figure 7.6 Steady-state temperature distribution in°C of a22× microhotplate array with two heat sources.

145 7 Model Reduction of Interconnected Systems

Figure 7.7 Steady-state temperature distribution in°C of a22× microhotplate array with three heat sources.

7.2 Block Arnoldi

A block Arnoldi method [98] offers a straightforward approach to reduce a model of an array structure with multiple heat sources, provided that the number of devices within an array remains moderate (see the discussion at the end of the section). With block Arnoldi the entire array structure is meshed and reduced in a similar way to a classical single-input-single-output Arnoldi process. Let us briefly explain this. The multiple-input-multiple-output heat transfer equation in discretized form is given by:

CT⋅ ˙ + KT⋅ = FQt⋅ () (7.1) T yt()= E ⋅ T

nm× np× whereFR∈ andER∈ are the input and the output matrix, and m and p denote the number of inputs and outputs, respectively. Unlike in (4.3) the inputs () []… T → m [ , ∞)→ p Qt = Q1 Q2 Qm R and the outputsy 0 R are both vector- valued functions. As a result, a transfer function of (7.1) is a matrix-valued ratio- pm× nal functionG:CC→ , which is analog to (5.5), given by:

146 7.2 Block Arnoldi

T –1 Gs()= E ⋅⋅()KsC+ Fs, ∈ C (7.2)

The block Arnoldi method obtains the information of the leading Taylor coeffi- () cients ofGs , i. e. moments around a chosen frequencys0 (zero in our case), in a similar manner to the standard SISO case. Before the block Arnoldi can be employed, the two matrices C and K have to be reduced to a single matrix, denoted by A in the following. This can be done by rewriting (7.2) as follows:

T –1 Gs()= –E ⋅⋅()IsA– ⋅ B (7.3)

–1 –1 whereAK= – C andBK= – F . m columns of the matrix []… Bb= 1 b2 bm are the starting vectors of the so-called block Krylov-subspace, which will be defined below. We start by defining the right block Krylov matrix as:

R (), 2 n – 1 K AB = BABAB … A B (7.4)

Note that the left block Krylov matrix, which is needed in two-sided approaches, T would have been induced byA and E, analogous to (3.26). Our goal is to define a sequences of ascending r-dimensional subspaces, r = 12,,… , that are spanned by the first r linearly independent columns of the R matrixK ()AB, . To properly define these subspaces, we need to delete the lin- j early dependent columns in (7.4). For example, if a columnA bi in (7.4) is linearly dependent on earlier columns, than this column and all its successive A-multipliers need to be deleted. Consequently, we obtain a so-called deflated block Krylov matrix:

defl 2 K ()AB, = B AB A B … B (7.5) 1 2 3 jmax

,,… , where for eachj = 12 jmax ,B j is a submatrix ofB j – 1 if deflation (the process of deleting linearly dependent vectors) occurred within the j-th Krylov

147 7 Model Reduction of Interconnected Systems

j – 1 blockA Bj . Here, for= 1 we setB1 = B . Hence, if we denote bym j the number of columns ofB j , we have:

mm= ≥ m ≥≥… m ≥1 (7.6) 1 2 jmax R The rth right block Krylov subspace induced by A and B,K ()AB, , is the subspace spanned by the first r columns of (7.5):

R ⎧⎫j – 1 K ()AB, = colspan⎨⎬ B , AB ,,… A B (7.7) r ⎩⎭1 2 j

… ≤≤ Therefore,rm= 1 + m2 + + m j with1 jjmax . We remark that, for a single starting vectorBb= block Krylov subspace (7.7) is identical to (5.3). The block Arnoldi process (Algorithm 7.1 [98]) extends the classical Arnoldi algo- R(), ≥ rithm to block Krylov subspacesK r AB ,r 1 . Some variations of the block- Arnoldi algorithm can be found in [168]. The reduction of the original system (7.1) is performed by projection, as in (5.4):

T T T V CV⋅ z˙ + V KV⋅ z = V FQt⋅ () (7.8) () T ⋅ yr t = E Vz

Let us go back to our case study. For the 2x2 microhotplate array with four heat T sources and four defined outputs, the multiple input vector Q(t) = [Q1 Q2 Q3 Q4] does not partake in the algorithm 7.1 and hence it is possible to switch each heat source on and off after the order reduction. Figure 7.8 and Figure 7.9 compare the step responses of the full-scale and reduced order 50 model, computed by block Arnoldi, for the case when two heat sources (each40mW ) are switched on. The observed outputs are located in the center of each hotplate. Figure 7.10 shows further that the reduced model accurately reproduces the crosstalk effect presented in the previous section.

148 7.2 Block Arnoldi

Algorithm 7.1 Inputs: AB and T Outputs: projection matrixVHV and the matrix of the reduced system = AV

0. Start: ,,… , Setvˆk = bk , fork = 12 m . ≈ Setmc = m and choose a deﬂation thresholddtol 0 .

For i = 12,,… ,r do: 1. Deﬂation: ≤ Ifvˆi dtol setmc = mc – 1 . Ifmc = 0 setii= – 1 and stop. ≠ ,,,… Ifmc 0 setvˆk = vˆk + 1 forkii= + 1 im+ c – 1 . Return to step 1.

2. Normalize: ⁄ vi = vˆi vˆi

3. Compute the next vector from the new block: vˆ = Av⋅ im+ c i 4. Orthogonalize: For k=1,2,,,i T seth = v vˆ and vˆ = vˆ – v h ki, k im+ im+ c im+ c k ki, For kim= – + 1, im–c + 2, …n – 1 Tc c seth , = v vˆ andvˆ = vˆ – v h . ik i km+ c km+ c km+ c i ik, End for i loop. One may ask the question if the block Arnoldi could have been avoided by using four times the standard Arnoldi with Krylov subspaces induced by single vectors instead. For example, one could generate scalar approximations for all sixteen coefficient functions of the44× matrix valued transfer function (7.2) via a (), (), (), suitable basis for standard Krylov subspacesK r Ab1 ,K r Ab2 , K r Ab3 (), andK r Ab4 . In this way instead of a single reduced model (7.8) with four heat () sources and output temperature vectoryr t , we would have four reduced models, each describing a complete output (speciality of Arnoldi process) for a single heat

149 7 Model Reduction of Interconnected Systems

350 300 250 200 150

Temperature (°C) Temperature 100 Full-scale model (100,934 DOF) Reduced model order 50 50 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Time(s) Figure 7.8 Step response of the full-scale and reduced order models in two output points when two heat sources of40mW each are switched on.

