Node based Method of Moments Solution to Combined Layer Formulation of Acoustic Scattering

Transcription

1 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 Node baed Method of Moment Solution to Combined Layer Formulation of Acoutic Scattering B. Chandraekhar 1 Abtract: In thi work, a novel numerical technique, baed on method of moment olution, i preented to olve the Combined layer formulation (CLF) to inure unique olution to the exterior acoutic cattering problem at all frequencie. A new et of bai function, namely, Node baed bai function are ued to repreent the ource ditribution on the urface of rigid body and the ame function are ued a teting function a well. Combined layer formulation (CLF) i defined by linearly combining the Single layer formulation (SLF) and Double layer formulation (DLF) with complex coupling parameter. The matrix equation for the SLF and DLF are derived and ame equation are extended to the CLF. The urface of the body i modeled by triangular patch modeling. Reult of numerical technique preented in thi paper, uing node baed bai function, are compared with the exact olution wherever available. Alo, condition number of the impedance matrice of SLF, DLF and CLF are plotted, for ome object of canonical hape, to how that only CLF i free from reonance problem. Keyword: Acoutic cattering, Method of moment, Node baed bai function, Boundary integral equation, Acoutic reonance problem, Non-uniquene. 1 Introduction The non-uniquene of the olution of exterior acoutic problem i widely dicued in the recent literature and ha been one of the intereting area of reearch for long time. Non-uniquene of olution i the breakdown of the formulation when the frequency of incident acoutic wave matche with the eigen-frequencie of the correponding interior problem. Though exterior acoutic cattering problem do not have any characteritic frequencie aociated with it, the integral equation do fail at certain frequencie due to the boundary integral equation formulation. More detail regarding the fictitiou frequencie can be found in the 1 Supercomputer Education and Reearch Center, Indian Intitute of Science, Bangalore. India

2 244 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 ref. [Chen and Chen (2006), Chen, Chen and Chen (2006)]. The adjoint of thee boundary integral equation repreent the interior acoutic problem which ha got well known characteritic frequencie. Hence boundary integral equation for the exterior acoutic problem fail when the incident wave frequency matche with the characteritic frequencie of the urface bounding the cattering volume. Two popularly known method to reolve the non-uniquene problem in the exterior acoutic problem are 1. Combined Helmholtz Integral Equation Formulation (CHIEF) and 2. Burton and Miller approach. CHIEF, propoed by Schenck [Schenck (1968)] i very popular among variou engineering application. CHIEF i baically addition of ome interior relation to the urface Helmholtz integral equation thu reulting in a over determined ytem of equation. Thi over determined ytem of equation may be olved by the leat quare technique. One of the major drawback of thi formulation i the election of CHIEF point to inure a uniquene of the olution. A good CHIEF point to be choen in uch a way that it hould be away from the neighborhood of nodal point of the correponding interior eigenmode. Although CHIEF i very popular, it i heuritic and prone to inaccuracie epecially at higher frequencie. Another popular method of reolving the non-uniquene iue i Burton and Miller (BM) approach. BM approach [Burton and Miller (1971)] baically ugget linearly combining the Helmholtz integral equation and it normal derivative with a complex coupling parameter. Lei [Lei (1965)], Panich [Panich (1965)], and, Brakhage and Werner [Brakhage and Werner (1965)] alo demontrated that by combining the ingle layer and double layer potential with a complex coupling parameter, the non-uniquene problem can be averted. Burton and Miller proved mathematically that, thi linearly combined integral equation inure unique olution at all frequencie. But, the normal derivative of the Helmholtz integral equation involve the evaluation of hyper-ingular integral. Many reearcher [Amini and Wilton (1986), Meyer, Bell, Zinn, and Stallybra (1978), Chien, Raliyah and Alturi (1990), Yan, Cui and Hung (2005)] attempted to evaluate the hyper-ingular integral which may be ued in the BM procedure to overcome the internal reonance problem. The uual procedure i to regularize the hyper ingular integral and the regularization technique i computationally very expenive and it i difficult to incorporate in a general-purpoe code. Alo, there are other method which reduce the hyper ingular kernel to a trongly ingular kernel and their olution i baed on Petrov-Galerkin cheme [Qian, Han, and Atluri (2004)] and collocationbaed boundary element method [Qian, Han, Ufimtev, and Atluri (2004)]. A deingularized boundary integral formulation i alo one of the recently propoed method [Callen, von Etorff, and Zaleki (2004)] to overcome the problem of ingularity.

