Hybrid finite element -- wave based method for steady-state acoustic ...

Hybrid finite element – wave based method
for steady-state acoustic analysis
B. van Hal 1 , C. Vanmaele 1 , W. Desmet 1 , P. Silar 2 , H.-H. Priebsch 2
1 K.U.Leuven, Dept. Mech. Engrg., Division PMA
Celestijnenlaan 300 B, B-3001, Heverlee, Belgium
e-mail: wim.desmet@mech.kuleuven.ac.be
2 ACC Acoustic Competence Center G.m.b.H.,
Inffeldgasse 25, A-8010 Graz, Austria
Abstract
This paper considers the extension of the frequency application range of the finite element method (FEM)
applied to steady-state acoustic problems. This is achieved by the hybrid coupling of the FEM with the wave
based method (WBM). The WBM is a computationally efficient alternative for the FEM, but it is restricted
to acoustic problems of moderate geometrical complexity. The hybrid finite element – wave based method
(HFE-WBM) benefits from the advantageous features of both methods. These are the high geometrical
flexibility of the FEM and the computational efficiency of the WBM. The potential of the HFE-WBM to
provide more accurate predictions for higher frequencies than the FEM is illustrated for the validation of a
numerical example of a 2D acoustic car cavity.
1 Extension of frequency application range of deterministic methods
The finite element method (FEM) is a commonly used deterministic numerical method for the analysis
of steady-state acoustic problems [1, 2, 3]. The finite element (FE) procedure consists of subdividing an
acoustic domain in a large number of small non-overlapping subdomains, i.e. the finite elements. Within
each element, a linear combination of simple (polynomial) basis functions approximates the exact pressure
distribution. The FEM exhibits a large geometrical flexibility since the FE subdivision is possible for most
practical engineering problems. However, the FEM is restricted to the low-frequency application range due
to the increasing model size and the subsequent computational efforts, which are required to keep the FE
approximation errors within reasonable limits.
The FE approximation errors consist of interpolation errors and dispersion errors [3]. Especially, the dispersion
errors gain importance at higher frequencies. The difference between the numerical wavenumber and
the physical wavenumber causes the dispersion errors. In order to control this type of error, the FE model
sizes and subsequent computational costs increase drastically for increasing frequencies.
A vast amount of research is currently spent on extending the frequency application range of the FEM. A
common feature of these extensions is that the computational costs for increasing frequencies is reduced by
controlling the dispersion errors more efficiently. Several classes of extensions exist, namely
• the stabilized methods, such as the Galerkin least-squares FEM [4] and the quasi-stabilized FEM [5],
• the generalized methods, such as the partition-of-unity FEM [6] and the element-free Galerkin
method [7],
• and the multi-scale methods, such as the residual-free bubbles method [8] and the discontinuous
enrichment method [9].
1629

1630 PROCEEDINGS OF ISMA2004
An alternative for the FEM is the class of indirect Trefftz methods [10, 11, 12]. These methods apply a
priori knowledge of the exact solution. The approximation solution satisfies the governing domain equations
and violates only the boundary conditions, so that the resulting numerical models are much smaller
than FE models. Moreover, the a priori knowledge of the exact solution includes knowledge of the physical
wavenumber such that the dispersion errors are reduced substantially compared with the FEM. Several indirect
Trefftz methods have emerged for structural-dynamic and acoustic applications, such as the spectral
FEM [13], the waveguide FEM [14], the Trefftz least-squares FEM [15], the hybrid-Trefftz FEM [16], the
wave based method (WBM) [17] and the variational theory of complex rays [18].
The reader is referred to [19] for a detailed overview of deterministic methods with an extended frequency
application range compared with the FEM. This paper focuses on the hybrid coupling between the FEM and
the WBM for acoustic applications. The resulting hybrid finite element – wave based method (HFE-WBM)
is proposed to benefit from the advantageous features of both methods [20, 21]. These are the geometrical
flexibility of the FEM and the computational efficiency of the WBM.
The paper is arranged as follows. Section 2 introduces the mathematical description of an acoustic problem.
Section 3 provides the theoretical background of the HFE-WBM. Section 4 illustrates the potential of the
HFE-WBM to provide more accurate approximations at higher frequencies than the FEM. Finally, section 5
summarizes the conclusions and provides some recommendations for further investigations.
2 Acoustic problem definition
Consider a 3-dimensional (3D) acoustic cavity, of which the dimension in the z-direction is infinite. If the
boundary conditions do not vary in the z-direction, then the 3D acoustic problem can be reduced to the 2D
problem shown in figure 1.