-0.2

-0.4 Relative Error (%) -0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 Time(s) Figure 7.9 Step response errors corresponding to Figure 7.8.

()…,, () source. Their outputs can be denoted withyr1 t yr4 t . In the case when, for example, the first and the fourth heat sources are on, the chosen output temperatures from (7.8) can be computed as a superposition of SISO models outputs:

() () () yr t = yr1 t + yr4 t (7.9)

150 7.2 Block Arnoldi

400

300

200 Temperature (°C) Temperature 100 Reduced model order 50

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Time(s) Figure 7.10 Step response of the reduced order 50 models in a single output point when one, two or three heat sources of40mW each are switched on. in each single node. The standard Lanzos algorithm, on the other hand, would have to be used sixteen times due to the approximation of a single output only. However, in [98] it is stated that block Krylov methods for the MIMO systems result in more efficient MOR than those based on standard Krylov subspaces. This issue certainly deserves more research. It seems so far that block Arnoldi offers a simple and effective solution for the model order reduction of interconnected MIMO systems. Its main advantage over a substructuring method (described in section 7.3) is that no master degrees of freedom have to be chosen and the minimal size of the reduced model is not influenced by the large number of interface nodes. As the output matrix E does not partake in the algorithm 7.1, the approximation of the complete output (EI= nn× ) is warranted as in the SISO case (see section 5.1.1). A further advantage is that the reduced system (7.8) can be very easily formally transferred into a HDL model, as will be demonstrated in the section7.2.1. The disadvantage of block Arnoldi becomes evident with the growing number of devices within an array. At some point, the system matrices C and K grow so large that the LU- or Cholesky decomposition and matrix vector multiplications needed for the construction of the projection matrix may become difficult or even prohibitive. In such cases, it is clearly better to search for an alternative to reduce each device on its own (using either projection or modal analysis) and then to couple them. How this can be done for-

151 7 Model Reduction of Interconnected Systems mally in general case, as well as in some special cases will be discussed in the following sections.

7.2.1 Interconnected System Behavioral Simulation We present here the HDL form of the system (7.8) and demonstrate that the crosstalk effect is described sufficiently well. In the system-level model it becomes obvious that the control of the heat sources is independent of the reduced model. Block Arnoldi projection is schematically shown in Figure 7.11. A T ⋅⋅ T ⋅⋅ reduced system is defined withKr = V KV ,Cr = V CV and T ⋅ Fr = V F . Figure 7.12 shows a HDL model for the 2 by 2 microhotplate model reduced by block Arnoldi. A template containing a reduced ODE system (7.8) is back coupled to each one of the four heaters through their input power ,,… ,,… Q1 Q4. The observed output temperaturesT out, 1 T out, 4 , placed in the ,,… center of four hotplates and the resistivities control temperaturesT 1 T 4 are computed as linear combinations of the reduced states:

∗ ⋅ T i = V z (7.10) T ⋅ T out, i = E Vz

The reduced model contains no nodes in the physical sense and hence the template region in Figure 7.12 is to be understood as a visualization of the equation system (7.8). A MAST template code for the22× microhotplate array and the netlist file can be found in the Appendix A.2. Figure 7.13 compares the numerical simulation results of the behavioral MAST model to those of the full-scale model integrated in ANSYS and the reduced order model (7.8) integrated in Mathematica. Again, the difference between Mathemat- ica and SABER solution totals to the numerical integration errors between the different solvers.

152 7.2 Block Arnoldi

Block Arnoldi

T = V z

CT + KT = F Q Q Q C z + K z = F Q Q Q [Q1 2 3 4] r r r [Q1 2 3 4]

Figure 7.11 Schematics of the Block Arnoldi projection for the microhotplate array model.

V 2 Q i = R i

R R R R 1 2 3 4 V

T2 T3 T T out,2 out,3 C z + K z = F Q Q Q Q r r r [ 1 2 3 4]

T1 T4 T T out,1 out,4

T ground Figure 7.12 Structure of the implemented HDL model of the microhotplate array.

153 7 Model Reduction of Interconnected Systems

350 300 250 200 150 100 Temperature (°C) Temperature Full-scale model (100,934 DOF) 50 Reduced model order 50 MAST HDL model 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) Figure 7.13 Step response of the full-scale (100,934 DOF) and reduced order 50 model in a single output node. The reduced system was integrated in Mathematica4.1 and SABER (MAST HDL model) and the full-scale system with ANSYS7.1.

7.3 Coupling of Reduced Order Models via Substructuring

This commercially available method [107] is based on the modification of the Guyan algorithm, which was described in chapter 3.3. Disadvantages of this algorithm when applied to order reduction of electro-thermal models have been demonstrated in section 5.2. Its major advantage, however, is that it allows the coupling of several reduced models into an array structure, due to the physical preservation of the shared surface nodes during reduction. The equation system which describes two thermal models, RM1 with orderr1 and RM2 with orderr2 , reduced by a modified Guyan algorithm using (3.50) and (3.58) is given as:

⋅ T˙ ⋅ T ⋅ () Cr1 e1 + K e1 = Fr1 Q1 t r1 (7.11) C ⋅ T˙ + K ⋅ T = F ⋅ Q ()t r1 e2 r2 e2 r2 2 × × ∈ r1 1 ∈ r2 1 whereT e1 R andT e2 R are terminal node vectors of RM1 and RM2 (see Figure 7.14). By performing a Guyan reduction of each submodel, the

154 7.3 Coupling of Reduced Order Models via Substructuring

chosen terminal nodesT e1 andT e2 must contain at least those surface nodes which are to be coupled to another model. Depending on the desired accuracy and efﬁciency, additional nodes can be chosen as terminals. In Figure 7.14 the shared nodes are noted as T e1, i andT e2, i .

