NUMERICAL ANALYSIS OF A COUPLED FINITE-INFINITE ELEMENT METHOD FOR EXTERIOR HELMHOLTZ PROBLEMS

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1 Journal of Computatonal Acoustcs, Vol. 14, No. 1 (2006) c IMACS NUMERICAL ANALYSIS OF A COUPLED FINITE-INFINITE ELEMENT METHOD FOR EXTERIOR HELMHOLTZ PROBLEMS JEAN-CHRISTOPHE AUTRIQUE LMS Internatonal, Interleuvenlaan 70 Researchpark Haasrode Z1, 3001 Leuven, Belgum jean-chrstophe.autrque@lms.be FRÉDÉRIC MAGOULÈS Unversté HenrPoncaré, Insttut Ele Cartan de Nancy BP 239, Vandoeuvre-les-Nancy Cedex, France frederc.magoules@ecn.u-nancy.fr Receved 1 May 2004 Revsed 1 September 2004 Coupled fnte-nfnte element computatons are very effcent for modelng large scale acoustcs problems. Parallel algorthms, lke sub-structurng and doman decomposton methods, have shown to be very effcent for solvng huge lnear systems arsng from acoustcs. In ths paper, a coupled fnte-nfnte element method s descrbed, formulated and analyzed for parallel computatons purpose. New numercal results llustrate the effcency of ths method for academc test cases and ndustral problems alke. Keywords: Infnte element; fnte element; parallel computng; acoustc scatterng; SYSNOISE. 1. Introducton The fnte element soluton of acoustc problems usually nvolves huge meshes snce the mesh sze should be proportonal to the frequency of the problem n order to have a good approxmaton of the soluton. So, the dscretzaton leads to an extremely large lnear system of equatons wth a sparse matrx. Ths becomes a crucal pont for acoustc scatterng problems where the doman around the scattered object s unbounded. If one wants to keep the sparsty of the matrx and reduce the number of unknowns of the lnear system, the nfnte element methods 1 3 are an effcent alternatve to the boundary element methods whch leads to a dense matrx, 4,5 or to the absorbng boundary condtons whch should be defned far enough from the object. 6 9 The accuracy of the nfnte element methods s lnked wth a parameter called the order of the nfnte element. The hghest ths order, the smallest the error between the approxmate soluton and the exact soluton. Unfortunately, Correspondng author. 21

2 22 J.-C. Autrque & F. Magoulès ncreasng the value of ths parameter tends to deterorate the condtonng number of the assembly matrx Addtonally the nfnte element methods can only be appled for convex objects. A remedy of ths drawback s to use a coupled fnte-nfnte element formulaton. The coupled method conssts of surroundng the object wth a convex envelope. The volume between the object and the convex envelope s meshed wth fnte elements, and nfnte elements are defned on the surface of the convex envelope. The teratve methods used to solve the lnear system of equatons arsng from the dscretzaton are very easy to program. 13 Precondtonng technques based on substructurng can addtonally be appled. 14 The doman decomposton methods for example, are based on a mesh parttonng of the global doman. Then the methods consst of solvng teratvely a lnear system defned at the nterface between the subdomans, and each teraton of the algorthm nvolves a drect soluton of an acoustc problem nsde each subdoman. Such methods are very well suted for dstrbuted parallel computng. In the case of a general mesh parttonng the nterface between the subdomans may have an nfnte length whch leads to some dffcultes to defne the absorbng boundary condtons at such nterfaces. In ths paper, a coupled fnte-nfnte element method s descrbed, formulated and analyzed for parallel computatons purposes. Ths method has been successfully mplemented n the SYSNOISE software, for solvng huge computatonal acoustc problems n parallel on hgh performance computers or on networks of PC s. Some numercal nvestgatons n unbounded domans, usng the SYSNOISE software, are presented to demonstrate the effcency and robustness of ths method. The scope of ths paper s as follows. Secton 2 descrbes the general scatterng problem analyzed n the followng. Then n Secs. 3.1 and 3.2 the fnte element methods and the nfnte element methods are remnded n an homogeneous formulaton. Then n Sec. 3.3 the couplng between nfnte and fnte elements s presented. Secton 4.1 presents the substructurng method followed n Sec. 4.2 by the nonoverlappng Schwarz method wth zeroth order absorbng boundary condtons. Some novel dscussons on the couplng between nfnte and fnte elements n a parallel computng context are nvestgated n Sec In Sec. 5, new numercal experments are presented on large computatonal acoustcs problems whch demonstrate the performance and robustness of the nonoverlappng Schwarz algorthm equpped wth zeroth order absorbng boundary condtons. Ths analyss nvestgates the dependency of the method upon dfferent parameters for general mesh parttonng. Both two dmensonal and three dmensonal analyss are performed on academc and ndustral test cases. The conclusons of our study are presented n Sec Mathematcal Formulaton A model radaton problem s consdered n an unbounded doman. The man motvaton for ths analyss s to determne the frequency response functons arsng from the vbratons of a structure. These vbratons can be caused by varous phenomena, lke a flud flow or a wave dffracton. In the followng the radaton of an object delmted by a boundary Γ N

