An iterative defect-correction type meshless method for acoustics

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2003; 58:000–000 (DOI: 10.1002/nme.757)

1

An iterative defect-correction type meshlessmethod for acoustics3

V. Lacroix1, Ph. Bouillard1;∗ and P. Villon2

1Universit�e Libre de Bruxelles; Continuum Mechanics Dept. CP 194=5; F.D. Roosevelt Av. 50;5B-1050 Brussels; Belgium

2Centre de Recherche de Royallieu; G�enie des Syst�emes M�ecaniques BP 20529 602057Compi�egne C�edex; France

SUMMARY9

Accurate numerical simulation of acoustic wave propagation is still an open problem, particularly formedium frequencies. We have then formulated a new numerical method better suited to the acoustical11problem: element-free Galerkin method (EFGM) improved by appropriate basis functions computed bya defect correction approach.13One of the EFGM advantages is that the shape functions are customizable. Indeed, we can construct

the basis of the approximation with terms that are suited to the problem which has to be solved.15Acoustical problems, in cavities with boundary �, are governed by the Helmholtz equation completedwith appropriate boundary conditions.17As the pressure p(x; y) is a complex variable, it can always be expressed as a function of cos �(x; y)

and sin �(x; y) where �(x; y) is the phase of the wave in each point (x; y).19If the exact distribution �(x; y) of the phase is known and if a meshless basis {1; cos �(x; y); sin �(x;

y)} is used, then the exact solution of the acoustic problem can be obtained.21Obviously, in real-life cases, the distribution of the phase is unknown. The aim of our work is to

resolve, as a �rst step, the acoustic problem by using a polynomial basis to obtain a �rst approximation23of the pressure �eld phI (x; y). As a second step, from phI (x; y) we compute the distribution of the

phase �hI (x; y) and we introduce it in the meshless basis in order to compute a second approximated25pressure �eld phII(x; y). From phII(x; y), a new distribution of the phase is computed in order to obtaina third approximated pressure �eld and so on until a convergence criterion, concerning the pressure or27the phase, is obtained. So, an iterative defect-correction type meshless method has been developed tocompute the pressure �eld in .29This work will show the e�ciency of this meshless method in terms of accuracy and in terms of

computational time. We will also compare the performance of this method with the classical �nite31element method. Copyright ? 2003 John Wiley & Sons, Ltd.

KEY WORDS: acoustics; Helmholtz equation; dispersion error; element-free Galerkin method; meshless33method

∗ Correspondence to: Ph. Bouillard, Universit�e Libre de Bruxelles, Continuum Mechanics Dept. CP 194=5,F.D. Roosevelt Av. 50, B-1050 Brussels, Belgium.

† E-mail: [email protected]

Contract=grant sponsor: R�egion WallonneContract=grant sponsor: Commissariat g�en�eral aux Relations Internationales (CGRI)

Received 17 October 2001

Revised 10 May 2002Copyright ? 2003 John Wiley & Sons, Ltd. Accepted 4 December 2002

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2 V. LACROIX, Ph. BOUILLARD AND P. VILLON

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1. INTRODUCTION1

The numerical solution of the Helmholtz equation, governing the wave propagation, is oneof the main problems that has not yet been properly addressed because of spurious phenom-3ena inherent to this di�erential operator. To compute the acoustic response several numericalmethods are used like the �nite element method (FEM) and the boundary element method5(BEM). Nevertheless, these methods present some disadvantages: the FEM su�ers from pol-lution (error on the amplitude) and dispersion (error on the phase) phenomena widely studied7among others by Babu�ska, Ihlenburg and Bouillard [1–4] (Figure 1); the BEM needs signif-icant computational time because it works with full, complex and non-symmetrical matrices.9More recently, Zienkiewicz [5] has classi�ed the short wave acoustic problems amongst thestill unsolved problem of the �nite element method (FEM).11Several authors have then suggested methods to stabilize the FEM: the Galerkin least square

