An integration algorithm for the temperature solution of the four-layer infinite plate structure

Abstract

An algorithm is presented for the effective and accurate integration of the temperature solution of a four-layer plate structure with infinite lateral boundaries. By using the method of images, the effect of the finite lateral boundaries of a rectangular structure can be taken into account, and the solution of the infinite plate structure can be utilized to represent exactly the solution of a rectangular structure. The solution for the rectangular structure is in the form of an infinite double Fourier cosine series. A large number of terms has to be summed for accurate temperature calculation, resulting in prohibitively long CPU time for structures with small heat sources. The solution of the infinite plate structure is an inverse double Fourier cosine integration whose integrand decreases very rapidly with the spatial frequencies alpha and beta . However, it is highly oscillatory, so that general-purpose integration routines require long CPU time and produce uncertain results. By using the integration algorithm developed, the authors were able to reduce the CPU time by a factor of 10 and at the same time obtain more accurate results. For the rectangular structure, the CPU time is reduced by a factor of 100 to 1000.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>