Finite Element Mesh Generation Finite Element Mesh Generation and its Applicationsand its Applications

S.H. LoS.H. Lo

Department of Civil EngineeringDepartment of Civil Engineering

The University of Hong KongThe University of Hong Kong

INTRODUCTIONINTRODUCTION The The Finite Element MethodFinite Element Method (FEM) has now (FEM) has now

become a general tool in solving become a general tool in solving engineering problems from solid structures engineering problems from solid structures and fluid dynamics to bio-mechanics and fluid dynamics to bio-mechanics systems. systems.

As the concept of the FEM is based on the As the concept of the FEM is based on the decomposition of a continuum into a finite decomposition of a continuum into a finite number of sub-regions (elements), an number of sub-regions (elements), an automatic procedure for the generation of automatic procedure for the generation of nodes and elements over an arbitrary nodes and elements over an arbitrary domain is crucial for the success of FE domain is crucial for the success of FE analysis.analysis.

Figure 3. Examples of FE applicationsFigure 3. Examples of FE applications

Figure 4. Examples of FE applicationsFigure 4. Examples of FE applications

Figure 5. Examples of FE applicationsFigure 5. Examples of FE applications

Mesh Generation Mesh Generation Problem:Problem:

Given a physical domain Given a physical domain and a node and a node spacing function spacing function defined over the entire defined over the entire domain domain , the task of mesh generation is to , the task of mesh generation is to discretize domain discretize domain into valid finite into valid finite elements with size consistent with the elements with size consistent with the specified node spacing function specified node spacing function . . In the more difficult case the boundary of In the more difficult case the boundary of has to be strictly respected. has to be strictly respected.

Figure 7 – Mesh of variable element size over a 2D domainFigure 7 – Mesh of variable element size over a 2D domain

Figure 8 – Surface meshFigure 8 – Surface mesh

Figure 9 – Finite element mesh of a buildingFigure 9 – Finite element mesh of a building

FINITE ELEMENT MESH FINITE ELEMENT MESH

GENERATIONGENERATION To divide a general domain into To divide a general domain into elements, essentially there are two ways: elements, essentially there are two ways:

(i) (i) fill the interior as yet unmeshed fill the interior as yet unmeshed region region with elements directly, andwith elements directly, and

(ii) (ii) modify an existing mesh that modify an existing mesh that already already covers the domain to be covers the domain to be meshed.meshed.

Figure 11. Fill the interior with elementsFigure 11. Fill the interior with elements

Meshed region Unmeshed region

Figure 12. Refining/Modifying an existing meshFigure 12. Refining/Modifying an existing mesh

Advancing Front Advancing Front

ApproachApproach The The advancing front approachadvancing front approach represents represents

mesh generation methods based on the first mesh generation methods based on the first idea. idea.

The generation front is defined as the The generation front is defined as the boundary between the meshed and the boundary between the meshed and the unmeshed parts of the domain. unmeshed parts of the domain.

The key step that must be addressed for the The key step that must be addressed for the advancing front methodadvancing front method is the proper is the proper introduction of new elements to the introduction of new elements to the unmeshed region and a consistent update of unmeshed region and a consistent update of the generation front as elements are formed.the generation front as elements are formed.

Figure 14. Figure 14. (a) Initial front (domain boundary); (a) Initial front (domain boundary); (b) current front; (b) current front; (c) updated front with new element included(c) updated front with new element included

(a) (b) (c)

Area of Area of application:application:

Triangular and tetrahedral meshes generated Triangular and tetrahedral meshes generated by the by the advancing frontadvancing front method are common, method are common, and the methods for generating quadrilateral and the methods for generating quadrilateral and hexahedral meshes by this approach are and hexahedral meshes by this approach are referred to as referred to as pavingpaving or or plasteringplastering techniques.techniques.

Mixed element type with better quality Mixed element type with better quality control control

Gradation and anisotropic meshGradation and anisotropic mesh Suitable for open boundary problemsSuitable for open boundary problems

Delaunay Triangulation Delaunay Triangulation

methodmethod Meshing by the second idea is the well-Meshing by the second idea is the well-known known Delaunay triangulation methodDelaunay triangulation method, , which provides a systematic approach which provides a systematic approach to modify and refine a triangular mesh to modify and refine a triangular mesh by adding first boundary nodes and by adding first boundary nodes and then interior nodes.then interior nodes.

