Rome, July 3, 2018 ; D - francescobonaldi.weebly.com · [Fischer and Gaul, 2005]: FEM{BEM coupling,...

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


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


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 ÀUptq Fptq, t P p0, T s,

Up0q U0


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]


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,vhqeÀehpuhptq,vhq À

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


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


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


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


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


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


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.


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.


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


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 ÀUptq Fptq, t P p0, T s,

Up0q U0.


fd@rm@0:

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