A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element...
Transcript of A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element...
![Page 1: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/1.jpg)
Intro SWE cG(1)cG(1) Tests
A Stable Equal Order Finite Element Discretization ofthe Shallow Water Equations of the Ocean
Erich L Foster
13 January 2014
E. L. Foster (BCAM) cG(1)cG(1) for SWE 1 / 13
![Page 2: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/2.jpg)
Intro SWE cG(1)cG(1) Tests
Characteristics of the Earth’s Oceans
Simulated Sea Surface Temperature
Absorbs energy from theSun and stores it.
Transports heat from theequator towards the poles.
71% of Eath’s surface iscovered by the oceans.
1000 times the heat capacityof the atmosphere.
Most of the Ocean’s KE iscontained in meso-scaleeddies (<100km).
E. L. Foster (BCAM) cG(1)cG(1) for SWE 2 / 13
![Page 3: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/3.jpg)
Intro SWE cG(1)cG(1) Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring severalthousand simulated years for “spin up.”
0.1
resolution or higher needed to capture the bulk of the energycontained in the meso-scale eddy field.
Large amounts of data, ∼1TB per simulated year for 0.1
gridresolution.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 3 / 13
![Page 4: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/4.jpg)
Intro SWE cG(1)cG(1) Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring severalthousand simulated years for “spin up.”
0.1
resolution or higher needed to capture the bulk of the energycontained in the meso-scale eddy field.
Large amounts of data, ∼1TB per simulated year for 0.1
gridresolution.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 3 / 13
![Page 5: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/5.jpg)
Intro SWE cG(1)cG(1) Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring severalthousand simulated years for “spin up.”
0.1
resolution or higher needed to capture the bulk of the energycontained in the meso-scale eddy field.
Large amounts of data, ∼1TB per simulated year for 0.1
gridresolution.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 3 / 13
![Page 6: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/6.jpg)
Intro SWE cG(1)cG(1) Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring severalthousand simulated years for “spin up.”
0.1
resolution or higher needed to capture the bulk of the energycontained in the meso-scale eddy field.
Large amounts of data, ∼1TB per simulated year for 0.1
gridresolution.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 3 / 13
![Page 7: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/7.jpg)
Intro SWE cG(1)cG(1) Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring severalthousand simulated years for “spin up.”
0.1
resolution or higher needed to capture the bulk of the energycontained in the meso-scale eddy field.
Large amounts of data, ∼1TB per simulated year for 0.1
gridresolution.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 3 / 13
![Page 8: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/8.jpg)
Intro SWE cG(1)cG(1) Tests
Shallow Water Equations (SWE)
Standard test problem for Ocean Modelling.
Like Navier-Stokes, suffers from spurious computational modes.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 4 / 13
![Page 9: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/9.jpg)
Intro SWE cG(1)cG(1) Tests
Shallow Water Equations (SWE)
Standard test problem for Ocean Modelling.
Like Navier-Stokes, suffers from spurious computational modes.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 4 / 13
![Page 10: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/10.jpg)
Intro SWE cG(1)cG(1) Tests
Shallow Water Equations (SWE)
Standard test problem for Ocean Modelling.
Like Navier-Stokes, suffers from spurious computational modes.
ηt + Θ−1H∇ · u = 0
ut + (u · ∇) u +Ro−1u⊥ + Fr−2Θ∇η −Re−1∆u = 0on Ω (1)
u · n = 0 on δΩ (2)
E. L. Foster (BCAM) cG(1)cG(1) for SWE 4 / 13
![Page 11: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/11.jpg)
Intro SWE cG(1)cG(1) Tests
Why finite elements?
Finite Difference grid of GIOMAS Finite Element mesh of SLIM
E. L. Foster (BCAM) cG(1)cG(1) for SWE 5 / 13
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Intro SWE cG(1)cG(1) Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad forNon-Mathematicians).
Complicated to program.
Use packages such as FEniCS, FreeFEM, OpenFOAM, etc.
Spurious computational modes for certain finite element pairs.(similar problem with finite differences)
Use a different formulation of the problem, e.g. Vorticity-Streamfunction form.Use Taylor-Hood or lesser known elements such as P1 − PNC
1 .Use a stabilization scheme.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 6 / 13
![Page 13: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/13.jpg)
Intro SWE cG(1)cG(1) Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad forNon-Mathematicians).
Complicated to program.
Use packages such as FEniCS, FreeFEM, OpenFOAM, etc.
Spurious computational modes for certain finite element pairs.(similar problem with finite differences)
Use a different formulation of the problem, e.g. Vorticity-Streamfunction form.Use Taylor-Hood or lesser known elements such as P1 − PNC
1 .Use a stabilization scheme.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 6 / 13
![Page 14: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/14.jpg)
Intro SWE cG(1)cG(1) Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad forNon-Mathematicians).
Complicated to program.
Use packages such as FEniCS, FreeFEM, OpenFOAM, etc.
Spurious computational modes for certain finite element pairs.(similar problem with finite differences)
Use a different formulation of the problem, e.g. Vorticity-Streamfunction form.Use Taylor-Hood or lesser known elements such as P1 − PNC
1 .Use a stabilization scheme.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 6 / 13
![Page 15: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/15.jpg)
Intro SWE cG(1)cG(1) Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad forNon-Mathematicians).
