Boundary integral equation for tangential

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2006; 66:334–363 Published online 12 December 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1563

Boundary integral equation for tangential derivative of flux in Laplace and Helmholtz equations R. Gallego∗,† and A. E. Martínez-Castro‡ Department of Structural Mechanics, University of Granada. Av. Fuentenueva, CP 18002 Granada, Spain

SUMMARY In this paper, the boundary integral equations (BIEs) for the tangential derivative of flux in Laplace and Helmholtz equations are presented. These integral representations can be used in order to solve several problems in the boundary element method (BEM): cubic solutions including degrees of freedom in flux’s tangential derivative value (Hermitian interpolation), nodal sensitivity, analytic gradients in optimization problems, or tangential derivative evaluation in problems that require the computation of such variable (elasticity problems in BEM). The analysis has been developed for 2D formulation. Kernels for tangential derivative of flux lead to high-order singularities (O(1/r 3 )). The limit to the boundary analysis has been carried out. Based on this analysis, regularization formulae have been obtained in order to use such BIE in numerical codes. A set of numerical benchmarks have been carried out in order to validate theoretical and practical aspects, by considering known analytic solutions for the test problems. The results show that the tangential BIEs have been properly developed and implemented. Copyright 䉷 2005 John Wiley & Sons, Ltd. KEY WORDS:

boundary integral equation; Cauchy principal value; Hadamard finite part; Laplace equation; Helmholtz equation; tangential derivatives; nodal sensitivity

1. TANGENTIAL BOUNDARY INTEGRAL EQUATIONS IN THE BEM Boundary integral equations (BIEs) for tangential derivatives of flux and potential is a high interest topic in the boundary element method (BEM) literature. Considering the standard BEM formulation (potential/displacement integral equation, u-BIE in what follows), tangential equations have been used in the literature in several fields: sensitivity analysis, error estimation,

∗ Correspondence

to: R. Gallego, Department of Structural Mechanics, University of Granada. Av. Fuentenueva, CP 18002 Granada, Spain. † E-mail: [email protected] ‡ E-mail: [email protected]

Copyright 䉷 2005 John Wiley & Sons, Ltd.

Received 4 March 2005 Revised 13 September 2005 Accepted 27 September 2005

BIE FOR FLUX’S TANGENTIAL DERIVATIVE

335

computation of tangential derivatives of potential (stress evaluation in elasticity problems), or new elements development based on cubic solutions. For fluxes, the hypersingular boundary integral equation (q-BIE) is required in specific problems (for instance, crack problems, mixed formulations, etc). It is necessary to establish a parallel scheme than the one set for the u-BIE: the BIEs for tangential derivatives of flux. The main difficulty involved in the derivative BIEs is the appearance of high-singular kernels. During the last years the number of papers dedicated to the evaluation of high-order singular integrals has increased. These singularities appear when the kernel functions of the standard BIE are differentiated at the collocation point. The review work by Tanaka et al. [1] shows the general techniques used in the BEM to solve the high-singularity integration (works up to 1994). One of them is the introduction of a Taylor series expansion. Aliabadi et al. [2] proposed the idea of a series expansion about the collocation point for singular kernels. Based on this work, Guiggiani [3–5] and Mantiˇc [6] presented a regularization method based on a Taylor series expansion. The regularization is done in the parametric plane. Gallego and Domínguez [7], based on Guiggiani’s idea, have used the radius variable (Euclid-distance) as the parameter of the series expansion. The work is done for 2D transient elastodynamics, for the antiplane problem. Sáez and Gallego [8] have extended the technique to 2D crack problems. Domínguez et al. [9] have applied this regularization method to three-dimensional potential and elasticity problems. The radial series expansion is used as a general technique, in which the regularization formulae are obtained as a result of a careful limit to the boundary analysis. Granados and Gallego [10] have extended this technique to nearly hypersingular integrals. Sensitivity analysis is a technique used for error estimation. The rate of change of two different BEM solutions obtained with two identical element meshes but different sets of collocation points leads to nodal sensitivities. Paulino et al. [11, 12], Guiggiani et al. [13], and Bonnet and Guiggiani [14] have developed theoretical and practical aspects of nodal sensitivity for the BEM. The standard BIE is used to solve the direct problem, and the tangential BIE is used in order to obtain the residual required to compute nodal sensitivities. Gallego and Martínez-Castro [15, 16] have presented sensitivity analysis for the hypersingular boundary integral equation of fluxes (qt -BIE) for potential problems (Laplace and Helmholtz equations). Recently, Muci-Küchler et al. have published a book on tangential derivatives for Hermitian solutions and error estimation [17]. In various works [18–22], the ut -BIE has been used in order to add an equation that leads to the cubic interpolation, by defining four degrees of freedom per element (as the usual Hermitian interpolation). Based on such solution, an error estimator is presented. In the current paper, tangential BIE for flux is obtained for 2D Laplace and Helmholtz equations. The qt -BIE is obtained through a careful limiting process. The limiting method is introduced previously by considering the tangential representation of potentials (ut -BIE). This method leads in a straightforward manner to the regularization formulae required in numerical codes. The radial series expansion has been used. Numerical tests show that practical and theoretical aspects have been correctly developed and implemented. Two types of tests have been considered: the direct computation of tangential derivatives and the comparison of nodal sensitivities computations considering two methods: finite perturbation approach (FPA) and direct differentiation approach (DDA).


Int. J. Numer. Meth. Engng 2006; 66:334–363

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R. GALLEGO AND A. E. MARTÍNEZ-CASTRO

2. BIE FOR TANGENTIAL DERIVATIVE OF POTENTIAL 2.1. ut -BIE for Laplace equation Let be an opened 2D domain, with boundary . For this domain, with reference to the Cartesian co-ordinate system {O; X1 , X2 } the problem considered is 2

2

* u(X1 , X2 ) * u(X1 , X2 ) + =0 *X12 *X22

in

u = u¯

on u

q = q¯

on q

(1)

with q = *u/*N. The mixed boundary conditions are applied to u and q . The union u ∪ q = , with u ∩ q = ∅. Let y be a smooth point of the boundary; thus y ∈ . Let be a domain such that = ∪v (see Figure 1). The domain v is defined through a circle centred at y with radius , noted as C , such that v = C − C ∩ . Thus, v ∩ = ∅. Let S be the part of the boundary of v exterior to . When the value of tends to 0, the domain tends to and the domain v tends to a semi-circle, if y is a smooth point of . Let y be a neighbourhood of next to the point y, such that it contains the common points S ∩ . Let s be the arc-length parameter and r the Euclid-distance between the collocation and observation points. This part of can be split into two intervals, defined by the radial values [0, R1 ] and [0, R2 ] (see Figure 1). For each interval, the application r to s must be single-valued. For a small interval it is possible to fulfil such a requirement. This condition is equivalent to dr = 0 ds

on y

(2)

The boundary can be split into two curves: ( − ) and , with the part of interior to (see Figure 1) = ( − ) ∪

(3)

A different decomposition can be done = ( − y ) ∪ y

(4)

Considering Figures 1 and 2, the following parameters are shown: r: y: x: : t: n:

distance between the collocation point and the observation point. collocation point. observation point. unit vector, pointing from y to x. tangential vector to the curve at x. outer normal to the curve at x.




