A Tetrahedral Spectral Element Method for Thermo-Mechanical Analysis of Electronic Devices

For the coupled thermo-mechanical simulation of complex electronic devices, we propose a tetrahedral spectral element method (TSEM), which truly combines the high accuracy of spectral methods with the strong geometric adaptability of finite element methods (FEMs). The TSEM employs a collapse transformation method and successfully converts the orthogonal Gauss-Lobatto–Legendre (GLL) basis functions from a hexahedron to a tetrahedral reference element. This ensures numerical accuracy and stability for arbitrary high-order interpolation. We also generate non-equidistant GLL spectral interpolation nodes for tetrahedral elements through Warp&Blend transformation, enabling direct nodal numerical integration calculation through Vandermonde matrices. This approach replaces the classical Gaussian quadrature in the FEM, not only significantly enhancing the computational efficiency of element matrix but also avoiding potential numerical dissipation and dispersion associated with Gaussian integration, thus achieving rapid convergence. In the TSEM thermo-mechanical coupling, we further construct a separate solver based on the implicit Newmark-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> method, which realizes unconditional stability for temporal integration for arbitrary time step increment. Moreover, it supports independent inversion of system matrices for temperature and stress fields, without increasing computational resource consumption due to the coupling of multiphysics equations. By quantitatively comparing with the COMSOL Multiphysics software, several typical case studies are used to demonstrate the significant advantages of the TSEM in numerical accuracy, efficiency, and stability.