3 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems 23 Ω e Γ N Fg. 1. Acoustc scatterng problem. mmersed n an unbounded doman Ω e, as shown n the Fg. 1 s consdered. Ths problem can be expressed as: u k 2 u =0 nω e u n = g on Γ N u r ku = O(1/r2 ) when r + (1) where g L 2 (Γ N ) s the prescrbed Neumann boundary condtons and k R + the wave number. The normal untary vector along the boundary Γ N s denoted by n,andr represents the radus n the sphercal coordnates. Equaton (1) s the Sommerfeld condton whch ensures the propagaton of the acoustc waves to nfnty. For the sake of smplcty the followng equatons are derved for the partcular case where Γ N s the untary sphere, but the general case does not lead to specal dffcultes as wll be demonstrated n the numercal experments. 3. Infnte and Fnte Element Methods 3.1. Fnte element method In summary, the fnte element method conssts of meshng the volume of a doman, for example wth hexahedra, and to dscretze the soluton n ths volume wth shape bass functons, for example wth Q 1 shape functons. 15, Problem defnton In the case of exteror acoustcs problems, the doman of nterest s unbounded and therefore cannot be meshed. The frst step of the fnte element method conssts of defnng a truncaton of the unbounded doman Ω e called Ω e γ as: Ω e γ =Ωe {x R 3 ; x <γ} where the artfcal boundary S γ (here, the sphere of radus γ>1) has been ntroduced. The doman Ω e γ s now bounded and can thus be meshed. An absorbng boundary condton s defned on the boundary S γ. The optmal dstance between the object and the artfcal

4 24 J.-C. Autrque & F. Magoulès boundary S γ wll be dependant upon the qualty of the absorbng boundary condton. The man motvaton s to avod the numercal reflectons of the wave on ths boundary. 7,17,6,9 The dffculty s that ncreasng the dstance between ths artfcal boundary and the object ncreases the number of elements of the mesh. In the followng a frst order approxmaton of the Sommerfeld boundary condton s appled on the boundary S γ. Our ntal expresson can now be reformulated usng the Sommerfeld boundary condton on the boundary S γ : u k 2 u =0 u r u n = g ku =0 ons nω e γ on Γ N γ where g L 2 (Γ N ) s the prescrbed Neumann boundary condtons Varatonal formulaton In the varatonal formulaton, the Helmholtz equaton s frst multpled by the complex conjugate of the test functon v (noted v). The ntegraton n the doman Ω e γ s then performed, and the Green formula s appled. The soluton u belongs to the space: H 1 (Ω e γ)={u : u 1 < } wth u 1 the norm assocated to the scalar product (u, v) 1 = u v dv+ Ω e γ Ω e γ uv dv where dv denotes the volume ntegraton. After substtuton of the Neumann boundary condton on Γ N and of the Robn boundary condton on S γ, the followng varatonal formulaton s obtaned: Fnd u H 1 (Ω e γ)suchas u v dv k 2 uv dv k uvds= gvds S γ Γ N Ω e γ Ω e γ for v H 1 (Ω e γ)andg L 2 (Γ N ), where ds denotes the surface ntegraton Dscretzaton In the cartesan coordnates system denoted by (x, y, z) n the current fnte element and by (ξ,η,ζ) n the reference fnte element, the approxmate soluton can be expressed n the form: n e u h (ξ,η,ζ) = a j N j (ξ,η,ζ) j=1

5 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems 25 Ω e γ Γ N S γ Fg. 2. Example of a fnte element mesh wth trangles. wth a j the n e unknown complex coeffcents assocated to the degree of freedom and N j the bass shape functons defned on the reference fnte element. For example n e =8nthe case of a dscretzaton wth Q 1 shape functons defned on hexahedra elements. Fgure 2 llustrates a fnte element mesh example. After dscretzaton of the varatonal formulaton, the followng lnear system s obtaned: Zu h = f where f s the rght hand sde, and Z the mpedance matrx. In the followng, the subscrpt fem denotes a dscretzaton wth fnte elements. If the degrees of freedom located nsde the volume Ω e γ and the degrees of freedom located on the boundary S γ are respectvely denoted by subscrpts and p, the lnear block matrx s obtaned: Z(fem) Z (fem) p Z (fem) p Z pp (fem) km R pp ( ) (fem) x = x (fem) p ( ) (fem) b b (fem) p where Z (fem) s the mpedance matrx equal to (K (fem) k 2 M (fem) )wthk (fem) the volume stffness matrx and M (fem) the volume mass matrx. The surface matrx M R arses from the Robn boundary condton defned on S γ. The fact that all these matrces are sparse s mportant to remember Infnte element method In summary, the nfnte element method conssts of meshng the surface of a convex object wth fnte elements and to extrude ths mesh to nfnte. The shape bass functons ncludes some classcal fnte elements shape functons and some shape functons ssue from the seres expanson of the Green functon. The method presented n the next secton s the one frst ntroduced n Ref. 1 and then reformulated and analyzed n Refs. 10 and 3.