3 Node baed Method of Moment Solution to Combined Layer Formulation 245 Recently, Chandraekhar and Rao [Chandraekhar and Rao (2004b)] extended concept of BM approach by linearly combining the integral equation baed on layer potential in contrat to combining the Helmholtz integral equation and it normal derivative a uggeted by Burton and Miller. They adapted the method of moment olution a a numerical technique to implement the combined layer formulation (CLF). The procedure [Chandraekhar and Rao (2004a)] that they followed neither regularize the hyper-ingular integral nor implement complex integration cheme. They alo extended the ame numerical procedure to the acoutic cattering from open bodie and interecting bodie [Chandraekhar and Rao (2005)]. The method of moment olution that they propoed wa baed on defining the bai function on the edge of the triangular patch modeling of the urface of the cattering body. A the method of moment olution reult in a full matrix, the ize of the reulting impedance matrix, baed on defining the bai function on the edge, i large. Recently, author [Chandraekhar (2005)] ha implemented the method of moment olution of ingle layer and double layer formulation baed on defining the bai function on the node in contrat to defining it on the edge. For a cloed body, the relationhip between the number of node N n, edge N e and patche N f in the triangular patch modeling i given by N n N e + N f = 2andN f = 2N e /3 [Oneill]; thi reult into N n = 2 +N e /3. Thu the order of the reulting matrix, defining the bai function on the node, i almot one third to that of defining the bai function on edge. A the torage matrix i maller, time required for the olution of imultaneou linear ytem of equation i maller and hence, much larger problem can be olved without increaing the olution time compared to the exiting olution baed on method of moment with face baed bai function [Raju, Rao, Sun (1991), Rao and Sridhara (1991), Rao, Raju, and Sun (1992), Rao and Raju, (1989)] and with edge baed bai function [Chandraekhar and Rao (2004a)]. In thi work, the non-uniquene problem i eliminated by implementing the combined layer formulation (CLF) uing method of moment olution procedure with node baed bai function. Traditionally, DLF i difficult to implement due to the preence of hyper-ingular kernel. In thi work imple vector calculu technique are ued to circumvent the problem of hyper-ingularity. The method developed in thi work can be ued in combination with other method like Meh Le Petrov-Galerkin cheme [Vavouraki, Sellounto and Polyzo (2006); Sladek, Sladek, Wen and Aliabadi (2006); Gao, Liu and Liu (2006); Zhang and Chen (2008); Sladek, Sladek, Solek and Wen (2008); Dang and Bhavani Sankar (2008); and Arefmaneh, Najafi and Abdi, (2008)] and boundary element formulation, but not limited to acoutic cattering [Owatiriwong, Phanri, and Park (2008); Criado, Ortiz, Manti c, Gray, and Pari (2007); Chandraekhar and Rao.

4 246 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 (2007); Zai You Yan (2006); and Soare Jr, and Vinagre (2008)]. 2 Organization of the paper In thi paper, next ection briefly decribe the method of moment olution procedure [Harrington (1968), Sun and Rao(1992)]. Mathematical formulation i laid in ection 4 for SLF, DLF and CLF. In ection 5, we decribe the numerical olution procedure and derive matrix equation for SLF, DLF and CLF. Numerical reult, baed on the development of new bai function are given in ection 6. Latly we preent ome important concluion drawn from the preent work. 3 Outline of Method of Moment Conider the determinitic equation Lf = g (1) where L i a linear operator, g i a known function and f i an unknown function to be determined. Let f be repreented by a et of known function f j, j = 1,2,...,N termed a bai function in the domain of L a a linear combination, given by f = N n=1 β j f j (2) where β j are calar coefficient to be determined. Subtituting Eq. 2 into Eq. 1, and uing the linearity of L, wehave N n=1 β j Lf j = g (3) where the equality i uually approximate. Let (w 1,w 2,w 3,...) define a et of teting function in the range of L. Now, taking the inner product of Eq. 3 with each w i and uing the linearity of inner product defined a f,g = f gd, we obtain a et of linear equation, given by N n=1 β j wi,lf j = wi,g i = 1,2,...,N. (4) The et of equation in Eq. 4 may be written in the matrix form a ZX = Y (5)

5 Node baed Method of Moment Solution to Combined Layer Formulation 247 which can be olved for X uing any tandard linear equation olution methodologie. The implicity, accuracy and efficiency of the method of moment lie in chooing proper et of bai/teting function and applying to the problem at hand. In thi work, we propoe a pecial et of bai function and a novel teting cheme to obtain accurate reult uing SLF, DLF and CLF. 4 Mathematical Formulation Conider an acoutic wave, with a preure and velocity ( p i,u i), incident on a threedimenional arbitrarily haped rigid body placed in a ource free homogeneou medium of denity ρ and peed of ound c through the medium. When the incident wave interact with the body, the acoutic wave get cattered with a preure and velocity (p,u ). Here, we note that, incident field are defined in the abence of the cattering body. Φ i the calar velocity potential atifying the Helmholtz differential equation 2 Φ + k 2 Φ = 0 for the time harmonic wave preent in the region exterior to the urface S of the body. One more condition on velocity potential i that it hould atify the appropriate boundary condition on the urface S of the body along with the Sommerfeld radiation condition. The preure and velocity field of acoutic wave i related to the calar velocity potential Φ a u = Φ and p = jωρφ. In thi paper, integral equation formulation are baed on potential theory and free pace Green function. The cattered velocity potential may be defined uing three different formulation baed on monopole and/or dipole ditribution. Formulation baed on monopole ditribution and dipole ditribution are called a ingle layer formulation (SLF), double layer formulation (DLF), repectively. Third formulation, namely Combined layer formulation (CLF) i a linear combination of SLF and DLF. Uing the potential theory and the free pace Green function, the cattered velocity potential Φ may be defined a Φ = σ ( r ) G ( r,r ) d (6) for SLF, Φ = σ ( r ) G(r,r ) d (7) for DLF, and Φ = σ ( r ) G ( r,r ) d +α σ ( r ) G(r,r ) d (8) for CLF.

