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*W. Teulings, I. Schmadlak, T. Hauck.*

**Thermal simulation of smart power devices on PCB with SPICE**.

International Exhibition and Conference for Power Electronics,
PCIM 2010, Nuernberg, 4 - 6 Mai 2010.

*E. B. Rudnyi.*

Effective Electrothermal Simulation for Battery Pack and Power Electronics in HEV/EV.

3rd Virtual Vehicle Symposium in Graz, GSVF, 6-7 May, 2010.

*T. Bechtold, D. Hohlfeld, E. B. Rudnyi, M. Guenther.*

**Efficient extraction of thin film thermal parameters from
numerical models via parametric model order reduction**.

J. Micromech. Microeng. v. 20, N 4, 045030, 2010.

Paper at JMM

*E. B. Rudnyi.*

Simulation of IGBT converter.

ANSYS Conference & 27. CADFEM Users Meeting,
18 - 20 November 2009, Congress Center Leipzig.

*L. Kostetzer.*

System Level Battery Thermal Behaviour Study.

ANSYS Conference & 27. CADFEM Users Meeting,
18 - 20 November 2009, Congress Center Leipzig.

*T. Hauck, I. Schmadlak, L. Voss, E. B. Rudnyi.*

Electro-Thermal Simulation of Multi-channel Power Devices: From Workbench to Simplorer by means of Model Reduction.

NAFEMS, Seminar: Multi-Disciplinary Simulations - The Future of Virtual Product Development, 9-10 November 2009, Wiesbaden, Germany, paper #5, 10 p.

*T. Hauck, W. Teulings, E. B. Rudnyi.*

Electro-Thermal Simulation of Multi-channel Power Devices on PCB with SPICE.

THERMINIC 2009, 7-9 October, Leuven, Belgium, 6 p.

*A. Augustin, T. Hauck.*

Transient Thermal Compact Models for Circuit Simulation.

Paper 2.5.3. 24th CADFEM Users Meeting 2006, International Congress on FEM Technology with 2006 German ANSYS Conference, October 25-27, 2006 Schwabenlandhalle Stuttgart/Fellbach, Germany.

A thermal problem is the easiest one for model reduction as the discretization of the heat transfer partial differential equation leads to a system of ordinary differential equations of the first order. In this case, one can use moment matching methods directly, as originally they have been developed for first order systems. The majority of examples presented below for the heat transfer are related to electro-thermal MEMS but the conclusions are applicable to other thermal applications as well.

The main conclusion is that model reduction is a very efficient way to find an accurate low-dimensional compact thermal model that approximates well dynamic behavior of the original high-dimensional problem. The results show that a thermal model with about hundred thousands nodes can be accurately described by a reduced system with about 30 generalized coordinates.

Full list of publications for thermal and electrothermal models in reverse chronological order.

*T. Bechtold, J. G. Korvink, E. B. Rudnyi,* **Fast Simulation of Electro-Thermal MEMS: Efficient Dynamic Compact Models.**

Book at Amazon or Springer.

*J. G. Korvink, E. B. Rudnyi,*

Keynote: Computer-aided
engineering of electro-thermal MST devices: moving from device to system
simulation.

EuroSimE'03, 4th international conference on thermal & mechanical
simulation and experiments in micro-electronics and micro-systems,
Aix-en-Provence, France, March 30 - April 2, 2003.

*T. Bechtold, E. B. Rudnyi, J. G. Korvink.*

**Dynamic electro-thermal simulation of microsystems: a review.**

Journal of Micromechanics and Microengineering 2005, v. 15, N 11, p. R17-R31.

Preprint, Final paper at IOP.

An overview of electro-thermal modeling of microsystems is presented. We consider the most important coupling between thermal and electrical phenomena, and then focus on the industry's central concern, that of Joule heating. A description of different solution approaches for the heat transfer partial differential equation, which constitutes the central part of electro-thermal simulation, is given. We briefly review the analytical solutions and consider further the numerical approaches, which are based on spatial discretization of the thermal domain. Lastly, we describe the final level of approximation, the dynamic compact thermal modeling. We emphasize the formal model order reduction methods, because they directly follow the spatial discretization, and thus preserve the investment into the finite element modeling.

*T. Bechtold, E. B. Rudnyi, J. G. Korvink.*

mor4ansys: Efficient Model Order Reduction of Finite Element Electro-Thermal MEMS Models.

Microsystemtechnik Kongress 2005, 10-12 October,
Freiburg, Germany, VDE Verlag GMBH, p. 649-652.

*T. Bechtold, E. B. Rudnyi and J. G. Korvink,*

Automatic Generation of
Compact Electro-Thermal Models for Semiconductor Devices.

IEICE Transactions on Electronics, 2003, v. E86C, N 3, pp. 459 - 465.

Final paper at IEICE.