(GLS) [6], proposed by Harari and Hughes, consists of a modi�cation of the variational prob-13lem in order to minimize the dispersion, the quasi-stabilized �nite element method (QSFEM)[7], by Babu�ska and Ihlenburg, modi�es the system matrix with the same goal, but is restricted15to regular meshes of square elements, and more recently a residual-free �nite element method(RFFEM) [8] was implemented for the Helmholtz equation by Franca et al. However, none17of these methods eliminates the dispersion in a general two-dimensional case, see Reference[9] for a complete analysis.19Moreover, on one hand Hughes has proposed another alternative to solve acoustic prob-

lems by formulating a multiscale FEM [10] which seems to give good results and on the21other hand Brandt and Livshits have formulated a multigrid method to solve the Helmholtzequation [11].23In order to decrease dispersion and pollution, several high-order formulations have also

been developed. Amongst them, the hp-FEM by Demkowicz and Gerdes [12], the reproducing25Kernel particle method (RKPM) by Liu and Christon [13, 14] and, simultaneously, Suleau andBouillard applied classical Element-free Galerkin method (EFGM) to acoustics [15, 16].27Nevertheless, everybody seems to agree that it is very advantageous to use a set of plane

wave solutions of the homogenized Helmholtz equation as the local function basis. A natural29and very e�cient way to achieve this is to use a meshless formulation. Babu�ska and Melenk[17] have developed the partition of unity method (PUM), Chadwick and Bettes suggest the31use of a set of plane wave to build the basis of the subspace [18], Farhat et al. have proposeda discontinuous generalized FEM [19] while Lacroix et al. formulated a new EFGM approach33showing very accurate results [20].The EFGM is based on the moving least square approximation (MLSA), �rst introduced35

by Lancaster et al. [21] in the �eld of surface and function smoothing. Then, the MLSA hasbeen extended by Nayroles et al. to develop the di�use element method [22]. Recently, the37EFGM has been extensively investigated by Belystchko et al. in the �elds of elasticity andcrack propagation problems [23, 24]. The main advantages of the formulation are well known39(no connections by nodes, easy pre- and postprocessing tasks). For the particular case of theHelmholtz equation, we also take advantage of the fact that the shape functions are non-41rational and the local basis can naturally contain terms which are solutions of the Helmholtzequation [25].43In our work, as the pressure is a complex variable, terms in cos �(x; y) and sin �(x; y) are

introduced in the meshless basis, where �(x; y) is the value of the phase of the pressure �eld45

Copyright ? 2003 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2003; 58:000–000

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MESHLESS METHOD FOR ACOUSTICS 3

Figure 1. Illustration of the dispersion.

in each point (x; y). Seeing that �(x; y) is a priori unknown, it has �rst to be evaluated by a1�rst computation of the pressure �eld with a classical polynomial basis. In this way, with thisnew meshless basis a second evaluation of �(x; y) can be obtained and so until convergence.3The paper is organized as follows. Sections 2 and 3 present the strong and variational forms

of the acoustic problem. In Section 4, the EFGM shape functions are de�ned and the method5is applied to acoustics. Principles and details of the iterative defect-correction type meshlessmethod (I2M) are presented in Section 5. Section 6 deals with numerical results showing7performances of the method.

2. STRONG FORMULATION OF THE ACOUSTIC PROBLEM9

Consider a uid inside a domain with boundary �, let c be the speed of sound in the uid and � the speci�c mass of the uid. If p′ denotes the �eld of acoustic pressure (small11perturbations around a steady uniform state), the equation of wave propagation (1) is derivedfrom the fundamental equations of continuum mechanics [2].13

�p′=1

c2@2p′

@t2(1)

If the phenomena are assumed to be steady harmonic, e.g.15

p′=p exp( j!t) (2)

where ! is the angular frequency, then the spatial distribution p of the acoustic pressure17(which now is a complex variable) inside , is the solution of Helmholtz equation

�p+ k2p=0 (3)19


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where the wave number k is de�ned by the ratio between the angular frequency and the speed1of sound:

k=!

c(4)3

Another important quantity of the acoustic analysis is the particle velocity (vector v) linkedto the gradient of the acoustic pressure through the equation of motion5

j�ckv+∇p=0 (5)

In order to completely address the acoustic problem, Helmholtz equation (3) is associated7with boundary conditions on �. The boundary can be split for interior problems into threeparts9

�=�D ∪�N ∪�R (6)

corresponding to di�erent types of boundary conditions11

• Dirichlet boundary conditions

p= �p on �D (7)13

• Neumann boundary conditions

vn= �vn or nt∇p=−j�ck �vn on �N (8)15

• Robin boundary conditions

n′∇p= − j�ckAnp on �R (9)17

where An is the admittance coe�cient modelling the damping.