337

Figure 1. Boundary = ∪ v .

Figure 2. General quadratic element.

T: tangential vector at y. N: normal vector at y. 0 : Curvature at y. The scalar and vector fields, described as functions of the radius r, can be expanded in a Taylor series, at the collocation point (r = 0). In Appendix B of this paper the expressions of such series, up to O(r 2 ) are shown. Copyright 䉷 2005 John Wiley & Sons, Ltd.


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The point y is interior to the domain . For this point, the standard boundary integral equation (u-BIE) is ∗ u(y) + q (y; x) u(x) d(x) = u∗ (y; x) q(x) d(x) (5) (− )+S

(− )+S

where d(x) denotes the arc differential over ( − ) + S . The kernel functions are u∗ (y; x) = −

1 ln(r) 2

(6)

1 q ∗ (y; x) = − ·n 2r The boundary integral equation for the derivative of u(X1 , X2 ) respect to T can be obtained by differentiation of Equation (5) with respect to T, leading to ut (y) + qt∗ (y; x)u(x) d(x) = u∗t (y; x)q(x) d(x) (7) (− )+S

(− )+S

where ut (y) =

*u(X1 , X2 ) *T

and u∗t (y; x) =

1 ·T 2r

(8)

qt∗ (y; x) =

1 (n · T − 2 · t · T) 2r 2

(9)

The limit to the boundary process leads to the tangential boundary integral equation (ut -BIE). Taking the limit of Equation (7) when → 0 one obtains ∗ ∗ ut (y) + lim qt (y; x)u(x) d(x) − ut (y; x)q(x) d(x) = 0 (10) →0

(− )+S

(− )+S

The boundary ( − ) + S can be decomposed as ( − y ) + (y − ) + S . The kernel functions are singular at r = 0, therefore in order to compute the limits of the integrals along the intervals (y − ) and S , a careful analysis of them has to be carried out. 2.1.1. Integral of the function qt∗ (y; x)u(x) over S . This integral is obtained by considering a local change of reference system. In Figure 1, the reference {y; x1 , x2 } is introduced. On such reference, a local polar co-ordinate system (r, ), is introduced, with ∈[−1 , + 2 ], 1 , 2 →0 for →0. Actually, since both 1 , 2 are O(2 ), it can be proved that for the cases Copyright 䉷 2005 John Wiley & Sons, Ltd.



339

considered in this paper, the contributions of the integrals over intervals [−1 , 0] and [, +2 ] at a smooth boundary point vanish when →0. Thus, in what follows the integration interval will be fixed to [0, ]. The following relations must be taken into account: x1 = sin() x2 = − cos() = (sin(), − cos()) = n T = (0, 1);

N = (1, 0) (11)

* * = *x1 *N * * = *x2 *T d = d The integral is obtained as 1 1 cos()u(x1 , x2 ) d (n · T − 2 · n · T)u(x) d(x) = 2 S 2r 0 2

(12)

The function u(x1 , x2 ) can be expanded in a Taylor series around the collocation point y *u *u u(x1 , x2 ) = u(y) + · sin() + · (− cos()) + O(2 ) (13) *x1 y *x2 y Only constant and linear terms must be considered. Terms of O(2 ) lead to an integral that vanishes when the limit is taken. The solutions of the integrals are u(y) cos() d = 0 (14) 2 0 1 *u cos() sin() d = 0 · 2 *x1 y 0 *u 1 *u 1 · cos()(− cos()) d = − 2 *x2 y 0 4 *x2 y

(15)

(16)

Considering now that the derivative with respect to x2 is equal to the derivative with respect to T, the limiting process leads to 1 ∗ lim qt (y; x)u(x) d(x) = − ut (y) (17) →0 4 S Copyright 䉷 2005 John Wiley & Sons, Ltd.


340


2.1.2. Integral of the function qt∗ (y; x)u(x) over (y − ). The integral is regular, as it is shown by considering the radial series expansion for the term n · T − 2( · n · T) 1 0 2 qt∗ = ∓ r − 2 ∓ ) = O(1) (18) r + O(r 0 2 2r 2 where a special notation is adopted, in which the double sign is written due to the change of sing in · t (see Appendix B). Kernel function is regular, O(1). At the limit, neither singular nor additional free terms appear for this regular integral. 2.1.3. Result of the integration of the function qt∗ (y; x)u(x). After the limit process, the integral over both intervals leads to a regular integral plus a free term ut (y) lim qt∗ (y; x)u(x) d(x) = qt∗ (y; x)u(x) d(x) − (19) →0 4 (y − )+S y 2.1.4. Integral of the function u∗t (y; x) q(x) over S . This integral is obtained by using the same change to polar variables 1 1 (20) · Tq(x) d(x) = · (− cos())q(x1 , x2 ) d S 2r 0 2 Function q(x) can be expanded in the new reference. Considering n = (nx1 , nx2 ), the expression for the derivative is q(x1 , x2 ) =

*u *u nx1 + nx *x1 *x2 2

(21)

Expanding the potential derivatives to O(r) *u *u = +O(r) = q(y) + O(r) *x1 *x1 y *u *u = +O(r) = ut (y) + O(r) *x2 *x2 y

(22)

the series for q(x) can be obtained q(x) = q(y) sin() − ut (y) cos() + O(r) The only integral that adds an additional free term is ut (y) 1 · (− cos())(−ut (y)) cos() d = 4 0 2 which, taking the limit as →0, leads to ut (y) ∗ lim ut (y; x)q(x) d(x) = →0 4 S Copyright 䉷 2005 John Wiley & Sons, Ltd.

(23)

(24)

(25)



341

2.1.5. Integral of the function u∗t (y; x) q(x) over (y − ). Considering Figure 1, the local change to the radial variable is introduced. Intervals I1 = [, R1 ] and I2 = [, R2 ] are considered. The Jacobian of the transformation is Jr (r), defined as

dX2 2 dX1 2 + (26) Jr (r) = dr dr Note that this value is positive in both intervals. Performing this change of variable the integral is transformed to 1 f (r) · Tq(r)J (r) dr = dr (27) u∗t (y; x)q(x) d(x) = (y − ) I1 ∪I2 2r I1 ∪I2 2r where f (r) = · Tq(r)Jr (r). In order to carry out the above written integral both intervals I1 and I2 are separately considered. • Integral over I1 : The function f (r) can be split into an O(r) function plus a constant f (r) = [f (r) − f (0)] + f (0)

(28)

and therefore the integral can be split into two adding terms: a regular and a singular integral R1 R1 [f (r) − f (0)] 1 dr + f (0) dr (29) 2r 2r In the limit →0, the first integral can be done in [0, R1 ] since the integrand is regular. The second integral produces a logarithmic singular term, for →0. f (0) R1 1 f (0) (30) dr = (ln(R1 ) − ln()) 2 r 2 The value f (0) can be obtained for this interval: f (0) = [ · Tq(r)Jr (r)]|r=0 = q(y) Thus, the integral over I1 leads to R1 R1 f (r) [f (r) − q(y)] q(y) dr = dr + (ln(R1 ) − ln()) 2r 2r 2

(31)

(32)

• Integral over I2 : The result is analogous to the one obtained for I1 . In this case there is a sign change due to the change of vector at the collocation point. The result is R2 R2 f (r) [f (r) + q(y)] q(y) dr = dr + (ln() − ln(R2 )) (33) 2r 2r 2 Adding the results in both intervals, (Equations (32) and (33)), it is seen that, although the integrals over each interval gives a singular term, the final result is finite. In this sense, Copyright 䉷 2005 John Wiley & Sons, Ltd.