The bounded 2D acoustic problem consists of the domain Ω surrounded by the closed boundary Γ. The
boundary Γ is composed of two non-overlapping parts (Γ = Γ v ∪ Γ Z ), namely the part Γ v with the prescribed
normal velocity ¯v n and the part Γ Z with the prescribed normal impedance ¯Z. The homogeneous Helmholtz
equation
(
∆ + k
2 ) p(r) = 0, ∀ r ∈ Ω (1)
governs the steady-state dynamic pressure p at position r in domain Ω, together with the natural boundary
conditions
v n (r) = ¯v n (r), ∀ r ∈ Γ v (2)
and the mixed boundary conditions
¯Z(r)v n (r) = p(r), ∀ r ∈ Γ Z . (3)
Figure 1: Bounded 2D acoustic problem

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1631
∆ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 represents the Laplace operator and k = ω/c the physical wavenumber with the
circular frequency ω and the speed of sound c. The normal velocity v n is given by
v n (r) = n T v(r), ∀ r ∈ Γ, (4)
where n represents the normal vector depicted in figure 1, with the transpose operator • T , and where v
represents the velocity vector, which is governed by the equation of motion
v(r) =
i ∇p(r), ∀ r ∈ Ω, (5)
ρω
with the unit imaginary number i = √ −1 and with the gradient operator ∇ = [∂/∂x ∂/∂y] T .
3 Theory on hybrid finite element – wave based method
3.1 Review of finite element method
The FEM applies the following strategy to approximate the exact solution of the 2D acoustic problem.
1. The problem domain Ω is subdivided in a large number of n e non-overlapping elements Ω e . Each
element boundary Γ e is composed of three non-overlapping parts Γ e = Γ e v ∪ Γ e Z ∪ Γe i , namely the
intersections with the problem boundaries Γ e v = Γ e ∩ Γ v and Γ e Z = Γe ∩ Γ Z and the common interface
of two adjacent elements Γ e i .
2. Within each element Ω e , a linear combination of simple (polynomial) basis functions, stored in the
row vector N, approximates the exact pressure field p
p(r) ≈ ˆp(r) = N(r) p e , ∀ r ∈ Ω e . (6)
The column vector p e contains the unknown contribution factors. These form the FE degrees of
freedom (DOFs), which are generally nodal values of the pressure approximation ˆp. The derived
velocity approximation ˆv satisfies the equation of motion (5).
3. The pressure approximation ˆp is continuous over element boundaries but it violates the governing
equations in (1) – (3). The violation of these relations are enforced to zero in an integral sense by
application of the following variational principle.
Consider the energy functional J e defined for each element Ω e
∫
J e = Π e + ˆp n T v i dΓ
Γ e i
∫
∫
∫
∫
with Π e = 1 2
eˆv T iρω ˆv dΩ + 1 ik
2
Ω Ωeˆp ρc ˆp dΩ + ˆp ¯v n dΓ + 1
Γ e 2
v
Γ e Z
ˆp ¯Z −1 ˆp dΓ,
where v i represents the unknown velocity at the interface Γ e i . The stationary point of J e enforces the
violation of (1) – (3) to zero on element level
∫
δJ e = δΠ e + δp n T v i dΓ = 0
Γ e i
∫
∫
with δΠ e = δv T iρω ˆv dΩ + δp ik ∫
Ω e Ω e ρc ˆp dΩ +
Γ e v
∫
δp ¯v n dΓ +
Γ e Z
δp ¯Z −1 ˆp dΓ.
The integral contributions with the interface velocity v i of two adjacent elements cancel out each other
at the time of assembling all element FE models into one global FE model. Therefore, this integral
(7)
(8)

1632 PROCEEDINGS OF ISMA2004
contribution is not considered further. The small perturbations δp and δv result from small arbitrary
perturbations δp e of the FE DOFs p e
δp(r) = N(r) δp e and δv(r) = i
ρω ∇N(r) δpe , ∀ r ∈ Ω e . (9)
4. The element FE model follows from (i) the substitution of ˆp, ˆv, δp and δv in the integral relation (8),
from (ii) multiplication with iρω and from (iii) the requirement that this relation holds for arbitrary
perturbations δp e .
5. The assembly of all element FE models results in the following FE model for 2D acoustic problems
(
−ω 2 M + iω C + K ) p E = −iω v n (10)
⋃n e ∫
⋃n e ∫
with M = N T c −2 N dΩ, C =
e=1 Ω e e=1
⋃n e ∫
⋃n e ∫
K = (∇N) T (∇N) dΩ and v n =
Ω e
e=1
e=1
Γ e Z
Γ e v
ρ N T ¯Z−1 N dΓ,
ρ N T ¯v n dΓ.
The model matrices M, C and K represent the acoustic mass, damping and stiffness matrix. The model
vector v n takes into account the acoustic excitation, which consists of a non-zero prescribed normal
velocity distribution. The column vector p E collects all FE DOFs.
6. The model matrices are generally sparse. This allows the application of efficient ’skyline’ solvers to
compute the FE DOFs in p E .
