A discontinuous Galerkin approach to theelasto-acoustic problem on polytopic grids
Francesco Bonaldijoint work with Paola F. Antonietti and Ilario Mazzieri
MOX, Dipartimento di Matematica
Politecnico di Milano
Rome, July 3, 2018
Discontinuous Galerkin Methods
Paola F. Antonietti
MOX, Dipartimento di MatematicaPolitecnico di Milano
International Doctoral School Gran Sasso Science Institute (GSSI), LAquila
22-25 February 2016
Paola F. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 1 / 48
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018
Motivations
Source: https://www.ictsmarhis.com/
Earthquake scenarios near coastal environments
Coupling of elastic and acoustic wave propagation
Requirements on the numerical scheme
Mesh flexibility
High-order accuracy
Suited to HPC techniques
Goal
Numerical treatment based on polytopic meshes
The dG method is well-suited to such meshes
UNSTRUCTURED GRID
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018 1 / 13
State-of-the-art
Minimal bibliography
[Komatitsch et al., 2000]: Spectral Elements
[Fischer and Gaul, 2005]: FEMBEM coupling, Lagrange multipliers
[Flemisch et al., 2006]: classical FEM on two independent meshes
[Brunner et al., 2009]: FEMBEM comparison
[Barucq et al., 2014]: Frechet differentiability of the elasto-acoustic field
[Barucq et al., 2014]: dG on simplices, curved edges on interface
[Peron, 2014]: asymptotic study, equivalent boundary conditions
[De Basabe and Sen, 2015]: Spectral Elements and Finite Differences
[Monkola, 2016]: Spectral Elements, different formulations
Our contribution
Well-posedness of the coupled problem in the continuous setting
Detailed analysis of a dG scheme on general polytopic meshes
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018 2 / 13
Elasto-acoustic coupling
Governing equations
$
&
%
e :u` 2e 9u` e2u divpuq fe in e p0, T s,puq Cpuq 0 in e p0, T s,
u 0 on eD p0, T s,
puqne a 9ne on I p0, T s,
up0q u0, 9up0q u1 in e,
c2 :4 fa in a p0, T s,
0 on aD p0, T s,
B{Bna 9una on I p0, T s,
p0q 0, 9p0q 1 in a
p Acoustic pressure exerted by the fluid onto the elastic body
p The normal component of va is continuous at I
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018 3 / 13
Well-posedness
Theorem (Existence and uniqueness)
Under suitable regularity hypotheses on initial data and source terms, there is a uniquestrong solution s.t.
u P C2pr0, T s;L2peqq X C1pr0, T s;H1Dpeqq X C0pr0, T s;H4C peq XH
1Dpeqq,
P C2pr0, T s;L2paqq X C1pr0, T s;H1Dpaqq X C0pr0, T s;H4paq XH1Dpaqq
H4C peq tv P L2peq : divCpvq P L2pequ,
H4paq tv P L2paq : 4v P L2paqu
Proof. Apply HilleYosida upon rewriting the system as
dUdtptq `AUptq Fptq, t P p0, T s,
Up0q U0
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Mesh
Assumptions
Nonconforming polytopic mesh Th T eh Y Tah
Arbitrary number of faces per element:
piq h d|F5 ||F |
, piiq
FBF5
Possible presence of degenerating faces
Consequences
Discrete trace inequality:
@ P Th, @v P Pppq, }v}L2pBq ph1{2 }v}L2pq
Approximation results in Pppq [Cangiani et al., 17]
[Antonietti et al., 17]
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Semi-discrete problem (SIP dG)
V eh tvh P L2peq : vh| P rPpe, pqsd, pe, 1 @ P T eh u,
V ah
h P L2paq : h| P Ppa, pq, pa, 1 @ P T ah(
Find puh, hq P C2pr0, T s;V eh q C2pr0, T s;V ah q s.t., for all pvh, hq P V
eh V
ah ,
pe :uhptq,vhqe ` pc2a :hptq, hqa ` p2e 9uhptq,vhqe ` pe
2uhptq,vhqe`Aehpuhptq,vhq `A
ahphptq, hq ` I
ehp 9hptq,vhq ` I
ahp 9uhptq, hq
pfeptq,vhqe ` pafaptq, hqa
Aehpu,vq pChpuq, hpvqqe xttChpuquu, rrvssyFeh
xrruss, ttChpvquuyFeh` xrruss, rrvssyFe
h@u,v P V eh ,
Aahp,q pah,hqa xttahuu, rrssyFah
xrrss, ttahuuyFah` xrrss, rrssyFa
h@, P V ah ,
Iehp,vq pane,vqI xane,vyFh,I @p,vq P Vah V
eh ,
Iahpv, q pavna, qI Iehp,vq @pv, q P V eh V ah
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018 6 / 13
Semi-discrete stability and error estimate
Define the following energy norm for W pv, q P C1pr0, T s;V eh q C1pr0, T s;V ah q:
}W ptq}2E }1{2e 9vptq}2e ` }
1{2e vptq}2e ` }vptq}
2dG,e ` }c
11{2a
9ptq}2a ` }ptq}2dG,a
Stability
For sufficiently large stabilization parameters,
}puhptq, hptqq}E }puhp0q, hp0qq}E ` t
0p}fepq}e ` }fapq}a qd
Energy-error estimate
Provided pu, q P C2pr0, T s;Hmpeqq C2pr0, T s;Hnpaqq, m pe ` 1, n pa ` 1,
suptPr0,T s
}peeptq, eaptqq}E CupT qhpe
pm3{2e
` CpT qhpa
pm3{2a
Proof. Properly use discrete trace inequality to bound interface contributions
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Numerical example I
Test case 1
We solve the elasto-acoustic problem one Y a, for T 1, t 104, for ahomogeneous isotropic elastic material, s.t.
upx, y; tq x2 cosp?