FE model

Guyan Te1 reduction Te2 RM1 RM2

Te1,bulk Te1,i Te2,i Te2,bulk

Figure 7.14 Two Guyan-based reduced models with terminal nodesT e1 and T e2 which can be coupled over the surface nodesT e1, i andT e2, i .

The coupling is done through the following set of2m constraint equations for the temperatures and fluxes on the shared interface:

, ≤≤ T , – T , = 01 im e1 i e2 i (7.12) ˙ ˙ , ≤≤ T e1, i – T e2, i = 01 im

155 7 Model Reduction of Interconnected Systems

(7.12) reduces the number of variables in (7.11) bym . The same coupling procedure can be used for coupling of two full-scale models or one reduced and one full-scale model. We now transform the equation system (7.11) into a block- matrix form:

˙ C11 C12 T 1 K11 K12 T 1 F1 ⋅ + ⋅ = (7.13) C C ˙ K K T F 21 22 T 2 21 22 2 2 where

T T 1 = e1, i (7.14) are the coupling nodes belonging to RM1 andT 2 are all other nodes belonging to both models:

∪∪ T 2 = T e1, bulk T e2, bulk T e2, i (7.15)

This assignment is arbitrary, i. e. it is only important thatm different nodes, out of2mT shared nodes are gathered within a single vector (1 in this case). For sim- () () plicity the input termsQ1 t andQ2 t have been incorporated in the load vector. ⋅ Let us further rewrite the constraint equations (7.12) asT 1 = XT2 and ⋅ T˙1 = XT˙2 or:

IX– ⋅ T = 0 (7.16) IX– ⋅ T˙ = 0

T × whereImm stands for the unity matrix of dimension andT = T 1 T 2 . In a general case the right hand side of (7.16) may be different than zero. The problem of solving (7.13) under the constraints (7.16) can be carried out by several methods [169], [170]. In ANSYS substructuring, the Lagrange multipliers

156 7.3 Coupling of Reduced Order Models via Substructuring adjunction method [171] is used with the goal of minimizing the energy function. The total energy of the thermal model is given as:

1 T 1 T T E = ---T˙ ⋅⋅CT˙ + ---T ⋅⋅KT– T⋅ F (7.17) 2 2 whereCK and are the system matrices and F is a load vector from (7.13). To impose the constraints (7.16), we adjoin2m Lagrange multipliers collected in λ λ vectors1 and2 and form a Lagrangian:

(),,˙ λ ,λ λT ⋅⋅˙ λT ⋅⋅ LT T 1 2 = E + 1 IX– T + 2 IX– T (7.18)

(), λ ˙ λ λ The minimization ofLT with respect toTT , ,1 and2 yields the multi- plier-augmented form of (7.13):

C C I 0 T˙ K K 0 I T F 11 12 1 11 12 1 1 T ˙ T T T 2 K K 2 F C21 C22 –X 0 ⋅ + 21 22 0 –X ⋅ = 2 (7.19) λ λ IX– 00 1 0000 1 0 λ λ 0 00 00 2 IX– 00 2

T λ λ By eliminating1 and1 + 2 from (7.19) the new equation system with the dimensionr1 + r2 – m , which describes two reduced coupled devices, is given through:

˜ ⋅⋅˜ ˜ C T 2 + K T 2 = F (7.20) where

157 7 Model Reduction of Interconnected Systems

T T ˜ ⋅⋅ ⋅ K ⋅ C = X C11 XX+ C12 + 21 XC+ 22 T T ˜ ⋅⋅K ⋅ K K ⋅ K (7.21) K = X 11 XX+ 12 + 21 X + 22 ˜ T ⋅ F = X F1 + F2

It is worth of noting that at this stage it is still possible to turn the heat sources on and off, i. e. to scale the load vectors by scalingFr1 andFr2 in (7.11) and hence,F1 andF2 .

After solving (7.20) it is possible to recover T 1 through (7.16) and gradually via (7.11) to split the nodes to the original reduced models, thus expanding ﬁnally the complete temperature field of a single device via (3.48). Let us go back to our case study. In order to accurately describe the heating area of the single hotplate during the reduction, we have preserved the 948 heater nodes additionally to the surface nodes. As a result, a reduced model of a single microhotplate (3.56) contains 2,140 ordinary differential equations. The lower limit of the reduced model when applying this method would have been the description of the interfaces, which is 1,192 ODEs. The coupling of four reduced models into an array structure according to equations (7.20) and (7.21) results in an equation system of 7,351 ODEs. Note once more that the full-scale FE model contains 100,934 equations. An important computational issue here is that the reduced system (7.20) is dense and hence its computation may not bring along a large decrease in CPU time versus computing the full-scale system, which is sparse. Figure 7.15 shows the assembled contour plots of four substructured and coupled microhotplates when two heat sources are switched on. The approximation of the thermal crosstalk effect, which takes place within a full-scale FE model (Figure 7.6), is excellent. Figure 7.16 compares further the step responses of the substructured array with the step response of the full-scale model and the step response of the block Arnoldi reduced model from Figure 7.8. The convenience of Guyan method is that it preserves the surface nodes in the physical sense, while using the projection matrix as defined in (3.52). Hence, it is

158 7.3 Coupling of Reduced Order Models via Substructuring

Figure 7.15 Expanded solution of four coupled reduced models when two heat sources of 40mW each are switched on.

10 Full-scale 2x2 array model (100,934DOF) Temperature (°C) Temperature 5 Reduced block Arnoldi model order 50 Substructured array model order 7,351 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Time(s) Figure 7.16 Single output step responses of the full-scale and reduced order models created by block Arnoldi and Guyan-based substructuring. Two heat sources of 40mW are switched on. possible to decouple an array, to reduce each device on its own and to couple several reduced models by using any of the methods for treating linear multifreedom constraints in finite elements systems [170]. The major disadvantage, however, is that the lower limit of the single reduced model is the description of the interfaces, which implies, with accurate meshing, relatively large model sizes. For the case

159 7 Model Reduction of Interconnected Systems studies presented in this thesis, several hundred nodes per single surface were accounted for. It is of course possible to reduce the number of coupling nodes by coarsening the mesh on the interface, but this results in a more difficult mesh generation and lower mesh quality. Besides, we may need to choose more bulk (non terminal) nodes in order to reach the desired accuracy (see the results in section 5.2). Hence, the best approach is to use the Krylov-subspace-based reduction of a single device because it guarantees smaller dimensions and higher precision than Guyan-based substructuring and then to couple several reduced models into an array structure. Unfortunately, this is possible only in some special cases or under special approximations, as will be shown in the next section.

