1. grid generation and refinement€¦ · 1. grid generation and refinement • numerical...
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Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
• main tasks: generatingand refininggrids or meshes; closely re-lated with discretization of PDE:
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
• main tasks: generatingand refininggrids or meshes; closely re-lated with discretization of PDE:
– point discretization, cf. finite differences
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
• main tasks: generatingand refininggrids or meshes; closely re-lated with discretization of PDE:
– point discretization, cf. finite differences
– cell discretization, cf. finite elements or volumes
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
• main tasks: generatingand refininggrids or meshes; closely re-lated with discretization of PDE:
– point discretization, cf. finite differences
– cell discretization, cf. finite elements or volumes
• objectives:
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
• main tasks: generatingand refininggrids or meshes; closely re-lated with discretization of PDE:
– point discretization, cf. finite differences
– cell discretization, cf. finite elements or volumes
• objectives:
– accuracy: accurate (and dense) enough to catch the essen-tial physical phenomena
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
• main tasks: generatingand refininggrids or meshes; closely re-lated with discretization of PDE:
– point discretization, cf. finite differences
– cell discretization, cf. finite elements or volumes
• objectives:
– accuracy: accurate (and dense) enough to catch the essen-tial physical phenomena
– boundary approximation: sufficiently detailed to representboundaries and boundary conditions
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
• main tasks: generatingand refininggrids or meshes; closely re-lated with discretization of PDE:
– point discretization, cf. finite differences
– cell discretization, cf. finite elements or volumes
• objectives:
– accuracy: accurate (and dense) enough to catch the essen-tial physical phenomena
– boundary approximation: sufficiently detailed to representboundaries and boundary conditions
– computational efficiency: small overhead for handling of datastructures, no loss of performance on supercomputers
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 1 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
1. Grid Generation and Refinement
• numerical treatment of PDE requires approximate descriptionand discretization of the resp. domain
• main tasks: generatingand refininggrids or meshes; closely re-lated with discretization of PDE:
– point discretization, cf. finite differences
– cell discretization, cf. finite elements or volumes
• objectives:
– accuracy: accurate (and dense) enough to catch the essen-tial physical phenomena
– boundary approximation: sufficiently detailed to representboundaries and boundary conditions
– computational efficiency: small overhead for handling of datastructures, no loss of performance on supercomputers
– numerical adequacy: features with a negative impact on nu-merical efficiency should be avoided (angles, distortions)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 2 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
2. Basic Types of Grids
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 2 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
2. Basic Types of Grids
• structured grids:
– construction of points or elements follows some regular pro-cess
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 2 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
2. Basic Types of Grids
• structured grids:
– construction of points or elements follows some regular pro-cess
– geometric(coordinates) and topological information (neigh-bour relations) can be derived
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 2 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
2. Basic Types of Grids
• structured grids:
– construction of points or elements follows some regular pro-cess
– geometric(coordinates) and topological information (neigh-bour relations) can be derived
• unstructured grids:
– completely irregular generation, even random choice is pos-sible
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 2 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
2. Basic Types of Grids
• structured grids:
– construction of points or elements follows some regular pro-cess
– geometric(coordinates) and topological information (neigh-bour relations) can be derived
• unstructured grids:
– completely irregular generation, even random choice is pos-sible
– explicit storage of basic geometric and topological informa-tion
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 3 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
3. Grid Manipulations
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 3 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
3. Grid Manipulations
• grid generation:
initial placement of grid points or elements
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 3 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
