Mesh generation in CFD
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The document discusses different types of meshes used in computational fluid dynamics (CFD) simulations. It explains that meshes are used to discretize spatial domains and store field variable values. Structured meshes include Cartesian, multi-block, and patched block grids, while unstructured meshes include Delaunay triangulations and advancing front methods. Various algorithms for generating meshes are also presented, such as inserting points sequentially or using grid-based, centroid-based, or advancing front approaches. Both benefits and challenges of different meshing methods are summarized.
![©ZeusNumerix
Nearest neighbour Property
In Delaunay triangulation an edge is obtained by joining the
nearest neighbour
But the nearest neighbour edge is one of the many edges
Properties of Delaunay Triangulation
Nearest neighbour
edge
Minimal roughness Property
Suppose a property fi is given at all vertices Vi , where i = 1 to N
Any triangulation will produce piecewise linear interpolation surface for this data
Delaunay triangulation minimises the integral given by
[(f/x)2 + (f/y)2 ] dx dy for any f
Non-nearest neighbour
edge](https://image.slidesharecdn.com/meshgenerationincfd-150112231350-conversion-gate01/85/Mesh-generation-in-CFD-15-320.jpg)
Mesh generation in CFD
- 1. ©ZeusNumerix Defense | Nuclear Power | Aerospace | Infrastructure | Industry Abhishek Jain abhishek@zeusnumerix.com Mesh Generation in CFD
- 2. ©ZeusNumerix Solving problems using CFD in 6 steps 3 2 Build Computational Domain Create suitable Mesh Boundary Conditions & Initial conditions Solution of discrete equationsPlot flow FieldInterpret solution
- 3. ©ZeusNumerix • The grid • Stores discrete values for the field variables • Helps in the evaluation of partial derivatives (structured meshes) • Helps in calculation of numerical flux / interpolations using basis functions (unstructured meshes) • It affects • Correctness of physics • Accuracy of solution • Overall efficiency of simulation such as CPU time • In fact, in terms of wall clock time and human efforts mesh generation is half simulation done! Importance of mesh
- 4. ©ZeusNumerix What meshes are required? Sr. No. Method Type of Meshes 1 Geometric Optics (GO) / GTD CAD surfaces defined as NURBS, surface quadrilaterals or triangles 2 Physical theory of Diffraction PO/PTD Surface quadrilaterals or triangles 3 Method of Moments Surface quadrilaterals or triangles 4 Finite Element Method Quadrilaterals or triangles in 2D and Hexahedrons or Tetrahedrons in 3D 5 Finite Difference Method No meshes are required in the basic method. Octree meshes in advanced FDTD 6 Finite Volume method Quadrilaterals or triangles in 2D and Hexahedrons or Tetrahedrons in 3D
- 5. ©ZeusNumerix Classification of Meshes Cartesian FDTD methods Multi-block structured for FVDT method Unstructured for MOM & FEM methods Hybrid meshes x = y = cons
- 7. ©ZeusNumerix Creating Structured Meshes Topological requirements in 2D Four vertices, four edges Opposite sides must have similar intervals / mesh points x y Topological requirements Eight vertices, twelve edges, six faces Opposite edges must have the same no of mesh points
- 8. ©ZeusNumerix Multi-block Structured Grid Patched Grids Domain is covered with sub-domain each with rectangular topology Grids in each blocks are created independently Grids are smoothened inside the block and also across the block Block boundary Each block has 4 edges Grid lines have C1 continuity
- 9. ©ZeusNumerix Patched Blocks with C1 Continuity Block No1 and Block No. 2 share, Block boundary completely, But grid lines do not have even C0 continuity Block 1 Block 1 Block 1 Block 1 Block 1 Block 1 Block No1 and Block No. 2 share, Block boundary completely, Grid lines have C0 continuity Block No1 and Block No. 2 share, Block boundary completely, Grid lines have C1 continuity C1 continuity across block boundaries gives best numerical results
- 10. ©ZeusNumerix Multi-block Structured Meshes – an example Identify the vertices, edges, faces and blocks in this figure
- 11. ©ZeusNumerix Unstructured Meshes : Delaunay and advancing front triangulations
- 12. ©ZeusNumerix Properties of Delaunay Triangulation Uniqueness : The Delaunay triangulation is unique, assuming that no four sites are circular. The circumcircle criteria : A triangulation for N > 2 site is Delaunay if and only if the circumcircle of every interior triangle is point-free A D C B Incircle test fails on the left. Swapping edges, as on right, diagonal can be used to produce Delaunay triangle A D C B Interior is not empty