7.4 Coupling of Reduced Order Models in the General Case

In sections 7.2 and 7.3 two state-of-the-art methods for MOR of interconnected thermal models were applied to the microhotplate array model. As usual, both have advantages and disadvantages. The main difference between them is that block Arnoldi reduces the entire system model and substructuring allows the model to be decoupled and reduces each subsystem. Hence, block Arnoldi may not be computationaly effective in cases when the number of subsystems is very large. Because substructuring preserves all shared nodes within the reduced system, it may result in unnecessarily large reduced models. In this section we discuss the possibilities of generalizing both approaches. The first possibility is to reduce each submodel without preserving the shared nodes, i. e. with Arnoldi, and to try to couple the reduced models back into a reduced array model. We will show that under some circumstances this is possible. The second possibility is to reduce each subsystem with Arnoldi without even decoupling the array model, which is of course, the best solution. We will show that the newest mathematical theories are developing in this direction.

7.4.1 Coupling by Fluxes If MOR of a single device model is done by projection, as it is the case in Krylov-subspace methods, the surface nodes are not preserved any more. Instead, we have generalized coordinatesz . In this case the Lagrange multipliers method

160 7.4 Coupling of Reduced Order Models in the General Case can not be applied any more and therefore the only way to couple the reduced models back into an array structure is over additional “flux inputs”. Let us explain this.

The two, still uncoupled models (M1 and M2 in Figure 7.17) are described through the equation system:

T1 T2 M1 M2

Φα = −Φβ Tα Tβ

Figure 7.17 Coupling via surface fluxes.

˙ K F ⋅ C11 0 T 1 11 0 T 1 1 u1 ⋅ + ⋅ = (7.22) 0 C ˙ 0 K T F ⋅ u 22 T 2 22 2 2 2

Let us, to begin with, suppose that M1 and M2 are the full-scale models with surface nodes vectors T α andT β . The heat fluxes between the models surfaces are introduced as:

qα = hT⋅ ()α – T β (7.23) qβ = hT⋅ ()β – T α

161 7 Model Reduction of Interconnected Systems where the heat transfer coefficient h can be chosen on the basis of, e. g. experimental measurements. The coupling of M1 and M2 can be done by adding the right hand side of (7.23) to (7.22) while first rewriting (7.23) into a matrix form:

qα T = H ⋅ 1 (7.24) qβ T 2

It should be noted that H is of low rank, i. e. only the rows and columns corresponding to the coupling nodes T α and T β are non-zero. This rows contain h and α β -h entries at thei andi locations:

α β i i 00… 00 α H = i 0 h … –h 0 (7.25)

β … i 0 –h h 0 00… 00

A coupled system is then given through:

˙ K T F ⋅ T C11 0 T 1 11 0 1 1 u1 1 ⋅ + ⋅ = + H ⋅ (7.26) 0 C ˙ 0 K T F ⋅ u T 22 T 2 22 2 2 2 2

Notice that we need a strategy for how to choose an optimal h, and that in this way the temperatures in the coupling nodesT α andT β , unlike the Lagrange multipliers adjunction method, are not exactly equal to each other during time evolution. It is now possible to treat the coupling flux terms as additional inputs by rewriting (7.26) as:

162 7.4 Coupling of Reduced Order Models in the General Case

C ⋅ T˙ + ()K –h ⋅ T = F ⋅ u + h ⋅ T 11 1 11 11 1 1 1 12 2 (7.27) ⋅ ˙ ()⋅ ⋅ ⋅ C22 T 2 + K22–h22 T 2 = F2 u2 + h21 T 1

,,… whereh11 h22 are the matrix blocks ofH . For convenience we will set (7.27) in the left hand side formulation:

A ⋅ T˙ = T + B ⋅ u + h ⋅ T 1 1 1 1 1 1 2 (7.28) ⋅ ˙ ⋅ ⋅ A2 T 2 = T 2 + B2 u2 + h2 T 2 with

–1 –1 ()⋅ ()K ⋅ A1 = – K11 – h11 C11,,A2 = – 22 – h22 C22 –1 –1 ()⋅ ()K ⋅ B1 = – K11 – h11 F1 ,,B2 = – 22 – h22 F2 –1 –1 ()⋅ ()K ⋅ h1 = – K11 – h11 h12 andh2 = – 22 – h22 h21 . (7.29)

The equation (7.28) is valid for the system of two coupled models when no MOR was performed. Let us emphasize once more that what we really want is to couple two reduced models in cases when no surface nodes T α andT β have been preserved. In order to be able to follow the reverse path, we project (7.28) using:

T = V ⋅ z 1 1 1 (7.30) T ⋅ 2 = V 2 z2 and get a coupled reduced system of the form:

T ⋅⋅⋅ T ⋅⋅ T ⋅⋅ ⋅ z˙1 = V 1 A1 V 1 z1 + V 1 B1 u1 + V 1 h1 V 2 z 2 (7.31) T T T z˙ = V ⋅⋅⋅A V z + V ⋅⋅B u + V ⋅⋅h V ⋅ z 2 2 2 2 2 2 2 2 2 2 1 1

