Tangential residual as error estimator in the boundary element method

Computers and Structures 83 (2005) 685–699 www.elsevier.com/locate/compstruc

Tangential residual as error estimator in the boundary element method Alejandro E. Martı´nez-Castro, Rafael Gallego

*

Department of Structural Mechanics, University of Granada, Ed. Polite´cnico Fuentenueva, Avda. Fuentenueva s/n, 18002 Granada, Spain Accepted 10 September 2004 Available online 21 December 2004

Abstract In this paper a new error estimator based on tangential derivative Boundary Integral Equation residuals for 2D Laplace and Helmholtz equations is shown. The direct problem for general mixed boundary conditions is solved using standard and hypersingular boundary integral equations. The exact solution is broken down into two parts: the approximated solution and the error function. Based on theoretical considerations, it is shown that tangential derivative Boundary Integral Equation residuals closely correlate to the errors in the tangential derivative of the solution. A similar relationship is shown for nodal sensitivities and tangential derivative errors. Numerical examples show that the tangential Boundary Integral Equation residual is a better error estimator than nodal sensitivity, because of the accuracy of the predictions and the lesser computational effort.
2004 Elsevier Ltd. All rights reserved. Keywords: Boundary element method; Error estimation; Boundary Integral Equation residual; Nodal sensitivity; Adaptivity; Mesh adaptation; Mesh refinement

1. Introduction The computation of accurate and effective error estimators is a key step in the development of automated procedures for mesh adaptation and refinement within the family of Boundary Element Methods (see [1] for a recent review). The adaptive process consists of three steps: error estimation, adaptive tactics, and mesh refinement. The error estimation is the most important of them since the whole process depends on its accuracy.

*

Corresponding author. Tel.: +34 958 24 89 55; fax: +34 958 24 99 59. E-mail addresses: [email protected] (A.E. Martı´nez-Castro), [email protected] (R. Gallego).

Error estimation schemes can be classified in five types: residual, interpolation error, boundary integral error, node-sensitivity and solution difference. In this classification, the procedure described in the present paper is a residual-type error estimator. Error estimators based on the approximation of the error function for potential or flux can lead to wrong error indicators. Two different displacement fields (potential) can be considered truly different if their tangential derivatives are different [2]. In this aspect, two types of error can be considered: • Error defined by the difference of two values at a point: the exact solution and the approximated one. • Error defined by the local interpolation space, through the shape functions.

0045-7949/$ - see front matter
2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.09.006

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A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

The aim of the error estimation techniques is to locate the sources of error due to wrong interpolation functions. In general, these error sources are not necessarily located at the elements with the worst computed values. Error estimators based on the error predicted at the variable values require especial norms to obtain an adequate error indicator. An easy example can illustrate this fact. For a fracture problem, Fig. 1 shows a schematic displacement field for the exact solution and the approximated solution. An error estimator based on the values of the displacements predicts the highest error in elements far to the crack tip. However, for this problem it is known that the behaviour of the displacement pffiffi field is Oð rÞ close to the crack tip (r = distance to the crack tip). The source of errors far from the tip is an incorrect shape function set close to the tip, but no the lack of refinement far from it. In order to obtain an error estimator for the approximated fields, sensitivity analysis has been proposed for different formulations of the Boundary Element Method (BEM): elastostatics [2,3], fracture problems [4,5], standard and hypersingular integral equations for potential and acoustic problems [6,7]. Based on interpolation ideas, nodal sensitivity can be related to tangential derivative error [4,5], but for a BEM procedure, a similar relationship is not obvious, because all the nodal values change with respect to nodal perturbations. Tangential derivatives of potential values have been considered in order to obtain an error estimator: one approach is based on two solution difference, with Hermitian and Lagrangian interpolations. For the Hermitian elements a boundary integral equation for tangential derivatives of potential (displacements) is used. This error estimator is applied to potential problems, 2D and 3D elasticity, and thermo-elastic problems [8– 12]. Recently, it has been presented an error estimation approach based on tangential derivatives [13,14]. This

work also presents the use of local error estimator in an adaptive mesh refinement procedure, based on a rrefinement approach. The mathematical analysis in the error estimation for the collocation BEM has been introduced in the last years. As cited above the mathematical relationship between sensitivity and tangential error has been established, but for interpolation theory. More recently, various mathematical relations between error and estimator for the residuals of the Boundary Integral Equations (BIEs) have been published [15,16]. The present paper describes a new error estimator based on tangential BIE residuals. A mathematical analysis is performed in order to show that the error function of the tangential derivative (flux or potential) is related to the residual of an appropriate BIE. A similar analysis is carried out for nodal sensitivities. Numerical tests validate the use of the proposed BIE residual as error estimator, and show that it is more accurate than nodal sensitivity, and requires less computational effort. This paper is focused on the critical study for the new error estimator proposed. This analysis is the most important of the process, because based on the information provided by the error estimator, several mesh adaptation techniques can be implemented (h-, p-, hp-, r-). The implementation of different adaptive techniques based on this error estimator is being carried out by the authors.

2. Basic equations Consider a 2D domain X whose boundary is C. Both the potential and acoustic problem, governed by the Laplace equation DuðxÞ ¼ 0

ð1Þ

and Helmholtz equation DuðxÞ þ k 2 uðxÞ ¼ 0

approximate solution exact solution

Fig. 1. Example: exact and approximated solution near a crack tip.

ð2Þ

respectively, are considered in parallel. In Eq. (2) k is the wave number. ou The flux is given by qðxÞ ¼ ou ¼ ox ni where n(x) is the on i outward normal to the boundary at x. In both problems, boundary conditions are provided, uðxÞ ¼ u on Cu, and qðxÞ ¼ q on Cq such that Cu [ Cq = C and Cu \ Cq = ;. For the Helmholtz problem u(x) and q(x) are complex numbers while they are real for the Laplace problem. For the sake of simplicity u(x) will be termed potential and q(x) flux, regardless of the problem considered, and no distinction is made between both problems, save when explicitly stated. For these problems, the potential Boundary Integral Equation, or u-BIE, is

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

Z 1 uðyÞ þ q ðx; yÞuðxÞ dCðxÞ 2 C Z ¼ u ðx; yÞqðxÞ dCðxÞ

ð3Þ

C

where y 2 C is a smooth boundary point; q*(x; y) and u*(x; y) are the fundamental solution flux and potential, respectively, for the problem under consideration (see Section 5). Likewise, the flux Boundary Integral Equation, or qBIE, at a smooth boundary point is Z 1 qðyÞ þ d ðx; yÞqðxÞ dCðxÞ 2 C s ðx; yÞuðxÞ dCðxÞ

