Problem Definition: Solution of PDE’s in Geosciences

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