Journal ArticleDOI

# A discontinuous Galerkin method with plane waves for sound absorbing materials

21 Dec 2015-International Journal for Numerical Methods in Engineering (John Wiley & Sons, Ltd)-Vol. 104, Iss: 12, pp 1115-1138
TL;DR: In this paper, a Discontinuous Galerkin method (DGM) with plane waves for poro-elastic materials using the Biot theory solved in the frequency domain is presented.
Abstract: Poro-elastic materials are commonly used for passive control of noise and vibration, and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper we present a Discontinuous Galerkin method (DGM) with plane waves for poro-elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly-oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave-based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave-based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material

### 1. INTRODUCTION

• The objective of this work is to develop a Discontinuous Galerkin Method (DGM) with plane waves to predict sound absorption in poro-elastic materials (PEM).
• The dynamic behaviour of porous materials is classically obtained from homogenized models and particularly from the Biot–Allard theory [1–4] which is based on a continuous field mechanics approach.
• The mid-frequencies correspond to situations where the standard FEM struggles with the size of the problem, but the statistical methods commonly used for high-frequency problems are not yet applicable.
• It was also shown that the DGM with numerical flux provides a unified framework to describe several wave-based methods [10, 15, 16], including the ultra-weak variational formulation, and the wave-based leastsquare method [17].
• In this paper the authors present a DGM using plane waves for poro-elastic materials modelled with Biot’s theory in the frequency domain.

### 2. GOVERNING EQUATIONS

• Throughout this paper, a harmonic time dependence e+iωt is assumed with the angular frequency ω.
• The numerical method and its applications are presented in two dimensions (x, y).
• The coefficient matrices Ax and Ay are assumed constant.
• For the porous material, these matrices are complex valued and vary with frequency ω.

### 2.1. First-order model for poro-elastic media

• There are several formulations of the Biot equations available in the literature.
• The right-hand side corresponds to elastic effects.
• Â andN are the elastic coefficients of the solid phase in vacuo which are directly obtained from the Lamé coefficients λ and µ.
• It should be noted that it is also possible to formulate a DGM starting from the non-conservative form (4), provided that the numerical flux is defined in a consistent way to ensure conservation of the field variables.
• One might think that the large number of unknowns introduced in this model implies that the computational cost of solving for all these variables will be high.

### 3. WAVE-BASED DGM

• The authors will now present the formulation and discretisation of the wave-based discontinuous Galerkin method of the conservative equations (1).
• The authors follow the same principles as in [10, 16], but the significant addition presented here is a general approach to incorporate a large class of boundary conditions (section 3.5), as well as coupling conditions between two different media (section 3.4).
• This approach relies heavily on the concept of characteristics which is introduced in section 3.2.

### 3.1. Variational formulation

• The variational formulation associated with the conservative form (1) is to find a solution u such that∑ ue = u|Ωe and ve = v|Ωe denote the restrictions of the solution and the test function to each element Ωe.
• The authors follow the usual idea from finite volume and discontinuous Galerkin methods of introducing a numerical flux on the interfaces between elements.
• Ωe and Ωe′ , and on this interface define the unit normal n pointing into Ωe′ .
• This aspect will be discussed in more detail in section 3.4.

### 3.2. Characteristics

• The concept of characteristics plays a central role in the analysis of partial differential equations of the form (1), see [24], and thus in the construction of numerical fluxes [25].
• Consider the boundary ∂Ω with unit normal vector n and tangential vector τ .
• For the Biot equations these eigenvalues are complex-valued, but the velocity and direction of propagation of each characteristic are easily obtained from the real part of the eigenvalue (which is consistent with group velocity).
• The general principle will guide the formulation of the boundary conditions and interface conditions in sections 3.4 and 3.5.

### 3.3. Numerical flux

• In the present work the authors use the upwind flux splitting (or exact Roe solver) which is a standard numerical flux for linear hyperbolic system.
• Λ± only contain the positive or negative eigenvalues.
• The two terms on the right correspond to the distinct contributions from the characteristics propagating in the positive and negative direction, respectively.
• If the authors now consider an interface Γee′ between two elements, the first term in (15) is associated with the characteristics travelling from element Ωe to Ωe′ and should therefore be calculated using ue.
• Conversely, the second term is associated with the characteristics travelling in the opposite direction and it is calculated using ue′ .

