# Thermal Model (38865)

### Boundary Condition Independent Thermal Model (38865)

E. B. Rudnyi, rudnyi@imtek.uni-freiburg.de

J. G. Korvink, korvink@imtek.uni-freiburg.de

One of important requirements for a compact thermal model is that it should be boundary condition independent. This means that a chip producer does not know conditions under which the chip will be used and hence the chip compact thermal model must allow an engineer to research on how the change in the environment influences the chip temperature. The chip benchmarks representing boundary condition independent requirements are described in [1].

Mathematically, the problem is that the thermal problem is modeled by the heat transfer partial differential equation when the heat exchange through device interfaces is modeled by convection boundary conditions. The latter contains the film coefficient, h_i, to describe the heat exchange for the i-th interface. After the discretization of both equations one obtains a system of ordinary differential equations as follows

E dT/dt = (A + Sum_i h_i A_i)T + B

where E and A are the device system matrices, A_i is the diagonal matrix due to the discretization of convection boudnary condition for the i-th interface, T is the vector with unknown temperatures.

In terms of the equation above, the engineering requirements read as follows. A chip producer specifies the system matrices but the film coefficient, h_i, is controlled later on by another engineer. As such, any reduced model to be useful should preserve h_i in the symbolic form. This problem can be mathematically expressed as parametric model reduction [2,3,4].

Unfortunately, the benchmark from [1] is not available in the computer readable format. For research purposes, we have modified a microthruster benchmark [5] (see Fig 1). In the context of the present work, the model is as a generic example of a device with a single heat source when the generated heat dissipates through the device to the surroundings. The exchange between surrounding and the device is modeled by convection boundary conditions with different film coefficients at the top, h_top, bottom, h_bottom, and the side, h_side. From this vewpoint, it is quite similar to a chip model used as a benchmark in [1]. The goal of parametric model reduction in this case is to preserve h_top, h_bottom, and h_side in the reduced model in the symbolic form.

We have used a 2D-axisymmetric microthruster model (T2DAL in [5]). The model has been made in ANSYS and system matrices have been extracted by means of mor4fem [6]. The benchmark contains a constant load vector. The input function equal to one corresponds to the constant input power of 15 mW.

The linear ordinary differential equations of the first order are written as:

E T=(A - h_top A_top - h_bottom A_bottom - h_side A_side) T + B u

y=Cx

where E and A are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), B is the load vector, C is the output matrix, A_top, A_bottom, and A_side are the diagonal matrices from the discretization of the convection boundary conditions and T is the vector of unknown temperatures.

The numerical values of film coefficients, h_top, h_bottom, and h_side can be from 1 to 1e9. Typical important sets film coefficients can be found in [1]. The allowable approximation error is 5 % [1].

The benchmark has been used in [7,8] where the problem is also described in more detail.

Download matrices in the Matrix Market format: (File 1), 223938 bytes. The matrix name is used as an extension of the matrix file. File T2DAL_BCI.C.names contains a list of ouput names written consecutively.

## Bibliography

1.
C. J. M. Lasance, *Two benchmarks to facilitate the study of compact thermal modeling phenomena*, IEEE Transactions on Components and Packaging Technologies, **24**, 559-565 (2001).

2.
D. S. Weile, E. Michielssen, E. Grimme, K. Gallivan, *A method for generating rational interpolant reduced order models of two-parameter linear systems*, Applied Mathematics Letters, **12**, 93-102 (1999).

3.
P. K. Gunupudi, R. Khazaka, M. S. Nakhla, T. Smy, and D. Celo, *Passive parameterized time-domain macromodels for high-speed transmission-line networks*, IEEE Transactions on Microwave Theory and Techniques, **51**, 2347-2354 (2003).

4. L. Daniel, O. C. Siong, L. S. Chay, K. H. Lee, and J. White, A Multiparameter Moment-Matching Model-Reduction Approach for Generating Geometrically Parameterized Interconnect Performance Models, IIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 23, 678-693 (2004).

5. E. B. Rudnyi, Micropyros Thruster, http://www.imtek.uni-freiburg.de/simulation/benchmark/.

6.
E. B. Rudnyi and J. G. Korvink,
*Model Order Reduction of MEMS for Efficient Computer Aided Design and System Simulation*, MTNS2004, Sixteenth International Symposium on Mathematical Theory of Networks and Systems, Katholieke Universiteit Leuven, Belgium, July 5-9, 2004.

7.
L. Feng, E. B. Rudnyi, J. G. Korvink, *Parametric Model Reduction to Generate Boundary Condition Independent Compact Thermal Model*, THERMINIC 2004, 10th International Workshop on Thermal Investigations of ICs and Systems, 29 September - 1 October 2004, Sophia Antipolis, CÂȘte d'Azur, France.

8.
L. Feng, E. B. Rudnyi, J. G. Korvink, *Preserving the film coefficient as a parameter in the compact thermal model for fast electro-thermal simulation*, 2004, to be submitted.