Rapidity and existence of triangulationRapidity and existence of triangulation Boundary integrityBoundary integrity

Voronoi/Dirichlet Voronoi/Dirichlet TessellationTessellation

Given a set S of n unique points in n-Given a set S of n unique points in n-dimensional space, associated with each dimensional space, associated with each point point

there exists a region Vthere exists a region Vii such that such that

The collection of regions The collection of regions

is called the Voronoi tessellation.is called the Voronoi tessellation.

SPi

},:{ ijPXPXXV jii

},1,{ niVi V

Convex PartitionConvex Partition

The region VThe region Vii can be shown to be convex can be shown to be convex intersection of the open half planes intersection of the open half planes separating the points Pseparating the points Pii and P and Pjj, and is the , and is the region in n-dimensional space closest to region in n-dimensional space closest to PPii than to any other points. than to any other points.

In two dimensions, the VIn two dimensions, the Vii are convex are convex polygons, and in three dimensions, they polygons, and in three dimensions, they are convex polyhedrons.are convex polyhedrons.

Figure 19. Voronoi TessellationFigure 19. Voronoi Tessellation

Figure 20. Delaunay triangulationFigure 20. Delaunay triangulation

Circum-sphere Circum-sphere containmentcontainment

In general, In general, Delaunay triangulationDelaunay triangulation generates n-dimensional simplexes with generates n-dimensional simplexes with the interesting property that a the interesting property that a circumscribing n-sphere contains no circumscribing n-sphere contains no points other than the n+1 points which points other than the n+1 points which form the n-dimensional simplex. form the n-dimensional simplex.

This property of circum-sphere This property of circum-sphere containment is the key to the various containment is the key to the various algorithms that construct algorithms that construct Delaunay Delaunay triangulationtriangulation for a given set of points. for a given set of points.

Figure 21. Figure 21. (a) Delaunay triangulation(a) Delaunay triangulation

(b) Non-Delaunay triangulation(b) Non-Delaunay triangulation

(a) (b)

Delaunay Delaunay triangulation:triangulation:

Existence:Existence: Proved by constructionProved by construction Uniqueness:Uniqueness: Unique when all points are in Unique when all points are in

general positiongeneral position Property:Property: For a given set of points in two For a given set of points in two

dimensions, dimensions, Delaunay Delaunay triangulationtriangulation maximizes the maximizes the minimum interior minimum interior angle of angle of the triangular mesh among the triangular mesh among all all possible triangulationspossible triangulations

Construction:Construction: Insertion algorithmInsertion algorithm

Figure 24.Figure 24.Packing circles Packing circles of variable size of variable size over a 2D over a 2D unbounded unbounded domaindomain

Figure 25.Figure 25.Triangular mesh Triangular mesh generated by generated by connecting centres connecting centres of circles based on of circles based on the advancing front the advancing front procedureprocedure

Figure 26. Packing ellipses along a curve Figure 26. Packing ellipses along a curve

Figure 27. Anisotropic mesh following grid linesFigure 27. Anisotropic mesh following grid lines

Figure 28. A magnified view Figure 28. A magnified view

Figure 29. Anisotropic mesh of a wavy surfaceFigure 29. Anisotropic mesh of a wavy surface

Figure 30. Mesh of an analytical curved surfaceFigure 30. Mesh of an analytical curved surface

Figure 31. Klein BottleFigure 31. Klein Bottle

Figure 32. Merging of meshed surfacesFigure 32. Merging of meshed surfaces

Figure 33. Intersection of an avion and a space shuttleFigure 33. Intersection of an avion and a space shuttle

Avion: 2891 elements

Columbia: 7087 elements

Intersection chains: 590 segments

Figure 34. A DNA molecule modeled by 160 spheresFigure 34. A DNA molecule modeled by 160 spheres

C: 49 spheres

O: 31 spheres

H: 55 spheres

N: 20 spheres

P: 5 spheres

Elements: 107520

Nodes: 54080

Loops: 693

Segments: 59999

Figure 35.Figure 35.