### 3.4. Boundary conditions

• As explained in [27], these boundary conditions should be used to specify the incoming characteristics in terms of the outgoing characteristics and the source terms, if any.
• Meth. (18) Separating the incoming characteristics from the others the authors get C̃−ũ− = s− C̃0+ũ0+ . (19) As mentioned in the previous section (and discussed in detail in [27]) a well-posed boundary condition specifies completely the incoming characteristics ũ−.
• The calculation of this matrix can be performed numerically in the implementation of the method, or can be done analytically beforehand.
• The authors now introduced some common boundary conditions used in practice for the Biot equations (2).
• There is also no tangential stresses on the surface since the porous material is allowed to slide along the surface.

### 3.5. Interface conditions

• The procedure described above for the boundary conditions can also be used for interface conditions between two different media.
• The unit normal on Γee′ points into Ωe. (25) A necessary condition is that the number of interface conditions corresponds to the total number of incoming characteristics on either sides of the interface.
• To fix ideas, two different sets of conditions are presented for the interface between a porous material and air.

### 3.6. Plane-wave discretization

• Finally the authors introduce the use of plane waves to discretise the variational formulations (12).
• Un of the plane waves are determined by requiring that these are solutions of the governing equations (1).
• In the case of the Biot equations (2–3) the calculation of the eigenvectors was found to be easier when using the non-conservative form (4).
• In addition, with other wave-based methods, in particular the partition-of-unity FEM, the presence of integrands involving polynomials and exponentials requires the use of numerical quadrature methods.

### 4.1. Plane wave propagating on a square

• This standard test case, already used in [10], is sufficiently simple to provide a detailed assessment of the performance of the method.
• The authors begin by considering the effect of the incident wave direction relative to other plane waves in the basis (33) as this has a profound impact on the accuracy of the method.
• The results are shown in figure 5 for a compression wave and the shear wave.
• These changes in actual element size induce the oscillations seen in the convergence curves for 1/h < 12.
• As expected, the numerical error decreases rapidly as the frequency is reduced (i.e. as the wavelength increases) until the conditioning deteriorates at levels of error of the order of 10−6 %.

### 4.2. Comparison with standard finite elements

• Numerical predictions for waves in porous material almost always rely on standard finite elements [29–32].
• The authors now compare the wave-based DGM against the standard finite element method with quadratic shape functions.
• Refining the mesh for the finite element method yields only a slow reduction in the absolute level of error but not a change in the rate of convergence (as expected from h-convergence).
• Similar conclusions can be obtained when considering the convergence with frequency presented in Figure 9.
• Several DGM results are presented corresponding to different numbers.

### 4.3. Sound absorption by a thin poroelastic layer

• The properties of the layer are given in Table I and correspond to Mat.
• The second resonance will be used to check the validity of the numerical scheme.
• While the mesh refinement is isotropic for the FEM, for the DGM a single element is used across the thickness of the layer and only the number of elements in the lateral direction is varied.
• Figure 11 shows the real part of the solutions of the pressure and the x component of the solid displacement, obtained with the FEM and the wave-based DGM.

### 4.4. Application to the scattering and absorption of sound by a porous cylinder

• The authors now consider a more complex test case, which involves the coupling of waves in air with the waves in a porous material.
• At the interface between the porous material and the surrounding volume of fluid, the two different conditions introduced in section 3.5 are considered.
• These two conditions can be formulated in the form of equation (25) and the methodology described in section 3.5 can be readily applied to implement these interface conditions in the variational formulation.
• A first remark on this figure is that for each solution is continuous which shows that each type of waves, taken individually, is discretised in a consistent manner by the numerical model.

### 5. CONCLUSIONS

• This paper described the first application of a plane-wave DGM for poroelastic materials modelled using the Biot equations.
• Using the charateristics of the governing equations, a general and systematic procedure was presented to include a large family of boundary conditions and interface conditions in the formulation of the numerical model.
• Interestingly it was found that the accuracy and the condition number are closely linked.
• This results in complex, three-dimensional geometries that will require a careful choice of plane-wave bases to maintain the improved efficiency of the method.
• Also, it might be worth considering hybrid model combining both wave-based elements and standard finite elements.