6 248 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 In the above three equation, σ i the ource denity function independent of r over the urface of the body, r and r are the poition vector of obervation and ource point, repectively, with repect to a global co-ordinate ytem O, and G/ i the normal derivative of Green function at ource point. Coefficient α i a complex coupling parameter, to be choen baed on the guideline given by Burton and Miller [Burton and Miller (1971)]. k real or imaginary Im(α) 0 (9) k complex Im(α)=0 (10) where k i the wave number in the medium. G(r,r ) i the free pace Green function, given by, G ( r,r ) = e jk r r 4π r r (11) G(r,r ) i the olution of the Helmholtz equation with a point ource inhomogeneity ( 2 +k 2) G(r,r )= δ ( r r ) (12) and can be interpreted a the olution at the obervation point r due to the preence of acoutic ource of unit trength located at the ource point r. For a rigid body, the normal derivative of total velocity potential, which i the um of incident and cattered velocity potential, with repect to the obervation point on the urface of the body vanihe. That i ( Φ i +Φ ) = 0 (13) Φ = Φi. Subtituting Eq. 6, 7 and 8. into Eq. 14, σ ( r ) G ( r,r ) d = Φi for SLF, σ ( r ) G(r,r ) d = Φi for DLF, and σ ( r ) G ( r,r ) d +α σ ( r ) G(r,r ) d = Φi (14) (15) (16) (17)

7 Node baed Method of Moment Solution to Combined Layer Formulation 249 for CLF. In the above equation, Φ i i the calar velocity potential of the incident wave. Eq. 15, the SLF can alo be re-written a σ (r ) 2 σ ( r ) G(r,r ) d = Φi. (18) The econd term in the above equation i the integration over the urface excluding the principal value term i.e. r = r. We note that, thi integral i a well behaved integral, although rapidly varying, which can be evaluated uing tandard integration algorithm. Following the procedure developed in [Maue (1949) and Mitzner (1966)], Eq. 16, the DLF, may be written a n n k 2 σ(r )G(r,r )d ( + n X σ ) (nx G)d = Φi where n and n are the unit normal vector at r and r, repectively. From Eq. 18 and Eq. 19, the CLF can be written a σ (r ) 2 σ ( r ) G(r,r ) d +α n n k 2 σ(r )G(r,r )d (19) ( + α n X σ ) (nx G)d = Φi. (20) In the following ection, a novel numerical technique i developed uing node baed bai function for SLF, DLF and CLF. SLF and DLF fail when the frequency of the incident field matche with the characteritic frequency of urface S. Burton and Miller [Burton and Miller (1971)] uggeted combining the Helmholtz integral equation and it normal derivative linearly with a complex coupling parameter to eliminate the internal reonance problem. However, the ame concept i extended in thi paper to combine the ingle and double layer formulation with a complex coupling parameter and hence the name Combined layer formulation (CLF). CLF do not fail at the characteritic frequencie and enure a unique olution at all incident frequencie. Thi paper focue on the numerical olution of ingle layer, double layer and combined layer formulation by uing a new et of bai function, namely, node baed bai function. The numerical olution of SLF and DLF i already reported in [Chandraekhar (2005)] and it i the endeavor of thi paper to extend the numerical olution to CLF a well to get the unique olution to the acoutic cattering

