Transactions on Modelling and Simulation vol 8, 1994 WIT Press, ISSN X

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1 Analysis of singular and near-singular integrals by the continuation approach D. Rosen & D.E. Cormack Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto M5S 1A4, Canada ABSTRACT We present a summary of the Continuation Approach for a broad class of singular and near-singular integrals. Insight is gained into the behaviour of these integrals, and general formulae for integration are obtained using conventional concepts in calculus. Moreover, conditions for boundedness of the singular integrals arise naturally, showing when a strongly singular integral coincides with the classical Cauchy Principal Value (and jump) or Hadamard Finite Part definitions. The continuation approach exploits the functional homogeneity shared by many Green's Functions, a property that has been mostly overlooked. Originally developed for homogeneous integrands on flat domains, the method has since been extended to more general integrands, as well as to curved surfaces and domains with corners. 1 Introduction. In this paper we present a brief overview of the Continuation Approach for analyzing a broad class of singular and near-singular integrals on domains of arbitrary dimension. It is our intent only to highlight the basic ideas and results.- A more detailed account of the method can be found in [lj-[4]. In BEM applications, we are interested in integrals of the form /(P) = f C(p-p)v(p)dQ(p) JQ where Q, is an M-dimensional finite surface in W + * (usually n = 1,2); the Green's function G is infinitely continuously differentiate for p#p, but is infinite when the two points coincide; and \ /(p) is the density function (usually given by a piecewise polynomial). The classical approach to render a finite value to a "strongly" singular integral that may arise in the limit when p approaches 1, usually involves placing p directly on Q. and adding a part of a sphere centered at p. The singular integral is then obtained in the limit as the radius of the sphere goes to zero [5]. This results in the Cauchy Principal Value interpretation (with a jump), for a Cauchy type integral, and as a Hadamard Finite Part, for a hypersingular integral. CD

2 310 Boundary Element Technology An alternative interpretation of singular integrals is to view them as limits or "continuations" of non-singular ones, in which the singularity originally lies outside the integration domain and is gradually moved towards it (c.f. [l]-[4] [6]-[9]). The surface is not modified, but an attempt is made to (semi)analytically integrate the kernel, or to map the integral to the contour of the surface. This interpretation is deeply rooted in the physical context of the problem and allows the treatment of singular and near-singular integrals in a unified way. To accomplish the mapping of the integral to the contour of the integration domain, the Continuation Approach further exploits the homogeneity of the Green's functions, an important property that has been largely overlooked. 2 Integration over Flat Surfaces. By exploiting the homogeneity of the integrand and the "continuation" interpretation of the integral, one obtains: i) simple, general formulae for singular and near-singular integrals, amenable to numerical evaluation. The integral is mapped to one performed exclusively on the contour of the domain. The singular integral simply appears as a special simplification of the more general (near-singular) formulation, ii) general boundedness, or "gauge", conditions for the singular integrals, that can readily be evaluated analytically or numerically, and Hi) an alternative view of these integrals that has considerable pedagogical value. Consider the integral Jo *^ (2) for some fixed value of ; where x = (x,, *2,...,* ), and Q is an /z-flat domain with volume element dv = dx^dx^ -dx*. In particular, let / be of the form /(;, ) = ^-^ (3) where r=t(*, ) = ( "*? + *) *". Notice that / is homogeneous of degree $ = I!li + m-k ; i.e. fq(.x,kz) = tff(x,z). The integrand can also be composed of a linear combination of terms of the form (2). Problems that involve the more general norm can be treated in the same way. Integrals of the type (2) are obtained from the more general integral (1), by placing the origin of a local coordinate system at the point p,,, directly below (or above) the singular FIGURE 1. Flat Intflqratinn Ftom«nt

3 Boundary Element Technology 311 point p, as shown in Figure 1. If the density \ / is not a polynomial in the local coordinates, then a simple expansion in Taylor series can be readily accomplished, to lead to integrals of the form (2) (see also section 4). 2.1 Near-Singular Integration. The property of homogeneity of / is exploited by applying Euler's identity for homogeneous functions By integrating both sides of (4), and subsequently applying Green's theorem to the left side, one obtains the continuation differential equation for /( ) : z /(z) - a/(z) = - where ds is the directed surface area element on the boundary of the integration domain 9Q ; for 2-D surface integrals, x ds -x^dx^-xax^. a = (3 + n is the degree of the singularity the integral generates when = 0. The case of a>0 corresponds to convergent singular integrals; a = 0 to Cauchy type; and a < 0 to hypersingular integrals. Equation (5) can be solved for /( ), as an initial value problem, to obtain n +/W (6) 7 ^30 J ZQ where /(ZQ) is the initial condition. Expression (6) is akeady well posed for numerical integration of near-singular integrals, since all function evaluations are now performed "far" from the singularity, as was shown in [1]. It follows from homogeneity that the second term on the right side of (6) vanishes if the initial condition is set at ZQ = ±<» (since f(x,z) = z*f(z~*x,\), and, hence, lim_^zo /(z<>) = 0). Moreover, the order of integration can be exchanged in (6). Thus, we define, the primitive boundary function F(x,z), by the indefinite integral )=/ (*, )<* (7) This allows (1) to be mapped to the contour of the integration domain: where = " F±_ = lirn^ _» ±_ F (z, Zg). The plus (minus) sign corresponds to a continuation from above (below) Q. F^ is always bounded for (jc, 0) e 3Q, and can be easily obtained analytically.

