A novel implicit material point method (MPM) using a cell-based integration scheme is proposed to solve large deformation static problems. An incremental weak form based on the updated Lagrangian approach is formulated for the implicit MPM. The volume integrals of the incremental weak form are evaluated at the integration points of grid cells instead of material points, which eliminates the cell-crossing error and reduces the integration error in MPM computations. Grid cells are equally sub-divided into grid cell sub-domains. The centers and the particle volumes of the grid cell sub-domains are, respectively, taken as the integration points and corresponding weights for the numerical integration of the incremental weak form. Particle information is transferred through grid nodes to the integration points of grid cells by using grid shape functions. A volume-weighted nodal averaging scheme is used for transferring the deformation gradient from material particles to grid nodes to correctly consider the particle volumes associated with the deformation gradient. Numerical results show that the present implicit MPM can effectively and efficiently solve large deformation static problems.

提出了一种使用基于单元的积分方案的新型隐式材料点方法（MPM）来解决大变形静态问题。为隐式 MPM 制定了基于更新拉格朗日方法的增量弱形式。增量弱形式的体积积分是在网格单元的积分点而不是材料点处计算的，这消除了单元交叉误差并减少了MPM计算中的积分误差。网格单元被同等地细分为网格单元子域。网格单元子域的中心和粒子体积分别作为增量弱形式数值积分的积分点和相应的权重。利用网格形状函数，粒子信息通过网格节点传递到网格单元的积分点。体积加权节点平均方案用于将变形梯度从材料粒子转移到网格节点，以正确考虑与变形梯度相关的粒子体积。数值结果表明，该隐式MPM能够有效且高效地解决大变形静力问题。