8 250 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 problem baed on the integral equation formulation. A the CLF i a linear combination of SLF and DLF, the matrix equation for SLF and DLF are alo derived in thi paper. 5 Numerical Solution Procedure Fig.1 how the triangular patch modeling of the urface of a arbitrarily haped three dimenional body. The triangular patch modeling i an approximation of the urface. The accuracy of approximation can be increaed by modeling the urface with large number of triangular patche of maller ize. The three geometric entitie that are preent in the triangular patch modeling are triangular patche, edge and node. Let N f, N e and N n repreent the number of triangular patche, number of edge and number of node, repectively, on the urface of triangulated body. For a cloed body, every edge i common to two adjacent triangular patche, and every node i common to at leat three triangular patche a well a at leat three edge. Figure 1: Triangular patch modeling of an arbitrarily haped three-dimenional body. Figure 2: Node baed bai function and geometric parameter aociated with the node. Conider a node n about which the bai function i defined. For illutration purpoe, conider one uch node a hown in Fig. 2, where there are five triangular patche attached to it. Let T 1,T 2,...,T 5 are the triangular patche and e 1,e 2,...,e 5 are the edge urrounding the node. The haded region, hown in the Fig. 2. a S n, i formed by joining the mid point of the edge, that are connected to the node n, to the centroid of adjacent triangular patche thu forming a cloed boundary around the node. Since the method of moment olution call for the definition of expanion/bai a well a teting/weighting function, expanion/bai function are defined on the ource node n and the teting/weighting function are defined on the field node m.

9 Node baed Method of Moment Solution to Combined Layer Formulation 251 Let there are p number of triangular patche attached to field node m and q number of triangular patche attached the ource node n. By joining the node on which the bai function or weighting function i defined, to the centroid of the attached triangular patche, the haded area on each triangular patch i divided into two ub triangle, reulting u number of ub triangle around field node m and v number of ub triangle around the ource node n. In thi paper, index x i ued to repreent the ub triangle attached to the field node and index y i ued to repreent the ub triangle attached to the ource node. The node baed bai function i defined over the haded area a follow. ( f n r ) { 1 r S n = (21) 0 Otherwie The ource denity function σ over the urface of the cattering object i approximated by σ ( r ) = N n n=1 β n f n where β n repreent the unknown coefficient to be determined and the equality ign i uually approximate. The bai function defined over the node ha all advantage a edge baed bai function [ee B. Chandraekhar and S.M. Rao (2004)]. In the numerical olution of all the three formulation, Galarkin approach i ued by defining the teting function in the ame manner a it i defined for the bai function. w m = { 1 r S m 0 Otherwie The next three ub-ection decribe the numerical olution procedure for the SLF, DLF and CLF. 5.1 Single Layer Formulation (SLF) Teting Eq. 18 with a teting function w m, reult in w m, σ (r ) w m, σ ( r ) G(r,r ) d = 2 w m, Φi (22) (23). (24) Uing the inner product definition decribed in Sec. 3, Eq. 24 can be written a 1 w m σ ( r ) d w m σ ( r ) G(r,r ) d Φ i d = w m d. (25) 2

10 252 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 Approximating the integration over the field node at the centroid of the attached ub triangle, and uing Eq. 23, the Eq. 25 become σ (r ) 2 u x=1 A x m u A x m x=1 σ ( r ) G(rm cx,r ) d = x m u x=1 A x Φ i m x m (26) where A x m i the area of ub-triangle attached to the field node. Thi approximation i jutified becaue the ub domain are ufficiently mall, which i a neceary condition to obtain accurate olution uing MoM. Subtitutingthe ource expanion defined in Eq. 22 into Eq. 26, reult in a ytem of linear equation, which can be repreented in the matrix form a Z l f X = Y (27) where Z l f i the impedance matrix of the ingle layer formulation of ize N n N n, X and Y are the column vector of ize N n. The element of Z l f, X and Y are given below. Z mn l f = and u 1 2 A x m x=1 u v x=1 y=1 A x m G(rm cx,rn cy ) x m d for m = n and x = y otherwie (28) Y m = u x=1 A x Φ i m x m (29) where rm cx i the poition vector to the centroid of the x th ub-triangle attached to field node, rn cy i the poition vector to the centroid of the y th ub-triangle attached to ource node and G/ x m i the normal derivative of Green function at the centroid of the x th ub-triangle attached to field node. Once the element of the impedance matrix Z l f and the forcing vector Y are determined, one may olve the linear ytem of equation, Eq. 27, for the unknown vector X. Here we note that, when the frequency of the incident wave i in the cloe vicinity of the characteritic frequency related to Dirichlet problem, the impedance matrix Z l f become highly ill-conditioned and the olution vector X turn out to be puriou reulting in unphyical value of ource ditribution σ.