¼

ð4Þ

C

where d*(x; y) and s*(x; y) are the normal derivatives of the kernels in the u-BIE (see section 5). In the rest of the paper, the variables x and y inside the integral sign are dropped out, for the sake of brevity. The right-hand side integral in (4) is understood in the sense of the Hadamard Finite Part.

numerical accuracy, at a given set of collocation points y. Then, at any collocation point where Dirichlet boundary conditions are prescribed, the q-BIE for the exact solution can be stated (4), and, taking into account (5) and (6) the q-BIE can be rearranged as 1 eq ðyÞ þ 2

ðd eq  s eu Þ dC C

1 ^ðyÞ  ¼ q 2

q  s ^ uÞ dC ðd ^

ð7Þ

C

where y is one of the collocation points. Next, the residual of the q-BIE, eq(y) is defined by the equation, 1 1 eq ðyÞ ¼  ^ qðyÞ  2 2

q  s ^ uÞ dC ðd ^ C

This residual is evaluated using the approximated solution, and its value is not nil, since the u-BIE, not the q-BIE, is fulfilled at the collocation point y 2 Cu to obtain the approximate solution. Thus, Eq. (7) is finally written as 1 eq ðyÞ þ 2

3. Error estimators based on BIE residuals

687

1 ðd eq  s eu Þ dC ¼ eq ðyÞ 2 C

ð8Þ

If the integral in the left-hand side were negligible, then, An error estimator based on BIE residuals was proposed for the collocation BEM almost two decades ago [17]. The authors use a residual type error estimator for 3D potential problems. The residual error estimator is latter extended to elastostatics by the same authors [18,19] and much latter the idea is further developed using the hypersingular or stress BIE for linear elasticity [4,20–22]. The error indicator is obtained recomputing the stresses with the hypersingular or stress BIE at the collocation points and comparing their value with the approximated solution. Based on this work, the hypersingular error estimator is extended to other related numerical method based on integral equations: Symmetric-Galerkin BEM [23], Meshless Boundary Node Method [24], Boundary Contour Method [25]. A recent review paper presents a thorough discussion of these ideas [26]. A brief explanation of the idea of hypersingular BIE residual as error estimator is presented here, since it is closely related to the estimator proposed in this paper. Firstly, the exact solutions u(x) and q(x) can be decomposed in two additive terms: approximated potential or flux (^uðxÞ,^qðxÞ) plus error for potential or flux (eu(x), eq(x)). Thus uðxÞ ¼ ^uðxÞ þ eu ðxÞ

ð5Þ

qðxÞ ¼ ^qðxÞ þ eq ðxÞ

ð6Þ

Using standard BEM an approximate solution ^uðxÞ, ^ðxÞ is obtained such that the u-BIE is fulfilled, within q

eq ðyÞ ’ eq ðyÞ Thus, the residual in the q-BIE can be used to approximate the error for the flux. This residual has a ‘‘symmetric’’ property [20]. In fact, at boundary points where Neumann conditions are prescribed, the q-BIE may be collocated to obtain the approximate solution and a new residual eu based on the u-BIE could be used to approximate eu. This scheme is used to estimate the error at interior points, as well. These BIE residuals, however, overestimate the exact errors, although they can be used as an error indicators [20]. The regression between this error estimators and the real errors is not 1:1, and their correlation factor is not close to 1.0. In the present work, using a similar approach, the residuals in the u-BIE and q-BIE are replaced by the residuals in the BIEs for the tangential derivative of potential or flux. These residuals are error estimators of the error in tangential derivative of potential and flux, respectively, as it is demonstrated in the following section.

4. Tangential boundary integral equation residuals as error estimator A stable second kind integral equation set is obtained by using the q-BIE at collocation points where the unknown is the flux, and the u-BIE at points where the unknown is the potential [28]. Thus, the approximate

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solution is computed using both BIEs in the same problem. For general mixed boundary conditions: • The u-BIE is collocated at points y 2 Cq (flux known, potential unknown). • The q-BIE at points y 2 Cu (potential known, flux unknown). With this information, the proposed error estimator for both the Neumann and Dirichlet parts of C can be described in a parallel scheme. 4.1. Sub–boundary with Neumann boundary conditions The boundary integral equation for the tangential derivative of the potential, or ut-BIE, at a smooth boundary point is [7] Z   1 ut ðyÞ þ --- q t ðx; yÞuðxÞ  u t ðx; yÞqðxÞ dCðxÞ ¼ 0 ð9Þ 2 C where ut(y) is the tangential derivative of the potential at the boundary point y, ou ou ¼ ut ðyÞ ¼ T i ¼ u,i T i oT oy i The kernels u t ðx; yÞ and q t ðx; yÞ are obtained from those of the u-BIE by differentiation, as shown in Section 5. The integral is understood in the sense of Cauchy Principal Value. The exact tangential derivative of the solution, ut, can be decomposed as ut ðyÞ ¼ ^ut ðyÞ þ etu ðyÞ

ð10Þ

where ^ut is the tangential derivative of the approximate o^u solution ^u, differentiating the shape functions, ^ut ¼ oT , t and eu ðyÞ is the true error in the potential tangential derivative. Considering Eqs. (10), (5) and (6), the ut-BIE (9) can be written as Z 1 t eu ðyÞ þ --- ðq t eu  u t eq Þ dC 2 C Z 1 ð11Þ ¼  ^ut ðyÞ  --- ðq t ^u  u t ^qÞ dC 2 C Defining the residual for the ut-BIE, etu , by the equation Z 1 t 1 eu ðyÞ ¼  ^ut ðyÞ  --- ðq t ^u  u t ^qÞ dC ð12Þ 2 2 C Eq. (11) can be re-written as Z 1 t 1 eu ðyÞ þ --- ðq t eu  u t eq Þ dC ¼ etu ðyÞ 2 2 C

ð13Þ

If the integral terms in the left-hand side were negligible, Eq. (13) leads finally to etu ðyÞ ’ etu ðyÞ

