424 boundary elements - wit press · todynamics, applications of triangular, constant and linear,...

A boundary element method for 3D

time-harmonic elastodynamics - numerical

aspects

J.J. Rego Silva, H. Power\ L.C. Wrobel

Wessex Institute of Technology, University of

Portsmouth, Ashurst Lodge, Ashurst,

Southampton S04 2AA, UK

Abstract

This paper presents an implementation of the boundary element methodfor 3D time-harmonic elastodynamics with higher order continuous/dis-continuous elements. In order to avoid the strong apparent singularitiesof the dynamic kernels, the singular integrations are carried out employ-ing a decomposition of the dynamic fundamental solutions into their staticcounterparts plus regular series. The static fundamental solutions are thennumerically evaluated using a technique appropriate to Cauchy principalvalue integrals for curved surface elements.

Introduction

The main objective of the present research work is the development of anadvanced boundary element method formulation for frequency-domain elas-todynamics which has a unique solution for all frequencies, i.e. which is freefrom fictitious eigen-frequencies (see Jones [1]). Since this formulation in-volves a hypersingular integral, it is essential that the numerical aspectsof its implementation are properly addressed because of the strong (realand apparent) singularities involved. As a first step we discuss herein thenumerical problems arising in a standard boundary element method imple-mentation for 3D elastodynamics using curved, higher order elements.

leave from Institute de Mecanica de Fluidos, Universidad Central de Venezuela.

Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

424 Boundary Elements

Boundary element methods were firstly applied to general transient elas-todynamics in 1968 by Cruse and Rizzo [2, 3], working in the Laplace trans-formed domain. Subsequent applications in the frequency domain have beenpresented by Niwa et al [4], Dominguez [5, 6], Manolis and Beskos [7], amongothers. Other researchers worked directly in the time domain, e.g. Cole etal [8], Mansur and Brebbia [9], Karabalis and Beskos [10] and Antes [11].

In this work, the family of elements presented in reference [12], con-sisting of continuous, discontinuous and transition type elements, is imple-mented for 3D time-harmonic elastodynamics. The first concept of familyof boundary elements was presented by Lachat [13] for elastostatics, follow-ing the ideas developed earlier by Cruse [14, 15] and Cruse and Vanburen[16]. Later on, this concept was well established by Lachat and Watson[17, 18], Rizzo and Shippy [19], Cruse and Wilson [20] and Watson [21].The first type of element used in 3D elastodynamics was the constant el-ement. In their pioneering work Cruse and Rizzo [2, 3] approximated thesurface data by piecewise-constant values. Despite its simplicity, the rect-angular constant element has also been successfully applied in the frequencydomain by Dominguez and Abascal [22] and Karabalis and Beskos [23] fordynamic soil-structure interaction. This type of element has the advantageof simplifying the evaluation of singular boundary integrals [24]. In 3D elas-todynamics, applications of triangular, constant and linear, elements can befound in Chang [25], who has also introduced a semi-analytical removal ofsingularities.

It is known, however, that analytical integration is applicable only toflat elements. In advanced implementations, employing higher order ele-ments, the evaluation of Cauchy principal value (CPV) integrals requiresmore attention. In this regard, Rizzo et al [26] developed a boundary inte-gral formulation which succeeded in eliminating the strong singularity viaregularization. Alternatively, Guiggiani [27] proposed another approach toevaluate Cauchy principal value integrals directly for curved surface ele-ments.

The asymptotic behaviour of the dynamic kernel functions is exactlyrepresented by their static counterparts [28]. However, another importantissue in time-harmonic elastodynamics is the apparent singularities thatmay arise in the kernels involved, independent on the technique applied to

• the evaluation of the CPV integral. This problem has been acknowledgedby Kobayashi [29] and Kitahara et al [30] who expanded the fundamentalsolutions into regular series when employing second order elements. Thisnumerical problem is also addressed in this paper and results will be pre-sented to justify the procedure implemented.


Boundary Elements 425

Mathematical Formulation

Governing equations

The equilibrium equations for a homogeneous, isotropic, linearly elasticbody, with domain fi (fl,- or Qe) and boundary 5, in the absence of bodyforces and under time-harmonic dynamic forces are

(cf - c])Ui + c]uj,a = -J*Uj (1)

known as Navier-Cauchy equations for steady-state elastodynamics, whereU{ — Ui(x,w) is the displacement vector at each point x G fi and u is thecircular frequency.