4 312 Boundary Element Technology 2.2 Singular Integrals and Gauge conditions. We define the continuation singular integral as 7(0) = Urn,- ^Q!(Z). Then, the existence of 7(0) can be summarize as follows (see [2] [4] for a detailed derivation): i) For a>0, 7(0) is always bounded. //; For a = 0, 7(0) is bounded iff ResI = \ /(jc,0)jc"da = 0, a = 0 Jan (9) ResI is called the general residue of the integral. ///) For a < 0, 7(0) is bounded iff )}jc^^ = 0, a<0 aa (10) We refer to Gfl as the gauge functional, and to expressions (9) and (10) as the gauge conditions of the integral. ResI and Gfl are independent of the shape and size of the domain. It can be shown that, for integrands of the form (2), (9) and (10) are equivalent to "at least one of the /, is odd". Notice also that (10) allows the initial condition at the surface since F(x,0) can substitute 7%. in (8). If the integral does not satisfy the gauge condition then it diverges as -» 0: ^K" a<0 ((Resl)\og\z\ a = 0 ^ We can show that if the gauge condition is satisfied then, when a # 0, the continuation singular integral is simply given by In this case, we_can also show that (12) gives the convergent singular integral for a > 0, and the finite part integral for a < 0. The case of the Cauchy singular integrals presents very interesting features, and is fully developed in [2]. Herein, we summarize the main results. It was shown in [ 10] that (9) is also necessary for the existence and uniqueness of the CPV. This condition is meaningful only when m = 0 (i.e. when f does not show in the numerator), since it is trivially satisfied otherwise. The continuation singular integrals are given by the following formulae, when a = 0: i) for m=0, if the gauge condition (9) is satisfied, 7(0) is equivalent to the CPV of the singular integral of /(x,0), and is given by /(O) =f(*,0)logpjt-ds, <x = 0, m=0 where p = jc (this is also the invariant imbedding formula for a CPV [10]). ii) for m odd, it leads to a jump, or discontinuity, at the surface of integration, given by 7(0*) -7(0") = 2 /<,, where /<, is the jump term given by

5 Boundary Element Technology (0") = ±/q = ± /%,(*)*'</,, a = 0, m odd Jao Jo is independent of the shape and size of the domain. Hi) for m even, it is unique and independent of the size and shape of Q. : 7(0) = liml- F(x,z)x*ds _>ov Jx* ot = 0, m even ' Example. As a simple example of the application of the present formulae, consider the hypersingular integral arising in 3-D potential problems: where Q. is a 2-D domain lying on the z/-^ plane. This integral generates a singularity a = -l when z > 0. It should be noted that the kernels in the hypersingular integrals in the BEM always arise from higher order derivatives of a Green's function, one of which is with respect to the normal of the surface. For these cases, the PBF is already known. Thus, in this example, the PBF is given by F(x^X2, z) = - 3r~V3z = zlr*. Then, the gauge condition (10) is satisfied, since F^ = FQ = 0, and the singular integral is bounded. The near-singular and singular integration formulae are where polar coordinates where introduced, and x ds = p'd6. Notice that 7(0) corresponds to the finite part of the integral of/ (%i,%2,0) = p~^. As a contrast, consider the integral 7(f ) = \^r~^ dx^dx^, (not a complete Green's function). In this case, F = f/(pv). Since Gfl = 2n * 0, the singular integral is truly unbounded. The near-singular integral is given by /( ) = 2n\ - f -^ z JaopV s (19) Thus, the integral diverges as z~* when z > 0. To assign a finite part to this continuation singular integral would be incorrect! 3 Integration on Corners. Problems where the boundary has corners sometimes present major difficulties for the BEM (as well as for other techniques), both conceptually and numerically. The reader is also referred e.g. to [11]-[14] for some comprehensive discussions on these issues.

