Welcome

May 25, 2010

Vincent Charles Betro’s Dissertation Proposal:

Fully Anisotropic Split-Tree Adaptive Refinement Mesh Generation Using

Tetrahedral Mesh Stitching

Outline

• Introduction

• Methodology

• Process

• Results

• Conclusions

• Future Work

IntroductionThe need exists for a robust Cartesian Mesh

Generation algorithm….– Needs to highly resolve geometry

– Needs to allow for anisotropic and isotropic generation, with deterministic results

– Needs to be constructed so that floating point tolerance issues are mitigated

– Needs to utilize adaptation as well as great control over initial generation

– Needs to retain tree for future dynamic integration

– Need to be able to generate viscous or inviscid meshes

IntroductionTypes of meshing• Overset (structured)• Extrusion (used to make

3-D slices or insert viscous layers)

• Delaunay (fully triangular or tetrahedral)

• Cartesian Hierarchical• FASTAR uses last three

types!

Introduction• Aspect Ratio is determined by taking the maximum length in x, y,

or z and dividing by the minimum length in x, y, or z

• An isotropic mesh (unit aspect ratio) has elements which all have an aspect ratio of one, and transition is done by gradation parameters.

• An anisotropic mesh (aspect ratio > 1.0) may have aspect ratios as high as desired by the user, which allows for significantly less elements and nodes that can capture the same physics.

Introduction• Oct-tree refinement can only do anisotropic

refinement, as seen in the third row in the diagram.

• Omni-tree refinement can do any of the seven types displayed in the figure, but it requires inheritance of children and grandchildren and can cause issues with tree coherency. P_HUGG runs in this mode.

• FASTAR uses a split-tree, which allows for the types of refinement seen in row one of the diagram.

Introduction

• The building block of a hierarchical Cartesian mesh is the voxel, which is short for “volumetric pixel”.

• Each voxel contains information pointing to its relational location in the mesh, but no physical coordinates

– parent index

– child indices

– High and low extent cornerpoints

– Split integer denoting direction of split (x, y, or z, or none)

Methodology: RMT• Riemannian metric tensors (RMT) allow the user to define target

element sizes in regions of the mesh.

• M is an spd matrix which is the product of a rotation matrix, R, which also represents the eigenvectors of M. λ is a scaling matrix into which the inverse squares of the desired values of x, y, and z spacing are entered, and is a diagonal matrix of the eigenvalues of M.

• The beauty of this approach is that if desired, through singular value decomposition, all of these attributes may be deduced from a given tensor without extra storage and be changed independently.

Methodology: RMT

• Using the metric tensor, M, and the length of a given edge in the current mesh, one constructs a metric length ( ) which determines refinement as a scalar.

Methodology: RMT• FASTAR uses the coordinates of the tetrahedron (P) and solves

the following system for , where M and P are defined below:

Methodology: RMT• System of equations which is solved using SVD and back substitution

to determine the Riemannian Metric Tensor to be applied to the geometry facet’s region of influence (extents).

Methodology: RMT

• In the adaptive spacing field code, the tensors created from the existing elements are decomposed, and the following equation is used to determine the new spacing parameters.

Methodology: Adaptation• Using adaptive refinement allows the user to capture the

physics of the problem being modeled by refining in areas of high gradient in one or more solution variables.

• This can allow meshes to be coarse in the far field and fine in areas of vortices, shocks, and other physical phenomena, thus reducing mesh time and speeding overall time to a converged, physically accurate solution.

• Effectively, since Riemanniann tensors are used, any kind of quantifiable adaptation can be done, including feature-based and adjoint-based.

Methodology: Viscous Insertion

• Since we are cutting away the voxels near the boundaries, the ability to extrude the boundaries to insert viscous layers is fortuitous.

• P_VLI allows difficult insertion without the perils of extrusion tangling.

• FASTAR allows ease of extrusion in situations with convex geometry.

Methodology: Neighbors• In order to best utilize the tree, save space on memory, and

allow nodes to be created at the end of the mesh generation process, a neighbor search algorithm was implemented.

• This allows the only tolerance to be of the minimum spacing.

Orange nodes: min_space

Red nodes: min_space

1

2 2

1

4

1

4

Process: Inputs• User-specified parameters:

– Geometry files– P_HUGG_geometry.params file– Adaptation spacing file– Geometry on/off– Minimum Spacing/Maximum Spacing– Maximum aspect ratio– Minimum metric length at which to refine– Minimum spacing between voxel front and geometry/viscous layers– Desired Normal Spacing– Restart/Output surface mesh file type– Internal tet generator– Viscous inertion/Layer size/Number of Layers/Viscous Boundaries

Process• Read in params file, geometry, and spacing file (if present), and create

root voxel around extents of geometry.

Here the root cell has been subdivided in x, the default direction

Process• The tensors are then passed down to the two children, and they refine.

Here the children have been subdivided in y

Process• Again, these children are subdivided, and this continues until the metric

length of 1.0 is reached for each tensor.

Here the root cell hasbeen subdivided in z

Process• Final Product: A mesh generated on a sphere using uniform tensors,

with no marking or quality constraints applied:

Process: Quality• Now, quality constraints are

applied:

– Desired constraints are applied first, in order to mitigate the need to apply required constraints in a biased fashion

• Desired Quality Constraints:

– Gradation parameter

– Neighbor refinement

– Aspect ratio

Opposite neighbors both at a higher level of refinement than the

current voxel

Tangential neighbor is more than twice as large

in the given direction

Process: Quality• Now, required quality constraints are applied:

– Faces and edges are checked such that a neighboring voxel may not be four times larger in the given normal direction.

– Opposite faces are checked for a crossbar situation.