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 2010; 00:1–26
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme
A Discontinuous Galerkin Method with Plane Waves
for Sound Absorbing Materials
G. Gabard
1
, O. Dazel
2
1
ISVR, University of Southampton, Southampton, UK
2
LAUM, UMR CNRS 6613, Universit
´
e du Maine, Le Mans, France
SUMMARY
Poro-elastic materials are commonly used for passive control of noise and vibration, and are key to
reducing noise emissions in many engineering applications, including the aerospace, automotive and energy
industries. More efﬁcient computational models are required to further optimise the use of such materials. In
this paper we present a Discontinuous Galerkin method (DGM) with plane waves for poro-elastic materials
using the Biot theory solved in the frequency domain. This approach offers signiﬁcant gains in computational
efﬁciency and is simple to implement (costly numerical quadratures of highly-oscillatory integrals are not
needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for
the formulation of the wave-based DGM. A key contribution is a general formulation of boundary conditions
as well as coupling conditions between different propagation media. This is particularly important when
modelling porous materials as they are generally coupled with other media, such as the surround ﬂuid or
an elastic structure. The validation of the method is described ﬁrst for a simple wave propagating through
a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy,
conditioning and computational cost of the method are assessed, and comparison with the standard ﬁnite
element method is included. It is found that the beneﬁts of the wave-based DGM are fully realised for the
Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid
attenuation of the waves in the porous material. Copyright
c
2010 John Wiley & Sons, Ltd.
KEY WORDS: porous material, Biot theory, discontinuous Galerkin method, plane wave
1. INTRODUCTION
The objective of this work is to develop a Discontinuous Galerkin Method (DGM) with plane waves
to predict sound absorption in poro-elastic materials (PEM). Such materials are commonly used
for passive control of noise and vibration. In practical applications, they are often combined as
layers attached or linked to a vibrating structure or to a ﬂuid cavity. When subjected to mechanical
or acoustical excitation, they can dissipate energy through viscous, thermal and structural effects,
making their computational modelling of particular importance for many engineering applications.
The dynamic behaviour of porous materials is classically obtained from homogenized models and
particularly from the Biot–Allard theory [1–4] which is based on a continuous ﬁeld mechanics
approach. The homogenized porous media is modelled as the combination of two continuous ﬁelds
whose inertial and constitutive coefﬁcients are given by phenomenological relations.
Prediction models for porous materials, by analogy with structures, are commonly classiﬁed
according to three categories. The ﬁrst corresponds to low frequencies for which the main
computational technique is the Finite Element Method (FEM). The third category is associated
Correspondence to: Email: olivier.dazel@univ-lemans.fr
c
2010 John Wiley & Sons, Ltd.
Prepared using nmeauth.cls [Version: 2010/05/13 v3.00]

2 G. GABARD, O. DAZEL
with high frequencies where the Transfer Matrix Method provides an efﬁcient representation of the
layers of materials mentioned above. Standing between these two extremes, the second category is
commonly referred to as the mid-frequency range and is currently the subject of active research.
The boundaries between these regimes are somewhat arbitrary, as they depend on the properties
and geometry of the porous material and of the structure it is attached to. The mid-frequencies
correspond to situations where the standard FEM struggles with the size of the problem, but the
statistical methods commonly used for high-frequency problems are not yet applicable. Most of the
computational methods for porous materials discussed in the literature are extensions of either the
low- or high-frequency methods. The method proposed in the present paper is designed for the low-
and mid-frequency regimes.
Discontinuous Galerkin Methods (DGM) have been actively developed for various branches
of science, especially for time-domain simulations of conservation equations, as these methods
directly provide high-order, explicit schemes [5]. They generally rely on polynomial interpolations
of the solution within each element, but recently the use of plane waves has been proposed.
This is part of the development of so-called wave-based, or Trefftz, methods where the use of
canonical solutions as basis functions improves signiﬁcantly the accuracy of the numerical model,
and offers exponential convergence when the number of plane waves is increased. Prominent
examples of this approach include the partition of unity ﬁnite element method (PUFEM) [6],
the ultra-weak variational formulation [7] and the discontinuous enrichment method [8]. The use
of the discontinuous Galerkin approach in this context was then proposed using either Lagrange
multipliers [9] or numerical ﬂux methods [10]. More recently a thorough analysis of the properties
of the wave-based DGM for the Helmholtz equation was conducted [11–14]. It was also shown
that the DGM with numerical ﬂux provides a uniﬁed framework to describe several wave-based
methods [10, 15, 16], including the ultra-weak variational formulation, and the wave-based least-
square method [17].
In this paper we present a DGM using plane waves for poro-elastic materials modelled with
Biot’s theory in the frequency domain. Relevant prior work include the use of a Trefftz wave-
based approach [18, 19] as well as the ultra-weak formulation for the lossy wave equation [20].
The PUFEM was recently applied to an equivalent ﬂuid model [21] and then to the full Biot
equations [22]. An issue with the PUFEM is the cost of calculating the element matrices associated
with numerical quadratures for highly-oscillatory integrands (it was reported in some cases that
the cost of calculating the element matrices is of the same order as solving the system of linear
equations). The present wave-based DGM does not suffer from this issue.
The Biot equations are introduced in the next section and it is shown that they can easily be cast
as a set of conservation equations which is needed for the present DGM. In section 3 the general
formulation of the wave-based DGM is recalled, and the emphasis is placed on the formulation
of boundary conditions as well as coupling conditions between two different propagation media
(typically between the porous medium and the surround ﬂuid). The validation of the method is
described in section 4, ﬁrst for a simple wave propagating through the PEM, and then for the
scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational
cost of the method are assessed.
2. GOVERNING EQUATIONS
Throughout this paper, a harmonic time dependence e
+iωt
is assumed with the angular frequency
ω. The numerical method and its applications are presented in two dimensions (x, y). To model the
propagation of waves in the porous material and in the surrounding ﬂuid, we will consider a general
system of linear conservation equations of the form:
iωu + A
x
u
x
+ A
y
u
y
= 0 , (1)
where u is the vector of physical ﬁeld variables whose number depends on the nature of the
propagation medium. Below we consider either a porous material, or a ﬂuid (using 8 or 3 ﬁeld
c
2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)
Prepared using nmeauth.cls DOI: 10.1002/nme

