Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used...
Transcript of Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used...
![Page 1: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/1.jpg)
HAL Id: hal-01416241https://hal.inria.fr/hal-01416241
Submitted on 14 Dec 2016
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Trefftz-DG Approximation for the Elasto-AcousticsHélène Barucq, Henri Calandra, Julien Diaz, Elvira Shishenina
To cite this version:Hélène Barucq, Henri Calandra, Julien Diaz, Elvira Shishenina. Trefftz-DG Approximation for theElasto-Acoustics. Workshop de DIP, l’action stratégique INRIA TOTAL, Oct 2016, Houston, UnitedStates. hal-01416241
![Page 2: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/2.jpg)
Trefftz-DG Approximation for the
Elasto-Acoustics
DIP workshop
H.Barucq1, H.Calandra2, J.Diaz1, E.Shishenina1
December 14, 2016
1Magique-3D, Inria Bordeaux-Sud-Ouest, 2Total S.A.
![Page 3: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/3.jpg)
Abstract
![Page 4: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/4.jpg)
Seismic survey
Figure 1: Schematic of overall field setup for a seismic survey1.
1 Park Seismic LLC. http://www.parkseismic.com. Internet resource.
3
![Page 5: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/5.jpg)
Basic numerical methods
Table 1: Generic properties of the most widely used numerical methods2.
Numerical Complex High-order Explicit semi- Conservation Elliptic
method geometries accuracy and discrete form laws problems
hp-adaptivity
FDM • • • • •
FVM • • • •
FEM • • • •
DG-FEM • • • •
2 J.S.Hesthaven T.Warburton. Nodal discontinuous galerkin methods. Algorithms, analysis, and
applications. Texts in Applied Mathematics, (54):1-370, 2007. 4
![Page 6: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/6.jpg)
Trefftz method
DG-FEM :
Adapted to the complex geometries ;
High-order accuracy and hp-adaptivity ;
Explicit semi-discrete form ;
Conservation laws ;
Higher number of degrees of freedom
comparing to the methods with continuous approximation.
5
![Page 7: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/7.jpg)
Trefftz method
Trefftz method 3 :
Given a region of an Euclidean space of some partitions of that region, a
Trefftz Method is any procedure for solving boundary value problems of
partial differential equations or systems of such equations, on such
region, using solutions of that differential equation or its adjoint,
defined in its subregions.
3 I.Herrera. Trefftz method: a general theory. Instituto de Investigaciones en Matematicas
Aplicadas y en Sistemas (IIMAS), pages 562-580, 2000.
6
![Page 8: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/8.jpg)
Trefftz method
Time-harmonic formulations :
O.Cassenat, B.Despres (1998);
C.Farhat, I.Harari, U.Hetmaniuk (2003); G.Gabard (2007);
T.Huttunen, P.Monk, J.P.Kaipo (2002);
R.Tezaur, C.Farhat (2006);
Time-domain formulations :
F.Kretzschmar, A.Moiola, I.Perugia (2015);
F.Kretzschmar, S.M.Schnepp, I.Tsukerman, T.Weiland(2014);
H.Egger, F.Kretzschmar, S.M.Schnepp, T.Weiland (2014).
7
![Page 9: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/9.jpg)
Trefftz method
Expected advantages of Trefftz method :
Better orders of convergence ;
Flexibility in the choice of basis functions ;
Low dispersion ;
Incorporation of wave propagation directions in the discrete space ;
Adaptivity and local space-time mesh refinement.
3 I.Herrera. Trefftz method: a general theory. Instituto de Investigaciones en Matematicas
Aplicadas y en Sistemas (IIMAS), pages 562-580, 2000.
8
![Page 10: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/10.jpg)
Mathematical formulation.
Fluid case
![Page 11: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/11.jpg)
Acoustic system. Problem equation
ΩF ⊂ Rn - space domain ;
I = (0,T ) - time domain ;
QF := ΩF × I ;
nQF= (nxQF
, ntQF) - o.p. unit normal vector on ∂QF ;
Acoustic system
1
c2f ρf
∂p
∂t+ divvf = f , in QF ;
ρf∂vf∂t
+∇p = 0, in QF ;
vf (·, 0) = vf 0, p(·, 0) = p0, on ΩF ;
vf = gDf , on ∂ΩF × I .
