Fig. Division of unbounded domain into closed interior domain and its eterior domain. Zienkiewicz [5, 6] Burnett [7, 8] [3] The conjugated Ast

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1 7 6 pp Rz; 43.4.Rj * 3 3 Unbounded problems, Finite element method, Infinite element, Hybrid variational principle, Fourier series. Boundary Element Method: BEM BEM Finite Element Method: FEMBEM FEM [, ] ρc Sommerfeld FEM Fourier-epansion hybrid-type infinite element for finite element analysis of nonaisymmetric sound fields in aisymmetric unbounded domains, by Toshiya Samejima Perfectly Matched Layer PML Fig. Dirichletto-Neumann DtN FEM FEM FEM FEM R k kr Bessel DtN FEM [3] Bettess Zienkiewicz [4] FEM

2 Fig. Division of unbounded domain into closed interior domain and its eterior domain. Zienkiewicz [5, 6] Burnett [7, 8] [3] The conjugated Astley-Leis formulation [9 ] Zienkiewicz [5, 6] Burnett [7, 8] FEM [3, 4] FEM [5, 6] 3 [7] 3 [8, 9] FEM []

3 637 z Ω F Ω C, Γ v, Γ z Π F π ( φ) rdsdϕ G F π k φ rdsdϕ G F π φ + φrdldϕ (5) n F Lv+Lz Fig. Aisymmetric closed sound field and its eterior sound field... Fig. (r, ϕ, z) z Ω F Ω I Ω F Ω I Ω C φ + k φ () φ k ep(jωt) () [8, 9] Π Π F + Π I () Π F ( φ) dv k φ dv Ω F Ω F φ + φds (3) Γ v+γ z n F Π I φ φ ds φ φ ds (4) n I n I Ωc Ωc ( )Γ v, Γ z Ω F φ/ n F n F Ω C Ω F n I Ω C Ω I φ Ω C (3)(4) G F Ω F G C Ω C G F L v Γ v G F L z Γ z G F Π I π Gc π Gc φ φ n I rdldϕ φ φ n I rdldϕ (6) δπ F δπ I FEM δπ F φ ϕ φ φ m cos(mϕ), φ m i N i φ mi Nφ m (7) FEM [ ] π [ ω M +jωc +(K r +K z +K m )]φ m π f m (8) M c G F N T NrdS C ρ Lz N T N Z rdl N K r GF T r N K z GF T z N r rds N z rds K m m G F N T N r ds f m Lv N T v m rdl N N i i φ m φ mi i Z Γ z c ρ Neumann m

4 m > v m Γ v v ϕ Ω F (6) Π I. (6) φ ϕ φ φ m cos(mϕ) (9) φ m G C φ m φ mi Ñi φ m i Ñ i φ mi Lφ m () φ φ Lφ m cos(mϕ) () L Ñi i Ω I φ (R, θ, ϕ) φ β nm h () n (kr)pn m (cos θ)cos(mϕ) nm U m β m cos(mϕ) () U m h () m+i (kr) P m m+i (cos θ) i β m β m+i,m i h () n ( ) Hankel Pn m ( ) Legendre Ω I n I r, z (n r,n z ) φ n I φ n I φ n r r + n φ z z n rr + n z z φ R R + n rz n z r φ R θ R m β m cos(mϕ) (3) R m (n rr + n z z) R U m R + (n rz n z r) U m R θ.3 ()()(3) (6) Π I Π I π U m β m cos(mϕ) m π m Gc R m β m cos(m ϕ)rdldϕ Gc Lφ m cos(mϕ) R m β m cos(m ϕ)rdldϕ [ π βt mh m β m βmg T m φ m ] (4) H m [QT m + Q m ], Q m RmU T m rdl (5) G m Gc Gc R T mlrdl (6) (4) Π I β m β m Hm G m φ m (4) [ ] π Π I φt mkmφ I m, Km I G T mhm G m (7) δπ I π Kmφ I m (8) Ω F Ω I FEM δπ F (8) δπ I (8) π [ ω M + jωc +(K r + K z +K m + Km) I ] π φ m f m (9) Km I (5)(7) FEM

5 639 Ñi Ñi L L ( ξ)/ (+ξ)/ } () R, θ (R,θ ), (R,θ ) L R LR, R } T R R () Fig. 3 Hybrid-type infinite element. θ Lθ, θ θ θ } T () r, z R, θ () K m I G C.4 H m G m Ω I ϕ () Ω I Km I [] [5] Ω I (4) Π I G C Ω I ϕ [6] Ω I Km I Fig. 3 G C 3 4, G F 3, 4 Ω I (5)(6) 3 4 () ξ( +) r R sin θ, z R cos θ (3) Q m i j q ij ξ q ij l 3 + (nr sin θ + n z cos θ)kh () m+i (kr) Pm+i (cos m θ)r sin θ (n r cos θ n z cos θ)h () m+i (kr) sin θpm+i (cos m θ) } h () m+j (kr)p m+j (cos m θ)dξ (4) G m i g i i g i ξ g i() l 3 + (nr sin θ+n z cos θ)kh () m+i (kr) Pm+i (cos m θ)r sin θ (n r cos θ n z cos θ)h () m+i (kr) sin θpm+i (cos m θ) } (+)ξ dξ (5) () 3 Ñi L L h () (kr)/h() (kr ) } (6) 3 (n r r + n z z), (n r z n z r) R (7) Q m i j q ij R

