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A matrix function solution for the scaled boundary finite-element equation in statics

机译：静力学中比例边界有限元方程的矩阵函数解

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摘要

The scaled boundary finite-element method is a fundamental-solution-less boundary element method based on finite elements. It leads to semi-analytical solutions for displacement and stress fields, which permits problems with singularities or in infinite domains to be handled conveniently and accurately. However, the present eigenvalue method for solving the scaled boundary finite-element equation requires additional treatments for multiple eigenvalues with parallel eigenvectors, which results from logarithmic terms in the solutions. A matrix function solution for the scaled boundary finite-element equation in statics is presented in this paper. It is numerically stable for multiple or near-multiple eigenvalues as it is based on real Schur decomposition. Power functions, logarithmic functions and their transitions as occurring in fracture mechanics, composites and two-dimensional unbounded domains, are represented semi-analyti-cally. No a priori knowledge on the types and orders of singularity are required when simulating stress singularities.
机译：比例边界有限元法是一种基于有限元的无基本解的边界元方法。它导致了位移场和应力场的半解析解，从而可以方便，准确地处理奇异或无限域的问题。但是，当前用于求解缩放边界有限元方程的特征值方法需要对具有平行特征向量的多个特征值进行额外处理，这是由解中的对数项引起的。本文提出了比例缩放边界有限元方程的矩阵函数解。它基于真实的Schur分解，对于多个或接近多个特征值在数值上稳定。幂函数，对数函数及其在断裂力学，复合材料和二维无界域中出现的跃迁以半解析表示。模拟应力奇异点时，无需先验知识即可了解奇异点的类型和顺序。

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