Transactions on Modelling and Simulation vol 12, 1996 WIT Press, ISSN X

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1 Simplifying integration for logarithmic singularities R.N.L. Smith Department ofapplied Mathematics & OR, Cranfield University, RMCS, Shrivenham, Swindon, Wiltshire SN6 SLA, UK Introduction Any implementation of the boundary element method requires consideration of three types of integral - simple non-singular integrals, singular integrals of two kinds and those integrals which are 'nearly singular' (Brebbia and Dominguez 1989). If linear elements are used all integrals may be performed exactly (Jaswon and Symm 1977), but the additional complexity of quadratic or cubic element programs appears to give a significant gain in efficiency, and such elements are therefore often utilised; These elements may be curved and can represent complex shapes and changes in boundary variables more effectively. Unfortunately, more complex elements require more complex integration routines than the routines for straight elements. In dealing with BEM codes simplicity is clearly desirable, particularly when attempting to show others the value of boundary methods. Non-singular integrals are almost invariably dealt with by a simple Gauss- Legendre quadrature and problems with near-singular integrals may be overcome by using the same simple integration scheme given a sufficient number of integration points and some restriction on the minimum distance between a given element and the nearest off-element node. The strongly singular integrals which appear in the usual BEM direct method may be evaluated using the row sum technique which may not be the most efficient available but has the virtue of being easy to explain and implement. This

2 234 Boundary Elements paper implements a recently developed one part Gauss integration scheme for singular logarithmic integrals which is both simpler to implement and more accurate than the usual two-part Gauss approach with separate Gauss- Legendre and logarithmically weighted Gauss schemes. We aim to present a description for the quadratic element which is self-contained and suitable as an introduction to this aspect of BEM. Quadratic elements Consider an isoparametric quadratic element defined on [0,1] as in figure #, # Figure 1: A straight quadratic element The variation of x, y, potential u arid potential derivative q du/dn are all expressed in the same form in terms of their nodal values, for example: 4-92^2 + %Ms (1) where the Lagrangian interpolation functions MI, A/2,A/3 are M, = (2f -!)( -!) Mi = 4f(l - f) -1) In one dimension, a typical boundary element integral of q over an element Si on the boundary has the form (omitting a l/(2ir) multiplier) / = / ln(r) q ds (2) J S, where ds is the element of arc length around the boundary S and r = i(x - xo)2 + (y - 3/0)2 is the distance between the source point (xo,yo) and the boundary point (,r,y) on the current element.

3 Boundary Elements 235 If g, r and ds are now all expressed in terms of the parameter using an isoparametric quadratic element then the integral (2) becomes (3) where J(^) is the Jacobian transforming ck to Jo If the point (xo,yo) coincides with the first node of the quadratic element, then r 0 at this point and the integral including MI above is weakly singular. The other two shape functions MI and M^ are zero at f = 0 and those integrals are not therefore singular for this position of the source node - although they will be singular for the source node at node 2 and node 3 respectively. We discuss here only the case for a singularity on node 1, since integrals singular on the other two nodes are simple transformations/combinations of the node 1 routine. Numerical integration Very simple elements allow the use of exact integration but for more complex elements, numerical integration is used. In general, an n-point Gauss- Legendre formula with abscissae x, and weights W{ (%)(& = ]T %,,/(%,) (4) i=i will give a result for the integral which is exact for all polynomials of degree 2n 1 (neglecting any rounding errors). The high efficiency of such integration schemes mean that they are almost invariably used for numerical integration in FEM and BEM. One weighted Gauss scheme which is particularly useful in the BEM is for integrals which include a log( ) term on the interval [0,1]. With abscissae Xi and weights W{ this scheme is written In order to incorporate this weighted Gauss scheme into the integral (3) we must isolate the part of? ( ). One way of doing this (Smith and Mason

4 236 Boundary Elements 1982) is as follows: When x is expressed as an isoparametric quadratic on [0,1] we can write where z = A^ 4- Bf + zi; % = Cf 4- Df + 2/1 C = 2%/i - 4?/2 + 2?/3, D = -3m + 4?/2-7/3 and if the element is singular on node 1 then r% = ^[(A" +C')(" + (2AB 4- The ln(r) integral singular on node 1 can thus be written as Jo This kind of separation by direct factorisation into the two components avoids division by small quantities or taking square roots and would thus seem a more robust approach than is sometimes used. The first integral on the right-hand side of (6) is evaluated using the weighted scheme (5) and the second non-singular integral using the simple quadrature (4). This two-part gauss quadrature with n points for each part is, of course, exact for polynomials of degree (2n 1) and therefore for integrals on straight quadratic elements when n > 1. An alternative scheme for the singular logarithmic integral is based on subtracting out the singularity after expanding the integrand using Taylor series (Aliabadi et al 1985) Whilst this approach is effective, it is substantially more difficult to explain or implement than a two-part Gauss quadrature. Crow (1993) developed a gauss quadrature scheme for a function of the form: which is exact where a and b are polynomials of degree at most N. Choosing a 0 and 6=0 separately leads to the equations: TV N 0 0 = / Jo i

