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 each membrane surface. These points provide a Lagrangian description of the membrane. To solve for the membrane motion, data carried by the material points are transferred to a background mesh where equations of motion are discretized and solved. Then the solution on the background mesh is used to update the membrane material points. This process of combining Lagrangian and Eulerian features is standard in MPM; the modiÞcation for membranes involves 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 substantially from that of the background mesh spacing. A general formulation is given, but the implementation is in a two-dimensional code that provides a proof-of-principle. Numerical examples including a spring, pendulum and a string with initial slack are used to illustrate the method. The string with slack uses an additional modiÞcation of the membrane constitutive equation that allows wrinkles to be modeled at low computational cost. Presented also are examples of two disks impacting, pinching a membrane and rebounding, a di¦cult problem for standard Þnite element codes. These simulations require a relaxation of the automatic no-slip contact algorithm in MPM. The addition of the capability to model membranes and the new contact algorithm provide a signiÞcant improvement over existing methods for handling an important class of problems. Copyright