3. Grid Manipulations
• grid generation:
initial placement of grid points or elements
• grid adaptation:
– need for grid points often becomes clear only during thecomputations
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 3 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
3. Grid Manipulations
• grid generation:
initial placement of grid points or elements
• grid adaptation:
– need for grid points often becomes clear only during thecomputations
– requires possibilities of both refinement and coarsening
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 3 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
3. Grid Manipulations
• grid generation:
initial placement of grid points or elements
• grid adaptation:
– need for grid points often becomes clear only during thecomputations
– requires possibilities of both refinement and coarsening
• grid partition :
standard parallelization techniques are based on some subdivi-sionor decompositionof the underlying domain
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 4 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
4. Structured Grids – Prototypes
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 4 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
4. Structured Grids – Prototypes
• rectangular meshes:
– rectangles (2D) or cuboids (3D)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 4 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
4. Structured Grids – Prototypes
• rectangular meshes:
– rectangles (2D) or cuboids (3D)
• triangular meshes:
– triangles (2D) or tetrahedra (3D)
• restricted with respect to complexity of domain
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 5 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
5. Composite Structured Grids
• subdivide (complicated) domain into subdomains of simpler formand use regular meshs there
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 5 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
5. Composite Structured Grids
• subdivide (complicated) domain into subdomains of simpler formand use regular meshs there
blockor patchedgrids:glue subregions togetheralong interfaces (with orwithout continuity)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 5 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
5. Composite Structured Grids
• subdivide (complicated) domain into subdomains of simpler formand use regular meshs there
blockor patchedgrids:glue subregions togetheralong interfaces (with orwithout continuity)
overlaidor chimeragrids:subdomain grids are com-pletely independent and do notfit together (overlap, interpola-tion)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 6 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
6. Block-Structured Grids
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 6 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
6. Block-Structured Grids
• subdivision into logically rectangular subdomains (with logicallyrectangular local grids)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 6 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
6. Block-Structured Grids
• subdivision into logically rectangular subdomains (with logicallyrectangular local grids)
• subdomains fit together in an unstructured way, but continuity isensured (coinciding grid points)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 7 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
7. Grid Transformation
• idea:
transformation of the unit square to the computational domain
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 7 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
7. Grid Transformation
• idea:
transformation of the unit square to the computational domain
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 7 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
7. Grid Transformation
• idea:
transformation of the unit square to the computational domain
• types:
– elliptic
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 7 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
7. Grid Transformation
• idea:
transformation of the unit square to the computational domain
• types:
– elliptic
– inverse elliptic
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 7 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
7. Grid Transformation
• idea:
transformation of the unit square to the computational domain
• types:
– elliptic
– inverse elliptic
– hyperbolic
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 7 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
7. Grid Transformation
• idea:
transformation of the unit square to the computational domain
• types:
– elliptic
– inverse elliptic
– hyperbolic
– algebraic
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 8 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
8. PDE Grid Generators 1
• elliptic:
– grid coordinates ξ(x, y) and η(x, y) are obtained as solu-tions of a system of elliptic PDE
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 8 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
8. PDE Grid Generators 1
• elliptic:
– grid coordinates ξ(x, y) and η(x, y) are obtained as solu-tions of a system of elliptic PDE
∆ξ(x, y) = 0 on ]0, 1[2,