342


taking the limit for →0, interval (y − ) tends to y , and the integral over such interval is interpreted as a Cauchy principal value (CPV). lim

→0

(y − )

u∗t (y; x)q(x) d(x)

=−

1 · Tq(x) d(x) = 2r y

+

0

R1 [f (r) − q(y)]

2r

dr + 0

R2 [f (r) + q(y)]

2r

q(y) (ln(R1 ) − ln(R2 )) 2

dr

(34)

2.1.6. Result of the integration of the function u∗t (y; x)q(x). The limit of the integral is a free term plus a CPV ut (y) ∗ ut (y; x)q(x) d(x) = (35) + − u∗t (y; x)q(y) d(x) lim →0 4 (y − )∪S y 2.1.7. Boundary integral equation for ut . Recalling the result for function qt∗ (y; x)u(x) given by Equation (19) and for function qt∗ (y; x)u(x) given by Equation (35), and including the boundary ( − y ) in this limit where the integrals are regular, one obtains the ensuing BIE for ut (ut -BIE) ut (y) (36) + qt∗ (y; x)u(x) d(x) = − u∗t (y; x)q(x) d(x) 2 Note that both limits in Equation (19) and Equation (35) exist independently. 2.2. ut -BIE for Helmholtz equation The Helmholtz problem we consider is u(X1 , X2 ) + k 2 u(X1 , X2 ) = 0

in

u = u¯

on u

q = q¯

on q

(37)

The parameter k is the wave number, related with the wavelength () through k = 2/. General solutions for u and q includes complex values. Let y be an internal smooth point to ; Equation (5) can be used, but in this case the kernels are 1 K0 (z) 2 −ik q ∗ (y; x) = K1 (z) · n 2 u∗ (y; x) =


(38)



343

where Kj (z) is the modified Bessel function of j th order, i is the imaginary unit, and z = ikr. Following the same approach done previously for the Laplace equation, the tangential boundary integral equation is obtained, in the form of Equation (36), but the tangential kernels are now ik K1 (z) · T 2 ik [K1 (z)n · T − (2K1 (z) + ikrK0 (z)) · n · T] qt∗ (y; x) = 2r u∗t (y; x) =

(39)

It can be proved that these kernels can be split into two adding kernels: the kernel for Laplace equation (singular) plus a regular increment kernel. Thus, no additional singularity analysis is required.

3. BIE FOR TANGENTIAL DERIVATIVE OF FLUX 3.1. qt -BIE for Laplace equation The qt -BIE is obtained following a similar limit to the boundary process than the one carried out for the ut -BIE. The BIE for the flux in direction N at y ∈ is given by q(y) + d ∗ (x, y) q(x) d(x) = s ∗ (x, y)u(x) d(x) (40) (− )+S

(− )+S

with d ∗ (y; x) =

−1 ·N 2r

−1 (n · N − 2 · n · N) s (y; x) = 2r 2

(41)

∗

To obtain the qt -BIE, the above written integral equation is differentiated with respect to arc-length parameter s of the curve . In order to clarify this, check Figure 3 where two types of derivatives are shown. The derivative with respect to T, where the source point moves from

Figure 3. Tangential derivatives. Copyright 䉷 2005 John Wiley & Sons, Ltd.


344


point y to point 2 (the normal direction changes to N2 ) is the derivative considered here. Note that this derivative is different than the one considered moving from point y to point 1, fixing the normal direction N1 at the first point. To obtain the tangential BIE at y, the derivative of Equation (40) with respect to T is taken dt∗ (y; x)q(x) d(x) = st∗ (y; x)u(x) d(x) (42) qt (y) + (− )+S

(− )+S

where qt (y) =

*q(y) *T

and 1 (r0 − 2 · N) · T 2r 2 1 st∗ (y; x) = − 3 · T(n · N − 4 · n · N) + n · T · N r

r0 − (n · T − 2 · T · n) 2

dt∗ (y; x) =

(43)

(44)

where 0 is the boundary curvature at point y. The qt -BIE can be obtained as a result of the limit ∗ ∗ qt (y) + lim dt (y; x)q(x) d(x) − st (y; x)u(x) d(x) = 0 (45) →0

(− )+S

(− )+S

The singularity analysis is required for each integral involved in order to compute the limit as →0. 3.1.1. Integral of the function dt∗ (y; x) q(x) over S . Considering the reference {y; x1 , x2 } (Figure 1), the integral is dt∗ (y; x)q(x) d(x) S

= 0

−0 cos()q(x1 , x2 ) d + 2

0

1 sin() cos()q(x1 , x2 ) d

(46)

Function q(x) can be written in the new reference, see Equation (21). The potential derivatives can be expanded as 2 2 *u * u * u *u = + (47) sin() + (− cos()) + O(2 ) *x1 y *x1 *x2 *x1 *x12 y y 2 2 *u *u * u * u = + sin() + (− cos()) + O(2 ) 2 *x2 *x2 y *x1 *x2 *x2 y y Copyright 䉷 2005 John Wiley & Sons, Ltd.

(48)


345


The first integral on the right-hand side in Equation (46) is regular, so its limit is easily computed using the first terms in expansions given by Equations (47) and (48). This integral produces a free term *u 0 0 2 d = (49) cos () ut (y) 4 *x2 y 0 2 For the second integral in Equation (46), the decomposition must include terms O(). The integral that includes O(2 ) terms vanishes after the limit; thus, the only integral to analyse can be written as 1 [sin() cos()(, ) sin() − sin() cos()(, ) cos()] d (50) 0 with (, ) and (, ) defined by 2 2 *u * u * u (, ) = + sin() + (− cos()) *x1 y *x1 *x2 *x12 y y

(51)

2 2 *u * u * u (, ) = + (− cos()) sin() + 2 *x2 y *x1 *x2 *x 2 y y

(52)

After the integration one obtains

0

2 1 1 * u 1 2 sin() cos()q(, ) d = − ut (y) − 3 4 *x1 *x2

(53)

y

Note that the integral over S produces two regular free terms, a singular free term, and a regular function O(); 2 0 1 * u 2 1 ∗ dt (y; x)q(x) d(x) = ut (y) − ut (y) + O() (54) − 4 4 3 *x *x 1 2 S y

3.1.2. Integral of the function dt∗ (y; x) q(x) over (y − ). Let us consider the series in Equations (102) and (104). In this domain, kernel dt∗ is O(1); therefore, the integral is regular dt∗ (y; x) =

1 0 2 r − 2 ) (±1 + O(r 2 )) = O(1) r + O(r 0 2 2r 2

(55)

Then, lim

→0

(y − )

dt∗ (y; x)q(x) d(x)