- 13. ©ZeusNumerix Delaunay Triangulation All triangles obey empty-circle (sphere) property Two triangles are highlighted Properties of Delaunay Triangulation
- 14. ©ZeusNumerix Circle shown by dotted lines does not satisfy empty circle (sphere) property Shaded triangle is not Delaunay Properties of Delaunay Triangulation
- 15. ©ZeusNumerix Nearest neighbour Property In Delaunay triangulation an edge is obtained by joining the nearest neighbour But the nearest neighbour edge is one of the many edges Properties of Delaunay Triangulation Nearest neighbour edge Minimal roughness Property Suppose a property fi is given at all vertices Vi , where i = 1 to N Any triangulation will produce piecewise linear interpolation surface for this data Delaunay triangulation minimises the integral given by [(f/x)2 + (f/y)2 ] dx dy for any f Non-nearest neighbour edge
- 16. ©ZeusNumerix Incremental Insertion Algorithms - Common Features (popular) New points are added to existing triangulation one by one New points are assumed to lie in the existing triangulation All incremental insertion algorithms start with a phantom triangle large enough to enclose all the given points The edges formed by these points, if given as a part of input data is initially ignored. These edges will be built later after all the points are inserted. Every insertion of new point requires locating (i) a triangle or (ii) circumcircles containing this point. In mesh adaptation the location is known. In initial triangulation ‘search is required Phantom triangle Points to be inserted Edge ignored
- 17. ©ZeusNumerix X Given a Delaunay Triangulation of N nodes and having inserted n nodes How do I insert the next given n+1 th node? Lawson Algorithm (not the global edge swapping!) Locate triangle containing X Subdivide triangle Recursively check adjoining triangles to ensure empty-circle property. Swap diagonal if needed Point Insertion Algorithm in Delaunay
- 18. ©ZeusNumerix Criteria • Insert node in the largest triangle • Insert node on the largest edge • Insert points at random • Insert points arranged as a lattice The boundary points of domain are rarely adequate for unstructured meshes. How do insert additional nodes in the domain. Point Insertion Algorithm in Delaunay X
- 19. ©ZeusNumerix Begin with Bounding Triangles (or Tetrahedra) Delaunay Triangulation in action Initial Triangulation
- 20. ©ZeusNumerix Insert boundary nodes using Delaunay method (Lawson or Bowyer- Watson) Initial Triangulation Delaunay Triangulation in action
- 21. ©ZeusNumerix Insert boundary nodes using Delaunay method (Lawson or Bowyer- Watson) Initial Triangulation Delaunay Triangulation in action
- 22. ©ZeusNumerix Insert boundary nodes using Delaunay method (Lawson or Bowyer- Watson) Initial Triangulation Delaunay Triangulation in action
- 23. ©ZeusNumerix Insert boundary nodes using Delaunay method (Lawson or Bowyer- Watson) Initial Triangulation Delaunay Triangulation in action
- 24. ©ZeusNumerix Insert boundary nodes using Delaunay method (Lawson or Bowyer- Watson) Initial Triangulation Delaunay Triangulation in action
- 25. ©ZeusNumerix Recover boundary Delete outside triangles Insert internal nodes Recovering Triangulation Delaunay Triangulation in action
- 26. ©ZeusNumerix Grid Based Node Insertion Nodes introduced based on a regular lattice Lattice could be rectangular, triangular, quadtree, etc… Outside nodes ignored h Delaunay Triangulation in action
- 27. ©ZeusNumerix Nodes introduced based on a regular lattice Lattice could be rectangular, triangular, quadtree, etc… Outside nodes ignored Grid Based Node Insertion Delaunay Triangulation in action
- 28. ©ZeusNumerix Centroid Based Node Insertion Nodes introduced at triangle centroids Continues until edge length, hl Delaunay Triangulation in action
- 29. ©ZeusNumerix Nodes introduced at triangle centroids Continues until edge length, hl l Centroid Based Node Insertion Delaunay Triangulation in action
- 30. ©ZeusNumerix Circumcenter (“Guaranteed Quality”) Node Insertion e.g , a strategy: Nodes introduced at triangle circumcenters Order of insertion based on minimum angle of any triangle Continues until minimum angle > predefined minimum )30( Delaunay Triangulation in action
