Dipartimento di Matematica

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1 Dipartimento di Matematica L. SCUDERI ON THE COMPUTATION OF NEARLY SINGULAR INTEGRALS IN 3D BEM COLLOCATION Rapporto interno N. 14, giugno 2006 Politecnico di Torino Corso Duca degli Abruzzi, Torino-Italia

2 On the computation of nearly singular integrals in 3D BEM collocation Letizia Scuderi Abstract In this paper we propose an efficient strategy to compute nearly singular integrals over plane triangles in R 3, arising in BEM collocation. The strategy is based on a proper use of various nonlinear transformations, which smooth, or move away or quite eliminate all the singularities close to the domain of integration. We will deal with singularities of the form 1/r, 1/r 2 and 1/r 3, being r = x y the distance between a near observation point x and a point y of a triangular element. Extensive numerical tests show that the approach here proposed is highly efficient and competitive. KEY WORDS: singular integrals; numerical quadrature; boundary element method 1 Introduction In the boundary integral equation methods for a linear partial differential equation Lu = 0, the fundamental solution G of L plays a fundamental role. For Laplace s equation u = 0 over a domain Ω R 3 the fundamental solution G is given by G L (x, y) := 1 1 4π x y, x, y R3 ; for the Helmholtz equation u + k 2 u = 0, where k is the wave number (in particular k = 2π/λ, being λ the wavelength of the electromagnetic radiation) the fundamental solution G is G H (x, y) := 1 4π exp ( ik x y ), x, y R 3, x y with i := 1. We explicitly remark that for static problems, i.e. for k = 0, G H G L. According to the single-layer or double-layer potential representation used for the solution This work was supported by the Ministero dell Università e della Ricerca Scientifica e Tecnologica of Italy. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy. letizia.scuderi@polito.it 1

3 G, there are two possibilities to transform the differential equation of a given Dirichlet boundary value problem into an integral equation, and the following two different integral kernels (1.1) K 0 (x, y) = G(x, y), K 1 (x, y) = G(x, y) n y are involved respectively. In (1.1) G(x, y)/ n y denotes the normal derivative of the solution with respect to y. By assuming that the bounded domain Ω may be approximated by a polytope whose boundary Γ approximates Ω, the resulting integral equation is defined on Γ. Since the faces of the polytope are polygons, we can assume that Γ consists of plane triangular pieces. Therefore in 3D boundary element method a key issue is the efficient and fast computation of integrals of the form (1.2) K(x, y)p(y) dy, where K(x, y) is one of the kernel functions in (1.1) and p(y) restricted to is a polynomial in y, e.g. a linear function on. In this paper we will assume that the observation point x is outside the integration domain, but very close to it, i.e. we deal with the so-called nearly singular integrals, which arise massively in BEM collocation. Typically the number of such integrals, which have to be evaluated for an N N system matrix, increases quadratically in N. Taking into account that the singular element integrals (i.e. those belonging to the main diagonal of the matrix) are only O(N), the efficiency of the BEM is greatly dependent on the accurate and efficient computation of the nearly singular boundary integrals. Taking into account that for G(x, y) = G L (x, y) and G(x, y) = G H (x, y) we have (1.3) K 1 (x, y) = K1 L (x, y) :=< n y, y G L (x, y) >= 1 < n y, x y > K 1 (x, y) = K H 1 (x, y) := 1 4π ( 1 x y 3 + 4π ik x y 2 respectively, the integral (1.2) involves integrals of the form f(x, x y ) (1.4) I := p(y) dy, x y l, x y ) 3 exp ( ik x y ) < n y, x y > where l = 1 if in (1.2) K = K 0, l = 3 if K = K1 L and l = 2, 3 if K = K1 H, and f is a smooth function depending on x y but it is not zero at x = y, so that l describes the order of the singularity of the integrand. Indeed, since integrals (1.4) are invariant under rotation and translation, the Cartesian co-ordinates system (t 1, t 2, t 3 ) in R 3 can be chosen such that the triangle lies in t 1, t 2 -plane and x has co-ordinates (0, 0, t 3 ) = t 3 n, being n = (0, 0, 1) T. From this configuration it immediately follows that < n y, x y > is constant. Therefore, in (1.4) f is equal to a constant or to a constant multiplied by exp ( ik x y ). If f is constant, then the computation of I can be partly or completely performed analytically; 2

4 otherwise it is necessary to proceed by numerical quadrature. Although x in (1.4) lies outside and, hence, the integrand function of I is regular, the computation of the nearly singular integrals is considered by some users the most difficult case. Indeed standard methods like Gauss formulae are less and less accurate as the distance of the poles of the integrand function from the domain of integration becomes smaller and smaller ([1], p. 312). Therefore, even if the integrands are regular, they require a large number of quadrature nodes to provide accurate enough results. In the literature this problem has been faced by using different strategies. Whenever possible, analytic integration is considered. However this approach may yield to strongly unstable formulae and needs a huge number of functional evaluations. Moreover, the analytic integration is ruled out when having curved triangles. Another type of strategy, proposed for the computation of the nearly singular integrals, is based on an adaptive element subdivision ([9], [11]). However by proceeding with subdivisions of the quasi singular element into a certain number of subelements, the highest degree of the polynomial, which can be integrated exactly, decreases drastically with respect to the total number of nodes. For this reason, in [12] it was considered a different approach based on a cubic variable transformation for each of the two variables describing the element, which automatically lumps the points towards the singularity. Variable transformations have been considered also in [2], [4]-[7], [10], in order to weaken or completely eliminate the nearly singularity, before applying Gauss quadrature. In [6], [7] (see also [5]) a variable transformation method is proposed for curved triangles; after introducing the local polar co-ordinates, it employs a transformation for the radial variable and, if the projection of the observation point on the element is close to an edge of this latter, another one for the angular variable. In [4] (see also [3]) the authors also use local polar co-ordinates, performing the inner integrals analytically and the outer integral by Gauss quadrature. Finally, in [2], [10] the authors consider planar triangles and, after introducing a proper change of variable which eliminates the nearly singular term in the inner integral, proceed by applying a Gauss quadrature to compute the (transformed) inner and the outer integral. However, in no one of the above cited papers where variable transformation methods are proposed, the authors take into account of the possible new nearly singularities arising from their previous changes of variable. Therefore, to reach a good accuracy they have to use Gauss quadrature rules with a large number of nodes. In this paper we propose a new variable transformation method, which takes into account of the original nearly singularities and of all the new singularities of the transformed integrand. To this aim for the inner integral we use the transformations considered in [6] and in [2], and for the outer integral, and possibly again for the inner, a polynomial transformation that we have already used in [14] (see also [13]). More precisely, we compute nearly singular integrals defined over planar triangles, by using first the local polar co-ordinates and then introducing in the inner integral a nonlinear transformation which eliminates exactly the nearly singularity. However, as we shall see, this is still not exhaustive because this manipulation gives rise to other real or complex singularities in the integrand itself or in its derivatives, which are at the boundary or outside the domain of integration but very close to it. Therefore, the computation of the inner and the outer integral by means of standard quadrature formulae could turn out to be fairly expensive. 3