10
![Page 12: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/12.jpg)
Acoustic system. Variational formulation
KF ⊂ QF (cf , ρf ≡ const. in KF ) ;
nKF= (nxKF
, ntKF) - o.p. normal vector on ∂KF ;
vf , p ∈ H1(KF ) ;
ωf , q ∈ H1(KF ) ;
Space-time integration
−∫KF
(vf · (ρf∂ωf
∂t+∇q) + p(
1
c2f ρf
∂q
∂t+ divωf ))dv+
+
∫∂KF
((pωf + vf q) · nxKF+ (
1
c2f ρf
pq + ρf vf · ωf )ntKF)ds =
∫∫KF
fqdv.
11
![Page 13: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/13.jpg)
Acoustic system. Space-time DG discretisation
Mesh and DG-notation
xxL xR
t
0
T
Th - mesh on QF := ΩF × I
Fh = ∪KF∈Th∂KF - mesh skeleton :
FΩFh := union of the internal Ω-like faces, (t ≡ const.) ;
F IFh := union of the internal I -like faces, (x ≡ const.) ;
F0Fh := ΩF × 0 ;
FTFh := ΩF × T ;
FDFh := ∂ΩF × [0,T ] - union of the Dirichlet boundary faces.
12
![Page 14: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/14.jpg)
Acoustic system. Space-time DG formulation
Seek (vf h, ph) ⊂ V(Th) ⊂ H1(Th)2, s.t. for all KF ∈ Th, (ωf , q) ⊂ V(Th)
it holds:
DG formulation
−∫KF
[ph(
1
c2f ρf
∂q
∂t+ divωf ) + vf h · (ρf
∂ωf
∂t+∇q)
]dv
+
∫∂KF
[(
1
c2f ρf
phq + ρf ˆvf h · ωf )ntKF+ (phωf + ˆvf hq) · nxKF
]ds =
∫KF
fqdv.
13
![Page 15: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/15.jpg)
Acoustic system. Trefftz-DG formulation
Trefftz space
TF (Th) :=
(ωf , q) ⊂ H1(Th)2, s. t. in all KF ∈ Th
ρf∂ωf
∂t+∇q = 0,
1
c2f ρf
∂q
∂t+ divωf = 0
.
14
![Page 16: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/16.jpg)
Acoustic system. Space-time Trefftz-DG formulation
Seek (vf h, ph) ⊂ V(Th) ⊂ H1(Th)2, s.t. for all KF ∈ Th, (ωf , q) ⊂ V(Th)
it holds:
Trefftz-DG formulation ( f ≡ 0 )
∫∂KF
[(
1
c2f ρf
phq + ρf ˆvf h · ωf )ntKF+ (phωf + ˆvf hq) · nxKF
]ds = 0
15
![Page 17: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/17.jpg)
Acoustic system. Space-time Trefftz-DG formulation
Numerical flux ˆvf h and ph :
on F IFh :
(ˆvf hph
):=
(vf h+ β[[ph]]x
ph+ α[[vf h]]x
),
on FΩFh :
(ˆvf hph
):=
(vf h−
ph−
),
on FTFh :
(ˆvf hph
):=
(vf hph
),
on F0Fh :
(ˆvf hph
):=
(vf 0
p0
),
on FDFh :
(ˆvf hph
):=
(ˆgDF
ph + α(vf h − gDF ) · nxKF
),
(α ∈ L∞(F IFh ∪ F
DFh ), β ∈ L∞(F IF
h ) - positive flux parameters)
16
![Page 18: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/18.jpg)
Acoustic system. Trefftz-DG formulation
Trefftz-DG formulation :
∫FΩF
h
[ 1
c2f ρf
ph−[[q]]t + ρf vf h
−[[ωf ]]t]ds
+
∫F IF
h
[ph[[ωf ]]x + vf h[[q]]x + α[[vf h]]x [[ωf ]]x + β[[ph]]x [[q]]x
]ds
+
∫FTF
h
[ 1
c2f ρf
phq + ρf vf h · ωf
]ds− 1
2
∫F0F
h
[ 1
c2f ρf
phq + ρf vf h · ωf
]ds
+
∫FDF
h
[ωf · (pnxKF
+ αvf h)]ds =
1
2
∫F0F
h
[ 1
c2f ρf
phq + ρf vf h · ωf
]ds +
∫FDF
h
[gDF
(αωf − q · nxKF)]ds.
