A New Integration Algorithm for Nearly Singular BIE Kernels T. A. Cruse Vanderbilt University Nashville, Tennessee 37235 USA
Overview
R. Aithal, Consultant Southwest Research Institute San Antonio, Texas 78228 USA
The boundary integral equation for the elasticity problem is written in terms of the boundary tractions ~ and boundary displacements uj in the usual manner [1]
c.u.(P) + J J T.(P,Q)u.(Q)dS(Q) - J J u..(P,Q)t.(Q)dS(Q) IJ J <s> IJ J <s> 'J J
(1)
where < S(Q) > denotes the principal value of the integrals on the boundary surface. The points Q(y) and P(x) respectively denote the integration point and the source point, corresponding to the point of application of the point load influence function. The tractions and displacements for the point load solution are written as Tjj(P,Q) and Ujj(P,Q), respectively. The Cjj matrix corresponds to the value of the jump in the first integral as the interior displacement evaluation point p(x) is taken to the boundary point P(x).
Following the usual procedures [2]1 for a numerical quadrature of the boundary integral equation (BIE), we replace the actual surface by a set of boundary elements, ASn ,
over which the boundary shape and boundary data are replaced by the usual quadratic shape functions and nodal values of the variables
(2)
The superscript (X has the range of six or eight depending on whether the boundary element is a triangle or a quadrilateral.
Each of the integrals in Eq (1) are represented by sums of integrals over each boundary element, which is illustrated for the traction kernel, as follows
The integration algorithm in this reference is adopted herein for illustration and numerical compari The comments developed herein apply to the totality of Gaussian integration strategies.
where J(~) is the Jacobian of the transformation to the reference (unit) area, A. The principal value notation will be imposed on any boundary element implementation of Eq (3) for which the boundary element contains the singular point P(x).
Eq (3) is normally integrated using Gaussian quadrature in boundary element codes. Unfortunately, the use of Gaussian quadrature does not produce exact results, since the integrated functions are singular to varying degrees depending on the kernel function and on the location of P(x) relative to the element. As a result, various numerical integration schemes have been used over the years in order to control the error of the numerical quadratures of the boundary integrals. Element subdivision, polar coordinates, and very highorder Gaussian integrations are generally used in these schemes.
The convergence characteristics of the Gaussian integration schemes is generally very poor for problems with graded meshes or thin sections, such that the source point P ... Q on the integration element. We take the distance to be close in the sense of distance of P(x) relative to the size of the boundary element. Codes with specific error control algorithms generally fail to converge within reasonable distances, while those codes without error control yield very inaccurate results. Such problems as fracture mechanics modeling and geometries with one thin dimension fall into this challenging category.
Recently, some very powerful concepts have been introduced in BEM implementations for the potential theory problem that eliminate the singular character of the kernels for Gaussian integration at P = Q. The approach taken by Lean and Wexler [3] is to regularize the singularity for P = Q through particular coordinate mappings that produce the desired, singularity-cancelling character in the mapping Jacobian. The modified mapping is applied to a new kernel, which is used to regularize the original BIB kernel. The numerical results were very encouraging for the P = Q case and indicated the important role of higher-order expansions of the mapping Jacobian.
A second approach published at the same time [4] is fundamentally different. The terms in the singular integrals for P = Q are individually expanded in a Taylor series manner. The leading singular terms are integrated exactly. Gaussian integration is applied only to terms which have been fully regularized. This approach has been recently extended to the elasticity case for P = Q [5]. The Taylor series expansion approach is taken in the current work for P ;II! Q.
Integration Algorithm
The proposed algorithm reduces the singular kernel functions to regular functions, for which the Gaussian integrations yield very accurate results for low orders of integration. The algorithm is based on Taylor series expansions of all terms in the integrands of Eq (3) such that explicit integrations of singular and weakly singular terms are performed analytically, and that numerical quadrature is only performed on the fully regularized terms.
The first step in the development of the new algorithm is to project the curved boundary element onto a flat plane. The flat plane is taken to be tangent to the curved boundary element (as distinct from the actual surface) at one of the boundary element nodes,
164
taken to be Qo, which is the node on the boundary element closest to the source point, P(x). The mapping from the boundary element to the flat plane is given by
dS(Q) - J(Q,Q,)dS«(t) - J'dS' (4)
where the prime denotes the flat projection plane.