Complicated to program.
Use packages such as FEniCS, FreeFEM, OpenFOAM, etc.
Spurious computational modes for certain finite element pairs.(similar problem with finite differences)
Use a different formulation of the problem, e.g. Vorticity-Streamfunction form.Use Taylor-Hood or lesser known elements such as P1 − PNC
1 .Use a stabilization scheme.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 6 / 13
![Page 16: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/16.jpg)
Intro SWE cG(1)cG(1) Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad forNon-Mathematicians).
Complicated to program.
Use packages such as FEniCS, FreeFEM, OpenFOAM, etc.
Spurious computational modes for certain finite element pairs.(similar problem with finite differences)
Use a different formulation of the problem, e.g. Vorticity-Streamfunction form.Use Taylor-Hood or lesser known elements such as P1 − PNC
1 .Use a stabilization scheme.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 6 / 13
![Page 17: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/17.jpg)
Intro SWE cG(1)cG(1) Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad forNon-Mathematicians).
Complicated to program.
Use packages such as FEniCS, FreeFEM, OpenFOAM, etc.
Spurious computational modes for certain finite element pairs.(similar problem with finite differences)
Use a different formulation of the problem, e.g. Vorticity-Streamfunction form.Use Taylor-Hood or lesser known elements such as P1 − PNC
1 .Use a stabilization scheme.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 6 / 13
![Page 18: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/18.jpg)
Intro SWE cG(1)cG(1) Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad forNon-Mathematicians).
Complicated to program.
Use packages such as FEniCS, FreeFEM, OpenFOAM, etc.
Spurious computational modes for certain finite element pairs.(similar problem with finite differences)
Use a different formulation of the problem, e.g. Vorticity-Streamfunction form.Use Taylor-Hood or lesser known elements such as P1 − PNC
1 .Use a stabilization scheme.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 6 / 13
![Page 19: A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element Discretization of the Shallow Water Equations of the Ocean Erich L Foster 13 January](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a8d0e1273b7c56fa5b7263/html5/thumbnails/19.jpg)
Intro SWE cG(1)cG(1) Tests
cG(1)cG(1) Finite Element
Spatial Discretization
Trial Functions - Piecewise linearTest Functions - Piecewise linear
Temporal Discretization
Trial Functions - Piecewise linearTest Functions - Piecewise constant
Weighted least squares stabilization
E. L. Foster (BCAM) cG(1)cG(1) for SWE 7 / 13
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Intro SWE cG(1)cG(1) Tests
Discretization of SWE
k−1n (un − un−1,v) +Ro−1(u⊥,v)− Fr−2Θ (η,∇ · v)
+ k−1n (ηn − ηn−1, χ) +H(∇ · u, χ)
+ δ1(R1(unh, η
nh), R1(v, χ))
+ δ2(R2(unh, η
nh), R2(v, χ))
(3)
where
unh =1
2(unh + un−1h ), ηnh =
1
2(ηnh + ηn+1
h )
and
R1(v, χ) = (u · ∇) v +Ro−1v⊥ + Fr−2Θ∇χR2(v, χ) = Θ−1∇ · v
are the linearized strong residuals while
δ1 =RoFr2 Θ−1
2(k−2n + |un|2h−2n )−1/2, δ2 =
Θ
2(k−2n + |ηn|2h−2n )−1/2.
E. L. Foster (BCAM) cG(1)cG(1) for SWE 8 / 13
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Intro SWE cG(1)cG(1) Tests
Linear Inviscid SWE
Compare the standard P1 − P1 finite element pair to cG(1)cG(1)applied to the Linear Inviscid SWE, i.e.
ηt + Θ−1H∇ · u = 0
ut +Ro−1u⊥ + Fr−2Θ∇η = 0on Ω (4)
u · n = 0 on δΩ (5)
Ro = 0.1Fr = 0.1Θ = 1H = 1.63Initial Condition:
u0 = 0
η0 = Ae−(x20+x
21)/(2∗σ2),
A = 1.0, σ = 5× 10−2
(6)
E. L. Foster (BCAM) cG(1)cG(1) for SWE 9 / 13
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Intro SWE cG(1)cG(1) Tests
Simulated Gaussian Drop for Linear Inviscid SWE, HeightLeft:P1 − P1, Right: cG(1)cG(1)
E. L. Foster (BCAM) cG(1)cG(1) for SWE 10 / 13
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Intro SWE cG(1)cG(1) Tests
Flow Around an Island
Compare the standard P1 − P1 finite element pair to cG(1)cG(1)
Re = 1 000
Ro = 0.1
Fr = 0.1
Θ = 1
H = 1.63
η = 1 at inflow and η = 0 at outflow.
(u0, η0) = (0, 0)
E. L. Foster (BCAM) cG(1)cG(1) for SWE 11 / 13
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Intro SWE cG(1)cG(1) Tests
Simulated flow around an Island for SWE, VelocityTop:P1 − P1, Bottom: cG(1)cG(1)
E. L. Foster (BCAM) cG(1)cG(1) for SWE 12 / 13
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Intro SWE cG(1)cG(1) Tests
Questions?
E. L. Foster (BCAM) cG(1)cG(1) for SWE 13 / 13