The Neumann boundary conditions correspond to vibrating panels while the Robin boundary19conditions correspond to the absorbant panels. Conditions (7)–(9) have been de�ned forinterior and exterior problems. For an in�nite medium, a non-re ecting wave is considered at21in�nity by the so-called Sommerfeld boundary condition.

3. VARIATIONAL FORMULATION OF THE ACOUSTIC PROBLEM23

The variational formulation corresponding to the strong form presented in Section 2 is wellknown and in the following, only the main aspects will be emphasized. For more details, see25Reference [13].The space of admissible trial functions p is de�ned as27

H 1D()= {p∈H 1() |p= �p on �D} (10)

and the space of homogeneous test functions w is29

H 10 ()= {w∈H 1() |w=0 on �D} (11)


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Both of them are subspaces of H 1(), the Sobolev space of functions square-integrable to-1gether with their �rst derivatives. Consider the functional �

�= 12a(p; p)− ’(p) (12)3

with

a(p; p) :H 1D×H 1

D→C|a(p; p) =∫

(∇p∇p− k2pp) d +∫

�R

j�ckAnpp d� (13)

’(p) : H 1D→C|’(p) =−

∫

�N

j�ckvnp d� (14)

where the notation • stands for the complex conjugate.5The variational form corresponding to Helmholtz equation (3) and boundary conditions

(7)–(9) is expressed by7

Find p∈H 1D|��=0 ∀�p∈H 1

0 (15)

It will be shown in Section 4 that, in the case of the EFGM, the approximation does not9interpolate the nodal values. The variational formulation has to be accordingly modi�ed totake into account Dirichlet boundary conditions (7) for instance by introducing any penality11method like Lagrange multipliers � in functional (12)

�∗=�+

∫

�D

�(p− �p) d� (16)13

and variational form (15) is reformulated as

Find p∈H 1|��∗=0 ∀�p∈H 10 ; ��∈H 0 (17)15

Note that the Dirichlet boundary conditions and their treatment by Lagrange multipliers haveonly been mentioned for completeness. In real-life acoustic problems, this kind of boundary17conditions seldom appears. This method of introducing the Dirichlet boundary conditions hasbeen developed in previous papers [32,33] but more recent techniques and more e�cient than19Lagrange multipliers exist for EFG [26, 27].

4. ELEMENT-FREE GALERKIN METHOD APPLIED TO ACOUSTICS21

4.1. Element-free shape functions: the moving least square approximation

A complete report on the construction of the shape functions de�ning the EFGM can be found23in References [23, 24, 28]. This paragraph only gives a brief overview of the main steps.The MLSA is de�ned on a cloud of n nodes, which are not connected by elements as25

required for the FEM. The nodes are located at xI inside (I =1; : : : ; n). For each node I ,we de�ne a domain of in uence characterized by a typical dimension size din (in 1D, the27domain is a segment and din is its half length while in 2D, the domain is a disc of radiusdin or a square of half lengthside din ). These domains are de�ned to connect the nodes: two29nodes are connected if their domains of in uence intersect.


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A weight function wI is also de�ned for each node to represent the in uence of the node1xI at a given point x. This weight function is equal to unity at the node, decreasing when thedistance to the node increases and zero outside the domain of in uence of the node. For all3the computations reported in this paper, we have used an exponential weight function, thatcan be de�ned either on a square domain of in uence as the product of two one-dimensional5weight functions

wI (x; y)=

e−

(

2x−xIdin ; I

)2

− e−41− e−4

e−

(

2y−yIdin ; I

)2

− e−41− e−4

(x6din ; I and y6din ; I)

= 0 (x¿din ; I or y¿din ; I)

(18)

7

or on a circular domain as a function of d, the distance between point x and node xI

wI (x; y)=e−

(

2d

din ; I

)2

− e−41− e−4 (d6din ; I)

=0 (d¿din ; I)

(19)

9

The construction of the MLSA and the corresponding shape functions is based on the choiceof a basis P(x) (dimension m) of functions which, in the case of 1D polynomials, are11

Pt(x) = {1; x} (linear basis; m=2) (20)

Pt(x) = {1; x; x2} (quadratic basis; m=3) (21)

