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

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

3. Motion equations
Tensor expression:
Engineering expression:

4. 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.

5. Motion equations (Variational form)
: arbitrary spatial function

6. Elastic Bar Dropping
Acceleration due to gravity

7. Particle Discretization? How to solve?

8. Direct solution
Solution:

9. Variational Form
: arbitrary spatial function

10. Particle Discretization?

11. Variational Equation
Considering free ends at a and b

12. Particle Discretization

13. Spatial grid

14. Shape function

15. Grid Interpolation

16. Grid Equation
Since is arbitrary:

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

18. Derivative of Shape Function

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