1. Discontinuous Galerkin Methods and Strand Mesh Generation
Andrey Andreyev
Advisor: James Baeder

2. Terminology: Structured Mesh Unstructured Mesh
Picture from:
Picture from:

3. Project Overview Two-Part Effort
Part 1: Implementation of the Discontinuous Galerkin (DG) Euler Equation Solver on a 2D Grid
Part 2: Implementation of a strand grid generator from a surface mesh
Image: from Bcmath.org
Image: Aaron Katz 3

4. The Euler Equations U is the conservative variable vector
F, G, and H are the Flux Vectors in the x, y, z direction respectively

5. Part 1: DG Method Finite Difference Methods Advantages:
Ease of Implementation
Easy to make higher order
Disadvantages:
Only applicable on structured grids
In general, methods in Computational Fluid Dynamics can be divided into three approaches:
Finite Element
Advantages:
Can be any order of accuracy
Based on variational methods
Applicable on unstructured grids
Disadvantages:
More complex
Not conservative!
Naturally implicit (can be explicit with modifications)
Finite Volume
Advantages:
Naturally Conservative (captures discontinuities in the flow field)
Many upwinding possibilities
Applicable on unstructured grids
Disadvantages:
Difficult to devise stable higher order scheme

6. Part 1: DG Method
What is the appeal behind Discontinuous Galerkin Methods?
Can be thought of as a combination between Finite Volume and Finite Element Methods
Can be implemented on unstructured grids
Can be made higher order
Does not enforce continuity in the shape functions of the element, continuity is enforced using numerical fluxes (seen in finite volume approaches)
Requires substantially less information about the neighboring elements making parallelization easier. (Result of not enforcing inter-element continuity

7. Part 1: DG Method Spatial Discretization
1. Start with the Euler Equation:
3. Multiply by weight function and integrate by parts
Note the boundary term has a different flux term. In normal finite element, the boundary terms need to enforce connectivity with neighboring elements. In Discontinuous Galerkin Methods the boundary fluxes are calculated using the Riemann Fluxes. This enforces connectivity and allows for discontinuities in the solution!
2. Discretize the spatial domain and assume and assume an approximate solution on a per-element basis

8. Part 1: DG Method Time Integration
Time integration of the equations will be carried out using a higher order Runge-Kutta technique.
The space discretization in the previous slide converted the PDEs into a system of ODEs in time.
Using Higher Order Runge-Kutta, we carry out the time integration on a per-element basis2

9. Part 1: DG Method Test Problems
The method will first be implemented on the one-dimensional version of the Euler Equations to test the methods accuracy. Sod’s Shock Tube Problem will be used as the test case since it has an exact solution and will test the scheme’s shock capturing ability.
Implemented using Fortran 95
Exact Solution
Image:
Image: Author Generated

10. Part 1: DG Method Test Problems
Two Dimensional Airfoil on a structured mesh. Mesh provided by Dr. Baeder.
Boundary Conditions:
Tangential Flow around the airfoil (Inviscid Wall)
Undisturbed flow at the domain edge (Farfield)
Run at a variety of mach numbers to create different flow regimes (subsonic, trans-sonic, supersonic) to steady state
Compare the results to experimental data (large database for many airfoils)
Compare results to other established computational tools
Implemented serially using Fortran 95, then parallelized using MPI

11. Part 1: DG Method Implementation Schedule
10/31/ One dimensional version. Apply to one dimensional problem with a known solution to test accuracy and shock capturing abilities. Sod shock tube problem Will validate the 1-D version (serial)
12/15/ Two dimensional version. Apply to 2-D airfoil problem using provided grids (serial)
02/15/ Validation of the two dimensional version using experimental airfoil results as well as the results published in literature
03/15/ Parallelization of the two dimensional. Validate using results from the serial version

12. Part 2: Strand Mesh Generation
One of the largest bottle necks in CFD based design is the generation of volume meshes for the flow solvers.
Picture: Katz, Wissink 3
Complex geometries make it difficult to resolve the surface mesh and extrude the volume mesh.
The strand mesh method seeks to eliminate the second problem of volume mesh extrusion by shooting out “strands” from the nodes of the surface mesh in prescribed manner making volume mesh generation automatic. The volume grid has prismatic which have good computational properties.

13. Part 2: Strand Mesh Generation
Approach:
Calculate the normal on each node of the mesh using an average of the normals at the neighboring cells
Apply a distribution function in the normal direction for node placement. Function of Reynold’s number
Apply vector smoothing at areas of high surface curvature
Katz, Wissink 3

14. Part 2: Strand Mesh Generation
Vector Smoothing
Katz, Wissink 3
Apply normal vector smoothing on areas with high curvature by minimizing the equation at each node using Lagrange multipliers and updated with Jacobi Iterations3.

15. Part 2: Strand Grid Generation Implementation Detail
Coding will be done in Fortran 95, unless a need for complex data structures arises, then it will be done in C
Parallelization of the code will be done in MPI.
Will require the use of an external library, Parmetis to partition the surface mesh
Very little communication required since only the neighboring nodes of the surface mesh are used for normal vector generation

16. Part 2: Strand Grid Generation Implementation Schedule
04/15/ Implementation of the strand mesh generation. Validation is trivial since the problem is geometric in nature and visual inspection of the resulting mesh will suffice.
Time Permitting- Integration of the strand methods into the DG Flow Solver
End of Semester- Final Report

17. Questions?

18. References:
S.-Y. Lin, Y.-S. Ching Quadrature-Free Implementation of the Discontinuous Galerkin Method for Hyperbolic Equations. San-Yih Lin Yan Shin-Ching. National Cheng Kung University, Tainan, Taiwan 70101, National Republic of China
B. Cockburn, C.W. Shu. Runge-Kutta Discontinuous Galerkin Methods for Convection-dominated Problems.
A. Katz, A. Wissink. Application of Strand Meshes to Complex Aerodynamic Flow Fields