5 We overcome this drawback by means of some proper changes of variable that, taking into account of the position of the nearly singularity, eliminate or move this latter away from the interval of integration. In this way, we obtain a very good accuracy by means of a Gauss-Legendre formula with a number of quadrature nodes significantly smaller than that used in [2], [6], [10]. Extensive numerical tests will emphasize the efficiency of our approach. We believe that it can handle efficiently also curved triangles; to this end we could start in a way similar to [6] and then proceed by introducing our transformations before applying the Gauss-Legendre quadrature rule. However, the transformations that in the case of plane triangles completely eliminate the nearly singularity due to the term 1/ x y l, l = 1, 2, 3, in the case of curved triangles can only weakener it, because of the previous introduction of the parametric representation of the curved surface on which lies the triangular element. Curved triangles, together with the computation of the nearly singular integrals arising in the Galerkin BEM, will be object of our next research. 2 The problem setting Since the triangle Γ is generally in an arbitrary position in the space R 3, we begin the treatment of the integral (1.4) by describing a transformation which maps the generic configuration of x and into the standard configuration defined below. Let us describe x by the Cartesian co-ordinates x = (x 1, x 2, x 3 ) T and denote the vertices of by d 1, d 2, d 3. By using the orthonormal basis (2.1) z 1 := d2 d 1 d 2 d 1, z3 := (d2 d 1 ) (d 3 d 1 ) (d 2 d 1 ) (d 3 d 1 ), z2 := z 3 z 1, we define the following affine transformation y t = (t 1, t 2, t 3 ) T : (2.2) t = Ay b or y = A T (t + b) with A := (z 1, z 2, z 3 ) T, b = (z 1 ) T x (z 2 ) T x (z 3 ) T d 1 which brings the general configuration into a standard one having the following properties: i) the transformed triangle T lies in the plane t 3 = 0 (this means that if t T, then t = (t 1, t 2, 0) T ); ii) the transformed observation point s := Ax b lies on the t 3 -axis (this means that s = (0, 0, c) T, c = (z 3 ) T (x d 1 ) R)., 4

6 By taking into account that the 3 3 matrix A is orthogonal and introducing the above transformation in (1.4), we obtain f(x, A(x y) ) f(x, Ax (t + b) ) I = p(y) dy = p(a T (t + b)) dt A(x y) (2.3) l T Ax (t + b) l g(s, s t ) = N(t) dt s t l T where s = (0, 0, c) T, t = (t 1, t 2, 0) T and g(s, s t ) := f(a T (s + b), s t ), N(t) := p(a T (t + b)). By recalling that the expression of f contains the scalar product < n y, x y > when in (1.2) K = K 1 (see (1.3)), now we examine how this latter changes after the introduction of the orthogonal transformation (2.2). By taking into account that orthogonal transformations of R n preserve angles between lines and that is a plane triangle, we have n y = n and < n, x y >=< An, A(x y) >=< n T, s t >= c. Therefore, from (1.2)-(1.4) it follows that g(s, s t ) in (2.3) is reduced to a constant or a constant multiplied by exp( ik s t ). More precisely, g(s, s t ) = 1/4π if in (1.2) K = G L, g(s, s t ) = c/4π if K = K1 L, g(s, s t ) = exp( ik s t )/4π if K = G H, g(s, s t ) = c exp( ik s t )/4π if K coincides with the first term of K1 H and, finally, g(s, s t ) = i k c exp( ik s t )/4π if K coincides with the second term of K1 H. Let us denote by v 0 = (0, 0, 0) T the projection of s on the t 3 = 0 plane and the vertices v 1, v 2, v 3 of T by the Cartesian co-ordinates. The indexing of the vertices v 1, v 2, v 3 will be chosen according to the orientation of Γ, i.e. z 3 defined in (2.1) is the outer normal of Γ on. The triangle T oriented in this way will be written as T = [v 1, v 2, v 3 ]. Taking into account of the orientation, for any integrand F, defined on all the domains involved, we can use the following formula F (t) dt = F (t) dt + F (t) dt + F (t) dt, [v 1,v 2,v 3 ] [v 0,v 1,v 2 ] [v 0,v 2,v 3 ] [v 0,v 3,v 1 ] where each integral is defined over an oriented triangle. For example, we can assume that the orientation is positive (negative), if the vertices of the triangle are disposed in the anticlockwise (clockwise) sense. By applying the above formula we split the integral over the triangle T = [v 1, v 2, v 3 ] into a sum of three integrals over triangles containing the origin as a corner point, i.e. I = I 1,2 + I 2,3 + I 3,1 where (2.4) I i,j := [v 0,v i,v j ] g(s, s t ) s t l N(t) dt. After having reduced the original problem to the computation of the integrals I i,j, we introduce in (2.4) the polar co-ordinates t 1 = ρ cos(ϑ), ϑ min ϑ ϑ max, t 2 = ρ sin(ϑ), 0 ρ R(ϑ), 5