ð14Þ

With this equation, the error in the tangential derivative (etu ) can be approximated by a residual of a BIE (etu ), which can be readily computed once the approximate solution is obtained. A different approach could be devised from (13): taku ing into account that, etu ¼ oe and that eu(y) = 0 on Cu oT and eq(y) = 0 on Cq, Eq. (13) can be discretized and solved using standard Boundary Element techniques, for the unknowns eu and eq. But this approach is not promising since, even if the values obtained for eu and eq were accurate, the errors in the potential and fluxes do not detect directly errors due to the interpolation functions, as mentioned in the introduction, which is the final aim of any error estimator for mesh adaptation. Eq. (14) implies that the error in the tangential derivative of the error on the boundary Cq is estimated by the amount by which the solution computed using u-BIE and q-BIE, fails to satisfy the ut-BIE. This approach has several advantages worth stressing: • The estimator can be computed at the nodal points or any other boundary point. • It does not involve any adjustable parameters. • It is defined by an analytical formula. • It is intrinsic to the Boundary Element Method. • Its evaluation can be performed at chosen points independently. • No new system of equations has to be solved. • It predicts errors due to the interpolation space. A similar approach with all these characteristics, except the last one, has been proposed [20]. This last property, however, is very important to rightly priorize the locations where the mesh refinement should be carried out. A theoretical argument can be provided to shown that the integral term in (13) is actually small. To do so, substituting in Z I ¼ --- ðq t eu  u t eq Þ dC C

u and eq ¼ q  ^ q, and rearranging the values of eu ¼ u  ^ terms, one obtains Z Z I ¼ --- ðq t u  u t qÞ dC  --- ðq t ^ u  u t ^ qÞ dC C

C

The first integral is simply  12 ut , from (9), i.e.,  12 times the exact value of the tangential derivative of the potential, while the second one is  12 ut jBIE where utjBIE is the recomputed value of the potential tangential derivative using the corresponding ut-BIE. Note that ut jBIE 6¼ ^ ut , since this last value is the tangential derivative of the approximate solution ^ u, and utjBIE 5 ut, as well, since the value in the left-hand side is computed using the approximate solution of the potential and flux, not the exact one.

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

689

Collecting these expressions one obtains Z 1 --- ðq t eu  u t eq Þ dC ¼  ðut  ut jBIE Þ 2 C

Again, if the integral term in the left-hand side were small,

It is well established that the approximation of the tangential derivative using the ut-BIE is much better than the one obtained by direct differentiation of the shape functions [16], and therefore

In conclusion, the residual for the qt-BIE can be used to approximate the value of the error in the tangential derivatives of the approximated solution for the flux. The reasoning to show that the integral term is actually negligible is similar to the one in the previous case, leading to Z 1 --- ðd t eq  s t eu Þ dC ¼  ðqt  qt jBIE Þ 2 C

etq ðyÞ ’ etq ðyÞ

etu ¼ ut  ^ut  ut  ut jBIE inequality that justifies the approximation in (14). 4.2. Sub-boundary with Dirichlet boundary condition On the part of the boundary where the potential u(y) is known, the q-BIE is collocated to compute the approximate solution. The boundary integral equation for the tangential derivative of flux, or qt-BIE, at a smooth boundary point, is [7] 1 q ðyÞ þ 2 t

C

ð15Þ The kernels in this equation are obtained by differentiation from the kernels in the q-BIE (see Section 5). The exact value of the flux tangential derivative, qt(y), can be written as ð16Þ

where, again, q^t is the tangential derivative of the approximate solution flux, and etq is the error in the flux tangential derivative. The integral equation (15) leads to a Hadamard Finite Part in the limit to the boundary [7]. To compute these Finite Parts is necessary to resort to regularization procedures to weaken the highly-singular kernels (up to O(r3) for 2D). In [7] these kernels have been employed to develop sensitivity analysis for the approximate solutions of the hypersingular boundary integral equation. Substituting now Eqs. (16), (5) and (6), in Eq. (15), and rearranging terms the following equation is obtained: 1 t e ðyÞ þ 2 q

ðd t eq  s t eu Þ dC C

1 ¼  q^t ðyÞ  2

q  s t ^uÞ dC ðd t ^

where qtjBIE is the value of the flux tangential derivative recomputed using the approximate solution in the qtBIE. Again, the inequality etq ¼ qt  ^ qt  qt  qt jBIE justifies the approximation in (20).

ðd t ðx; yÞqðxÞ  s t ðx; yÞuðxÞÞ dCðxÞ ¼ 0

qt ðyÞ ¼ ^qt ðyÞ þ etq ðyÞ

ð20Þ

ð17Þ

C

4.3. Tangential derivative BIE residuals, nodal sensitivities, and errors in the tangential derivative Eqs. (13) and (19) summarize the theoretical considerations that leads to a relationship between tangential derivative BIE residuals and the true errors in the tangential derivative. In this section, the relationship between nodal sensitivities and tangential derivative errors is shown. Nodal sensitivities are defined as the rate of change of nodal solutions when the coordinates of one or a set of collocation nodes suffer an infinitesimal variation. The sensitivity formulation for the u-BIE in elastostatics can be found in [2,5] while the formulation for the sensitivity of the u-BIE and the q-BIE for potential and acoustic problems is developed in [7,6]. In all cases, nodal sensitivity values of potential (displacement) and flux (traction) are obtained using the Direct Differentiation Approach (DDA) [2]. Thus, from the u-BIE, the following BIE is obtained: Z 1 ~ uðyÞ þ --- ðq ~ u  u ~ qÞ dC 2 C Z 1 ut ðyÞ  --- ðq t ^ u  u t ^ qÞ dC ¼ ^ 2 C

The residual of the qt-BIE is defined by the equation

where ~ uðxÞ, and ~ qðxÞ are the potential and flux sensitivity, respectively. and, likewise, from the q-BIE

1 t 1 e ðyÞ ¼  ^qt ðyÞ  2 q 2

1 ~ qðyÞ þ 2

ðd t ^q  s t ^uÞ dC

ð18Þ

C

1 ^ ðyÞ  q  s ~ uÞdC ¼  q ðd ~ 2 t C

q  s t ^ uÞdC ðd t ^ C

and therefore Eq. (15) can be written as 1 t e ðyÞ þ 2 q

1 ðd t eq  s t eu Þ dC ¼ etq ðyÞ 2 C

ð19Þ

Considering the expressions of the residuals in Eqs. (12) and (18), the BIEs for the sensitivity can be written as

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

690

Z 1 1 ~ uðyÞ þ --- ðq ~u  u ~ qÞ dC ¼ etu ðyÞ 2 2 C

ð21Þ

1 ~ qðyÞ þ 2

ð22Þ

1 ðd ~q  s ~uÞ dC ¼ etq ðyÞ 2 C

In these equations, the residuals in the right-hand side can be computed once the approximate solution is obtained. After discretization of (21) and (22) by standard Boundary Element techniques the nodal sensitivities are computed. Note that the kernels in these equations are those of the u-BIE and q-BIE, and therefore, the ensuing algebraical system of equations has the same system matrix. To show the relationship between nodal sensitivities and tangential BIE errors, the value of the residuals in terms of the actual errors given in (13) and (19), is substituted in these equations, leading to Z Z 1 t 1 eu ðyÞ ¼ ~uðyÞ þ --- ðq ~u  u ~qÞ dC  --- ðq t eu  u t eq Þ dC 2 2 C C ð23Þ Z Z 1 t 1 q  s ~ uÞ dC  --- ðd t eq  s t eu Þ dC eq ðyÞ ¼ ~ qðyÞ þ --- ðd ~ 2 2 C C ð24Þ