PLANE-WAVE DGM FOR POROELASTIC MATERIALS 3
variables, respectively). The coefﬁcient matrices A
x
and A
y
are assumed constant. For the porous
material, these matrices are complex valued and vary with frequency ω. The conservation equations
(1) represent the basis for most discontinuous Galerkin methods, and we will also use this as a
starting point to formulate the proposed wave-based DGM in section 3.
2.1. First-order model for poro-elastic media
There are several formulations of the Biot equations available in the literature. In the present work
we will use the Biot equations as formulated in [23] as the simplicity of this formulation greatly
facilitates the algebra when deriving the plane wave basis. The only difference with [23] is that the
velocity is used here instead of the displacement. The equations of motion involve the velocity v
s
of the solid phase of the porous material, as well as the total velocity v
t
of the porous material:
iωeρ
s
v
s
x
+ iωeρ
eq
eγv
t
x
=
ˆσ
xx
x
+
ˆσ
xy
y
, (2a)
iωeρ
s
v
s
y
+ iωeρ
eq
eγv
t
y
=
ˆσ
xy
x
+
ˆσ
yy
y
, (2b)
iωeγ eρ
eq
v
s
x
+ iωeρ
eq
v
t
x
=
p
f
x
, (2c)
iωeγ eρ
eq
v
s
y
+ iωeρ
eq
v
t
y
=
p
f
y
. (2d)
The left-hand side of these equations is related to visco-inertial terms. The PEM equivalent densities
˜ρ
s
, ˜ρ
eq
and the coupling factor ˜γ are deﬁned in [23]. The right-hand side corresponds to elastic
effects. The pressure in the ﬂuid phase is denoted p
f
. The tensor
ˆ
σ corresponds to the stresses in
the solid phase of the porous material in the absence of ﬂuid (i.e. in vacuo). These are deﬁned as
follows:
iωp
f
=
e
K
eq
v
t
x
x
+
v
t
y
y
, (3a)
iωˆσ
xx
=
ˆ
A
v
s
y
y
+ (
ˆ
A + 2N)
v
s
x
x
, (3b)
iωˆσ
xy
= N
v
s
x
y
+
v
s
y
x
, (3c)
iωˆσ
yy
= (
ˆ
A + 2N)
v
s
y
y
+
ˆ
A
v
s
x
x
. (3d)
e
K
eq
is the compressibility of the ﬂuid.
ˆ
A and N are the elastic coefﬁcients of the solid phase in vacuo
which are directly obtained from the Lam
´
e coefﬁcients λ and µ, as shown in [23] and recalled in
Appendix A.
From equations (2) and (3) it is clear that can ﬁrst be cast into a non-conservative form
iωMu + B
x
u
x
+ B
y
u
y
= 0 , (4)
with the following deﬁnitions:
u =
v
s
x
v
s
y
v
t
x
v
t
y
ˆσ
+
ˆσ
xy
ˆσ
p
f
, M =
eρ
s
0 eγeρ
eq
0 0 0 0 0
0 eρ
s
0 eγeρ
eq
0 0 0 0
eγeρ
eq
0 eρ
eq
0 0 0 0 0
0 eγeρ
eq
0 eρ
eq
0 0 0 0
0 0 0 0 (
ˆ
A + N)
1
0 0 0
0 0 0 0 0 N
1
0 0
0 0 0 0 0 0 N
1
0
0 0 0 0 0 0 0
e
K
1
eq
, (5)
c
2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)
Prepared using nmeauth.cls DOI: 10.1002/nme