163 7 Model Reduction of Interconnected Systems

wherez1 andz2 are the generalized coordinates. It should be noted that (7.31) is formally correct, regardless of the rank of the matricesh1 andh2 and the choice ofV 1 andV 2 . Hence, if we know the projection matrices and the heat transfer coefficients at the shared surfaces, the equation (7.31) couples the reduced models. The key question becomes how to find good projection subspaces, especially if the number of coupling nodes is large, as is usually the case after the spatial discretization of the thermal domain. The authors in [164] suggest a modal approach, which is to construct the projection matricesV 1 andV 2 by using the eigenvectors of the original system –1 ⋅ –1 ⋅ matricesC11 K11 andC22 K22 of (7.22). This approach offers the advantage of not having to recreate the reduced model corresponding to the new coupling conditions, but provides no guidelines on how to choose the important eigenvectors. It is also possible to reduce a number of shared interface nodes by coarsening the mesh. In such a case, the rank of the matricesh1 andh2 gets low. Hence, it is convenient to apply the block Arnoldi projection to each subsystem in (7.31) with the starting vectors defined as (non-zero) columns ofB1 h1 andB2 h2 . V 1 andV 2 are computed as the basis for the block Krylov subspaces:

⎧⎫⎧⎫ K ⎨⎬A , B h andK ⎨⎬A , B h (7.32) r1⎩⎭1 1 1 r2⎩⎭2 2 2

However, the reduction of shared node numbers at meshing level requires additional knowledge of the ANSYS meshing tool and makes a process non-automatic. It is further possible to consider only a sum of columns ofh1 andh2 by constructing (7.32) and then to compute (7.31) as written above. This is equivalent to focusing the entire heat flow between two devices through a single point. Both of the last approaches require a sacrifice of precision at the model’s surfaces, but in cases when there is only a small temperature gradient over the surface nodes, this might be an acceptable approximation. Let us conclude this section with the statement that the coupling equation (7.31) works regardless of the number of coupling nodes and that the choice of the projection subspaces requires further research.

164 7.4 Coupling of Reduced Order Models in the General Case

7.4.2 Structure Preserving Model Order Reduction The methods presented in sections 7.3 and 7.4.1 were both based on decoupling the array structure, reducing each single device model and then coupling the reduced models back into an array model. In this way a number of equations corresponding to the shared surface nodes was doubled. By back coupling with Lagrange multipliers during substructuring this number was reduced again, whereas by coupling via surface fluxes this was not the case. It would clearly be best not to decouple an array at all, but to still be able to reduce only its parts (different than by block Arnoldi in section 7.2). A structure-preserving model reduction technique, which has recently received a lot of attention by mathematicians [173]-[175], seems to offer some possibilities in this direction.

x1 x2

x3 Figure 7.18 Interconnected system whose structure should be preserved during order reduction.

Let us assume the interconnected system is built out of two subsystems (Figure 7.18) and described through the equation system:

x˙ A 0 A x 1 11 13 1 b1 ˙ = 0 A A ⋅ x + ⋅ ut() (7.33) x2 22 23 2 b2 ˙ A31 A32 A33 x3 0 x3

165 7 Model Reduction of Interconnected Systems

For thermal systems, a diagonal heat capacity matrix was assumed in (7.33). We denote the number of the DOFs of both subsystems and the interface byn1 , n2 andn3 respectively. The goal of structure-preserving MOR is to replace (7.33) with a smaller system which has the same block structure:

z˙ A 0 A z 1 r11 r13 1 br1 ˙ = 0 A A ⋅ z + ⋅ ut() (7.34) z2 r22 r23 2 br2 ˙ Ar31 Ar32 Ar33 z3 0 z3 () < The number of DOFs of the state vectorzt isr1 ++r2 r3 , withr1 n1 , < < r2 n2 andr3 n3 . Furthermore, each sub-block in Ar should be a direct reduction from the corresponding sub-blocks in the original system, e. g.Ar11 from A11 . This can be accomplished by picking a block diagonal projection matrix:

V 1 00

V = 0 V 2 0 (7.35)

00V 3

T ⋅⋅ T ⋅ and then constructing the new model withAr = V AV andbr = V B . In [174] and [175] it is stated that if V is computed by first computing a basis for the Krylov subspace:

–1 ()˜ ⎛⎞–1, span V = Kr⎝⎠A A b (7.36) and then partitioningV˜ as:

˜ n1 V 1 V˜ ˜ = n2 V 2 (7.37) n ˜ 3 V 3

166 7.4 Coupling of Reduced Order Models in the General Case

˜ and orthogonalizing theV i ’s using the QR method to get:

˜ T V i = V iRi ,V i V i = I r (7.38) then the transfer functions of (7.33) and (7.34) match in the first r moments around arbitrary frequency. Let us observe a special case when the moment matching about infinity is aimed, i. e.

()˜ (), span V = Kr Ab (7.39) is computed [156]. In this case, it is not difficult to see that due to A12 ===A21 b3 0 in (7.33) we have:

()˜ ≈ (), span V1 Kr A11 b1 and (7.40)

()˜ ≈ (), span V2 Kr A22 b2 hhh (7.41) because

A ⋅ b 2 ⋅ 11 1 A11 b1 + Sum 2 Ab ⋅ A b = A22 b2 ,= 2 ⋅ , etc. (7.42) A22 b2 + Sum A ⋅ b + A ⋅ b 31 1 32 2 Sum and hence,V 1 andV 2 may be approximated directly from the submatrices A11 and A22, i. e. from the submodels without shared interface nodes. However, it is not clear how to approximateV 3 . If we assume that the number of interface nodes is to be preserved during reduction, we may construct a reduced order matrix and the load vector of the interconnected system as:

167 7 Model Reduction of Interconnected Systems

r1 r2 n3

T T r V ⋅⋅A V 0 V ⋅⋅A I Ar = 1 1 11 1 1 13 and (7.43) T T r2 ⋅⋅ ⋅⋅ 0 V 2 A22 V 2 V 2 A23 I n3 ⋅⋅ ⋅⋅ ⋅⋅ IA31 V 1 IA32 V 2 IA33 I

T ⋅ V 1 b1 b r = T ⋅ (7.44) V 2 b2 0

(), without having to computeK r Ab at all, but rather computing only (7.40) and (7.41). Unfortunately, in this way only the moments about infinity could be approximately matched and even this is not guaranteed, for the lastn3 rows of (), K r Ab are not necessarily a unity matrix, as assumed in (7.39). Hence, we can offer no proof at present for (7.39). Besides, the problem of having to preserve the dimension of the interface block, which was already present in substructuring, remains. Nevertheless, this method deserves further research.

