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In this article, we present an efficient transient electromagnetic–thermal co-simulation solver based on the hybrid implicit–explicit finite-difference time-domain (HIE-FDTD) method. This solver is designed to model the multiscale structure and multiphysics field effects of 3-D microwave integrated circuits (ICs). For the first time, we introduce the HIE-FDTD method to solve the heat conduction equation and establish a unified 3-D electromagnetic–thermal co-simulation framework based on the HIE-FDTD algorithm. In addition, this article provides a detailed explanation of the thermal continuity condition, which is rooted in the principle of energy conservation. It addresses the significant variation in thermal conductivity among the different layers of advanced package structures and highlights the potential for substantial errors if the thermal continuity condition is not considered. The stability condition of the proposed method is also given. Numerical results demonstrate that the proposed algorithm achieves the same level of calculation accuracy as the COMSOL software. Thanks to the incorporation of the HIE-FDTD algorithm, our approach significantly enhances computational efficiency compared to the traditional FDTD method and COMSOL software. Furthermore, the proposed algorithm holds the potential for applications in electromagnetic compatibility (EMC) assessments and signal/power integrity (SI/PI) evaluations in radio frequency ICs (RFICs) and advanced packaging.

An efficient space mapping method (SMM) is proposed to simulate electromagnetic (EM) propagation in 3-D multilayered lossy media. In regard to the spatial geometry discretization, the 3-D spatial layered structure is projected as an axial stratified model, and then, the finite element method (FEM) is utilized for mesh generation. This process reduces the complexity of model discretization compared to traditional numerical algorithms. Furthermore, to avoid the generation of large-scale sparse matrices in FEM implementation, the tangential electric field partial differential equations (TEF-PDEs) are established in fully anisotropic medium. This is achieved by transforming the spatial partial differential operator tensor in the 3-D vector wave equation into an antisymmetric tensor. Thus, the dimension of coefficient matrix can be effectively reduced compared to those based on the omnidirectional electric field equations. To validate the accuracy, efficiency, universality, and stability of this proposed method, several numerical cases are carried out, and the results as well as the method performance are compared with analytical method and commercial software COMSOL.

This article introduces a computational approach that utilizes the stochastic finite-difference time-domain (S-FDTD) method to solve problems involving both electromagnetic and thermal fields with uncertainty parameters. This stochastic solver not only considers electrical parameters such as permittivity and permeability as random variables, but also takes into account thermal parameters such as specific heat capacity and thermal conductivity. In addition, two different simulation strategies were proposed for the specific implementation of electromagnetic-thermal coupling. As a result, the proposed algorithm achieves statistical estimation of uncertainty in the multiphysical co-simulation of electromagnetic fields and thermal fields. This article presents the procedure and intricacies of formula derivation for the proposed method. Furthermore, the precision and feasibility of the proposed method are verified through simulations of layered tissue models with reference to Monte Carlo results. The simulation of a complex human head model demonstrates that the proposed method is capable of handling complex objects and meeting practical application requirements. This method has wide applications in biomedical applications and electromagnetic protection and provides theoretical and technical support for statistical estimation in these areas.

The numerical superiority of the unconditionally stable fundamental locally one-dimensional finite-difference time-domain (FLOD-FDTD) method has been proved in some studies. In this article, the FLOD-FDTD method is extended to simulate general dispersive media in combination with the auxiliary differential equation (ADE) scheme with formulation based on the electric field ( ${E}$ ) and electric polarization ( ${P}$ ). The relative permittivity of dispersive media is described by the arbitrary multi-term modified Lorentz model which can also represent Debye and Drude models. Thus, this method is able to include different types of dispersion in a unified formulation. The correctness and efficiency of the proposed method are demonstrated by some numerical cases.