17
![Page 19: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/19.jpg)
Acoustic system. Trefftz-DG formulation
Seek (vf h, ph) ⊂ V(Th) ⊂ TF (Th) s.t. for all (ωf , q) ⊂ V(Th) it holds:
Trefftz-DG formulation
ATDGF((vf h, ph); (ωf , q)) = `TDGF
(ωf , q).
18
![Page 20: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/20.jpg)
Acoustic system. Analysis of Trefftz-DG formulation
Trefftz-DG formulation in terms of L2(Th)-norms
ATDGF((ωf , q); (ωf , q)) :=
1
2
∥∥∥(1
c2f ρf
)1/2 [[q]]t
∥∥∥2
L2(FΩFh )
+1
2
∥∥∥ρ1/2
f [[ωf ]]t
∥∥∥2
L2(FΩFh )
+∥∥∥α1/2 [[ωf ]]x
∥∥∥2
L2(F IFh )
+∥∥∥β1/2 [[q]]x
∥∥∥2
L2(F IFh )
+1
2
∥∥∥(1
c2f ρf
)1/2q
∥∥∥2
L2(FTFh )
+1
2
∥∥∥ρ1/2
f ωf
∥∥∥2
L2(FTFh )
+∥∥∥α1/2ωf
∥∥∥2
L2(FDFh ).
19
![Page 21: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/21.jpg)
Acoustic system. Analysis of Trefftz-DG formulation
Norm ||| · |||TDGFin TF (Th)
|||(ωf , q)|||2TDGF:=
1
2
∥∥∥(1
c2f ρf
)1/2 [[q]]t
∥∥∥2
L2(FΩFh )
+1
2
∥∥∥ρ1/2
f [[ωf ]]t
∥∥∥2
L2(FΩFh )
+∥∥∥α1/2 [[ωf ]]x
∥∥∥2
L2(F IFh )
+∥∥∥β1/2 [[q]]x
∥∥∥2
L2(F IFh )
+1
2
∥∥∥(1
c2f ρf
)1/2q
∥∥∥2
L2(FTFh )
+1
2
∥∥∥ρ1/2
f ωf
∥∥∥2
L2(FTFh )
+∥∥∥α1/2ωf ‖2
L2(FDFh ).
20
![Page 22: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/22.jpg)
Acoustic system. Analysis of Trefftz-DG formulation
Coercivity of Trefftz-DG formulation
ATDGF((ωf , q); (ωf , q)) = |||(ωf , q)|||2TDGF
, ∀(ωf , q) ∈ TF (Th).
21
![Page 23: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/23.jpg)
Acoustic system. Analysis of Trefftz-DG formulation
Add-on norm ||| · |||TDG∗Fin TF (Th)
|||(ωf , q)|||2TDG∗F
:=|||(ωf , q)|||2TDGF
+‖ρ1/2
f ω−f ‖2
L2(FΩFh )
+ ‖( 1
c2f ρf
)1/2q−‖2
L2(FΩFh )
+‖β−1/2ωf ‖2
L2(F IFh )
+ ‖α−1/2q‖2
L2(F IFh )
+‖α−1/2ωf ‖2
L2(FDFh ).
22
![Page 24: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/24.jpg)
Acoustic system. Analysis of Trefftz-DG formulation
Continuity of Trefftz-DG formulation
|ATDGF((vf , p); (ωf , q))| ≤ 2 |||(vf , p)|||TDG∗
F|||(ωf , q)|||TDGF
,
|`TDGF(ωf , q)| ≤
√2[‖ρ
1/2
f vf 0‖2
L2(F0Fh )
+ ‖( 1
c2f ρf
)1/2p0‖2
L2(F0Fh )
]1/2
.
23
![Page 25: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/25.jpg)
Acoustic case. Analysis of Trefftz-DG formulation
Well-posedness
|||(vf − vf h, p − ph)|||TDGF≤ 3 inf
(ωf ,q)∈V(Th)|||(vf − ωf , p − q)|||TDG∗
F.
24
![Page 26: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/26.jpg)
Mathematical formulation.
