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Moving grid methods and Multi-mesh methods

Moving grid methods and Multi-mesh methods. Tao Tang Hong Kong Baptist University Inter national Workshop on Frontiers in Scientific Computing June 10-13, 2008, Wuyi Mountain , China. Outline:. Basic ideas of moving mesh method Applications and results Why using multi-mesh methods?

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Moving grid methods and Multi-mesh methods

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  1. Moving grid methods and Multi-mesh methods Tao Tang Hong Kong Baptist University International Workshop on Frontiers in Scientific Computing June 10-13, 2008, Wuyi Mountain, China

  2. Outline: • Basic ideas of moving mesh method • Applications and results • Why using multi-mesh methods? • Conclusions and future work Collaborators: Ruo Li, Huazhong Tang, Pingwen Zhang (Peking University) Yana Di (Chinese Academy of Sciences) Heyu Wang, Xianliang Hu (Zhejiang University)

  3. Different types of adaptive grid refinement • h-refinement: • # of grid points not constant • adds (or deletes) grid points • grid equations are not coupled to physical PDEs )=> extra interpolation procedure needed • p-refinement: • varies degree of piecewise polynomials (FE’s) • often in combination with h-refinement • r -refinement: • # of grid points constant • r e-locates (moves) grid points • grid equations uncoupled or coupled with physical PDE

  4. Physical vs. computational coordinates

  5. Viscous Shock Problem

  6. Key ingredients of the moving mesh methods • Mesh equation -- determine a one-to-one mapping from a parameter space to a physical space. • Monitor function -- used to guide the mesh redistribution. • Interpolation -- may be required to pass the solution information on the old mesh to the newly generated mesh.

  7. The ‘grid-energy ’ Can be taken to represent the energy of a system of springs with spring constants spanning each interval. The non-uniform grid point distribution resulting from the equidistribution principle thus represents the equilibrium state of the spring system, i.e., the state of minimum energy.

  8. The adaptive grid seen as a system of springs

  9. Winslow’s method (1960s)

  10. Regularity of the transformation in 2D

  11. Harmonic mapping Let dij and rαβ be metric tensors in some local coordinates . Define the energy for a map as (1) where d=det(dij), (dij)=(dij)-1. The Euler-Lagrange equations, whose solution minimizes the above energy, are given by (2) a. The inverse of (Gij) is called monitor functions. b. Solutions to (2) are harmonic functions giving a continuous and one-to-one mapping. c. Solutions to (2) minimizes the energy (1). Dvinsky (JCP, 1991): (2) may provide a general framework for mesh generation.

  12. We solve the constrained optimization problem: (3) Note that the boundary values ξb are not fixed, instead they are unknowns in the same way as the interior points.

  13. The whole moving mesh algorithm can be packed in a black box which requires the following inputs:  the current solution of the underlying PDEs,  the algorithm for solving the mesh PDEs,  and an interpolation algorithm. Such a black box has been implemented in the adaptive finite element package AFEPack (Ruo Li and Wenbin Liu) http://circus.math.pku.edu.cn/AFEPack

  14. Level set / Moving mesh Approach Recently, methods that couple two different schemes have been developed for simulating fluid flows with moving interfaces. Examples are • The coupled level set and volume-of-fluid (VOF) method • The hybrid particle level set method • The mixed markers and VOF method A coupled method takes advantage of the strengths of each of the two methods, and is therefore superior to either method alone.

  15. The 2D incompressible Navier-Stokes equations separated by a free surface: After a standard non-dimensionalization procedure,

  16. Solve the N-S eqns:

  17. Solution interpolation can be realized by solving the system The following monitor function is proposed:

  18. Impact of water drop with an 1002 moving grid. The parameters used are Re = 28144, Fr = 204, We = 1760, g /  l =1 / 816, g /l = 1; t = 0.25, 0.3, 0.46, 0.67.