If the integral terms in the right-hand sides of these equations were negligible, then etu ðyÞ  ~uðyÞ etq ðyÞ

 ~qðyÞ

ð25Þ ð26Þ

Numerical experiments shown in Section 6 confirm that the integral terms in (23) and (24) are small, and therefore nodal sensitivities are truly error estimators for the tangential derivative errors. However, it has not been possible to justify theoretically that the first integral term in Eqs. (23) and (24) is always small, in comparison with the terms 12 etu ðyÞ and 12 etq ðyÞ. Nevertheless, this error estimator is computationally more expensive than the tangential BIE residual, since it involves the computation of this one and the solution of a system of equation, even though the system matrix were factorized. On the other hand, it will be shown in the numerical applications that the sensitivities overpredict the values of the tangential derivative errors, and is less closely correlated to the true error than the tangential BIE residuals.

5. Tangential derivative kernels In this section, the expressions of the kernels in the ut-BIE and qt-BIE, both for the Laplace and Helmholtz problems are shown. In order to calculate the tangential BIE residuals, regularization techniques must be employed. The development of the theoretical consider-

ations that leads to the tangential derivative BIEs, their singular behaviour and its regularization is not covered in this paper (see [7]). 5.1. Kernels in the ut-BIE The kernels in the ut-BIE, u t ðx; yÞ and q t ðx; yÞ are obtained from the kernels of the u-BIE, u*(x; y) and q*(x; y) by the equations u t ðx; yÞ ¼

ou ou ou ¼ Ti ¼  T i ¼ u ,i T i oT oy i oxi

q t ðx; yÞ ¼

oq oq oq ¼ Ti ¼  T i ¼ q ,i T i oT oy i oxi

where T is the unit vector tangent to the boundary at the collocation point y. 5.1.1. Kernels in the ut-BIE for the Laplace problem The kernels in the standard u-BIE for the Laplace equation (1) are well known in the BEM literature: u ðx; yÞ ¼ 

1 ln r; 2p

q ðx; yÞ ¼ 

1 qn 2pr

where r = jx  yj; n is the outward normal vector to the boundary at x; and q is the unit vector in the xy direction. The kernels in the ut-BIE are readily obtained, u t ðx; yÞ ¼

1 q  T; 2pr

q t ðx; yÞ ¼

1 ðn  T  2q  tq  T Þ 2pr2

where t is the unit vector tangent to the boundary at x. The kernel q t ðx; yÞ is regular, while u t ðx; yÞ is singular O(r1), and its Cauchy Principal Value can be evaluated by a simple zero order regularization. 5.1.2. Kernels in the ut-BIE for the Helmholtz problem The kernels in the u-BIE for the Helmholtz problem are u ðx; yÞ ¼

1 K 0 ðikrÞ; 2p

q ðx; yÞ ¼

ik K 1 ðikrÞq  n 2p

where Kj(z) is the modified Bessel function of jth order. These kernels have the same order of singularity than the kernels for LaplaceÕs equation. Actually, they can be decomposed in their singular part (LaplaceÕs kernels) plus a regular part (dynamic increment) for their numerical integration, and therefore no new regularization procedures are required. The kernels in the ut-BIE for the Helmholtz problem are u t ðx; yÞ ¼

ik K 1q  T 2p

q t ðx; yÞ ¼

ik ½K 1 n  T  ð2K 1 þ ikrK 0 Þq  nq  T  2pr

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

The variable z = ikr in the Bessel functions is dropped out for the sake of brevity. Again, u t ðx; yÞ is singular O(r1) and q t ðx; yÞ is regular.

From these kernels, differentiating with respect to the tangential vector at y, the kernels in the qt-BIE are obtained, d t ðx; yÞ ¼

5.2. Kernels in the qt-BIE The kernels in the qt-BIE, d t ðx; yÞ and s t ðx; yÞ are obtained from the kernels of the q-BIE, d*(x; y) and s*(x; y), respectively, by the equations d t ðx; yÞ ¼

od od od ¼ Ti ¼  T i ¼ d ,i T i oT oy i oxi

s t ðx; yÞ ¼

os os os ¼ Ti ¼  T i ¼ s ,i T i oT oy i oxi

691

ik ½j0 rK 1  ð2K 1 þ ikrK 0 Þq  N q  T 2pr h ik ð2K 1 þ ikrK 0 Þ q  Tðn  N  4q  nq  NÞ 2pr i rj0 ðn  T  2q  Tq  nÞ þ n  Tq  N  2 k 2 j0 K 0 ik 3 K 1 þ nT  q  nq  Tq  N 4p 2p

s t ðx; yÞ ¼ 

6. Applications

where T is the unit vector tangent to the boundary at y. 5.2.1. Kernels in the qt-BIE for the Laplace problems The kernels in the q-BIE for the Laplace problem have the following expression: d ðx; yÞ ¼

1 q  N; 2pr

s ðx; yÞ ¼

1 ðn  N  2q  nq  N Þ 2pr2

where N is the outward normal to the boundary at the point y. The kernel d*(x; y) is regular, while s*(x; y) is hypersingular O(r2), and especial regularization formulas are required for the computation of its Finite Part. The kernels for the tangential q-BIE are obtained by differentiation leading to d t ðx; yÞ ¼

1 ðj0 r  2q  NÞq  T 2pr2 1 h q  Tðn  N  4q  nq  NÞ pr3 i rj0 þ n  Tq  N  ðn  T  2q  Tq  nÞ 2

s t ðx; yÞ ¼ 

where j0 is the boundary curvature at point y. The kernel d t ðx; yÞ is regular and s t ðx; yÞ is highly singular, O(r3), so special regularization formulas are required to compute its Finite Part. 5.2.2. Kernels in the qt-BIE for the Helmholtz problem The kernels in the q-BIE and in qt-BIE are highly singular functions. Such kernels have the same order of singularity than the kernels for the q-BIE and qt-BIE for the Laplace problem. The expressions of the kernels in the q-BIE are d ðx; yÞ ¼ 

