Challenges%20in%20the%20Generation%20of%203D%20Unstructured%20Mesh%20for%20Simulation%20of%20Geological%20Processes

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Problem Definition Solution of PDEs in
Geosciences

• Finite elements and finite volume require
• 3D geometrical model
• Geological attributes and
• Numerical meshes

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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)

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Attributes

• Horizons and faults are the building blocks
• They have attributes, such as age and type
• Attributes supply boundary conditions for PDEs
• 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

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How to Generate Layers Automatically?

• A 2.5D fence diagram
• Two faults
• Seven horizons

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A Block Depicting Five Layers

• Generally a layer is defined by two horizons, the
  eldest being at the bottom
• Salt may cut several layers

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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

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A 3D Model With Four Layers

• The blue layer is a salt diapir
• All layers have been detected automatically

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Automatic Mesh Generation

• Three main families of algorithms
• Octree methods
• Delaunay based methods
• Advancing front methods

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Delaunay Advantages

• Simple criteria for creating tetrahedra

• Unconstrained Delaunay triangulation requires
  only two predicates
• Point-in-sphere testing
• Point classification according to a plane

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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

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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

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Bravais Lattices

• Hexagonal and Cubic-F (diamond) generate perfect
  tetrahedra in the nature

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Challenges

• 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

• Size of a 3D triangulation
• Each vertex may generate in average 7 tets
• Multi-domain meshing
• Implies that each simplex has to be classified

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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

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Geological Scale

• The scale may vary from hundred of kilometers in
  X and Y

• To just a few hundred meters in Z

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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

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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

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A 45 Degree Cut of the Gulf of Mexico

• 7 horizons
• Bathymetri
• Neogene
• Paleogene
• Upper Cretaceous
• Lower Cretaceous
• Jurassic
• Basement

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Cross Section of the Gulf of Mexico

• Numbers
• 2706 triangles
• 4215 edges
• 1210 vertices

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Simplex Classification

• 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

• Faces, edges and vertices on the boundary of the
  model are marked

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Gulf of Mexico Basin

• Numbers
• 6 regions
• 63704 faces
• 95175 edges
• 31431 vertices

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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 lt 3.55
• 2257 out of 2715 tets with 4 vertices on
  constrained faces

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Detail Showing Small Dihedral Angles
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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, 104 are not
  uncommon

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Breaking the Egg

• The ability of slicing a model reveals its
  internal structure.

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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

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Conclusions

• The size of a tetrahedral mesh can be quite large
• For a moderate size problem a laptop is enough

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