6 314 Boundary Element Technology The limit of many BEM integrals, when the singularity approaches a corner of a single element, may not exist. However, when the integration domain is composed of a set of connected flat surfaces and the singularity approaches a corner at the intersection of the set, boundedness (and uniqueness) may arise from integrating simultaneously over all the elements. The continuation formulae of section 2, which describe the behaviour of an integral as the singularity moves along the normal to the surface, can be generalized to an arbitrary direction of approach. Then, the singularity is placed a small distance from the corner, and these integration formulae are applied; on each surface separately. This effectively transfers all function evaluations away from the singularity. For the analysis, we express the integrand in terms of local coordinate systems aligned with each surface. When the origin is placed at a different point than p^, say p^ (see Figure 1), the integrand becomes a function of the form /(*,*( ), ( )). Following a similar procedure as in section 2, the generalization of equation (3) to describe the behaviour of the integral along the line joining p to po is given by e /(c) - a/(e) = - (20) where 9/3E is the directional derivative along the line joining the two points. The solution to (20) is similar to expression (6), but with 8 substituting. The analysis thereafter follows that in the previous section, with the exception that the PBFs for this case are more complicated and difficult to obtain. Thus, it is convenient to leave the final near-singular integration formulae as double integrals: along E and along 3Q. The integration formulae derived from (20), simplify considerably when p,, is FIGURE 2. Integration Domain with a Comer. placed on the corner of the element (formed by two straight lines), since x ds = 0 along those lines. Hence, the integration is only performed on the "outer" boundary of the element; i.e. far from the singularity. Thus, when the singularity is close to a corner formed by a set of elements, the integration formulae involve only integration on the outer boundary of all the elements (see 3Q in Figure 2). In particular, the formulae for near-singular integration is simply given by equation (6) with replaced by E, and 3Q by 3Q. Singular integration formulae and boundedness conditions, similar to those obtained in section 2, also follow from this, by taking the singularity to the corner. The gauge conditions provide a very useful tool for setting the continuity requirements of the densities at a corner [14], and for studying problems with truly singular fields.

7 Boundary Element Technology 315 Numerical examples, including the direct computation of jump terms, CPVs and hypersingular integrals, are given elsewhere [3] [4]. 4 Integration of More General Functions over Curved Elements. Many advanced applications of the BEM require integration on curved elements, where the continuation formulae presented in the previous sections are not directly applicable. In any case, the present concepts can be extended to derive practical numerical integration formulae of i) integrals that contain non-polynomial density functions, ii) integrals where the Green's Functions are not themselves homogeneous or even real valued, and Hi) integrals performed on curved elements. The application of the continuation approach to these broader classes of integrals can be made by directly applying the Taylor Series expansion/subtraction concepts, common in the BEM literature (c.f. [15]). A complete treatment of these cases, with numerical examples, can be found in [3] [4]. In what follows, we briefly describe the main ideas. As usual, we start by placing the singularity a small distance away from the surface. Then, for curved integration surfaces, it is convenient to map first the integral to the exact projection of the surface on the tangent plane at the singularity. A Taylor series expansion of the integrand is then performed (since this is done while the singularity is away from the surface, the integrand is sufficiently continuous). A particular feature is that the kernel is first expanded in the normal direction to the surface; only if necessary, is it further expanded in the tangent coordinates. The density is expanded also in the tangential coordinates. Overall, the expansion proposed is, both conceptually and algebraically, simpler that those more commonly performed after the integral is mapped to the parametric element. The series expansion/subtraction leads to a sum of integrals over a flat integration domain (the projection of the original domain on the tangent plane) of homogeneous functions that retained the stronger singularities (given by the first few terms in the expansion), plus a weakly (near-)singular integral. The former can be treated by the continuation formulae of section 2, and the latter by more conventional means. Similar ideas for this expansion were independently developed by Cruise and Aithal [17]. However, the approach presented therein only dealt,with singular and near-singular integrals with weak and Cauchy type singularities. Also, it should be pointed out that the final formulae for singular integration are similar to those obtained by Guiggiani et al. (cf. [18] [19]) through a very different limiting process. 5 Concluding Remarks. In the case of flat surfaces, all the final formulae for any specific example can also be obtained by transforming the original integral to polar coordinates, and then integrating analytically in the radial direction. The singular integration formulae can be obtained subsequently by taking the analytical limit of the resulting expression, as z ->0 (c.f. [9] [12] [113] [19]). This requires, however, more effort and does not directly exploit common characteristics of this type of integral. In this sense, the Continuation Approach can be regarded as a systematic method of semi-analytically computing singular and near-singular integrals. However, the approach also leads to a general and unified description of these integrals. From it, we can derive general "laws"

