Academia.eduAcademia.edu

Figure from:Liquid simulation on lattice-based tetrahedral meshes

See full PDFdownloadDownload figure
Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes

Figure 1 Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes

Related Figures (7)

Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes Figure 2: Each time step uses one surface triangulation and one tetrahedral mesh. A new time step’s surface S;+ is created by semi-Lagrangian advection: tracing paths back- ward in time through the velocity field on M; to locate a poini relative to S;. The isosurface stuffing algorithm generate: a tetrahedral mesh M;., (with coarser surface resolution) from S;41. Velocities are interpolated/extrapolated from M,.
Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes Figure 2: Each time step uses one surface triangulation and one tetrahedral mesh. A new time step’s surface S;+ is created by semi-Lagrangian advection: tracing paths back- ward in time through the velocity field on M; to locate a poini relative to S;. The isosurface stuffing algorithm generate: a tetrahedral mesh M;., (with coarser surface resolution) from S;41. Velocities are interpolated/extrapolated from M,.
Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes Figure 3: Top: A wave of liquid flowing over a hemispher- ical obstacle. Center: Cutaway view of the tetrahedra en- closed by the green box in the top image. Bottom: Portions of the surface triangulation and tetrahedral mesh, respec- tively, enclosed by the yellow box in the top image.
Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes Figure 3: Top: A wave of liquid flowing over a hemispher- ical obstacle. Center: Cutaway view of the tetrahedra en- closed by the green box in the top image. Bottom: Portions of the surface triangulation and tetrahedral mesh, respec- tively, enclosed by the yellow box in the top image.
Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes Figure 4: A comparison of images taken from animations generated with our thickening scheme (top) and without (bottom). The example with no thickening suffers from visible holes and artifacts where thin structures vanish. Observe the difference in liquid volume between the last frames of the two sequences: without thickening, the simulation has lost appreciable volume.
Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes Figure 4: A comparison of images taken from animations generated with our thickening scheme (top) and without (bottom). The example with no thickening suffers from visible holes and artifacts where thin structures vanish. Observe the difference in liquid volume between the last frames of the two sequences: without thickening, the simulation has lost appreciable volume.
Figure 5: Finding points that lie approximately on the me- dial axis. In this example, my is used but mz is rejected. Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes
Figure 5: Finding points that lie approximately on the me- dial axis. In this example, my is used but mz is rejected. Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes
Simulation domains with sinuous shapes make it diffi- cult to geometrically coarsen a mesh in a way suitable for multigrid—clustering unstructured tetrahedra does not cre- ate larger tetrahedra. This problem can occur even for regu- lar hexahedral grids on irregularly shaped domains. We cir- cumvent this problem with algebraic multigrid, which uses only the finite volume matrices, and not the geometry, to construct the prolongation and restriction operators. From the prolongation and restriction operators the system matri- ces can be defined by Ay_1 = Ry—1,xAgPhk-1- V-cycle multigrid begins by applying a relaxation oper- ator S(Ax,X,,b,), which returns an approximation solution (usually with poor accuracy) to the system A,x, = by. We use two Gauss-Seidel iterations. Next, it computes the resid- ual of the system, restricts it to the next coarser grid, and recursively solves the coarse problem on the restricted resid- ual. It prolongs the coarse solution and adds it, as a correc- tive term, to the fine solution. Finally, the relaxation operator is applied again. We use full-cycle multigrid, which uses V- cycle multigrid as a subroutine as follows. VCvecr eMintricein(y, h. /L)
Simulation domains with sinuous shapes make it diffi- cult to geometrically coarsen a mesh in a way suitable for multigrid—clustering unstructured tetrahedra does not cre- ate larger tetrahedra. This problem can occur even for regu- lar hexahedral grids on irregularly shaped domains. We cir- cumvent this problem with algebraic multigrid, which uses only the finite volume matrices, and not the geometry, to construct the prolongation and restriction operators. From the prolongation and restriction operators the system matri- ces can be defined by Ay_1 = Ry—1,xAgPhk-1- V-cycle multigrid begins by applying a relaxation oper- ator S(Ax,X,,b,), which returns an approximation solution (usually with poor accuracy) to the system A,x, = by. We use two Gauss-Seidel iterations. Next, it computes the resid- ual of the system, restricts it to the next coarser grid, and recursively solves the coarse problem on the restricted resid- ual. It prolongs the coarse solution and adds it, as a correc- tive term, to the fine solution. Finally, the relaxation operator is applied again. We use full-cycle multigrid, which uses V- cycle multigrid as a subroutine as follows. VCvecr eMintricein(y, h. /L)
Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes Figure 6: A sequence beginning with a fluid configuration in the shape of an angel. Once the simulation begins, the structure falls, creating a complex splash pattern.
Chentanez, Feldman, Labelle, O’Brien, and Shewchuk / Liquid Simulation on Lattice-Based Tetrahedral Meshes Figure 6: A sequence beginning with a fluid configuration in the shape of an angel. Once the simulation begins, the structure falls, creating a complex splash pattern.
figure 7: A sequence wherein a block of liquid is released at one side of a closed container. The wave of water flows over an »bstacle and crashes against the far side of the container.
figure 7: A sequence wherein a block of liquid is released at one side of a closed container. The wave of water flows over an »bstacle and crashes against the far side of the container.