6 26 J.-C. Autrque & F. Magoulès Varatonal formulaton The frst step conssts of defnng a truncaton of the unbounded doman Ω e called Ω e γ followng a smlar approach to that ntroduced n Sec For the partcular case where the unbounded doman s the exteror of the untary sphere, an annulus s obtaned: Ω e γ = {x R3 ;1< x <γ}. After the multplcaton of the Helmholtz equaton by the complex conjugate of the test functon v, the applcaton of the Green formula, and applyng the Neumann boundary condton on Γ N, we obtan: u vdv k 2 Ω e γ Ω e γ uvdv S γ u r vds= gvds. Γ N The Sommerfeld condton Eq. (1) can be expressed as: u r = ku + φ where φ = O(1/r 2 ) s an unknown functon. After substtuton n the varatonal formulaton (because u/ n = u/ r), the equaton can be rewrtten as: u vdv k 2 uvdv k uvds= gvds+ φvds. Ω e γ Ω e γ S γ Γ N S γ The second steps consst of takng the lmt of the prevous expresson when γ tend to nfnty. The Atknson Wlcox results 11 shows that the leadng term of the soluton u s of the form e kr /r. As a consequence, u and u can no longer be ntegrated to nfnty over L 2. The dea conssts of usng specal shape functons of the form O(1/r 3 ). Ths helps to consder the prevous ntegral as Lebesgue ntegral. Wth ths choce, the ntegral on S γ wth φ vanshes when γ tends to nfnty. The problem s that the ntegral on S γ wth u vanshes too. In other words, ths partcular choce of the tests functons does not allow to keep the Sommerfeld condton n the varatonal formulaton. An dea proposed n Ref. 18 conssts of ntroducng the Sommerfeld condton drectly n the defnton of the space. The soluton u belongs to the Sobolev weghted space: Hw 1,+ (Ω e )={u : u + 1,w < } wth u + 1,w the norm assocated to the scalar product ( ) ( ) u v (u, v) + 1,w Ω = w u vdv + wuvdv + e Ω e Ω e r ku r kv dv. Two common weghts are of nterest, w =1/r 2 and the dual weght w = r 2.Wththese notatons, the varatonal formulaton can be wrtten: Fnd u Hw 1,+ (Ω e )suchas u vdv k 2 uvdv = gvds Ω e Ω e Γ N for v H 1,+ w (Ωe )andg L 2 (Γ N ).

7 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems Dscretzaton A complete overvew of the nfnte element methods can be obtaned n Refs. 12 and 11. The exact soluton n the sphercal coordnates system (r, θ, ϕ) of the current nfnte element can be expended n the form (Atknson Wlkox): u(r, θ, ϕ) =e kr κ=1 G κ (θ, ϕ, k) r κ. (3) Ths seres converges for r > γ. Consderng only the frst m termsofthsseres,and expressng these terms n the coordnates systems (ξ,η,r) of the reference nfnte element leads to the approxmate soluton: u h (ξ,η,r) =e kr m µ=1 G µ (ξ,η,k) r µ where the functons G µ (ξ,η,k) are defned by: n G µ (ξ,η,k) = Q ν,µ (k) N ν (ξ,η) ν=1 wth n an nteger defned below. After substtuton, the followng expresson s obtaned: n e u h (ξ,η,r) = a j N j (ξ,η,r) j=1 wthn j (ξ,η,r) =N ν (ξ,η) N µ (r) for ν =1,...,n, µ =1,...,m, n e = n m,andwtha j the n e unknowns complex coeffcents assocated to the degree of freedom. The shapes functons N j are defned on the reference nfnte element: N ν denotes the angular functons wth a total number of n and N µ the radal functons wth a total number of m. Thentegerm s called the order of the nfnte element. The hghest ths order, the smallest the error between the approxmate soluton and the exact soluton. Fgure 3 llustrates an nfnte element mesh example. The lnear system ssue from the dscretzaton s the followng: Zu h = f where f s the rght hand sde, and Z the mpedance matrx. In the followng, the subscrpt fem denotes a dscretzaton wth nfnte elements. If the degrees of freedom located outsde the object,.e. n the doman Ω e, and the degrees of freedom located on the boundary of the object Γ N are respectvely denoted by subscrpts and p, the lnear block matrx s

