1. Grids generation methods and adaptive meshes Paweł Cybułka Finite element method

2. Plan presentation 1.Grid generation –mesh types, –grids generation methods. 2.Adaptive finite element method –ph–adaptivity, –error estymator, –hierarchical grids.

3. 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.

4. 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.

5. 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

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

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

8. 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.

9. Typical grid generation methods Structured mesh: simple mappings; multiblock. Unstructured mesh: quadtree(2D) and octree(3D); the advancing front technique (AFT); Delaunay triangulation.

10. 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

11. 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.

12. 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.

13. 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 angle between two following sides from the front.

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

15. 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

16. Convergence of FEM Using the FEM computation approximate results are received. The accuracy of the approximation can be computed using the formula: ||| u – u a ||| < Ch p || u || u – accurate solution, u a – 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.

17. Influence of the number of elements and the degree of approximation of functions on the accuracy of the FEM calculations

18. 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.

19. 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.

20. 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.

21. 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 E i may be defined as the second derivative normalized after medium gradient test variable value. U i - test variable value c n – depends on chosen algorithm for the solution of the physical problem

22. 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

23. 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.

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

25. 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 obtained at the smallest cost computation.

26. 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. Banaś K. Metoda Elementów Skończonych, Seminarium BJT CM UJ, 2006.

27. Thanks for your attention