adaptive mesh refinement
DESCRIPTION
Adaptive mesh refinement. for discontinuous Galerkin method on quadrilateral non-conforming grids Michal A. Kopera PDE’s on the Sphere 2012. Motivation. Cut the number of elements down to a minimum necessary to sufficiently well resolve the problem - PowerPoint PPT PresentationTRANSCRIPT
Adaptive mesh refinement
for discontinuous Galerkin method on quadrilateral non-conforming grids
Michal A. Kopera
PDE’s on the Sphere 2012
Motivation
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• Cut the number of elements down to a minimum necessary to sufficiently well resolve the problem
• Tackle problems previously difficult or impossible to solve due to limited computational resources
Source: NASA
• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
• Each parent element always replaced by four children
• At most 2:1 size ratio of face-neighboring elements
Non-conforming quad-based DG
Non-conforming quad-based DG
level 0
• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
Non-conforming quad-based DG• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
level 0
level1
Non-conforming quad-based DG
level 0
level1
level 2
• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
Non-conforming quad-based DG• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
• Each parent element always replaced by four children
• At most 2:1 size ratio of face-neighboring elements
Non-conforming quad-based DG• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
• Each parent element always replaced by four children
• At most 2:1 size ratio of face-neighboring elements
• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
• Each parent element always replaced by four children
• At most 2:1 size ratio of face-neighboring elements
Non-conforming quad-based DG
Non-conforming quad-based DG• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
• Each parent element always replaced by four children
• At most 2:1 size ratio of face-neighboring elements
• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
• Each parent element always replaced by four children
• At most 2:1 size ratio of face-neighboring elements
Non-conforming quad-based DG
!
!
• Non-conforming flux computation handled by the DG solver
• Forest of quad-trees approach
• Each parent element always replaced by four children
• At most 2:1 size ratio of face-neighboring elements
Non-conforming quad-based DG
How to compute flux?
1) Scatter data from the parent edge to children edges
How to compute flux?
1) Scatter data from the parent edge to children edges2) Compute flux on children edges like in a conforming case
How to compute flux?
1) Scatter data from the parent edge to children edges2) Compute flux on children edges like in a conforming case
+3) Gather fluxes from children edges to the parent edge
How to compute flux?
1) Scatter data from the parent edge to children edges2) Compute flux on children edges like in a conforming case3) Gather fluxes from children edges to the parent edge
4) Apply fluxes like in a conforming case
+
How to move data through an interface?
Let us define the space for both parent and child faces:
with mappings
Expanding variables yields
For each children face we require
Substitution of expansions and reorganizing the terms yields
Let
+
We require that
After splitting the integrals, plugging-in extensions, reorganizing and variable change we arrive at:
Refinement criteriumQuickTime™ and a
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Refinement criteriumWhat are the benefits and costs?
What are the benefits and costs?
What are the benefits and costs?
thresholdfront position
[m]
0.001 14,754
0.1 14,754
1.0 14,754
4.0 14,754
Analyzing mountain
cases
Multi-rate time-
stepping
CG AMR
GPU 3D + MPI
Multigrid ?
Outlook
Optimized data
structures
Shallow water
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Shallow Water Equations2D wave with 2D bathymetry
Linear hydrostatic mountain