Material Point Method Grid Equations Mass matrix Lumped mass matrix Monday, 10/7/2002

Elastic Bar Dropping Acceleration due to gravity

Variational Form : arbitrary spatial function

Particle Discretization?

Grid Equations of Motion External force Internal force Mass matrix Shape function between node n and particle p:

Shape Functions (1D)

Elastic Bar Dropping (coordinates) Particles: Nodes:

Elastic Bar Dropping (particle mass) Mass density: Cross section area:

Elastic Bar Dropping (mass matrix)

Elastic Bar Dropping (grid mass) Mass matrix

Grid mass (1,1)

Grid mass (1,2)

Grid mass (2,1)

Grid mass (2,2)

Grid equations

Mass matrix Compact mass matrix: Only matrix elements close to diagonal are not zero.

Lumped Mass Matrix Lumped grid mass :

Mass matrix vs. lumped mass matrix