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

2. Elastic Bar Dropping
Acceleration due to gravity

3. Variational Form
: arbitrary spatial function

4. Particle Discretization?

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

6. Shape Functions (1D)

7. Elastic Bar Dropping (coordinates)
Particles:
Nodes:

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

9. Elastic Bar Dropping (mass matrix)

10. Elastic Bar Dropping (grid mass)
Mass matrix

11. Grid mass (1,1)

12. Grid mass (1,2)

13. Grid mass (2,1)

14. Grid mass (2,2)

15. Grid equations

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

17. Lumped Mass Matrix
Lumped grid mass :

18. Mass matrix vs. lumped mass matrix