The propagation velocities of longitudinal and transverse waves are de-fined as

2 A + 2Et 2 EtC —— _^____ /» — __/ "~ t ""0 £

in which p is the mass density, A and Et are the Lame's constants:

^ = 2(TTO *=(! +.0(1-2*)

with EI being the longitudinal elastic modulus and v the Poisson's coeffi-cient.

The corresponding constitutive equation is given by

and the stress vector on a differential element with normal n, is

/o\Pj = (TijHi . (6)

Boundary integral formulation

The Navier-Cauchy equations for steady-state elastodynamics, equation (1),can be expressed in integral form as

- f p*ij(z,y)ui(y)<*s, , (4)J S

for every x G fli, known as the transformed dynamic equivalent to 's identityof elastostatics. In this equation, u*j(x,y) and p*j(x,?/) are the fundamental



solution of equation (1) and its associated traction tensor, respectively, givenby

-Xr^) (5)

with

, 1 1 \e-""r c?/ 1 12 I ~- 2~2 / (G)r cf \iKir -*-* '

/ , 3 \ a-'""" c? / 3 3 \ e-'««' ,„.X = 1 + j-r I j 1 + J-T ) (7)

* ««H/ r cf \ t/cjr K^r^y r

and

with

d$ _( . 2 3 3 \ e-', — I %/C( - r -f- 5 r Iar \ r z/CfH /c/r / r

3" - + ' -c r

. _ 4

dr V * r

c? /. 4 9 9 \ e- "" .^+ -L[iKt+- + -.—j - -J-T ' , (10)

GI \ r z/c/r K^r / r

where the wave numbers %& are defined as /c^ = w/c^, with t, = /, £. In theabove equations, the outward normal is taken as positive and r is definedas|x — y|.

The stress state at a point x G ft, can be obtained by combining thederivatives of equation (4), with respect to the coordinates of x, and thensubstituting the result into Hooke's law, as follows

,;(*)= /< ( ,2Js



where

(12)\ar 7- / J

and

\ dr* r dr r dr r*/

5 d\ 7

" r^~~ H

( 3 C / Y 6

y2 r c(r r cfr r* J '* * '*

^ Id0 ld\ 2+2 ~ "IT " —r + "Tr dr r dr r*

with

7 12 12 \ e-'"' H—r 4""—T —

/ / I r\t-t I l\>4 I / I

and

o 5i/ct 17 36 36 \ e~^"dr'* r

-2— . (15)r

Equation (1) can also be written in the following form

Py tf, 2/)«;(j/) = / «r.(f , y)n(y)dS, , (16)

obtained from equation (4) by taking its limit when the point x G 0, tendsto the point <f € 5. The integral on the left-hand side is to be evaluated inthe sense of Cauchy principal value (CPV).



The coefficient C,j(f) in equation (16) is a non-complex constant whichdepends on the boundary geometry and its orientation with respect to theglobal coordinate system [31]. When the boundary is smooth at point £ thevalue of Ct-j is given by

C«(0 = f - (17)

A special treatment can be given to the fundamental solutions. Expand-ing the exponential term e~""" in series of the form

n=0

the functions (6) and (7) can be written as

</> = </>, + I'd (19)

with

andX = X, + Xd (22)

with

Their first and second derivatives, with respect to r, can be obtained directlyfrom the above expressions.

Substitution of , and %, and their derivatives into equations (5) and(8) results in the Kelvin fundamental solutions for the static case

(26)



Thus, the terms given by equations (21) and (24) represent the dynamiceffects.

The straightforward conclusion from this result is that the steady statedynamic solution will tend to the static one when w tends to zero [28]. Itcan also be seen that the singularity of the kernels u*j(x, y) and p (z, y) areof O(l/r) and O(l/r*) respectively, as in the static case, since the dynamiccounterparts are regular at u\r = 0.

Evaluation of the CPV integral

In equation (16), the Cauchy principal value of the singular integral willbe evaluated applying the technique proposed by Guiggiani [27] (see alsoGuiggiani and Gigante [32]). This technique is the numerical implemen-tation of the formal mathematical definition given by Mikhlin [33] for aCauchy principal value integral as follows

lim / p*A£,y)uj(y) dSy (27)

where £ 6 e« G S.