8 28 J.-C. Autrque & F. Magoulès e Ω Γ N Fg. 3. Example of an nfnte element mesh. obtaned: Z(fem) Z (fem) p Z (fem) p Z pp (fem) ( (fem) ) x = x (fem) p ( (fem) ) b b (fem) p where Z (fem) s the mpedance matrx equal to (K (fem) k 2 M (fem) )wthk (fem) the volume stffness matrx and M (fem) the volume mass matrx. It s mportant to pont out that all these matrces are sparse matrces Coupled fnte-nfnte element method The coupled fnte-nfnte element method conssts of surroundng the object wth a convex envelope. The volume between the object and the convex envelope s meshed wth fnte elements and nfnte elements are defned on the surface of the convex envelope, as shown n Fg. 4. Ths approach s mandatory f ones want to use nfnte elements for nonconvex objects, lke a submarne for example. Indeed Eq. (3) s not vald anymore f the surface of the object s nonconvex. The soluton s then dscretzed wth fnte elements bass shape functons nsde the volume between the object and the envelope and wth nfnte elements bass shape functons outsde the envelope. The lnear system can be expressed as: Zu h = f where f s the rght hand sde, and Z the mpedance matrx. In the followng, the subscrpt fem and fem denotes a dscretzaton wth fnte elements or wth nfnte elements respectvely. If the degrees of freedom located n the doman between the object and the convex envelope, then the degrees of freedom located outsde the convex envelope, and fnally the degrees of freedom located on the convex envelope are respectvely numbered, the lnear

9 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems 29 Γ N Fg. 4. Coupled fnte-nfnte element mesh. block matrx s obtaned: Z (fem) 0 Z (fem) p 0 Z (fem) Z (fem) p Z (fem) p Z (fem) pp Z (fem) p + Z (fem) pp x (fem) x (fem) x p = b (fem) p b (fem) b (fem) + b (fem) p where the above mentoned matrces have been defned n the prevous secton. The prevous numberng of the degrees of freedom s very smlar to the sub-structurng methodology, as presented n the followng secton. 4. Parallel Computng 4.1. Sub-structurng methods Let us now consder n detal a number of algorthms to solve the lnear system Zu h = f effcently on parallel computers. The followng dscretzaton scheme s presented for a decomposton of a general doman Ω nto two subdomans Ω (1) and Ω (2) wth an nterface Γ as shown n Fg. 5. The doman Ω s meshed wth fnte elements only. The degrees of freedom located nsde subdoman Ω (s), s =1, 2 and on the nterface Γ are denoted by subscrpts and p. Wth ths notaton the contrbuton of subdoman Ω (s), s =1, 2tothe mpedance matrx and to the rght-hand sde can be wrtten as n Refs. 19 and 20: Z (s) = Z(s) Z (s) p Z (s) p Z pp (s), b (s) = ( (s) ) b, s =1, 2. b (s) p

10 30 J.-C. Autrque & F. Magoulès Ω (1) Ω (2) Fg. 5. Nonoverlappng doman splttng. Γ The global problem s a block system obtaned by assemblng the local contrbutons from each subdoman: Z (1) 0 Z (1) p 0 Z (2) Z (2) p Z (1) p Z (2) p Z pp x (1) x (2) x p = b (1) b (2) b p. (4) The matrces Z pp (1) and Z pp (2) represent the nteracton matrces between the nodes on the nterface obtaned by ntegraton on Ω (1) andonω (2). The block Z pp s the sum of these two blocks. In a same way the term b p = b (1) p + b (2) p s obtaned by local ntegraton of the rght hand sde over each subdoman and the summaton on the nterface. In order to solve ths lnear system wth an teratve method, a matrx vector product of the matrx Z by a descent drecton vector w =(w (1),w (2),w p ) T should be computed at each teraton. Ths matrx vector product can be performed usng the prevous sub-structurng expresson n two successve steps: Computaton of local matrx vector product n each subdoman: ( ) (1) v = Z(1) Z (1) p v p (1) Z (1) p Z pp (1) ( ) (1) w, w p Assembly of the vectors on the nterface: v p = v p (1) + v p (2) ( ) (2) v = Z(2) Z (2) p v p (2) Z (2) p Z pp (2) ( ) (2) w. w p whch gves the vector v =(v (1),v (2),v p ) T. In the case of a general mult-doman mesh splttng, addng the contrbutons of the local dot products wll ntroduce a weghtng factor per node n the dot product equal to the number of subdomans the node belongs to. A weghtng vector on the nterface must be ntroduced n order to avod havng to consder multple contrbuton of the vector component at such cross ponts.