11 Node baed Method of Moment Solution to Combined Layer Formulation Double Layer Formulation (DLF) Teting Eq. 19 with the function defined in Eq. 23, w m, n n k 2 σ(r )G(r,r )d + Taking the firt term for evaluation, w m, n n k 2 σ(r )G(r,r )d = = ( w m, n x σ ) (nx G)d w m u x=1 A x m nx m = n n k 2 σ(r )G(r,r )d d w m, Φi. (30) n k 2 σ(r )G(r,r )d (31) where n x m repreent the unit normal vector of the ub triangle connected to the m th node. We alo note that the double urface integration in Eq. 31 i converted to a ingle urface integral by approximating the integrand at the centroid of the ub triangle attached to m th node and multiplying by the area of the ub triangle a before. To evaluate the econd term in the Eq. 30, we follow the imilar procedure mentioned in [Chandraekhar and Rao (2004)], but node baed bai function are ued in contrat to the edge baed bai function. The advantage of uing the node baed bai function are already mentioned in the earlier ection. Let u define, J = n X σ and the vector J i again approximated by a new et of bai function. We have, J = N f g f (r ). f =1 (32) where g f i pread over the triangular patch and aumed to be contant. Since g f i not an independent quantity, a relationhip between β n, f n,andg f may be derived a follow: Conider the triangular patch T f with aociated non-boundary edge l 1,l 2,l 3 and node n 1,n 2,n 3 a hown in the Fig. 3. Then, uing well-known Stoke theorem and imple vector calculu, J = n X σ may be re-written a J d = n X σd = σ dl = β 1 f 1 (l 2 +l 3 ) C e 2 + β 2 f 2 (l 3 +l 1 ) + β 3 f 3 (l 1 +l 2 ) 2 2 = β 1 f 1 l 1 +β 2 f 2 l 2 +β 3 f 3 l 3 2 (33)

12 254 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 Figure 3: Triangular patch T f and aociated edge and node. where C e i the contour bounding the triangle T f and l i, i = 1,2,3 repreent the edge vector a hown in the Fig. 3. Noting that, J d = g f d (34) = g f A fn where A fn i the area of the triangular patch attached to the ource node. g f = 1 J d A fn (35) ubtituting Eq. 33 into Eq. 35, we get g f = 1 2A fn (β 1 f 1 l 1 +β 2 f 2 l 2 +β 3 f 3 l 3 ). (36) Thi i the required relationhip between β n, f n and g f. We have, J (nx G)=n GXJ = n ( X [ GJ ] G XJ ). (37) The term G XJ vanihe becaue the curl operation i on the unprimed variable only. Thu, J (nx G)d =n X J Gd. (38) Let A = J Gd, then the above equation can be written a J (nx G)d =n XA. (39)

13 Node baed Method of Moment Solution to Combined Layer Formulation 255 Taking the teting operation, w m, J (nx G)d = w m (n XA)d = A dl Cm = l m 2 ( p x=1 A ( rm cfp ) ) (40) where p i the number of triangular patche attached to the field node and rm cfp i the poition vector to the centroid of triangular patch attached to the field node. The expreion for the teting operation on right hand ide of the Eq. 30 i ame a that of ingle layer formulation, i.e. Eq. 25. Subtituting Eq. 31, Eq. 40 and RHS of Eq. 26 into Eq. 30, we get u x=1 A x m nx m ( n k 2 σ(r )G(rm cx,r )d + l m p 2 A ( rm cfp,r )) = x=1 u x=1 Φ i. (41) A x m x m Subtituting the ource expanion defined in Eq. 21 and Eq. 22, reult into a ytem of linear equation which can be expreed in matrix form a Z dl f X = Y (42) where Z dl f i the impedance matrix for double layer formulation of ize N n N n, X and Y are column vector of ize N n. Z mn dl f = u v x=1 y=1 k 2 A x mn x m n y n G(r cx m,r cy n )d + l m 2 l n q ( p 2A fn x=1 y=1 f G ( r cfp m,rn cfq ) d Note that the integration on the firt term of right hand ide of the above equation i on the ub triangle attached to the ource node and the integration on the econd term i on the triangular patch attached to the ource node. Y m = u A x m x=1 Φ i (r cx m ) x m where A x m i the area of the ub triangle attached to the field node, A fn i the area of the triangular patch attached to the ource node, rm cx i the poition vector to the centroid of the x th ub triangle attached to field node, rn cy i the poition vector to the centroid of the y th ub triangle attached to ource node, rm cfp i the poition vector ). (43) (44)

14 256 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 to the centroid of the p th triangular patch attached to field node, and rn cfq i the poition vector to the centroid of the q th triangular patch attached to ource node. Here we again note that, a in the cae of SLF, DLF alo uffer from the reonance problem and the impedance matrix Z dl f become highly ill-conditioned when the frequency of the incident wave i in the cloe vicinity of the characteritic frequency related to Neumann problem. The olution vector X turn out to be puriou reulting in unphyical value of ource ditribution σ. 5.3 Combined Layer Formulation (CLF) Teting Eq. 20 with the function defined in Eq. 23 w m, σ (r ) w m, 2 σ ( r ) G(r,r ) + α w m, Thi can be expreed in matrix form a d +α w m, n n k 2 σ(r )G(r,r )d ( n x σ ) (nx G)d = w m, Φi. (45) Z mn cl f X = Y m (46) where Z mn cl f = Zmn l f +αzmn dl f (47) a the LHS of the above equation i a linear combination of SLF and DLF. The impedance matrix of CLF i free from ill-conditioning at all frequencie and enure a unique olution. Integral appearing in Eq. 28 and Eq. 43 may be evaluated uing the technique mentioned in [Wilton, Rao, Glion, Schaubert, Al-Bundak, and Bulter (1984) and Hammer, Marlowe, and Stroud (1956)] for an accurate olution,a the former contain ingular kernel. Here one may note that the number of integration to carried for the cae of CLF i ix (three each for SLF and DLF) for each triangle in cae of edge baed olution [Chandraekhar and Rao (2004b)], where a one may need twelve integration for each triangular patch in cae of Node baed olution of CLF (ix integration for each SLF and DLF). However, extra time that the node baed olution take in generating the moment matrix i negligible compared to the time it ave in the olution of the linear ytem of equation. For a plane wave incidence, we et Φ i = e jkˆk r (48)

