Considerations about level-set methods: Accuracy, interpolation and reinitialization
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1 Considerations about level-set methods: Accuracy, interpolation and reinitialization Gustavo C. Buscaglia Coll: E. Dari, R. Ausas, L. Ruspini Instituto de Ciências Matemáticas e de Computação Universidade de São Paulo, São Carlos, Brasil
2 Level set method Osher & Sethian, 1988 Books: Sethian, 1996, 1999 (over 1800 citations) Osher & Fedkiw, 2003 (over 700 citations)
3 Level set method Flexible Easy to code Preferred method in engineering applications Pumping station. Yang, Garcia, Buscaglia, 2005.
4 Level set method Flexible Easy to code Preferred method in engineering applications Spillway. Abad, Garcia, Buscaglia, 2005.
5 Level set method in fluid dynamics Sussman, Smereka, Osher, 1994 General algorithm: Each time step perform the following operations 1. Pre-process the level set function. 2. Transport the level set function. 3. Correct the level set function. 4. Detect the interface. 5. Solve for velocities & pressures with the updated interface location. 6. Back to 1.
6 Level set method in fluid dynamics Reinitialization 1. Pre-process the level set function. 2. Transport the level set function. 3. Correct the level set function. 4. Detect the interface. 5. Solve for velocities & pressures with the updated interface location. 6. Back to 1. Idea: Smooth the level set function without modifying its zero-level set. Make it approximate the signed distance function. Difficulty: It is impossible. Available methods: Pseudo-transient redistancing Sussman, Osher,... Fast-marching, geometric Osher & Fedkiw, 2003 Mut, Buscaglia & Dari, 2006
7 Level set method in fluid dynamics Computing the distance function: Examples 1. Pre-process the level set function. 2. Transport the level set function. 3. Correct the level set function. 4. Detect the interface. 5. Solve for velocities & pressures with the updated interface location. 6. Back to 1.
8 Level set method in fluid dynamics 1. Pre-process the level set function. 2. Transport the level set function. 3. Correct the level set function. 4. Detect the interface. 5. Solve for velocities & pressures with the updated interface location. 6. Back to 1. Reinitialization: Preserving the volume (Mut, Buscaglia & Dari, 2006) Volume preservation depends on the nodal values adjacent to S. Mission: Given a level-set function that defines S, modify the nodal values leaving S unchanged IMPOSSIBLE, even the (interpolant of) the exact distance moves S. Mission (version 2):
9 Level set method in fluid dynamics 1. Pre-process the level set function. 2. Transport the level set function. 3. Correct the level set function. 4. Detect the interface. 5. Solve for velocities & pressures with the updated interface location. 6. Back to 1. Transport Main criticism to LS method: Numerical dissipation when solving Eq. (*). High-order methods are advocated: State of the art for LS transport is HJ-WENO in space (5th order) and TVD-RK in time (3rd order). Losasso et al., Comp. & Fluids, Semi-Lagrangian methods seem natural (Strain, JCP, 1999), however they are too diffusive. A Lagrangian (SPH, particle) method seems natural too. Hieber & Koumoutsakos, JCP, 2005.
10 Level set method in fluid dynamics 1. Pre-process the level set function. 2. Transport the level set function. 3. Correct the level set function. 4. Detect the interface. 5. Solve for velocities & pressures with the updated interface location. 6. Back to 1. Correction of the level set function Enright et al, JCP, 2002, Hybrid Particle-LS method Escaped particles in light blue identify the needed correction. Enright et al, Comput. & Struct., 2005 Same idea with semi-lagrangian transport. 256 x 256 mesh ,000 initial particles!!
11 Level set method in fluid dynamics Detect the interface Given the nodal values of the LS function, where is the interface? Interpolation In the end, what counts is the difference between the exact and calculated interface positions. This position is then used to evaluate material properties, surface tension forces, etc., and proceed to the fluid solver. Some reported orders for several problems (circle, Zalezak, vortex, etc.) Sussman'99, ENO3 = 1-3 Enright'02, HJ-WENO = 1-3 Enright'02, HJ-WENO + part = 1.5 Enright'05, HJ-WENO + part = 1.4 Enright'05, SemiLag + part =1.0 Gómez'05, WENO =2.2 Olsson'05, TVD = Pre-process the level set function. 2. Transport the level set function. 3. Correct the level set function. 4. Detect the interface. 5. Solve for velocities & pressures with the updated interface location. 6. Back to 1.
12 Level set method in fluid dynamics 1. Pre-process the level set function. 2. Transport the level set function. 3. Correct the level set function. 4. Detect the interface. 5. Solve for velocities & pressures with the updated interface location. 6. Back to 1. Some recent work Quantitative assessment of redistancing. Curvilinear grids. Comparison of several TVD/ENO schemes for LS transport Detection How good standard Eulerian methods are, or can get?
13 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Numerical methods in 1D Reconstruction Slopes
14 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Numerical methods in 1D Reconstruction TVD ENO
15 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Numerical methods in 1D Transport with u = c/x
16 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. TVD/ENO schemes, typical behavior: Transport with u = exp(-x/2) Redistancing: Improves robustness (errors on coarse meshes are smaller) but not accuracy. We also considered the following question: Given that each TVD/ENO scheme reconstructs the solution differently, should one use this reconstruction to detect the surface? Numerical tests have shown that the best choice to detect the surface (with 2nd order accuracy) is simple linear interpolation.
17 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. TVD/ENO schemes, 2D/3D:
18 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Assessment of redistancing in 2D/3D: Is it needed? (it will cause errors) If yes, how to do it? Semi-Lagrangian RK4, dt=0.01, after 44 steps
19 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Assessment of redistancing in 2D/3D: Is it needed? (it will cause errors) If yes, how to do it? Method: ENO, CFL=0.9
20 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Assessment of redistancing in 2D/3D: In classical test cases, TVD/ENO second-order methods seem not to need redistancing if mesh is good.
21 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Assessment of redistancing in 2D/3D: 120x120x120
22 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Assessment of redistancing in 2D/3D: Sometimes redistancing deteriorates the accuracy In any case, the exact distance is a bad choice.
23 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Assessment of redistancing in 2D/3D: 90x90x90, curvilinear, CFL=0.9, ENO Without With
24 Assessment of redistancing. Some recent work Curvilinear grids. TVD/ENO schemes. Detection. Some conclusions: Best available choice: ENO, redistancing (robustness), (bi/tri) linear detection. Accuracy in multidimensions insufficient, no improvement seems to be expectable from higher-order schemes or sophisticate detection within Eulerian framework.
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26 Strain, JCP, 1999 Semi-Lagrangian methods for Level Set equations
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