Solid case
![Page 27: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/27.jpg)
Elastodynamic system. Problem equation
ΩS ⊂ Rn - space domain ;
I = (0,T ) - time domain ;
QS := ΩS × I ;
nQS= (nxQS
, ntQS) - o.p. unit normal vector on ∂QS ;
Elastodynamic system
∂σ
∂t− C ε(vs) = 0 in QS ;
ρs∂vs
∂t− divσ = 0 in QS ;
vs(·, 0) = vs 0, σ(·, 0) = σ0
on ΩS ;
vs = gDSon ∂ΩS × I .
26
![Page 28: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/28.jpg)
Elastodynamic system. Problem equation
ΩS ⊂ Rn - space domain ;
I = (0,T ) - time domain ;
QS := ΩS × I ;
nQS= (nxQS
, ntQS) - o.p. unit normal vector on ∂QS ;
Elastodynamic system
A∂ σ
∂t− ε(vs) = 0 in QS ;
ρs∂vs
∂t− divσ = 0 in QS ;
vs(·, 0) = vs 0, σ(·, 0) = σ0
on ΩS ;
vs = gDSon ∂ΩS × I .
27
![Page 29: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/29.jpg)
Elastodynamic system. Variational formulation
KS ⊂ QS (A, ρs ≡ const. in KS) ;
nKS= (nxKS
, ntKS) - o.p. unit normal vector on ∂KS ;
vs , σ ∈ H1(KS) ;
ωs , ξ ∈ H1(KS) ;
Space-time integration
−∫KS
[σ : (A
∂ ξ
∂t− ε(ωs)) + vs · (ρs
∂ωs
∂t− divξ)
]dv
+
∫∂KS
[(Aσ : ξ + ρsvs · ws) · ntKS
− (vs · ξ + σ · ωs) · nxKS
]ds = 0.
28
![Page 30: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/30.jpg)
Elastodynamic system. Space-time Trefftz-DG formulation
Trefftz space
TS(Th) :=
(ωs , ξ) ⊂ H1(Th)2, s. t. in all KS ∈ Th
ρs∂ωs
∂t− divξ = 0, A
∂ξ
∂t− ε(ωs) = 0
.
29
![Page 31: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/31.jpg)
Elastodynamic system. Space-time Trefftz-DG formulation
Seek (vs h, σh) ⊂ V(Th) ⊂ H1(Th)2, s.t. for all KS ∈ Th, (ωs , ξ) ⊂ V(Th)
it holds:
Space-time Trefftz-DG formulation
∫∂KS
[(A σ
h: ξ + ρs ˆvs h · ws) · ntK − ( ˆvs hξ + σ
hωs) · nxK
]ds = 0
30
![Page 32: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/32.jpg)
Elastodynamic system. Space-time Trefftz-DG formulation
Trefftz-DG formulation
∫FΩS
h
[Aσ
h− : [[ξ]]t + ρsvs h
−[[ωs ]]t]ds
−∫F IS
h
[σ
h[[ωs ]]x + vs h[[ξ]]x − γ[[vs h]]x [[ωs ]]x − δ[[σ
h]]x [[ξ]]x
]ds
+
∫FTS
h
[Aσ
h: ξ + ρsvs h · ωs
]ds− 1
2
∫F0S
h
[Aσ
h: ξ + ρsvs h · ωs
]ds
−∫FDS
h
[ξ(vs h · nxKS
− δσh)]ds =
1
2
∫F0S
h
[Aσ
h: ξ + ρsvs h · ωs
]ds +
∫FDS
h
[gDS
(ωs · nxKS+ δξ)
]ds.
31
![Page 33: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/33.jpg)
Elastodynamic system. Space-time Trefftz-DG formulation
Seek (vs h, σh) ⊂ V(Th) ⊂ TS(Th) s.t. for all (ωs , ξ) ⊂ V(Th) it holds:
Trefftz-DG formulation
ATDGS((vf h, σh
); (ωs , ξ)) = `TDGS(ωs , ξ).
32
![Page 34: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/34.jpg)
Elastodynamic system. Analysis of Trefftz-DG formulation
Norms in TS(Th)
|||(ωs , ξ)|||2TDGS:=
1
2
∥∥∥(A)1/2 [[ξ]]t
∥∥∥2
L2(FΩSh )
+1
2
∥∥∥ρ1/2s [[ωs ]]t
∥∥∥2
L2(FΩSh )
+∥∥∥γ1/2 [[ωs ]]x
∥∥∥2
L2(F ISh )
+∥∥∥δ1/2 [[ξ]]x
∥∥∥2
L2(F ISh )
+1
2
∥∥∥(A)1/2ξ∥∥∥2
L2(FTSh )
+1
2
∥∥∥ρ1/2s ωs
∥∥∥2
L2(FTSh )
+∥∥∥δ1/2ξ
∥∥∥2
L2(FDSh ).