15 Node baed Method of Moment Solution to Combined Layer Formulation 257 where the propagation vector ˆk i given by, ˆk = inθ 0 coφ 0 a x +inθ 0 inφ 0 a y +co θ 0 a z (49) (θ 0,φ 0 ) define the angle of arrival of the plane wave in the conventional pherical co-ordinate ytem and a x, a y and a z are the unit vector along the x, y and z axe, repectively. The normal derivative of the incident field may be written a Φ i = n Φi = jkn ˆke jkˆk r. (50) Once the element of the impedance matrix Z and the forcing vector Y are determined, one may olve the linear ytem of equation, Eq. 27, 42 and 46, for the unknown vector X uing any tandard matrix inverion technique. 6 Numerical Reult In thi ection, the numerical olution developed in the above ection i validated againt exact olution wherever available and againt the edge baed olution wherever exact olution are not available for the cae of SLF, DLF and CLF. Alo the invere condition number are plotted in order to demontrate the exitence of the internal reonance problem for the SLF and DLF cae and the elimination of the reonance problem for the cae of CLF. The geometrie conidered here are imple canonical hape like Sphere, Cube, Cylinder and a Cone. For all thee cae, the body i placed at the center of the co-ordinate ytem and a plane wave, traveling along Z axi i incident on the body. Here, we note that, no convergence tudy i carried out to acertain the optimum number of node required to obtain certain degree of accuracy. The cattering cro ection i defined by S = 4π Φ 2 Φ i 1 4π N n n=1 [ v 2 β n A y nn y n rne y jkny n rn] y. (51) y=1 In all the formulation, a a firt cae, a phere of radiu 1m and 2m are conidered with triangular patch modeling. The modeling i done by dividing the θ and φ direction equal egment each and the complete modeling procedure i decribed in [Chandraekhar and Rao (2004)]. Alo the geometrie with harp corner and harp edge uch a cube, cylinder and cone are conidered in order to demontrate the capability of the node baed bai function baed on method of moment olution.

16 258 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , Single layer formulation (SLF) In thi ection, numerical reult baed on the SLF numerical olution procedure i preented to validate the reult with the exact olution for the cae of phere of radii 1m and 2m. Fig. 4 how the far field cattering cro ection of the phere veru the polar angle baed on the SLF for the cae when the phere of radiu 1m i excited by an acoutic plane wave with k = 1 rad/m, traveling along Z axi. It i evident from the figure that, the method of moment olution baed on the node baed bai function matche well with the exact olution [Bowman, Senior and Ulenghi (1969)]. In order to demontrate the convergence of the olution, phere i modeled with 58 node, 112 face and 168 edge on the urface of the phere; 156 node, 308 face and 462 edge; a well a with 212 node, 420 face and 630 edge. The reult for thee three cae are plotted againt the exact olution for the ake of comparion. S X Z q Y 58 N 156 N 212 N Exact θ Figure 4: Scattering cro ection veru polar angle for an acoutically rigid phere of radiu 1m, ubjected to an axially incident plane wave of k = 1 rad/m, baed on Single layer formulation (SLF). S N 382 N Exact X θ Figure 5: Scattering cro ection veru polar angle for an acoutically rigid phere of radiu 2m, ubjected to an axially incident plane wave of k = 1 rad/m, baed on Single layer formulation (SLF). Z q Y A a econd cae, the phere of radiu 2m modeled with 212 node and 382 node are alo excited by a plane wave with k = 1 rad/m and the cattering cro ection i plotted in Fig. 5. The dicrepancy between the two olution can be attributed to inufficient number of unknown and the fact that the urface area of the triangulated phere i le than that of the actual phere. The olution for SLF cae can be further improved by increaing the number of unknown.

