A Review of Mathematical Models of Elasticity Theory Based on the Methods of Iterative Factorizations and Fictitious Components

Abstract

This review analyzes articles on mathematical modeling of elasticity theory using iterative factorizations and fictitious components. To carry out this study, various methods are developed, for example, an approximate analytical method of iterative factorizations for calculating the displacements of a rectangular plate, and a modified method of fictitious components for calculating the continuous displacement of plates. The performed calculations confirm the effectiveness of these methods. Descriptions of solutions to problems of elasticity theory and possible applications of the considered mathematical models and methods are given. An overview of the methods used to solve these problems is given. Particular attention is paid to problematic issues that arise in solving these problems. Techniques used to reduce complex problems to the solution of simple problems are given, for example, the lowering of the order of differential equations and the reduction of solutions in complex domains to solutions in a simple domain. For the first approach, iterative factorization methods are often used, and for the second, methods of the fictitious component type are often used. The main presentation in this review is focused on the approximate solution of elliptic boundary value problems. The works considered in the review raise questions about the development of methods in research on fictitious domains, fictitious components, and iterative factorizations.