3.2 Review of wave based method
The WBM applies the following strategy to approximate the exact solution of the 2D acoustic problem.
1. The problem domain Ω is subdivided in a small number of n w non-overlapping convex subdomains
Ω w . Similar to the FEM, each subdomain boundary Γ w is composed of three non-overlapping parts
Γ w = Γ w v ∪ Γ w Z ∪ Γw i .
2. Within each subdomain Ω w , a linear combination of Trefftz basis functions, stored in the row vector
Φ, approximates the exact pressure field p
p(r) ≈ ˆp(r) = Φ(r) q w , ∀ r ∈ Ω w . (11)
The column vector q w contains the unknown wave based (WB) contribution factors, which are not
nodal values of the pressure approximation ˆp as is the case for the FEM. The Trefftz basis functions
belong to the following set of propagating and evanescent wave functions
Φ rs (r) = cos(k r x) cos(k s y) e −ikrsz , with k r = rπ , k s = sπ , k rs = ± √ k
L x L 2 − kr 2 − ks,
2
y
Φ tu (r) = e −iktux cos(k t y) cos(k u z), ” k t = tπ
L y
, k u = uπ
L z
, k tu = ±
Φ vw (r) = cos(k w x) e −ikvwy cos(k v z), ” k v = vπ
L z
, k w = wπ
L x
, k vw = ±
√
k 2 − kt 2 − k2 u,
√
k 2 − kv 2 − kw.
2
The numerical wavenumbers k • depend on the dimensions (L x , L y , L z ) of the (preferable) smallest
rectangular domain, which encloses the actual subdomain Ω w . The r, s, t, u, v and w represent integer
sets [0, n • ], which are limited from above by the following truncation numbers n •
(12)
with a user defined truncation number T .
n r
L x
≈ n s
L y
≈ n t
L y
≈ n u
L z
≈ n v
L z
≈ n w
L x
≥ T k π
(13)

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1633
3. The pressure approximation satisfies the Helmholtz equation (1), but violates the boundary conditions
in (2) and (3) and the pressure and velocity continuity across WB subdomain boundaries. A variational
principle enforces the violations of these relations to zero in an integral sense.
The WBM applies a direct approach to couple WB subdomains. In case of two WB subdomains Ω 1 and
Ω 2 , the continuity conditions are considered as prescribed essential and natural boundary conditions
as follows
pressure continuity: p 1 = p 2 (essential BC † for Ω 1 ),
(14)
velocity continuity: vn 2 = −vn 1 (natural BC for Ω 2 ),
where superscripts • 1 and • 2 refer to subdomains Ω 1 and Ω 2 . The extension for n w > 2 subdomains
is straightforward and, therefore, is not considered further.
The following variational principle is employed. Consider the energy functional E 1 and E 2
∫
∫
E 1 = Π 1 + ˆv n 1 (ˆp 1 − ˆp 2 ) dΓ and E 2 = Π 2 − ˆp 2 ˆv n 1 dΓ (15)
Γ 1 i
with the energy functional Π w (w = 1, 2) according to (7) and where the approximation of the normal
velocity ˆv n w satisfies relation (4) and the equation of motion (5). The stationary points of E 1 and E 2
enforces the above violations to zero
∫
δE 1 = δΠ 1 +
Γ 1 i
∫
δE 2 = δΠ 2 − δp 2 ˆv n 1 dΓ,
Γ 2 i
Γ 2 i
∫
δvn 1 (ˆp 1 − ˆp 2 ) dΓ + δp 1 ˆv n 1 dΓ,
Γ 1 i
with δΠ w according to (8). The small perturbations δp w and δv w n result from small arbitrary perturbations
δq w of the WB DOFs q w
δp w (r) = Φ(r) δq w , ∀ r ∈ Ω w and δv w n(r) = i
ρω nT ∇Φ(r) δq w , ∀ r ∈ Γ w . (17)
The application of several mathematical manipulations ‡ transforms the integral formulations in (16)
into the boundary integral formulations
∫
∫
∫
δE 1 = δp 1 (ˆv n 1 − ¯v n ) dΓ+ δp 1 (ˆv n 1 − ¯Z −1 ˆp 1 ) dΓ− δvn 1 (ˆp 1 − ˆp 2 ) dΓ = 0,
∫
δE 2 =
Γ 1 v
Γ 2 v
∫
δp 2 (ˆv n 2 − ¯v n ) dΓ+
Γ 1 Z
Γ 2 Z
∫
δp 2 (ˆv n 2 − ¯Z −1 ˆp 2 ) dΓ+
Γ 1 i
Γ 2 i
δp 1 (ˆv 1 n + ˆv 2 n) dΓ = 0.