33
![Page 35: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/35.jpg)
Elastodynamic system. Analysis of Trefftz-DG formulation
Norms in TS(Th)
|||(ωs , ξ)|||2TDG∗S
:=|||(ωs , ξ)|||2TDGS
+‖ρ1/2s ω−s ‖2
L2(FΩSh )
+ ‖(A)1/2ξ−‖2
L2(FΩSh )
+‖δ−1/2ωs‖2
L2(F ISh )
+ ‖γ−1/2ξ‖2
L2(F ISh )
+‖δ−1/2ξ‖2
L2(FDSh ).
34
![Page 36: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/36.jpg)
Elastodynamic system. Analysis of Trefftz-DG formulation
Coercivity of Trefftz-DG formulation
ATDGS((ωs , ξ); (ωs , ξ)) = |||(ωs , ξ)|||2TDGS
, ∀(ωs , ξ) ∈ TS(Th).
35
![Page 37: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/37.jpg)
Elastodynamic system. Analysis of Trefftz-DG formulation
Continuity of Trefftz-DG formulation
|ATDGS((vs , σ); (ωs , ξ))| ≤ 2 |||(vs , σ)|||TDG∗
S|||(ωs , ξ)|||TDGS
,
|`TDGS(ωs , ξ)| ≤
√2[‖ρ
1/2s vs0‖2
L2(F0Sh )
+ ‖A1/2σ
0‖2
L2(F0Sh )
]1/2
.
36
![Page 38: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/38.jpg)
Elastodynamic system. Analysis of Trefftz-DG formulation
Well-posedness
|||(vs − vs h, σ − σh)|||TDGS
≤ 3 inf(ωs ,ξ)∈V(Th)
|||(vs − ωs , σ − ξ)|||TDG∗S.
37
![Page 39: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/39.jpg)
Mathematical formulation.
Fluid-solid case
![Page 40: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/40.jpg)
Coupled system
Representation of the acoustic medium as a limit case of an
elastic isotropic medium with shear modulus µ
tending or equal to 0 ?
39
![Page 41: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/41.jpg)
Coupled system
Advantages of the numerical coupling:
Computing 1 unknown scalar pressure instead of 6 components
of stress tensor ;
Avoiding presence of the numerical artifacts caused of
the slow S-waves appearance.
40
![Page 42: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/42.jpg)
Coupled system. Problem equation
Acoustic system
1
c2f ρf
∂p
∂t+ divvf = f , in QF ;
ρf∂vf∂t
+∇p = 0, in QF ;
vf = gDF , on ∂ΩF × I ;
Elastodynamic system
A∂ σ
∂t− ε(vs) = 0 in QS ;
ρs∂vs∂t− divσ = 0 in QS ;
vs = gDS on ∂ΩS × I ;
41
![Page 43: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/43.jpg)
Coupled system. Problem equation
Fluid-solid transmission conditions through the ΓFS
vf · nxΩ = vs · nxΩ on ΓFS ;
σnxΩ = −pnxΩ on ΓFS .
42
![Page 44: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/44.jpg)
Coupled system. Space-time Trefftz-DG formulation
Trefftz space
T(Th) :=
(ωf , q, ωs , ξ) ⊂ H1(Th)4, s. t.
ρf∂ωf
∂t+∇q = 0,
1
c2f ρf
∂q
∂t+ divωf = 0, ∀KF ∈ Th,
ρs∂ωs
∂t− divξ = 0,
∂A ξ
∂t− ε(ωs) = 0, ∀KS ∈ Th
.
