1. Edge and Face Meshing

2. Meshing - General • To reduce overall mesh size, confine smaller cells to areas where they are needed • Locations of large flow field gradients. • Locations of geometric details you wish to resolve. • Controlling cell size distribution • Edges, faces, and volumes can be meshed directly. • A uniform mesh is generated unless pre-meshing or size functions are used. • Pre-meshing • Edge meshes can be graded (varying interval size on edge) • A graded edge mesh can be used to control the cell size distribution of a face mesh. • Controlling distribution of cell size on face mesh also controls the cell size distribution of the volume mesh. • Size functions and boundary layers • Allow direct control of cell size distribution on edges, faces and volumes directly for automatic meshing.

3. Edge Meshing • Edge mesh distribution is controlled through the spacing and grading parameters on the Mesh Edges form. • Picking • Temporary graphics • Links, Directions • Grading/Spacing • Special characteristics • Apply and Defaults • Invert and Reverse • Options

4. Edge Meshing • Sense • Sense is used to show direction of grading • Every picked edge will show its sense direction using an arrow • The sense can be reversed by a shift+middle-click on the last edge picked (this is in addition to the “next” functionality) or by clicking the Reverse button • Edge mesh preview • When you pick an edge, the edge mesh is displayed using white nodes. • This is a temporary mesh that has not been applied to the edge. • Displayed edge mesh is based on current grading and spacing parameters • If you modify the grading or spacing, the temporary mesh will be updated immediately. • Meshing the edge • The edge mesh is generated by clicking the Apply button. • The nodes will then be displayed in blue.

5. Single-sided grading Symmetric grading Asymmetric grading Grading • Controls mesh density distribution along an edge. • Grading can produce single-sided or double-sided mesh • Doubled-sided mesh can be symmetric or asymmetric. • Symmetric schemes produce symmetric mesh about edge center. • Asymmetric schemes can produce asymmetric mesh about edge center. • Single-sided grading: • Uses a multiplicative constant, R, to describe the ratio of the length of two adjacent mesh elements: • R can be a user-specified value (Successive Ratio) or calculated by GAMBIT. • GAMBIT also uses edge length and spacing information to determine R.

6. Double Sided Grading • Double-sided grading can be generated by activating the double sided option in the Mesh Edge form. • Asymmetric grading is possible when the double-sided option is used with: • Successive Ratio, First Length, Last Length, First-Last Ratio, and Last-First Ratio • The mesh is symmetric if R1 = R2 • The mesh is asymmetric if R1 ≠ R2. • Edge center is determined automatically. • Some schemes implicitly generate double sided grading that is symmetric.

7. Soft Links • Picking and soft links • Pick with links • By enabling this option, soft-linked edges can be selected in a single pick • Linked edges share the same information and can be picked in a single pick • Modifying soft links • At any time, you can • Form links • Break links • Maintain links • By default, GAMBIT will form links between unmeshed edges that are picked together • By default, GAMBIT will maintain links between meshed edges that are picked together

8. Spacing • In all meshing forms, the following spacing functions can be specified: • Interval Count (recommended for edge mesh only) • Example – Entering a value of 5 will create 5 intervals along the selected edge(s) (6 nodes, including end nodes) • Interval Size (default setting) • Requires input of distance between nodes. • Edge is meshed with “average” interval size if grading is used. • Example: An edge length of 10 and a value of 2 creates 5 intervals on the edge • Shortest Edge % • Meshes the selected edge according to a percentage of the length of the shortest edge in the model. • Example – Shortest edge in model has length of 1. Entering a value of 20 will create a mesh with interval size 0.2.

9. First Edge Settings • Use First Edge Settings option • If enabled: • First edge selected in pick list updates all entries in the form. • This mode is useful to copy settings from one meshed edge to other edge(s). • If disabled: • Use this setting any time you pick two or more meshed edges where there is a difference in type or spacing. • The local Apply button for that option will be turned off • This allows you to maintain pre-existing grading and/or spacing settings for each edge. • Enforce a change in grading and/or spacing by enabling Apply button.

