A general integration algorithm for the inverse Fourier transform of four-layer infinite plate structures

An integration algorithm is presented for the effective and accurate integration of the thermal and electrostatic solutions 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. As a result, the solution of the infinite plate structure can be utilized to represent exactly the solution of a rectangular structure. The rectangular structure solution is an infinite double Fourier cosine series. A large number of terms has to be summed for accurate temperature calculation, resulting in a prohibitively long CPU time for structures with small sources. The solution of the infinite plate structure, on the other hand, is an inverse double Fourier cosine integration. The integrand decreases very rapidly with the spatial frequencies α and β. However, it is also highly oscillatory, as the location of temperature calculation is remote from the source. Consequently, general-purpose integration routines require long CPU time and produce uncertain results. By using the integration algorithm developed, it is possible to reduce the CPU time by a factor of 10 and at the same time obtain more accurate results. In comparison with the infinite Fourier series solution of the rectangular structure, the CPU time is reduced by a factor of 100 to 1000 with the use of the integral solution incorporated with the integration algorithm