6963_MPM.ppt.pptVIP

* * * * * * * * * * * * * * * * * * * * * The Generalized InterpolationMaterial Point Method Compaction of a foam microstructure Tungsten Particle Impacting sandstone 2. Overlying mesh defined 1. Lagrangian material points carry all state data (position, velocity, stress, etc.) 5. Particle positions/velocities updated from mesh solution. 6. Discard deformed mesh. Define new mesh and repeat 1 2 3 4 5 The Material Point Method (MPM) 3. Particle state projected to mesh, e.g.: 4. Conservation of momentum solved on mesh giving updated mesh velocity and (in principal) position. Stress at particles computed based on gradient of the mesh velocity. 6 Interpolation function - MPM Interpolation function - GIMP Cell width Particle width Interpolation function In 2D, shape functions are formed by the products of the x and y function: Gradients involve PARTIAL derivatives, so for instance: Steps in an MPM code Initialize particles and create (logically) a grid Project (integrate) particle data to grid (mass, velocity, etc.) Set boundary conditions on velocity Compute internal force from divergence of stress Compute acceleration on the grid (a=F/m) Integrate velocity on the grid (v* = v + a*dt) Set boundary conditions on v* and a. Compute Stress, update volume, compute dt_new Update particle position and velocity, t = t+ dt, dt = dt_new “Reset” grid and return to step 2 and repeat while tt_final There are two basic methods for determining particle locations Acquire them from a file (e.g. image data) Use geometric primitives to describe geometry, and inside/outside tests to determine particle placement. Step 1: Particle Initialization Day 8 Step 1: Particle Initialization At t=0, the particles need to be initialized with the following data Position, xp Volume, vp Mass, (mp=density*volume) Velocity, Vp Stress, ?p = 0 Deformation Gradient (Fp =Identity) Size Lxp, Lyp (dx/PPC) Step 2: Particle Data to Grid Compute mass and velocity on the gr