Use of constant, linear and quadratic boundary elements in ...

Use of constant, linear and quadratic boundary elements in 3D wavediffraction analysis

A. Tadeu* , J. Antonio

Department of Civil Engineering, Faculty of Science and Technology, University of Coimbra, 3030 Coimbra, Portugal

Received 8 September 1999; accepted 29 November 1999

Abstract

The performance of the Boundary Element Method (BEM) depends on the size of the elements and the interpolation function used.However, improvements in accuracy and efficiency obtained with both expansion and grid refinement increases demand on the computa-tional effort. This paper evaluates the performance of constant, linear and quadratic elements in the analysis of the three-dimensionalscattering caused by a cylindrical cavity buried in an infinite homogeneous elastic medium subjected to a point load. A circular cylindricalcavity for which analytical solutions are known is used in the simulation analysis. First, the dominant BEM errors are identified in thefrequency domain and related to the natural vibration modes of the inclusion. Comparisons of BEM errors are then made for different types ofboundary elements, maintaining similar computational costs. Finally, the accuracy of the BEM solution is evaluated when the nodal pointsare moved inside linear and quadratic discontinuous elements.q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Boundary elements; Constant; Linear and quadratic elements; Wave propagation

1. Introduction

The solution of how waves propagate between a sourceand a receiver placed below the ground has occupiedresearchers for years. Some of the first analytical studieson wave diffraction and scattering were concerned withthe problem of wave motion and reverberations in alluvialbasins of regular shape [1,2], and with wave scatteringinduced by cavities [3–6]. More recently, semi-analyticalmethods have been used to analyse wave diffraction causedby geological irregularities of arbitrary shape withinglobally homogeneous media [7–9]. By contrast, the appli-cation of purely numerical methods (i.e. finite elements ordifferences combined with boundaries) has been restricted,for the most part, to situations where the response isrequired only within localised irregular domains, such assoil–structure interaction problems [10–12]. Discretemethods have also occasionally been used to model largealluvial basins, but only in plane-strain [13]. Finally, hybridmethods involving a combination of finite elements, tomodel the interior domain containing the inhomogeneities,and semi-analytical representations for the exterior domainhave been used [14].

The application of these numerical methods has mostlybeen restricted to situations where the solution is requiredwithin two-dimensional (2D) domains. The evaluation ofthe full scattering wave field generated by sources placedin the presence of three-dimensional (3D) propagationmedia requires the use of computationally demandingnumerical schemes.

The solution becomes much simpler if the medium is 2D,even if the dynamic source remains 3D, a point load, forexample. Such a situation is frequently referred to as a two-and-a-half-dimensional problem (2-1/2-D), for whichsolutions can be obtained by means of a two spatial Fouriertransform in the direction in which the geometry does notvary. This requires solving a sequence of 2D problems withdifferent spatial wavenumberskz: Then, using the inverseFourier transform, the 3D field can be synthesized.

This solution is known in closed form for inclusions withsimple geometry, such as a circular cylinder, for which thewave equation is separable. However, if the inclusion has anirregular cross-section it is more difficult to obtain thesolution. In this case, the Boundary Element Method(BEM) is possibly the best tool for analyzing wave propa-gation in unbounded media, because it automaticallysatisfies the far-field radiation conditions and allows acompact description of the medium in terms of boundaryelements placed at the material’s discontinuities [15–18].

Engineering Analysis with Boundary Elements 24 (2000) 131–144

0955-7997/00/$ - see front matterq 2000 Elsevier Science Ltd. All rights reserved.PII: S0955-7997(99)00064-8

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* Corresponding author. Tel.:1 351-239-797202; fax: 1 351-239-797190.

The BEM solution at each frequency is expressed interms of waves with varying wavenumberkz; (with zbeing the direction in which the geometry does not vary),which is subsequently Fourier transformed into the spatialdomain. The wavenumber transform in discrete form isobtained by considering an infinite number of virtual pointsources equally spaced along thez-axis and at a sufficientdistance from each other to avoid spatial contamination[19]. In addition, the analyses are performed using complexfrequencies, shifting the frequency axis down, in thecomplex plane, in order to remove the singularities on (ornear) the axis, and to minimize the influence of theneighboring fictitious sources [20].