=

y

dt∗ (y; x)q(x) d(x)

(56)


346


3.1.3. Integral of the function st∗ (y; x) u(x) over S . Following the same procedure, this integral is done by a local expansion of u(x), in polar co-ordinates. The results of the integral in S are: one singular free term, two constant free terms and an O() regular function

2 1 * 4 1 0 u st∗ (y; x)u(x) d(x) = ut (y) + − ut (y) + O() 3 4 *x1 *x2 4 S

(57)

y

3.1.4. Integral of the function st∗ (y; x) u(x) over (y − ). The kernel function st∗ (y; x) can be split into three kernels ∗(1)

st∗ (y; x) = st

∗(2)

(y; x) + st

∗(3)

(y; x) + st

(y; x)

(58)

where 1 · Tn · N r 3

(59)

(y; x) =

2 · n · T · N r 3

(60)

(y; x) =

1 (4 · n · T · N − 2n · T · N) 2r 3

∗(1)

(y; x) = −

∗(2)

∗(3)

st st

st

−

1 (−0 n · T + 20 · T · n) 2r 2

(61)

The singularity order of each term is different. Using the expansions in Appendix B, it is shown that

5 2 3 ±1 ∓ r + O(r ) 8 20 2 2 2 ∗(2) st (y; x) = 3 · n · T · N = 3 ∓ r + O(r 3 ) 4 r r ∗(1) st (y; x) =

∗(3)

st

(y; x) =

−1 −1 · Tn · N= 3 3 r r

∗(1)

∗(2)

(y; x) is O(r −3 ), st

(63)

1 2 2 2 2 3 ∓ r ± r + O(r ) 0 0 2r 3 −

Then, st

(62)

1 (±20 r ∓ 20 r + O(r 2 )) = O(1) 2r 2 ∗(3)

(y; x) is O(r −1 ) and st


(64)

(y; x) is O(1).


347


Let I1 and I2 be the intervals [, R1 ], and [, R2 ] in the radial variable (see Figure 2). The integral over (y − ) can be done over I1 ∪ I2 ∗(1)

• Integral over ( − y ) for st This integral is evaluated for a generic interval [, Ri ], i = 1, 2

Ri

−

1 · T n · Nu(r)Jr (r) dr = r 3

Ri

−

1 f (r) dr r 3

(65)

where f (r) is defined as f (r) = · T n · Nu(r)Jr (r). The following terms are obtained for the series expansion of f (r): f (0) = ±u(y) df (r) du(r) =± = ut (y) dr r=0 dr r=0 d2 u d2 u d2 f (r) 2 = ± 2 ∓ u(y)0 = ± 2 ∓ u(y)20 dr 2 r=0 dr 0 ds y

(66) (67)

(68)

The subtracting and adding procedure leads to a regular kernel plus three singular terms which can be analytically computed — Term O(r −3 )

Ri

−

1 f (0) f (0) u(y) u(y) f (0) dr = − =± ∓ 2 2 2 2 r 3 2 2 2Ri 2Ri

(69)

Note that collecting the integrals over both intervals I1 and I2 , the singular terms are cancelled out. — Term O(r −2 )

Ri −1 df (r) ut (y) 1 1 (70) r dr = − Ri r 3 dr r=0 Considering the intervals I1 and I2 , the integral leads to

ut (y) 1 1 2 −1 u (y) dr = + − 2 t R1 R2 (y − ) r

(71)

— Term O(r −1 )

Ri

Ri −1 d2 f (r) r2 −1 d2 u dr = ± 2 ∓ u(y)20 ln 3 2 2 r dr ds y r=0 2

(72)

The singularity in interval I1 is cancelled out with the singularity in interval I2 . Copyright 䉷 2005 John Wiley & Sons, Ltd.


348


The integral of kernel st∗ (y; x) over (y − ) leads to a singular term, shown in Equation (71). ∗(2) • Integral over (y − ) for st (y; x) Evaluating the integral in an interval [, Ri ], we have Ri Ri 2 · n · N · Tu(r) 2 f (r) dr (73) Jr dr = 2 r r r In this integral, only the consideration of the first term in f (r) is required f (0) = ∓

20 u(y) 4

The integral of the singular term is Ri 2 2 f (0) dr = ∓ u(y) 0 (ln(Ri ) − ln()) r 2

(74)

(75)

Therefore, the singular terms shown in Equation (75) and the previous equations are cancelled out when both intervals, I1 and I2 , are considered together. 3.1.5. Boundary integral equation for qt . The integral over (y − ) of the function st∗ (y; x) u(x) leads to a singular term, Equation (71). This term vanishes when the free singular terms in S for st∗ (y; x) u(x), Equation (54), and dt∗ (y; x) q(x), in Equation (57), are considered. In this sense, the limiting integral over (y − ) can be interpreted as a finite part, in the sense of Hadamard (HFP). The decomposition done here shows that such consideration is ∗(1) ∗(2) the integral is a strictly necessary for the integral that involves kernel st . For kernel st ∗(3) Cauchy principal value. For st , the integral is regular. The full integral with kernel st∗ can be interpreted as a HFP. 2 1 ∗ lim = = st∗ (y; x)u(x) d(x) st (y; x)u(x) d(x) + ut (y) (76) →0 (y − ) y In order to obtain the limit, all the free terms in Equations (54), (57), and (71) can be moved to the first term in Equation (45). Singular terms are cancelled out and two regular free terms are obtained. Taking into account the expression for the tangential derivative of flux, on the local reference 2 * u qt (y) = (77) − 0 ut (y) *x1 *x2 y

the only free term is 1 2 qt (y)

(78)

Taking the limit as → 0 the qt -BIE is obtained 1 ∗ qt (y) + dt (y; x)q(x) d(x) = = st∗ (y; x)u(x) d(x) 2 Copyright 䉷 2005 John Wiley & Sons, Ltd.

(79)



349

3.2. qt -BIE for Helmholtz equation The qt -BIE for the Helmholtz problem has the same structure than Equation (79). For this problem, each kernel function can be split into two functions: the kernel for the Laplace equation (singular) plus a regular function. Singularities have been solved for Laplace equation, thus only the kernel expressions are required. Kernels for the q-BIE can be written as d ∗ (y; x) = −

ik K1 (z) · N 2

(80)

s ∗ (y; x) = −

ik [K1 (z)n · N − (2K1 (z) + ikrK0 (z)) · n · N] 2r

(81)

The kernel functions for the qt -BIE are dt∗ (y; x) =

ik [0 rK1 (z) − (2K1 (z) + ikrK0 (z)) · N] · T 2r

(82)

ik (2K1 (z) + ikrK0 (z)) · T(n · N − 4 · n · N) 2r

r0 +n · T · N − (n · T − 2 · T · n) 2

st∗ (y; x) = −

+

k 2 0 K0 (z) ik 3 K1 (z) n·T− · n · T · N 4 2

(83)

4. NUMERICAL TESTS A set of numerical tests have been designed in order to validate the proposed tangential BIEs. The tangential BIEs are discretized using standard boundary elements techniques. In this work, quadratic isoparametric elements are employed. These tests show that both, theoretical and practical aspects of the tangential formulation have been correctly developed. The boundary integral equations for the tangential derivatives are employed for computing two types of values: tangential derivative and nodal sensitivity. The first value only requires a mesh in which a known solution is to be interpolated in a fine mesh; on the contrary, the second test requires its use in a boundary element code, once a direct solution using a coarse mesh is computed. Figure 4 represents the disc sector domain considered, for which analytic solutions can be obtained. Tangential BIEs are evaluated at selected points of the mesh. This domain allows the validation for both curved or linear elements. The benchmarks include mixed boundary conditions, as well. An additional test considering Dirichlet boundary conditions has been proposed in order to validate the tangential formulations for flux in curved elements. Copyright 䉷 2005 John Wiley & Sons, Ltd.