2tq cos
2x
sinpyq pu,
px, y; tq x2 sinp?
2tq sinpxq sinpyq,
with pu p1, 1q
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220.240.26
10-3
10-2
||u-uh||dG,e|| - h||dG,ahp
1 1.5 2 2.5 3 3.5 4 4.5 5
10-6
10-5
10-4
10-3
10-2
10-1||u-uh||dG,e|| - h||dG,a
}u uh}dG,e and } h}dG,a vs. h(top) and p (bottom) at T 1
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Numerical example II
Test case 2 [Monkola, 16]
We solve the elasto-acoustic problem one Y a, for T 0.8, t 104, for ahomogeneous isotropic elastic material, s.t.
upx, y; tq
cos4x
cp
, cos4x
cs
cosp4tq,
px, y; tq sinp4xq sinp4tq,
cp
d
` 2e
, cs c
e
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220.240.2610-3
10-2
10-1
100 ||u-uh||dG,e|| - h||dG,ahp
1 1.5 2 2.5 3 3.5 4 4.5 5
10-5
10-4
10-3
10-2
10-1
100||u-uh||dG,e|| - h||dG,a
}u uh}dG,e and } h}dG,a vs. h(top) and p (bottom) at T 0.8
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Physical example
t }upx, y; tq}2 and t |px, y; tq|
Physical example
Point seismic source in the acoustic domain:
fapx, tq 2`
1 2pt t0q2
eptt0q2px x0q,
x0 P a, t0 P p0, T s,
x0 p0.2, 0.5q, t0 0.1
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Conclusions & perspectives
Conclusions
We proved that the elasto-acoustic problem is well-posed in the continuous setting
We proved and validated hp-convergence for a dG method on polytopic meshes
We used the method to simulate an example of physical interest
Perspectives
Simulating 3D scenarios, using SPEED (http://speed.mox.polimi.it/)
Considering the case of totally absorbing boundary conditions
Inferring error estimates for the fully discrete problem
Enriching the model by considering a viscoelastic material response:
pupx, tq; tq Cpx, 0qpupx, tqq t
0
BCBspx, t sqpupx, sqqds
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http://speed.mox.polimi.it/
References I
P. F. Antonietti, F. Bonaldi, and I. Mazzieri.
A high-order discontinuous Galerkin approach to the elasto-acoustic problem.
Preprint arXiv:1803.01351 [math.NA], submitted, 2018.
A. Cangiani, Z. Dong, E. H. Georgoulis, and P. Houston.
hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes.
SpringerBriefs in Mathematics, Springer International Publishing, 2017.
P. F. Antonietti, P. Houston, X. Hu, M. Sarti, and M. Verani.
Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methodson polygonal and polyhedral meshes.
Calcolo, 54 (2017), pp. 11691198.
S. Monkola.
On the accuracy and efficiency of transient spectral element models for seismicwave problems.
Adv. Math. Phys., (2016).
J. D. De Basabe and M. K. Sen.
A comparison of finite-difference and spectral-element methods for elastic wavepropagation in media with a fluid-solid interface.
Geophysical Journal International, 200 (2015), pp. 278298.
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018 12 / 13
https://arxiv.org/abs/1803.01351
References II
H. Barucq, R. Djellouli, and E. Estecahandy.
Characterization of the Frechet derivative of the elasto-acoustic field with respectto Lipschitz domains.
J. Inverse Ill-Posed Probl., 22 (2014), pp. 18.
H. Barucq, R. Djellouli, and E. Estecahandy.
Efficient dG-like formulation equipped with curved boundary edges for solvingelasto-acoustic scattering problems.
Int. J. Numer. Meth. Engng, 98 (2014), pp. 747780.
V. Peron.
Equivalent boundary conditions for an elasto-acoustic problem set in a domainwith a thin layer.
ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 14311449.
B. Flemisch, M. Kaltenbacher, and B. I. Wohlmuth.
Elastoacoustic and acousticacoustic coupling on non-matching grids.
Int. J. Numer. Meth. Engng, 67 (2006), pp. 17911810.
D. Komatitsch, C. Barnes, and J. Tromp.
Wave propagation near a fluid-solid interface: a spectral-element approach,
Geophysics, 65 (2000), pp. 623631.
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018 13 / 13
Thanks for your attention
Discontinuous Galerkin Methods
Paola F. Antonietti
MOX, Dipartimento di MatematicaPolitecnico di Milano
International Doctoral School Gran Sasso Science Institute (GSSI), LAquila
22-25 February 2016
Paola F. Antonietti (MOX-PoliMi) Discontinuous Galerkin Methods GSSI Phd School 1 / 48
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018
Application of HilleYosida
Let w 9u, 9, and U pu,w, , q. We introduce
H H1Dpeq L2peq H1Dpaq L
2paq,
with scalar product
pU1,U2qH pe2u1,u2qe ` pCpu1q, pu2qqe` pew1,w2qe ` pa1,2qa ` pc
2a1, 2qa .
Then, we define the operator A : DpAq H H by
AU `
w, 2w ` 2u 1e divCpuq, , c24
@U P DpAq,
DpAq !
U P H : u PH4C peq, w PH1Dpeq, P H
4paq, P H1Dpaq;
pCpuq ` aIqne 0 on I, p`wqna 0 on I)
.
Finally, let F p0, 1e fe, 0, c2faq.
For F P C1pr0, T s;Hq and U0 P DpAq,find U P C1pr0, T s;HqXC0pr0, T s;DpAqq s.t.
dUdtptq `AUptq Fptq, t P p0, T s,
Up0q U0.
F. Bonaldi (MOX,Polimi) SIMAI 2018 July 3, 2018
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