The accuracy of the BEM solution depends on thenumber of the boundary elements used to discretize thematerial discontinuities and on the nodes inside eachelement [21,22]. The BEM solution improves as the orderof the element increases and its size decreases. However, the

improvement in accuracy and efficiency that can beobtained by using higher order elements is offset by theincreased computational cost in CPU time. Thus, whilethe response improves with the number of nodes perelement, this is not necessarily useful, because of theincreased computational expense that these more accuratemodels entail.

The present work assesses the benefit of using constant,linear and quadratic elements to calculate the displacementfield around a circular cylindrical cavity buried inside an elas-tic medium, for which the solution is known in closed form.

This paper first formulates the problem and brieflypresents the equations required to solve the BEM problem,while its analytical solution is addressed in Appendix A.Then, the BEM errors occurring in the 3D scatteringanalysis of a cylindrical circular cavity are identified inthe frequency domain and correlated to those related tothe natural vibration modes. In these analyses, differentnumbers of constant, linear and quadratic elements areused, according to different ratios of the incident wavewavelength to the length of the boundary elements. Resultscomputed with a similar number of nodal points arecompared, thus keeping the computational cost essentiallyconstant. Thereafter, the performance of discontinuouslinear and quadratic elements is analyzed when the positionsof the nodal points inside the boundary elements are movedaround in the vicinity of those used in the Gauss–Legendrenumerical integration.

2. Problem formulation

Consider a cylindrical irregular cavity of infinite extent,buried in a spatially uniform elastic medium (Fig. 1),subjected to a harmonic dilatational point source at position�x0;0;0�; oscillating with a frequencyv: The incident field

A. Tadeu, J. Anto´nio / Engineering Analysis with Boundary Elements 24 (2000) 131–144132

Y

X

Zr

θ

y

x

z

O

r0

(x,y,z)

(Xo,0,0)

z

α

ρ

Fig. 1. Geometry of the problem.

1

+ 1 -10

φξ

ξ= 0 .00

1

-10

φ1

φ21

ξ

+ 1

ξ +-= 0 .57735

0 -1+ 1

1

1φ3

φ

1

2

φ1

ξ

ξ +-= 0 .77459

ξ= 0 .00

Fig. 2. Interpolation functions and position of the nodal points.

can be expressed by means of the now classical dilatationalpotentialf .

finc � Aei va �at2��x2x0�21y21z2p

��x 2 x0�2 1 y2 1 z2p �1�

in which the subscriptinc denotes the incident field,A is thewave amplitude,a is the compressional wave velocity ofthe medium, andi � ��

21p

:

Defining the effective wavenumbers

ka ��v2

a2 2 k2z

s; Im ka , 0 �2�

by means of the axial wavenumberkz; and Fourier-trans-forming Eq. (1) in thez direction, one obtains

f inc�v; x; y; kz� � 2iA2

H�2�0 �ka��x 2 x0�2 1 y2

q� �3�

in which theH�2�n �…� are second Hankel functions of ordern.If one considers an infinite number of virtual point

sources equally spaced along thez direction at a sufficientdistance,L, from each other to avoid spatial contamination[19], the incident field may be written as

pinc�v; x; y; z� � 2pL

X∞m�2 ∞

pinc�v; x; y; kz� e2ikzmz �4�

with kzm� �2p=L�m: This equation converges and can beapproximated by a finite sum of terms.

3. Boundary element formulation

The BEM is used to obtain the 3D field generated by acylindrical cavity subjected to spatially sinusoidal harmonicline loads defined by Eq. (3). The fundamental equationsunderlying the application of boundary elements to wavepropagation are well known [23]. It is therefore enough tostate here that the application of the method in the frequencydomain requires for the type of scattering problem presented

herein the evaluation of the integral

Hklil �

ZCl

fHil �xk; xs; ns� dCl �i; l � 1; 2;3� �5�

in which Hil �xk; xs;ns� are the traction components at thepoint xk in direction i caused by a concentrated load actingat the source pointxs in direction l. Also, ns is the unitoutward normal for thelth boundary segmentCl and fare the interpolation functions.

