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

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

4. Motion equations Tensor expression: Engineering expression:

5. 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.

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

7. Elastic Bar Dropping Acceleration due to gravity

8. Particle Discretization? How to solve?

9. Direct solution Solution:

10. Variational Form  : arbitrary spatial function

11. Particle Discretization?

12. Variational Equation Considering free ends at a and b

13. Particle Discretization

14. Spatial grid

15. Shape function

16. Grid Interpolation

17. Grid Equation Since is arbitrary:

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

19. Derivative of Shape Function

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