Mixed Barycentric Interpolation Collocation Method Of Displacement-pressure For Incompressible Plane Problems

The finite element method is the main method for numerical analysis of elasticity.However,the volume locking phenomenon occurs when classical finite element methods(such as linear elements)are used to solve(nearly)incompressible problems.It is necessary to use a special numerical solution method for(nearly)incompressible plane elastic problem.Defining an artificial pressure variable,the elastic constitutive equations are expressed by the form of stress-strain-pressure.The boundary value problem of coupled partial differential equation system of displacement-pressure equations and equations of(nearly)incompressible condition for plane strain elastic problems are established.For regular(nearly)incompressible plane elastic materials,partial differential matrices on calculating nodes are obtained by using the tensor-product type barycentric Lagrange interpolation to approximate binary functions.Incompressible elastic governing equations are discretized by using collocation method.The matrix-vector form of the governing equations are established directly by using partial differential matrices.Discrete boundary conditions are obtained by using barycentric interpolation.By applying additional method,boundary conditions are imposed and over-constrained linear system of algebra equations are obtained.By using least-square method,the over-constrained system of algebra equations are solved.For irregular(nearly)incompressible plane elastic materials,the irregular materials are embedded into a regular rectangular domain.The dispersion of the elastic control equations is the same as the regular discrete method.The discrete displacement and stress boundary conditions of the irregular domain can be obtained by barycentric Lagrange interpolation.The function values on the regular domain can be worked out by the regular domain collocation method.By applying barycentric Lagrange interpolation,the function values in each interpolation node on the irregular domain can be solved.When the beam equations are solved by barycentric interpolation collocation method,computational accuracy will decline gradually as the number of nodes increases.The research on barycentric interpolation collocation method based on depression of order,can provide the new method that has a good numerical stability and high computational accuracy for beam equations.In chapter 5,based on barycentric Lagrange interpolation and its differential matrices,the formula of barycentric interpolation collocation method based on depression of order is derived.Numerical examples are given to verify the effectiveness of depression of order method.The calculation programs are written by MATLAB.In this paper,7(nearly)incompressible and 2 depression of order numerical examples demonstrate the effectiveness and calculation accuracy of the mixed barycentric interpolation collocation method of displacement-pressure and depression of order method.