43
![Page 45: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/45.jpg)
Coupled system. Space-time Trefftz-DG formulation
Seek (vf h, ph, vs h, σh) ⊂ V(Th) ⊂ T(Th), s. t. for all KF ,KS ∈ Th,
(ωf , q, ωs , ξ) ⊂ V(Th) it holds :
Trefftz-DG formulation ( f ≡ 0 )
∫∂KF
[(
1
c2f ρf
phq + ρf ˆvf h · ωf )ntKF+ (phωf + ˆvf hq) · nxKF
]ds = 0,
∫∂KS
[(A σ
h: ξ + ρs ˆvs h · ws) · ntK − ( ˆvs hξ + σ
hωs) · nxK
]ds = 0
44
![Page 46: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/46.jpg)
Coupled system. Space-time DG formulation
Numerical flux ˆvf h, vsh, ph and σh
through F FSh :
on FFSh :
ˆvf h · nx
KF
ph
vs hσh· nx
KS
:=
vs h · nx
KF
p + α(vf h − vs h) · nxKF
vs h − δ(σh− ph) · nx
KS
−phnxKS
45
![Page 47: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/47.jpg)
Coupled system. Space-time Trefftz-DG formulation
Seek (vf h, ph, vs h, σh) ⊂ V(Th) ⊂ T(Th) s.t. ∀ (ωf , q, ωs , ξ) ⊂ V(Th) it
holds:
Trefftz-DG formulation
ATDG ((vf h, ph, vsh, σh); (ωf , q, ωs , ξ)) = `TDG (ωf , q, ωs , ξ)
46
![Page 48: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/48.jpg)
Coupled system. Analysis of Trefftz-DG formulation
Norms in T(Th)
|||(ωf , q, ωs , ξ))|||2TDG := |||(ωf , q)|||2TDGF+ |||(ωs , ξ)|||2TDGS
+ 2‖δ1/2ξ‖2
L2(FFSh
)
|||(ωf , q, ωs , ξ)|||2TDG∗ :=|||(ωf , q)|||2TDG∗F
+ |||(ωs , ξ)|||2TDG∗S
+1
2‖δ−
1/2ξ‖2L2(FFS
h)
47
![Page 49: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/49.jpg)
Coupled system. Analysis of Trefftz-DG formulation
Coercivity of Trefftz-DG formulation
ATDG ((ωf , q, ωs , ξ); (ωf , q, ωs , ξ)) = |||(ωf , q, ωs , ξ)|||2TDG
Continuity of Trefftz-DG formulation
|ATDG ((vf , p, vs , σ); (ωf , q, ωs , ξ))| ≤ 2 |||(vf , p, vs , σ)|||TDG∗ |||(ωf , qωs , ξ)|||TDG ,
|`TDG (ωf , q, ωs , ξ)| ≤√
2[‖ρ
1/2f vf 0‖2
L2(F0Fh
)+ ‖( 1
c2f ρf
)1/2p0‖2
L2(F0Fh
)
+ ‖ρ1/2s vs0‖2
L2(F0Sh
)+ ‖A
1/2σ0‖2
L2(F0Sh
)
]1/2
.
48
![Page 50: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/50.jpg)
Coupled system. Analysis of Trefftz-DG formulation
Well-posedness
|||(vf − vf h, p − ph,vs − vs h, σ − σh)|||TDG
≤ 3 inf(ωf ,q,ωs ,ξ)∈V(Th)
|||(vf − ωf , p − q, vs − ωs , σ − ξ)|||TDG∗ .
49
![Page 51: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/51.jpg)
Implementation of the algorithm.
Acoustic system
![Page 52: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/52.jpg)
Implementation of the algorithm
xxL xR
t
0
T
hx
ht KF
Figure 2: Rectangular mesh on QF := [xL, xR ]× [0,T ].
51
![Page 53: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/53.jpg)
Implementation of the algorithm
Nx
Nt j
i
1 2 · · · Nx
Nx + 1 · · · K K = (j − 1)Nx + i
Figure 3: Element numbering.
52
![Page 54: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/54.jpg)
Implementation of the algorithm
Seek (vf h, ph) ⊂ V(Th) ⊂ TF (Th) s.t. for all (ωf , q) ⊂ V(Th) it holds:
Treftz-DG formulation
∫FΩF
h
[ 1
c2f ρf
ph−[[q]]t + ρf vf h
−[[ωf ]]t]ds
+
∫F IF
h
[ph[[ωf ]]x + vf h[[q]]x + α[[vf h]]x [[ωf ]]x + β[[ph]]x [[q]]x
]ds
+
∫FTF
h
[ 1
c2f ρf
phq + ρf vf h · ωf
]ds =
∫F0F
h
[ 1
c2f ρf
p0q + ρf vf 0 · ωf
]ds.