∆η(x, y) = 0 on ]0, 1[2
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 8 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
8. PDE Grid Generators 1
• elliptic:
– grid coordinates ξ(x, y) and η(x, y) are obtained as solu-tions of a system of elliptic PDE
∆ξ(x, y) = 0 on ]0, 1[2,
∆η(x, y) = 0 on ]0, 1[2
– boundary conditions:(ξ, η) shape of the computational domain’s boundary
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 8 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
8. PDE Grid Generators 1
• elliptic:
– grid coordinates ξ(x, y) and η(x, y) are obtained as solu-tions of a system of elliptic PDE
∆ξ(x, y) = 0 on ]0, 1[2,
∆η(x, y) = 0 on ]0, 1[2
– boundary conditions:(ξ, η) shape of the computational domain’s boundary
– ensures very smooth grids, even if boundaries are not smooth
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 8 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
8. PDE Grid Generators 1
• elliptic:
– grid coordinates ξ(x, y) and η(x, y) are obtained as solu-tions of a system of elliptic PDE
∆ξ(x, y) = 0 on ]0, 1[2,
∆η(x, y) = 0 on ]0, 1[2
– boundary conditions:(ξ, η) shape of the computational domain’s boundary
– ensures very smooth grids, even if boundaries are not smooth
– explicit grid control (exact position of points) is difficult
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 8 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
8. PDE Grid Generators 1
• elliptic:
– grid coordinates ξ(x, y) and η(x, y) are obtained as solu-tions of a system of elliptic PDE
∆ξ(x, y) = 0 on ]0, 1[2,
∆η(x, y) = 0 on ]0, 1[2
– boundary conditions:(ξ, η) shape of the computational domain’s boundary
– ensures very smooth grids, even if boundaries are not smooth
– explicit grid control (exact position of points) is difficult
– in nonconvex case, lines may leave domain
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 9 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
9. PDE Grid Generators 2
• inverse elliptic:
– Laplacians are defined on the computational (curvilinear)domain
∆x(ξ, η) = 0 on Ω,∆y(ξ, η) = 0 on Ω
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 9 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
9. PDE Grid Generators 2
• inverse elliptic:
– Laplacians are defined on the computational (curvilinear)domain
∆x(ξ, η) = 0 on Ω,∆y(ξ, η) = 0 on Ω
– boundary conditions:(x, y) shape of the unit square’s boundary
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 9 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
9. PDE Grid Generators 2
• inverse elliptic:
– Laplacians are defined on the computational (curvilinear)domain
∆x(ξ, η) = 0 on Ω,∆y(ξ, η) = 0 on Ω
– boundary conditions:(x, y) shape of the unit square’s boundary
– no external lines, but now a more complicated system to besolved
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 9 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
9. PDE Grid Generators 2
• inverse elliptic:
– Laplacians are defined on the computational (curvilinear)domain
∆x(ξ, η) = 0 on Ω,∆y(ξ, η) = 0 on Ω
– boundary conditions:(x, y) shape of the unit square’s boundary
– no external lines, but now a more complicated system to besolved
• hyperbolic:
– solve hyperbolic system
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 9 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
9. PDE Grid Generators 2
• inverse elliptic:
– Laplacians are defined on the computational (curvilinear)domain
∆x(ξ, η) = 0 on Ω,∆y(ξ, η) = 0 on Ω
– boundary conditions:(x, y) shape of the unit square’s boundary
– no external lines, but now a more complicated system to besolved
• hyperbolic:
– solve hyperbolic system
– for physically unbounded domains
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 10 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
10. Algebraic Grid Generators
• interpolation-based, no extra PDE to be solved
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 10 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
10. Algebraic Grid Generators
• interpolation-based, no extra PDE to be solved
• most famous representative:
transfinite interpolation or Coons patch
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 10 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
10. Algebraic Grid Generators
• interpolation-based, no extra PDE to be solved
• most famous representative:
transfinite interpolation or Coons patch
F1(x, y) = (1 − x) · c(0, y) + x · c(1, y)
F2(x, y) = (1 − y) · c(x, 0) + y · c(x, 1)
F12(x, y) = (1 − x)(1 − y) · c(0, 0) + x(1 − y) · c(1, 0) +
(1 − x)y · c(0, 1) + xy · c(1, 1)
TF (x, y) = (F1 + F2 − F12)(x, y)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 10 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
10. Algebraic Grid Generators
• interpolation-based, no extra PDE to be solved
• most famous representative:
transfinite interpolation or Coons patch
F1(x, y) = (1 − x) · c(0, y) + x · c(1, y)
F2(x, y) = (1 − y) · c(x, 0) + y · c(x, 1)
F12(x, y) = (1 − x)(1 − y) · c(0, 0) + x(1 − y) · c(1, 0) +
(1 − x)y · c(0, 1) + xy · c(1, 1)
TF (x, y) = (F1 + F2 − F12)(x, y)
• interpolation of boundary curves into interior, same in 3D
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 10 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