7 where for the triangle represented in Figure 1 and for v i = (v1, i v2, i 0) T and v j = (v1, j v2, j 0) T, the limits of integration are ( ) v j 2 arctan if v j 1 0 ϑ min = ϑ (i,j) min := ϑ max = ϑ (i,j) max := π 2 arctan π 2 v j 1 ( ) vi 2 v1 i otherwise, if v i 1 0 otherwise. Figure 1: The triangle [v 0, v i, v j ]. Moreover, sin(ϑ) a (i,j) cos(ϑ) (2.5) R(ϑ) = R (i,j) v1 i (ϑ) := cos(ϑ) b (i,j) sin(ϑ) b (i,j) if v j 1 v i 1 if v j 1 = v i 1 if v j 2 = v i 2 where We then rewrite I i,j in (2.4) as follows a (i,j) = vj 2 v2 i v j, b (i,j) = vj 1v2 i v2v j 1 i 1 v1 i v j. 1 v1 i (2.6) I i,j = ϑmax ϑ min R(ϑ) g(c, ρ dϑ 2 + c 2 ) 0 ( N(ρ cos(ϑ), ρ sin(ϑ)) ρ dρ. ρ 2 + c 2 ) l 6

8 As basis function N, we can consider local Lagrange basis for polynomials of degree p = 1, 2,.... Example 1 Recalling that the Lagrange linear basis functions in the case of the unit triangle σ = [(0, 0) T, (1, 0) T, (0, 1) T ] are given by N σ 1 ( t 1, t 2 ) = t 1, N σ 2 ( t 1, t 2 ) = t 2, N σ 3 ( t 1, t 2 ) = 1 t 1 t 2, by using the affine invertible mapping t = Bt with ( v j 1 v1 B = i v j 2 v2 i we deduce the corresponding basis function for the triangle [v 0, v i, v j ] N 1 (t 1, t 2 ) = d(v i 2t 1 v i 1t 2 ), N 2 (t 1, t 2 ) = d(v j 1t 2 v j 2t 1 ), ), N 3 (t 1, t 2 ) = 1 d[(v i 2 v j 2)t 1 (v i 1 v j 1)t 2 ], where d = 1/det(B). Therefore in the linear case we have to compute the following integrals (2.7) I 1 i,j := I 2 i,j := I 3 i,j := ϑmax ϑ min ϑmax ϑ min ϑmax ϑ min R(ϑ) dϑ 0 cos(ϑ) dϑ sin(ϑ) dϑ g(c, ρ 2 + c 2 ) ( ρ dρ, ρ 2 + c 2 ) l R(ϑ) 0 R(ϑ) 0 g(c, ρ 2 + c 2 ) ( ρ 2 + c 2 ) l ρ 2 dρ, g(c, ρ 2 + c 2 ) ( ρ 2 + c 2 ) l ρ 2 dρ. In the following two subsections we will describe the numerical strategy adopted for the efficient computation of the inner and the outer integral of (2.6), respectively. 2.1 On the computation of the inner integral in (2.6) In this subsection we will focus our attention on the computation of the inner integral in (2.6). In the following treatment we will consider three different cases according to the type of singularity and, within them, we will distinguish the case of g equal to a constant, for example g 1, and the case of g function of x and r := x y. This because the above cases are treated by means of different and tailored strategies. Therefore, by taking 7

9 (1.4), (2.3) and (2.4) into account, we will consider singularities of type 1/r, 1/r 2 and 1/r 3. In particular, we will deal with the following integrals (2.8) S p,l (ϑ, c) := if g 1 in (2.6), and (2.9) D p,l (ϑ, c) := R(ϑ) 0 R(ϑ) 0 ρ p ( dρ, p 1, l = 1, 2, 3, ρ 2 + c 2 ) l ρ p ( ρ 2 + c 2 ) l g(c, ρ 2 + c 2 ) dρ, p 1, l = 1, 2, 3, if g 1. The integrals S p,l can be computed by means of a recursion formula, which can be easily derived by integration by parts. Explicit formulae to evaluate the integrals S p,l for every p and l are also given in [4]. The integrals D p,l defined in (2.9) must be computed numerically. But a straightforward application to the integral D p,l of a Gauss-Legendre quadrature rule gives a poor accuracy because of the nearly singularity associated with the 1/( ρ 2 + c 2 ) l term. We note explicitly that the accuracy worsens as c decreases, because the smaller c is, the closer the singularity is to the interval of integration. Therefore, by following the idea developed in [6] and in the recent paper [2] (see also [10]), to make the Gauss formula more efficient, as described below, we first introduce a proper change of variable to eliminate the above-mentioned nearly singularity. Then, if there are no other singularities arising from the previous transformation that need to be weakened, we apply to the transformed integral a Gauss- Legendre quadrature rule. 1/r singularity For the computation of the integral S p,1 we use the following formula ( R(ϑ) + ) [R(ϑ)] S 0,1 (ϑ, c) = log 2 + c 2, c (2.10) S 1,1 (ϑ, c) = [R(ϑ)] 2 + c 2 c, S p,1 (ϑ, c) = [R(ϑ)]p 1 p [R(ϑ)]2 + c 2 p 1 p c2 S p 2,1 (ϑ, c), p 2. For the computation of the integral D p,1, before applying a Gauss-Legendre quadrature rule, we introduce the following change of variable ( ) ρ 1 (2.11) ρ = c sinh(x) x = asinh, dx = c ρ2 + c dρ, 2 to cancel the nearly singular term 1/ ρ 2 + c 2 in D p,1. Hence we have (2.12) D p,1 (ϑ, c) = asinh( R(ϑ) c ) 0 ( [ c sinh(x)] p g c, c ) [sinh(x)] dx. 8

