Material Point Method (MPM)

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Presentation transcript:

Material Point Method (MPM) Variational principle Particle discretization Grid interpolation Wednesday, 10/2/2002

Lennard-Jones Potential Handout: Molecular dynamics of simple systems Error, page 7 potential force

Motion equations Tensor expression: Engineering expression:

Discretization in Material Point Method (MPM) Solid line is an outline of the body analyzed. Black dots are the material points. Dashed lines show a regular, background grid for calculation.

Motion equations (Variational form) : arbitrary spatial function

Elastic Bar Dropping Acceleration due to gravity

Particle Discretization? How to solve?

Direct solution Solution:

Variational Form : arbitrary spatial function

Particle Discretization?

Variational Equation Considering free ends at a and b

Particle Discretization

Spatial grid

Shape function

Grid Interpolation

Grid Equation Since is arbitrary:

Grid Equation (F=ma) External force Internal force Mass matrix

Derivative of Shape Function

Summary Variational principle Particle discretization Grid interpolation Next class: MPM procedure