10. Meshing Options • Mesh • This option is useful in cases where you want to impose a scheme without prescribing the number of intervals • The higher level meshing scheme will decide (and match) the intervals • Remove old mesh • Deletes old mesh • When selected, option to also delete lower geometry mesh appears. • Ignore size function • Toggle to either obey or ignore size functions • Size function takes precedence when this option is disabled.

11. Meshing Options – Example 1 Specify interval size, no grading, apply without meshing Face Mesh Generated Using Quad Pave Scheme (Pave face meshing schemes require an even number of elements on edge meshes) 3 Generate face mesh. Specify grading only, apply without meshing 2 Face Mesh Generated Using Submap Scheme

12. Face Meshing • Mesh Faces form • Upon picking a face: • GAMBIT automatically chooses quad elements • GAMBIT chooses the type based on the solver/face vertex types • Available element/scheme type combinations • Quadrilaterial: Map, Submap, Tri-Primitive, Pave • Triangular: Pave • Quad/Tri (hybrid): Map, Pave, Wedge • Quad-to-tri conversion utility.

13. Face Meshing - Quad Examples • Quad: Map • Quad: Submap • Quad: Tri-Primitive • Quad: Pave

14. Face Meshing - Quad/Tri and Tri Examples Quad/Tri Map Quad/Tri Wedge Face must be split to generate more than one cell across Tri Pave Quad/Tri Pave Triangular cell Quad cells Triangular cell

15. Deleting Old Mesh • Existing mesh must be removed before remeshing. • Mesh can be deleted using delete mesh form. • Lower topology mesh can also be deleted (default) • Alternatively, existing mesh can be deleted by selecting the Remove Old Mesh option • Remove old mesh alone will leave all lower topology mesh • Remove old mesh + remove lower mesh will delete all lower topology mesh that is not shared with another entity • Undo after any meshing operation also works.

16. E E E R S E E Face Vertex Types • All vertices that are connected to a face are assigned initial face vertex types based on the angle between the edges connected to the vertex. • Vertices shared by multiple faces can have multiple types, depending on which face you are considering. • The combination of vertex types describes the topology of the face. • Face vertex types are used automatically to determine all quad face meshing schemes (except quad pave and tri pave).

17. E E E S S S C C C R Face Vertex Types End (E) • zero internal grid lines Side (S) • One internal grid line Corner (C) • Two internal grid lines Reverse (R) • Three internal grid lines R

18. Modifying Face Vertex Types • Face vertex types can be changed from their default settings: • Automatically • By enforcing certain meshing schemes in face and volume meshing. • Can sometimes result in undesirable mesh. • Manually • By direct modification in the Face Vertex Type form. • Select Face • symbols appear in graphics window • Select New Vertex Type • Select Vertices to be affected • Vertex Types can be applied to just Boundary Layers as option. • A vertex can have multiple types; one for each associated face. • For a given set of face vertex types, GAMBIT will choose which meshing scheme to use based on predefined formulae. E S

19. E E S R E E E E E E E R E E E E E E R E E S E Formula for Submap Scheme • A face can be made submappable by manually changing vertex types • Consider which vertex should be changed to type S (side) • In the Set Face Vertex Type form, change vertex type to S by enforcing the submap scheme. • In the Face Mesh form, change the scheme from default to submap and click Apply. • GAMBIT will attempt to change the vertex types so that the scheme is honored. • User has less control – the resulting mesh may be undesirable! Which vertex to change? E

20. S E + E E E E E E E Formula for Map Scheme Map Periodic Map Project intervals can be specified for more control.

21. E E C C E E E E S S S S E E E E E E E E How to Make a Face Mappable • Enforce the Map scheme (most common method) • In the Face Mesh form, change the scheme from default to “Map” and click Apply. • GAMBIT will attempt to change the vertex types so that the chosen scheme is honored • Manually change the vertex types • In Set Face Vertex Type form, change vertices (default) to "Side“. • Open the Face Mesh form and pick the face. • GAMBIT should automatically select the map scheme) E S E E

22. S E E E E C E C C S E C C E E E C C C C C E C E Formula for Submap Schemes Submap: (additional terms when interior loops exist) Periodic Submap where m > 2. (additional terms when interior loops exist) E E C C E E E E S S