Expressions for the tensions may be obtained from the 2-1/2-D fundamental solution Gil ; by taking partial derivativesto deduce the strains and then applying Hooke’s law toobtain the stresses. The displacement functions that applyto the present case can be written as follows:

Gxx � B0

(�2ikz�2H�2�0 �kbr�1 k2

b

�"

2 H�2�0 �kbr�1B1

kbr2

2r2x

2r2x

B2

#)�6�

Gyx � Gxy � B0 2k2b

2r2x

2r2y

B2

� �� 7�

Gzx� Gxz� B0 kb2r2x

B1

� �� 2ikz� �8�

Gyy � B0

(�2ikz�2H�2�0 �kbr�1 k2

b

�"

2 H�2�0 �kbr�1B1

kbr2

2r2y

2r2y

B2

#)�9�

Gzy� Gyz� B0 kb2r2y

B1

� �� 2ikz� �10�

Gzz� B0b 2 k2bH�2�0 �kbr�1 �2ikz�2H�2�0 �kar�c �11�

with kb ��v2

b2 2 k2z

s; r �

��x2 1 y2

q; j � ka

kb;

B0 � i4rv 2 ; Bn � �H�2�n �kbr�2 j nH�2�n �kar��

n� 1;2

After mathematical manipulation of the integralequations, combined and subjected to the continuity con-ditions at the interface between the two media, and discre-tized appropriately, a system of equations is obtained thatcan be solved for the nodal displacements. The requiredintegrations in Eq. (5) are performed using Gaussian quad-rature when the element to be integrated is not the loadedelement. For the loaded element, the existing singular inte-grands are carried out in closed form [18].

The system of equations obtained is fully populated and

A. Tadeu, J. Anto´nio / Engineering Analysis with Boundary Elements 24 (2000) 131–144 133

Y

X

Zrθ y

x

z

O

(x,y,z)

(0,0,0)

z

5.0m

1 .0m

s/m4208=α

s/m2656=β

3m/kg2140=ρ

Fig. 3. Circular cylindrical cavity in an unbounded elastic medium. Mediumproperties.


-0.10

-0.05

0

0.05

0.10

0 1000 2000 3000 4000

RealImaginary

Frequency (Hz)

Am

plitu

de

10-6

10-4

10-2

100

0 1000 2000 3000 4000

Frequency (Hz)

Err

or

10-6

10-4

10-2

100

0 1000 2000 3000 4000

Frequency (Hz)

Err

or

10-6

10-4

10-2

100

0 1000 2000 3000 4000

Frequency (Hz)

Err

or

normal modesR

3R

2R

1

a) Analytical Solution

b) BEM Solution

Constant elements

c) BEM Solution

Linear elements

d) BEM Solution

Quadratic elements

Fig. 4. Scattering response at receiver 1�kz � 1 rad=�m=s��:

in general not symmetric. As a consequence, the globalcomputational cost of the BEM mainly represents the timerequired to solve the system of equations. In this paper,different types of boundary elements are used. The perfor-mance of the different solutions is evaluated by comparingthe results obtained for similar computational cost; that is,involving equation systems of the same size (i.e. using asimilar number of nodal points).

The displacement and stress variations within a boundaryelement are defined as a function of the nodal values.Discontinuous boundary elements are used to deal withthe traction discontinuity existing at the corner betweentwo boundary elements. This consists of moving to theinside the nodes that would meet at the corner [24]. Fig. 2presents the interpolation functions used to model the inclu-sions used in the simulation analysis.

4. Analytical solution

The scattering field produced by a circular cylindricalcavity, placed in a homogeneous elastic medium, subjectedto a point dilatational load can be defined in a circularcylindrical coordinate system (r, u , z) and evaluated by


Table 1Relationsl=L used in the calculations

Interpolation function R1 R2 R3

Constant 6 12 18Linear 3 6 9Quadratic 2 4 6

-3-2

-10

12

3 2

4

6

80

0.1

0.2

0.3

0.4

0.5

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de

a) 29 elements b)

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de

15 elements c) 10 elements d)

Fig. 5. Horizontal displacements�kz � 1 rad=�m=s��: Frequency of excitation—2058.6 Hz: (a) analytical solution; (b) modulus of the BEM error (R1)—constant elements; (c) modulus of the BEM error (R1)—discontinuous linear elements; and (d) modulus of the BEM error (R1)—discontinuous quadraticelements.

using the separation of variables method, as brieflydescribed in Appendix A [25,26].