Polynomial bases are not the only choice: non-polynomial bases can also be chosen, intro-ducing better suited functions for solving the Helmholtz equation as will be seen further.13The unknown ph (acoustic pressure, the upper h standing for numerical solution) of the

problem is interpolated from15

ph(x)=Pt(x)a(x) (22)

where the a(x) coe�cients are non-constant and are determined by minimizing a L2 norm17(see References [22, 23]), leading to

a(x) = A−1(x)B(x)p (23)19

where p is the array of the nodal values pI . A(x) and B(x) are the matrices de�ned by

A(x) =n(x)∑

I=1

wI (x)P(xI)Pt(xI) (24)

B(x) = [w1(x)P(x1); : : : ; wn(x)P(xn)] (25)21


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where n(x) is the number of nodes in uencing the point (x). Equation (21) can then be1written as

ph(x)=N(x)p (26)3

where N(x) contains the shape functions and is de�ned by

N(x)=Pt(x)A−1(x)B(x) (27)5

At this point, we have to underline the fact that the m∗m matrix A(x) is the sum of matricesof rank 1. As wI (x) is zero for all nodes that do not in uence point x;A(x) is the sum of7only n(x) matrices of rank 1, where n(x) is the number of nodes in uencing x. The rankof A(x) must be equal to m since (27) needs the computation of A−1(x). This leads to the9necessary (but not su�cient) condition of existence of the MLSA: n(x)¿m, i.e. each pointof has to be in uenced by at least as many nodes as there are functions in the basis P(x).11

4.2. Application to the acoustic problem

The application of the EFG to the acoustic problem formulated in Sections 2 and 3 is com-13pletely detailed in Reference [28]. We choose to approximate the acoustic pressure �eld andits variation by15

ph=Np �ph=NTp (28)

while the Lagrange multipliers and their variation are chosen to be17

�h=N�� h=N�T� (29)

where N� is a Lagrange interpolant de�ned on the boundary.19Introducing (28)–(29) into variational form (17), a linear system of equations, similar to

the system obtained for a problem of structural dynamics, is obtained21

K+ j�ckC− c2k2M Kp�

Ktp� 0

{

p

�

}

=

{−j�ck f

b

}

(30)

where the matrices and vectors are de�ned as follows23

• the ‘sti�ness’ matrix K

K=

∫

(∇N)t(∇N) d (31)25

• the ‘damping’ matrix C (Robin boundary conditions)

C=

∫

�R

NtNAn d� (32)27

• the ‘mass’ matrix M

M =1

c2

∫

NtN d (33)29


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• the vector p of nodal pressure unknowns1• the vector � of nodal Lagrange multipliers unknowns• the matrix Kp�, coupling both kinds of unknowns3

Kp�=

∫

�D

NtN� d� (34)

• the vector f , containing the prescribed normal velocities (Neumann boundary conditions)5

f =

∫

�N

Nt �vn d� (35)

• the vector b, containing the prescribed values of the pressure (Dirichlet boundary con-7ditions)

b=

∫

�D

Nt� �p d� (36)9

5. ITERATIVE DEFECT-CORRECTION TYPE MESHLESS METHOD

5.1. Introduction of the phase in the meshless basis11

As mentioned in the introduction, the purpose of our work is to take the phase of the waveinto account to build the meshless basis. As the pressure is a complex variable, we can always13write in each point (x; y)

p(x; y)= �P(x; y)[cos �(x; y) + j sin �(x; y)] (37)15

where �P(x; y) is the amplitude of the wave and �(x; y) the phase.Therefore, if the distribution of the phase is exactly known over the whole domain and if17

the basis

Pt(x; y)= {1; cos �(x; y); sin �(x; y)} (38)19

is used, the obtained meshless solution is dispersion-free if the errors due to the numericalintegration are not considered.21Obviously, for real-life cases, the distribution of �(x; y) is a priori unknown. Thus, in the

latter, �(x; y) will be approximated by a distribution �h(x; y) obtained by a �rst computation23of the pressure �eld using, for instance, a linear polynomial meshless basis.