4 G. GABARD, O. DAZEL
and
B
x
=
0 0 0 0 1 0 1 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
, B
y
=
0 0 0 0 0 1 0 0
0 0 0 0 1 0 1 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
.
(6)
The reason for introducing the non-conservative form (4) is that deriving closed-form expression
of the plane-wave basis for the solution and the test functions is easier when using (4), since it
provides simple links between the direct and adjoint plane wave bases, as will be shown in section
3.6. This stems from the fact that the matrices B
x
and B
y
are real and symmetric, while the complex-
valued matrix M is complex symmetric (but not Hermitian).
The conservative form (1) is easily recovered from equation (4) using
A
x
= M
1
B
x
, A
y
= M
1
B
y
. (7)
For convenience we have introduced σ
+
= (ˆσ
xx
+ ˆσ
yy
)/2 and σ
= (ˆσ
xx
ˆσ
yy
)/2 in the vector
u as it simpliﬁes the expression of the mass matrix M and the calculation of its inverse in (7).
It should be noted that it is also possible to formulate a DGM starting from the non-conservative
form (4), provided that the numerical ﬂux is deﬁned in a consistent way to ensure conservation
of the ﬁeld variables. This approach has been derived and implemented by the authors, but it is
not described in the present paper as this leads to weak forms equivalent to that obtained from the
well-established DGM based on (1).
One might think that the large number of unknowns introduced in this model implies that the
computational cost of solving for all these variables will be high. This is true for standard ﬁnite
element methods where each variable is discretised independently, but it does not apply here. With
the present wave-based method the degrees of freedom are the amplitudes of the plane waves in
each element and their number is completely independent of the number of variables introduced in
the governing equations.
2.2. Acoustic waves in air
To describe the acoustic waves in the ﬂuid around the porous material we use the standard Helmholtz
equation which can be written directly in the conservative form (1) by introducing the acoustic
pressure p
a
and linearised momentum ρ
0
v
a
as ﬁeld variables:
u =
p
a
ρ
0
v
a
x
ρ
0
v
a
y
, A
x
=
0 c
2
0
0
1 0 0
0 0 0
, A
y
=
0 0 c
2
0
0 0 0
1 0 0
, (8)
where ρ
0
is the mean density and c
0
is the sound speed. This corresponds to the same set of equations
used in [16].
3. WAVE-BASED DGM
We will now present the formulation and discretisation of the wave-based discontinuous Galerkin
method of the conservative equations (1). We follow the same principles as in [10, 16], but the
signiﬁcant addition presented here is a general approach to incorporate a large class of boundary
conditions (section 3.5), as well as coupling conditions between two different media (section 3.4).
This approach relies heavily on the concept of characteristics which is introduced in section 3.2.
c
2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)
Prepared using nmeauth.cls DOI: 10.1002/nme