- 31. ©ZeusNumerix Nodes introduced at triangle circumcenters Order of insertion based on minimum angle of any triangle Continues until minimum angle > predefined minimum )30( Circumcenter (“Guaranteed Quality”) Node Insertion Delaunay Triangulation in action
- 32. ©ZeusNumerix “Front” structure maintained throughout Nodes introduced at ideal location from current front edge A B C Advancing Front Node Insertion Delaunay Triangulation in action
- 33. ©ZeusNumerix “Front” structure maintained throughout Nodes introduced at ideal location from current front edge Advancing Front Node Insertion Delaunay Triangulation in action
- 34. ©ZeusNumerix Nodes introduced at midpoint of segment connecting the circumcircle centers of two adjacent triangles Voronoi-Segment Node Insertion Delaunay Triangulation in action
- 35. ©ZeusNumerix Nodes introduced at midpoint of segment connecting the circumcircle centers of two adjacent triangles Voronoi-Segment Node Insertion Delaunay Triangulation in action
- 36. ©ZeusNumerix Nodes introduced at along existing edges at l = h Check to ensure nodes on nearby edges are not too close h h h Edges Delaunay Triangulation in action
- 37. ©ZeusNumerix Nodes introduced at along existing edges at l = h Check to ensure nodes on nearby edges are not too close Edges Delaunay Triangulation in action
- 38. ©ZeusNumerix Boundary Constrained Boundary Intersection Nodes and edges introduced where Delaunay edges intersect boundary Delaunay Triangulation in action
- 39. ©ZeusNumerix Boundary Intersection Nodes and edges introduced where Delaunay edges intersect boundary Boundary Constrained Delaunay Triangulation in action
- 40. ©ZeusNumerix Boundary Constrained Local Swapping Edges swapped between adjacent pairs of triangles until boundary is maintained Delaunay Triangulation in action
- 41. ©ZeusNumerix Local Swapping Edges swapped between adjacent pairs of triangles until boundary is maintained Boundary Constrained Delaunay Triangulation in action
- 42. ©ZeusNumerix Local Swapping Edges swapped between adjacent pairs of triangles until boundary is maintained Boundary Constrained Delaunay Triangulation in action
- 43. ©ZeusNumerix Boundary Constrained Local Swapping Edges swapped between adjacent pairs of triangles until boundary is maintained Delaunay Triangulation in action
- 44. ©ZeusNumerix Boundary Constrained Local Swapping Edges swapped between adjacent pairs of triangles until boundary is maintained Delaunay Triangulation in action
- 45. ©ZeusNumerix Algorithm (in 2D) • Boundary is discretised based upon the points per wavelength (ppw) criteria • Initialize front as a set of line segments defining the boundary completely • The front is advanced in to the domain producing triangles as it advances. • Front advances in a variety of ways depending on the angle between two consecutive edges of the front: • Three possibilities arise : as discussed in the next slide Advancing front Algorithm A B Advancing front Domain : /ppw
- 46. ©ZeusNumerix Case Deleting vertices Deleting edges Adding vertices Adding edges Included angle Case A 1 2 Nil 1 < 30 Case B 1 2 1 2 30 < < 120 Case C Nil 1 1 2 > 120 New front Trial point Advancing front Algorithm • Determine whether element formed with the trial point crosses any edge. • If yes, select a new trial point and try again. • Add the new point, edges and triangles to the respective lists. Delete base edge from the front and add new edge(s) to the front till front is empty Case A Case B Case C
- 47. ©ZeusNumerix Advancing Front A B C Step 1: Create a list of vertices, edges and triangles Consider boundary as the initial front. Advance the front by creating a triangle in the domain with an edge (say AB). Locate vertex C (trial point) for the purpose. If point C is acceptable form a triangle in the domain Trial point initial front
- 48. ©ZeusNumerix A’ B’ C’ r Step 2: Delete one edge, add one vertex, add two edges and add one triangle from / to the respective lists Choose a new advancing front (A’ B’). Determine if trial point C’ is within radius r (= /ppw) , if not, accept D. D Advancing Front (contd.) Edge deleted Triangle added Vertex added
- 49. ©ZeusNumerix Step 3 : Delete two edges, add one edge and add one triangle to the respective lists Note that “Book-Keeping” of the vertices, edges and triangles is important D Advancing Front (contd.) Triangle 1 added Triangle 2 added Edge deleted Edge deleted
- 50. ©ZeusNumerix Step 4 : Delete two edges, add one edge and add one triangle to the respective lists Advancing Front (contd.)
