Strong form meshless analysis of solids using partially constrained polynomial finite-difference operators and overdetermined equilibrium

Abstract

This paper proposes a new strong-form meshless method for arbitrary (non-rectangular) grids that combines a new class of finite-difference (FD) differential operators with overdetermined equilibrium. The proposed new class of FD operators combine short-range interpolation with long-range approximation over their support. Polynomial approximation is realized using weighted least squares, whereas polynomial interpolation is enforced using Lagrange multipliers. This partially constrained polynomial (PCP) FD operator is implemented through a proposed algorithm that automatically selects the stencil size and short range radius, such that the numerical error of the stencil weights due to potential ill-conditioning is below a selected tolerance. Using the PCP-FD operators, a meshless method for 2D elasto-static problems was formulated. Instabilities that are often inherent in strong form methods are addressed by introducing additional equations of field equilibrium over a set of additional points within the solid/domain. The resulting overdetermined set of equations is solved as a constrained least squares problem using Lagrange multipliers to enforce traction BCs, which bypasses the need for scaling of the traction BC equations, as is often the case in weighted collocation methods. The proposed meshless method is evaluated in several examples and is shown to achieve high accuracy that is usually better than that of the finite element method.