4. The substitution of ˆp w , ˆv n, w δp w and δvn w in the boundary integral formulations, together with the
requirement that the boundary integral formulation holds for arbitrary perturbations δq w , results in the
following WB model for 2D acoustic problems with two convex subdomains
[
(A
1
v + A 1 Z + Q1 b ) ] { } { }
Q1 2 q
1 b
1
v
Q 2 1 (A 2 v + A 2 Z + Q2 b ) =
q 2 b 2 v
i
∫
ρω nT ∇Φ dΓ, b w v = Φ T ¯v n dΓ,
Γ w v
( )
i
Φ T
Γ w ρω nT ∇Φ − ¯Z −1 Φ dΓ,
Z
∫
Q 1 b = − i (
n T ∇Φ ) ∫
T
Φ dΓ, Q
1
ρω
2 =
∫
with A w v = Φ T
Γ w v
∫
A w Z =
Q 2 b = ∫
Γ 2 i
Γ 1 i
Φ T
i
ρω nT ∇Φ dΓ, and Q 2 1 =
∫
Γ 1 i
Γ 2 i
i
( ) n 1 T T
∇Φ
1 Φ 2 dΓ,
ρω
Φ 2 T
i
ρω n1 T
∇Φ 1 dΓ.
† boundary condition
‡ the equation of motion (5), integration by parts, the divergence theorem, the Helmholtz equation (1) and multiplication by -1
(16)
(18)
(19)

1634 PROCEEDINGS OF ISMA2004
A w v and b w v take into account the acoustic excitation and A w Z the mixed boundary conditions. Qw b
represent the back-coupling matrices and Q 1 2 and Q 2 1 the coupling matrices.
5. The model matrices are small compared with FE models, but fully populated, frequency dependent
and complex. Only direct methods can be employed to solve the linear system in (19). The resulting
WB DOFs in q 1 and q 2 are not nodal pressure approximations as is the case for the FEM. The WB
pressure approximation ˆp at discrete positions r follows from an additional post-processing step using
relation (11).
3.3 Comparison of finite element method and wave based method
This paragraph compares briefly the advantages and disadvantages of the FEM and the WBM. The features
of interest are the model properties, the accuracy and the practical applicability.
• Comparison of model properties:
– The FEM requires substantially larger models than the WBM in order to obtain the same level
of accuracy. However, the FE model matrices are sparsely populated, which allows the use of
efficient ’skyline’ solvers. The WB model matrices are dense such that only direct solvers are
applicable.
– The solution of the FE model is computationally more expensive than its generation. In case of
the WBM, the major part of the computational burden is involved with the WB model generation.
• Comparison of accuracy:
The WBM is computationally more efficient than the FEM, certainly for increasing frequencies [17,
22, 23, 24]. The WBM suffers less from dispersion errors. Consequently, it exhibits an improved
convergence behaviour as compared to the FEM.
• Comparison of practical applicability:
The FEM is well suited for most practical engineering problems due to its highly geometrical flexibility.
However, it is limited to the low-frequency range due to vastly growing computational costs
necessary to control the dispersion errors at higher frequencies. The WBM is computationally more
efficient than the FEM in case of a WB domain subdivision with only a small number of subdomains.
However, if the number of subdomains increases for reasons of geometrical complexity, the computational
efficiency is less pronounced. This feature restricts the application of the WBM to acoustic
problems of moderate geometrical complexity.
The comparison of the practical applicability shows some advantageous and disadvantageous features of both
methods, which form the motivation to investigate the hybrid coupling between the FEM and the WBM.
The HFE-WBM, which incorporates this hybrid coupling, is proposed in order to extend the frequency
application range of the FEM without loss of geometrical flexibility as is the case for the WBM. The next
paragraph presents the theoretical background of this new method.
3.4 Hybrid FE-WB modelling based on equivalent velocity frame
The HFE-WBM employs an indirect coupling approach based on the Lagrange multiplier technique (LMT)
[1]. The LMT allows the coupling of two nonconforming approximation fields at the expense of an additional
interface approximation. The procedure is as follows.

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1635
1. A frame Γ F is placed at the common interface between the FE submodel Ω E and the WB submodel
Ω W , on which the additional interface field or frame field is introduced. Assume that the WB submodel
consists of one convex WB subdomain Ω w for the sake of simplicity. The frame Γ F is subdivided in
n f non-overlapping frame elements Γ f . The frame elements Γ f match geometrically with the edges
of adjacent finite elements Ω e . Figure 2 illustrates this modelling step graphically.
2. The LMT enforces weakly the following continuity conditions at the frame Γ F
pressure continuity:
equivalent velocity continuity:
ˆv e n(r) −
ˆp e (r) = p f (r) = ˆp w (r),
−1 ¯Z
f
ˆp e (r) = ˆv eq(r) f = −ˆv n(r) w −
−1 ¯Z
f
ˆp w (r),
(20)
where p f represents the unknown frame pressure and ˆv f eq represents the approximation of the equivalent
frame velocity, i.e. the additional frame approximation. The equivalent velocity continuity is the
combination of the velocity continuity and the pressure continuity divided by a user defined frame
impedance ¯Z f . The application of this continuity condition stabilizes the FE submodel and the WB
submodel in order to avoid singularity problems at subdomain eigenfrequencies.