- 51. ©ZeusNumerix Advancing Front (contd.) Step 5 : Delete one edge, add two edges and add one triangle to the respective lists Note that so far only one vertex has been added to the list of vertices
- 52. ©ZeusNumerix Advancing Front (contd.) Step 6 : Delete four edges, delete four vertices and add two triangles Where multiple choices are available, use best quality (closest shape to equilateral) triangle Reject any triangle that intersects existing front
- 53. ©ZeusNumerix A B C Step 7: To recover , reject any inverted triangles AB X AC < 0) r Advancing Front (contd.)
- 54. ©ZeusNumerix Advantages: Method can be used for surfaces and volume though a lot of modification is required Anisotropic grids can be produced Disadvantages Grid quality is not high Advancing fronts collide to produce poor grids Front Advancing in the process and triangulation completed Advancing Front Algorithm
- 56. ©ZeusNumerix Hybrid Methods Structured mesh + tet meshes Image courtesy of acelab, University of Texas, Austin, http://acelab.ae.utexas.edu Image courtesy of Roy P. Koomullil, Engineering Research Center, Mississippi State University, http://www.erc.msstate.edu/~roy/ When two or more meshes / methods are used
- 57. ©ZeusNumerix Hybrid Mesh Through Advancing Front Discretize Boundary
- 58. ©ZeusNumerix Hybrid Mesh Through Advancing Front Concave Corner Convex Corner
- 59. ©ZeusNumerix Closure • Mesh generation consists of • 3D Mesh Generation : Dividing volume where electromagnetic field is required to be calculated in to tetrahedrons or hexahedrons • 2D/ Surface Mesh Generation : Diving surfaces which limits the electromagnetic field or material boundaries into triangles or quadrilaterals • Mesh generation affect the accuracy of solution • Mesh Generation requires domain experience • Mesh generation affects the duration and cost of simulation • Zeus Numerix uses it proprietary software GridZ™ for the purpose.
- 60. ©ZeusNumerix Thank You! 3 November 2014 60
Editor's Notes
- #3: Iggest headache: interpretation Common mistake: e.g presuure differene But user may use absolute values. Say 10 values of pr at inlet, 10 values of pr at outlet. In fact equations govern only the gradient. So absolute value may be floating ( subtract pr) Another mistake Habit of listing the variables and their beautiful plots In fact ranges of values given to visualizer will show gradients where values may be ( in absolute terms) very much same ( contour plot of density: interpret density) ( incompressible flow should not show any variation, let alone beautiful plot )
- #4: If mesh is too coarse : behavior of field as if dispersion occurs( numerical dispersion) when actually in physical world there is no dispersion some mesh choice may distort the physics e.g traingles Vs rectangles in case of stiffness of thin walled structure
- #5: GTD : Geometric Theory of Diffraction Because the size of objects as compared to the wavelengths is such (so small) that the behaviour of electromagnetic waves is like optical rays i.e only reflection PTD : FDTD : Finite Difference Time Domain
- #10: C0 : curves meet at boundary C1 : Curves have same slopes at boundary
- #39: We have to tolerate non-delaunay triangles to accommodate geometry of boundary else we may end up distorting the geometry too much Getting this right takes months of effort sometimes ICEM : software to only generate meshes . Zeus also has developed a competetive alternative.
- #40: Unsymmetric mesh for symmetric object! That is life. However if mesh is adequately designed, i.e grid-indpendent study is done, the results can be acceptably accurate.