10 As we can see in Tables 9 and 10, the computation of (2.12) by means of a Gauss-Legendre rule is less and less accurate as c becomes smaller and smaller. Therefore, when c is very small, for example c < 0.01, we suggest the use of the following different change of variable (2.13) ρ = x 2 c 2 x = ρ 2 + c 2, dx = ρ ρ2 + c 2 dρ, which, similarly to (2.11), allows to eliminate the nearly singular term 1/ ρ 2 + c 2 through the Jacobian and, on the contrary of (2.11), allows to obtain a good accuracy by means of the n-point Gauss-Legendre rule already for small values of n. We then have [R(ϑ)] 2 +c 2 ( ) p 1 (2.14) D p,1 (ϑ, c) = x2 c 2 g (c, x) dx. c As we can see, when p is even, the change of variable (2.13), unlike (2.11), introduces a singularity at c in the derivatives of the integrand function and this makes the computation of the transformed integral by means of a Gauss-Legendre quadrature formula fairly expensive. For this reason, before applying a quadrature rule we introduce a further change of variable given by (2.15) x = c + t q, q > 1 t = (x c ) 1/ q. By choosing q even, (2.15) allows to eliminate exactly the above mentioned singularity, otherwise it only allows to make the transformed integrand function smoother than the original one. By introducing (2.15) in (2.14), we have (2.16) D p,1 (ϑ, c) = ( [R(ϑ)] 2 +c 2 c ) 1/ q 0 qt q 1 ( t q (t c )) p 1 g (c, c + t q ) dt. In the numerical tests we choose q = 2 when c = 0.1, 0.01 and q = 4 when c = 0.001, For c > the changes (2.11) and (2.13) are quite equivalent, while for c (2.13) (possibly combined with (2.15)) is more effective than (2.11). As we shall see in Subsection 2.2, changes of variable of type (2.15) play a fundamental role in the computation of the outer integral in (2.6). 1/r 2 singularity For the computation of the integral S p,2 we use the following formula ( ) [R(ϑ)]2 + c S 1,2 (ϑ, c) = log 2, (2.17) c ( R(ϑ) S 2,2 (ϑ, c) = R(ϑ) c atan c S p,2 (ϑ, c) = [R(ϑ)]p 1 p 1 ), c 2 S p 2,1 (ϑ, c), p 3. 9

11 For the computation of the integral D p,2, before proceeding numerically we introduce the following change of variable ( ) x (2.18) ρ = exp c c 2 x = c log(ρ 2 + c 2 ), dx = 2 c ρ ρ 2 + c dρ 2 to remove the singularity associated with the term 1/(ρ 2 + c 2 ). However, in this way the derivatives of the transformed integrand function are singular exactly at an endpoint of the corresponding interval of integration. Indeed, we have (2.19) D p,2 (ϑ, c) = c log([r(ϑ)] 2 +c 2 ) c log(c 2 ) 1 2 c ( exp ( ) ) p 1 ( ( )) x x c c 2 g c, exp dx. 2 c Therefore, when p is even we have to smooth the singularity of the integrand function at c log(c 2 ); to this end we introduce the following change of variable (2.20) x = c log(c 2 ) + t q, q > 1 t = ( x c log(c 2 ) ) 1/ q and we get (2.21) ( c log([r(ϑ)] 2 +c 2 ) c log(c 2 )) 1/ q D p,2 (ϑ, c) = 0 2 c ( ( )) c log(c 2 ) + t q g c, exp dt. 2 c qt q 1 ( ( ) ) p 1 c log(c2 ) + exp t q c c 2 In the numerical tests, as smoothing exponent of the polynomial transformation (2.20), we choose q = 2 and, for improving further on the accuracy, we also split the interval of integration of (2.21) about its midpoint. 1/r 3 singularity For the computation of the integral S p,3 we use the following formula S 1,3 (ϑ, c) = 1 c 1 [R(ϑ)]2 + c, 2 ( R(ϑ) (2.22) S 2,3 (ϑ, c) = [R(ϑ)]2 + c + log R(ϑ) + ) [R(ϑ)] 2 + c 2, 2 c S p,3 (ϑ, c) = [R(ϑ)] p 1 (p 2) [R(ϑ)] 2 + c 2 p 1 p 2 c2 S p 2,3 (ϑ, c), p 3. When l = 3 the computation of the integral D p,3 in (2.9) is a little more involved than the integrals D p,1 and D p,2. To remove the nearly singularity associated with the term 1/(ρ 2 + c 2 ) 3/2 we introduce the following change of variable ( c ) 2 (2.23) ρ = c x 2 x = c ρ2 + c, dx = c ρ dρ 2 (ρ 2 + c 2 ) 3/2 10

