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Problem Definition: Solution of PDE’s in Geosciences. Finite elements and finite volume require: 3D geometrical model Geological attributes and Numerical meshes. Model Creation. 3D objects are defined by polygonal faces Polygonal surfaces are input and intersected - PowerPoint PPT Presentation
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21/07/2000
Challenges in The Generation of 3DUnstructured Mesh for Simulation of
Geological ProcessesPaulo Roma Cavalcanti
Ulisses T. Mello
Universidade Federal do Rio de JaneiroIBM T. J. Watson Research Center
Problem Definition: Solution of PDE’s in Geosciences
Finite elements and finite volume require: 3D geometrical model Geological attributes and Numerical meshes
Model Creation
3D objects are defined by polygonal faces Polygonal surfaces are input and intersected A spatial subdivision is created
We require only the topological consistency of the input polygons
Vertices, edges and faces are constrained for meshing (internal and external boundaries)
Attributes
Horizons and faults are the building blocks They have attributes, such as age and type Attributes supply boundary conditions for PDE’s
The setting of attributes is not a simple task Each vertex, edge, face has to know their horizons A set of regions may correspond to a single layer
How to Generate Layers Automatically?
A 2.5D fence diagram Two faults Seven horizons
A Block Depicting Five Layers
Generally a layer is defined by two horizons, the eldest being at the bottom
Salt may cut several layers
The Algorithm
All regions have inward normals We use the visibility of horizons from an outside
point
The top horizon defines the layer It has a negative volume and the greatest
magnitude
A 3D Model With Four Layers
The blue layer is a salt diapir
All layers have been detected automatically
Automatic Mesh Generation
Three main families of algorithms Octree methods Delaunay based methods Advancing front methods
Delaunay Advantages
Simple criteria for creating tetrahedra
Unconstrained Delaunay triangulation requires only two predicates Point-in-sphere testing Point classification according to a
plane
Delaunay Disadvantages
No remarkable property in 3D Does not maximize the
minimum angle as in 2D
Constraining edges and faces may not be present (must be recovered later)
May produce “useless” numerical meshes Slivers (“flat” tetrahedra)
must be removed
Background Meshes
The Delaunay criterion just tells how to connect points - it does not create new points
We use background meshes to generate points into the model Based on crystal lattices 20% of tetrahedra are perfect, even using the
Delaunay criteria
Bravais Lattices
Hexagonal and Cubic-F (diamond) generate perfect tetrahedra in the nature
Challenges
Size of a 3D triangulation Each vertex may
generate in average 7 tets
Multi-domain meshing Implies that each
simplex has to be classified
Mesh quality improvement Resulting mesh has to be useful in
simulations
Remeshing with deformation If the problem evolve over the time,
the mesh has to be rebuilt as long as topology change
Robustness Geological scale
Robustness Automatic mesh generation requires robust
algorithms Robustness depends on the nature of the geometrical
operations We have robust predicates using exact arithmetic
Intersections cause robustness problems Necessary to recover missing edges and faces When applied to slivers may lead to an erroneous
topology
Geological Scale
The scale may vary from hundred of kilometers in X and Y
To just a few hundred meters in Z
Non-uniform Scale
Implies bad tetrahedra shape. The alternative is either to: Insert a very large number of points
into the model, or Refine the mesh, or Accept a ratio of at least 10 to1
Multi-domain Models
We have to triangulate multi-domain models Composed of several 3D internal regions One external region
We have to specify the simplices corresponding to surfaces defining boundary conditions This is necessary in finite element applications
A 45 Degree Cut of the Gulf of Mexico
7 horizons Bathymetri Neogene Paleogene Upper Cretaceous Lower Cretaceous Jurassic Basement
Cross Section of the Gulf of Mexico
Numbers 2706 triangles 4215 edges 1210 vertices
Simplex Classification
Faces, edges and vertices on the boundary of the model are marked
A point-in-region testing is performed for a single tetrahedron (seed) All tetrahedra reached from the seed
without crossing the boundary are in the same region
tetrahedra in the external region are deleted
Gulf of Mexico Basin
Numbers 6 regions 63704 faces 95175 edges 31431 vertices
Triangulation of a Single Region
Numbers 146373 tetrahedra 1173 points
automatically inserted DA: [0.001241, 179.9] Sa: [0.0, 359.2] 2715 (1.854%) tets
with min DA < 3.55 2257 out of 2715 tets
with 4 vertices on constrained faces
Detail Showing Small Dihedral Angles
Conclusions
The use of a real 3D model opens a new dimension Permits a much better understanding of geological
processes
Multi-domain models are created by intersecting input surfaces Must handle vertices closely clustered Vertices in the range [10-7, 10+4] are not uncommon
Breaking the Egg
The ability of slicing a model reveals its internal structure.
Conclusions
Generation of 3D unconstrained Delaunay triangulation is straightforward Hint: use an exact arithmetic package The complicated part is to recover missing constrained
edges and faces
Attributes must be present in the final mesh We have a coupling during the mesh generation with the
model being triangulated
Conclusions
The size of a tetrahedral mesh can be quite large For a moderate size problem a laptop is enough