Adaptive Mesh Refinement Gauges Benchmarks Randall J. LeVeque - - PowerPoint PPT Presentation

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Adaptive Mesh Refinement Gauges Benchmarks Randall J. LeVeque - - PowerPoint PPT Presentation

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Adaptive Mesh Refinement Gauges Benchmarks Randall J. LeVeque Applied Mathematics University of Washington R. J. LeVeque Gene Golub SIAM Summer School, 2012 Malpasset Dam Failure Catastrophic failure in 1959 R. J. LeVeque Gene Golub SIAM

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SLIDE 1

Adaptive Mesh Refinement Gauges Benchmarks

Randall J. LeVeque Applied Mathematics University of Washington

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 2

Malpasset Dam Failure

Catastrophic failure in 1959

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 3

Malpasset Dam Failure

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 4

Modeling work by David George, using GeoClaw

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 5

Modeling work by David George, using GeoClaw

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

slide-6
SLIDE 6

Modeling work by David George, using GeoClaw

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

slide-7
SLIDE 7

Modeling work by David George, using GeoClaw

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

slide-8
SLIDE 8

Modeling work by David George, using GeoClaw

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

slide-9
SLIDE 9

Modeling work by David George, using GeoClaw

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

slide-10
SLIDE 10

Modeling work by David George, using GeoClaw

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 11

Malpasset survey locations

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 12

Malpasset survey locations

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 13

Grid convergence study

Water depth gauge at location P2 computed with two different resolutions (using 4 levels or only 3):

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 14

Adaptive Mesh Refinement (AMR)

  • Cluster grid points where needed
  • Automatically adapt to solution
  • Refined region moves in time-dependent problem

Basic approaches:

  • Cell-by-cell refinement

Quad-tree or Oct-tree data structure Structured or unstructured grid

  • Refinement on “rectangular” patches

Berger-Colella-Oliger style (AMRCLAW and CHOMBO-CLAW)

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 15

Nested AMR grids

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 16

AMR Issues

  • Refinement in time as well as space
  • Conservation at grid interfaces
  • Accuracy at interfaces, Spurious reflections?
  • Refinement strategy, error estimation
  • Clustering flagged points into rectangular patches
  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 17

Time stepping algorithm for AMR

  • Take 1 time step of length k on coarse grid with spacing h.
  • Use space-time interpolation to set ghost cell values on

fine grid near interface.

  • Take L time steps on fine grid.

L = refinement ratio, ˆ h = h/L, ˆ k = k/L.

  • Replace coarse grid value by average of fine grid values on

regions of overlap — better approximation and consistent representations.

  • Conservative fix-up near edges.

Q0

j

Q1

j

ˆ h ˆ Q0

m

ˆ Q1

m

ˆ Q2

m

ˆ Q0

m−1

ˆ Q1

m−1

ˆ Q2

m−1

ˆ Q0

m−2

tn tn + ˆ k tn + 2ˆ k

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 18

Flagging Cells for Refinement

Every kcheck time-steps at each level (except finest), check all grid cells and flag those needing refinement. Use one or more of the following flagging criteria:

  • Richardson estimation of truncation error.

Compare result after last two time steps on this grid with

  • ne time step on a coarsened grid.
  • Estimate spatial gradient of one or more components of

solution.

  • Check for regions where refinement is user-forced to some

level.

  • Problem-specific, e.g. near shore for tsunami simulation.
  • Other user-supplied criterion set in flag2refine.f.
  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 19

Clustering Flagged Cells for Refinement

Use Berger-Rigoutsos algorithm [IEEE Trans. Sys. Man & Cyber.] 21(1991), p. 1278] Clusters flagged points into a set of rectangular patches. Tradeoff between:

  • Many small patches cover flagged points with minimal

refinement of unflagged points.

  • But.... increases overhead associated with each patch,

e.g. boundary values: ghost cell values set by copying or interpolation from other grids, B-G algorithm has cut-off paramter: require that this fraction of refined cells be flagged (usually set to 0.7).

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 20

Refinement of topography

Topography should be consistent between different levels. Bℓ

1 = 1

2(Bℓ+1

1

+ Bℓ+1

2

)

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 21

Refinement of topography

Topography should be consistent between different levels. Bℓ

1 = 1

2(Bℓ+1

1

+ Bℓ+1

2

) Important to interpolate surface, not depth, as in...

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 22

Refinement of topography near shore

Again need to maintain flat surface before wave arrives: Mass cannot always be conserved!

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 23

Refinement of topography near shore

Again need to maintain flat surface before wave arrives: Mass cannot always be conserved!

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 24

Chesapeake Bay and Anapolis

Cannot conserve mass when refining near shore!

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 25

Chesapeake Bay and Anapolis

Cannot conserve mass when refining near shore!

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 26

Gauges in GeoClaw

Set gauge locations in setrun.py, e.g. DART location: # == setgauges.data values == geodata.gauges = [] # for gauges append lines of the form # [gaugeno, x, y, t1, t2] geodata.gauges.append([32412, \

  • 86.392, -17.975, 0., 1.e10])

Can add additional lines of this form.

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 27

Gauges in GeoClaw

Set gauge locations in setrun.py, e.g. DART location: # == setgauges.data values == geodata.gauges = [] # for gauges append lines of the form # [gaugeno, x, y, t1, t2] geodata.gauges.append([32412, \

  • 86.392, -17.975, 0., 1.e10])

Can add additional lines of this form. Useful for comparison with observations or lab measurements. Also useful for quantitatively comparing different grid resolutions, parameter choices, etc.

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 28

Radial ocean verification study

From: Berger, George, RJL, Mandli, Adv. Water Res. 2011,

www.clawpack.org/links/awr11/

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 29

Radial ocean verification study

From: Berger, George, RJL, Mandli, Adv. Water Res. 2011,

www.clawpack.org/links/awr11/

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 30

Radial ocean verification study

From: Berger, George, RJL, Mandli, Adv. Water Res. 2011,

www.clawpack.org/links/awr11/

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

slide-31
SLIDE 31

Radial ocean verification study

From: Berger, George, RJL, Mandli, Adv. Water Res. 2011,

www.clawpack.org/links/awr11/

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

slide-32
SLIDE 32

Radial ocean verification study

From: Berger, George, RJL, Mandli, Adv. Water Res. 2011,

www.clawpack.org/links/awr11/

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 33

Radial ocean verification study

Comparison of Gauges 1 and 2 from Test 1 and 2:

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 34

Radial ocean verification study

Comparison of Gauges 1 and 2 with more refined grids (Test 1):

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 35

Benchmarking project

National Tsunami Hazard Mitigation Program set of 9 benchmark problems.

  • One-dimensional waves on beach: analytic and wavetanks
  • Waves around conical island (wave tank)
  • Okushiri Island tsunami of 1993
  • Wave tank model of Monai Valley
  • Wave tank experiments of submarine landslides

Recently solved by several teams and comparisons soon to appear. Our results available at

www.clawpack.org/links/nthmp-benchmarks/

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 36

Monai Valley wave tank experiment

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 37

Monai Valley wave tank experiment

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 38

Monai Valley wave tank experiment

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 39

Monai Valley wave tank experiment

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012

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SLIDE 40

Monai Valley wave tank experiment

  • R. J. LeVeque

Gene Golub SIAM Summer School, 2012