The material point method for simulation of thin membranes

The material-point method (MPM) is extended to handle membranes, which are discretized by a collection of unconnected material points placed along eac h membrane surface. These points provide a Lagrangian description of the me mbrane. To solve for the membrane motion, data carried by the material poin ts are transferred to a background mesh where equations of motion are discr etized and solved. Then the solution on the background mesh is used to upda te the membrane material points. This process of combining Lagrangian and E ulerian features is standard in MPM; the modification for membranes involve s merely an implementation of the constitutive equation in a local, normal- tangential coordinate system. It is shown that this procedure does, in fact , provide adequate resolution of membranes with thicknesses that can vary s ubstantially from that of the background mesh spacing. A general formulatio n is given, but the implementation is in a two-dimensional code that provid es a proof-of-principle. Numerical examples including a spring, pendulum and a string with initial s lack are used to illustrate the method. The string with slack uses an addit ional modification of the membrane constitutive equation that allows wrinkl es to be modeled at low computational cost. Presented also are examples of two disks impacting, pinching a membrane and rebounding, a difficult proble m for standard finite element codes. These simulations require a relaxation of the automatic no-slip contact algorithm in MPM. The addition of the cap ability to model membranes and the new contact algorithm provide a signific ant improvement over existing methods for handling an important class of pr oblems. Copyright (C) 1999 John Wiley & Sons, Ltd.