8 316 Boundary Element Technology describing the behaviour of a large class of integrals as the singularity is close to, or on, the integration domain; the conditions that need to be satisfied for the singular integral to be bounded; how fast the integral diverges if these conditions are not met; uniqueness of the singular integrals, etc. Moreover, more complicated homogeneous functions, that may preclude analytical integration can still be treated numerically using formulae such as (6). This is a major advantage of the method in treating problems with geometrical comers. The continuation approach also presents an alternative, intuitive view of singular and near-singular integrals that has a considerable pedagogical value. Other researchers have also attempted to map the singular and near-singular integrals to the contour of the integration domain. In particular, the approach described in [7] [9], using Taylor series expansions and Stokes theorem, has many similarities to the present work. REFERENCES [ 1 ] Vijayakumar S., Cormack D.E., A New Concept in Near-Singular Integral Evaluation: The Continuation Approach, SIAM J. Appl. Math, 1989,49 (5), pp [2] Rosen D., Cormack D.E., Singular and Near Singular Integrals in the BEM: A Global Approach, SIAM J. Appl. Math., 1993, 53 (2), pp [3] Rosen D., Cormack D.E., Boundedness and Integration formulae for Singular and Near-Singular Integrals on Finite Domains, SIAM J. Appl. Math., 1992 submitted [4] Rosen D., Cormack D.E., The Continuation Approach for Singular and Near-Singular Integration, Engrg. Analysis with Boundary Elements, 1994, in press. [5] Chen G., Zhou J., Boundary Element Methods, Computational Mathematics and Applications, Academic Press, N.Y., 1992 [6] Krishnasamy G., Schemerr L.W., Rudolphi T.J., Rizzo F.J., Hypersingular Boundary Integral Equations: some Applications in Acoustic and Elastic Wave Scattering, Transactions of the ASME, 1990, 57, pp [7] RudolphiTJ., Krishnasamy G.,SchtmQnL.W.,RizzjoF.].,On the Use of Strongly Singular Integral Equations for Crack Problems, Boundary Elements X, Vol 3, Computational Mechanics Publications, Southampton Boston, Springer Verlag N.Y., 1988, pp [8] Gray L.J., Martha L.F., Ingraffea A.R., Hypersingular Integrals in Boundary Element Fracture Analysis, Int. J. Num. Meth Engrg., 1990, 29, pp [9] Krishnasamy G., Rizzo F.J., Liu Y., Boundary Integral Equations for Thin Bodies, Int. J. Num. Meth Engrg., 1994, 29, pp [10] Vijayakumar S., Cormack D.E., An Invariant Imbedding Method for Singular Integral Evaluation On Finite Domains, SIAM J. Appl. Math., 1988, 48 (6), pp [11]-Rudolphi Tr, Agarwal R., Mitra A., Coupling Boundary Integral Equations with Nonsingular Functional Equations by Exterior Collocation, Engrg. Analysis With Boundary Elements, 1991, 8 (5), pp [12] Gray L.G., Lutz E., On the Treatment of Corners in the Boundary Element Method, J. Comput. Appl. Math., 1990, 32, pp [13] Gray L.J., Marine L.L., Rudolphi T., Agarwal R., Mitra A., Hypersingular Integrals at a Corner, Engrg. Analysis With Boundary Elements, 1993, 11, pp [14] Rosen D., Cormack D.E., Corner Analysis in the BEM by The Continuation Approach, Engrg. Analysis with Boundary Elements. 1994, submitted. [15] Aliabadi M.H., Hall W.S., Taylor Expansionsfor Singular Kernels in the Boundary Element Method, Int. J. Num. Meth. Engrg., 1985, 21, pp [16] Cruse T.A. Aithal R., Non-Singular Boundary Integral Equation Implementation, Int. J. Num. Meth. Engrg., 1993, 36, pp [ 17] Guiggiani M., The Evaluation of Cauchy Principal Value Integrals in the Boundary Element Method-A Review, Mathl. Comput. Modelling, 1991,15, pp [18] Guiggiani M., Krishnasamy G., Rudolphi TJ., Rizzo F.J., A General Algorithm/or the Numerical Solution of Hypersingular Boundary Integral Equations, Transactions of the ASME, 1992, 59, pp [ 19] Krishnasamy G., Rizzo F.J., Rudolphi T.J., Continuity Requirements for Density Functions in the Boundary Integral Equation Method, Comp. Mech., 1992, 9, pp