17 Node baed Method of Moment Solution to Combined Layer Formulation Double layer formulation (DLF) In thi ection, numerical reult baed on the DLF numerical olution procedure i preented to validate the reult with the exact olution for the cae of phere of radii 1m and 2m. Fig. 6 how the reult baed on the DLF for the ame geometrie conidered in the cae of SLF viz 58 node, 156 node and 212 node. Next, the DLF numerical olution procedure i again teted and the reult are hown for a plane wave incidence on a phere of radiu 2m. Thee reult alo very well match with the exact olution and the convergence of the olution i clearly evident in the Fig. 7 a the number of node i increaed on the urface of the phere. S X Z q Y 58 N 156 N 212 N Exact θ Figure 6: Scattering cro ection veru polar angle for an acoutically rigid phere of radiu 1m, ubjected to an axially incident plane wave of k = 1 rad/m, baed on Double layer formulation (DLF). S N 382 N Exact X θ Figure 7: Scattering cro ection veru polar angle for an acoutically rigid phere of radiu 2m, ubjected to an axially incident plane wave of k = 1 rad/m, baed on Double layer formulation (DLF). Z q Y The reult of the DLF numerical olution procedure hown in Fig. 7 well matche with the exact olution a compared to the SLF numerical olution procedure for the cae of ka = Combined Layer Formulation (CLF) In thi ection, numerical reult baed on the CLF numerical olution procedure i preented to validate the reult with the exact olution for the cae of phere of radii 1m and 2m. The complex coupling parameter conidered in the CLF olution i α = 0.1 j, though thi value i not critical. In order to demontrate the capability of the node baed SLF, DLF and CLF olution, geometrie with harp edge and harp corner are conidered in the following paragraph.

18 260 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 S X Z q Y 58 N 156 N 212 N Exact θ Figure 8: Scattering cro ection veru polar angle for an acoutically rigid phere of radiu 1m, ubjected to an axially incident plane wave of k = 1rad/m, baed on Combined layer formulation (CLF). S Node 382 Node Exact X θ Figure 9: Scattering cro ection veru polar angle for an acoutically rigid phere of radiu 2m, ubjected to an axially incident plane wave of k = 1rad/m, baed on Combined layer formulation (CLF). Z q Y Firt, we conider the cae of a cube with ide length l = 1m. The cae of a cube preent a challenging tak of handling harp edge and corner. To obtain a triangular patch model, each ide of the cube i divided into 4 equal egment reulting in 96 quare patche and 98 node on the cube. By joining the diagonal, we get 192 triangular patche and 288 edge. Fig. 10 how the cattering cro ection S a a function of Θ. It i evident from the figure that the contant bai function defined over a node for SLF, DLF and CLF (α = 0.1 j) compare very well with that of edge baed SLF (288 edge) olution. The edge baed SLF olution i already validated in [Chandraekhar and Rao (2004)]. Alo, note that even though the bai function are defined on harp corner, the numerical reult are fairly accurate. Finally, we note that the node-baed conventional boundary integral method [Schuter and Smith (1985),.Schuter (1985), Seybert, Soenarko, Rizzo, and Shippy (1985), Malbequi, Candel, Rignot (1987)] need complex calculation to obtain accurate reult for thi geometry. A a next example, we conider the cae of a finite cylinder of height 2.0m and 1.0m radiu. The triangular patch modeling of the cylinder i obtained by dividing the length and circumference into 8 and 10 uniform egment, repectively, reulting in 80 rectangular patche. By joining the diagonal, we obtain 160 triangular patche. The cylinder i cloed on both end by circular dik which are modeled by and additional70triangularpatche each. Thu, intotalwe have 152node, 300patche and 450 edge for thi geometry. Fig. 11 how the cattering cro ection S a a function of Θ. For thi cae alo, we note good comparion for SLF, DLF and CLF

19 Node baed Method of Moment Solution to Combined Layer Formulation 261 S DLF-98 N CLF-98 N SLF-98 N Edge Sol θ Figure 10: Scattering cro ection veru polar angle for an acoutically rigid cube of length 1m, ubjected to an axially incident plane wave of k = 1rad/m, baedonslf,dlfandclf. S DLF-152 N CLF-152 N SLF-152 N Edge Sol θ Figure 11: Scattering cro ection veru polar angle for an acoutically rigid cylinder of radiu 1m and height 2m, ubjected to an axially incident plane wave k = 1 rad/m, baed on SLF, DLF and CLF. 0` (α = 0.1 j) reult with the edge baed SLF (450 edge) reult. Latly, we conider a cone of 1m height and 1m radiu. The cone ha a harp corner at the tip which i difficult to handle uing node-baed BIE method. Uing a imilar dicretization cheme a in the cae of finite cylinder, the cone i divided into 112 node, 220 patche and 330 edge. Fig. 12 how the cattering cro ection S for SLF, DLF and CLF (α = 0.1 j) a a function of Θ which compare very well with the edge baed SLF (330 edge) reult. 6.4 Elimination of ill-conditioning problem in CLF In thi ection the preence of ill-conditioning of the impedance matrice in the cae of SLF and DLF are preented along with the non-exitence of the ill-conditioning forthecaeofclf. Fig. 13 how the abolute value of logarithm of the invere condition number of the impedance matrix of the SLF, DLF and CLF for the cae of a phere with a radiu of 1m veru the wave number k. It i evident from the Fig that matrix become highly ill-conditioned when the frequency of the incident acoutic wave i cloe to or matche with the reonance frequency of cavity formed by the urface of the phere. The matrix become ill-conditioned when k = 3.19 and k = 2.1 for the cae of SLF and DLF repectively. However no uch phenomenon exit for the cae of CLF. The correponding theoretical value are around k = 3.14 and k = 2.08 which are the root of the Spherical Beel function for the Dirichlet