The new algorithm applied herein is based on integrations of the singular terms in a local coordinate system, shown in Fig. 1. The I'l> I'2 coordinates denote points in the plane containing the reference field point, Qo, and the tangent to the boundary element at that point. The normal to the plane is easily computed from the Jacobian elements, evaluated at Qo. The I'I coordinate is taken for convenience to be aligned with the isoparametric integration direction given by ~ I' The origin of the local coordinate system is taken to be the projection of the source point P(x) onto the I'1-I'2 plane. Coordinates of the integration point Q(y) are given in this coordinate system by
I'I - pcos(O) I'2 - psin(O)
I'3 - ~(I'1'I'2)
(5)
This coordinate system was first used by Cruse [6] for exact integration of the BIE formulation for flat boundary elements and linear data interpolations. The earlier analytical integrations are applied in the current work.
We begin the regularization process by expanding the Jacobian of the transformation in terms of the kI'z directions, relative to the value of the Jacobian at the reference point Qo. The first-order expansion terms are given by
The variable ~ is the distance of the integration point Q(x) from the flat integration surface, that is I'3(Q). For quadratic isoparametric elements, the value of ~ is proportional to fl, where 0 is the projected distance of the integration point from the reference point, Qo. Thus, we see that the explicit expansion is to terms of order oz.
The boundary data for the quadratic isoparametric problem is given by Eq 2. The linear part of the boundary data is given by the following form, illustrated for the boundary displacements
165
The use of the linear expansion of the displacements has been previously used to regularize the traction integral for the linear element case [7]. This earlier regularization led directly to the explicit derivation of the surface stress term in BIE analysis. The difference term for the quadratic variation is shown to be of order A, or fl.
If we now expand the traction (or displacement) kernel in Eq 1 with respect to the integration point in the mapped plane A,' rather than the mapped boundary element area, A, the following is obtained
(8)
where the truncation term is of the order of A divided by the distance r(P,Q). Now substitute Eq 6-8 into Eq 3 for the flat integration element, A,' to obtain
AI - f f TiP,Q')Lo (u) Lo (J')pdpd() n 6 p (6)
(9)
The p«() integral in Eq 8 can be integrated analytically, using the approach in [7]. When the element has straight sides in the projected plane, the integral with () may also be done in closed form. 2
A major point to be made is that the singular nature of the kernel functions is entirely removed by the radial integration process. The remaining integral with respect to () is totally regular. In the work reported herein, the () integration is computed numerically.
The original integral, Eq ~, may be fully regularized by subtracting Eq 9. The difference integral is of the following form
The forms of these integrals and the analytical results are available from the first author.
166
such that Eq 3 is given by the sum of Eq 10 and Eq 9. The fact that Eq 10 is fully regular is seen by carrying out the expansion of all the terms with the following result
~{I I } - I I [L(u·(~o»O (~)J(~o) + I;,(P,Q') O(~)J(~) Il.S A..{' I r ~ (11)
+ I;iP,Q')L(Ui(~J)O(~) + O(~2)]dA'
The result in Eq 11 is of the order of~. As P -+ Q, the terms are totally regular, so long as the polar form [2] of the Gaussian integration is used to cancel the r(P,Q) in the denominator of the term from the kernel function expansion. The limit for P = Q also exists, and is regular. This result is consistent with that of [5]. Thus, Eq 11 may be integrated with standard, low-order Gaussian integration. No element subdivisions or other elaborate error correction system is required. As will be shown in the examples, low order Gaussian integration is sufficient to produce nearly exact results in most cases. While the above discussion is for the traction kernel, the extension to the displacement kernel results in the same conclusions.
Numerical Results
The numerical evaluation of the above algorithm has been made by computing the traction and displacement kernels for a single element with the source point P(x) approaching the element along a normal to the corner or mid side node. The element is taken to be flat as well as curved. The integral results for the constant, linear and quadratic boundary data cases are also computed. The data shown are taken only from the Ui! and Tu terms in the kernels, but these are representative of all other terms. A more comprehensive set of combinations of source point and integration element would add nothing to the conclusions we can draw from these numerical results.