5. Analytical versus boundary element solution

The benefit of using higher order elements is evaluated bycalculating the displacement field around a cylindricalcircular cavity buried in an infinite homogeneous space.The cavity is struck byP waves elicited by a dilatationalpoint load applied at pointO, as in Fig. 3.

Simulation analyses have been performed for a broadrange of frequencies and inclusions. However, given theimpossibility of presenting all the results, a restrictednumber of simulations will be used to illustrate themain findings.

6. Position of the BEM errors in the frequency versuswavenumber domain

Fig. 4 gives some of the scattering results obtained at onereceiver placed atx� 2:5 m and y� 4:0 m; hereinafterdesignated receiver 1. The results are computed for 1000frequencies, in the range 4.0–4000 Hz. The response wascalculated both analytically and by using the BEM,discretizing the boundary with constant, linear and quadraticdiscontinuous elements, as shown in Fig. 2. The positions ofthe nodal points are the same as those used in Gauss–Legendre numerical integration (Fig. 2). Displacements inx, y and z directions were computed for a wide range ofwavenumberskz; which were then be used to obtain the3D solution by means of (fast) inverse Fourier transforminto space. To illustrate the results, only the horizontal


-3-2

-10

12

3 2

4

6

80

0.1

0.2

0.3

0.4

0.5

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de

a) 58 elements

b)

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de



displacements forkz � 1:0 rad=�m=s� will be given. Fig. 4agives the exact values for the horizontal displacement. Fig.4b–d, illustrate the error occurring with the BEM solutionwhen constant, linear and quadratic elements are used. Toenhance the difference in the responses the error is displayedon a logarithmic scale. Three different relations (R1, R2, R3)between the wavelength (l ) of the shear waves and thelength of the boundary elements (L) were considered, aslisted in Table 1. In all cases, the number of the nodal pointsis at least 24. The positions of the lower natural modes,given by solving Eq. (A6), are also displayed.

Analysis of the results shows that the BEM accuracy ismuch poorer at frequencies in the vicinity of the naturalmodes, for which big peaks of error occur. The constantelements are conspicuously bad in these localized frequencydomains. Outside these domains, all the responses improve

as the number of the nodal points increases. This waspredictable since an element with a greater number ofnodes can approximate the response better, because it canmodel variations in displacement and/or traction much morerealistically. It can further be observed that, as the numberof boundary elements increases, the position of the errorpeaks moves towards the values of the frequenciessatisfying Eq. (A6). This behavior can be explained by thefact that the increased number of elements allows the BEMmodel to approach the definition of the circular cylinderbetter, with respect to its dynamic behavior.

In addition, it can be observed that for higher relations ofl=L (R2 andR3), the BEM errors obtained, using linear andquadratic elements, have a global tendency to decrease asthe frequency increases. That is to say, a better degree ofaccuracy is therefore expected for the higher frequencies


-3-2

-10

12

3 2

4

6

80

0.1

0.2

0.3

0.4

0.5

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de

a) 87 elements b)

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.05

0.1

0.15

0.2

y(m)x(m)

Am

plitu

de



because, to satisfy a specific relation ofl=L; more boundaryelements are required for those excitation frequencies thanfor lower ones. The number of the elements itself seems tobe an important factor, in addition to the number of elementsrequired to satisfy a specific relation ofl=L:

7. Behavior of BEM solution in the vicinity of the naturalmodes

In order to get a better picture of the behavior of theinfluence of the natural modes on the BEM solution, thescattering response was subsequently calculated over afine grid, placed around the cavity. In the following examplethe inclusion is struck by a pulsating source vibrating at asingle frequency atkz � 1:0 rad=�m=s�; which is either 2058.6or 2450 Hz. These two frequencies correspond to one of thenatural modes of the real system (Eq. (A6)) and a frequency

outside the vicinity of any eigen-frequency, respectively. Theinclusion is modeled with discontinuous elements, usingconstant, linear and quadratic interpolation functions. Thenodal points are not at the end of these elements, but rathershifted inside them, to allow nodal points in which only thestresses relative to the element itself to be considered.

