Extended B‐spline‐based implicit material point method

Abstract

AbstractAn implicit material point method (MPM) is enhanced by extended B‐splines with the aim of achieving numerical stability and properly imposing boundary conditions. The standard B‐spline basis functions capable of suppressing the cell‐crossing error are used for the approximation over the domain, whereas extended B‐splines (EBS) are active for boundary cells, each of which is occupied by a small physical domain. The basis consisting of EBS plays a central role to avoid both stress oscillation arising from inaccurate numerical integration and ill‐conditioning of their resulting tangent matrices, both of which are caused by the boundary cells containing smaller number of material points than interior ones. Besides the standard material points for the interior region, boundary points are introduced to explicitly represent the boundary geometries and their movements, and to weakly impose an arbitrary boundary condition with the Nitsche's method. The Nitsche's terms in the linearized weak form are decomposed into normal and tangential directions to deal with both slip and nonslip boundary conditions. Four case studies are made to demonstrate the performance and capability of the proposed method, named EBS‐MPM, in stably solving the quasi‐static equilibrium problems for hyperelastic bodies and properly applying arbitrary boundary conditions.