A weighted collocation on the strong form with mixed radial basis approximations for incompressible linear elasticity

A weighted strong form collocation framework with mixed radial basis approximations for the pressure and displacement fields is proposed for incompressible and nearly incompressible linear elasticity. It is shown that with the proper choice of independent source points and collocation points for the radial basis approximations in the pressure and displacement fields, together with the analytically derived weights associated with the incompressibility constraint and boundary condition collocation equations, optimal convergence can be achieved. The optimal weights associated with the collocation equations are derived based on achieving balanced errors resulting from domain, boundaries, and constraint equations. Since in the proposed method the overdetermined system of the collocation equations is solved by a least squares method, independent pressure and displacement approximations can be selected without suffering from instability due to violation of the LBB stability condition. The numerical solutions verify that the solution of the proposed method does not exhibit volumetric locking and pressure oscillation, and that the solution converges exponentially in both L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{2}$$\end{document} norm and H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{1}$$\end{document} semi-norm, consistent with the error analysis results presented in this paper.