Figure 2 and 3 show the numerical integration results for the U u and T u terms for a flat element ten units square, considering the boundary data to be constant over the element. The legend indicates the distance of the source point from the element. The new algorithm described above gives constant values of the integral results versus the integration order, since Eq 10 for this case gives a·zero result, even with the source point within 0.3% distance from the element. These figures simply illustrate the fact that the use of the Gaussian integration algorithm requires significant integration order for accuracy as P -+ Q, especially for the traction kernel (an expected result!).
The results in Figure 4 and 5 are obtained by letting the integration element be curved into a cylindrical shape, with the displacement of the midside node relative to the corners given by Delta. Thus, in this first case the element is covering more than a 135° arc .. The source point is taken from a distance of 10% of the element size from the element, but along the normal to a corner node in Figure 4, and a midside node in Figure 5. Clearly, the new algorithm far outperforms Gaussian integration.
Figure 6 is a more realistic case of element curvature. The element curvature is roughly equivalent to four elements on a 90° arc. The distance of the source point from the element is again 10% of the element size. The standard Gaussian integration scheme is not
167
much better than before, while the new algorithm provides very accurate answers with very low integration orders. The Gaussian integration algorithm used in these examples retains the square array of Gauss points which is likely leading to the oscillatory behavior of all of these results.
Figure 7 considers the case of linear boundary data and a flat boundary element. Again, the new algorithm gave fully converged results, independent of Gaussian integration order, as expected. The standard Gaussian system is seen to be very slowly convergent, for the source point distances selected.
Figure 8 is for the case of quadratic boundary data and compares the new algorithm directly with the standard Gaussian integration scheme. Even for points 5 % of the distance from the element, the standard scheme requires nearly the full complement of Gauss points to converge. The new algorithm converges with excellent accuracy within a 4x4 integration order.
Conclusions
The results confirm that the use of a semi-analytical approach to integrating the boundary integral equation kernels eliminates the need for higher order Gaussian integration, element subdivision, and elaborate error control schemes. Continuing work places emphasis now on developing a fast implementation of the new algorithm, as well as the logic for mixing the new algorithm and the standard algorithm, for the highest possible code efficiency. Application of the new algorithm to problems with steeply graded meshes is also planned.
Rererences 1. Cruse, T. A., Boundary Element Analysis in Computational Fracture Mechanics,
Kluwer Academic Publishers, The Netherlands (1988).
2. F. J. Rizzo and D. J. Shippy, An Advanced Boundary Integral Equation Method for Three-Dimensional Thermoelasticity, Int. J. Num. Meth. Eng. 11, 1753-1768 (1977).
3. Meng H. Lean and A. Wexler, Accurate Numerical Integration of Singular Boundary Element Kernels over Boundaries with Curvature, Int. J. Numer. Meth. Eng., 21,211-228 (1985).
4. M. H. Aliabadi, W. S. Hall, and T. G. Phemister, Taylor Expansions for Singular Kernels in the Boundary Element Method, Int. J. Numer. Meth. Eng., 21, 2221-2236 (1985).
5. M. Guiggiani and A. Gigante, A General Algorithm for Multidimensional Cauchy Principal Value Integrals in the Boundary Element Method, submitted for publication.
6. T. A. Cruse, An Improved Boundary-Integral Equation Method for Three Dimensional Elastic Stress Analysis, Compo & Struct., 4, 741-754 (1974).
7. T. A. Cruse, Three-Dimensional Elastic Stress Analysis of a Fracture Specimen with an Edge Crack, Int. J. Fract. Mech., 7, 1-15 (1971).
168
Figure 1:
VllfUll(12x12)
Figure 2:
P(x)
/ /
e /r(P,Q) /
'/
s
~I
Transformation of Isoparametric Element Area to Flat Integration Element
V-Integrals vs. Integration Order
1.05
.1 = .50
0.95 I •• 25
0.9 *' .0<i2S
, {\'.03125
0.85
0.8 +----1---+---+----+---+----+---+---+---+--
3 4 6 7 10 11 12
Integration Order
Gaussian Integration Results for U 11 Kernel Function: Element Size Variable Distance of P(x) from Element
lOx 10;
169
T -Integrals vs. Integration Order
1.2
.4.0
02.0
0.8 • 1.0
Tllrrll(12x12) 0.6 <, .50
'*.250
Figure 3:
0.4
2 6 7 9
Integration Order
10 11 12
i'1 .125
/- .0625
;< .03125
Gaussian Integration Results for T 11 Kernel: Element Size = lOx 10; Variable Distance of P(x) from Element