A high power dissipation density in today's miniature electronic/mechanical systems makes on-chip thermal management very important. In order to achieve quick to evaluate, yet accurate electro-thermal models, needed for the thermal management of microsystems, a model order reduction is necessary. In this paper, we present an automatic, Krylov-subspace-based order reduction of a electro-thermal model, which we illustrate by a novel type of micropropulsion device. Numerical simulation results of the full finite element model and the reduced order model, that describes the transient electro-thermal behavior, are presented. A comparison between Krylov-subspace-based order reduction, order reduction using control theoretical approaches and commercially available reduced order modeling has been performed. A Single-Input-Single-Output setup for the Arnoldi reduction algorithm was proved to be sufficient to accurately represent the complete time-dependent temperature distribution of the device.

*T. Bechtold, J. Hildenbrand, J. Woellenstein and J. G.
Korvink,*

Model Order Reduction of
3D Electro-Thermal Model for a Novel Micromachined Hotplate Gas Sensor.

Proceedings of 5th International conference on thermal and mechanical
simulation and experiments in microelectronics and microsystems, EuroSimE2004
May 10-12, 2004, Brussels, Belgium, p. 263-267.

Final paper at IEEE

*C. Bohm, T. Hauck, E. B. Rudnyi, J. G. Korvink*,

Compact Electro-thermal
Models of Semiconductor Devices with Multiple Heat Sources.

Proceedings of 5th International conference on thermal and mechanical
simulation and experiments in microelectronics and microsystems, EuroSimE
2004, May 10-12, 2004, Brussels, Belgium, p. 101 - 104.

Final paper at IEEE

*T. Bechtold, D. Hohlfeld, E. B. Rudnyi, H. Zappe, J. G. Korvink.*

Inverse Thermal Problem via Model Order Reduction: Determining Material Properties of a Microhotplate .

THERMINIC 2005, 11th International Workshop on Thermal Investigations of ICs
and Systems, 27 - 30 September 2005, Belgirate, Lake Maggiore, Italy,
p. 146 - 150.

*L. H. Feng, E. B. Rudnyi, J. G. Korvink.*

**Preserving the film coefficient as a parameter in the compact thermal
model for fast electro-thermal simulation.**

IEEE Transactions on Computer-Aided Design of Integrated Circuits and
Systems, December, 2005, v. 24, N 12, p. 1838-1847.

Final paper at IEEE.

Compact thermal models are often used during joint electro-thermal simulation of MEMS and circuits. Formal model reduction allows us to generate compact thermal models automatically from high-dimensional finite element models. Unfortunately, it requires us to fix a film coefficient employed to describe the convection boundary conditions. As a result, compact models produced by model reduction do not comply with the requirements of being boundary condition independent. In the present paper, we suggest an approach of successive series expansion with respect to the film coefficient as well as to the frequency during model reduction, which allows us to overcome the problem and keep the film coefficient as a symbolic parameter in the reduced model. The approach is justified with a numerical example of electro-thermal simulation of a microthruster unit.

*E. B. Rudnyi, L. H. Feng, M. Salleras, S. Marco, J. G. Korvink.*

Error Indicator to Automatically Generate Dynamic Compact Parametric Thermal Models .

THERMINIC 2005, 11th International Workshop on Thermal Investigations of ICs
and Systems, 27 - 30 September 2005, Belgirate, Lake Maggiore, Italy,
p. 139 - 145.

*L. H. Feng, E. B. Rudnyi, J. G. Korvink, C. Bohm, T. Hauck.*

Compact Electro-thermal
Model of Semiconductor Device with Nonlinear Convection Coefficient.

In: Thermal, Mechanical and Multi-Physics Simulation and Experiments in
Micro-Electronics and Micro-Systems. Proceedings of EuroSimE 2005, Berlin,
Germany, April 18-20, 2005, p. 372-375.

Final paper at IEEE

*T. Bechtold, E. B. Rudnyi, J. G. Korvink, M. Graf, A. Hierlemann.*

**Connecting heat transfer macromodels for MEMS-array
structures.**

Journal of Micromechanics and Microengineering 2005, v. 15, N 6, p.
1205-1214.

Preprint, Final paper at IOP.

Different methodologies to extract a dynamic compact thermal model of a microelectronic or MEMS device have been developed in recent years. They include strategies based on data fitting, a time-constant spectrum, modal analysis and finally formal model reduction. Researchers seek compact thermal multiport representation for system level simulation. However, thermal flux is not lumped by nature as electrical flow and, as a matter of fact, there appears to be very few works on how to couple dynamic compact thermal models with each other. In the present work, we take a finite element model of a MOS-transistor-based microhotplate array made in ANSYS as a case study. We consider two available techniques to make the model reduction. First, we employ the block Arnoldi algorithm that makes model reduction of the whole array at once. Second, we use the modified Guyan algorithm for a single hotplate and couple reduced models via substructuring. We compare both techniques with each other and discuss the possibility of combining the best parts of the two approaches.

Evgenii B. Rudnyi

Designed by

Masha Rudnaya