PLANE-WAVE DGM FOR POROELASTIC MATERIALS 5
3.1. Variational formulation
We consider a domain which is represented by a set of N
e
elements
e
. We allow for the solution
u to be discontinuous at the interfaces between the elements. The variational formulation associated
with the conservative form (1) is to ﬁnd a solution u such that
X
e
Z
e
v
T
e
iωu
e
+ A
x
u
e
x
+ A
y
u
e
y
dΩ = 0 , v , (9)
where
T
denotes the Hermitian transpose. u
e
= u|
e
and v
e
= v|
e
denote the restrictions of the
solution and the test function to each element
e
.
After integrating by parts on each element and rearranging terms we get:
X
e
Z
e
iωv
e
+ A
T
x
v
e
x
+ A
T
y
v
e
y
T
u
e
d +
X
e
Z
e
v
T
e
F
e
u
e
dΓ = 0 , v , (10)
where we have introduced the matrix F
e
= A
x
n
x
+ A
y
n
y
which represents the normal ﬂuxes
across the boundary of the element
e
. The unit normal n = (n
x
, n
y
) on the element boundary
e
points out of the element.
A key aspect of the wave-based DGM is to use test functions v whose restrictions v
e
on each
elements are solutions of the adjoint problem deﬁned on each element:
iωv
e
+ A
T
x
v
e
x
+ A
T
y
v
e
y
= 0 , (11)
which is readily identiﬁed from equation (10). With this choice of test functions the integral over
each element
e
vanishes and one is left with integrals on the interfaces between elements and on
the boundary of the domain.
Secondly, we follow the usual idea from ﬁnite volume and discontinuous Galerkin methods
of introducing a numerical ﬂux on the interfaces between elements. Consider the interfaces Γ
ee
0
between elements
e
and
e
0
, and on this interface deﬁne the unit normal n pointing into
e
0
.
The ﬁeld variables satisfy the conservation equations (1), and this implies that the ﬂux Fu across
this interface should be continuous. It follows that we can deﬁne a numerical ﬂux f
ee
0
such that
f
ee
0
(u
e
, u
e
0
) = F
e
u
e
= F
e
0
u
e
0
. We will discuss the choice of numerical ﬂux in more details in
section 3.3.
Finally we arrive at the following formulation of the wave-based discontinuous Galerkin methods:
X
e,e
0
<e
Z
Γ
ee
0
(v
e
v
e
0
)
T
f
ee
0
(u
e
, u
e
0
) dΓ +
Z
v
T
Fu = 0 , v . (12)
The boundary integrals are then modiﬁed to implement the different boundary conditions. This
aspect will be discussed in more detail in section 3.4.
3.2. Characteristics
The concept of characteristics plays a central role in the analysis of partial differential equations of
the form (1), see [24], and thus in the construction of numerical ﬂuxes [25]. The basic deﬁnitions
and notations are deﬁned in this section to support the discussion of the numerical ﬂux, boundary
conditions and interface conditions in the following sections.
Consider the boundary with unit normal vector n and tangential vector τ . Through a simple
change of variables we can write the governing equations (1) as
iωu + F
u
n
+ T
u
τ
= 0 , (13)
where F is the ﬂux matrix deﬁned above and the matrix T = A
x
n
y
+ A
y
n
x
corresponds to the
ﬂux tangential to boundary.
c
2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)
Prepared using nmeauth.cls DOI: 10.1002/nme

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##### References
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Book
01 Jan 1974
TL;DR: In this paper, a general overview of the nonlinear theory of water wave dynamics is presented, including the Wave Equation, the Wave Hierarchies, and the Variational Method of Wave Dispersion.
Abstract: Introduction and General Outline. HYPERBOLIC WAVES. Waves and First Order Equations. Specific Problems. Burger's Equation. Hyperbolic Systems. Gas Dynamics. The Wave Equation. Shock Dynamics. The Propagation of Weak Shocks. Wave Hierarchies. DISPERSIVE WAVES. Linear Dispersive Waves. Wave Patterns. Water Waves. Nonlinear Dispersion and the Variational Method. Group Velocities, Instability, and Higher Order Dispersion. Applications of the Nonlinear Theory. Exact Solutions: Interacting Solitary Waves. References. Index.

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TL;DR: In this article, a theory for the propagation of stress waves in a porous elastic solid containing compressible viscous fluid is developed for the lower frequency range where the assumption of Poiseuille flow is valid.
Abstract: A theory is developed for the propagation of stress waves in a porous elastic solid containing compressible viscous fluid. The emphasis of the present treatment is on materials where fluid and solid are of comparable densities as for instance in the case of water‐saturated rock. The paper denoted here as Part I is restricted to the lower frequency range where the assumption of Poiseuille flow is valid. The extension to the higher frequencies will be treated in Part II. It is found that the material may be described by four nondimensional parameters and a characteristic frequency. There are two dilatational waves and one rotational wave. The physical interpretation of the result is clarified by treating first the case where the fluid is frictionless. The case of a material containing viscous fluid is then developed and discussed numerically. Phase velocity dispersion curves and attenuation coefficients for the three types of waves are plotted as a function of the frequency for various combinations of the characteristic parameters.

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Book
01 Jan 2002
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Abstract: Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

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• ...Prominent examples of this approach include the partition of unity FEM (PUFEM) [6], the ultra-weak variational formulation [7] and the discontinuous enrichment method [8]....

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• ...El Kacimi A, Laghrouche O. Improvement of PUFEM for the numerical solution of high-frequency elastic wave scattering on unstructured triangular mesh grids....

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• ...Engng 2015; 104:1115–1138 DOI: 10.1002/nme wave-based methods, in particular the PUFEM, the presence of integrands involving polynomials and exponentials requires the use of numerical quadrature methods....

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