fluid structure interaction simulation in marine renewable ...€¦ · fluid structure interaction...

Fluid structure interaction simulation in marinerenewable energy

Barbel HolmComputational Science and TechnologyKTH Royal Institute of Technology, Stockholm, Sweden

Havsenergiforum 2016, Smogen Hafvsbad

Overview

Motivation

Discretization

Model problem

Locally modified finite element method

Analysis of the locally modified finite element method

Numerical experiments

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Simulation of Fluid-Structure interaction problems

Ph.D project by Jeannette Spuhler.Simulation of blood flow in the human heart.

Large deformations of the structure.

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Simulation of Fluid-Structure interaction problems

Ph.D project by Jeannette Spuhler.Simulation of blood flow in the human heart.

Large deformations of the structure.

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Simulation of turbines

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Rotation of structures

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Rotation of structures

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Rotation of structures

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Rotation of structures

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Rotation of structures

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Rotation of structures

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Rotation of structures

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Overview

Motivation

Discretization

Model problem

Locally modified finite element method

Analysis of the locally modified finite element method

Numerical experiments

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Discretization of the domain

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Discretization of the domain

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Remeshing

Create a new mesh in each step.

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Remeshing

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Remeshing

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Remeshing

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Remeshing

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Remeshing

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Remeshing

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Arbitrary Lagangian Eulerian (ALE)

Move the existing vertices at the interface in each step.

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ALE

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ALE

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ALE

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ALE

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ALE – cells get distorted

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Nonconforming mesh

Use the same mesh in each step.Do not align the mesh with the interface.Allow cells to be cut.

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Nonconforming mesh

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Nonconforming mesh

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Nonconforming mesh

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Nonconforming mesh

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Nonconforming mesh

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Nonconforming mesh

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Nonconforming mesh

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Overview

Motivation

Discretization

Model problem

Locally modified finite element method

Analysis of the locally modified finite element method

Numerical experiments

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Goal

Fixed mesh, flexible interface

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Flexible interface

Model problem

−∇ · (κi∇u) = 1 on Ωi , i = 1, 2, [u] = 0, [κ∂nu] = 0 on I,

depending on diffusion parameters κi . By

[u](x) := lims↓0

u(x + sn)− lims↑0

u(x + sn), x ∈ I

we denote the jump at the interface.

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Exact solution

0

1

0

1

0

0.04

0.08

u(x) = 1/κ(1/16− 1/4‖x‖), κ =

1, ‖x‖ < 1/4,

1/10, else.

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Exact solution

u(x) = 1/κ(1/16− 1/4‖x‖), κ =

1, ‖x‖ < 1/4,

1/10, else.

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Suboptimal convergence

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1

O(h1/2)

O(h)

∥∥∇(u − uh)∥∥

‖u − uh‖

mesh size h

P1-elements

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Suboptimal convergence - even with higher orderelements

0.00001

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1

O(h1/2)

O(h)

∥∥∇(u − uh)∥∥

‖u − uh‖

mesh size hP1-elementsP2-elements

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Bad news

If the interface I cannot be resolved by the mesh, the overall errorfor a standard finite element ansatz will be 1, 2∥∥∇(u − uh)

∥∥ = O(h1/2).

1I. Babuska “The finite element method for elliptic equations withdiscontinuous coefficients” Computing 5, No. 3, pp. 207–213, 1970.

2R.J. Mackinnon, G.F. Carey “Treatment of material discontinuities in finiteelement computations” International Journal for Numerical Methods inEngineering 24, No. 2, pp. 393–417, 1987.

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Representation of the interface II Overlapping mesh methods.I XFEM methods.I Cut cells.

I Introduction of new local degrees of freedom.I Problems for load balancing.

I CutFEM.I Additional terms as in DG methods.I Choice of stabilization parameters.

I Local interpretation of degrees of freedom and localquadrature. 3

I Number of degrees of freedom stay the same.I Connectivity of the matrix is kept.

I Many more.3S. Frei, T. Richter “A locally modified parametric finite element method

for interface problems” Journal on Numerical Analysis 52, No. 5, pp.2315–2334, 2014.

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Overview

Motivation

Discretization

Model problem

Locally modified finite element method

Analysis of the locally modified finite element method

Numerical experiments

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Interface through the domain

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Piecewise linear approximation

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Identify subdomains

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Special treatment on cells at the interface

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Let’s focus on two cells at the interface

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Let’s focus on two cells at the interface

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Degrees of freedom

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Degrees of freedom

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Reinterpretation

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Cut through a vertex

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Reinterpretation if a vertex is cut

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Define modified finite element space

BP(xi ) = xPi , i = 1, . . . 6.