10. Algebraic Grid Generators
• interpolation-based, no extra PDE to be solved
• most famous representative:
transfinite interpolation or Coons patch
F1(x, y) = (1 − x) · c(0, y) + x · c(1, y)
F2(x, y) = (1 − y) · c(x, 0) + y · c(x, 1)
F12(x, y) = (1 − x)(1 − y) · c(0, 0) + x(1 − y) · c(1, 0) +
(1 − x)y · c(0, 1) + xy · c(1, 1)
TF (x, y) = (F1 + F2 − F12)(x, y)
• interpolation of boundary curves into interior, same in 3D
• cheap, easy to control
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 10 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
10. Algebraic Grid Generators
• interpolation-based, no extra PDE to be solved
• most famous representative:
transfinite interpolation or Coons patch
F1(x, y) = (1 − x) · c(0, y) + x · c(1, y)
F2(x, y) = (1 − y) · c(x, 0) + y · c(x, 1)
F12(x, y) = (1 − x)(1 − y) · c(0, 0) + x(1 − y) · c(1, 0) +
(1 − x)y · c(0, 1) + xy · c(1, 1)
TF (x, y) = (F1 + F2 − F12)(x, y)
• interpolation of boundary curves into interior, same in 3D
• cheap, easy to control
• non-smooth grids
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 10 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
10. Algebraic Grid Generators
• interpolation-based, no extra PDE to be solved
• most famous representative:
transfinite interpolation or Coons patch
F1(x, y) = (1 − x) · c(0, y) + x · c(1, y)
F2(x, y) = (1 − y) · c(x, 0) + y · c(x, 1)
F12(x, y) = (1 − x)(1 − y) · c(0, 0) + x(1 − y) · c(1, 0) +
(1 − x)y · c(0, 1) + xy · c(1, 1)
TF (x, y) = (F1 + F2 − F12)(x, y)
• interpolation of boundary curves into interior, same in 3D
• cheap, easy to control
• non-smooth grids
• leaving lines possible
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 11 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
11. Unstructured Grids – Delauny Triangluation
• closely related to FEM, typically triangles/tetrahedra
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 11 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
11. Unstructured Grids – Delauny Triangluation
• closely related to FEM, typically triangles/tetrahedra
• Delaunay triangulations:
– suppose you already have grid points – how to define ele-ments?
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 11 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
11. Unstructured Grids – Delauny Triangluation
• closely related to FEM, typically triangles/tetrahedra
• Delaunay triangulations:
– suppose you already have grid points – how to define ele-ments?
– go back to Dirichlet and Voronoi and define Voronoi regionsaround each given grid point:
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 11 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
11. Unstructured Grids – Delauny Triangluation
• closely related to FEM, typically triangles/tetrahedra
• Delaunay triangulations:
– suppose you already have grid points – how to define ele-ments?
– go back to Dirichlet and Voronoi and define Voronoi regionsaround each given grid point:
Vi = P : ‖P − Pi‖ < ‖P − Pj‖ ∀j 6= i
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 11 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
11. Unstructured Grids – Delauny Triangluation
• closely related to FEM, typically triangles/tetrahedra
• Delaunay triangulations:
– suppose you already have grid points – how to define ele-ments?
– go back to Dirichlet and Voronoi and define Voronoi regionsaround each given grid point:
Vi = P : ‖P − Pi‖ < ‖P − Pj‖ ∀j 6= i– result is called a Voronoi diagram(subdivision of domain into
polygons or polyhedra, resp.)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 11 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
11. Unstructured Grids – Delauny Triangluation
• closely related to FEM, typically triangles/tetrahedra
• Delaunay triangulations:
– suppose you already have grid points – how to define ele-ments?
– go back to Dirichlet and Voronoi and define Voronoi regionsaround each given grid point:
Vi = P : ‖P − Pi‖ < ‖P − Pj‖ ∀j 6= i– result is called a Voronoi diagram(subdivision of domain into
polygons or polyhedra, resp.)
– draw mid-lines between all pairs of neighbouring points;leads to set of disjoint triangles or tetrahedra
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 11 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
11. Unstructured Grids – Delauny Triangluation
• closely related to FEM, typically triangles/tetrahedra
• Delaunay triangulations:
– suppose you already have grid points – how to define ele-ments?
– go back to Dirichlet and Voronoi and define Voronoi regionsaround each given grid point:
Vi = P : ‖P − Pi‖ < ‖P − Pj‖ ∀j 6= i– result is called a Voronoi diagram(subdivision of domain into
polygons or polyhedra, resp.)
– draw mid-lines between all pairs of neighbouring points;leads to set of disjoint triangles or tetrahedra
– very widespread
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 12 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
12. Point Generation
• point generation: how to get grid points?
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 12 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
12. Point Generation
• point generation: how to get grid points?
