Method for Calculating a Thermal-Expansion-Induced Mechanical Stress in Three-Dimensional Solid-State Structures Using Mathematical Modeling

Abstract

At the end of the 20th century, the demand for more efficient methods for solving large, sparse, unstructured linear systems of equations increased dramatically. Classical single-level methods had already reached their limits, and new hierarchical algorithms had to be developed to provide efficient solutions to even larger problems. The efficient numerical solution of large systems of discrete elliptic partial differential equations (PDEs) requires hierarchical algorithms that ensure a fast reduction of both shortwave and longwave components in the error vector expansion. The breakthrough, and certainly one of the most important advances of the last three decades, came through the multigrid principle. Any appropriate method works with a grid hierarchy specified a priori by coarsening the given sampling grid in a geometrically natural way (a geometric multigrid method). However, defining a natural hierarchy can become very difficult for very complex, unstructured meshes, if possible at all. This article proposes an algorithm for calculating the deformation that occurs under the action of a thermal expansion force in three-dimensional solid models based on a grid approximation of the problem by hexagonal 8-node cells. The operation of the algorithm is illustrated by solving three problems.