P = ϕ ∈ C (P), ϕ|Ti∈ span1, x , y,T1, . . . ,T4

Vh = ϕ ∈ C (Ω), ϕ B−1P |P ∈ P for all patches P ∈ Th

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Overview

Motivation

Discretization

Model problem

Locally modified finite element method

Analysis of the locally modified finite element method

Numerical experiments

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Do we get better convergence now?

A priori error analysis

For a Lagrangian interpolation operator

Lh : H2(T ) ∩ C (T )→ Vh

to satisfy ∥∥∥∇k(v − Lhv)∥∥∥T≤ ch2−k

T ,max

∥∥∥∇2v∥∥∥T

we need a maximum angle condition to be satisfied 4.

4T. Apel “Anisotropic Finite Elements: Local Estimates and Applications”Advances in Numerical Mathematics, Teubner, Stuttgart, 1999.

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Limit cases

Maximum angle condition cannot be satisfied.

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Modification to ensure maximum angle condition

s

q

r

I r and q determined by interface

I s free to choose

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Modification to ensure maximum angle condition

s

q

r

I q determined by interface

I s and r free to choose

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Maximum angle condition

LemmaWith the modification, all angles of the triangles are bounded by135 independent of r , s, q ∈ (0, 1).

Proof

s 1− s

1−r

r

q1−

q

γ

βα

s 1− s

1−r

q1−

q

rγ1

α2

γ2

β2

α1

β1

s 1− s

1−r

r

q1−

q

γ4β4

α4β3α3

γ3

Estimation of the angles by basic geometric analysis depending onthe parameters r , s, q ∈ (0, 1).

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A priori error estimate

TheoremLet Ω ∈ R2 be a domain with convex polygonal boundary. Weassume that the interface I admits a C 2-parametrization and thatit splits the domain into Ω = Ω1 ∪ I ∪ Ω2 such that the solutionu ∈ H1

0 (Ω) satisfies a stability estimate

u ∈ H10 (Ω) ∩ H2(Ω1 ∪ Ω2), ‖u‖H2(Ω1∪Ω2) ≤ cs‖f ‖ .

Then the estimate for the modified finite element solution uh ∈ Vh∥∥∇(u − uh)∥∥

Ω≤ Ch‖f ‖ , ‖u − uh‖Ω ≤ Ch2‖f ‖

holds.

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A priori error estimate

TheoremLet Ω ∈ R2 be a domain with convex polygonal boundary. Weassume that the interface I admits a C 2-parametrization and thatit splits the domain into Ω = Ω1 ∪ I ∪ Ω2 such that the solutionu ∈ H1

0 (Ω) satisfies a stability estimate

u ∈ H10 (Ω) ∩ H2(Ω1 ∪ Ω2), ‖u‖H2(Ω1∪Ω2) ≤ cs‖f ‖ .

Then the estimate for the modified finite element solution uh ∈ Vh∥∥∇(u − uh)∥∥

Ω≤ Ch‖f ‖ , ‖u − uh‖Ω ≤ Ch2‖f ‖

holds.

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A priori error estimate – sketch of the proof

Sketch of the proof

I Derive a perturbed bestapproximation property

‖∇eh‖2 ≤c‖∇eh‖∥∥∇(u − Lhu)

∥∥+

2∑i=1

‖δκi∇uh‖Ωi\Ti,h

∥∥∇(Lhu − uh)∥∥

Ωi\Ti,h.

I Derive estimation for∥∥∇(u − Lhu)

∥∥.

I Apply duality argument to derive a bound for the error inL2(Ω).

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Overview

Motivation

Discretization

Model problem

Locally modified finite element method

Analysis of the locally modified finite element method

Numerical experiments

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Circular cut

The cells are only cut for visualization purposes.

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Circular cut

The cells are only cut for visualization purposes.

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Circular cut

The cells are only cut for visualization purposes.

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Optimal convergence in L2(Ω)

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

0.001 0.01 0.1 1

‖u−uh‖ L

2(Ω

)

mesh size h

O(h)

O(h2)

mod P1-elementsP1-elementsP2-elements

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Optimal convergence in H1(Ω)

10−4

10−3

10−2

10−1

100

0.001 0.01 0.1 1

∥ ∥ ∇(u−uh)∥ ∥ L

2(Ω

)

mesh size h

O(h1/2)

O(h)

mod P1-elementsP1-elementsP2-elements

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Future work

Extension to tetrahedra.

Time dependent problems.Parallelization.

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Thank you very much for listening

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