Grids generation methods and adaptive meshes

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Grids generation methods and adaptive meshes
Pawel Cybulka Finite element method
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Plan presentation

• Grid generation
• mesh types,
• grids generation methods.
• Adaptive finite element method
• phadaptivity,
• error estymator,
• hierarchical grids.

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Mesh types

• Mesh types are varied as the numerical
  methodologies they support, and can be classified
  according to
• conformality
• surface or body alignment
• topology
• element type.

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Conformality

• Conformal meshes are characterized by a perfect
  match of edges and faces between neighbouring
  elements.
• Non-conforming meshes exhibit edges and faces
  that do not match perfectly between neighbouring
  elements, giving rise to so-called hanging nodes
  or overlapped zones.
• Figure 1. a) conforming mesh, b) non-conforming
  mesh.

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Surface or body alignment

• Surface or body alignment is achieved in those
  meshes whose boundary faces match the surface of
  the domain to be gridded perfectly. If faces are
  crossed by the surface, the mesh is denote as
  being non-aligned.
• Figure 2. a)surface aligned, b)non-surface aligned

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Mesh topology

• Mesh topology denote the structure or order of
  the elements. There are three possibities
• Micro-structured, each points has the same number
  of neighbours .
• Micro-unstrutured, each point can have arbitrary
  number of neighboures
• Macro-unstrutured, micro-structured, where the
  mesh is assembled from groups of micro-structured
  subgrids

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Element type

• Typical element types for 2D domains are
  triangles and quads, and tetrahedra, prisms and
  brick for 3D domains.
• Figure 3. Element types

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Description of the domain to be gridded

• There are two possible ways of describing the
  surface of a computational domain.
• Using analytical functions. This is the preferred
  choice if a CAD-CAM database exists for the
  description of the domain. Typical data types
  splines, B-splines, non-unifom rational B-splines
  (NURBS) surfaces. Important characteristic of
  this approach is that the surface is continuous,
  there are no holes in the information.
• Via discrete data. When we get a cloud of points
  or an already existing surface triangulation
  describes the surface of the computational
  domain. Examples are remote sensing data, medical
  imaging data, data sets from computer games.

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Typical grid generation methods

• Structured mesh
• simple mappings
• multiblock.
• Unstructured mesh
• quadtree(2D) and octree(3D)
• the advancing front technique (AFT)
• Delaunay triangulation.

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Simple mappings

• The computational domain can be mapped into the
  unit square or cube. The distribution of points
  and elements in space is controlled either by an
  algebraic function, or by solution of a partial
  differential equation in the transformed space.
• Figure 4. Structured meshes built on the base of
  various coordinate system a) Cartesian
  coordinate system, b) cylindrical system, c)
  combination of various coordinate system

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Multiblock grid

• Multiblock grid is based on division of
  difficult to discretization area into several
  areas which are simpler to discretization and,
  then proper connection of these areas. There are
  some variations of this strategy including
  overset method, patched multiblock, composite
  multiblock. These methods differ depending on the
  way of connection of the subareas into the whole.
• Figure 5. Grids generated by various multiblock
  methods a) overset mesh combination b) composite
  mesh combination.

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Quadtree and octree mesh methods

• Quadtree and octree are a simple method where all
  domain is mapping by quads(2D) or bricks(3D). In
  the next step all quads containing the boundary
  points are divided into four parts whereas bricks
  are divided into eight parts. This process is
  repeated by the moment when all boundary points
  are closed in the least quads or bricks. The size
  of the least quads is given by a user. In the
  last step all quads are transformed into
  triangles.
• Figure 6. Scheme of grid generation by quadtree
  method, a)quadtree grid after thicken, b) quatree
  grid after division of quads into triangles.

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The advancing front technique (AFT)

• The principle of this method is based on the
  so-called front created with the points located
  on discretized boundary of domain. Properly
  connected points form sides so that continuous
  area boundary is replaced by a set of sides ( the
  line segments in the case of 2D and triangles in
  the case of 3D) creating a closed loop. Then,
  elements are built in accordance with the
  established direction in loop on the basis of
  existing set of sides and possibly added points.
  How will create another element (figure 7)
  depends on a angle between two following sides
  from the front.

