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Finite difference schemes for heat conduction analysis in integrated circuit design and manufacturing
Author(s) -
Shen Yijiang,
Wong Ngai,
Lam Edmund Y.,
Koh ChengKok
Publication year - 2011
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.675
Subject(s) - tridiagonal matrix , finite difference , computation , crank–nicolson method , finite difference method , boundary value problem , thermal conduction , computer science , partial differential equation , mathematics , algorithm , mathematical optimization , mathematical analysis , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material
The importance of thermal effects on the reliability and performance of VLSI circuits has grown in recent years. The heat conduction problem is commonly described as a second‐order partial differential equation (PDE), and several numerical methods, including simple explicit, simple implicit and Crank–Nicolson methods, all having at most second‐order spatial accuracy, have been applied to solve the problem. This paper reviews these methods and further proposes a fourth‐order spatial‐accurate finite difference scheme to better approximate the PDE solution. Moreover, we devise a fourth‐order accurate approximation of the convection boundary condition, and apply it to the proposed finite difference scheme. We use a block cyclic reduction and a recently developed numerically stable algorithm for inversion of block‐tridiagonal and banded matrices to solve the PDE‐based system efficiently. Despite their higher computation complexity than direct computation in a sequential processor, we make it possible for the very first time to employ a divide‐and‐conquer algorithm, viable for parallel computation, in heat conduction analysis. Experimental results prove such possibility, suggesting that applying divide‐and‐conquer algorithms, higher‐order finite difference schemes can achieve better simulation accuracy with even faster speed and less memory requirement than conventional methods. Copyright © 2010 John Wiley & Sons, Ltd.