基于离散几何法的多物理场耦合问题研究

Study of multiphysics problem based on discrete geometric method

Tonti图揭示并高度概括了多种物理场所共同具备的数学结构,为求解不同物理场提供相同的拓扑算子以构建刚度矩阵。离散几何法是基于离散空间中互为正交的原始和对偶网格,利用其几何尺度直接导出本构矩阵,从而快速建立代数方程的数值解法,它的形式结构直观简单,便于计算。本文根据Tonti图将离散几何法应用于三维集成电路中TSV阵列的电-热-力场的耦合问题,并将计算结果与基于有限元法的商业软件COMSOL对比。实验结果表明:在多物理耦合问题上,离散几何法精度可靠,从而有望应用推广至电子设计自动化工具之中。

The Tonti diagrams disclose a general common mathematical structure for several physical fields, and hence provide uniform topological operators to establish global stiffness matrices for various field solvers in differential form. The discrete geometric method implements the Hodge discretization of constitutive laws through the division of geometric dimensions of mutually orthogonal primal and dual meshes. Complying with Tonti diagrams, it serves as an elegant solution for multiphysics problems intuitively and practicably. The stationary electro-thermo-meehanical coupled problem, including building of the constitutive equations and treating of the coupling terms, is studied in this paper. A typical problem with regard to electro-thermo-mechanical stress induced mainly by resistive losses in the TSV structures is investigated. The proposed method COMSOL based on the finite element method, and design automation. offers accurate solutions as the commercial software is expected to applied in the utilities of electronic