The number of elements was chosen to allow comparisonwith results evaluated at similar computational cost (i.e. thesame number of nodal points). In these calculations threedifferent ratios between the wavelength of the shear wavesand the length of the boundary elements were considered, aslisted in Table 1.

Figs. 5–7 illustrate the analytical response and themodulus of the error occurring with the BEM solution fora frequency of 2058.6 Hz, when different relationsl=L areused (R1,R2, andR3). As can be seen, the agreement betweenthe BEM and the analytical solution is not good whenconstant elements are used to model the inclusion (Figs.


-3-2

-10

12

3 2

4

6

80

0.1

0.2

0.3

0.4

0.5

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.01

0.02

0.03

0.04

0.05

y(m)x(m)

Am

plitu

de

a) 104 elements b)

-3-2

-10

12

3 2

4

6

80

0.01

0.02

0.03

0.04

0.05

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.01

0.02

0.03

0.04

0.05

y(m)x(m)

Am

plitu

de


Fig. 8. Horizontal displacements�kz � 1 rad=�m=s��: Frequency of excitation—2450 Hz: (a) analytical solution; (b) modulus of the BEM error (R3)—constantelements; (c) modulus of the BEM error (R3)—discontinuous linear elements; and (d) Modulus of the BEM error (R3)—discontinuous quadratic elements.

5b, 6b and 7b). Notice that the BEM error is significant, withvalues similar to those given by the analytical solution. Itcan further be observed that the BEM solution does notimprove as refinements are made to the boundary elements(i.e. changing the number of elements). Indeed, when theinclusion is modeled using 87 constant elements the errorincreases (see Fig. 7b). Notice that as the number ofboundary elements increases, the BEM model defines thebehavior of the circular inclusion better, approaching, inthis case, one of its natural modes� f � 2058:6 Hz�: As theconstant boundary elements perform poorly in the vicinity ofthe natural modes, they cause the error to increase, because itapproaches the behavior of one of the natural modes. If thenumber of the boundary elements is further increased (notshown here), at greater computational cost, the error startsdecreasing. For a low number of nodal points (30) the quad-ratic elements seem to perform poorly in the vicinity of theinclusion (see Fig. 5d). As the number of elements increases,

the quadratic elements improve rapidly, and when theresponse is calculated with 87 nodal points its performanceis similar to that of the linear elements (see Fig. 7c and d).

Fig. 8 displays the analytical response and the BEM errorwhen the vibrating source is excited with a frequency of2450 Hz, and the number of the boundary elements usedto model the inclusion satisfies the values ofl=L listed inTable 1, columnR3. Again, each plot results from computa-tions performed with a similar number of nodal points,which is assumed to require similar computational cost.There are several interesting features to be noted in theseresults, which are strikingly different from the previous ones(i.e. those evaluated for a frequency of 2058.6 Hz). Theconstant elements now perform much better. However, theuse of linear interpolating functions is still advantageous(see Fig. 8c).

The examples described depict the behavior of the BEMwhen a source with a specific frequency and wavenumber is


Fig. 9. Horizontal displacements at receiver 1 in the frequency versus wavenumber domain: (a) analytical solution; (b) modulus of the BEM error (R1)—constant elements; (c) modulus of the BEM error (R1)—discontinuous linear elements; and (d) modulus of the BEM error (R1)—discontinuous quadraticelements.

excited. Indeed, as presented earlier, the 3D solution infrequency requires the calculation of a broad range of wave-numbers (see Eq. (4)), while the time solution additionallyrequires the response for a set of frequencies. To illustratethe performance of the BEM in the resolution of thisproblem, simulation analyses are next performed tocalculate the full 3D response at receiver 1. Computationsare performed in the frequency range (40–4000 Hz), with afrequency increment of 40 Hz. The spatial period con-sidered in the analysis isL � 2Ta � 210m: The cavity ismodeled with a number of boundary elements that changeswith the frequency excitation of the harmonic load satisfy-ing the ratiosR1 andR3 (see Table 1). The minimum numberof nodal points is kept at a minimum of 24. Fig. 9a displaysthe amplitude of the analytical horizontal displacement inthe frequency versus wavenumber domain at receiver 1.Notice, that values ofkz in excess tov=a correspond toinhomogeneous, evanescent waves which decay rapidly in