11 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems Doman decomposton methods The prevous sub-structurng method requres at each teraton a local matrx vector product, whch s computed n parallel n each subdoman, and assembly of the vectors on the nterface between the subdoman. Ths method s very easy to mplement, 13 but the convergence may be dffcult to acheve n the case of large acoustc problems wthout precondtonng technques. In order to mprove the convergence speed of the teratve algorthm, some precondtonng technques based on doman decomposton methods are an effcent alternatve. The nonoverlappng Schwarz method for example nvolves an teratve method (performed on the degrees of freedom located on the nterface) and a local matrx factorzaton (on the degrees of freedom located nsde each subdoman). At each teraton a local forward backward substtuton s nvolved n each subdoman, and assembly on the nterface. Ths algorthm s based on the followng theorem. Theorem 4.1. Under a splttng of the form Z pp = Z pp (1) + Z pp (2) and b p = b (1) p + b (2) p, for all matrces A (1),A (2) there s one and only one assocated value λ (1),λ (2) such as the followng coupled problems: Z(1) Z (1) p Z (1) p Z (1) pp + A (1) Z(2) Z (2) p Z (2) p Z (2) pp + A (2) ( (1) ) x = x (1) p ( (1) ) b b (1) p + λ (1) (5) ( ) ( ) (2) (2) x b = x (2) p b (2) p + λ (2) (6) x (1) p x (2) p =0 (7) λ (1) + λ (2) A (1) x (1) p A (2) x (2) p =0 (8) are equvalent to the problem (4). Proof. The admssblty condton (7) derves from the relaton x (1) p = x (2) p = x p. If x (1) p = x (2) p = x p, the frst rows of local systems (5) and (6) are the same as the two frst rows of the global system (4), and addng the last rows of the local systems (5) and (6) gves: Z (1) p x(1) + Z (2) p x(2) + Z pp x p b p = λ (1) + λ (2) A (1) x (1) p A (2) x (2) p. So, the last equaton of global system (4) s satsfed only f: λ (1) + λ (2) A (1) x (1) p A (2) x (2) p =0. Conversely, f x (1) p, x (2) p and x p are derved from the global system (4), then the local systems (5) and (6) defne λ (1) and λ (2) n a unque way. The complete nonoverlappng Schwarz algorthm conssts of searchng teratvely for the value of (λ (1),λ (2) ) T such as the value of (x (1) p,x (2) p ) T satsfy Eqs. (7) and (8). The only

12 32 J.-C. Autrque & F. Magoulès restrcton mposed on the matrces A (1) and A (2) n the prevous theorem s that for a gven rght hand sde the local sub-problems defned n Eqs. (5) and (6) have an unque soluton. The elmnaton of x (1) and x (2) n favor of x (1) p and x (2) p n the prevous equatons leads to the followng lnear system: ( I I (A (1) + A (2) )[S (1) + A (1) ] 1 ( (A (1) + A (2) )[S (2) + A (2) ] 1 c (2) ) p = (A (1) + A (2) )[S (1) + A (1) ] 1 c (1) p )( ) I (A (1) + A (2) )[S (2) + A (2) ] 1 λ (1) where S (q) = Z pp (q) Z (q) p [Z(q) ] 1 Z (q) p s the condensed matrx and c(q) p = b (q) p Z (q) p [Z(q) ] 1 b (q) s the condensed rght hand sde, for q =1, 2. Ths lnear system s solved wth an teratve method, and each teraton nvolves a soluton of an Helmholtz sub-problem n each subdoman. The choce of the matrces A (1) and A (2) has a strong nfluence on the convergence speed of the nonoverlappng Schwarz algorthm. Dfferent choce of these matrces has been nvestgated n Refs In the followng the matrces A (1) and A (2) are obtaned from a Taylor zeroth order approxmaton of the Steklov Poncaré operator and from an optmzed zeroth order approxmaton of the Steklov Poncaré operator for nternal acoustcs problems dscretzed wth fnte elements, as ntroduced n Ref. 23. These matrces are equal to I λ (2) (9) A (1) := αm Γ, A (2) := αm Γ where α s equal to k for a Taylor zeroth order approxmaton and obtaned from the soluton of a mnmzaton problem for an optmzed zeroth order approxmaton. 24 The matrx M Γ s a surface mass matrx defned on the nterface between the subdomans Couplng fnte and nfnte element When a general mesh parttonng of the global doman s performed, the nterface jons some (fnte or nfnte) elements sharng a common edge on the nterface and belongng to dfferent subdomans. Three possbltes may appear: two fnte elements sharng an edge on the nterface, or one fnte element and one nfnte element sharng an edge on the nterface, or two nfnte elements sharng an edge on the nterface. In ths last case the length of the nterface s nfnte. If some Lagrange fnte elements are consdered, for example P 1 -fnte elements, the degrees of freedom of an element corresponds to the nodes of the trangle. Defnng the Lagrange multplers at the nodes of the fnte element helps to apply the sub-structurng methodology descrbed Sec Fgure 6 shows the defnton of the degrees of freedom and of the Lagrange multplers for two fnte elements sharng one edge on the nterface. In the second case, the Lagrange multplers should be defned at the element nodes as shown n Fg. 7. Indeed n ths case, the restrcton on the edge of the angular bass functons

13 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems 33 Fg. 6. Defnton of the degree of freedom (whte and black bullets) of the elements and of the Lagrange multplers (black bullets) between two fnte elements. Fg. 7. Defnton of the degree of freedom (whte and black bullets) of the elements and of the Lagrange multplers (black bullets) between an nfnte element and a fnte element. of the nfnte element s smlar to the restrcton of the P 1 -fnte element bass functons. As a consequence, there s no dfference between ths case and the prevous one. In the thrd case, the Lagrange multplers should be defned at the element nodes and at the Gauss ponts of the nfnte elements as shown n Fg. 7. These Gauss ponts correspond to the degree of freedom of the nfnte element and are used to compute the ntegrals of Sec Increasng the order of the nfnte element mples ncreasng the number of Gauss ponts and so far the number of Lagrange multplers. As a consequence the sze of the lnear system defned Eq. (9) becomes much bgger. A second consequence s that ncreasng the order of the nfnte element tends to deterorate the condtonng number of the assembly matrx. In summary, the Lagrange multplers are smply defned on the degrees of freedom. Ths can be the nodes of the elements (for fnte elements) or the Gauss ponts (for the nfnte elements). If zeroth order absorbng boundary condtons are consdered n the nonoverlappng Schwarz algorthm, a surface mass matrx should be computed on the nterface between the subdoman. Ths matrx s of the form: M Γ = uv ds. Γ In the case of an nterface between two fnte elements, the coeffcents of the matrx M Γ are computed as: [M Γ ] lm = N l N m ds Ω (1) Ω (2)

