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Development of Boundary Condition Independent Reduced Order Models using POD
Keywords: IC packages, Proper Orthogonal Decomposition (POD), Finite Volume Method (FVM)
Compact Thermal Models (CTM) to represent IC packages has been most commonly developed using the DELPHI methodology. This network-based CTM, that provides the junction temperature of specific components, does not provide any other information such as the temperature of other components or the temperature gradient across the nodal surface. Also, its use poses a number of challenges when an attempt is made to extend the methodology to multi-heat source components. If there is a model that provides the complete information at a high level of accuracy with less computational resources, then such a model can be effectively used in system level simulations. Such a model that offers many orders of reduction in computations, and thus processing time, is a reduced order model. One way to obtain reduced order models is to statistically process the data that captures the complete characteristics of the system. Proper Orthogonal Decomposition (POD) is one such method that can be used to reduce the order of the degree of freedom (DOF) or variables of the computations. The application of this method has been suggested in [1] [2] [3]. POD along with the Galerkin projection allows us to create reduced order models that reproduce the characteristics of the system with a considerable reduction in computational resources while maintaining a high level of accuracy. The 1D transient heat equation is solved using a Finite Volume Method (FVM) using a fully implicit formulation. The reason for using FVM is that the major CFD software packages that are used in electronics cooling (Flotherm and Icepak) are FVM based codes. Once the methodology for generating reduced order models for complex 3d objects is established, it can be easily integrated into such codes. The result from the CFD solution is validated with analytical solutions for a particular set of boundary and initial conditions. A transient case is considered as it adds more complexity than a steady state 1D heat equation, which has linear temperature profiles. The POD method is applied to this 1D case to show that a reduced order model can easily replicate the temporal behavior of the temperature variable for the same set of boundary conditions used for generating the CFD solution. The process is then extended to show that a reduced order model can be developed that captures the characteristics of the data over different boundary conditions. References 1. Shapiro, B., “Creating Compact Models of Complex Electronic Systems: An Overview and Suggested Use of Existing Model Reduction and Experimental System Identification Tools”, IEEE Trans. on Components and Packaging Technologies, Vol. 26, No.1, March 2003. 2. Astrid, P., “Reduction and Predictive Control Design for a Computational Fluid Dynamics Model”, Proc. 41st IEEE Conf on Decision and Control, Nevada, 2002. 2. Nie, Q., and Joshi, Y., “Multi-scale Thermal Modeling Methodology for Electronic Cabinets”, IEEE, 2006.
Arun P. Raghupathy, Doctoral Student
Electronic Cooling Solutions Inc.
Mountain View, CA

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