3. A linear combination of simple basis functions, stored in the row vector N f , approximates the exact
equivalent frame velocity v f eq
v f eq(r) ≈ ˆv f eq(r) = N f (r) v f , ∀ r ∈ Ω f . (21)
The frame basis functions in N f belong to the set of hierarchical polynomial functions based on midside
nodes, which are indicated by ⋄ in figure 2. Consequently, the unknown contribution factors in the
column vector v f do not equal the nodal values of the equivalent frame velocity approximation ˆv f eq.
4. Separate variational principles are employed for the FE submodel and the WB submodel.
(a) The energy functional J e in (7) is augmented as follows for the finite elements adjacent to the
frame Γ F ∫
L e = J +∫
e ˆv eq f (ˆp e − p f ) dΓ + 1
{Γ e ∩Γ f 2
ˆp e ¯Z−1 f
ˆp e dΓ. (22)
}
{Γ e ∩Γ f }
The stationary point of L e enforces the violation on the Helmholtz equation (1), on the boundary
conditions in (2) and (3) and on the continuity conditions in (20) to zero
∂J e ∂J
e
⎫ ⎧ ∫
∫
δp +
∂ ˆp ∂ˆv δv = 0 ⎪⎬ ⎪⎨ δJ e + δp ˆv f −1
eq dΓ + δp ¯Z
f
ˆp e dΓ = 0,
{Γ
∂J e
→
e ∩Γ f }
{Γ e ∩Γ f }
∫
(23)
δveq f = 0⎪⎭
⎪⎩ δveq f (ˆp e − p f ) dΓ = 0.
∂ˆv f eq
{Γ e ∩Γ f }
Figure 2: Hybrid FE-WB modelling based on equivalent velocity frame

1636 PROCEEDINGS OF ISMA2004
The small perturbations δp and δv are defined in (9) and the small perturbation δv f eq result from
small arbitrary perturbations δv f of the frame DOFs v f
δv f eq(r) = N f (r) δv f , ∀ r ∈ {Γ e ∩ Γ f }. (24)
(b) Recall that the WB submodel consists of a single convex subdomain Ω w . The associated energy
functional E w in (15) reduces to Π w , which is augmented as follows
∫
∫
F w = Π w − ˆv eq f (ˆp w − p f ) dΓ − 1 2
ˆp w ¯Z−1 f
ˆp w dΓ. (25)
{Γ w ∩Γ F }
{Γ w ∩Γ F }
The stationary point of F w enforces the violation on the boundary conditions in (2) and (3) and
on the continuity conditions in (20) to zero
∂F w ∂F
w
⎫ ⎧ ∫
∫
δp +
∂ ˆp ∂ˆv δv = 0 ⎪⎬ ⎪⎨ δΠ w − δp ˆv f −1
eq dΓ − δp ¯Z
f
ˆp w dΓ = 0,
{Γ
∂F w
→
w ∩Γ F }
{Γ w ∩Γ F }
∫
(26)
δveq f = 0⎪⎭
⎪⎩ δveq f (p f − ˆp w ) dΓ = 0.
∂ˆv f eq
{Γ w ∩Γ F }
The application of the same mathematical manipulations, used in paragraph 3.2, transforms the
first integral formulation into the boundary integral formulation
∫
∫
∫
δp (ˆv n w − ¯v n ) dΓ+ δp w (ˆv n w − ¯Z −1 ˆp w ) dΓ− δp (ˆv n w −1
+ ¯Z
f
ˆp w + ˆv eq) f dΓ = 0. (27)
Γ w v
Γ 1 Z
{Γ w ∩Γ F }
The small perturbations δp and δv f eq are defined in (17) and (24).
5. The hybrid FE-WB model follows from
• the generation of the element FE models according to step 4 in paragraph 3.1,
• the assembly of all element FE models according to step 5 in paragraph 3.1,
• the WB model generation according to step 4 in paragraph 3.2
• and the assembly of the FE submodel and the WB submodel, where the contributions with the
unknown frame pressure p f cancel out each other.