14 34 J.-C. Autrque & F. Magoulès Fg. 8. Defnton of the degree of freedom (whte and black bullets) of the elements and of the Lagrange multpler (black bullets) between two nfnte elements. where N l and N m are the fnte element shape functons assocated wth node l and node m on the common edge on the nterface between subdomans Ω (1) and Ω (2).Inthecaseof an nterface between one fnte element and one nfnte element, the fnte element shape functons on the common edge s smlar to the angular nfnte element shape functon.e. the functons N ν. When two nfnte elements share a common edge, the ntegral along ths nfnte edge only nvolves the radal shape functons.e. the functons N µ,andthentegral s computed usng the Gauss ponts along the nfnte edge. 5. Numercal Experments 5.1. Radaton of an nfnte cylnder In ths secton the convergence propertes of the parallel teratve GMRES precondtoned by the dagonal versus the nonoverlappng Schwarz method are analyzed. The behavor of these methods upon dfferent parameters s nvestgated. The test case consst of a mult-pole radaton of an nfnte cylnder of radus a. Dueto the symmetry of the geometry, only one half cross secton s consdered for the analyss. The radaton of the cylnder s generated by the vbraton of the surface. These vbraton are modeled by a normal acceleraton of the partcles along the surface. The normal velocty dstrbuton s defned by the relaton V n (θ) = V cos(pθ) whereθ denotes the angle n cylndrc coordnates and where p =0, 1, 2...for a mult-pole of order 0, 1, 2...An artfcal boundary s defned on an nfnte cylnder of radus 1.5a. The volume between the cylnder and the artfcal boundary s meshed wth quadrlateral fnte elements. Infnte elements are defned on the surface of the artfcal boundary. Because of the order p of the multpole, the order of the nfnte elements should be at least equal to m = p +1, see Ref. 3. The sx elements per wavelength crtera s ensured over all the mesh presented n Fg. 9. The doman s then splt n subdomans wth a geometrc based algorthm, n such a way that each subdoman has at most two neghborng subdomans as shown n Fg. 10. The mesh parttonng software ensures a load balancng dstrbuton of the degree of freedom

15 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems 35 Fg. 9. Radaton of an nfnte cylnder: Fnte element mesh. Fg. 10. Radaton of an nfnte cylnder: Mesh parttonng. n each subdoman. Ths decomposton has frst the advantage of reducng the numercal error by ensurng that the nterfaces between the subdomans are parallel to the cylnder. Secondly, ths decomposton presents the advantage of collectng all the nfnte elements n the same subdoman. The acoustc soluton of a mult-pole of order four s presented n Fg. 11. The parameters ndcated are the radus of the nfnte cylnder a, the mesh sze h, the wave number k, the order of the nfnte element m, the order of the mult-pole p, and the number of subdomans N s, respectvely. Fg. 11. Radaton of an nfnte cylnder: Acoustc pressure.

16 36 J.-C. Autrque & F. Magoulès The parallel teratve GMRES precondtoned by the dagonal and the nonoverlappng Schwarz method have been mplemented n the SYSNOISE software for tral purpose. The condensed nterface problem of the nonoverlappng Schwarz algorthm s solved wth the GMRES algorthm, and local Crout factorzatons are performed n each subdoman. The CPU tme ndcated for the nonoverlappng Schwarz algorthm s the total CPU tme, ncludng the factorzaton of the matrx. The convergence s analyzed wth the followng stoppng crtera Zu h f L f L2 where f L2 denotes the module of the complex number ff. The numercal smulaton are performed on a SGI Orgn 200 wth four processors. As expected from the theory, the convergence speed of the nonoverlappng Schwarz algorthm s weakly dependent upon the mesh sze, see Table 1. On the contrary the parallel GMRES precondtoned by the dagonal presents a strong dependance upon ths parameter. As already reported for nternal acoustc problems, 24 the nonoverlappng Schwarz algorthm performs up to 35% better wth an optmzed zeroth order (OO0) absorbng boundary condtons than wth a Taylor zeroth order (TO0) absorbng boundary condtons. The results presented n Table 2 llustrate the dependence upon the wave number. Once agan, the good convergence propertes of the nonoverlappng Schwarz method wth zeroth order absorbng boundary condtons can be notced. Snce the number of teratons of the precondtoned GMRES does not depend upon the number of subdomans resultng from the mesh parttonng, the results reported n Table 3 may appear dsappontng. However, ncreasng the number of subdomans n the GMRES method ncreases the number of data exchange between the processors, and each teraton requres more tme. For ths reason the nonoverlappng Schwarz algorthm s stll very compettve. Fnally the results reported n Table 4 llustrate the dependence of the methods upon the order of the nfnte element. Snce all the nfnte elements are collected n the same subdomans, and because the nonoverlappng Schwarz algorthm nvolves a drect soluton nsde each subdomans, the dependence s very weak. Ths dependence even dsappears Table 1. Number of teratons versus the mesh sze parameter for the radaton problem. The mult-pole order s equal to p = 2, the wave number equal to ka = 20, and the order of the nfnte element equal to m = 3. A total number of N s = 4 subdomans have been used for the smulaton. GMRES wth Dag. Prec. Schwarz wth TO0 Schwarz wth OO0 h # Iteratons CPU # Iteratons CPU # Iteratons CPU 1/ sec sec sec 1/ sec sec sec 1/ sec sec sec

