Edge and Face Meshing

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.

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

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.

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

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.

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

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.

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.

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.

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. 2 Specify grading only, apply without meshing Face Mesh Generated Using Submap Scheme

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.

Face Meshing - Quad Examples Quad: Map Quad: Submap Quad: Tri-Primitive Quad: Pave

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

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.

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

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

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

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

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

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

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

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

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.

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.

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.

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.

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

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

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.

Appendix

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

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.

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

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

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

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.

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

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