5.2. Iterative computations: principle25

In this section, the acoustic problem iterative resolution based on a �-adaptive meshless basisis presented step by step.27

5.2.1. First step: computation of �h(x; y). To introduce �(x; y) in the meshless basis wecompute a �rst approximation phI (x; y) of the pressure �eld by using, for instance, a classical29


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linear meshless basis1

Pt(x; y)= {1; x; y} (39)

because this solution is already very accurate [26] and needs a small CPU time.3Thus we obtain in each point of the domain, by splitting the real and imaginary parts of

the pressure5

phI (x; y)=phI; r(x; y) + jp

hI; i(x; y) (40)

From the general expression of the pressure given in (37) it immediately comes7

phI (x; y)= �PhI (x; y)[cos �hI (x; y) + j sin �

hI (x; y)] (41)

where9

cos �hI =phI; r

√

(phI; r)2 + (phI; i)

2and sin �hI =

phI; i√

(phI; r)2 + (phI; i)

2(42)

5.2.2. Second step: EFGM resolution with local basis. Consider now the meshless basis11de�ned by

Pt(x; y)= {1; cos �hI (x; y); sin �hI (x; y)} (43)13

with cos �hI (x; y) and sin �hI (x; y) coming from the �rst computation.

A new approximated pressure �eld phII (x; y) is computed by a EFGM with local basis (43).15Of course, this method can be iterated: a third approximation of the pressure can be computedby building a basis of type (43) with Equation (42) but by using phII (x; y) instead of p

hI (x; y)17

and so on until a convergence criterion is obtained, for instance, at iteration i√

∫

(pi − pi−1) (pi − pi−1) d

∫

pi−1pi−1 d

6U (44)19

or, if the convergence of �(x; y) is preferred, the following criterion is used√

∫

(�i − �i−1)2 d∫

�2i−1 d

6U (45)21

5.2.3. I2M algorithm. So, an iterative defect-correction type meshless method (I2M) has beendeveloped to compute the pressure �eld in . Figures 2(a) and 2(b) illustrate this method by23its algorithm according to the convergence criterion (44) or (45).

6. NUMERICAL RESULTS25

In this section, I2M is compared from the accuracy point of view with other methods alreadyused to solve acoustic problems like FEM, classical EFGM, etc. The behaviour of this new27method in relation to the acoustic dispersion phenomenon is also studied. But �rst, one hasto demonstrate that the iterative scheme of the method is well-founded.29


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Figure 2. (a) Iterative algorithm using convergence criterion (44); and(b) iterative algorithm using convergence criterion (45).


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Figure 3. Square cavity: plane wave propagation.

6.1. Square cavity1

This �rst numerical test deals with a square cavity (Figure 3) in which a plane wave propa-gates. The analytical solution of this problem is known and given by (46)3

p(x; y)= cos k(x cos �+ y sin �) + j sin k(x cos �+ y sin �) (46)

where � is the propagation angle. For information, the linear system (30) is solved here with5a Gauss–Jordan algorithm because of the Dirichlet b.c. enforced by Lagrangian multipliers.

6.1.1. Behaviour of the error with iterations. For this result, two discretizations are con-7sidered: 441 nodes (21× 21) and 1681 nodes (41× 41). Moreover, in order to analyse thein uence of the numerical integration error, two quadrature schemes will be used for the9�rst discretization: integration cells with 3× 3 Gauss points and with 10×10 Gauss points(size of the cells= h). The evolution of the L2 norm in relation to the number of iterations11(frequency=550 Hz) is represented in Figure 4.One can observe that for a given discretization the error decreases with the number of13

iterations until a saturation value depending on the considered discretization and the qualityof the quadrature scheme. Therefore, this example shows that the iterative principle of the15method is well-founded. Moreover, the error decreases when a re�ned discretization is usedi.e. I2M converges when h→ 0.17

6.1.2. Frequency response function. The second numerical test on the square cavity dealswith the frequency response function (FRF) in the middle of the square cavity. The FRF is19computed with linear FEM, linear basis meshless method and I2M limited to one iteration.The analytical FRF is also represented. These curves are shown in Figure 5 for the real part21of the pressure. The lower and upper bounds of the frequencies are 100 and 1500 Hz. Theresponse is given in dBa (ref. 2×10−5).23One can notice that I2M presents a good behaviour when the frequency increases over the

numerical description limit of the wave with linear FEM [2] i.e. h= �=√12. For information25


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Figure 4. Error in L2 norm is relative to the number of iterations.

the frequency corresponding to the classical rule of thumb for linear FEM [29] has also been1plotted i.e. h= �=6.