space. The BEM errors, illustrated in Figs. 9b–d and 10b–d,when the relationl=L assumes the valuesR1 and R3,respectively, have features similar to those observed in theprevious cases. The position of the peak errors in these plotsagrees with the solutions given by Eq. (A6), when thenumber of boundary elements increases. The quadraticelements perform poorly when the number of elements isdefined by the ratioR1 (see Fig. 9d). Again, the linearelements outperform constant and quadratic boundaryelements, especially at low spatial wavenumbers (see Figs.9c and 10c).

8. Position of nodal points inside linear and quadraticelements

In the above examples, the nodal points placed along thediscontinuous elements were chosen to coincide with those


Fig. 10. Horizontal displacements at receiver 1 in the frequency versus wavenumber domain: (a) analytical solution; (b) modulus of the BEM error (R3)—constant elements; (c) modulus of the BEM error (R3)—discontinuous linear elements; and (d) modulus of the BEM error (R3)—discontinuous quadraticelements.

in a Gauss–Legendre numerical integration. Simulationsperformed using different positions for the nodal pointssuggest that this is a good choice. To demonstrate thisassertion, consider the wave propagation problem describedearlier. The scattered field is again computed for acylindrical circular cavity over a fine grid placed around

the inclusion. Fig. 11 displays the BEM error for thehorizontal displacement for a frequency of 2058.6 Hz, anda relation of l=L set to R3 (see Table 1), for differentlocations of nodal points around those given by theGauss–Legendre numerical integration, and when theinclusion is modeled with linear and quadratic elements,


a)

-3-2

-10

12

3 2

4

6

80

0.005

0.01

0.015

0.02

y(m)x(m)

Am

plitu

de

b)

-3-2

-10

12

3 2

4

6

80

0.01

0.02

0.03

y(m)x(m)

Am

plitu

de

ξ=±0.7 ξ=±0.85

-3-2

-10

12

3 2

4

6

80

0.005

0.01

0.015

0.02

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.01

0.02

0.03

y(m)x(m)

Am

plitu

de

ξ=±0.57735 ξ=±0.77459

-3-2

-10

12

3 2

4

6

80

0.005

0.01

0.015

0.02

y(m)x(m)

Am

plitu

de

-3-2

-10

12

3 2

4

6

80

0.01

0.02

0.03

y(m)x(m)

Am

plitu

de

ξ=±0.5 ξ=±0.7

Fig. 11. Horizontal displacements�kz � 1 rad=�m=s��: when the nodal points are placed at different positions inside of each element. Frequency of excitation—2058.6 Hz: (a) modulus of the BEM error (R3)—discontinuous linear elements; (b) modulus of the BEM error (R3)—discontinuous quadratic elements.

respectively. In all the analyses, the interior nodal point ofthe quadratic elements is placed at the center of theboundary segment.

The accuracy obtained with the linear elements variesconspicuously with the position of the nodal points. It canbe further observed that the solution improves as the nodalpoints are moved to the positions used in the Gauss–Legendre integration. The quadratic elements’ results donot appear to be greatly affected when the nodal pointsare moved around the vicinity of the Gauss–Legendrenumerical integration points. However, as the nodal pointsmove to the extremity of the boundary element, the BEMsolution clearly loses accuracy.

Simulation analysis performed when the source vibratesat 2450 Hz (not illustrated here) demonstrates similarbehavior. The BEM error was subsequently tested fordifferent sized cylindrical circular cavities, for a broadrange of frequencies and spatial wavenumbers, whichdemonstrated similar conduct.

9. Conclusions

This paper evaluates the benefits gained by using discon-tinuous elements, (in which the nodal points are inside eachelement), to model 3D elastic environments with constant,linear and quadratic interpolation functions. Simulationanalyses were performed using circular cylindrical models,namely, cavities. The number of boundary elements usedchanged according to the different relations between thewavelength (l ) of the incident waves and the length ofthe boundary elements (L).

Comparison of the BEM results with those obtained byanalytical solution revealed the importance of using linearelements instead of constant and quadratic elements tocalculate the displacement field. It has been illustrated that

the linear boundary elements outperform the constant andquadratic elements, especially at frequencies in the vicinityof natural modes.

