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 Edge mesh preview Meshing the edge
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. Grading Controls mesh density distribution along an edge.
Single-sided 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.
Symmetric grading
Asymmetric grading

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 Modifying soft 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 Remove old mesh Ignore size function
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 – Example1
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. 2
Specify grading only, apply without meshing
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. 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). E R S

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

19. 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! R E
Which vertex to change? R E S E

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

21. 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 C S E E E S E E

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

23. Formula for Tri-Primitive Sheme
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 E E S

24. 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 T E T C N T E

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
(best) 1
(worst)
Quad/Hex
Tri/Tet

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. 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:
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
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. 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.
Tri
N = 3
Quad
N = 4
Tet
N = 6
Pyramid
N = 8
Hex
N = 12
Wedge
N = 9

36. 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.
Actual element
Area = S
0 < QEVS < 1
Equilateral element
Area = Seq
QEVS = 0

37. 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:
Bisectors

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. 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
Bisectors
Corner
node T T1 T2
Element edge
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
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