17 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems 37 Table 2. Number of teratons versus the wave number parameter for the radaton problem. The mult-pole order s equal to p = 2, the mesh sze parameter equal to h =1/40, and the order of the nfnte element equal to m =3.A total number of N s = 4 subdomans have been used for the smulaton. GMRES wth Dag. Prec. Schwarz wth TO0 Schwarz wth OO0 ka # Iteratons CPU # Iteratons CPU # Iteratons CPU sec sec sec sec sec sec sec sec sec sec sec sec Table 3. Number of teratons versus the number of subdomans for the radaton problem. The mult-pole order s equal to p = 2, the mesh sze parameter equal to h =1/40, the wave number equal to ka = 40, and the nfnte element order parameter equal to m =3. GMRES wth Dag. Prec. Schwarz wth TO0 Schwarz wth OO0 N s # Iteratons CPU # Iteratons CPU # Iteratons CPU sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec Table 4. Number of teratons versus the nfnte element order for the radaton problem. The mult-pole order s equal to p = 2, the mesh sze parameter equal to h =1/40, the wave number equal to ka = 40. A total number of N s = 4 subdomans have been used for the smulaton. GMRES wth Dag. Prec. Schwarz wth TO0 Schwarz wth OO0 m # Iteratons CPU # Iteratons CPU # Iteratons CPU sec sec sec sec sec sec sec sec sec sec sec sec when the nonoverlappng Schwarz algorthm s equpped wth an optmzed zeroth order absorbng boundary condtons Acoustc scatterng In ths secton a three dmensonal acoustc scatterng problem where the obstacle has the shape of a submarne s analyzed. The length of the submarne s equal to 76 meters,

18 38 J.-C. Autrque & F. Magoulès the heght equal to 9.25 meters and the dameter equal to 7.5 meters. The characterstc of the ocean are a densty equal to 1000 kg/m 3 and a sound speed equal to c = 1500 m/s. The goal of ths analyss conssts of evaluatng the frequency response functons generated by the vbraton of the structure of the submarne ssue from the scatterng of an ncdent wave. The computng steps can be expressed as the followng sequence: An ncdent planar wave s defned n the ocean and strkes the submarne. A coupled flud-structure computaton s performed. The flud s dscretzed wth boundary elements and the structure of the submarne s dscretzed wth shell fnte element. The soluton,.e. the acoustc pressure for the flud and the dsplacement for the structure, of ths coupled problem are obtaned for dfferent frequences. An acoustc computaton s then performed. The ocean around the submarne s dscretzed wth coupled fnte-nfnte elements. An ellpsod s defned around the submarne and the volume between the submarne and the ellpsod s meshed wth fnte elements. Infnte elements are defned on the surface of the ellpsod. The crtera of sx nodes per wavelength s satsfed over all the mesh. The fnal mesh s composed wth nodes, tetrahedra fnte elements and nfnte elements. Usng the dsplacement of the structure of the submarne gven by the flud-structure problem as the boundary condtons of the acoustc problem, the acoustc pressure can be obtaned for dfferent frequences. The flud-structure computaton s performed wth the MSC-NASTRAN software. The acoustc problem s solved wth the SYSNOISE software equpped wth the nonoverlappng Schwarz method. Fgure 12 shows the shape of the submarne, whlst Fg. 13 shows the fnte element mesh of the volume between the submarne and the ellpsod. Two examples of mesh parttonng are presented n Fgs. 14 and 15. These two mesh parttonngs generate load balancng subdomans. Fgure 14 presents a geometrc based mesh parttonng. In ths case all the nfnte elements are located n the same subdoman. The couplng between the only subdoman wth all the nfnte elements and the only neghborng subdoman wth only fnte elements becomes smlar to the couplng between two subdomans wth only fnte elements. Fgure 15 presents a mesh parttonng performed wth the METIS software. 25,26 In ths case, the mesh parttonng generates subdomans whch can share a common nfnte nterface. Fgure 16 shows the acoustc pressure n decbel n the ocean around the Fg. 12. Submarne acoustc problem: Geometry.