6.2. Bidimensional section of a car3

The second numerical example deals with a real-life problem in order to show the e�ciencyof I2M in cases where the solutions are no more plane but common waves. The problem5(Figure (6a)) is a 2D-section in the bodywork of a car [30]. The air inside the cabin is excitedby the vibrations due to the engine through the front panel (Neumann boundary conditions).7The roof is covered with an absorbent material (Robin boundary conditions). The acousticresponse inside the car is studied at a frequency of 200Hz with a discretization of 777 nodes9(Figure 6(b)). For this example, to solve the linear system (30), a QMR-type algorithm isused which is more suited for an e�cient resolution.11

6.2.1. Analysis of dispersion phenomenon. In order to analyse the behaviour of I2M in rela-tion to the dispersion phenomenon, four computations have been performed on this distribution13of nodes: two classical meshless computations with linear basis and cubic basis, I2M with oneiteration and a linear FEM computation. In order to compare the results, we use as reference15a FEM solution on a highly re�ned mesh (17859 nodes).Figure 7 presents the three computations for the distribution of the real part of the acoustic17

pressure inside the car at a frequency of 200Hz. The results are along the straight line de�nedin Figure 8.19First, one can immediately notice that the linear FEM result is subject to the dispersion

whereas this phenomenon is hardly reduced by meshless computations.21Moreover, one can observe that the I2M solution is more close by the reference than the

others i.e. I2M presents less pollution error with only one iteration. Nevertheless, the gain of23accuracy between the cubic basis and the I2M seems to be not very signi�cant in comparison


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Figure 5. FRF for the real part of the pressure in the middle of the square and zoom.

with all the intermediate steps and computations to obtain a I2M solution. To justify the using1of I2M, the computational time of I2M has to be analysed.

6.2.2. Computational time of I2M. Finally, the total computational time of the I2M solution3(computational time of linear basis solution added to the computational time of the secondsolution with the new basis) is lower than the one for the cubic basis solution as illustrated5in Figure 9.


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Figure 6. (a) Model problem; and (b) distribution of modes.

Figure 7. Real part of the pressure at 200 Hz along the straight line.

Figure 8. De�nition of the straight line.

One can explain that fact by considering that we work with Lagrangian polynomial sub-1spaces. Thus, the used cubic basis contains ten monomial terms whereas both linear basis andI2M-basis de�ned in (38) contain only three terms. Hence the domains of in uence of the3cubic basis method will be much more large than those of I2M to ensure the existence of theshape function matrix as mentioned in Section 4.1. It implies that the time needed to compute5the sti�ness and the mass matrices and to solve (30) (larger bandwidth) will be much moreimportant for the cubic basis than the one of I2M.7


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Figure 9. Comparison of the computational time between 13M and cubic basis.

In this example, I2M presents better performances in terms of accuracy and computational1time than the classical polynomial meshless method.

7. CONCLUDING REMARKS3

This paper has presented the research and the development of an iterative defect-correctiontype meshless method (I2M) to solve acoustic problems governed by the Helmholtz equation.5The idea of the I2M consists in introducing in the meshless basis the local phase of the

wave which is an intrinsic property of this acoustic wave. The way to build this new basis7can be extended to an iterative algorithm in order to compute more accurately the distributionof the phase and the pressure �eld.9The e�ciency of I2M has been demonstrated on a square cavity model example and on a

real-life case problem.11The �rst example demonstrates that the iterative principle of the method is well-founded i.e.

improvement of the solution with iteration. The FRF computed on this example has shown a13very good behaviour of I2M when the frequency increases and a weak sensitivity to the �rstapproximation of the pressure �eld is needed to compute the distribution of the phase. Indeed,15one can notice that the error on the �rst evaluation of the pressure �eld does not prevent onefrom obtaining a better approximation with the new basis.17The I2M presents very accurate solution in terms of dispersion and pollution error as

presented by the bidimensional section of a car example. And, in terms of computational19time, for a prescribed accuracy, I2M resolution is faster than classical meshless resolutionusing polynomial basis.21Finally, one must emphasize that the I2M can be easily extended to the resolution of 3D

problems because it is only based on the fact that the acoustic pressure is a complex variable.23

ACKNOWLEDGEMENTS25

The �rst author is supported by the R�egion Wallonne under grant SIVA and by the Commissariat g�en�eralaux Relations internationales (CGRI—Communaut�e fran�caise de Belgique). Free Field Technologies(FFT) is gratefully acknowledged for its collaboration.


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REFERENCES1

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An iterative defect-correction type meshless method for acoustics