Problem Definition: Solution of PDE’s in Geosciences  Finite elements and finite volume require: u 3D geometrical model u Geological attributes and u Numerical meshes

Model Creation  3D objects are defined by polygonal faces u Polygonal surfaces are input and intersected u 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 u They have attributes, such as age and type u Attributes supply boundary conditions for PDE’s  The setting of attributes is not a simple task u Each vertex, edge, face has to know their horizons u A set of regions may correspond to a single layer

How to Generate Layers Automatically?  A 2.5D fence diagram u Two faults u 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 u We use the visibility of horizons from an outside point  The top horizon defines the layer u 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 u Octree methods u Delaunay based methods u Advancing front methods

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

Delaunay Disadvantages  No remarkable property in 3D u 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 u 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 u Based on crystal lattices u 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 u Each vertex may generate in average 7 tets  Multi-domain meshing u Implies that each simplex has to be classified  Mesh quality improvement u Resulting mesh has to be useful in simulations  Remeshing with deformation u 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 u Robustness depends on the nature of the geometrical operations u We have robust predicates using exact arithmetic  Intersections cause robustness problems u Necessary to recover missing edges and faces u 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: u Insert a very large number of points into the model, or u Refine the mesh, or u Accept a ratio of at least 10 to1

Multi-domain Models  We have to triangulate multi-domain models u Composed of several 3D internal regions u One external region  We have to specify the simplices corresponding to surfaces defining boundary conditions u This is necessary in finite element applications

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

Cross Section of the Gulf of Mexico  Numbers u 2706 triangles u 4215 edges u 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) u All tetrahedra reached from the seed without crossing the boundary are in the same region u tetrahedra in the external region are deleted

Gulf of Mexico Basin  Numbers u 6 regions u faces u edges u vertices

Triangulation of a Single Region  Numbers u tetrahedra u 1173 points automatically inserted u DA: [ , 179.9] u Sa: [0.0, 359.2] u 2715 (1.854%) tets with min DA < 3.55 u 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 u Permits a much better understanding of geological processes  Multi-domain models are created by intersecting input surfaces u Must handle vertices closely clustered u Vertices in the range [10-7, 10+4] are not uncommon

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

Conclusions  Generation of 3D unconstrained Delaunay triangulation is straightforward u Hint: use an exact arithmetic package u The complicated part is to recover missing constrained edges and faces  Attributes must be present in the final mesh u We have a coupling during the mesh generation with the model being triangulated

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