The hybrid FE-WB model is given by
⎡
(−ω 2 M + iω (C + C E f ) + K) iω ⎤ ⎧
QE f 0
p ⎢
⎣ iω Q E T
⎪⎨
E ⎧ ⎫
⎫⎪
−iω v n
⎬ ⎪⎨ ⎪⎬
f
0 −iρω Q w f T ⎥
⎦ v F = 0
0 Q w f (A w v + A w Z + Qw fb ) ⎪⎩ ⎪ ⎭ ⎪⎩ ⎪⎭
q w
b w v
(28)
n f
with C E f = ⋃
f=1
Q w fb = ∫
∫
{Γ e ∩Γ f }
{Γ w ∩Γ F }
n f
ρ N T ¯Z−1 f
N dΓ, Q E f = ⋃
∫
( i
Φ T ρω nT ∇Φ −
f=1
{Γ e ∩Γ f }
¯Z
−1
f Φ )
dΓ and Q w f = ∫
ρ N T N f dΓ,
{Γ w ∩Γ F }
Φ T N f dΓ,
where C E f , QE f , Qw fb and Qw f represent the frame damping matrix, the FE-frame coupling matrix, the
frame back-coupling matrix and the WB-frame coupling matrix. The column vector v F collects all
frame DOFs.

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1637
4 Numerical validation example
4.1 Numerical models for 2D acoustic car cavity
Figure 3 shows the bounded 2D acoustic problem of a car cavity filled with air (ρ = 1.225 [kg/m 3 ] and
c = 340 [m/s]). The walls are rigid (¯v n =0 [m/s]) except for (i) the roof, where mixed boundary conditions are
imposed ( ¯Z = 2000 [Pa.s/m]), and for (ii) the firewall, where a unit normal velocity distribution (¯v n =1 [m/s])
excites the acoustic cavity. The 2D acoustic car cavity has a rather complex shaped boundary.
This paper considers the following four types of models (see figure 3).
• A quadratic FE model with 32982 DOFs serves as reference model. It is derived from the original FE
model by subdividing each element in 36 quadratic triangular elements.
• FE models with linear triangular elements are derived from the original FE model by subdividing each
element in smaller elements.
• WB models of 18 subdomains are considered for various values of the user defined truncation parameter
T .
• FE subdomains replace the small WB subdomains in the above WB models, which results in the hybrid
FE-WB models. These models use a fixed truncation parameter of T = 2 for the WB submodel,
whereas their FE submodels are refined gradually. Furthermore, the user defined frame impedance is
fixed to ¯Z f = ρ c.
4.2 Pressure spectrum analysis
It is common practice to predict the pressure response at a discrete position for a broad frequency band. This
paragraph considers such a pressure spectrum analysis for the position near the driver’s ear indicated by in
figure 3. Figure 4 shows the prediction results obtained with
• the quadratic FE model (reference),
• a linear FE model with 1007 DOFs (FEM),
• a WB model with T = 2 (WBM)
p
Z
n
original FE model
of 449 elements
WB model
of 18 subdomains
hybrid FE-WB model
Figure 3: Problem definition and numerical models of 2D acoustic car cavity

1638 PROCEEDINGS OF ISMA2004
• and a hybrid FE-WB model with T = 2 (maximum of 218 WB DOFs at 1000 [Hz]), with 303 FE
DOFs and with 36 frame DOFs (HFE-WBM).
The following phenomena are observed.
• The FEM suffers from dispersion errors from approximately 400 [Hz] onwards. This causes the overestimation
of the resonance frequencies.
• The WBM suffers hardly from any dispersion error. at upper end of the However, some erroneous
predictions occur for the low frequencies. At those frequencies, the WBM has not converged sufficiently
with T = 2. This confirms that the WBM computational efficiency is less pronounced for
rather complexly shaped domains.
• The HFE-WBM suffers from dispersion errors but less pronounced as the FEM. It predicts some
erroneous results in the low frequency range but for different frequencies than the WBM since the WB
submodel has not converged yet. The WB domain subdivision is not optimal in the sense that most
WB subdomains are still irregularly shaped.
The pressure spectrum analysis illustrates that the HFE-WBM is capable of providing more accurate approximations
for higher frequencies than the FEM. However, with the current models, the FEM is more accurate
in the low-frequency range. Another WB domain subdivision may improve the HFE-WBM accuracy. This
is topic of further research.
10 3 reference
10 2
10 1
FEM
|| p || in [Pa]
10 3
10 2
10 1
reference
WBM
10 3
reference
10 2
10 1
HFE-WBM
0 200 400 600 800 1000
frequency in [Hz]
Figure 4: Pressure spectrum prediction for 2D acoustic car cavity

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1639
4.3 Convergence analysis
In order to quantify the computational efficiency, an a posteriori error analysis is performed at three discrete
frequencies, namely at 215 [Hz], 525 [Hz] and 955 [Hz]. Figure 5 shows the corresponding convergence
curves, which give the averaged relative pressure error 〈ɛ〉 as function of the model size. The averaged
relative pressure error 〈ɛ〉 is defined as
〈ɛ〉 =
n rp
∑
j=1
A j
A
‖ˆp j − ˆp ref,j ‖
, (29)
‖ˆp ref,j ‖
where ˆp j represents the predicted pressure at response point r j , ˆp ref,j the corresponding reference solution,
A j the area surrounding response point r j and A the total area. The n rp = 279 response points are indicated
by + in figure 3.