Process: Marking• A two-front mesh on the sphere within a box. Note that the outmost

and innermost layers of voxels have been removed to make way for the tetrahedral mesh.

Process: Mesh Creation

• Distinct nodes are created.

• The voxel front is triangulated.

• The space between viscous layers/geometry and voxel front is tetrahedralized.

• The final polyhedral mesh is stitched to the tetrahedral mesh.

Process: Mesh Creation• Below is a close up of the inner and outer triangular surface meshes

and the resulting tetrahedra from the Pointwise® mesher. These exist on the outer tetrahedral mesh of the sphere case.

Stitching boundary

Geometry boundary

Process: Mesh Creation• These are the tet meshes that fill the yellow areas on previous slide.

(Not to Scale)

Inner tetrahedral mesh Outer tetrahedral mesh

Process: Mesh Creation• The surface meshes

created on such geometries as the Seafighter jets are very fine and stairstepped in order to conform to the geometry.

• This causes Pointwise®, Gridgen® , and TetGen difficulties in not only creating domains but also in recovering the boundaries.

Process: Mesh Creation• Finally, here is the stitched mesh with polyhedral elements.

Process: Mesh Creation• Here is the same mesh converted to the four basic types by

Conv.LINUX64.

Results: Cube• Simply generated cube polyhedral mesh, with aspect ratio of 10

(anisotropic) and 1 (isotropic).

Results: Cube• Isotropic cube (four basic types) with viscous layers inserted (1.0e-03)

Results: Cube-in-Cube• Simply generated polyhedral cube in a cube mesh, with aspect ratio

constrained to a maximum of 10 (anisotropic).

Results: Cube-in-Cube• Simply generated polyhedral cube in a cube mesh, with aspect ratio

constrained to a maximum of 1 (isotropic).

Results: Spacing Field• Use of a spacing field on the cube mesh, with the z direction given

three orders of magnitude smaller spacing and aspect ratio constrained to a maximum of 10 (anisotropic) or 1 (isotropic...academic only as does extra work on mesh to “undo” spacing field).

Aspect Ratio = 10.0 Aspect Ratio = 1.0

Results: NACA 0015• NACA 0015, with 13° angle of attack.

• NOTE: Adapted meshes in progress!

Results: NACA 0015• Initial Solution on NACA 0015, inviscid

Results: Sea FighterSeafighter geometry

Results: Sea Fighter• Seafighter polyhdedral mesh with defined normal spacing (1e-03),

including tetrahedralization.

Results: Sea Fighter• Seafighter polyhdedral mesh with defined normal spacing (1e-03), including

tetrahedralization. Close up pictures of jets, showing two-layer construction.

Conclusions• FASTAR has proven to be capable of generating large meshes on complex

geometries, while allowing for anisotropic mesh creation, which is important in a diverse set of arenas, including the hydrodynamics cases presented in this dissertation and for generally keeping the node and element counts as low as possible, making rapid solution generation easier.

• FASTAR retains the exact tessellation supplied in the user’s CAD geometry file, allowing for the final mesh to conform to the exact shape and still be decoupled from the flow solver being used.

• FASTAR allows for adaptive refinement to occur, helping the user capture the exact physics of their flow phenomena with repeated, automated generation in between solver runs.

• FASTAR allows for insertion of viscous layers without the need to smooth the mesh to do so, possibly losing Cartesian alignment with solution.

• FASTAR works with P_OPT, P_VLI, and P_REFINE, all of which were designed by Vince Betro and Dr. Steve Karman.

Future Work• Allow for dynamic mesh generation and interfacing with UX by

using the tree to move the geometry and remesh in appropriate spots.

• An interesting study will include the comparison of the robustness of this method versus using the newly created P_SMOOTH (written by Dr. Steve Karman, and based on the two-dimensional work of Dr. Mandar Sahasrabudhe) to move and resmooth the existing mesh via Winslow Smoothing with Virtual Control Volumes.

• This is not part of the current work, but has been allowed for!

Bibliography• [Berger et al., 1998] Berger, M.J., and Aftosmis, M.J. (1998). Aspects (and Aspect Ratios) of

Cartesian Mesh Methods. In Proceedings of the 16th International Conference on Numerical Methods in Fluid Dynamics (Springer, Heidelberg).

• [Domel et al., 2000] Domel, N.D., Karman, Jr., S.L. (2000). Splitflow: Progress in 3D CFD with Cartesian Omni-tree Grids for Complex Geometries (2000-1006). At 38th Aerospace Sciences Meeting and Exhibit (AIAA, Reno, NV January 10-13, 2000).96

• [Huang, 2004] Huang, Weizhang. (2004). Metric tensors for anisotropic mesh generation. In Journal of Computational Physics, Volume 204 (Academic Press, Amsterdam 2004), pp. 633-655.

• [Karman et al., 2007] Karman, Jr., S.L., Anderson, W.K., and Sahasrabudhe, M. (2007). Mesh Generation Using Unstructured Computational Meshes and Elliptic Partial Differ- ential Equation Smoothing (2007-0559). AIAA American Institute of Aeronautics and Astronautics Journal, Volume 44, Number 6 (AIAA, Reston, VA 2007), pp. 1277-1286.

• [Luo et al., 2008] Luo, H., Chen, G., and Lohner, R. (2008). A Hybrid Grid Genera- tion Method for Complex Geometries. At 46th Aerospace Sciences Meeting and Exhibit (AIAA, Reno, NV January 7-10, 2008).

• [Si, 2006] Si, Hang. (2006). TetGen: A Quality Tetrahedral Mesh Generator and Three Dimensional Delaunay Triangulator (Version 1.4 User’s Manual). http://tetgen.berlios.de. January 18, 2006.