7.5 Conclusion

We have presented several methods for a MOR of interconnected thermal systems. Presently, we are able to apply the block Arnoldi method and the Guyan- based substructuring. Block Arnoldi is straightforward, reduces the entire array model and offers high accuracy. Its main disadvantage is that it is not easily scal- able to a large number of devices within an array. Substructuring requires the decoupling of the array model and physical preservation of all shared nodes during reduction with modified Guyan’s algorithm. This allows easy back coupling of the reduced models afterwards, but results in an

168 7.5 Conclusion

full-scale FE array model

single hotplate

do not decouple decouple

MOR don't preserve interface nodes preserve interface nodes (projection ?) (substructuring) block Arnoldi structure preserving MOR ?

input fluxes Lagrange multipliers

reduced array model

Figure 7.19 Possibilities for MOR of interconnected systems.

169 7 Model Reduction of Interconnected Systems unnecessarily large reduced array model. Hence, both methods require alternatives for use in engineering practice. We have further discussed the possibility of ﬁnding a “mixture” of both methods, i. e. to reduce each subsystem by projection and then to couple the reduced models into an array, which could work in a general case. Equation (7.31) couples the reduced models (via surface fluxes) regardless of the number of coupling nodes and regardless of the choice of the projection subspaces. The question which remains open is how to find good projection matrices. Due to the large number of shared nodes (several hundreds), it appears that the Krylov subspace methods have reached their limits. However, by observing the numerical simulation results for the microhotplate array, we find almost no temperature gradient over the model’s surface nodes. This can also be expected for other thermal MEMS models, which by design allow a heat transfer mostly in a horizontal direction. In such cases it may be acceptable to focus the heat ﬂux on one or several points, which would allow us to consider the coupling flux terms as additional inputs. Hence, the subspaces (7.32) could be easily computed and block Arnoldi could be applied to the reduction of each subsystem. The back coupling can be done afterwards using equation (7.31). Figure 7.19 schematically summarizes the main characteristics of the discussed methods.

An alternative method which would not involve decoupling of the array model could be connected with structure preserving MOR. This method certainly deserves more attention.

170 8CONCLUSION AND OUTLOOK

In this thesis we have developed a methodology for applying mathematical model order reduction (MOR) to the automatic generation of dynamic compact thermal models for MEMS. Various MOR methods for linear systems have been researched and successfully tested on several novel MEMS devices. Presently, we can recommend the Arnoldi algorithm for the practical use in model order reduction of very large scale electro-thermal MEMS models. The software tool mor4ansys enables automatic Arnoldi-based reduction of the 3D finite element thermal models that are made and meshed in ANSYS. We have shown that with Arnoldi it is possible to reduce linear thermal ODE systems of around 100,000 equations to orders between 50 and 100 with only a minimal loss of accuracy. This increases computational efficiency by more than 10 times in the case of a microhotplate gas sensor and in general reduces computational time to the time needed for a single stationary system solution. Further advantages of Arnoldi-based reduction are the approximation of the complete output and the reduction of models with temperature dependent heating power. Its main disadvantage was the fact that no error estimate between the original and the reduced models exists. We have suggested three heuristic strategies for error estimation. At the present stage, the convergence of relative error and sequential model order reduction can be recommended for practical use. They are both straightforward to implement. We have researched the possibilities for model order reduction of MEMS array structures. Presently, we are able to apply Block Arnoldi and Guyan-based substructuring. Block Arnoldi can be recommended in cases of a moderate number of devices within an array. However, when the number of interconnected devices grows, both methods need alternatives.

171 8 Conclusion and Outlook

The next step in model order reduction of thermal systems could be the treatment of temperature dependent material properties, as well as the solution of the so-called inverse problem. Often, it happens that a MEMS designer has the measured data, but is uncertain about the material parameters (for example the heat capacity and thermal conductivity of doped silicon). In such a case, it is possible to use model order reduction within a fast design alteration cycle (see Figure 8.1) to extract the true parameter values via curve ﬁtting (parameter optimization).

put funct in ion a lt FEM e r n

MOR DCTM p

J a

M O R

fast design alteration K Figure 8.1 Application of MOR to parameter extraction in MEMS design process.

Even quicker design change could be achieved by the further development of parametric MOR. It is connected to the preservation of, for example, film coefficient as a parameter in compact thermal models, which allows it to be changed after the order reduction (fast solution loop in Figure 8.1). Preliminary work on this subject exists already [176]. Coupling of Arnoldi-based reduced thermal models also requires further research. Focusing the heat flux through only several points between two thermal MEMS devices seems to make sense from the physical point of view. The newest mathematical developments of so-called structure preserving model order reduction are

172 also very promising. They seem to open the possibility of reducing a single device model without even decoupling the array. Lastly, the search for an optimal model reduction method is far from being fin- ished. As already mentioned in chapter 2, a new group of methods called SVD- Krylov are presently being developed. They are supposed to combine mathematical advantages of control theory methods and computational advantages of Krylov-subspace methods. Their application to thermal problems could also be of interest for the electro-thermal MEMS community.

173 8 Conclusion and Outlook

174 APPENDIX

A.1 Material Properties The following material parameters have been used for solid modeling in ANSYS:

Si SiO SiNx poly-Si SOG Igniting Microthruster (Figure 4.2) 2 (spin on glas) fuel Density (kg/m3) 2328 2270 2270 3280 2270 1526

Speciﬁc heat Cp (J/kg K) 702 1000 1000 700 1000 1852

Thermal conductivity 100 1.4 49 100 1.4 0.25 κ()W/mK Electrical resistivity -- -10-5 -- ()Ωm Table A.1: Material properties of the materials used in the 2D axi-symmetrical model of microthruster. All values are given at room temperature.

Membrane layers Platinum heater on the Optical filter (Figure 4.6) membrane layers –4 –3 Density (kg/m3) 44.06⋅ 10 14.95⋅ 10 Specific heat Cp (J/kg K) 731.9 309.5 –6 –6 Thermal conductivity κ()W/mK 3.05⋅ 10 38.05⋅ 10 –14 Electrical resistivity (Ωm ) - 5.3⋅ 10 Table A.2: Material properties of the materials used in the 2D model of optical filter. All values are given at room temperature.

