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Partitioned methods for time-dependent thermalfluid-structure interaction
Azahar MongePhilipp Birken
Lund University
DeustoTech, Bilbao, January 17th, 2019
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 1 / 24
Outline
1 A fast solver for thermal FSI
2 Limitations of the Dirichlet-Neumann method
3 A multirate approach
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 2 / 24
Outline
1 A fast solver for thermal FSI
2 Limitations of the Dirichlet-Neumann method
3 A multirate approach
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 3 / 24
Applications of thermal FSI
Figure: Left: Vulcain engine for Ariane 5. Right: Sketch of the cooling system;Pline, Wikimedia Commons
Thermal interaction between fluid and structure needs to be modelled
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 4 / 24
Dirichlet–Neumann coupling
Partitioned approach for the solution of the coupled problem.
Fluid Model : Compressible Navier–Stokes - FVM, DLR-TAU-Code
Structure Model : Nonlinear heat equation - FEM, NATIVE inhousecode
Θk+1Γ = S(F(Θk
Γ)), Interface Temperature Iteration
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 5 / 24
Outline
1 A fast solver for thermal FSI
2 Limitations of the Dirichlet-Neumann method
3 A multirate approach
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 6 / 24
Limitations of the Dirichlet–Neumann method
Convergence rates FSI application
log( "t)10 -8 10 -6 10 -4 10 -2 10 0 10 2
Con
v. R
ate
10 -8
10 -6
10 -4
10 -2
10 0
10 2
CR Plate1D | '|-
/
Limitations
The subsolvers are sequential.
Same time integration for bothfields.
Use a different method!
More at: A. Monge, P. Birken, Computational Mechanics, 2017
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 7 / 24
Limitations of the Dirichlet–Neumann method
List of wishes
Two independent time integration schemes.
High order resolution (at least 2nd order).
To be able to insert time adaptivity in the framework.
Option 1
Exchange fixed point iteration withtime recursion
Option 2
Use a different domaindecomposition method
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 8 / 24
Limitations of the Dirichlet–Neumann method
Option 1
Dirichlet-Neumann WaveformRelaxation (DNWR) algorithm
+ Computationally cheap
– Sequential method
Option 2
Neumann-Neumann WaveformRelaxation (NNWR) algorithm
+ Parallel method
– Computationally expensive
DNWR and NNWR introduced by Gander and Kwok 2016: Constantcoefficients and one single time integration scheme.
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 9 / 24
Outline
1 A fast solver for thermal FSI
2 Limitations of the Dirichlet-Neumann method
3 A multirate approach
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 10 / 24
Model Problem: Coupled heat equations
αm∂um(x, t)
∂t−∇ · (λm∇um(x, t)) = 0,
t ∈ [t0, tf ], x ∈ Ωm ⊂ Rd , m = 1, 2
um(x, t) = 0, x ∈ ∂Ωm\Γu1(x, t) = u2(x, t), x ∈ Γ
λ2∂u2(x, t)
∂n2= −λ1
∂u1(x, t)
∂n1, x ∈ Γ
um(x, 0) = gm(x) x ∈ Ωm
where αm = λm/Dm.
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 11 / 24
Dirichlet-Neumann waveform relaxation (DNWR)
(D)
α1
∂uk+11 (x,t)∂t −∇ · (λ1∇uk+1
1 (x, t)) = 0, x ∈ Ω1,
uk+11 (x, t) = 0, x ∈ ∂Ω1\Γ,
uk+11 (x, t) = gk(x, t), x ∈ Γ,
uk+11 (x, 0) = u0
1(x), x ∈ Ω1.
(N)
α2
∂uk+12 (x,t)∂t −∇ · (λ2∇uk+1
2 (x, t)) = 0, x ∈ Ω2,
uk+12 (x, t) = 0, x ∈ ∂Ω2\Γ,λ2
∂uk+12 (x,t)∂n2
= −λ1∂uk+1
1 (x,t)∂n1
, x ∈ Γ,
uk+11 (x, 0) = u0
1(x), x ∈ Ω1.
(U) gk+1(x, t) = Θuk+12 (x, t) + (1−Θ)gk(x, t), x ∈ Γ.
How to choose the relaxation parameter Θ properly?
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 12 / 24
Neumann-Neumann waveform relaxation (NNWR)
(Dm),m = 1, 2
αm
∂uk+1m (x,t)∂t −∇ · (λm∇uk+1
m (x, t)) = 0, x ∈ Ωm,
uk+1m (x, t) = 0, x ∈ ∂Ωm\Γ,
uk+1m (x, t) = gk(x, t), x ∈ Γ,
uk+1m (x, 0) = u0
1(x), x ∈ Ωm.
(Nm),m = 1, 2
αm
∂ψk+1m (x,t)∂t −∇ · (λm∇ψk+1
m (x, t)) = 0, x ∈ Ωm,
ψk+1m (x, t) = 0, x ∈ ∂Ωm\Γ,λm
∂ψk+1m (x,t)∂n1
= λ1∂uk+1
1 (x,t)∂n1
+ λ2∂uk+1
2 (x,t)∂n2
, x ∈ Γ,
ψk+1m (x, 0) = 0, x ∈ Ωm.
(U) gk+1(x, t) = gk(x, t)−Θ(ψk+11 (x, t) + ψk+1
2 (x, t)), x ∈ Γ.
