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Accurate and Efficient Methods for Multiscale and Multiphysics Analysis
thesisposted on 14.05.2019, 17:30 by Kaiyuan Zeng
Multiscale and multiphysics have been two major challenges in analyzing and designing new emerging engineering devices, materials, circuits, and systems. When simulating a multiscale problem, numerical methods have to overcome the challenges in both space and time to account for the scales spanning many orders of magnitude difference. In the finite-difference time-domain (FDTD) method, subgridding techniques have been developed to address the multiscale challenge. However, the accuracy and stability in existing subgridding algorithms have always been two competing factors. In terms of the analysis of a multiphysics problem, it involves the solution of multiple partial differential equations. Existing partial differential equation solvers require solving a system matrix when handling inhomogeneous materials and irregular geometries discretized into unstructured meshes. When the problem size, and hence the matrix size, is large, existing methods become highly inefficient.
In this work, a symmetric positive semi-definite FDTD subgridding algorithm in both space and time is developed for fast transient simulations of multiscale problems. This algorithm is stable and accurate by construction. Moreover, the method is further made unconditionally stable, by analytically finding unstable modes, and subsequently deducting them from the system matrix. To address the multiphysics simulation challenge, we develop a matrix-free time domain method for solving thermal diffusion equation, and the combined Maxwell-thermal equations, in arbitrary unstructured meshes. The counterpart of the method in frequency domain is also developed for fast frequency-domain analysis. In addition, a generic time marching scheme is proposed for simulating unsymmetrical systems to guarantee their stability in time domain.