23. E S E E E S E E E E E E Formula for Tri-Primitive Sheme Tri-Primitive • To mesh a rectangular face with the tri-primitive scheme: • Manually change one of the vertex types to "Side" in this example • The Tri Primitive scheme can not be enforced

24. T T T E C N E E T Meshing Faces with Hybrid Quad/Tri Schemes • Quad/Tri: Tri-Map • The face vertex types must be changed manually to Trielement (T) • The Tri-Map scheme must be selected. • Quad/Tri: Pave • All vertex types are ignored except Trielement (T) and Notrielement (N) • Trielement (T) will force a triangular element. • Notrielement (N) will avoid a triangular element. • Quad/Tri: Wedge • Used for creating cylindrical/polar type meshes • The Vertex marked (T) is where rectangular elements are collapsed into triangles

25. Hard Links • Mesh linked entities have identical mesh • Created for periodic boundary conditions • Applicable to edge, face, and volume entities • Best to use soft links for edge meshing • To link volume meshes, all faces must be hard linked first. • Hard links for faces • Select faces and reference vertices • The sense of each edge appears. • Reverse orientation on by default • Periodic option should be used for periodic boundary conditions, which creates a matched mesh even if the edges are split differently. • Meshing one of the faces either before or after hard linking will generate an identical mesh on the linked face. • Multiple pairs of hard links can be created.

26. Mesh Smoothing • Smoothing can increase mesh quality beyond that of the default meshing algorithms • Most noticable in complicated geometry. • May have little or no effect in simple geometries. • Mesh smoothing algorithms adjust interior node locations to obtain marginal improvement in mesh quality. • Boundary meshes are not altered. • The mesh at the boundary is not altered. • Face and volume meshes are smoothed using a default scheme. • Different schemes can be selected and applied after meshing. • Face mesh smoothing • Length-weighted Laplacian: Uses the average edge length of the elements surrounding each node to adjust the nodes. • Centroid Area: Adjust node locations to equalize areas of adjacent elements. • Winslow (quad meshes only): Optimizes element shapes with respect to perpendicularity. • Volume mesh smoothing • Length-weighted Laplacian: same as for face mesh smoothing • Equipotential: Adjusts node locations to equalize the volumes of the mesh elements surrounding each node.

27. Examining the Mesh • Display Type • Plane/Sphere • View mesh elements that fall in plane or sphere. • Range • View mesh elements within quality range. • Histogram shows quality distribution. • Show worst element zooms the view to the worst element • Select 2D/3D and element type • Select Quality Type • Display Mode – Change cell display attributes • Show Worst Element – Automatically zooms the display to the worst element (based on current settings). • Update button – Will update values reported in the panel when options are changed.

28. Assessing Mesh Quality • GAMBIT has several methods for assessing mesh quality. • Aspect Ratio • Diagonal Ratio • Edge Ratio • EquiAngle Skew • EquiSize Skew • MidAngle Skew • Size Change • Stretch • Taper • Volume • The most important of these quality metrics are EquiAngle Skew and Size Change.

29. Mesh Quality – EquiAngle Skew • The most important mesh quality metric is EquiAngle Skew (QEAS). • Range of EquiAngle Skew values • QEAS = 0 describes a perfectly orthogonal element • QEAS = 1 describes a degenerate element Quad/Hex Tri/Tet 0 (best) 1 (worst)

30. Mesh Quality – Size Change • Another important mesh quality metric is Size Change (QSC). • This metric applies only to 3D elements. • By definition, QSC > 0. • QSC = 0 describes an element whose neighbor elements have exactly the same volume as the element of interest (i.e. uniform mesh). 3D Example QSC,i = Vj=1/Vi since j=1 has largest volume ratio

31. Striving for Quality • A poor quality grid can cause inaccurate solutions and/or slow convergence. • Minimize EquiAngle Skew: • Hex, Tri, Quad: Skewness for all/most cells should be less than 0.85. • Tetrahedral: Skewness for all/most cells should be less than 0.9. • All elements: Size Change for cells in regions of interest should be less than 2 • Minimize local variations in cell size, such as large jumps in size between adjacent cells. • If Examine Mesh shows such violations: • Determine the reason(s) for the violations • Differences in spacing and grading on adjacent edges • Geometry with small features or other defects • Geometric complexity and size • Mesh that grows too rapidly • Delete mesh completely or partially. • Clean and/or decompose geometry, premesh edges and faces or adjust meshing parameters • Remesh the domain.