53
![Page 55: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/55.jpg)
Implementation of the algorithm
Treftz-DG formulation. 1st layer
[vK−1 · Lvα + pK−1 · Lp
]+[vK · Cvα + pK · Cp
]+[vK+1 · Rvα + pK+1 · Rp
][vK−1 · Lv + pK−1 · Lpβ
]+[vK · Cv + pK · Cpβ
]+[vK+1 · Rv + pK+1 · Rpβ
]
Lvα
Lv
Lp
Lpβ
”Left” block
Cvα
CpβCv
Cp
”Central” block
Rvα
Rpβ
Rp
Rv
”Right” block
54
![Page 56: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/56.jpg)
Implementation of the algorithm
Treftz-DG formulation. 1st layer
[vK−1 · Lvα + pK−1 · Lp
]+[vK · Cvα + pK · Cp
]+[vK+1 · Rvα + pK+1 · Rp
][vK−1 · Lv + pK−1 · Lpβ
]+[vK · Cv + pK · Cpβ
]+[vK+1 · Rv + pK+1 · Rpβ
]
L
”Left” block
C
”Central” block
R
”Right” block
55
![Page 57: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/57.jpg)
Implementation of the algorithm
C R L· · ·
L C R
L C R. . .
...
L C R
...
R L C· · ·
·
U1
U2
U3
UN−2
UN−1
...=
U01
U02
U03
U0N−2
U0N−1
...
56
![Page 58: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/58.jpg)
Implementation of the algorithm
· =M
32Nx × 32Nx
U
1 × 32Nx
U0
1 × 32Nx
57
![Page 59: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/59.jpg)
Implementation of the algorithm
1.Initialization
invMM ; ;; U0
2.Propagation
For t = 1 : Nt
invM · U0=U ;
U=U0 ;
end
58
![Page 60: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/60.jpg)
Implementation of the algorithm
Figure 4: Exact and approximate
solution for velocity
vf (x , t), t = ∆t × 1, on the first
time-layer (α = β = 0.5).
Figure 5: Exact and approximate
solution for velocity
vf (x , t), t = ∆t × 1, on the first
time-layer (α = β = 0.5).
59
![Page 61: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/61.jpg)
Implementation of the algorithm
Figure 6: Time propagation of the
approximate solution for velocity
vf (x , t) (α = β = 0.5).
Figure 7: Time propagation of the
approximate solution for velocity
vf (x , t) (α = β = 0.5).
60
![Page 62: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/62.jpg)
Implementation of the algorithm
Figure 8: Convergence of numerical
solution for velocity vf (x , t) in
logarithmic scale (α = β = 0.5).
Figure 9: Convergence of numerical
solution for velocity vf (x , t) in
logarithmic scale (α = β = 0.5).
61
![Page 63: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/63.jpg)
Figure 10: Mesh for heterogeneous case (bilayer medium).
∆x1 = 0.01, ∆t1 = 0.01, c1 = 1.0, ∆x2 = 0.02, ∆t2 = 0.01, c1 = 2.0.
62
![Page 64: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/64.jpg)
Implementation of the algorithm
Figure 11: Approximate solution for
velosity v(x , t).
Figure 12: Approximate solution for
velocity v(x , t), t = ∆t × 51.
63
![Page 65: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/65.jpg)
Conclusion
![Page 66: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/66.jpg)
Conclusion
On-going work :
Numerical implementation of the method for 1D AS ;
Different choice of the basis functions ;
Analysis of efficiency of the method / algorithm / code.
65
![Page 67: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/67.jpg)
Conclusion
Perspectives :
Numerical implementation of the method for ES 1D,2D,3D ;
Numerical coupling 1D,2D,3D ;
Coupling Trefftz-DG in solid with FVM in fluid,
integrating of the transmission layer.
66
![Page 68: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/68.jpg)
References
J.S.Hesthaven T.Warburton. Nodal discontinuous galerkin methods. Algorithms,
analysis, and applications. Texts in Applied Mathematics, (54):1370, 2007.
F.Kretzschmar A.Moiola I.Perugia. A priori error analysis of space-time trefftz
discontinuous galerkin methods for wave problems. arXiv:1501.05253v2(math.NA),
2015.
O.C.Zienkiewicz. Trefftz type approximation and the generalized finite element
method - history and developement. Comp Ass Mech and Eng Sci, (4):305316, 1997.