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The advancing front technique (AFT)

• How to combine elements depending on a angle
• a lt 90 a new element is built (created from
  existing points),
• 90 lt a lt 120 a new point is added and two
  elements are created,
• 120 lt a a new point is added and one element is
  created.
• Figura 7. The principle of conduct during the
  construction of the grid by AFM

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Delaunay triangulation

• Delaunay algorithm for triangulation starts by
  forming the super triangle enclosing all the
  points from set V that has to be triangulated.
  Then, incrementally, a process of inserting the
  points p into the set V is performed. After every
  insertion step a search is made to find the
  triangles whose circumcircles enclose p.
  Identified triangles are then deleted from the
  set. As a result, an insertion polygon containing
  p is created. Edges between the vertices of the
  insertion polygon and p are inserted and form the
  new triangulation.
• Figure 8. The Delaunay triangulation technique

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Convergence of FEM

• Using the FEM computation approximate results are
  received. The accuracy of the approximation can
  be computed using the formula
• u ua lt Chp u
• u accurate solution,
• ua FEM solution ( approximate),
• h the size of elements,
• p the degree of approximation.
• Therefore the accuracy of the FEM solution
  depends on the
• size of elements,
• the degree of the function approximation.

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Influence of the number of elements and the
degree of approximation of functions on the
accuracy of the FEM calculations
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Adaptive finite element method

• The idea behind AFEM is to make local
  hp-adaptivity based on local error analysis.
• The aim is to obtain sufficient accuracy of the
  result at the smallest computation cost.

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H - adaptivity

• H - adaptation is the process of changing the
  concentration of elements in the area calculation
  in order to change the accuracy of the
  computation carried out there.
• Typically, h-adaptation is associated with
  thickening of areas of high variability of the
  analyzed qualities by what the calculation error
  is minimized in this area.
• Figure 9. H-adaptation elements on the grain
  boundaries.

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P - adaptivity

• The accuracy of the results obtained in the
  domain increases or decreases by increasing or
  decreasing the degree of approximation of
  function of shape in the elements.
• In the case of p-adaptivity we need to draw
  attention to the proper connection of elements
  with higher degree of approximation with
  neighbouring elements with a lower degree of
  approximation. Provided the correct computation
  is the continuity of approximation. Therefore,
  approximation of the function corresponding to
  the side of element adjacent to the element with
  a higher degree of approximation should be raised
  to the same degree.

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Error estimation

• Error estimation is a way to evaluate the error
  occurs in a given computation domain.
• Error estimation includes a criterion defining
  the degree of adaptivity that must be used in
  order to obtain the assumed accuracy of
  computation.
• The criterion of adaptation Ei may be defined as
  the second derivative normalized after medium
  gradient test variable value.
• Ui - test variable value
• cn depends on chosen algorithm for the
  solution of the physical problem

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Error estimation

• One of the simpler and more frequently used error
  estimations is estimation as shown in Figure 10.
  In the first step of the algorithm gradients
  value in each element is computed. Then we
  compare the values of the adjacent elements. If
  the difference between neighbouring elements
  exceeds the determined threshold the elements are
  divided.
• Figure 10. Schematics of a simple error estimator
  

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Hierarchical grids

• Hierarchical grids were formed for adaptive
  finite element methods. Their structure
  corresponds to all needs associated with
  hp-adaptivity.
• The algorithm of hierarchical grids reminds
  quadtree (2D) and octree (3D) methods.
• The starting point for hierarchical grids are
  grids created by the grid generator. The elements
  of this grid are called parents. Each of the
  parent can be divided into proper number of
  identical in terms of the shape children. Each
  child can be a parent. Thus we have possibility
  of any compacting the grid.
• Each element in the hierarchical grid knows its
  parent and its children by what the grid can be
  easily thicken and thin.

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Hierarchical grid

• Figure 11. The schema of the hierarchical
  structure of an adaptive numerical grid

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Summary

• The grids are a very important element in the
  computation using finite element method. They
  determine the accuracy of the computations
  carried out and the time of their performance.
• Adaptive finite element method based on
  hierarchical meshes and local error estimation
  enables to carry out of an approximation only in
  these areas where it is required. Thanks to it
  accurate results are obtainedat the smallest
  cost computation.

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LITERATURE

• Löhner R., 2008, Applied Computational Fluid
  Dynamics Techniques An Introduction Based on
  Finite Element Methods, Second Edition, John
  Wiley Sons, Ltd, Chichester.
• Joe F. Thompson, Bharat K. Soni, Nigel P.
  Weatherill, Handbook of Grid Generation, CRC
  Press, 1998.
• Banas K. Metoda Elementów Skonczonych, Seminarium
  BJT CM UJ, 2006.

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Thanks for your attention