Here, eg is the subregion of 5 containing the singularity, defined as eg ={£ G 5 | r < c}, and the shape e« must be preserved throughout thelimiting process.

Considering 8S as a subregion of 5 containing eg the limit in (27) maybe written as

PL(WKW <#„ 4- lim (f,Z/KW <#, (28)(S-6S) *-~"J(8S-e,)

Following Guiggiani [27], 6S in (28) is chosen as the portion of the surfaceS (set of elements) that contains the singular point <f and the limiting pro-cess is then carried out in such portion, already mapped to the intrinsiccoordinate system (7/1,7/2), after the boundary discretization.

Considering that the boundary geometry is represented by shape func-tions # in terms of nodal coordinates, each singular element of SS is hencemapped into a regular element R (quadrilateral or triangular). Accordingly,the subregion eg in the real space is mapped into a subregion <7g about thesource point £.

Defining a polar coordinates system (/?, 0) centered at the source point

f, i.e.m = 6 +



7/2 = 6 + psm(0)

dr]id7]2 = pdpdO , (29)

the Cauchy principal value integral is therefore evaluated as

= / pytf, »)«,-J(s-ss)

Jo Jo [ ' P J Jo

after the discretization of the boundary, considering that the surface dataare approximated over each element by interpolations functions $ and theirnodal values.

In equation (30), function E(p,0) represents the singular integrand inthe following form

E(P,9) = P'a(t,y)*\J\P (31)

with J being the jacobian of the mapping and p the jacobian of the transfor-mation to polar coordinates, that is expanded in terms of its correspondingTaylor's series for small values of p as

E(p,6) = + 0(l), (32)

where the most singular part of the integrand is given by

which corresponds to the most singular part of the static counterpart.

The terms that need to be expanded are 1/r^, r,,, <& and the productsn,-J, rij J; only the first term of each series is needed. The Taylor's seriesexpansions of these functions are given in references [27] and [32].

All integrals in equation (30) are in polar coordinates defined in the localplane, allowing for their evaluation by standard quadrature rules.

Numerical approach

Although the singular behaviour of the dynamic kernels are exactly de-scribed by their static counterparts when r—»0, apparent singularities of

0 (f Tpr) and 0 (J JT), m u£(f,y) and p*j(t,y) respectively, will appear

when K^r — » 0 i f expressions (6) and (7) are employed. Therefore, their



numerical calculations is a reason of concern for small K.J. Alternatively, inthis case, expressions (21) and (24) can be employed once the series are notsingular. According to our results, convergence of the series (21) and (24)is very quick when K^r is small. However, in spite of being regular even forlarge /c^r (except for K^r —+ oo), convergence of these series is slow for largeK^r. Hence, the use of equations (6) and (7) seems to be more suitable forhigher frequencies.

In the present work, the procedure adopted was to apply the seriesexpansions only for the integrations over the elements that contain the sin-gularity. Doing so, their convergence can then be assured and the CPVintegral is evaluated locally with no need to evaluate the static counterpartover the whole domain, as done in references [29] and [30]. This procedurehas shown to be efficient even for large values of wave numbers.

Numerical examples

Cantilever Beam

In this example, a cantilever beam is subject to a dynamic concentrated loadP(XZ) = 3000e""' applied at its free end. The Poisson's ratio was taken as0, the Young modulus as 2.1 x 1CF and the density as 2. Figure 1 shows theproblem geometry and the mesh employed. The cantilever was discretizedusing 18 quadrilateral quadratic isoparametric elements, with h = 3, 6 = 1and / = 10.

Figure 1: Cantilever beam - boundary element mesh.

Firstly the static case was analysed. Two different configurations for theboundary element mesh were considered: the first employed semi-continuouselements to avoid discontinuities in the normal direction, corresponding to



126 nodes; in the second, fully discontinuous elements were used, corre-sponding to 162 nodes. Figure 2 shows the displacement 1*3 along the xiaxis in both cases. It can be seen that the results improved with fully dis-continuous elements, as expected, since it is known that bending problemsgenerally require finer discretizations.

analytical solutionOOOOO mesh 1: 126 nodes (720 sec)ooooo mesh 2: 162 nodes (970 sec)

0 2.0 4.0 6.0 8.0 10.0 12.0-0.025

Figure 2: Displacement 1*3 along x\ axis - static case

Fully discontinuous elements were then employed for the dynamic case.The displacements u$ along the x\ axis obtained with frequencies equal tow = Irad/s, w = IQrad/s and w = IQQrad/s are plotted in figures 3, 4and 5, respectively.

