Barycentric Interpolation Collocation Method Based On Mixed Displacement-stress Formulation For Solving Plane Elastic Problems

The problems of elastic mechanics can be reduced to the boundary value problem of second order coupled elliptic partial differential equations. Most of the problems encountered in the project are of difficult to get its analytical solutions. In order to solve the elastic mechanics equations, the numerical solution technique is widely used in engineering practice.Firstly in this paper, the barycentric interpolation collocation method based on mixed displacement-stress formulation for solving plane elastic problems is proposed. The governing equations of elastic are expressed as a coupled system of partial differential equations with displacements and stresses variables. Both displacements and stresses are approximated via tensor-product type barycentric Lagrange interpolation. The matrix-vector form expressions of the governing equations for plane elasticity problems are obtained by using barycentric interpolation differentiation matrices. Discrete boundary conditions of the displacements and stresses are obtained by barycentric interpolation. The additional method is applied to impose boundary conditions, and an over-constrained linear system of algebraic equations for elastic plane problem is constructed. Numerical solutions of displacements and stresses for plane elasticity problem are solved by using least-square method.Secondly this paper describes a regular domain collocation method based on the barycentric Lagrange interpolation for solving the plane elastic problems. Embedded the irregular domain into a regular domain region, the barycentric lagrange interpolation in tensor form is used to approximate unknown functions. By using the collocation method, the differential equation is forced to be established precisely at the discrete nodes, and a mixed system of displacement-stress in the regular domain region is obtained. The constraint algebraic equation of boundary conditions can come from the unknown function value of interpolation nodes of the barycentric interpolation of the regular domain by taking a number of nodes on the boundary of the irregular domain region. The mixed displacement-stress formulation and the constraint algebraic equation of boundary condition are combined into new constraint algebraic equations. Numerical solutions of displacements and stresses for plane elasticity problem are solved by using least-square method.In this theme, the five numerical examples of regular domain and four numerical examples of irregular domain are shown. Application of barycentric Lagrange interpolation collocation method and barycentric interpolation regular domain method, which can effectively solve the problem of plane elastic in regular and irregular domain. The calculation formula of the method is simple, node adaptability and program of it is convenient, the calculation precision is very high.