Volume Meshing and the Sizing Function

1. Volume Meshingand theSizing Function

2. Approach • To potentially reduce discretization errors, and to reduce cell count, a ‘high’ quality hex mesh is preferred. • For a hex mesh, complicated geometries (volumes) typically need to be decomposed into simpler ones so that one of the hex meshing schemes can be used. • In some instances, some geometries may be too complex and decomposition for hex meshing is impractical or impossible. In these instances use a tet/hybrid mesh.

3. Volume Meshing Form: Upon picking a Volume GAMBIT will automatically choose a Type based on the solver selected and the combination of the face Types of the volume. In ambiguous cases, GAMBIT chooses the Tet/Hybrid: TGrid combination Available element/scheme type combinations Hex Map Submap Tet-Primitive Cooper Stairstep Hex/Wedge Cooper Tet/Hybrid TGrid Volume Meshing

4. Hex: Map Hex: Submap Hex: Tet-Primitive Volume Meshes - Hex Examples • Hex: Cooper • Hex: Stairstep

5. Hex/Wedge and Tet/Hybrid Examples • Hex/Wedge: Cooper • Tet/Hybrid: TGrid

6. Volumes that are mappable by default: A logical cube All faces map-able (or Submap-able) and mesh is matching Map Scheme Hex Meshing - Map mesh mesh

7. Volumes that are Submap-able by default: All faces map-able or submap-able Topological matching of opposite faces Submap Scheme Hex Meshing - Submap mesh mesh

8. All hex elements in a four-sided (tet) volume Volumes directly meshable using Tet-Primitive scheme How the Tet Primitive Scheme works Connect center points on edges, faces and the volume Map the four sub-volumes Tet-Primitive scheme Hex Meshing - Tet-Primitive Mesh Tet Primitive

9. The Cooper Scheme, in essence, projects or extrudes a face mesh (or a set of face meshes) from one end of a volume to the other and then divides up the extruded mesh to form the volume mesh. The projection direction is referred to as the Cooper direction. Faces topologically perpendicular to this direction are called Source faces. Source faces do not have to be premeshed. In practice, at least one source face must not be meshed and must span across the entire cross section. Faces that intersect the source faces are referred to as Side faces. Side faces must be Mappable or Submappable. Source Faces Side Faces Cooper direction Hex Meshing - Cooper

10. Cooper Scheme - permissible geometries A volume with multiple source faces on multiple sides Holes or “wells” are allowed Permissible Cooper Geometries source faces source faces source faces

11. (a) A B C (c) (b) Impermissible Cooper Geometries • Cooper Scheme - impermissible geometries • (a) Cannot construct logical cylinder, the side faces of which is mappable • (b) All source faces are meshed • (c) Cannot use Cooper (readily) with multiple source faces on opposing ends.

12. Cooper scheme - Application When the Cooper scheme is selected, a source face list box appears in the panel. GAMBIT will automatically select all source faces for direct Cooper-able volumes (scheme comes up as default). Cooper Application • If GAMBIT chooses the sources faces, you should check the source face list and visually check for an intelligent selection. Change, if necessary. • If GAMBIT fails to pick a set of source faces, you can either manually change the vertex types on the side faces or manually select the source faces.

13. The Stairstep scheme creates a single-block structured mesh. The Stairstep scheme creates and meshes a faceted volume the shape of which approximates the volume to be meshed. The original volume is not meshed. Faceted volume: is generated as a result of the meshing process is not connected to original volume. Assignment of continuum and boundary zonetypes must be applied to faceted volume. The Stairstep scheme can be used for quickmesh when boundary mesh is less important. ‘Body-fitted’ boundary mesh willbe implemented in future. Hex Meshing - Stairstep

14. Tetrahedral/Hybrid Meshing • Tetrahedral/Hybrid Mesh Scheme - TGrid • Automatic - most volumes can be meshed without decomposition. • Use boundary layers to create hybrid grids (prism layers on boundaries to capture important viscous effects). • Using on volumes that are adjacent to volumes that have been meshed with hex elements will automatically result in a transitional layer of pyramids. Tet mesh second Hex mesh first

15. low quality pyramid prism layer acute angle Tet/Hybrid Meshing: Troubleshooting • Quality of the tetrahedral mesh is highly dependent on the quality of the triangular mesh on the boundaries. • Initialization process may fail or highly skewed tetrahedral cells may result if there exists: • highly skewed triangles on the boundaries. • large cell size variation between adjacent boundary triangles. • small gaps that are not properly resolved with appropriate sized triangular mesh. • Difficulties may arise in generation of hybrid mesh. • Cannot grow pyramids from high aspect-ratio faces. • Prism and pyramid generation may not work properly between surfaces forming very acute angles.

16. Sizing Functions • Sizing Function controls mesh distribution in a region of space (Edges, Faces, and Volumes) in a manner analogous to the way grading controls mesh distribution on edges. • Sizing Function accessed through Toolbar: Without a Size Function With a Size Function

17. Sizing Function Types • Sizing Function requires the specification of Type, Entities, and Parameters. • Sizing Function ‘Type’ controls method by which scope of sizing function is obeyed. • Fixed • Scope is defined as a fixed region about a source. • Curvature • Scope is defined as a region near highly curved surfaces. • Proximity • Scope is defined as a region within a specified distance from objects.

18. Sizing Function Definition • Each Sizing Function Type requires the specification of: • Entities • Source entity defines shape and location of the ‘origin’ of affected region. • Attachment entities host the mesh that will be affected. • Parameters • Defines growth rate of cells in affected region for Fixed and Curvature Type. • Defines number of cells in gaps for Proximity Type. • Defines extent of affected region in Attachment entities. • Defines maximum cell size allowed in affected region.

19. Fixed Sizing Function - Source • Source • Can be vertices, edges, faces, or volumes • Can be internal or external to attachment entities • Source entity defines shape of scope

20. Sizing Function - Attachments • The attached entities host mesh to be affected.

21. Fixed Sizing Function - Parameters • Start size • Size adjacent to the source • Growth rate • Ratio of two adjacent mesh-element edge size • Distance • Determine boundary of size function • Size limit • Maximum allowable size for attachment entity

22. Curvature Sizing Function • Modifies size according to geometric curvature • Sources can only be face entities • Parameters • Angle - Maximum allowable angle between outward pointing normals for any two adjacent mesh elements located immediately adjacent to the surface of a source • Others are as with Fixed

23. Proximity Size Function • Specifies number of cells in face gap (3D) and edge gap (2D) • Parameters • Cells per gap - number of mesh layers in the gap. • Distance - maximum distance from the source at which size function applies. • Size limit • Limitations • Becomes slow on large models • Improper use may result in abrupt change in size • Solutions • Use multiple size functions • Specify large value for distance • Increase resolution by changing the defaults for background grids