5 Boundary Elements 237 The choice of basis polynomials for this problem is arbitrary since the unknown abscissae are not the roots of orthogonal polynomials but Legendre polynomials P*(x] on [0,1] produce lower condition numbers than powers of x in the following scheme: E o giving 2N+2 equations for the 2N+2 unknowns: Given an initial estimate for, the first equation above is solved for w and the second equation for in an iterative fashion. The computation of quadrature points in this way seems to be particularly ill-conditioned and data is only available for N up to N = 7. The quadrature points for the case N 6 are given below for convenience: The above quadrature scheme X w X w Table 1: Gauss quadrature points covers the requirements for the BEM ln(r) singularity without the need to explicitly form the two components of the integrand in the singular case. Results The two-part log procedure and the single gauss quadrature are now compared for several elements with different profiles. Since exact solutions are not generally available, accurate comparison values were obtained from the adaptive numerical integration provided by the symbolic algebra package. The other results were obtained using single precision arithmetic (c± 7dp) on a DEC MICRO VAX and include a straightforward 6-point Gauss-Legendre

6 238 Boundary Elements rule which takes no account of the singularity. Both the singular rules also use 6 points, the two-part rule with 3 log -f 3 Legendre points. The integrals from all 3 nodes are treated in the same way although strictly only the first node integral is singular. nodes 1,2,3 x-coord 6-pt Gauss Two part log Crow 6-pt Table 2: log integrals on a straight element Table 2 gives integrals for a linear geometry quadratic element which can be evaluated exactly since the Jacobian is a constant. The singular numerical schemes should also be 'exact' to within numerical errors since the unaccounted for polynomial in the integrand is just the quadratic shape function of degree n = 2 and these schemes should therefore be exact with (2n 1) > 2. This is borne out by table 2 which shows all 3 methods apart from the simple Gauss rule agreeing to machine precision. Observe that although the integrals from node 2 and node 3 are non-singular, the singular rules appear to give significantly better estimates than simple Gauss- Legendre. Once the element deviates from a pure linear element, exact nodes 1,2,3 x-coord 6-pt Gauss Two part log Crow 6-pt Table 3: log integrals - small curvature integration is no longer generally possible and tables 3 and 4 show that the Crow rule performs significantly better than the other schemes; accurate to 5 or 6 places of decimals compared to 3 or 4 otherwise. As the element curvature increases, integrals become less accurate and in table 5 accuracy may be only 3 decimal places but the Crow scheme is still better than other rules considered. Although it is usual in two-part schemes to use the same number of points in each part, there is no particular reason for this, other than convenience. Indeed, the second non-singular part of the

7 Boundary Elements 239 nodes 1,2.3 x-coord 6-pt Gauss Two part log Crow 6-pt Table 4: log integrals - moderate curvature nodes 1,2,3 x-coord 6-pt Gauss Two part log Crow 6-pt Table 5: log integrals - high curvature integrand must represent a more complex expression because it includes an explicit log term which is represented implicitly by the weight in the singular part. Table 6 confirms this as 4 Gauss-Legendre (g) and 3 logarithmically weighted points (1) are much better than the converse. A Crow style scheme should be possible which takes into account the different numbers of points required. nodes 1,2,3 x-coord Two part log 3g+31 Two part log 3g+41 Two part log 4g-f Table 6: log integrals - unequal points

8 240 Boundary Elements Conclusions The special gauss quadrature used here for the evaluation of singular logarithmic integrals seems to be the simplest to teach and to program, as well as being more accurate than the conventional two-part gauss quadrature. However, abscissae and weights for larger numbers of integration points are needed for higher accuracy on quadratic elements or for integration on higher order elements. The logarithmic integrals on the other nodes of an element which are not strictly speaking singular, are nevertheless evaluated more accurately by the singular schemes used here than by a straightforward Gauss-Legendre scheme. More points will generally be needed for the nonsingular part of the integration scheme than for the singular part. Acknowledgements I would like to thank G J Fernandes for computing some example data. References Aliabadi M H, Hall W S and Phemister T G, (1985) Taylor expansions for singular kernels in the boundary element method. Int J Num Meth Engg, 21, Brebbia, C.A. and Dominguez, J. Boundary Elements: An Introductory Course, Computational Mechanics Publications, Southampton, Crow, J.A. (1993) Quadrature of integrands with a logarithmic singularity. Math. Comp. 60, 201, \ J as won, M.A. and Symm, G.T. (1977) Integral Equation Methods in Potential Theory and Elastostatics. Academic Press, London. Smith R.N.L. and Mason J C (1982) A Boundary element method for curved crack problems in two dimensions, Proc. 4th Int Conf. on Boundary Elements (Ed. Brebbia C A), Springer-Verlag Berlin, pp