finite element mesh generation and its applications
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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 !
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