350


Figure 4. Disc sector domain.

4.1. Computation of tangential derivatives In this set of tests the direct computation of tangential derivatives has been explored. In each problem three values have been obtained at the mid-nodes of the elements: 1. Analytic tangential derivative computed analytically from the exact solution. 2. Approximed tangential derivative computed by taking derivatives of the shape functions, from the solution obtained with BEM. 3. Approximed tangential derivative by using the tangential boundary integral equation (ut -BIE or qt -BIE). In order to compute ut and qt using the corresponding BIEs, the values of u and q will be computed numerically using the BEM, given exact quadratic functions as boundary conditions. The reason to use a BEM solution is because no closed-form are available, but series. These series shows slower convergence for the flux than the potential. It is well known that, if the mesh is fine enough, the BEM gives good quality results, requiring less computational time than the series. To obtain the BEM solution a mixed formulation is used: u-BIE is used at points in which the unknown value is the potential, and the hypersingular q-BIE is used at points in which the unknown value is the flux. A stable second-kind integral equation is employed at each point, as pointed out by Ingber and Mondy [23]. In all the examples, regularized quadrature formulae have been employed. Twenty points in Gauss quadratures have been employed. This elevated number of points may be reduced by an advanced integration algorithm that reduces this number when the integration element is Copyright 䉷 2005 John Wiley & Sons, Ltd.



351

far from the collocation point; this aspect has not been explored in this paper, as the aim of the tests is the validation of the proposed formulation. In order to avoid the problems related with quasi-singular integration, double number of Gauss points have been considered when the ratio R/L (radius to length of the element) is less than 0.1. Finally, the collocation points at the corners have been shifted a fixed 0.2 value in the interpolation parameter of the shape functions, . The dimensions of the radii in the disc sector domain (Figure 4) are R1 = 50, R2 = 100. 4.1.1. Laplace equation: mixed boundary conditions. In this test, the suitability and reliability of the procedure and its implementation for curved and linear elements are assessed. The geometry and variation of the prescribed boundary conditions are shown in Figure 4. The straight edges 1 and 3 are both meshed with 50 equal-sized elements and 2 and 4 with 100 elements each. On the edges 1 and 3 the potential is prescribed to u = 0, while the flux varies quadratically along 2 and 4 , as shown in the figure, with 2 = 0.1, and 4 = 0.2. The analytic solution for this problem is u(r, ) =

∞

(An r 2n + Bn r −2n ) sin(2n)

(84)

n=1

where An R22n−1 − Bn R2−2n−1 = n An R12n−1 − Bn R1−2n−1 = −n The right-hand side terms are computed by n =

2 8n

n = 4 8n

/2

1−

0

16 2 sin(2n) d − 4 2

/2

0

2 16 1− 2 − sin(2n) d 4

Tangential derivatives have been computed by using qt -BIE on 1 and 3 ; ut -BIE is used on 2 and 4 , which are curved elements. Table I shows the results in three columns. It is shown that tangential BIEs produces a value close to the analytic solution, more accurate than the same value obtained with the ITD approach. 4.1.2. Helmholtz equation: mixed boundary conditions. Boundary conditions have been prescribed in potential and flux values. The potential is prescribed to u = 0 on 1 and 3 , and the flux on 2 and 4 with 2 = 0.2 and 4 = 0.1i. Boundaries 1 and 3 have been divided into 50 quadratic elements; boundaries 2 and 4 require 100 elements each. The wavelength is = 250. Copyright 䉷 2005 John Wiley & Sons, Ltd.


352


Table I. Analytical tangential derivative (ATD), interpolated tangential derivative (ITD) and tangential boundary integral equation (TBIE) for disc sector with mixed boundary conditions. Laplace equation. Boundary

ATD

ITD

TBIE

1

1.627489E−002 1.467767E−002 1.331443E−002 1.212122E−002 1.106301E−002 1.011676E−002 9.265628E−003 8.496565E−003 7.799056E−003 7.164428E−003

1.629154E−002 1.468751E−002 1.332149E−002 1.212695E−002 1.106757E−002 1.012072E−002 9.269089E−003 8.499503E−003 7.801622E−003 7.166892E−003

1.627257E−002 1.467789E−002 1.331457E−002 1.212129E−002 1.106305E−002 1.011678E−002 9.265645E−003 8.496577E−003 7.799064E−003 7.164434E−003

2

1.774254E−001 1.771044E−001 1.765067E−001 1.756791E−001 1.746411E−001 1.733966E−001 1.719557E−001 1.703337E−001 1.685388E−001 1.665734E−001

1.774303E−001 1.771069E−001 1.764927E−001 1.756700E−001 1.746331E−001 1.733812E−001 1.719527E−001 1.703179E−001 1.685361E−001 1.665683E−001

1.774182E−001 1.771013E−001 1.765045E−001 1.756770E−001 1.746390E−001 1.733945E−001 1.719538E−001 1.703318E−001 1.685369E−001 1.665716E−001

The analytic solution for this problem can be obtained by a polar series expansion in terms of Bessel functions ∞

u(r, ) =

(An J2n (kr) + Bn Y2n (kr)) sin(2n)

(85)

n=1

The coefficients in this series are the solution of the equations

An J2n (kR2 ) + Bn Y2n (kR2 ) = n

An J2n (kR1 ) + Bn Y2n (kR1 ) = −in

where n =

2 4k

n = 4 4k

/2

1−

0

0

/2

2 16 − sin(2n) d 4 2

2 16 1− 2 − sin(2n) d 4

(86)

The derivatives of the Bessel functions are taken with respect to z = kr. Tables II and III show the values obtained for the real and the imaginary part for tangential derivatives. Copyright 䉷 2005 John Wiley & Sons, Ltd.