How to choose the relaxation parameter Θ properly?
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 13 / 24
Choices
Space discretization: 1D and 2D finite elementsTime discretization: Implicit Euler and SDIRK2Matching space grid at the interface, unknowns on interfaceNonmatching time grids at the interface, linear interpolation
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 14 / 24
Multirate 1D solution
Iteration 1
0.5
t02
1x
0
0.8
1
0.6
0.4
0.2
0
uh1(x
,t)
Iteration 3
0.5
t02
1x
00
1
0.8
0.6
0.4
0.2
uh3(x
,t)Iteration 2
0.5
t02
1x
0
0.8
1
0.6
0.4
0.2
0
uh2(x
,t)
Iteration 20
0.5
t02
1x
0
1
0.8
0.6
0.4
0.2
0
uh20
(x,t)
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 15 / 24
Relaxation parameter
Q: How to choose the relaxation parameter Θ?
A: Find Θ s.t. minimizes the spectral radius of Σ w.r.t uΓ(Tf ),
uk+1,TfΓ = Σuk,Tf
Γ +2∑
i=2
(ψk+1,τi + ψk,τi
).
Procedure to find the iteration matrix
1 Isolate u(m),k+1,Tf
I from the Dirichlet solver.
2 Isolate u(2),k+1,Tf
I or ψ(m),k+1,Tf
Γ from the Neumann solver.
3 Use the update step to get Σ w.r.t uΓ(Tf ).
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 16 / 24
Iteration Matrix
After lengthly derivations one gets,
DNWR Algorithm
Σ = I−Θ(I + S(2)−1
S(1)),
NNWR Algorithm
Σ = I−Θ(
2I + S(1)−1S(2) + S(2)−1
S(1)),
with
S(m) = (M(m)ΓΓ + ∆tA
(m)ΓΓ )− (M
(m)ΓI + ∆tA
(m)ΓI )(Mm + ∆tAm)−1(M
(m)IΓ + ∆tA
(m)IΓ )
for m = 1, 2.
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 17 / 24
Iteration Matrix
A closer look at the iteration matrix Σ:
Problem: Matrices Mm + ∆tAm are sparse but (Mm + ∆tAm)−1 aredense.
Solution: Use their eigendecompositions to compute the desiredentries of the Toeplitz matrices: (Mm + ∆tAm)−1 = VΛ−1
m V.
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 18 / 24
Optimal Relaxation Parameter
Space discretization: 1D equidistant FE/FE.
Time integration: nonmultirate Implicit Euler.
DNWR Algorithm
Θopt =
(1 +
S(1)
S(2)
)−1
,
NNWR Algorithm
Θopt =
(2 +
S(1)
S(2)+
S(2)
S(1)
)−1
,
with
S(m) = (6∆x(αm∆x2 + 3λm∆t)− (αm∆x2 − 6λm∆t)2sm).
Θopt gives the optimal parameter for any coupled materials!
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 19 / 24
Comparison DNWR and NNWR
1 2 3 4 5 6 7 8 9 10
10−9
10−8
10−7
10−6
10−5
10−4
tol
number of iterations k
resi
dual
=m
ax|u
k−1
Γ−
uk Γ| DN, MR5/5
NN, MR5/5DN, MR5/2NN, MR5/2
Nonmultirate: NNWR performs half the iterations than DNWR.
Nonmultirate: DNWR uses half the resources than NNWR.
Multirate: DNWR performs better than NNWR.
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 20 / 24
Convergence Rates: Air-Water coupling
DNWR (1D case)
0 0.2 0.4 0.6 0.8 110
−4
10−3
10−2
10−1
100
101
Θ
Con
v. R
ates
Σ(Θ)IE (C−C)IE (C−F)IE (F−C)
NNWR (1D case)
0 0.002 0.004 0.006 0.008 0.01
10−10
10−5
100
ΘC
onv.
Rat
es
Σ(Θ)IE (C−C)IE (C−F)
Θopt is more sensitive for NNWR than for DNWR.
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 21 / 24
Convergence Rates: NNWR in 2D
Air-Steel
0 0.5 1 1.5 2
x 10−3
10−4
10−3
10−2
10−1
100
101
Θ
Con
v. R
ates
Σ(Θ)IE (C−C)IE (C−F)SDIRK2 (C−C)SDIRK2 (C−F)
Air-Water
0 0.2 0.4 0.6 0.8 1
x 10−3
10−4
10−2
100
Θ
Con
v. R
ates
Σ(Θ)IE (C−C)IE (C−F)SDIRK2 (C−C)SDIRK2 (C−F)
The 1D Θopt is very good for 2D examples.
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 22 / 24
Summary and Further Work
Multirate parallel method for coupled parabolic problems (NNWR).
Multirate sequential method for coupled parabolic problems (DNWR).
We have performed a 1D analysis to find Θopt .
Θopt is dependent on ∆t, ∆x , λm and αm, m = 1, 2.
Θopt is more sensitive for NNWR than for DNWR.
Θopt works for 2D, multirate and time adaptivity.
Investigate the time adaptive extension.
Apply to FSI test cases.
Load balancing.
More at: A. Monge, P. Birken, arXiv:1805.04336
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 23 / 24
Thank you!
Azahar Monge (Lund University) Partitioned methods for thermal FSI October 10th, 2018 24 / 24
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