Finite Element Mesh Generation and its Applications

1
Finite Element Mesh Generation and its
Applications
S.H. Lo Department of Civil Engineering The
University of Hong Kong
2
INTRODUCTION

• The Finite Element Method (FEM) has now become a
  general tool in solving engineering problems from
  solid structures and fluid dynamics to
  bio-mechanics systems.
• As the concept of the FEM is based on the
  decomposition of a continuum into a finite number
  of sub-regions (elements), an automatic procedure
  for the generation of nodes and elements over an
  arbitrary domain is crucial for the success of FE
  analysis.

3
Figure 3. Examples of FE applications
4
Figure 4. Examples of FE applications
5
Figure 5. Examples of FE applications
6
Mesh Generation Problem

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

7
Figure 7 Mesh of variable element size over a
2D domain
8
Figure 8 Surface mesh
9
Figure 9 Finite element mesh of a building
10
FINITE ELEMENT MESH GENERATION

• To divide a general domain into elements,
  essentially there are two ways
• (i) fill the interior as yet unmeshed region
  with elements directly, and
• (ii) modify an existing mesh that already
  covers the domain to be meshed.

11
Figure 11. Fill the interior with elements
Meshed region
12
Figure 12. Refining/Modifying an existing mesh
13
Advancing Front Approach

• The advancing front approach represents mesh
  generation methods based on the first idea.
• The generation front is defined as the boundary
  between the meshed and the unmeshed parts of the
  domain.
• The key step that must be addressed for the
  advancing front method is the proper introduction
  of new elements to the unmeshed region and a
  consistent update of the generation front as
  elements are formed.

14
Figure 14. (a) Initial front (domain boundary)
(b) current front (c) updated front with
new element included
(a) (b) (c)
15
Area of application

• Triangular and tetrahedral meshes generated by
  the advancing front method are common, and the
  methods for generating quadrilateral and
  hexahedral meshes by this approach are referred
  to as paving or plastering techniques.
• Mixed element type with better quality control
• Gradation and anisotropic mesh
• Suitable for open boundary problems

16
Delaunay Triangulation method

• Meshing by the second idea is the well-known
  Delaunay triangulation method, which provides a
  systematic approach to modify and refine a
  triangular mesh by adding first boundary nodes
  and then interior nodes.
• Rapidity and existence of triangulation
• Boundary integrity

17
Voronoi/Dirichlet Tessellation

• Given a set S of n unique points in
  n-dimensional space, associated with each point
  
• there exists a region Vi such that
• The collection of regions
• is called the Voronoi tessellation.

18
Convex Partition

• The region Vi can be shown to be convex
  intersection of the open half planes separating
  the points Pi and Pj, and is the region in
  n-dimensional space closest to Pi than to any
  other points.
• In two dimensions, the Vi are convex polygons,
  and in three dimensions, they are convex
  polyhedrons.

19
Figure 19. Voronoi Tessellation
20
Figure 20. Delaunay triangulation
21
Circum-sphere containment

• In general, Delaunay triangulation generates
  n-dimensional simplexes with the interesting
  property that a circumscribing n-sphere contains
  no points other than the n1 points which form
  the n-dimensional simplex.
• This property of circum-sphere containment is
  the key to the various algorithms that construct
  Delaunay triangulation for a given set of points.

22
Figure 21. (a) Delaunay triangulation (b)
Non-Delaunay triangulation
(a) (b)
23
Delaunay triangulation

• Existence Proved by construction
• Uniqueness Unique when all points are in
  general position
• Property For a given set of points in two
  dimensions, Delaunay triangulation
  maximizes the minimum interior angle of
  the triangular mesh among all possible
  triangulations
• Construction Insertion algorithm

24
Figure 24.Packing circles of variable size over
a 2D unbounded domain
25
Figure 25.Triangular mesh generated by
connecting centres of circles based on the
advancing front procedure
26
Figure 26. Packing ellipses along a curve
27
Figure 27. Anisotropic mesh following grid lines
28
Figure 28. A magnified view
29
Figure 29. Anisotropic mesh of a wavy surface
30
Figure 30. Mesh of an analytical curved surface
31
Figure 31. Klein Bottle
32
Figure 32. Merging of meshed surfaces
33
Figure 33. Intersection of an avion and a space
shuttle
Avion 2891 elements Columbia 7087
elements Intersection chains 590 segments
34
Figure 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
35
Figure 35.Intersection of two Bunny hares
36
Figure 36. Tree modeled by quadrilateral elements
37
Figure 37. Rendered image of the tree model
38
Figure 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
39
Figure 39. Magnified views of the Hands model
40
Figure 40. Tetrahedral mesh of an airplane
41
Figure 41. Tetrahedral mesh of an elephant
42
Figure 42. Cross-section of the elephant model
43
Figure 43. Packing spheres of variable size
44
Figure 44. Packing spheres over a space curve
45
Figure 45. Tetrahedral mesh over a space curve
46
Figure 46. Tetrahedral elements along a space
curve
47
Transitional quadrilateral and hexahedral elements
48
Adaptive Mesh Coarsening of Hexahedral Meshes
49
Anisotropic mesh adaptation based on a functional
50
(No Transcript)
51
Parallel Delaunay Refinement
52
Meshing by Automatic intersection of solid 3D
elements
53
Curved boundary layer meshing for viscous flow
54
Advancing Front Technique for filling space with
arbitrary objects
55
Hybrid mesh generation for reservoir flow
simulation
56
Modeling of Nanostructured Materials
57
Size Gradation Control of Anisotropic Meshes
58
Generation of 3D elements by Mapping
59
Parametric Surface Meshing
60

• Finite Elements in Analysis and Design
• Volume 46, Issues 1-2, Pages 1-228
  (January-February 2010)
• Mesh Generation - Applications and Adaptation
• Edited by S.H. Lo and H Borouchaki

1. Adaptive meshing and analysis using
transitional quadrilateral and hexahedral
elements, 2-16, S.H. Lo, D. Wu, K.Y. Sze 2.
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 Si 4. Hybrid mesh
smoothing based on Riemannian metric
non-conformity minimization, 47-60, Yannick
Sirois, Julien Dompierre, Marie-Gabrielle Vallet,
François Guibault 5. An anisotropic mesh
adaptation method for the finite element solution
of variational problems, 61-73, Weizhang Huang,
Xianping Li 6. Boundary recovery for Delaunay
tetrahedral meshes using local topological
transformations, 74-83, Hamid Ghadyani, John
Sullivan, Ziji Wu
61
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. Chrisochoides 9. Adaptive mesh
generation procedures for thin-walled tubular
structures, 114-131, C.K. Lee, S.P. Chiew, S.T.
Lie, T.B.N. Nguyen 10. Curved boundary layer
meshing for adaptive viscous flow simulations,
132-139, O. Sahni, X.J. Luo, K.E. Jansen, M.S.
Shephard 11. Advancing front techniques for
filling space with arbitrary separated objects,
140-151Rainald Löhner, Eugenio Oñate 12. Hybrid
mesh generation for reservoir flow simulation
Extension to highly deformed corner point
geometry grids, 152-164, T. Mouton, H.
Borouchaki, C. Bennis 13. Numerical modeling of
nanostructured materials, 165-180, Azeddine
Benabbou, Houman Borouchaki, Patrick Laug, Jian
Lu 14. Size gradation control of anisotropic
meshes, 181-202, F. Alauzet 15. Sweeping of
unstructured meshes over generalized extruded
volumes, 203-215, Daniel Rypl 16. Some aspects of
parametric surface meshing, 216-226, Patrick Laug
62
Thank you !