12 and we have (2.24) D p,3 (ϑ, c) = 1 c p 2 c [R(ϑ)] 2 +c 2 ( 1 x 2 x ) p 1 ( g c, c ) dx. x Notice that in this way the transformed integrand function must be smoothed for p even at 1 where the derivatives of 1 x 2 are singular and at 0, which is close to c / [R(ϑ)] 2 + c 2 for c small, where 1/x is singular. We face the above mentioned singularities separately, and therefore we first rewrite the integral as sum of two integrals over the intervals of integration [ c / [R(ϑ)] 2 + c 2, ( c / [R(ϑ)] 2 + c 2 +1)/2] and [( c / [R(ϑ)] 2 + c 2 +1)/2, 1] and then we introduce in the first integral the change of variable x = t q 1 and in the second integral the change x = 1 t q 2, with q 1, q 2 > 1. Thus, we get (2.25) D p,3 (ϑ, c) = [( ) ] 1/ q1 c [R(ϑ)] 2 +c 2 +1 /2 ( ) 1/ q1 c p 2 q t q c [R(ϑ)] 2 +c 2 [ ( ) ] 1/ q2 c 1 [R(ϑ)] + 2 +c 2 +1 /2 c p 2 q t q ( 1 t 2 q 1 t q 1 ( t q t q 2 ) p 1 ( g c, c ) t q 1 ) p 1 g ( c, dt ) c dt. 1 t q 2 Notice that in the first integral the change of variable x = t q 1 with a large q 1 moves the left endpoint of the interval of integration away from the singularity at zero; in the second integral the change x = 1 t q 2, with q 2 even, completely eliminates the singularity at 1. In the numerical tests we will choose q 1 = 3 and q 2 = 2 when c = 0.1 and q 1 = 6 and q 2 = 4 when c = 0.01, 0.001, These latter choices are purely experimental and allow to reach a good enough accuracy with a few quadrature nodes. 2.2 On the computation of the outer integral in (2.6) In this subsection we will focus our attention to the computation of the outer integral in (2.6). In particular, we will consider the following integrals (2.26) O S := and (2.27) O D := ϑmax ϑ min ϑmax ϑ min S p,ϑ (ϑ, c)s(ϑ) dϑ D p,ϑ (ϑ, c)s(ϑ) dϑ, where s(ϑ) is a smooth function. For instance, if we consider the linear basis functions of Example 1, then p = 1, 2 and s(ϑ) = 1, sin(ϑ), cos(ϑ) (see (2.7)). Generally for the computation of the integrals (2.26) and (2.27) the straightforward application of the Gauss-Legendre quadrature rule (see [2], [4], [10]) is used. However this 11

13 might be unsatisfactory because of the presence of singularities off but close to the interval of integration, which may affect adversely the accuracy of a standard quadrature rule. Indeed, as we have already remarked, this latter could request too many quadrature nodes in order to reach a good accuracy, and this fact is not acceptable for the collocation method, and even more so far the Galerkin method, where many integrals must be computed or approximated. Since in our case it is possible to determine the location of the singularities, we have worked out a special approach to compute the above integrals by using a few quadrature nodes. It is based on proper changes of variable, possibly associated with a splitting of the interval of integration. From the expression of the functions S p,ϑ in (2.10), (2.17), (2.22) and of the upper limit of integration of the integrals D p,ϑ in (2.14) or (2.16), (2.21) and in (2.25), it follows that to compute accurately and efficiently the outer integral in (2.6), it is necessary to adopt a special strategy to compute, accurately and with a few quadrature nodes, the functions i) [R(ϑ)] p, ii) [R(ϑ)] p [R(ϑ)] 2 + c 2, iii) R(ϑ)R (ϑ)/([r(ϑ)] 2 + c 2 ), iv) [R(ϑ)] p / [R(ϑ)] 2 + c 2, with a non negative integer p. From (2.5) and (b (i,j) ) 2 + c 2 [sin(ϑ) a (i,j) cos(ϑ)] 2 if v j [sin(ϑ) a (i,j) cos(ϑ)] 2 1 v1 i (2.28) [R(ϑ)] 2 + c 2 (v i = 1) 2 + c 2 [cos(ϑ)] 2 if v j [cos(ϑ)] 2 1 = v1 i (b (i,j) ) 2 + c 2 [sin(ϑ)] 2 if v j [sin(ϑ)] 2 2 = v2 i it follows that the functions i)-iv) are singular at ϑ = atan(a (i,j) ) + mπ, m Z, in the case v j 1 v1, i at ϑ = π/2 + mπ in the case v j 1 = v1 i and at ϑ = mπ in the case v j 2 = v2. i A simple computation shows that ϑ represents also the real part of the complex and conjugate zeros of the analytic function at the numerators of (2.28). More precisely, we have that the function (b (i,j) ) 2 + c 2 [sin(ϑ) a (i,j) cos(ϑ)] 2 is zero at ϑ = atan(a (i,j) ) + mπ ± 1/4i log((1 β) 2 + α 2 )/(1 + β) 2 + α 2 )), where α = a (i,j) c 2 /((b (i,j) ) 2 + c 2 ) and β = b (i,j) (b (i,j) ) 2 + c 2 + (a (i,j) c) 2 /((b (i,j) ) 2 + c 2 ), the function (v1) i 2 + c 2 [cos(ϑ)] 2 vanishes at ϑ = π/2 + mπ ± i log(v1/c i (v1/c) i 2 ) and, finally, the function (b (i,j) ) 2 + c 2 [sin(ϑ)] 2 vanishes at ϑ = mπ ± i log(b (i,j) /c (b (i,j) /c) 2 ). Hence we deduce that the functions i)-iv) have a singular behavior at the points above denoted with ϑ. When one of these points is close to the interval of integration, the above mentioned functions become large in their vicinity, thus reducing the accuracy of the numerical computation of the integral. In these cases we obtain a considerable improvement 12

