SUMMARYThe 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 modification 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 modification 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 difficult problem for standard finite 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 significant improvement over existing methods for handling an important class of problems.

The material point method (MPM) uses unconnected, Lagrangian, material points to discretize solids, #uids or membranes. All variables in the solution of the continuum equations are associated with these points; so, for example, they carry mass, velocity, stress and strain. A background Eulerian mesh is used to solve the momentum equation. Data mapped from the material points are used to initialize variables on the background mesh. In the case of multiple materials, the stress from each material contributes to forces at nearby mesh points, so the solution of the momentum equation includes all materials. The mesh solution then updates the material point values. This simple algorithm treats all materials in a uniform way, avoids complicated mesh construction and automatically applies a noslip contact algorithm at no additional cost. Several examples are used to demonstrate the method, including simulation of a pressurized membrane and the impact of a probe with a pre-in#ated airbag.

SUMMARYThe material point method (MPM) uses unconnected, Lagrangian, material points to discretize solids, #uids or membranes. All variables in the solution of the continuum equations are associated with these points; so, for example, they carry mass, velocity, stress and strain. A background Eulerian mesh is used to solve the momentum equation. Data mapped from the material points are used to initialize variables on the background mesh. In the case of multiple materials, the stress from each material contributes to forces at nearby mesh points, so the solution of the momentum equation includes all materials. The mesh solution then updates the material point values. This simple algorithm treats all materials in a uniform way, avoids complicated mesh construction and automatically applies a noslip contact algorithm at no additional cost. Several examples are used to demonstrate the method, including simulation of a pressurized membrane and the impact of a probe with a pre-in#ated airbag.

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