ik K1q  N 2p

s ðx; yÞ ¼ 

ik ½K 1 n  N  ð2K 1 þ ikrK 0 Þq  nq  N  2pr

In order to validate the formulation proposed and to show some properties of both the tangential BIE residual and nodal sensitivity as error estimators, several numerical examples are presented here. The aims of the different tests are the following: • To show that residuals in the tangential derivative BIEs can be correlated to actual errors in the tangential derivatives of the solution. Using exact solutions, the computed residuals are compared to the values of the exact error in the tangential derivative. The slope of the linear regression between the exact error and the residual, and the correlation factor (R2) are computed. • To validate that nodal sensitivities are truly error estimators. Nodal sensitivities have been obtained and contrasted with errors in tangential derivatives for mid-points of the quadratic elements. Again, the slope of the regression and the correlation factor are computed to test the accuracy of the sensitivity as error estimator. For all of these applications there are various common numerical aspects: (1) Element type: Quadratic isoparametric elements. (2) Collocation for u-BIE and q-BIE is always performed at smooth points. At corners, outof-node collocation is employed, shifting the collocation points inside the element but close to the end node (collocation points n = ±0.8, where n is the natural coordinate). Nevertheless, the interpolation nodes are at the end nodes (conforming elements). (3) Except at corners, it is assumed that the boundary normal at inter-element nodes is continuous, so these nodes are treated as smooth boundary points. It is well known in the literature that a spurious logarithmic singularity remains if the q-BIE is

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692

collocated in such nodes. This effect is dependent on the difference of tangential vectors at the inter-element nodes, and on the difference of potential tangential derivatives. The effect is very important for isoparametric linear elements, but can be neglected for isoparametric quadratic elements if using a fine enough mesh. Nevertheless, an out-of-node collocation approach can be adopted for these nodes, avoiding altogether this minor source of computational error. (4) Tangential BIE residuals and nodal sensitivities are collocated only at the element mid-nodes, so all continuity requirements for the boundary and the variables are fulfilled. (5) Numerical integration with Gauss quadrature is performed with 10–15 points. For singular and hypersingular terms, after regularization, double number of points is employed. (6) For the computation of the approximate solution a mixed formulation is used: u-BIE at nodes where the potential is unknown and q-BIE where the flux is. The following numerical applications are included in this section: (1) Laplace equation: (a) Rectangular plate with Dirichlet boundary conditions (Fig. 2, a = 100, b = 50). (b) Rectangular plate with mixed boundary conditions (Fig. 2, a = 100, b = 50). (c) Disk sector with mixed boundary conditions (Fig. 3, R1 = 50, R2 = 100). (2) Helmholtz equation: (a) Rectangular plate with mixed boundary conditions (Fig. 2, a = 100, b = 50). (b) Disk sector with mixed boundary conditions (Fig. 3, R1 = 50, R2 = 100). In order to evaluate the exact errors, analytical solutions of these problems are provided. To leave out avoidable sources of error, the boundary conditions vary quadratically at most along the boundary, so they are exactly represented with the quadratic elements.

γ2 Γ3 Γ2

R2 Γ4 γ4

Γ1

R1 Fig. 3. Disk sector domain.

For each case, two results are displayed: (1) Table of values: at mid-nodes, the following values are listed: • Exact value of Error in Tangential Derivative (ETD). • Tangential Boundary Integral Equation Residual corresponding to the unknown at the point (TBIER). • Nodal Sensitivity for the unknown (NS). (2) X–Y graphic: The exact error (X axis) vs. the error estimators (Y axis) are shown. The regression lines for exact tangential error vs. residual, and exact tangential error vs. nodal sensitivities are plotted as well. The better the error estimator, the closer to 1.0 the regression line slope will be. However even if the slope is not 1.0 the closer the correlation factor is to 1.0, the better the expression as error indicator.

6.1. Error estimators: application for Laplace problems Γ3 γ4

Γ4

Γ2

Γ1 a

Fig. 2. Rectangular domain.

γ2

b

6.1.1. Rectangular plate with Dirichlet boundary conditions The domain in the first application is a rectangular plate a · b with Dirichlet boundary conditions. At the upper and lower sides u = 0 while at the left and right side the prescribed potential varies quadratically, attaining its maximum value c2 = c4 = 1.0 at the center of each side, as shown in Fig. 2. The analytical solution for this problem can be computed in series form:

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

uðx,yÞ ¼

1 X 

 An ekn x þ Bn ek n x sinðk n yÞ

5.E-03

ð27Þ

4.E-03

n¼1

3.E-03

The coefficients An and Bn are obtained solving

An e

k n a

þ Bn e

where an ¼

bc4 2

bc bn ¼ 2 2

2.E-03

TBIER, NS

An þ Bn ¼ an kn a

693

¼ bn

1.E-03 0.E+00 -1.E-03 -2.E-03 TBIER NS Lineal (TBIER) Lineal (NS)

-3.E-03

Z

b

" 1

0

Z

b 0

 2 # 4 b sinðk n yÞ dy y  2 b2

"

 2 # 4 b sinðk n yÞ dy 1 2 y 2 b

and k n ¼ np , n = 1, 2, . . . b The edges C1 and C3 have been divided into six quadratic elements, and C2 and C4 have been divided into three quadratic elements. Table 1 lists the results at the mid nodes of the elements on C1 and C2. In Fig. 4, the slope of the regression line between the Tangential BIE Residual and the exact tangential derivative error is 1.069 while the correlation factor is 0.9958; the regression slope between the exact tangential derivative error and the nodal sensitivity is 1.319, and the correlation factor is 0.9939.

-4.E-03 -5.E-03 -5.E-03 -4.E-03 -3.E-03 -2.E-03 -1.E-03 0.E+00 1.E-03

2.E-03

3.E-03

4.E-03

5.E-03

ETD

Fig. 4. Exact Error in Tangential Derivative vs. estimators for rectangular plate with Dirichlet boundary conditions (LaplaceÕs problem).

An  Bn ¼ an An ekn a  Bn ekn a ¼ bn where bc an ¼ 4 2k n bc bn ¼ 2 2k n

Z

b

"

b

"

0

Z 0

 2 # 4 b 1 2 y sinðk n yÞ dy 2 b  2 # 4 b 1 2 y sinðk n yÞ dy 2 b

np , b

6.1.2. Rectangular plate with mixed boundary conditions For this test, the rectangular domain in Fig. 2 is subject to mixed boundary conditions. At the upper and lower edge the potential is again fixed to u = 0 while on the right and left side, the flux is prescribed. The flux varies quadratically along both edges attaining its maximum at the center, as shown in figure. For this application c2 = 0.1 and c4 = 0.2. The analytical solution for this test is given again by (27), but with coefficients An and Bn computed by the following equations:

and k n ¼ n = 1, 2, . . .. The boundaries C1 and C3 have been divided into six quadratic elements and C2 and C4 into three. Table 2 lists the results along boundaries C1, C2. Note that on C1 the error is in the tangential derivative of q, while on C2 the error is in the tangential derivative of u. Therefore, on each boundary, the corresponding residual and sensitivity is computed. In Fig. 5 the value of the exact error vs. both error estimators, plus the corresponding regression lines are shown. The slope of the regression line between the exact error and the tangential BIE residual is 1.0047 and

