adaptive mesh refinement

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