  19. Multiphase flows in 3D • Dimension-independent moving mesh method • Multi-grid speed up • Grid redistribution • Solution interpolation • Monitor function

  20. A phase field model for two incompressible fluids with initial conditions and appropriate boundary conditions. (t) is the Lagrange multiplier corresponding to the constant volume constraint:

  21. Finite element scheme The discretization of the time derivative is given by the backward difference formula: resulting in a semi-implicit second-order approximation. Introduce the notation N(u, )=uu  (  )+g(x). v ()2 and qH1(), Then update  k+1 by: H1()

  22. Coalescence of two kissing bubbles,  = 0.04,  = 0.1,  = 0.1,  = 0.1, 32  32  64 grid in an 1  1  2 domain. t = 0, 0.1, 0.3, 0.6, 0.8. The bottom figures are the computational meshes (t = 0.8). The box regions are magnified in the next figure.

  23. Rising bubble example: (a) the bubble evolution: t = 1, 4, 7 (b) the rising Reynolds number, obtained with two fixed grids (16  16 32,32  32  64) and one moving grid (16  16 32).

  24. Nonaxisymmetric merging of two viscous gas bubbles ( = 0.04,  =  / We = 0.02,  = 1/503/4,  = 0.01 and Fr = 1). The moving tetrahedral mesh contains 32  32  64 nodes. t = 0.0, 1.0, 2.0, 3.0, 4.0. Nonaxisymmetric merging of two viscous gas bubbles: the moving tetrahedral mesh contains 32  32  64 nodes. t = 3.0.

  25. We compared the efficiency of the moving mesh methods for the nonaxisymmetric merging of two bubbles. • To reach similar accuracy, the fixed mesh simulation requires at least 64  64  128 cells to match that for a moving mesh with 32  32  64 cells. • To reach t = 4, the simulation on the moving grid with 32  32  64 cells takes about 20h 30min, while the simulation on the 64  64  128 fixed mesh takes about 106h 20min.

  26. Dentritic Growth with MovingFinite Element Methods The mesh redistrbution is realized by • solving an elliptic boundary control problem • a nonlinear multigrid algorithm With a particularly designed monitor function, the qualitiy of the redistributed mesh grids is improved significantly. Dendritic growth has been a central problem in pattern formation and metallurgy

  27. The quantitative phase field model of the dendritic crystallization of a pure melt in two and three dimensions takes the form

  28. where , are the test functions and The weak formulation of the phase field equations mentioned in the previous page takes the form

  29. with

  30. Based on the interaction between the phase field variable and the thermal field, we defined a new variable which satisfies and reformulated the monitor function of the form * where

  31. Multi-mesh adaptation(motivation …) • The heuristic error indicator can be selected as where [] means the jump on the interface and h is the length of edge e. • The fixed threshold rule is adopted as the strategy for mesh adaptation: ensuring the element error indicators T satisfying where tol is the prescribed tolerance and are two constants. • Practically, the tolerance is often selected as • The difference between the system and a scalar equation;

  32. The solving procedure Algorithm: multi-mesh adaptive finite element method for the phase field model. Prepare the background mesh 0; Set the initial value for  and u and obtain the initial mesh for both variables   and  u; whilet < Tdo Solve PDEs; t = t + t; Ifthe meshes have not been updated for N (being constant) steps then Update mesh   and _h; Update mesh  u and u_h; end end

  33. 2D complex dendritic structures Figure: The simulation result with parameters  = 0.70, D = 2, t=0.1, = 0.05 and  is chosen to simulate  = 0. Clockwisely from upper right quarter in both figures are the contour of u, contour of  = 0,  and u at 104 time steps. The right figure is zoomed in from the part inside the black box in left figure. 

  34. 2D complex dendritic structures Figure:The meshes and the profiles for complex dendritic structures when =0.65, D=3 at 1000 time steps(left). The right one is the magnification at the tip show in black box.

  35. 2D complex dendritic structures Figure:The comparison of the meshes for u(left part) and (right part) when =0.65, D=3 at 1000 time steps. It is the magnification of the left plot in Fig. 5 withincenter black box.

  36. 3D simulations: DOF comparison Figure:The comparison of degree of freedoms between  and u as thefunction of the time. The left figure is for the case of=0.55, D=1, and the right one is for the case of =0.45, D=1.4.

  37. Concluding Remark

  38. http://www.math.hkbu.edu.hk/~ttang/MMmovie Collaborators: Yana Di, Chinese Academy of Sciences Ruo Li, Pingwen Zhang of Peking University; Heyu Wang, Hong Kong Baptist University Thanks

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