14 of the behaviour of the integrand function in the interval of integration by introducing the following simple change of variable (2.29) ϑ = ϑ + t q, q > 1. Thanks to it the transformed integrand function need to be evaluated at fewer points than those requested by the computation of the original integral. Indeed, the transformation (2.29) offers the primary advantage of moving the singularity away from the interval of integration. Therefore, for the computation of the outer integral, when ϑ is close to the interval of integration we use the change of variable (2.29), with q = 3; this latter has turned out the best choice in the innumerable numerical tests done. We use (2.29) whenever the distance of ϑ (for some m Z) from the interval of integration is less than a certain value d. On the ground of the numerical tests, we have set d = 0.3. Notice that when ϑ max ϑ min tends to π, there is a singular behavior close to both of the endpoints of the interval of integration. In this case we rewrite the integral as sum of two integrals, the first over [ϑ min, (ϑ min + ϑ max )/2] and the second over [(ϑ min + ϑ max )/2, ϑ max ], and then we introduce in each of them the change (2.29) with the respective ϑ. Very often we apply a splitting of the interval of integration to reach a good enough accuracy by few quadrature nodes. More precisely, in our algorithm we proceed as follows: we first check if ϑ min is close to ϑ for some m; if this is the case and if ϑ max ϑ min is less than a certain value a, we introduce (2.29) and then we apply a n-point Gauss-Legendre rule. Likewise to d, we have signed to a an experimental value and we have set a = If (ϑ max ϑ min ) > a we first split the interval of integration by the midpoint (ϑ min + ϑ max )/2 and then we introduce (2.29) in the integral over [ϑ min, (ϑ min + ϑ max )/2] and, if ϑ max is close to ϑ for m = m + 1, also in the integral over [(ϑ min + ϑ max )/2, ϑ max ]. Successively, we apply the n-point Gauss-Legendre rule to both the integrals. In the case that ϑ min is not close to ϑ, we check if by chance ϑ max is close to ϑ for some m Z, and then we proceed as for ϑ min. In the case that none of the endpoints ϑ min and ϑ max are close to ϑ, we proceed by applying directly a n-point Gauss-Legendre rule. To show the efficacy of the change of variable (2.29) we use it in the computing of the following integrals (2.30) (2.31) 1 ε 1 ε [R(ϑ)] dϑ, R(ϑ) [R(ϑ)] 2 + c 2 dϑ, 1 R(ϑ)R (2.32) (ϑ) ε [R(ϑ)] 2 + c dϑ, 2 1 [R(ϑ)] p (2.33) [R(ϑ)]2 + c dϑ, 2 ε with R(ϑ) = b/ sin(ϑ) and varying ε; the constants b, c and p are specified in the caption of the corresponding Tables 1-5. In these tables the choice q = 1 corresponds to the straightforward application of the n-point Gauss-Legendre quadrature rule, without using 13

15 any transformation. In the columns q = 3 (q = 5) we report the relative errors obtained by introducing the change of variable ϑ = t 3 (ϑ = t 5 ). In Tables 1-3, where we report the relative errors for the integrals (2.30)-(2.32) respectively, the comparison q = 1 and q > 1 is striking: for the same value of n the relative errors for q > 1 are very much smaller than those for q = 1. Moreover, for q > 1 the accuracy increases very fast. Notice that for ε = by choosing q = 5 the relative accuracy improves with respect to q = 3; however, bigger values of q give a sensible increase of the accuracy only when using higher order Gauss-Legendre quadrature rules. ε = 0.1 ε = 0.01 ε = n q = 1 q = 3 q = 1 q = 3 q = 1 q = 3 q = E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 14 Table 1: Relative errors for the integral (2.30) with b = 0.1. ε = 0.1 ε = 0.01 ε = n q = 1 q = 3 q = 1 q = 3 q = 1 q = 3 q = E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 14 Table 2: Relative errors for the integral (2.31) with b = 0.1, c = 0.1. ε = 0.1 ε = 0.01 ε = n q = 1 q = 3 q = 1 q = 3 q = 1 q = 3 q = E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 12 Table 3: Relative errors for the integral (2.32) with b = 0.1, c = 0.1. The complex poles are ϑ ±i When the singular behavior is mainly due to the presence of complex and conjugate poles ϑ close to the interval of integration, an improvement in accuracy of the n-point Gauss-Legendre rule can be also obtained by means of the following transformation (2.34) ϑ = R(ϑ ) + I(ϑ )sinh(t), where R(ϑ ) and I(ϑ ) denote the real and the imaginary part of ϑ, respectively. This transformation comes from that used in [8]. Notice that the Jacobian I(ϑ )cosh(t) of the transformation (2.34), unlike that of (2.29), is not zero but tends to zero at the nearly singular point as I(ϑ ) tends to zero, thus reducing the peaked nature of the integrand. 14

16 Moreover, unlike (2.29) the transformation (2.34) does not depend on an arbitrary parameter, even if sometimes we have found convenient to use (2.34) in the following form (2.35) ϑ = R(ϑ ) + q I(ϑ )sinh(t), where the real number q is chosen small properly. For example in Table 5, q = ε represents a better choice than q = 1, because in (2.33) for p = 1 the dominant nearly singularity arises from the term R(ϑ). Incidentally, we notice that the transformation (2.35) may be used also to improve the behavior of an integrand function with only a nearly real singularity ϑ, but in this case (2.35) must be rewritten as follows (2.36) ϑ = ϑ + q sinh(t). In Tables 4 and 5 we report a comparison of the transformations (2.29) and (2.35) in the computation of the integral (2.33). ε = 0.1 ε = 0.01 ε = n q = 1 q = 3 q = 1 q = 1 q = 3 q = 1 q = 1 q = 5 q = E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 08 Table 4: Relative errors for the integral (2.33) with b = 0.01, c = 0.1, p = 0. The complex poles are ϑ ±i ε = 0.1 ε = 0.01 ε = n q = 1 q = 3 q = ε q = 1 q = 3 q = ε q = 1 q = 5 q = ε E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 10 Table 5: Relative errors for the integral (2.33) with b = 0.1, c = 0.1, p = 1. The complex poles are ϑ ±i The numerical results show that by using the sinh transformation the accuracy increases especially when the integrand function have only complex poles very close to the interval of integration. However, the transformation (2.29) seems to be generally more effective. Finally, we remark that the numerical results obtained for the other two expressions of R(ϑ) (see (2.5)) are similar to those of Tables 1-5 and, for this reason, we do not report them. 3 Numerical tests In this last section we present some of the innumerable and various tests on the numerical approach proposed in Section 2. We first consider the case of an observation point x, 15

