A simple meshfree method based on Trefftz attitude for 2D and 3D elasticity problems

A simple meshfree method is developed for the solution of 2D and 3D elasticity problems in potentially heterogeneous media. The rationale of the method follows that of Trefftz approaches to apply the Partial Differential Equation (PDE) and the boundary conditions in separate steps, while the basis functions can satisfy the PDE. The solution domain is discretized by a regular nodal grid including DOFs as the displacement components. The boundary is also discretized by some boundary points independent from the nodes, making the method applicable for arbitrarily shaped domains without imposing irregularity to the nodal grid. Each node corresponds to a cloud that contains some of its adjacent nodes as well. The overlap of the clouds integrates the displacement and stress components throughout the domain. The governing elasticity PDEs in heterogeneous media have nonconstant coefficients, preventing Trefftz techniques to be applicable. The present method, based on equilibrated basis functions, satisfies the PDE in weighted residual approach to extract some bases capable of its approximately satisfaction. The weighting may remove the boundary integrals, so the boundary conditions are simply collocated. The integrals are composed by combination of 1D predefined ones, then removing the numerical quadrature from the solution procedure.