32. Appendix

33. Tri/Tet f is a scaling factor R and r are radii of circles (tri elements) or spheres (tet elements) that inscribe and circumscribe the mesh element. f = 1/2 for tri elements and f = 1/3 for tet elements. Quad/Hex ei is the average length of edges in a coordinate system local to the element. N is the number of coordinate directions associated with the element N = 2 for quad elements N = 3 for hex elements Mesh Quality – Aspect Ratio • The Aspect Ratio metric (QAR) applies to tri, tet, quad, and hex elements and is defined differently for each element type. The definitions are as follows: Inscribed circle Circumscribed circle

34. Mesh Quality – Diagonal Ratio • The Diagonal Ratio metric (QDR) applies only to quad and hex elements and is defined as follows: • The di are the diagonals of the element. • N is the total number of diagonals for a given element • N = 2 for quad elements • N = 4 for hex elements.

35. Tet N = 6 Tri N = 3 Quad N = 4 Wedge N = 9 Hex N = 12 Pyramid N = 8 Mesh Quality – Edge Ratio • The Edge Ratio quality metric (QER) is defined as follows: • The si are the edge lengths of the element. • N is the total number of edges for the element of interest.

36. Actual element Area = S 0 < QEVS < 1 Equilateral element Area = Seq QEVS = 0 Mesh Quality – EquiSize Skew • The EquiSize Skew metric (QEVS) applies only to quad and hex elements and is defined as follows: • S is the area (2D) or volume (3D) of the element of interest. • Seq is the maximum area (2D) or volume (3D) of an equilateral cell the circumscribing radius of which is identical to that of the mesh element.

37. Bisectors Mesh Quality – MidAngle Skew • The MidAngle Skew (QMAS) applies only to quadrilateral and hexahedral elements. • Defined by the cosine of the minimum angle formed between the bisectors of the element edges (quad) or faces (hex). • For quad elements: • For hex elements:

38. Mesh Quality – Stretch • The Stretch quality metric (QS) applies only to quadrilateral and hexahedral elements and is defined as follows: • di is the length of diagonal i • sj is the length of the element edge j, • n and m are the total numbers of diagonals and edges, respectively. • Quad elements: n = 2, m = 4, and K = 2; • Hex elements: n = 4, m = 12, and K = 3. • By definition, 0 < QS < 1. • QS = 0 describes an equilateral element • QS = 1 describes a completely degenerate element.

39. T1 T T2 Corner node Bisectors Element edge Mesh Quality – Taper • The Taper quality metric (QT) applies only to quadrilateral and hexahedral mesh elements and is defined as follows: • For any quadrilateral (or hexahedral) mesh element, it is possible to construct a parallelogram (or parallelepiped) such that the distance between any given corner of the parallelogram (or parallelepiped) and its nearest element corner node is a constant value. • As a result, any vector, T, constructed from an element corner node to the nearest corner of the parallelogram (or parallelepiped) possesses a magnitude identical to that of all other such vectors. • Each vector T can be resolved into components, Ti, that are parallel to the bisectors of the mesh element. • Quad elements: two components • Hex elements: three components • The Taper quality metric is defined as the normalized maximum of all such components for the element. • By definition, 0 < QT < 1. • QT = 0 describes an equilateral element • QT = 2 describes a degenerate element

40. Mesh Quality – Volume and Warpage • Volume • The Volume quality metric (QV) applies only to 3D elements and represents quality in terms of element volume. • Warpage • The Warpage (QW) applies only to quad elements and is defined as follows: • Z is the deviation from a best-fit plane that contains the element • a and b are the lengths of the line segments that bisect the edges of the element. • By definition, 0 < QW < 1 • QW = 0 describes an equilateral element • QW = 1 describes a degenerate element. Bisectors Element edge Best-fit plane