It was also observed that the position of the nodalpoints inside the linear elements significantly influencesthe accuracy of the displacement solution at frequenciesin the vicinity of the natural modes of vibration.Locating the nodal points to coincide with the Gauss–Legendre numerical integration points was found to be agood choice.

Appendix A

This appendix presents in condensed form the closed-form solution to evaluate the 3D field generated by adilatational point source in the presence of a circularcylindrical cavity of infinite length submerged in anelastic medium.

Consider a spatially uniform elastic medium of infiniteextent, having a cylindrical cavity (Fig. 12). Decomposingthe homogeneous wave equations for elastic media bymeans of the now classical dilatational potentialf andshear potentialsc , x , one is led to the three scalar waveequations in these potentials, with associated wavepropagation velocitiesa , b , and b , respectively. For aharmonic dilatational point source at an off-center position�x0;0; 0� that is oscillating with a frequencyv , the scalarwave equations go over into three Helmholtz equationswhose solution can be expressed in terms of the singledilatational potential for theincidentwaves, together witha set of potentials forscattered waves in the elasticmedium.

The incident dilatational potential, after beingFourier-transformed Eq. (A1) in thez direction, can beexpressed by

finc�v; x; y;kz� � 2iA2

H �2�0 �ka

��x 2 x0�2 1 y2

q� �A1�

This equation can be written in terms of waves centered atthe origin,

finc�v; r ; u; kz� � 2iA2

X∞n�0

�21�nenH�2�n �kar0�Jn�kar� cosnu

whenr , r0 (A2)

finc�v; r ; u;kz� � 2iA2

X∞n�0

�21�nenH�2�n �kar�Jn�kar0� cosnu

whenr . r0 (A3)

in which theJ�2�n �…� are Bessel functions of ordern, u is the


Y

X

Zrθ

y

x

z

O

r0

(x,y,z)

a

(Xo,0,0)

z

α

ρ

Fig. 12. Geometry of the problem.

azimuth, and

en �12

if n � 0

1 if n ± 0

8><>: �A4�

r ��x2 1 y2

q� radial distance to the receiver

r0 ��x2

0

q� ux0u � radial distance to the source

cosu � x=r; sinu � y=r

In the frequency-axial-wavenumber domain, the scatteredfield can be expressed in a form similar to that of theincident field, namely

fssca�v; r ; u; kz� �

X∞n�0

AnH�2�n �kar� cosnu �A5�

c ssca�v; r ; u; kz� �

X∞n�0

BnH�2�n �kbr� sin n u

xssca�v; r ; u; kz� �

X∞n�0

CnH�2�n �kbr� cosn u

in which the subscriptscadenotes the scattered field,An, Bn,and Cn, are as yet unknown coefficients to be determinedfrom appropriate boundary conditions, namelysrr � sru �srz � 0; at r � a: The imposition of the three statedboundary conditions for each summation indexn thenleads to a system of three equations in the three unknownconstants. Having obtained the constants, we may computethe motions associated with the scattered field by meansof the well-known equations relating potentials anddisplacements.

Natural modes—The frequency and spatial wavenumberposition of the natural modes are found to be important tohow the BEM performs. The position of these natural modescan be evaluated, assuming that they are both incoming andoutgoing cylindrical waves propagating to and from thecenter of the cylinder. Imposing the resulting boundaryconditions,ur � uu � uz � 0 at r � a; in the presence ofthis standing field, the response is other than zero if thefollowing determinant is set to be zero. The solution ofthe resulting equation gives the required position of thenatural modes.

d11 d12 d13

d21 d22 d23

d31 d32 d33

�� 0 �A6�

where

d11 � na

Jn�a1a�2 a1Jn11�a1a��

d12 � na

Jn�b1a�

d13 � 2ikzna

Jn�b1a�2 b1Jn11�b1a��

d21 � 2na

Jn�a1a�

d22 � 2na

Jn�b1a�1 b1Jn11�b1a��

d23 � na

ikzJn�b1a�

d31 � 2ikzJn�a1a�

d32 � 0

d33 � b 21 Jn�b1a�:

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