19 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems 39 Fg. 13. Submarne acoustc problem: Fnte element mesh. Fg. 14. Submarne acoustc problem: Geometrc based mesh parttonng. Fg. 15. Submarne acoustc problem: METIS mesh parttonng.

20 40 J.-C. Autrque & F. Magoulès Fg. 16. Submarne acoustc problem: Acoustc pressure n decbel. Fg. 17. Submarne acoustc problem: Comparson of frequency responses functon for a soluton computed wth coupled fnte-nfnte element versus a soluton computed wth boundary element. submarne for a frequency equal to 10 Hz. Fgure 17 represents the accuracy of the coupled fnte-nfnte elements soluton compared to the soluton computed wth boundary elements. An nfnte element order equal to three s mandatory n order to ensure the same accuracy between the coupled fnte-nfnte element computaton, and the boundary element computaton. Bearng n mnd that ncreasng the frequency requres a fner mesh and wll ncrease the dmenson of the dense matrx ssued from the boundary element method. For such hgh frequences, the coupled fnte-nfnte element, whch keeps the sparsty of the matrx, s defntely a good alternatve to the boundary element method.

21 A Coupled Fnte-Infnte Element Method for Exteror Helmholtz Problems 41 Tables 5 and 6 present the convergence results for a frequency equal to 47 Hz and a stoppng crtera equal to The frst table consder a mesh parttonng based on a geometrc algorthm, and the second table consder a mesh parttonng wth METIS. In Table 5, smlar propertes than n the prevous subsecton can be notced. The GMRES algorthm s not presented here snce for ths smulaton ths algorthm would requre more than 1000 teratons and more than 3600 seconds CPU tme, compared to 423 seconds CPU tme for the nonoverlappng Schwarz algorthm wth an optmzed zeroth order absorbng boundary condtons. Table 5. Number of teratons for dfferent number of subdomans and dfferent nfnte element order for the submarne acoustc problem. The wave number s equal to ka =0.2, and the mesh parttonng s based on a geometrc algorthm. Schwarz wth TO0 Schwarz wth OO0 N s m # Iteratons CPU # Iteratons CPU sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec sec Table 6. Number of teratons for dfferent number of subdomans and dfferent nfnte element order for the submarne acoustc problem. The wave number s equal to ka =0.2, and the mesh parttonng s obtaned wth METIS. Schwarz wth TO0 Schwarz wth OO0 N s m # teratons # teratons

22 42 J.-C. Autrque & F. Magoulès The results presented n Table 6 show the dependence of the nonoverlappng Schwarz algorthm upon the number of subdomans and upon the order of the nfnte element for a general mesh parttonng. 6. Conclusons In ths paper, a revew of the fnte element method and of the nfnte element method s frst presented. Then the coupled fnte-nfnte element method s descrbed n detal. Ths coupled method s nterestng for solvng acoustc scatterng problems n unbounded doman nvolvng nonconvex scattered objects. The descrpton of two parallel algorthms mplemented n the SYSNOISE software s then presented. The frst algorthm conssts of a parallel precondtoned teratve method. The second algorthm conssts of a parallel nonoverlappng Schwarz method wth absorbng boundary condtons defned on the nterface between the subdomans. The defnton of these absorbng boundary condtons n the case of a fnte and/or an nfnte nterface s analyzed. Then the parallel precondtoned teratve method and the parallel nonoverlappng Schwarz method wth zeroth order absorbng boundary are compared. A wde range of numercal experments are studed for computatonal acoustcs scatterng problems n unbounded domans that demonstrate the performance and robustness of the nonoverlappng Schwarz method wth zeroth order absorbng boundary condtons. Acknowledgments The authors would lke to acknowledge J.-P. Coyette and K. Gerdès for the useful dscussons, comments and remarks on ths paper. References 1. J. Astley, J. Macaulay and J. P. Coyette, Mapped wave envelope elements for acoustcal radaton and scatterng, J. Sound and Vbraton 170(1) (1994) D. Burnett, A 3-d acoustc nfnte element based on a generalzed multpole expanson, J. Acoust. Soc. Am. 96(5) (1994) K. Gerdes, Soluton of the 3D Laplace and Helmholtz equaton n exteror domans of arbtrary shape usng hp-fnte-nfnte elements, PhD thess, Unversty of Texas at Austn (1996). 4. G. Chen and J. Zhou, Boundary Element Methods (Academc Press London, 1992). 5. M. Costabel, Symmetrc methods for the couplng of fnte elements and boundary elements, n Boundary Elements IV, Vol. 1, Comput. Mech. (Breber, Southampton, 1987), pp B. Engqust and A. Majda, Absorbng boundary condtons for the numercal smulaton of waves, Math. Comp. 31(139) (1977) L. Halpern, Absorbng boundary condtons for the dscretzaton schemes of the one dmensonal wave equaton, Math. Comp. 38 (1982) D. Gvol, Hgh-order non-reflectng boundary condtons wthout hgh-order dervatves, J. Comp. Phys. 170 (2001) D. Gvol and B. Neta, Hgh-order non-reflectng boundary condtons for dspersve waves, Wave Moton 37 (2003)

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