Intersection of Intersection of two Bunny two Bunny

hareshares

FFigure 36. igure 36. TTree modeled by quadrilateral elementsree modeled by quadrilateral elements

Figure 37. Rendered image of the tree modelFigure 37. Rendered image of the tree model

Figure 38. Hands modeled by triangular elementsFigure 38. Hands modeled by triangular elements

Elements: 1309332

Nodes: 654646

Loops: 16

Intersection segments: 54966

Neighbor time: 8.953s

Grid time: 6.520s

Intersection time: 46.257s

Overall: 61.730s

Figure 39. Magnified views of the Hands modelFigure 39. Magnified views of the Hands model

Figure 40. Tetrahedral mesh of an airplaneFigure 40. Tetrahedral mesh of an airplane

Figure 41. Tetrahedral mesh of an elephantFigure 41. Tetrahedral mesh of an elephant

Figure 42. Cross-section of the elephant modelFigure 42. Cross-section of the elephant model

Figure 43. Packing spheres of variable sizeFigure 43. Packing spheres of variable size

Figure 44. Packing spheres over a space curveFigure 44. Packing spheres over a space curve

Figure 45. Tetrahedral mesh over a space curveFigure 45. Tetrahedral mesh over a space curve

Figure 46. Tetrahedral elements along a space curveFigure 46. Tetrahedral elements along a space curve

Transitional quadrilateral and hexahedral elementsTransitional quadrilateral and hexahedral elements

Adaptive Mesh Coarsening of Hexahedral Meshes

Anisotropic mesh adaptation based on a Anisotropic mesh adaptation based on a functionalfunctional

Parallel Delaunay RefinementParallel Delaunay Refinement

Meshing by Automatic intersection of solid 3D elementsMeshing by Automatic intersection of solid 3D elements

Curved boundary layer meshing for viscous flowCurved boundary layer meshing for viscous flow

Advancing Front Technique for filling space with arbitrary objectsAdvancing Front Technique for filling space with arbitrary objects

Hybrid mesh generation for reservoir flow simulationHybrid mesh generation for reservoir flow simulation

Modeling of Nanostructured MaterialsModeling of Nanostructured Materials

Size Gradation Control of Anisotropic MeshesSize Gradation Control of Anisotropic Meshes

Generation of 3D elements by MappingGeneration of 3D elements by Mapping

Parametric Surface MeshingParametric Surface Meshing

Finite Elements in Analysis and Finite Elements in Analysis and DesignDesign

Volume 46, Issues 1-2, Pages 1-228 (January-Volume 46, Issues 1-2, Pages 1-228 (January-February 2010) February 2010)

Mesh Generation - Applications and AdaptationMesh Generation - Applications and Adaptation

Edited by S.H. Lo and H BorouchakiEdited by S.H. Lo and H Borouchaki1. Adaptive meshing and analysis using transitional quadrilateral and hexahedral

elements, 2-16, S.H. Lo, D. Wu, K.Y. Sze2. Adaptive mesh coarsening for quadrilateral and hexahedral meshes, 17-32, Jason F.

Shepherd, Mark W. Dewey, Adam C. Woodbury, Steven E. Benzley, Matthew L. Staten, Steven J. Owen

3. Constrained Delaunay tetrahedral mesh generation and refinement, 33-46, Hang Si4. Hybrid mesh smoothing based on Riemannian metric non-conformity minimization, 47-

60, Yannick Sirois, Julien Dompierre, Marie-Gabrielle Vallet, François Guibault5. An anisotropic mesh adaptation method for the finite element solution of variational

problems, 61-73, Weizhang Huang, Xianping Li6. Boundary recovery for Delaunay tetrahedral meshes using local topological

transformations, 74-83, Hamid Ghadyani, John Sullivan, Ziji Wu

7. Improved 3D adaptive remeshing scheme applied in high electromagnetic field gradient computation, 84-95, Houman Borouchaki, Thomas Grosges, Dominique Barchiesi

8. A template for developing next generation parallel Delaunay refinement methods, 96-113

Andrey N. Chernikov, Nikos P. Chrisochoides9. Adaptive mesh generation procedures for thin-walled tubular structures, 114-131, C.K.

Lee, S.P. Chiew, S.T. Lie, T.B.N. Nguyen10. Curved boundary layer meshing for adaptive viscous flow simulations, 132-139, O.

Sahni, X.J. Luo, K.E. Jansen, M.S. Shephard11. Advancing front techniques for filling space with arbitrary separated objects, 140-151

Rainald Löhner, Eugenio Oñate12. Hybrid mesh generation for reservoir flow simulation: Extension to highly deformed

corner point geometry grids, 152-164, T. Mouton, H. Borouchaki, C. Bennis13. Numerical modeling of nanostructured materials, 165-180, Azeddine Benabbou,

Houman Borouchaki, Patrick Laug, Jian Lu14. Size gradation control of anisotropic meshes, 181-202, F. Alauzet15. Sweeping of unstructured meshes over generalized extruded volumes, 203-215, Daniel

Rypl16. Some aspects of parametric surface meshing, 216-226, Patrick Laug

TThank you !hank you !