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Parallel Mesh Refinement with Optimal Load Balancing

Explore Discontinuous Galerkin Method and Adaptive Parallel Software for Conservation Laws. Learn about higher-order equations and computing derivatives, with focus on optimal load balancing in parallel algorithms. Develop skills in spatial and time discretization, numerical fluxes, and implementation of local time stepping for better efficiency. Discover solutions to parallel issues and dynamic load balancing for large-scale computations, tackling complex geometries and challenging problems.

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Parallel Mesh Refinement with Optimal Load Balancing

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  1. Parallel Mesh Refinement with Optimal Load Balancing Jean-Francois Remacle, Joseph E. Flaherty and Mark. S. Shephard Scientific Computation Research Center

  2. Scope of the presentation • The Discontinuous Galerkin Method (DGM) • Discontinuous Finite Elements • Spatial discretization • Time discretization • DG for general conservation laws • Adaptive parallel software • Adaptivity • Parallel Algorithm Oriented Mesh Datastructure

  3. The DGM for Conservation Laws • Find such that • Weighted residuals + integration by parts • Spatial discretization

  4. The DGM for Conservation Laws • Discontinuous approximations • Conservation on every element

  5. The DGM for Conservation Laws • Numerical Flux • Choices for numerical fluxes • Lax Friedrichs • Roe linearization with entropy fix • Exact 1D Riemann solution (more expensive) • Monotonicity not guaranteed • Higher-order limiters

  6. Higher order equations • Discontinuous approximations needs regularization for gradients

  7. Computing higher order derivatives • How to compute when u is not even C0 ? • Stable gradients : find such that • Or weakly

  8. Computing higher order derivatives • Solution of the weak problem: w=u. Weak derivatives are equal, then fields are equal. • If we choose a constrained space for w with no average jumps on interfaces i.e. with • We have • With • And

  9. Computing higher order derivatives • For higher order derivatives:

  10. Time discretization • Explicit time stepping • Efficient in case of shock tracking e.g. • Method of lines may be too restrictive due to • Mesh adaptation (shock tracking) • Real geometry's (small features) • Local time stepping, use local CFL • The key is the implementation • Important issues in parallel

  11. Local time stepping

  12. Local time stepping • Grouping elements

  13. Example: muzzle break mesh Speedup around 50

  14. Parallel Issues • Good practice in parallel • Balance the load between processors • Minimize communications/computations • Alternate communications and computations • Local time stepping • Elementary load depends on local CFL • Not the mostly critical issue

  15. Parallel issues • Example, load is balanced when • Proc 0 : 2000(1dt) + 1000 (2dt) • Proc 1 : 3000(1dt) + 500 (2dt) • Total Load : 4000  dt • If synchronization after every sub-time steps • Proc 0 waits 1000  dt at the first sync. • Proc 1 waits 1000  dt at the first sync • Maximum parallel speedup = 4/3

  16. Parallel issues • Solution • Synchronization only after the goal time step • Non blocking sends and receive after each sub-time step • Inter-processor faces store the whole history • Some elements may be “retarded”

  17. A Parallel Algorithm Oriented Datastructure

  18. Objectives of PAOMD • Distributed mesh • Partition boundaries treated like model boundaries • On processor : serial mesh • Services • Round of communication • Parallel adaptivity • Dynamic load balancing

  19. Dynamic load balancing

  20. Example

  21. 2D Rayleigh Taylor

  22. Four contacts

  23. Higher order equations • Navier-Stokes • Von Karman vortices • Re = 200 • Numerics • use of p=3 • no limiting • filtering • In parallel

  24. Large scale computations 128 processors of Blue Horizon 108 dof’s 64 processors of the PSC alpha cluster 1 106 to 2.0107 dof’s

  25. Muzzle break problem • Process • Input: ProE CAD file • MeshSim: Mesh gen. • Add surface mesh for force computations • Choice of parameters • Orders of magnitude • 1 day (single proc., no adaptation, LTS) • will need ~ 100 procs. for adaptive computations

  26. Force computations • Conservation law • Integral of fluxes • Numerical issues • Geometric search

  27. 2D computations • Importance of • adaptivity • 2nd order method • Influence of the muzzle

  28. 2D computations

  29. 3D computations • Challenging • Large number of dof’s • Complex geometry

  30. Discussion • The issue • Large scale computations • Explicit time stepping, • Load balancing • In progress • Semi implicit and implicit schemes • Higher order limiters improved

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