353


Table II. Analytic tangential derivative (ATD), interpolated tangential derivative (ITD) and tangential boundary integral equation (TBIE) for disc sector with mixed boundary conditions. Helmholtz equation, real part. Boundary

ATD

ITD

TBIE

1

2.199979E−002 2.016785E−002 1.851942E−002 1.703477E−002 1.569659E−002 1.448972E−002 1.340076E−002 1.241791E−002 1.153072E−002 1.072993E−002

2.200753E−002 2.017549E−002 1.852625E−002 1.704088E−002 1.570207E−002 1.449464E−002 1.340519E−002 1.242190E−002 1.153433E−002 1.073319E−002

2.200096E−002 2.016785E−002 1.851943E−002 1.703478E−002 1.569661E−002 1.448973E−002 1.340077E−002 1.241792E−002 1.153073E−002 1.072993E−002

2

7.699408E−001 7.690324E−001 7.672606E−001 7.646736E−001 7.612927E−001 7.571244E−001 7.521819E−001 7.464842E−001 7.400437E−001 7.328678E−001

7.699086E−001 7.689967E−001 7.672271E−001 7.646409E−001 7.612602E−001 7.570924E−001 7.521508E−001 7.464536E−001 7.400134E−001 7.328379E−001

7.699316E−001 7.690239E−001 7.672525E−001 7.646654E−001 7.612844E−001 7.571162E−001 7.521738E−001 7.464762E−001 7.400358E−001 7.328599E−001

Table III. Analytic tangential derivative (ATD), interpolated tangential derivative (ITD) and tangential boundary integral equation (TBIE) for disc sector with mixed boundary conditions. Helmholtz equation, imaginary part. Boundary

ATD

ITD

TBIE

1

3.068052E−002 2.809645E−002 2.584563E−002 2.385375E−002 2.207665E−002 2.048322E−002 1.904948E−002 1.775606E−002 1.658682E−002 1.552811E−002

3.069808E−002 2.811006E−002 2.585633E−002 2.386266E−002 2.208429E−002 2.048986E−002 1.905532E−002 1.776123E−002 1.659142E−002 1.553222E−002

3.068211E−002 2.809641E−002 2.584562E−002 2.385375E−002 2.207666E−002 2.048322E−002 1.904949E−002 1.775606E−002 1.658682E−002 1.552811E−002

2

5.747165E−001 5.741481E−001 5.730118E−001 5.713088E−001 5.690407E−001 5.662100E−001 5.628194E−001 5.588723E−001 5.543727E−001 5.493252E−001

5.746962E−001 5.741245E−001 5.729883E−001 5.712854E−001 5.690174E−001 5.661868E−001 5.627963E−001 5.588494E−001 5.543500E−001 5.493027E−001

5.747108E−001 5.741426E−001 5.730063E−001 5.713033E−001 5.690353E−001 5.662046E−001 5.628140E−001 5.588670E−001 5.543674E−001 5.493199E−001



354


4.1.3. Helmholtz equation: Dirichlet boundary conditions. In order to test the qt -BIE at curved elements, a problem with Dirichlet boundary conditions has been tested. The q-BIE has been used at each point. The condition for boundaries 1 and 3 has been u = 0. The same discretization as in the previous tests have been employed. The value = 250 has been considered. The analytic solution is given by the series in Equation (85), with Equation (86), but now the values An and Bn are defined by An J2n (kR2 ) + Bn Y2n (kR2 ) = n

(87)

An J2n (kR1 ) + Bn Y2n (k R1 ) = in

(88)

Tables IV and V show the values obtained for the real and the imaginary part for tangential derivatives. 4.2. Computation of nodal sensitivities Direct differentiation approach (DDA) leads to nodal sensitivities. These values are defined as the limit of the rate of change of the approximated solution when a set of collocation points are perturbed in a boundary element mesh, keeping the same geometry and element mesh. Paulino et al. [12] and Guiggiani et al. [24] have established nodal sensitivities for the BEM. The relation with exact errors can be found in Reference [25]. A way to validate the results obtained with DDA is its contrast with the finite perturbation approach (FPA). In FPA, the rate of change is evaluated by considering two solutions, shifting a finite and fixed value for the perturbation points. The computation of nodal sensitivities requires the evaluation of the tangential residual. The geometry, boundary conditions and wavelengths considered in this section are the same as the ones considered in Section 4.1. The only novel geometrical aspect is that coarser meshes are used, in order to obtain large enough errors that cause nodal sensitivities. The following points are common for all the tests: 1. Finite perturbations described by an arc increment of 0.01 in the position of the mid-points of the quadratic boundary elements for FPA. 2. In order to avoid collocation points at the corners, a 0.2 shift in the parameter has been fixed. 3. Numerical integration with Gauss quadratures is carried out by using 10–15 points. This number might be reduced for collocation nodes far from the integration element but this aspect has not been explored in depth. 4. Quasi-singular integration is minimized by using the double number of Gauss points when R/L < 0.1, with R the minimum distance from element nodes to the collocation point, and L the maximum distance between the extreme points of the integration element. 5. The first 10 values for each boundary have been tabulated. 4.2.1. Laplace equation: mixed boundary conditions. For boundaries 1 and 3 , 5 elements have been considered at each curve; boundaries 2 and 4 have been meshed with 10 elements each. Table VI shows the comparison of DDA vs FPA. Copyright 䉷 2005 John Wiley & Sons, Ltd.


355


Table IV. Analytical tangential derivative (ATD), interpolated tangential derivative (ITR) and tangential boundary integral equation (TBIE) for disc sector with Dirichlet boundary conditions. Helmholtz equation, real part. Boundary

ATD

ITR

TBIE

1

−1.115655E−004 −1.052741E−004 −9.952168E−005 −9.424828E−005 −8.940185E−005 −8.493698E−005 −8.081401E−005 −7.699816E−005 −7.345891E−005 −7.016938E−005

−1.115757E−004 −1.052953E−004 −9.954115E−005 −9.426587E−005 −8.941768E−005 −8.495124E−005 −8.082687E−005 −7.700979E−005 −7.346944E−005 −7.017893E−005

−1.115597E−004 −1.052729E−004 −9.952123E−005 −9.424810E−005 −8.940180E−005 −8.493701E−005 −8.081409E−005 −7.699828E−005 −7.345904E−005 −7.016952E−005

2

−1.149181E−004 −1.147871E−004 −1.145254E−004 −1.141337E−004 −1.136129E−004 −1.129643E−004 −1.121893E−004 −1.112899E−004 −1.102679E−004 −1.091259E−004

−1.149253E−004 −1.147836E−004 −1.145211E−004 −1.141292E−004 −1.136083E−004 −1.129597E−004 −1.121847E−004 −1.112853E−004 −1.102635E−004 −1.091216E−004

−1.149238E−004 −1.147878E−004 −1.145254E−004 −1.141334E−004 −1.136125E−004 −1.129638E−004 −1.121888E−004 −1.112893E−004 −1.102674E−004 −1.091254E−004

Table V. Analytical tangential derivative (ATD), interpolated tangential derivative (ITR) and tangential boundary integral equation (TBIE) for disc sector with Dirichlet boundary conditions. Helmholtz equation, imaginary part. Boundary

ATD

ITD

TBIE

1

8.710161E−004 6.945616E−004 5.913491E−004 5.197647E−004 4.652928E−004 4.215789E−004 3.853093E−004 3.545261E−004 3.279663E−004 3.047618E−004

8.828859E−004 6.971280E−004 5.925238E−004 5.204263E−004 4.657188E−004 4.218780E−004 3.855319E−004 3.546987E−004 3.281041E−004 3.048744E−004

8.716579E−004 6.943318E−004 5.912735E−004 5.197347E−004 4.652788E−004 4.215717E−004 3.853053E−004 3.545238E−004 3.279650E−004 3.047611E−004