– superimpose a regular grid and refine (quadtree, octree)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 12 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
12. Point Generation
• point generation: how to get grid points?
– superimpose a regular grid and refine (quadtree, octree)
– or:
* start with some boundary point distribution,
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 12 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
12. Point Generation
• point generation: how to get grid points?
– superimpose a regular grid and refine (quadtree, octree)
– or:
* start with some boundary point distribution,
* generate Delaunay triangulation,
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 12 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
12. Point Generation
• point generation: how to get grid points?
– superimpose a regular grid and refine (quadtree, octree)
– or:
* start with some boundary point distribution,
* generate Delaunay triangulation,
* continue with subdivision following suitable rules
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 12 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
12. Point Generation
• point generation: how to get grid points?
– superimpose a regular grid and refine (quadtree, octree)
– or:
* start with some boundary point distribution,
* generate Delaunay triangulation,
* continue with subdivision following suitable rules
– if helpful, add point or lines sources (singularities, bound.layers)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
– choose an edge on the current front, say PQ
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
– choose an edge on the current front, say PQ
– create a new point R at equal distance d from P and Q
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
– choose an edge on the current front, say PQ
– create a new point R at equal distance d from P and Q
– determine all grid points lying within a circle around R, ra-dius r
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
– choose an edge on the current front, say PQ
– create a new point R at equal distance d from P and Q
– determine all grid points lying within a circle around R, ra-dius r
– order these points w.r.t. distance from R
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
– choose an edge on the current front, say PQ
– create a new point R at equal distance d from P and Q
– determine all grid points lying within a circle around R, ra-dius r
– order these points w.r.t. distance from R
– for all these points, form triangles with P and Q, select one
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
– choose an edge on the current front, say PQ
– create a new point R at equal distance d from P and Q
– determine all grid points lying within a circle around R, ra-dius r
– order these points w.r.t. distance from R
– for all these points, form triangles with P and Q, select one
– if accepted (no intersections etc.), add to list of points
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
– choose an edge on the current front, say PQ
– create a new point R at equal distance d from P and Q
– determine all grid points lying within a circle around R, ra-dius r
– order these points w.r.t. distance from R
– for all these points, form triangles with P and Q, select one
– if accepted (no intersections etc.), add to list of points
– change triangulation: add new edges
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 13 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
13. Advancing front methods
• start from the boundary (starting front) and advance step by stepinto the interior:
– choose an edge on the current front, say PQ
– create a new point R at equal distance d from P and Q
– determine all grid points lying within a circle around R, ra-dius r
– order these points w.r.t. distance from R
– for all these points, form triangles with P and Q, select one
– if accepted (no intersections etc.), add to list of points
– change triangulation: add new edges
– change front line: add new edges, remove old edges
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 14 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
14. Further Grid Generation Methods
• spacetreewith boundary fitted closure:
– create a quadtree/octree of given accuracy, add triangles ortetrahedra, resp., at boundary
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 14 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
14. Further Grid Generation Methods
• spacetreewith boundary fitted closure:
– create a quadtree/octree of given accuracy, add triangles ortetrahedra, resp., at boundary
• hybrid grids (structured-unstructured grids):
– many mixed forms to achieve optimum compromise betweenregularity and flexibility
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
– block-wise refinement, interpolation at hanging nodes
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
– block-wise refinement, interpolation at hanging nodes
– hierarchical approach (quadtrees, octrees)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
– block-wise refinement, interpolation at hanging nodes
– hierarchical approach (quadtrees, octrees)
• adaptivitiy with unstructured grids: more popular
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
– block-wise refinement, interpolation at hanging nodes
– hierarchical approach (quadtrees, octrees)
• adaptivitiy with unstructured grids: more popular
– global error estimator:tells us whether the result computed so far is sufficientlyaccurate
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
– block-wise refinement, interpolation at hanging nodes
– hierarchical approach (quadtrees, octrees)
• adaptivitiy with unstructured grids: more popular
– global error estimator:tells us whether the result computed so far is sufficientlyaccurate
– refinement criterion: determines the aim of refinement:
* balancing of error over grid points,
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
– block-wise refinement, interpolation at hanging nodes
– hierarchical approach (quadtrees, octrees)
• adaptivitiy with unstructured grids: more popular
– global error estimator:tells us whether the result computed so far is sufficientlyaccurate
– refinement criterion: determines the aim of refinement:
* balancing of error over grid points,
* keeping error below some threshold everywhere,
* . . .