17 whose projection onto the plane defined by is within. Therefore, by introducing the transformation (2.2) we get a triangle T, which lies onto the plane t 3 = 0 and contains the projection v 0 = (0, 0, 0) T of the transformed point s of x. The triangle T is then split into three subtriangles about the projection v 0. The geometry of the splitting into subtriangles is shown in Figure 2, with varying x x=(0.1,0.89,0.1) T x=(0.1,0.89,0.1) T x=(0.1,0.89,0.1) T x=(0.1,0.89,0.1) T Figure 2: The three subtriangles of the transformed triangle T of = [(0, 0, 0) T, (1, 0, 0) T, (0, 1, 0) T ], with varying x. Notice that, although the original triangle has a proper shape, the three subtriangles may have rather inconvenient shapes depending on the position of s. In such a case, the convergence of the direct use of the Gauss-Legendre quadrature may become extremely slow. For this reason we have worked out the new procedure described in Section 2; it performs very efficiently also on integrals over nearly degenerate subtriangles. To test in a meaningful way our numerical approach we consider the triangle = [(0, 0, 0) T, (1, 0, 0) T, (0, 1, 0) T ] and an observation point x whose projection on is close to the right angle of (x = (0.1, 0.1, c) T, c R), far away from the edges and the vertices of (x = (0.2, 0.4, c) T ) and close to an edge but far away from a vertex (x = (0.4, 0.49, c) T ) and, finally, close to an edge and to a vertex (x = (0.1, 0.89, c) T ). The real constant c represents the effective distance from ; we choose c = 0.1, 0.01, even if values of c smaller than are admissible. Indeed, in the case of 1/r singularity, for comparing our results with those obtained in [10] we consider c = By taking into account what has been remarked on the integral (1.4) in Section 1, we apply our approach 16

18 to the following integrals: (3.1) (3.2) (3.3) (3.4) 1 x y l dy y 1 x 1 x y l dy exp( ik x y ) x y l dy (y 1 x 1 ) exp( ik x y ) x y l dy, where x = (x 1, x 2, x 3 ) T, y = (y 1, y 2, y 3 ) T, l = 1, 2, 3 and in (3.3) and (3.4) k = 2π/λ with λ = 10. This latter choice is done to compare some of our results with those in [2] and in [10]. By using the transformation (2.2), where in our case A is the identity matrix of order 3 and b = (x 1, x 2, 0) T, and the splitting into subtriangles, each of the above integrals is rewritten as the sum of three integrals over the subtriangles depicted in Figure 2. Hence, we first introduce the polar co-ordinates in each of these latters and then the transformations, described in Subsections 2.1 and 2.2, into the inner and the outer integral, respectively. Notice that the inner integral of (3.1) and (3.2) can be evaluated analytically. In Table 6 we report a detailed description of the numerical computation of the inner and the outer integral in the case l = 1, according to our approach. INNER INTEGRAL OUTER INTEGRAL subtriangles of T subtriangle 1 of T subtriangle 2 of T subtriangle 3 of T x = (0.1, 0.1, c) T smoothing (q = 3) at the right n-point Gauss-Legendre smoothing (q = 3) at the left c = 0.1, 0.01, analytical ((2.10)) endpoint with splitting endpoint with splitting x = (0.2, 0.4, c) T or numerical n-point Gauss-Legendre n-point Gauss-Legendre n-point Gauss-Legendre c = 0.1, 0.01, ((2.12) or (2.16) with with splitting with splitting x = (0.4, 0.49, c) T q = 2 for c = 0.1, 0.01, n-point Gauss-Legendre smoothing (q = 3) at the n-point Gauss-Legendre c = 0.1, 0.01, q = 4 for c = 0.001) endpoints with splitting x = (0.1, 0.89, c) T n-point Gauss-Legendre smoothing (q = 3) at the smoothing (q = 3) at the right c = 0.1, 0.01, endpoints with splitting endpoint with splitting Table 6: The main details on the computation of the inner and outer integral, which arise from (3.1)-(3.4) with l = 1, according to our approach. In the cases l = 2 and l = 3 the computation of the inner integral varies according to its nearly singular term. More precisely, for l = 2 (l = 3) we proceed analytically by using (2.17) ((2.22)) or numerically by using (2.21) with q = 2 ((2.25) with q 1 = 3 and q 2 = 2 when c = 0.1 and q 1 = 6 and q 2 = 4 when c = 0.01, 0.001). On the contrary, in the cases l = 2 and l = 3 the computation of the outer integral is substantially unchanged. We consider only a variation in the cases l = 3 and x = (0.2, 0.4, c) T for the subtriangles 2 and 3; in those cases we do not split the interval of integration before applying the n- point Gauss-Legendre rule, since this splitting is useless. In the setting of our strategy, 17

19 in particular in the choice of the smoothing parameters q (or q 1, q 2 ) and q, our main aim has been that of obtaining a relative accuracy smaller than 10 5 by using the n-point Gauss-Legendre formula with n = 6. Very often and precisely, as already remarked in Subsection 2.2, when the length of the interval of integration is greater than 1.80, to achieve this objective we split the interval of integration about its midpoint and apply to each subinterval the n-point Gauss-Legendre rule. We prefer to split rather than use the 2n-point Gauss-Legendre rule for two reasons: the main one is that in this way, by halving the length of the interval of integration, the closeness of the complex poles is less adverse to the performance of a Gauss-Legendre rule and the secondary reason is to avoid the computation of the weights and nodes of the n-point and 2n-point Gauss-Legendre rules. As remarked in Subsection 2.1, the further change of variable depending on the parameter q (or q 1, q 2 in the case of 1/r 3 singularity) for the integrals D p,l is necessary whenever p is even and l = 1, 2 or p is even, l = 3 and an endpoint of the interval of integration is 1 or, finally, l = 3 and an endpoint of the interval of integration is close to 0. When no one of the above-mentioned cases happens, the above change is useless; however one could introduce all the same it by setting the smoothing parameter q (or q 1, q 2 ) equal to 1. Even if in this way the numerical results are slightly more accurate than those with q (or q 1, q 2 ) greater than 1, for simplicity we do not distinguish the various cases and we make only a choice for the smoothing parameter of the polynomial transformation of the outer integral, depending on the distance c. In each of the following Tables 7-16, we compare our approach, whose results are reported in the columns labelled at the top with q = 3, with the approach which uses only the transformation eliminating the nearly singular term in the inner integral. The numerical results of this latter are reported in the columns labelled with q = 1. To compare the approaches q = 1 and q = 3 in a fair way from the point of view of the total number of quadrature nodes required, in the case q = 1 we use the 2n-point Gauss-Legendre rule every time that in the case q = 3 we operate the splitting and apply the n-point Gauss- Legendre rule to each of the two resulting integrals. From this comparison, it appears that the smoothing strategy associated to the splitting outperforms the straightforward application of a Gauss-Legendre rule even if the latter uses twice the number of quadrature nodes. Our approach seems to be highly competitive especially when the projection of the observation point x is close to the boundary of the triangle. In Table 7 we report the absolute errors for the integrals I 1,2, I 2,3 and I 3,1 (see (2.4)), which arise from (3.1) with l = 1 and defined over the subtriangles 1, 2 and 3 of Figure 2, respectively. Notice, that the subtriangles, in which the special strategy is required, are those nearly degenerate; the more the triangle is degenerate, the more our approach is efficient and competitive. In particular, the comparison with q = 1 is remarkable for the subtriangle 2 when x = (0.1, 0.89, 0.1) T. 18

