Maximum-Entropy Meshfree Method for Linear and Nonlinear Elasticity
By Alejandro A. Ortiz
Office of Graduate Studies,
University of California, Davis, California
A Galerkin-based maximum-entropy meshfree method for linear and nonlinear elastic media is developed. The standard displacement-based Galerkin formulation is used to model compressible linear elastic solids, whereas the classical u-p mixed formulation for near-incompressible linear elastic media is adopted to formulate a volume-averaged nodal technique in which the pressure variable is eliminated from the analysis. This results in a single-field formulation that is devoid of volumetric locking. A modified Gauss integration technique that alleviates integration errors in meshfree methods with guaranteed patch test satisfaction to machine precision is devised. The performance of the maximum-entropy meshfree method is assessed for problems in compressible and near-incompressible linear elastic media using three-node triangular and four-node tetrahedral background meshes. Both structured and unstructured meshes are considered to assess the accuracy, performance and stability of the maximum-entropy meshfree method by means of various numerical experiments, which include patch tests, bending dominated problem, combined bending-shear problem, rigid indentation, Stokes flow and numerical stability tests.
An extension of the volume-averaged nodal technique is proposed for the analysis of near-incompressible nonlinear elastic solids in two dimensions. In the nonlinear version, the volume change ratio of the dilatational constraint, namely J, is volume-averaged around nodes leading to a locking-free displacement-based formulation. The excellent performance of the maximum-entropy meshfree method for problems in near-incompressible nonlinear elastic solids is demonstrated via three standard two-dimensional numerical experiments—a combined bending-shear problem, a plane strain compression of a rubber block and a frictionless indentation problem. Three-node structured and unstructured triangular background meshes are employed and the results are compared to two finite element methods that use such meshes, namely, the linear displacement/constant pressure triangle and the linear displacement/linear pressure triangle enriched with a displacement bubble node (MINI element). The two-dimensional nonlinear simulations reveal that the maximum-entropy meshfree method effectively improves the poor performance of linear triangular meshes in the analysis of near-incompressible solids at finite strains.