An adaptive local maximum entropy point collocation method for linear elasticity. (November 2021) - Fan, Lei; Coombs, William M.; Augarde, Charles E.
Highlights: Combined rh -adaptivity is applied to the local maximum entropy point collocation method for the first time. Collocation points are relocated driven by material force residuals in r -adaptivity. Collocation and source points are added in local regions with relatively high errors determined by physical equilibrium residuals in h -adaptivity. Combined rh -adaptivity exhibits greater efficiency than pure adaptive strategies for linear elasticity problems. Abstract: Point collocation methods are strong form approaches that can be applied to continuum mechanics problems and possess attractive features over weak form-based methods due to the absence of a mesh. While various adaptive strategies have been proposed to improve the accuracy of weak form-based methods, such techniques have received little attention for strong form-based methods. In this paper, combined rh -adaptivity, in which r - and h -adaptivities are adopted iteratively, is applied to the local maximum entropy point collocation method for the first time to solve linear elasticity problems. Material force residuals act as driving forces in r -adaptivity to relocate collocation points, reducing the error associated with a given point distribution. Physical equilibrium residuals are used as the error estimator in h -adaptivity to determine the insertion locations for new points, diminishing the error caused by inadequate degrees of freedom. Issues arising in mesh-based methods, such as mesh distortion and hanging nodes, are entirely absent from the proposed method. The paper introduces the approach for the first time and the study is therefore confined to 2D domains. Numerical examples are presented to demonstrate the performance of the proposed adaptive strategies, comparing convergence rates and computational costs using uniform refinement, pure r -, h - and combined rh -adaptivities.