# Challenges%20in%20the%20Generation%20of%203D%20Unstructured%20Mesh%20for%20Simulation%20of%20Geological%20Processes - PowerPoint PPT Presentation

Title:

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

Description:

### We require only the topological consistency of the input polygons ... Upper Cretaceous. Lower Cretaceous. Jurassic. Basement. Cross Section of the Gulf of Mexico ... – PowerPoint PPT presentation

Number of Views:54
Avg rating:3.0/5.0
Slides: 30
Provided by: pauloromac
Category:
Tags:
Transcript and Presenter's Notes

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

1
(No Transcript)
2
Problem Definition Solution of PDEs in
Geosciences
• Finite elements and finite volume require
• 3D geometrical model
• Geological attributes and
• Numerical meshes

3
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)

4
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

5
How to Generate Layers Automatically?
• A 2.5D fence diagram
• Two faults
• Seven horizons

6
A Block Depicting Five Layers
• Generally a layer is defined by two horizons, the
eldest being at the bottom
• Salt may cut several layers

7
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

8
A 3D Model With Four Layers
• The blue layer is a salt diapir
• All layers have been detected automatically

9
Automatic Mesh Generation
• Three main families of algorithms
• Octree methods
• Delaunay based methods
• Advancing front methods

10
• Simple criteria for creating tetrahedra
• Unconstrained Delaunay triangulation requires
only two predicates
• Point-in-sphere testing
• Point classification according to a plane

11
• 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

12
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

13
Bravais Lattices
• Hexagonal and Cubic-F (diamond) generate perfect
tetrahedra in the nature

14
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

15
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

16
Geological Scale
• The scale may vary from hundred of kilometers in
X and Y
• To just a few hundred meters in Z

17
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

18
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

19
A 45 Degree Cut of the Gulf of Mexico
• 7 horizons
• Bathymetri
• Neogene
• Paleogene
• Upper Cretaceous
• Lower Cretaceous
• Jurassic
• Basement

20
Cross Section of the Gulf of Mexico
• Numbers
• 2706 triangles
• 4215 edges
• 1210 vertices

21
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

22
Gulf of Mexico Basin
• Numbers
• 6 regions
• 63704 faces
• 95175 edges
• 31431 vertices

23
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

24
Detail Showing Small Dihedral Angles
25
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

26
Breaking the Egg
• The ability of slicing a model reveals its
internal structure.

27
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

28
Conclusions
• The size of a tetrahedral mesh can be quite large
• For a moderate size problem a laptop is enough

29
(No Transcript)