Table 1 Exact Error in Tangential Derivative (ETD), Tangential Boundary Integral Equation Residual (TBIER) and Nodal Sensitivity (NS) for rectangular plate with Dirichlet boundary conditions (LaplaceÕs problem) Boundary

ETD

TBIER

NS

C1

3.325636298E3 4.453948290E4 9.422882186E5 9.422479854E5 4.453854413E4 3.325858325E3

3.374462729E3 4.367052738E4 9.160686815E5 9.160759406E5 4.366966657E4 3.374680633E3

4.106028936E3 5.938090616E4 1.347382349E4 1.347642537E4 5.938263493E4 4.106276202E3

C2

2.703529272E3 1.278448867E7 2.703324411E3

3.124706058E3 5.150470520E8 3.124737954E3

3.910138585E3 5.495116541E8 3.910177098E3

694

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

Table 2 Exact Error in Tangential Derivative (ETD), Tangential BIE Residual (TBIER) and Nodal Sensitivity (NS) for rectangular plate under mixed boundary conditions (LaplaceÕs problem) Boundary

ETD

TBIER

NS

C1

4.53663888E4 1.34142697E4 5.18417442E5 3.73564477E5 7.03220870E5 2.15149554E4

4.14285845E4 1.34086832E4 5.17010282E5 3.73191354E5 7.08433254E5 2.00368683E4

3.18110221E4 1.69998034E4 1.00673969E4 8.41290820E5 1.05712373E4 1.58525040E4

C2

2.46090409E3 1.17265264E7 2.46103689E3

2.45290635E3 7.58885011E8 2.45309681E3

3.32952188E3 1.06284234E7 3.32965020E3

7.E-03 6.E-03 5.E-03 4.E-03

TBIER, NS

3.E-03 2.E-03 1.E-03 0.E+00 -1.E-03 -2.E-03 -3.E-03 TBIER NS Lineal (TBIER) Lineal (NS)

-4.E-03 -5.E-03 -6.E-03 -7.E-03 -5.E-03 -4.E-03 -3.E-03 -2.E-03 -1.E-03 0.E+00 1.E-03

2.E-03

3.E-03

4.E-03

5.E-03

6.1.3. Disk sector with mixed boundary conditions In this test, the suitability and reliability of the procedure and its implementation for curved elements is assessed. The geometry and variation of the prescribed boundary conditions is shown in Fig. 3. The edges C1 and C3 are both meshed with four equal size elements and C2 and C4 with ten elements each one. On the straight edges C1 and C3 the potential is prescribed to u = 0, while the flux varies quadratically along C2 and C4, as shown in the figure,with c2 = 0.1, and c4 = 0.2. The analytical solution for this problem is

ETD

Fig. 5. Exact Error in Tangential Derivative vs. estimators for rectangular plate with mixed boundary conditions (LaplaceÕs problem).

the correlation factor 0.9999; the slope of the regression line between the exact error and the nodal sensitivity is 1.2859 while its correlation factor is 0.9977. An enlargement of this graphics for small values of the error is presented in Fig. 6.

4.E-04 3.E-04 2.E-04 1.E-04 0.E+00 -1.E-04 -2.E-04

TBIER NS Lineal (TBIER) Lineal (NS)

-3.E-04 -4.E-04

-5.E-04 -5.E-04 -4.E-04 -3.E-04 -2.E-04 -1.E-04 0.E+00 1.E-04 2.E-04 3.E-04 4.E-04 5.E-04 ETD

Fig. 6. Exact Error in Tangential Derivative vs. estimators for rectangular plate with mixed boundary conditions: range of small values (LaplaceÕs problem).

1 X 

 An r2n þ Bn r2n sinð2nhÞ

ð28Þ

n¼1

where  Bn R2n1 ¼ an An R2n1 2 2 An R2n1  Bn R2n1 ¼ bn 1 1 The right side terms are computed by  Z p2  pc 16 p 2 an ¼ 2 sinð2nhÞ dh 1 2 h p 4 8n 0 bn ¼

5.E-04

TBIER, NS

uðr,hÞ ¼

pc4 8n

Z p2  0

1

 16 p 2 sinð2nhÞ dh h  p2 4

Table 3 lists the results over boundaries C1, C2 for midnodes at each element while in Fig. 7 the same values and the regression lines between exact error and estimators are shown. The regression slopes between the exact error and the residual estimator is 0.765, while it is 1.24 for the exact error vs. sensitivity; the correlation factors are 0.9932 and 0.6085, respectively. 6.2. Error estimators: applications for Helmholtz problems Numerical tests with Helmholtz problems have several common points:

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699 Table 3 Exact Error in Tangential Derivative (ETD), Tangential BIE Residual (TBIER) and Nodal Sensitivity (NS) for disk sector with mixed boundary conditions (LaplaceÕs problem) Boundary

ETD

TBIER

NS

C1

4.646418E4 1.402840E4 4.612122E5 2.441525E6

4.583388E4 1.398047E4 4.638241E5 1.310575E5

3.764549E4 1.245958E4 3.661558E5 9.718736E6

C2

6.836445E4 5.580177E4 3.883472E4 2.309783E4 7.715245E5 7.723740E5 2.309644E4 3.884978E4 5.582355E4 6.823407E4

5.254651E4 4.108137E4 2.715217E4 1.568586E4 5.161977E5 5.168742E5 1.568204E4 2.716298E4 4.109436E4 5.238649E4

3.085253E4 4.394386E4 4.039168E4 2.699831E4 9.379958E5 9.388060E5 2.701790E4 4.039921E4 4.395501E4 3.104894E4

695

(1) Mixed boundary conditions. (2) Value of k = 2p/k; k = 250. This case is intermediate between low and high frequencies, and is representative of the behaviour of both error estimators in Helmholtz equation. (3) The real part and imaginary part of the results are presented separately.

