Technique for Solving Finite Element Systems of High-Order Linear Algebraic Equations Describing the Stress–Strain State of One-Dimensional and Two-Dimensional Structures

The article discusses the problems that arise when applying the finite element method to design models of complex structures and structures represented by design schemes with a large number of finite elements. The development of an algorithm for reducing finite element systems of high-order equations describing the stress–strain state of one-dimensional and two-dimensional structures based on linear interpolation, which allows to significantly reduce the order of these systems, while maintaining sufficient accuracy of calculations, is described. The article presents a mathematical model for a two-dimensional plate and describes a method for solving a system of equations based on linear interpolation at the nodes of a fine grid. To test the effectiveness of the algorithm, numerical experiments were carried out comparing the results obtained using the proposed method with the analytical and results obtained on the basis of the FEM in the form of a classical mixed method. The results of numerical experiments have shown that the proposed algorithm has a sufficiently high accuracy and efficiency in solving problems of solid mechanics. The proposed approach can be applied to reduce systems of high-order algebraic equations describing the stress–strain state of one-dimensional and two-dimensional structures, and to speed up calculations in large and complex problems of solid mechanics, which is an urgent problem in engineering practice.