Variational Statements and Discretization of the Boundary-Value Problem of Elasticity Where Stress at the Boundary is Kn
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SYSTEMS ANALYSIS VARIATIONAL STATEMENTS AND DISCRETIZATION OF THE BOUNDARY-VALUE PROBLEM OF ELASTICITY WHERE STRESS AT THE BOUNDARY IS KNOWN N. A. Vareniuk,1† E. F. Galba,1‡ and I. V. Sergienko1††
UDC 517.95:519.63
Abstract. The equations of elastic equilibrium of bodies in displacements with the stresses specified at the surface of the body are considered. Such a problem does not have a unique solution in the whole space of vector functions where it exists. Two variational problems for the considered static problem of the theory of elasticity with a unique solution in the whole space are proposed and investigated. The mathematical apparatus of the study is one of the variants of the Korn inequality that is proved in the paper. Discretization of these variational problems by the finite-element method and convergence of discrete solutions is considered. Keywords: elasticity problem, variational statements, existence of a unique solution in function spaces, discrete problems, methods for solving discrete problems. INTRODUCTION Mathematical modeling of many established processes of various physical nature often leads to boundary-value problems for partial differential equations of elliptic type, which have a unique solution in the subspace of functions from the function space where it exists. These are stationary problems of thermal conductivity, filtration, diffusion, electrostatics, magnetostatics, stress–strain problems of various objects and structures. Typically, these problems are solved with the use of the finite difference method (FDM) and the finite element method (FEM), which are discussed in numerous papers and monographs. This study is devoted to the construction and analysis of new variational and discrete problems for the static problem of the theory of elasticity with stresses given at the boundary of the domain. As a result of discretization of this problem by means of FDM or FEM without regard for the conditions that define its unique solution, systems of linear algebraic equations (SLAE) with symmetric, positive semidefinite matrices and right-hand sides that can or cannot satisfy the solvability conditions are obtained. The question arises in the first case, about finding one of the solutions of the SLAE, and in the second case, about calculating the normal pseudosolution. One of the solutions or pseudosolutions of these SLAEs can usually be found by direct or iterative methods developed for systems with arbitrary matrices (see, for example, [1–3] and references therein). However, taking into account the properties of matrices, more efficient methods are used to solve such systems. The emergence of these problems was an impetus for the development of iterative processes that coincide on a subspace. A lot of studies are devoted to iterative methods on subspaces (see, for example, [4–11]). They mostly addressed the issue of adapting iterative processes developed for SLAEs with positive definite symmetric matrices to solving SLAEs with symmetric positive semi-definite matrices. For example, for the d
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