6.2.1. Rectangular plate with mixed boundary conditions For this test, considering the notation in Fig. 2, u = 0 has been prescribed on C1 and C3, while a quadratically varying flux is prescribed on C2 and C4 where c2 = 0.1 and c4 = 0.2i, an imaginary value; these boundary conditions assure that the solution is complex, with non-zero real and imaginary parts. Analytical solution for this case can be obtained again in series form: 1 X uðx,yÞ ¼ ðAn fn ðxÞ þ Bn gn ðxÞÞ sinðxn yÞ ð29Þ n¼1

where the coefficients An and Bn are obtained from the system of equations:

2.0E-03 1.5E-03

An fn0 ðaÞ þ Bn g0n ðaÞ ¼ an

TBIER, NS

1.0E-03

An fn0 ð0Þ þ Bn g0n ð0Þ ¼ ibn

5.0E-04 0.0E+00 -5.0E-04 -1.0E-03

TBIER NS Lineal (TBIER) Lineal (NS)

-1.5E-03 -2.0E-03 -2.0E-03

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

The right-hand side terms in the equations above are given by  2 # Z b" bc 4 b an ¼ 2 sinðxn yÞ dy 1 2 y 2 2 0 b

2.0E-03

ETD

Fig. 7. Exact Error in Tangential Derivative vs. estimators for disk sector with mixed boundary conditions (LaplaceÕs problem).

bc bn ¼ 4 2

Z 0

b

"

 2 # 4 b 1 2 y sinðxn yÞ dy 2 b

where xn ¼ np ,n ¼ 1,2, . . . b

Table 4 Exact Error in Tangential Derivative (ETD), Tangential BIE Residual (TBIER) and Nodal Sensitivity (NS) for rectangular plate with mixed boundary conditions: real part Boundary

ETD

TBIER

NS

C1

6.40312971E6 7.38853426E7 2.50572621E6 5.48595092E6 1.14136432E5 2.38644136E5 5.39390835E5

3.68496821E6 1.12052185E6 2.60945539E6 5.48959532E6 1.13754866E5 2.38147139E5 5.38833649E5

2.25227392E5 2.45438364E5 2.42221900E5 2.03683743E5 1.19885496E5 3.30906839E6 3.29961499E5

C2

1.60055280E4 1.86972233E3 6.22569242E4 6.22569238E4 1.86972232E3

1.41391825E4 1.33739291E3 4.42370519E4 4.42370519E4 1.33735047E3

9.24150426E5 1.30993438E3 4.23996357E4 4.24002981E4 1.30989821E3

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696

Table 5 Exact Error in Tangential Derivative (ETD), Tangential BIE Residual (TBIER) and Nodal Sensitivity (NS) for rectangular plate with mixed boundary conditions: imaginary part Boundary

ETD

TBIER

NS

C1

3.10705826E4 1.07283346E4 4.75630123E5 2.28292554E5 1.11242299E5 5.44638126E6 2.77020090E6 7.03164174E6

2.77030678E4 1.07715119E4 4.76217007E5 2.27504096E5 1.09840009E5 5.23610617E6 2.34887522E6 4.79536267E6

1.84500130E4 1.10999428E4 6.45451372E5 4.07683531E5 2.64478627E5 1.69506943E5 1.03471497E5 1.15653790E5

C2

3.83913326E5 1.58262783E5 1.58262782E5 3.83913325E5

3.36266634E4 1.16612943E4 1.16612943E4 3.36274824E4

1.82730614E4 5.45252415E5 5.45139710E5 1.82729337E4

The expansion functions fn(x) and gn(x) are harmonic or exponential functions, depending on the value of k and n. For k < 2b/n, and k 5 xn they are

For values k > 2b/n the functions are fn ðxÞ ¼ ekn x ;

gn ðxÞ ¼ ekn x ;

3.E-03 2.E-03

TBIER, NS

fn ðxÞ ¼ cosðk n xÞ; gn ðxÞ ¼ sinðk n xÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k n ¼ k 2  x2n

4.E-03

where k n ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2n  k 2

Boundaries C1 and C3 are divided into eight quadratic elements, and boundaries C2 and C4 into four elements. Tables 4 and 5 lists the results over boundaries C1 and C2, for real and imaginary parts . Figs. 8 and 9 shows the exact errors in tangential derivatives vs. both error estimators. The regression slopes and correlation factors between the true error and the estimators are shown in Table 6.

1.E-03 0.E+00 -1.E-03 -2.E-03

TBIER NS Lineal (TBIER) Lineal (NS)

-3.E-03 -4.E-03 -4.E-03

-3.E-03

-2.E-03

-1.E-03

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

ETD

Fig. 9. Exact Error in Tangential Derivative vs. estimators for rectangular plate with mixed boundary conditions: imaginary part (Helmholtz problem).

Table 6 Regression slopes and correlation factors between Tangential BIE Residual (TBIER) vs. Exact Error in Tangential Derivative (ETD) and Nodal Sensitivity (NS) vs. Exact Error (ETD) for rectangular plate with mixed boundary conditions

2.E-03 2.E-03

TBIER, NS

1.E-03 5.E-04

Regression slope (real part) Correlation factor (real part) Regression slope (imaginary part) Correlation factor (imaginary part)

0.E+00 -5.E-04 -1.E-03

NS vs. ETD

0.716 0.9995 0.712 0.9864

0.679 0.9983 0.902 0.9973

TBIER NS Lineal (TBIER) Lineal (NS)

-2.E-03 -2.E-03 -2.E-03

TBIER vs. ETD

-2.E-03

-1.E-03

-5.E-04

0.E+00

5.E-04

1.E-03

2.E-03

2.E-03

ETD

Fig. 8. Exact Error in Tangential Derivative vs. estimators for rectangular plate with mixed boundary conditions: real part (Helmholtz problem).

6.2.2. Disk sector with mixed boundary conditions The domain of this test is shown in Fig. 3. Boundary conditions have been prescribed in potential and flux values. The potential is prescribed to u = 0 on C1 and C3, and the flux on C2 and C4 with c2 = 0.2 and c4 = 0.1i.

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

697

Table 7 Exact Error in Tangential Derivative (ETD), Tangential BIE Residual (TBIER) and Nodal Sensitivity (NS) for disk sector with mixed boundary conditions: real part Boundary

ETD

TBIER

NS

C1

1.35975456E4 3.65006182E4 1.66097016E4 7.31456726E5 3.62011589E5 1.68833323E3

1.84863377E4 3.46669457E4 1.60424847E4 7.72138864E5 2.80021256E5 1.76042915E3

7.55331899E4 1.35731544E4 5.62965759E5 1.72057565E5 1.12568921E5 1.53806043E3

C2

1.463047400E2 1.163801095E2 4.168842970E3 4.16884391E3 1.16380111E2 1.46304777E2

8.061796610E3 7.299859848E3 2.640025793E3 2.64075973E3 7.30041893E3 8.06196319E3

2.974887786E3 3.344471756E3 1.115705317E3 1.11864714E3 3.34666445E3 2.97567954E3

Table 8 Exact Error in Tangential Derivative (ETD), Tangential BIE Residual (TBIER) and Nodal Sensitivity (NS) for disk sector with mixed boundary conditions: imaginary part Boundary