2

−5.745903E−005 −5.739353E−005 −5.726270E−005 −5.706686E−005 −5.680646E−005 −5.648214E−005 −5.609466E−005 −5.564493E−005 −5.513397E−005 −5.456296E−005

−5.744008E−005 −5.738840E−005 −5.725887E−005 −5.706352E−005 −5.680340E−005 −5.647926E−005 −5.609192E−005 −5.564230E−005 −5.513145E−005 −5.456053E−005

−5.744870E−005 −5.739066E−005 −5.726101E−005 −5.706562E−005 −5.680547E−005 −5.648130E−005 −5.609392E−005 −5.564427E−005 −5.513337E−005 −5.456241E−005



356


Table VI. Direct differentiation approach (DDA) and finite perturbation approach (FPA) for disc sector with mixed boundary conditions. Laplace equation. Boundary

DDA

FPA

1

5.554420E−004 −2.653766E−004 3.259886E−005 −1.117646E−004 1.011975E−005 −4.331738E−005 6.020516E−006 −1.496472E−005 7.503761E−006 −8.832607E−006

5.549053E−004 −2.650631E−004 3.257280E−005 −1.117442E−004 1.011012E−005 −4.331361E−005 6.015307E−006 −1.496286E−005 7.502470E−006 −9.048771E−006

2

−6.559288E−004 −6.349155E−005 −6.175158E−004 −1.801300E−004 −5.385671E−004 −1.970175E−004 −3.862392E−004 −1.387298E−004 −2.001351E−004 −4.952447E−005

−6.561806E−004 −6.356073E−005 −6.174242E−004 −1.802484E−004 −5.382971E−004 −1.964182E−004 −3.859009E−004 −1.386715E−004 −1.997824E−004 −4.879682E−005

Table VII. Direct differentiation approach (DDA) and finite perturbation approach (FPA) for disc sector with mixed boundary conditions. Helmholtz equation, real part. Boundary

DDA

FPA

1

9.115220E−004 −2.189642E−004 8.474223E−005 −1.466077E−004 3.259630E−005 −5.529259E−005 2.430145E−005 −6.191238E−006 3.471344E−005 −3.043721E−005

9.009870E−004 −2.167779E−004 8.528889E−005 −1.466727E−004 3.190242E−005 −5.664864E−005 2.198016E−005 −1.003259E−005 2.797929E−005 −4.337096E−005

2

−2.180758E−003 −4.981641E−004 −2.293961E−003 −5.137371E−004 −2.147136E−003 −5.615636E−004 −1.630841E−003 −4.108523E−004 −8.766644E−004 −1.492292E−004

−2.376904E−003 −1.304245E−004 −2.303644E−003 −2.232442E−004 −2.151402E−003 −3.525211E−004 −1.633621E−003 −2.860963E−004 −8.797773E−004 −1.098984E−004




357

Table VIII. Direct differentiation approach (DDA) and finite perturbation approach (FPA) for disc sector with mixed boundary conditions. Helmholtz equation, Imaginary part. Boundary

DDA

FPA

1

8.175162E−004 −3.915367E−004 8.241104E−005 −1.862725E−004 3.589633E−005 −6.998535E−005 2.060827E−005 −2.774204E−005 6.844587E−006 −8.968386E−005

8.074102E−004 −3.853729E−004 8.489566E−005 −1.848109E−004 3.685232E−005 −6.916844E−005 2.145943E−005 −2.653252E−005 8.975979E−006 −8.647519E−005

2

−1.190909E−003 9.882359E−004 −7.634449E−004 1.002105E−003 −7.534075E−004 7.092646E−004 −5.891009E−004 4.233272E−004 −3.212609E−004 1.406628E−004 −1.176048E−007

−1.031058E−003 8.307932E−004 −6.808515E−004 9.036010E−004 −6.749625E−004 6.511196E−004 −5.280014E−004 3.948638E−004 −2.862280E−004 1.350410E−004 4.097511E−006

Table IX. Direct differentiation approach (DDA) and finite perturbation approach (FPA) for disc sector with Dirichlet boundary conditions. Helmholtz equation, real part. Boundary

DDA

FPA

1

−2.507334E−006 2.010288E−009 −8.907731E−007 −2.088786E−007 −7.084193E−007 −3.263600E−007 −5.574454E−007 −5.419387E−008 −2.387969E−007 3.173458E−006

−2.512707E−006 −1.525132E−009 −8.931161E−007 −2.110434E−007 −7.102803E−007 −3.281687E−007 −5.592341E−007 −5.605004E−008 −2.410380E−007 3.174086E−006

2

2.599059E−007 −2.819494E−006 1.952111E−007 4.432414E−007 2.095404E−007 4.817173E−007 1.555892E−007 3.400642E−007 8.173454E−008 1.039349E−007

2.560744E−007 −2.813099E−006 1.939766E−007 4.460191E−007 2.079871E−007 4.829564E−007 1.538133E−007 3.398650E−007 7.978401E−008 1.023543E−007



358


Table X. Direct differentiation approach (DDA) and finite perturbation approach (FPA) for disc sector with Dirichlet boundary conditions. Helmholtz equation, imaginary part. Boundary

DDA

FPA

1

−7.248933E−006 −3.869507E−005 −2.473065E−006 −6.293793E−006 −1.055092E−006 −2.444009E−006 −8.626031E−007 −1.574354E−006 −1.112217E−006 −1.834096E−006

−7.265037E−006 −3.867294E−005 −2.473482E−006 −6.292678E−006 −1.055425E−006 −2.444012E−006 −8.629606E−007 −1.574549E−006 −1.112523E−006 −1.833286E−006

2

3.979185E−006 2.178468E−006 1.883312E−006 1.414490E−006 1.281882E−006 9.443280E−007 8.027202E−007 5.479695E−007 3.872613E−007 1.784205E−007

3.978316E−006 2.178584E−006 1.882987E−006 1.414263E−006 1.281739E−006 9.443002E−007 8.027916E−007 5.481275E−007 3.874908E−007 1.786798E−007

4.2.2. Helmholtz equation: mixed boundary conditions. Curves 1 and 3 have been divided into 5 elements; curves 2 and 4 into 10 elements. Tables VII and VIII show the comparison of DDA vs FPA. 4.2.3. Helmholtz equation: Dirichlet boundary conditions. Edges 1 and 3 have been divided into 5 elements each; curves 2 and 4 have been divided into 10 elements, each. Tables IX and X show the comparison for two boundaries between DDA and FPA.

5. CONCLUSIONS Tangential boundary integral equation, for fluxes in Laplace and Helmhotz equations are presented. As a previous stage, the tangential BIE is obtained for potential. The limit to the boundary process reveals that the qt -BIE is hypersingular, interpreted in the sense of Hadamard Finite Part. Such a process produces the regularization formulae in a straightforward manner, required when numerical computations based on such BIEs are developed. Based on the ut -BIE and qt -BIE, a set of numerical tests have been proposed. The results show that both theoretical and practical aspects have been successfully developed. Tangential BIEs can be used to solve problems in the BEM in several fields: • Sensitivity analysis: When the hypersingular formulation must be employed (crack problems), or as a general procedure when the q-BIE is used. Copyright 䉷 2005 John Wiley & Sons, Ltd.