20 262 Copyright 2008 Tech Science Pre CMES, vol.33, no.3, pp , 2008 S DLF-112 N CLF-112 N SLF-112 N Edge Sol θ Figure 12: Scattering cro ection veru polar angle for an acoutically rigid cone of radiu 1m and height 1m, ubjected to an axially incident plane wave k = 1 rad/m, baed on SLF, DLF and CLF. N SLF DLF CLF k Figure 13: Condition number veru the frequency for an acoutically rigid phere of radiu 1m, ubjected to an axially incident plane wave k = 1rad/m, baedonslf,dlfandclf N SLF DLF CLF k Figure 14: Condition number veru the frequency for an acoutically rigid cube of ide length 1m, ubjected to an axially incident plane wave k = 1 rad/m, baed on SLF, DLF and CLF. and Neumann problem. The difference between the numerical reult and the theoretical value may be attributed to the approximation of the phere with the triangular patch modeling. Fig. 14 how the abolute value of logarithm of the invere condition number of the impedance matrix of the SLF, DLF and CLF for the cae of a cube with a ide length of 1m veru the wave number k. Fig how the SLF and DLF fail around

21 Node baed Method of Moment Solution to Combined Layer Formulation 263 k = 5.4andk = 3.0 and CLF produce a conditioned matrix at all frequencie. 7 Concluion In thi work, a novel numerical technique i preented, which implement the olution propoed by Burton and Miller in order to eliminate the reonance problem aociated with the boundary integral equation formulation of the exterior acoutic cattering problem. The SLF and DLF do uffer the ill-conditioning of the reulting impedance matrix which doe not exit in the cae of CLF. The CLF along with SLF and DLF are olved uing the method of moment olution and by defining the bai function on the node. Though traditionally, DLF i difficult to olve due to the preence of hyper-ingular kernel, the DLF i olved, in thi work, uing imple vector calculu technique to circumvent the problem of hyper-ingularity and by defining the bai function on the node. Node baed bai function are defined to reduce the ize of impedance matrix and hence the time required for olving the linear ytem of equation. Triangular patch modeling i ued to approximate the urface of the cattering body. The olution of SLF, DLF and CLF are validated with repect to the exact olution wherever available and edge baed olution wherever exact olution are not available. However, no comparion i made in thi paper on the accuracy of node baed veru edge baed cheme. It i clearly evident from the numerical reult that, cheme baed on node baed bai function converge toward the exact olution with higher number of node and have advantage over the edge baed bai function from the computational complexity perpective. Reference Amini, S.; Wilton, D.T. (1986): An invetigation of Boundary element method for the exterior acoutic problem. Computational Method in Applied Mechanical Engineering, vol 54, pp Arefmaneh, A.; Najafi, M.; Abdi, H. (2008): Mehle Local Petrov-Galerkin Method with Unity Tet Function for Non-Iothermal Fluid Flow. CMES: Computer Modeling in Engineering and Science, vol. 25, No. 1, pp Bowman, J.J.; Senior, T.B.A.; Ulenghi, P.L.E. (1969): Electromagnetic and Acutic Scattering by Simple Shape North-holland, Amterdam. Brakhage, H.; Werner, P. (1965); U ber da Dirichletche Auenraumproblem fu r die Helmholtzche Schwingunggleichung, Arch. Math., vol 16, Burton, A.J.; Miller, G.F. (1971): The application of integral equation method to the numerical olution of ome exterior boundary value problem. Proceeding of Royal Society, London, vol A323, pp

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25 Node baed Method of Moment Solution to Combined Layer Formulation 267 vol AP-32, pp Yan, Z.Y.; Cui, F. S.; Hung, K. C. (2005): Invetigation on the Normal Derivative Equation of Helmholtz Integral Equation in Acoutic. CMES: Computer Modeling in Engineering & Science, Vol. 7, No. 1, pp Zai You Yan (2006): Treatment of Sharp Edge/Corner in the Acoutic Boundary Element Method under Neumann Boundary Condition. CMES: Computer Modeling in Engineering & Science, vol. 13, No. 2, pp Zhang, Y.Y.; Chen, L. (2008): A Simplified Mehle Method for Dynamic Crack Growth, CMES: Computer Modeling in Engineering & Science, vol. 31, No. 3, pp

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