1. 1 Point-implicit diffusion operator in a moving particle method (MPPM) Yao-Hsin Hwang Department of Marine Engineering National Kaohsiung Marine University

2. 2 Diffusion operator Time-step constraint for explicit scheme Implicit scheme: Iteration Numerically stable without iteration

3. 3 Features of existing particle methods SPH, MPS, FVPM…. Material particles (mass point, global mass conservation) Operators realized with randomly particle cloud

4. 4 Disadvantages of existing schemes Inaccurate (inconsistent) operator realization (particle smoothing) —continuity constraint Assignment of boundary condition Particle management (impractically clustered or dispersive) Computationally inefficient

5. 5 Essentials of MPPM (1) Inserted pressure mesh to realize pressure- related operators Background mesh and length scale Local mass conservation Particle as observation point

6. 6 Essentials of MPPM (2) No particle management constraint Feasible non-uniform distribution No special boundary treatment Constant and diagonally dominant coefficient matrix of pressure equation

7. 7 Diffusion Procedure Equation Laplacian,

8. 8 Explicit Operator Scheme Time-step constraint

9. 9 Implicit Operator Scheme Iteration required

10. 10 Point-implicit Operator Operator

11. 11 Point-implicit Operator Scheme A Scheme B

12. 12 Effective-Time Step Scheme A Scheme B

13. 13 Stability Limits Scheme A Scheme B

14. 14 Three-Point Analysis(1) 1. Boundary conditions: 2. Solution at 3. Results

15. 15 Three-Point Analysis(2) Results

16. 16 Pure Diffusion(1) Particle distribution

17. 17 Pure Diffusion(2)

18. 18 Pure Diffusion(3)

19. 19 Cavity Flow(1)

20. 20 Cavity Flow Re=10

21. 21 Cavity Flow(2)

22. 22 Cavity Flow(3)

23. 23 Cavity Flow(4)

24. 24 Cavity Flow(5)

25. 25 Backward-Facing Step Flow(1)

26. 26 Backward-Facing Step Flow(2)

27. 27 Backward-Facing Step Flow(3)

28. 28 Backward-Facing Step Flow(4)

29. 29 Backward-Facing Step Flow(5)

30. 30 Backward-Facing Step Flow(6)