359

• Generation of C 1 solutions: Considering a classical Hermitian polynomial interpolation (nodal function values and tangential derivatives) a C 1 solution can be obtained. The extension to the qt -BIE is possible by considering the tangential BIE presented in this paper. • Optimization problems and inverse problems: Computing geometrical sensitivities when the q-BIE is used requires the evaluation of the qt -BIE. • Computation of tangential residuals: The tangential residual can be used as an error estimator in the BEM, as shown in Reference [25].

APPENDIX A: NUMERICAL INTEGRATION In order to implement Equations (36) and (79) in a numerical code, regularization formulae are proposed. The analysis performed for the limiting process gives the regular integrals and the analytic terms directly, considering the domain y as the singular element. The condition shown in Equation (2) for the elements must be stressed. If the collocation point is between two adjacent elements, two singular elements are considered, and the regularization scheme is the same. In this case it must be remarked that the mesh has to be fine enough to avoid an additional error due to the presence of small corners at the boundary (due to the inexact isoparametric representation of a general curve) and the difference between tangential values due to the C 0 potential and traction representation. In the discrete version, the solution is interpolated at each element by introducing a parameter ; ∈ [−1, 1]. Let ( ) be a generic shape function. Let c be the value of the parameter at the collocation point. In this appendix the final expressions required to compute the integrals, PVC and HFP, which appear in the formulation, taking into account the series expansions in Appendix B, are shown. A.1. Numerical integration for the potential ut -BIE The integral that involves the kernel function qt∗ can be computed using a standard Gauss quadrature. For kernel function u∗t , the following regularization formula is proposed: 1 ∗ ut d · T( ) d( ) y y 2r =

c

−1

[f ( ) + ( c )] dr d + 2r d

1

c

[f ( ) − ( c )] dr ( c ) R1 d + ln 2r d 2 R2

(A1)

where f ( ) = · T( )|Jr ( )|. Note that f ( ) is not continuous. In fact lim f ( ) = − ( c )

→ − c

and

lim f ( ) = ( c )

→ + c

A.2. Numerical integration for the flux qt -BIE For the qt -BIE, only the integral that involves the kernel st∗ (y; x) requires regularization formulae. The integral containing kernel dt∗ (y; x) is regular, as Equation (55) shows. Copyright 䉷 2005 John Wiley & Sons, Ltd.


360


Regularization formulae for numerical implementations are a result of the limiting process carried out. Kernel st∗ (y; x) can be split into three sub-kernels, as shown in Equation (58). ∗(1) The interval y is, in the discrete version, the singular element. Only kernels st (y; x) and ∗(2) st (y; x) require regularization formulae. The formulae are shown for each interval. • Integral over [0, R1 ] The interval [0, R1 ] is applied onto the interval [ c , 1]. The integral can be done for each sub-kernel. ∗(1)

— Kernel st

1

(y; x)

f ( ) dr d r 3 d c 1 2 dr 1 d( ) d2 ( ) 2 r d − 3 f ( ) − ( c ) − r − − ( ) = c 0 2 ds 2 d r ds c c c −

d( ) 1 1 d2 ( ) ( c ) 2 + − − ( c )0 ln(R1 ) + ds c R1 2 ds 2 c 2R12

(A2)

with f ( ) = · T n · N( )Jr ( ). ∗(2) — Kernel st (y; x)

1

c

dr 2 f ( ) d = r d

1

c

2 r

2 ( c ) f ( ) + 0 4

2 ( c ) dr d − 0 ln(R1 ) d 2

(A3)

with f ( ) = r12 · n · N · T( )Jr ( ). • Integral over [0, R2 ] The interval [0, R2 ] is applied onto the interval [−1, c ]. For = − 1, r = R2 ; = c , r = 0. ∗(1)

— Kernel st

c

(y; x)

f ( ) dr d r 3 d −1 c 2 1 d( ) d2 ( ) dr 2 r = − 3 f ( ) + ( c ) − r + − ( ) d c 0 2 ds c 2 d r ds −1 c −

d( ) 1 1 d2 ( ) ( c ) 2 + + − ( c )0 ln(R2 ) − ds c R2 2 ds 2 c 2R22

(A4)

with f ( ) = · T n · N( )|Jr ( )|. Copyright 䉷 2005 John Wiley & Sons, Ltd.



361

∗(2)

— Kernel st (y; x) c c 20 ( c ) dr 2 ( c ) 2 2 dr f ( ) d = f ( ) − d + 0 ln(R2 ) d 4 d 2 −1 r −1 r

(A5)

with f ( ) = r12 · n · N · T( )|Jr ( )|. APPENDIX B: SERIES’ EXPANSIONS A set of series’ expansions in the radial variable r are presented in this appendix. Information about a special notation adopted is presented previously. The limits of some of the functions of the radius r, f (r) when the radius tends to zero, from the right and the left sides, may have different values. This fact leads to this special notation for the consideration of the two values in the same expression. Let sc be the value of the arc-length parameter s at the collocation point. For a smooth boundary point, it may be written as lim f (r(s)) = ± L

(B1)

s→sc±

With this expression, both limits are written in a more compact form. In general, both limits will have different values because the unitary radial vector is opposite on the left side at each element, considering left and right sides in the interpolation parameter space. Thus, the sign of dr/ds changes when s moves from sc− to sc+ . Note that we are only considering the case where both limits have the same absolute value L (at a smooth boundary point), but signs may be different. For some functions f (r), the limits will have the same values. Thus, it will be written as lim f (r(s)) = L

(B2)

s→sc±

The series’ expansions are summarized as 0 N 20 T r 2 0 N r+ ± ∓ + O(r 3 ) = ±T + 2 3 4 2 r2 t = T ± 0 Nr + 0 N − 20 T + O(r 3 ) 2 n = N ∓ 0 Tr − ( 0 T + 20 N)

r2 + O(r 3 ) 2

(B3)

(B4)

(B5)

· t = ±1 ∓

20 r 2 + O(r 3 ) 4 2

(B6)

· T = ±1 ∓

20 r 2 + O(r 3 ) 4 2

(B7)



362


·n=− ·N=

0 2 r2 r ∓ 0 + O(r 3 ) 2 3 2

r 2 0 r± 0 + O(r 3 ) 2 3 2

n · N = 1 − 20

r2 + O(r 3 ) 2

n · T = ∓0 r − 0

r2 + O(r 3 ) 2

(B8)

(B9)

(B10)

(B11)

where (·) stands for d(·)/ds. Series’ expansion for a generic shape function, ( ), with a local interpolation parameter, is 1 d (r) = ( c ) ± r d = c J ( c )

+

d2 d 2

= c

d Jˆ ( c ) r 2 1 − + O(r 3 ) d c J 3 ( c ) 2 J 2 ( c )

(B12)

where Jˆ stands for dJ /d . Finally, the Jacobian Jr in terms of the radial variable admits the following derivatives, used in Equation (68): Jr =

±1 ·t

Jr (0) = 1 dJr (r) =0 dr r=0 20 d2 Jr (r) = 4 dr 2 r=0

(B13) (B14) (B15)

(B16)

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