20 x = (0.1, 0.1, 0.1) T subtriangle 1 of T subtriangle 2 of T subtriangle 3 of T I 1,2 I1,2 n I 2,3 I2,3 n I 3,1 I3,1 n n q = 1 n q = 3 n q = 1 n q = 3 n q = 1 n q = E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E x = (0.2, 0.4, 0.1) T subtriangle 1 of T subtriangle 2 of T subtriangle 3 of T I 1,2 I1,2 n I 2,3 I2,3 n I 3,1 I3,1 n n q = 1 n q = 3 n q = 1 n q = 3 n q = 1 n q = E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 13 x = (0.4, 0.49, 0.1) T subtriangle 1 of T subtriangle 2 of T subtriangle 3 of T I 1,2 I1,2 n I 2,3 I2,3 n I 3,1 I3,1 n n q = 1 n q = 3 n q = 1 n q = 3 n q = 1 n q = E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 14 x = (0.1, 0.89, 0.1) T subtriangle 1 of T subtriangle 2 of T subtriangle 3 of T I 1,2 I1,2 n I 2,3 I2,3 n I 3,1 I3,1 n n q = 1 n q = 3 n q = 1 n q = 3 n q = 1 n q = E E E E E E E E E E E E E E E E E E E E E E E E E E E E Table 7: Absolute errors for the three integrals I 1,2, I 2,3 and I 3,1, which arise from (3.1) with l = 1 and defined over the subtriangles 1, 2 and 3 of Figure 2, respectively. 19

21 In Table 8 and following, while n denotes the number of quadrature nodes of the used Gauss-Legendre quadrature rule, n tot denotes the total number of the quadrature points required in the computation of the integral over the original triangle. The computation of n tot easily results from Table 6 and its respective comments. In all the tables the sign means that full relative accuracy (i.e. 14 significant digits in our case) has been achieved. x = (0.1, 0.1, 0.1) T x = (0.1, 0.1, 0.01) T x = (0.1, 0.1, 0.001) T I I I E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 12 x = (0.2, 0.4, 0.1) T x = (0.2, 0.4, 0.01) T x = (0.2, 0.4, 0.001) T I I I E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 13 x = (0.4, 0.49, 0.1) T x = (0.4, 0.49, 0.01) T x = (0.4, 0.49, 0.001) T I I I E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 10 x = (0.1, 0.89, 0.1) T x = (0.1, 0.89, 0.01) T x = (0.1, 0.89, 0.001) T I I I E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 12 Table 8: Relative errors for the integral (3.1) with l = 1, whose inner integral is evaluated by (2.10). We wish to remark that we obtain practically the same results of Table 8 if we apply our numerical approach for computing the inner integral instead of proceeding analytically. 20

22 In Tables 9 and 10 we report the relative errors obtained for the integrals (3.3) and (3.4) with l = 1, respectively. Notice that the approach q = 3 outperforms q = 1 when c = 0.1, 0.01, with a few exceptions for x = (0.2, 0.4, c) T. As already remarked in Subsection 2.1 when c is equal or less than 0.001, within our approach it is more efficient to use the transformation (2.13) instead of (2.11). This is evident in Figures 3, 4 e 5. x = (0.1, 0.1, 0.1) T x = (0.1, 0.1, 0.01) T x = (0.1, 0.1, 0.001) T R(I) R(I) R(I) I(I) I(I) I(I) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 12 x = (0.2, 0.4, 0.1) T x = (0.2, 0.4, 0.01) T x = (0.2, 0.4, 0.001) T R(I) R(I) R(I) I(I) I(I) I(I) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E e E 12 x = (0.4, 0.49, 0.1) T x = (0.4, 0.49, 0.01) T x = (0.4, 0.49, 0.001) T R(I) R(I) R(I) I(I) I(I) I(I) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 12 x = (0.1, 0.89, 0.1) T x = (0.1, 0.89, 0.01) T x = (0.1, 0.89, 0.001) T R(I) R(I) R(I) I(I) I(I) I(I) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 09 Table 9: Relative errors for the integral (3.3) with l = 1, whose inner integral is computed by (2.12). The relative errors of Table 9 and those of Figure 4 for x = (0.1, 0.1, c) T with c = 0.1, 0.01 can be compared with those of Figure 9 in [10]. To this end we recall that in [10] the authors do not use the polar co-ordinates and introduce an asinh transformation to eliminate the nearly singular term in the inner integral; then, they split the subtriangles along the radial direction at two points depending on the distance c of the observation point from the triangle and use the same Gauss-Legendre scheme in each of the resulting three regions and for all three subtriangles. By using about 1000 quadrature points in [10] they achieve 11 significant digits for c = 0.1, 8 digits for c = 0.01 and 5 digits for c = , while by means of the same number of quadrature points and q = 3 we achieve the full relative accuracy (i.e. 14 significant digits) for all values of c = 0.1, 0.01, Notice that our approach with q = 1 already provides an accuracy of the order when c = 0.1, 0.01,