The 2D elastostatic problem in inhomogeneous anisotropic bodies by the meshless analog equation method (MAEM)

Abstract

The meshless analog equation method (MAEM) is employed to solve the 2D elastostatic problem for inhomogeneous anisotropic bodies. In this case, the response of the body is governed by two coupled PDEs of second order with variable (position-dependent) coefficients, which are solved using the new meshless method developed by Katsikadelis for solving PDEs. The method is based on the concept of the analog equation of Katsikadelis, hence its name (MAEM), which converts the original coupled PDEs into two uncoupled Poisson's equations, the analog equations, under fictitious sources. The fictitious sources are represented by MQ-RBFs. Integration of the analog equations allows the approximation of the sought solution by new RBFs. Then inserting the solution into the PDEs and BCs and collocating at the mesh-free nodal points yields a system of linear equations, which permit the evaluation of the expansion coefficients. The method exhibits key advantages over other RBF collocation methods as it is highly accurate and the matrix of the resulting linear equations is always invertible. The accuracy is increased using optimal values of the shape parameters of the multiquadrics and of the integration constants of the analog equation by minimizing the total potential of the elastic body. Several examples are studied, which demonstrate the efficiency and high accuracy of the solution method.