0.025

0.020

0.015

0.010

0.005

0.000

OOOOO dynamic fundamental solution (2070 sec)ooooo static solution + series (1300 sec)

a

,00

>°oO°'

0.0 2.0 4.0 6.0 8.0 10.0 12.0Z

Figure 3: Displacement \u^\ along x\ axis - w = Irad/s



0.025 -a OOOOO dynamic fundamental solution (2070 sec)ooooo static solution + series (1300 sec)

0.020

0.015

0.010

0.005

0.000

Figure 4

iofti0 2.0 4.0 6.0 8.0 10.0 12.0

Z

Displacement \u^\ along x\ axis - w = IQrad/s

0.120 q ooooo dynamic fundamental solution (2070 sec)OOOOO static solution + series (1300 sec)

Oo

Oo

oo

oo

oo

oo

oo

o

0.100

0.080

0.060

0.040

0.020

0.000 nO"0.0 2.0 4.0 6.0 8.0 10.0 12.0

Z

Figure 5: Displacement \u^\ along x\ axis - w = 100rad/s

From the results obtained, it can be concluded that the value w — 1is small enough to cause numerical problems when expressions (6) and (7)are applied. This is only due to the strong apparent singularities whichcannot be properly treated with standard algorithms for CPV integrals.For large values of k^r the results are in good agreement; however, theseries expansions are less time consuming.



The times shown in figures 3 to 5 are the average run-times for theseproblems. The series have converged to a tolerance of 10" . The differ-ence in run-time between the two approaches appears to be due to thedynamic fundamental solutions being complex and more complicated, sincethe necessary number of integration points for the evaluation of the singularintegrals is very large, but the same in both cases. The regular series requiremuch less integration points. All problems were run in double precision ina SUN sparcstation with 32Mb RAM memory.

Scattering by a Spherical Cavity

For scattering problems, the total exterior displacement field is given by

^(4 = W + <W , (34)

where uf(x) is the incident displacement field and uf (x) is the scatteredone, for every x 6 fie-

In this case, the boundary integral equation (16) may be extended to ex-terior domains since the Sommerfeld's radiation condition is automaticallysatisfied by the fundamental solution [34, 35]. This gives

(35)

for every f G 5.

In this problem, it is not necessary to evaluate the single layer termbecause the traction vanishes on the cavity surface. Equation (35) is thusreduced to

C.XfNf) 4- f%(W%,W = %f(f) . (36)

The mesh adopted in this example, shown in figure 6, consists of 16 fullycontinuous isoparametric quadratic elements: 4 triangles and 12 quadrilat-erals, corresponding to 73 nodes. Due to the symmetry of the problem, onlyone quarter of the sphere needs to be discretized. Symmetry is consideredby reflection and condensation with no discretization of the symmetry axes[36].

The radial and tangential displacements obtained for a plane longitu-dinal incident wave, travelling along the 23 direction with unit amplitudeu*(x] = e**'*3 , scattered by a spherical cavity of radius a, is shown in figure7 for /c/a = 0.3, «/a = 0.7 and K/a = 1.3. The Poisson's ratio was takenas v — 0.25. The results are in very good agreement with the analyticalsolution given in [37].


Boundary Elements

u

435

Figure 6: Spherical cavity - boundary element mesh.

o ko = 0.3A ko = 0.7o ka = 1.3

Figure 7: \Ur\ and \Ut\ for *,a = 0.3, /c/a = 0.7, fc/a = 1.3.



Conclusion

This paper has presented an implementation of the BEM for 3D time-harmonic elastodynamics using higher order elements. Based on our ex-periments the most efficient strategy appears to be to use the dynamic fun-damental solutions for all regular integrations, and its decomposition intostatic fundamental solutions plus regular series for all singular integrals.

We are now modifying our formulation for external problems to makeit free from fictitious eigenfrequencies. The new implementation, in whichapparent singularities are even stronger, will be based on the experiencedescribed in this paper.

Acknowledgment

The first author would like to acknowledge the financial support of CNPq,Brazil.

References

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