The convergence analysis is based on
• the quadratic FE model, which provides the reference solution ˆp ref,j ,
• the linear FE models (FEM),
• the WB models (WBM)
• and the hybrid FE-WB model with T = 2 (HFE-WBM).
The convergence behaviour of the three methods is frequency dependent. The following phenomena are
observed.
10 0
WBM
215 [Hz]
10 -1
HFE-WBM
averaged relative pressure error [-]
10 -2
10 0
10 -1
10 -2
10 0
WBM
WBM
HFE-WBM
FEM
525 [Hz]
FEM
955 [Hz]
10 -1
HFE-WBM
FEM
10 -2
10 2 10 3 10 4
number of DOFs
Figure 5: Convergence for 2D acoustic car cavity

1640 PROCEEDINGS OF ISMA2004
• At the low frequency of 215 [Hz]:
– The WBM convergence curve exhibits a non-monotonic behaviour, which does not allow to draw
firm conclusions on the computational efficiency of the WBM compared with the FEM.
– The HFE-WBM convergence curve exhibits a more monotonic behaviour, but it converges to the
wrong solutions. The WB submodel size is fixed by using T = 2. However, at this low frequency
the WB part of the model is not converged yet with T = 2. Refinement of the WB submodel will
restore the initial convergence rate.
• At the mid frequency of 525 [Hz]:
– The WBM convergence curve is less oscillatory than at 215 [Hz]. The WBM is computationally
more efficient than the FEM in the sense that (i) the same accuracy is obtained with substantially
smaller models and that (ii) it exhibits on average a higher convergence rate.
– The HFE-WBM is computationally more efficient than the FEM in the sense that the same accuracy
is obtained with smaller models.
• At the high frequency of 955 [Hz]:
– The WBM is computationally most efficient.
– Less data is used for the FEM convergence curve, because the FEM suffers severely from dispersion
errors. Consequently, large FE models are necessary to control these errors.
– The computational efficiency of the HFE-WBM lies between those of the WBM and the FEM.
The same accuracy as the FEM is obtained with smaller HFE-WBM models. However, the
convergence rate of the HFE-WBM is comparable with the one of the FEM. This suggest that
the FE submodel governs the overall accuracy.
The convergence analysis illustrates that the HFE-WBM provides more accurate approximations for higher
frequencies as compared to the FEM. Obviously, this improved computational efficiency at the higher frequencies
is obtained without loss of geometrical flexibility, since the regions near complexly shaped boundaries
can be modelled by the FEM and the interior, more regularly shaped regions by the WBM.
5 Conclusions and recommendations for future research
This paper considers the extension of the frequency application range of the deterministic FE methods for
steady-state acoustic problems. Several improvements of the FEM and several indirect Trefftz methods
have been proposed in literature to achieve this. This paper focuses on the hybrid coupling of the FEM
with the WBM, which is a promising alternative for the FEM. The resulting HFE-WBM benefits from the
advantageous features of both methods, which are the high geometrical flexibility of the FEM and the high
computational efficiency of the WBM. This paper reviews the theory on the FEM and WBM and it provides
a theoretical foundation for the new HFE-WBM.
A pressure spectrum analysis and a convergence analysis have been performed on a numerical validation
example of a 2D acoustic car cavity. The results of these analyses illustrate that the HFE-WBM is capable
to provide more accurate results at higher frequencies than the FEM. Furthermore, the HFE-WBM exhibits
more geometrical flexibility than the WBM.

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1641
The promising results form the motivation for the further development of the HFE-WBM. Future research
will focus on
• the influence of the domain subdivision on the low frequency accuracy, especially the influence of the
WB domain subdivision,
• the influence of the user defined frame impedance ¯Z f
• and the feasibility of a direct hybrid coupling approach, which does not involve an additional frame
approximation.
Acknowledgements
The research work of Bas van Hal and Caroline Vanmaele is financed by scholarships of the Institute for
the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen). Part of the
presented work is performed during the research stay of Petra Silar at K.U.Leuven in the framework of the
EDSVS - European Doctorate in Sound and Vibration Studies, which is supported by the European Union
through the Marie Curie Multi-Partner Training Site EC program.
References
[1] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method - Volume 1: Basic formulation and linear
problems, McGraw-Hill (1989).
[2] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method - Volume 2: Solid and Fluid Mechanics,
Dynamics and Non-Linearity, McGraw-Hill (1991).
[3] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Applied Mathematical Sciences, Vol. 132,
Springer (1998).
