Discontinuous Galerkin Methods and Strand Mesh Generation

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Discontinuous Galerkin Methods and Strand Mesh
Generation
Andrey Andreyev (andreyev_at_umd.edu) Advisor
James Baeder (baeder_at_umd.edu)
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Terminology
Structured Mesh
Unstructured Mesh
Picture from http//www.cgl-erlangen.com/download
s/Manual/ch09s16s01.html
Picture from http//ta.twi.tudelft.nl/users/wesse
lin/projects/unstructured.html
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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
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The Euler Equations
U is the conservative variable vector F, G, and
H are the Flux Vectors in the x, y, z direction
respectively
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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 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

• 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)

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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

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Part 1 DG MethodSpatial 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
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Part 1 DG MethodTime 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
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Part 1 DG MethodTest Problems
The method will first be implemented on the
one-dimensional version of the Euler Equations to
test the methods accuracy. Sods Shock Tube
Problem will be used as the test case since it
has an exact solution and will test the schemes
shock capturing ability. Implemented using
Fortran 95
Exact Solution
Image http//en.wikipedia.org/wiki/FileSodShockT
ubeTest_Regions.png
Image Author Generated
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Part 1 DG MethodTest 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
http//www.salome-platform.org/forum/forum_10/2133
29959/view
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Part 1 DG Method Implementation Schedule
10/31/12- 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) 12/15/13-
Two dimensional version. Apply to 2-D airfoil
problem using provided grids
(serial) 02/15/13- Validation of the two
dimensional version using experimental airfoil
results as well as the results published in
literature 03/15/13- Parallelization of the
two dimensional. Validate using results from the
serial version
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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.

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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 Reynolds number Apply vector
smoothing at areas of high surface curvature
Katz, Wissink 3
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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.
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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

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Part 2 Strand Grid Generation Implementation
Schedule
04/15/13- 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
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Questions?
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• 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