175 Gas sensor(Figure 4.9) Si SiO SiNx Pt PSG Glass Air

Density (kg/m3) 2330 2200 3200 21450 2200 2200 1.293

Speciﬁc heat Cp (J/kg K) 700 700 700 133 730 730 1005

Thermal conductivity 150 1.5 3.7 71.6 0.614 1.15 0.02454 κ()W/mK Electrical resistivity (Ωm ) - - - 10-7 -- -

Table A.3: Material properties of the materials used in the 3D model of microhotplate gas sensor. The conductivity of PSG (Phosphor Silicat Glass) is given at255°C . All other values are at room temperature.

176 A.2 MAST Templates Below the shortened listings are presented. The original source code can be found on the thesis CD.

TEMPLATE GASSENSORNETLIST.SIN

#Netlist for the microhotplate gas sensor #Author: Tamara Bechtold #Date: April 2004 #Description: Netlist for microhotplate gas sensor with back #coupled temperature- dependent heater (according to #Figure 5.14) #Hierarchy: lower level: GasSensor.sin, r_tc_own.sin; upper #level: none ###########################################################################

GasSensor.1 pow(r_tc_own.Rheater) TN Tout 0 r_tc_own.Rheater B A TN = r0=274.94, alpha=1.469e-3 r.Rlead A 0 = 148.13 #lead resistance v.1 B 0 =dc = 14

TEMPLATE GASSENSOR.SIN

#Reduced order model of the microhotplate gas sensor #Author: Tamara Bechtold #Date: April 2004 #Description: Reduced order 10 model is given as an ODE system #with heater’s power as input function (according to equations #(5.18) and (5.19)) #Hierarchy: lower level: none; upper level: GasSensorNetlist.sin ########################################################################### template GasSensor cp TN Tout Tground ref p cp #reference variable - power over the thermal resistor thermal_c TN, Tout, Tground { #BEGIN BODY var tc z1, z2, z3, z4, z5, z6, z7, z8, z9, z10

177 var p heat_flow_out_ground #help through variable in order to #fix Tout var p heat_flow_heater_ground #help through variable in order to #fix TN control_section { initial_condition(z1, 0) … initial_condition(z10, 0) } #ODE System z1 = d_by_dt(1/-9.9387*z1)--10.9897/-9.9387*z2\ --38.8233/-9.9387*z3--72.0347/-9.9387*z4--47.2934/-9.9387*z5\ --74.1997/-9.9387*z6--102.914/-9.9387*z7--36.7739/-9.9387*z8\ --40.699/-9.9387*z9--23.3836/-9.9387*z10\ --881110/-9.9387*abs(cp) … z10 = d_by_dt(1/-7.81642*z10)--0.271456/-7.81642*z1\ --0.350085/-7.81642*z2--1.7525/-7.81642*z3--\ 3.78266/-7.81642*z4--2.86592/-7.81642*z5--7.01618/-7.81642*z6\ --11.9542/-7.81642*z7--6.46862/-7.81642*z8\ --8.80007/-7.81642*z9--24065.7/-7.81642*abs(cp) p(Tout->Tground) += heat_flow_out_ground heat_flow_out_ground: tc(Tout)=-0.010882*z1\ -0.00422874*z2-0.0031295*z3-0.00395612*z4-0.00240095*z5\ -0.00336211*z6-0.0045475*z7-0.00163377*z8\ -0.00178964*z9-0.00108029 *z10 p(Theater->Tground) += heat_flow_heater_ground heat_flow_heater_ground: tc(Theater)=-0.0109511*z1\ -0.0042771*z2 -0.00333204*z3-0.00431827*z4-0.00284783 *z5\ -0.00430855 *z6 -0.00626694 *z7-0.00265075*z8-0.00310066 *z9\ -0.00240555 *z10 } #END BODY

178 ELEMENT TEMPLATE R_TC_OWN.SIN

#Description: r_tc_own.sin is a modified thermal resistor (element template r_tc.sin created by Analogy, Inc.) between two #electrical nodes and a thermal node. This template generates #power as a system variable (pow), which allows it to be exported #and used as a reference variable in template GasSensor.sin

TEMPLATE MICROHOTPLATEARRAYNETLIST.SIN

#Netlist for the microhotplate array #Author: Tamara Bechtold #Date: July 2004 #Description: Netlist for microhotplate array with four back coupled temperature-dependent heaters (according to Figure 7.12) #Hierarchy: lower level: MicrohotplateArray.sin, r_tc_own.sin; upper level: none ########################################################################### MicrohotplateArray.1 pow(r_tc_own.R1) pow(r_tc_own.R2) pow(r_tc_own.R3) pow(r_tc_own.R4)\ T1 T2 T3 T4 Tout1 Tout2 Tout3 Tout4 0 r_tc_own.R1 A B T1 = r0=625, alpha=0 switch.1 B 0 = condition = closed r_tc_own.R2 A C T2 = r0=625, alpha=0 switch.2 C 0 = condition = open r_tc_own.R3 A D T3 = r0=625, alpha=0 switch.3 D 0 = condition = closed r_tc_own.R4 A E T4 = r0=625, alpha=0 switch.4 E 0 = condition = open v.1 A 0 =dc = 5

TEMPLATE MICROHOTPLATEARRAY.SIN

#Reduced order model of the 2x2 microhotplate array #Author: Tamara Bechtold #Date: July 2004

179 #Description: Reduced order 50 model is given as an ODE system with four heaters’ power as input functions (according to equations (7.8) and (7.10)); template body is analog to the body of GasSensor.sin #Hierarchy: lower level: none; upper level: MicrohotplateAr- rayNetlist.sin ########################################################################### template MicrohotplateArray cp1 cp2 cp3 cp4 T1 T2 T3 T4 Tout1 Tout2 Tout3 Tout4 Tground ref p cp1, cp2, cp3, cp4 #reference variables - power over #thermal resistors thermal_c T1 T2 T3 T4 Tout1 Tout2 Tout3 Tout4 Tground

{ #BEGIN BODY (analog to GasSensor.sin, except with four inputs and four outputs) } #END BODY

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197 198 PUBLICATIONS LIST

The scientific results of this thesis have been partly published in the following journal papers and conference proceedings:

JOURNAL PUBLICATIONS T. Bechtold, E. B. Rudnyi, Markus Graf, Andreas Hierlemann, J. G. Korvink, “Connecting heat transfer macromodels for MEMS array structures”, Journal of Micromechanics and Microengineering, 15(6), pp. 1205-1214, (2005).