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
– block-wise refinement, interpolation at hanging nodes
– hierarchical approach (quadtrees, octrees)
• adaptivitiy with unstructured grids: more popular
– global error estimator:tells us whether the result computed so far is sufficientlyaccurate
– refinement criterion: determines the aim of refinement:
* balancing of error over grid points,
* keeping error below some threshold everywhere,
* . . .
– local error estimator or indicator:tells us during computation where to refine the grid locally
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 15 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
15. Adaptive Refinement
• adaptivity with structured grids:
– block-wise refinement, interpolation at hanging nodes
– hierarchical approach (quadtrees, octrees)
• adaptivitiy with unstructured grids: more popular
– global error estimator:tells us whether the result computed so far is sufficientlyaccurate
– refinement criterion: determines the aim of refinement:
* balancing of error over grid points,
* keeping error below some threshold everywhere,
* . . .
– local error estimator or indicator:tells us during computation where to refine the grid locally
– refinement procedure:determines the technical process of refinement (centroid,red, green)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
– free surface problems
* injection moulding,
* melting,
* freezing
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
– free surface problems
* injection moulding,
* melting,
* freezing
– multiphase flows
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
– free surface problems
* injection moulding,
* melting,
* freezing
– multiphase flows
– fluid-structure interactions
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
– free surface problems
* injection moulding,
* melting,
* freezing
– multiphase flows
– fluid-structure interactions
• common strategies:
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
– free surface problems
* injection moulding,
* melting,
* freezing
– multiphase flows
– fluid-structure interactions
• common strategies:
– front tracking methods:
* describe boundary or interface explicitly,
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
– free surface problems
* injection moulding,
* melting,
* freezing
– multiphase flows
– fluid-structure interactions
• common strategies:
– front tracking methods:
* describe boundary or interface explicitly,
* update of geometry due to movements,
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
– free surface problems
* injection moulding,
* melting,
* freezing
– multiphase flows
– fluid-structure interactions
• common strategies:
– front tracking methods:
* describe boundary or interface explicitly,
* update of geometry due to movements,
* accurate, but expensive,
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 16 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
16. Varying Geometries 1
• applications:
– free surface problems
* injection moulding,
* melting,
* freezing
– multiphase flows
– fluid-structure interactions
• common strategies:
– front tracking methods:
* describe boundary or interface explicitly,
* update of geometry due to movements,
* accurate, but expensive,
* topology changes?
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 17 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
17. Varying Geometries 2
• common strategies (continued):
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 17 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
17. Varying Geometries 2
• common strategies (continued):
– front capturing methods:
* follow interface indirectly (some global quantity)
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 17 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
17. Varying Geometries 2
• common strategies (continued):
– front capturing methods:
* follow interface indirectly (some global quantity)
* less precise, but more straightforward
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 17 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
17. Varying Geometries 2
• common strategies (continued):
– front capturing methods:
* follow interface indirectly (some global quantity)
* less precise, but more straightforward
* examples: Volume-of-Fluid, Marker-and-Cell
Grid Generation and . . .
Basic Types of Grids
Grid Manipulations
Structured Grids – . . .
Composite Structured . . .
Block-Structured Grids
Grid Transformation
PDE Grid Generators 1
PDE Grid Generators 2
Algebraic Grid Generators
Unstructured Grids – . . .
Point Generation
Advancing front methods
Further Grid . . .
Adaptive Refinement
Varying Geometries 1
Varying Geometries 2
Page 17 of 17
Introduction to Scientific Computing
13. GridsMiriam Mehl
17. Varying Geometries 2
• common strategies (continued):
– front capturing methods:
* follow interface indirectly (some global quantity)
* less precise, but more straightforward
* examples: Volume-of-Fluid, Marker-and-Cell
– sliding mesh techniques:
* part of the grid moves, the other part is fixed (cf. ALEapproach)