I.Herrera. Trefftz method: a general theory. Instituto de Investigaciones en
Matematicas Aplicadas y en Sistemas (IIMAS), pages 562580, 2000.
A.Maciag. Wave polynomials in elasticity problems. Department of Mathematics,
Kielce University of Technology. Al.1000-lecia P.P.7, 2006
67
![Page 69: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/69.jpg)
Thank you for your attention!
68
![Page 70: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/70.jpg)
Questions?
69
![Page 71: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/71.jpg)
Implementation of the algorithm
Polynomial waves basis functions
• Generating functions g v (x , t) and gp(x , t) - solution of wave equation if
a2 = b2
g v (a, b, x , t) = e i(ax+bcf t), gp(a, b, x , t) = −cf e i(bx+acf t).
• Taylor expansions for g v (x , t), gp(x , t) (a2 = b2):
e i(ax+bcf t) =∞∑n=0
n∑k=0,k<2
Qvn−k,k(x , t)an−kbk ,
−cf e i(bx+acf t) =∞∑n=0
n∑k=0,k<2
Qpn−k,k(x , t)an−kbk .
Rvn−k,k(x , t) = <(Qv
n−k,k(x , t)), I vn−k,k(x , t) = =(Qvn−k,k(x , t))
Rpn−k,k(x , t) = <(Qp
n−k,k(x , t)), I pn−k,k(x , t) = =(Qpn−k,k(x , t)).
70
![Page 72: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/72.jpg)
Implementation of the algorithm
Polynomial waves basis functions 1D
φv1 = 0 φv2 = 1 φv3 = x φv4 = cf t
φv5 = − x2
2− c2
f t2
2φv6 = −cf xt φv7 = − x3
6− xc2
f t2
2φv8 = − c3
f t3
6− x2cf t
2
φp1 = −cf φp2 = 0 φp3 = −c2f t φp4 = −cf x
φp5 = c2f xt φp6 = cf ( x2
2+
c2f t
2
2) φp5 = cf (
c3f t
3
6+ x2cf t
2) φp6 = cf ( x3
6+
xc2f t
2
2).
71
![Page 73: Trefftz-DG Approximation for the Elasto-AcousticsTable 1:Generic properties of the most widely used numerical methods2. Numerical Complex High-order Explicit semi- Conservation Elliptic](https://reader030.fdocuments.in/reader030/viewer/2022040612/5f039fbe7e708231d409f81a/html5/thumbnails/73.jpg)
Implementation of the algorithm
Polynomial waves basis functions 2D
p = 0
φvx1 = 0 φvy
1 = 0 φp1 = −cf
φvx2 = 1 φvy
2 = 1 φp2 = 0
p = 1
φvx3 = x φvy
3 = y φp3 = −cf (cf t)
φvx4 = cf t φvy
4 = x φp4 = −cf x
φvx5 = y φvy
5 = cf t φp5 = −cf y
p = 2
φvx7 = − x2
2 −y2
2 −c2f t
2
2 φvy7 = −xy φp
7 = cf (cf tx)
φvx8 = −xy φvy
8 = − x2
2 −y2
2 −c2f t
2
2 φp8 = cf (cf ty)
φvx9 = −cf ty φvy
9 = −cf tx φp9 = cf (xy)
φvx10 = −cf tx φvy
10 = −cf ty φp10 = cf ( x2
2 + y2
2 +c2f t
2
2 )
p = 3
φvx11 = − x2y
2 −c2f t
2y
2 − y3
6 φvy11 = − xy2
2 −c2f t
2x
2 − x3
6 φp11 = cf (cf txy)
φvx12 = −cf txy φvy
12 = − c3f t
3
6 −cf tx
2
2 − cf ty2
2 φp12 = cf ( x2y
2 +c2f t
2y
2 + y3
6 )
φvx13 = − c3
f t3
6 −cf tx
2
2 − cf ty2
2 φvy13 = −cf ty φp
13 = cf ( xy2
2 +c2f t
2x
2 + x3
6 )
φvx14 = − xy2
2 −c2f t
2x
2 − x3
6 φvy14 = − x2y
2 −c2f t
2y
2 − y3
6 φp14 = cf (
c3f t
3
6 +cf tx
2
2 +cf ty
2
2 )
72