T. Bechtold, E. B. Rudnyi, J. G. Korvink, “Error indicators for fully automatic extraction of heat-transfer macromodels for MEMS”, Journal of Micromechanics and Microengineering, 15(3), pp. 430-440, (2005). T. Bechtold, E. B. Rudnyi, J. G. Korvink, “Automatic Generation of Compact Electro-Thermal Models for Semiconductor Devices”, IEICE Transactions on Electronics, E86-C, pp. 459-465, (2003). B. Salimbahrami, B. Lohmann, T. Bechtold, J. G. Korvink, “A two-sided Arnoldialgorithm with stopping criterion and MIMO selection procedure”, Math- ematical and Computer Modelling of Dynamical Systems, 11(1), pp. 79-93, (2005). J. Hildenbrandt, T. Bechtold, “Microhotplate Gas Sensor” , in Benner, P.,Golub, G., Mehrmann, V., Sorensen, D. (eds) “Dimension Reduction of Large-Scale Systems”, Springer, 45, (2005). D. Hohlfeld, T. Bechtold, “Tunable Optical Filter”, in P. Benner, V. Mehrmann, D. Sorensen, “Dimension Reduction of Large-Scale Systems”, Springer, 45, (2005). T. Bechtold, E. B. Rudnyi, J. G. Korvink, “Dynamic Electro-Thermal Simulation of Microsystems - a Review”, submitted to Journal of Micromechanics and Microengineering.

199 CONFERENCE PUBLICATIONS M. Salleras, T. Bechtold, L. Fonseca, J. Santander, E. B. Rudnyi, J. G. Korvink, S. Marco, “Comparision of Model Order Reduction Methodologies for Thermal Problems”, Proc. EUROSIME, pp. 60-65, (2005). T. Bechtold, J. Hildenbrand, J. Woellenstein, J. G. Korvink, “Model Order Reduction of 3D Electro-Thermal Model for a Novel Micromachined Hotplate Gas Sensor”, Proc. EUROSIME, pp. 263-267, (2004). T. Bechtold, E. B. Rudnyi and J. G. Korvink, “Error Estimation for Arnoldi-based Model Order Reduction of MEMS”, Proc. NanoTech, (2004). T. Bechtold, E. B. Rudnyi, J. G. Korvink, C. Rossi, “Efficient Modelling and Simulation of 3D Electro-Thermal Model for a Pyrotechnical Microthruster”, Proc. PowerMEMS, (2003). T. Bechtold, B. Salimbahrami, E. B. Rudnyi, B. Lohmann, J. G. Korvink, “Krylov- Subspace-Based Order Reduction Methods Applied to Generate Com- pact Thermo- Electric Models for MEMS”, Proc. NnanoTech, (2003). B. Salimbahrami, B. Lohmann, T. Bechtold, J. G. Korvink, “Two-sided Arnoldi Algorithm and Its Application in Order Reduction of MEMS”, Proc. Mathmod, pp. 1021-1028, (2003). T. Bechtold, E. B. Rudnyi, J. G. Korvink, “Automatic Order Reduction of Thermo- Electric Models for MEMS: Arnoldi vs. Guyan”, Proc. ASDAM, (2002). T. Bechtold, E. B. Rudnyi, J. G. Korvink, “Automatic Order Reduction of Thermo- Electric Models for Micro-Ignition Unit”, Proc. SISPAD, pp. 131-134, (2002). E. B. Rudnyi, T. Bechtold, J. G. Korvink, C. Rossi, “Solid Propellant Micro- thruster: Theory of Operation and Modelling Strategy”, Proc. NanoTech 2002- 5755, (2002).

200 ACKNOWLEDGEMENTS

First of all I would like to thank my supervisor Prof. Dr. Jan Gerrit Korvink for giving me the chance to perform this interesting work in his group. His unique ideas and visions were always a great source of motivation for me. Furthermore, his words: “Ph.D. student is not a job, it’s a style of life” changed my attitude to work forever. I also wish to thank Prof. Dr. Christoph Ament for co-examining this thesis. Dis- cussions with him contributed very much to my better understanding of the model order reduction topic and to improving the manuscript.

I am deeply grateful to my group leader and second supervisor Dr. Evgenii Boris- ovich Rudnyi for most energetic support all the way through my Ph.D. work. His endless patience and good advises helped me to never give up. I feel honoured to have had a chance to work together with this people. I owe special thanks to Prof. Dr. Boris Lohmann for a number of nice discussions during our cooperation. I would also like to thank Dr. Marcus Graff for explaining me a microhotplate array, to Jürgen Hildenbrandt for his nice Master thesis about the gas sensor device and to all my past and present colleagues of the simulation group for their diverse contributions to the accomplishment of this work: Dr. Andreas Greiner, Dr. Zhenyu Liu, Jan Lienemann, Oliver Rubenkönig, Christian Moosmann, Dr. Jens Müller, Dr. Ricardo Osorio, Darius Koziol, Dr. Kaiping Zeng, David Kauzlaric, Martin Gayer, Lars Pastewka, Dr. Jeong Sam Han and Dr. Lihong Feng. Finally, my deepest gratitude goes to my beloved husband Dr. Dennis Hohlfeld. His support in both professional and private life is unequaled. Whatever may hap- pen, my gratitude and admiration for him will never fade away. This work was partly funded by the EU (European Community), by the DFG (Deutsche Forschungsgemeinschaft) and by the Women’s Representative of the University of Freiburg through a grant for women in science.

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