[4] I. Harari, T.J.R. Hughes Galerkin/least-squares finite element methods for the reduced wave equation
with non-reflecting boundary conditions in unbounded domains, Computer methods in applied mechanics
and engineering, Vol. 98 (1992), pp. 411–454.
[5] I. Babu˘ska, F. Ihlenburg, E.T. Paik, S.A. Sauter, A Generalized Finite Element Method for solving the
Helmholtz equation in two dimensions with minimal pollution, Computer methods in applied mechanics
and engineering, Vol. 128 (1995), pp. 325–359.
[6] J.M. Melenk, I. Babu˘ska, The partition of unity finite element method: Basic theory and applications,
Computer methods in applied mechanics and engineering, Vol. 139 (1996), pp. 289–314.
[7] Ph. Bouillard, S. Suleau, Element-Free Galerkin solutions for Helmholtz problems: formulation and
numerical assessment of the pollution effect, Computer methods in applied mechanics and engineering,
Vol. 162 (1998), pp. 317–335.
[8] L.P. Franca, C. Farhat, A.P. Macedo, M. Lesoinne, Residual-free bubbles for the Helmholtz equation,
International Journal for Numerical Methods in Engineering, Vol. 40 (1997), pp. 4003–4009.
[9] C. Farhat, I. Harari, L.P. Franca, The discontinuous enrichment method, Computer methods in applied
mechanics and engineering, Vol. 190 (2001), pp. 6455–6479.
[10] E. Trefftz, Ein Gegenstück zum Ritzschen Verfahren, in Proceedings of the 2 nd International Congress
of Applied Mechanics, Zürich (1926), pp. 131–137.

1642 PROCEEDINGS OF ISMA2004
[11] E. Kita, N. Kamiya, Trefftz method: an overview, Advances in Engineering Software, Vol. 24 (1995),
pp. 3–12.
[12] J. Jirousek, W. Wróblewski, T-elements: State of the Art and Future Trends, Archives of Computational
Methods in Engineering – State of the art reviews, Vol. 3 (1996), pp. 323–434.
[13] S. Finnveden, Exact spectral finite element analysis of stationary vibrations in a rail way car structure,
Acta Acustica, Vol. 2 (1994), pp. 461–482.
[14] U. Orrenius, S. Finnveden, Calculation of wave propagation in rib-stiffened plate structures, Journal
of Sound and Vibration, Vol. 198 (1996), pp. 203–224.
[15] M. Stojek, Finite T-elements for the Poisson and Helholtz equations, Ph.D. thesis, École Polytechnique
Fédéral de Lausanne, Lausanne (1996).
[16] J.A. Teixeira de Freitas, Hybrid-Trefftz displacement and stress elements for elastodynamic analysis
in the frequency domain, Computer Assisted Mechanics and Engineering Sciences, Vol. 4 (1997),
pp. 345–368.
[17] W. Desmet, A wave based prediction technique for coupled vibro-acoustic analysis, Ph.D. thesis,
Katholieke Universiteit Leuven, Leuven (1998).
[18] L. Arnaud, P. Ladevèze, C. Blanzé, First application of the variational theory of complex rays to forced
vibrations of plates in the medium-frequency range, in B.H.V. Topping, editor, Computational Mechanics:
Techniques and Developments (2000), pp. 121–128.
[19] W. Desmet, Mid-frequency vibro-acoustic modelling: challenges and potential solutions, in Proceedings
ISMA2002 – Noise and Vibration Engineering, Leuven (2002), pp. 835–862.
[20] B. van Hal, W. Desmet, D. Vandepitte, P. Sas, A Coupled Finite Element – Wave Based Method for
the Steady-State Dynamic Analysis of Acoustic Systems, Journal of Computational Acoustics, Vol. 11
(2003), pp. 285–303.
[21] B. van Hal, W. Desmet, D. Vandepitte, P. Sas, Hybrid Finite Element – Wave Based Method for Acoustic
Problems, Computer Assisted Mechanics and Engineering Sciences, Vol. 10 (2003), pp. 479–494.
[22] W. Desmet, B. van Hal, P. Sas, D. Vandepitte, A computational efficient prediction technique for the
steady-state dynamic analysis of coupled vibro-acoustic systems, Advances in Engineering Software,
Vol. 33 (2002), pp. 527–540.
[23] B. Pluymers, W. Desmet, B. van Hal, D. Vandepitte, P. Sas, An object-oriented implementation scheme
for an efficient wave based prediction technique for steady-state vibro-acoustic problems, in Proceedings
of the Ninth International Congress on Sound and Vibration, Orlando (2002).
[24] B. van Hal, A. Hepberger, H.-H. Priebsch, W. Desmet, P. Sas, High performance implementation and
conceptual development of the wave based method for steady-state dynamic analysis of acoustic problems,
in Proceedings ISMA2002 – Noise and Vibration Engineering, Leuven (2002), pp. 817–826.