ETD

TBIER

NS

C1

1.80215040E5 8.57339842E5 3.45717287E5 1.26655401E5 6.65351863E6 4.22183945E4

2.84461682E5 1.05816923E4 4.77841849E5 2.34732487E5 1.11773770E5 4.04794515E4

9.86661980E5 8.20370456E5 4.94533552E5 4.96651360E5 7.54518361E5 5.11981375E4

C2

4.35548319E3 3.43596072E3 1.24913938E3 1.24914015E3 3.43596122E3 4.35548287E3

3.97337720E3 2.96073347E3 1.04151211E3 1.04096167E3 2.96031002E3 3.97321860E3

4.68831258E3 3.81298739E3 1.38782512E3 1.38871256E3 3.81363026E3 4.68857111E3

The analytical solution for this problem can be obtained by a series expansion in terms of Bessel functions using polar coordinates: uðr,hÞ ¼

1 X

ðAn J 2n ðkrÞ þ Bn Y 2n ðkrÞÞ sinð2nhÞ

The derivatives of the Bessel functions are taken with respect to z = kr.

ð30Þ

2.5E-02 2.0E-02

n¼1

1.5E-02

An J 02n ðkR2 Þ þ Bn Y 02n ðkR2 Þ ¼ an An J 02n ðkR1 Þ þ Bn Y 02n ðkR1 Þ ¼ ibn

bn ¼

pc4 4k

5.0E-03 0.0E+00 -5.0E-03 -1.0E-02 TBIER NS Lineal (TBIER) Lineal (NS)

-1.5E-02

where pc an ¼ 2 4k

1.0E-02

TBIER, NS

The coefficients in this series are the solution of the equations:

 Z p2  16 p 2 1 2 h sinð2nhÞ dh p 4 0 Z p2  0

 16 p 2 sinð2nhÞ dh 1 2 h p 4

-2.0E-02

-2.5E-02 -2.5E-02 -2.0E-02 -1.5E-02 -1.0E-02 -5.0E-03 0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02

ETD

Fig. 10. Exact Error in Tangential Derivative vs. estimators for disk sector with mixed boundary conditions: real part (Helmholtz problem).

A.E. Martı´nez-Castro, R. Gallego / Computers and Structures 83 (2005) 685–699

698 8.E-03 6.E-03

TBIER, NS

4.E-03 2.E-03 0.E+00 -2.E-03 TBIER NS Lineal (TBIER) Lineal (NS)

-4.E-03 -6.E-03 -8.E-03 -8.E-03

-6.E-03

-4.E-03

-2.E-03

0.E+00

2.E-03

4.E-03

6.E-03

8.E-03

ETD

Fig. 11. Exact Error in Tangential Derivative vs. estimators for disk sector with mixed boundary conditions: imaginary part (Helmholtz problem).

Table 9 Regression slopes and correlation factors between Tangential BIE Residual (TBIER) vs. Exact Error in Tangential Derivative (ETD) and Nodal Sensitivity (NS) vs. Exact Error (ETD) for disk sector with mixed boundary conditions

Regression slope (real part) Correlation factor (real part) Regression slope (imaginary part) Correlation factor (imaginary part)

TBIER vs. ETD

NS vs. ETD

0.694 0.9854 0.561 0.8893

0.581 0.8497 0.516 0.6921

All boundaries have been divided into six quadratic elements. Tables 7 and 8 show the results obtained for mid-nodes of each element. Figs. 10 and 11 show the values and regression lines, for real and imaginary parts. The regression slopes and correlation factors between the true error and the estimators are shown in Table 9.

7. Conclusions In this paper a new error estimator based on tangential BIE residual (TBIER) is proposed. Theoretical considerations lead to its relationship to the true error in the tangential derivative and several numerical applications show that the values predicted by the TBIER closely correlate to the exact error in the tangential derivative. The estimator is developed both for Laplace and Helmholtz problems, and both for the error in the potential and flux derivative. The TBIER estimates the error in the tangential derivative, which is more significant than error in the variable itself for mesh refinement, since it avoids the refinement of parts of the boundary where the error in the variable stems from a coarse representation somewhere else in the model. Mesh adaptation schemes based on this error estimator can be devised in a general

mesh adaptation solution, and various schemes can be used. Note that for LaplaceÕs problem on a domain with straight edges, the correlation between the TBIER and the exact error is almost perfect. In other cases there are differences, but the source of the discrepancies may be traced to numerical inaccuracies due to the integration of the hypersingular kernels over curved elements and procedures for the dynamic kernel regularization which can be improved further. It is also shown that nodal sensitivities are error truly estimators for the error in the tangential derivative. Again both Laplace and Helmholtz problems and both potential and flux sensitivities are considered. This error estimator, however, is more expensive than the TBIER since it requires the computation of the TBIER itself at perturbation points and the solution of a system of equations to obtain the sensitivities. Furthermore, the TBIER is shown to be a more accurate error estimator for tangential derivative errors than the nodal sensitivities, and therefore there is no reason to use the nodal sensitivities for error estimation. This method could be straightforwardly applied to other BEM formulations: 3D potential problems, elasticity problems, etc, and readily extended to other methods based on boundary integral equations: symmetric Galerkin, standard meshless, hypersingular boundary node method, and boundary contour method, as it has been done with the hypersingular residual estimator. References [1] Kita E, Kamiya N. Error estimation and adaptive mesh refinement in boundary element method, an overview. Eng Anal Bound Elem 2001;25:479–95. [2] Guiggiani M. Sensitivity analysis for boundary element error estimation and mesh refinement. Int J Numer Methods Eng 1996;39:2907–20. [3] Bonnet M, Guiggiani M. Tangential derivative of singular boundary integrals with respect to the position of collocation points. Int J Numer Methods Eng 1998;41:1255–75. [4] Paulino GH. Novel formulations of the boundary element method for fracture mechanics and error estimation. PhD thesis, Cornell University, August 1995. [5] Paulino GH, Shi F, Mukherjee S, Ramesh P. Nodal sensitivities as error estimates in computational mechanics. Acta Mech 1997;121:191–312. [6] Gallego R, Martı´nez-Castro A, Sua´rez J. Sensitivity analysis of approximate solutions for the hypersingular boundary element method. In: Proceedings of the Boundary Elements Techniques Conference. London: M.H. Aliabadi; 1999. p. 311–9. [7] Martı´nez Castro AE. Adaptive boundary elements. Application in fracture mechanics [in Spanish]. Proyecto Fin de Carrera. Grupo de Meca´nica de So´lidos y Estructuras, Granada, 1999. [8] Muci-Ku¨chler KH, Miranda-Valenzuela JC, Soriano-Soriano S. Use of the tangent derivative boundary integral

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