6 q ij h () m+i (kr)sin θ Pm+i (cos m θ ) R h () m+j (kr)p m+j (cos m θ )dr sin θ Pm+i (cos m θ )Pm+j (cos m θ ) I(m + i,m+ j,r ) (8) I(n, m, a) a h () n (kr)h () m (kr)dr.5 G m i g i L g i i g i R g i R h () m+i (kr)sin θ P m m+i (cos θ ) h() (kr) h () (kr ) dr sin θ Pm+i (cos m θ ) h () (kr ) I(m + i,,r ) (9) (3) 4 3 L L h () (kr)/h() (kr ) } (3) (n r r + n z z), (n r z n z r) R (3) q ij g i q ij sin θ Pm+i (cos m θ )Pm+j (cos m θ ) I(m + i,m+ j,r ) (3) g i sin θ Pm+i (cos m θ ) h () (kr ) I(m + i,,r ) (33) L g i.5 I(n, m, a) (8)(9)(3)(33) R I(n, m, a) ( kr) I(n, m, a) k π k h () n ()h () m ()d H() ()H () ()d n+ m+ (34) H n () ( ) Hankel n, m () n m [3, 4] H() n+ H () n++ ()H () ()d m+ ()H () H () ()H () () n+ m+ n + m + () n m H() n+ () H () ()H () m+ n+ ( n+ ) ( m+ ) ()H () ()d m+ H () (3) n m> [3, 4] H() n+ n+ r [ m++ } () (35) ()} d (36) ()H () ()d m+ H () n } + + H () +r() } d+ () H () H () ] } +n() } () (37) (36)(37) H n () ( ) / H () ()} d } j π e j d [ π cos()d j 4 [ } cos() + si() π }] sin() j ci() si() Si() ci() sin t dt Si() π t sin t dt t Ci() γ +ln + cos t dt Ci() t cos t dt t sin()d (38) γ si() Si() ci() Ci() Si() ]

7 64 Fig. 4 Geometry of the calculated sound field model. Fig. 5 Finite and infinite element division. Ci() [5] 3. Fig. 4 [9] z ( s,,z s ) φ E Q e jkl + Q e jkl (39) 4πL 4πL Fig. 6 Relative error of the proposed FEM. Q L L Q s z s. z z Fig. 5 r z r, z m, Hz / (7)(9) ϕ m M () θ n N Ω C p E p ds Ωc ε (4) p E ds Ωc p E (39) p Fig. 6 (M,N) ε θ n N ε ϕ m M ε M ε (M,N) (8, 8) Fig. 7, Hz [Pa]

8 Fig. 7 Sound pressure distribution in z plane (, Hz). z y (a).4, z m [Pa] (M,N) (, ) (b) (M,N) (8, 8) (c) Fig. 8, Hz [Pa] z.3m y.4, y m [Pa] Fig. 7 (M,N) (, ) (b) (M,N) (8, 8) (c) Ω C Fig. 6 θ ϕ Fig. 8 Sound pressure distribution in y plane (, Hz). ε Fig. 7, 8

9 643 (4) (M,N) FEM BEM FEM [],,,. [], /,, 983. [ 3 ] K. Gerdes, A review of infinite element methods for eterior Helmholtz problems, J. Comput. Acoust., 8, 43 6 (). [ 4 ] P. Bettess and O.C. Zienkiewicz, Diffraction and refraction of surface waves using finite and infinite elements, Int. J. Numer. Methods Eng.,, 7 9 (977). [ 5 ] O.C. Zienkiewicz, C. Emson and P. Bettess, A novel boundary infinite element, Int. J. Numer. Methods Eng., 9, (983). [ 6 ] O.C. Zienkiewicz, K. Bando, P. Bettess, C. Emson and T.C. Chiam, Mapped infinite elements for eterior wave problems, Int. J. Numer. Methods Eng.,, 9 5 (985). [ 7 ] D.S. Burnett, A three-dimensional acoustic infinite element based on a prolate spheroidal multipole epansion, J. Acoust. Soc. Am., 96, (994). [ 8 ] D.S. Burnett and R.L. Holford, An ellipsoidal acoustic infinite element, Comput. Methods Appl. Mech. Eng., 64, (998). [ 9 ] R.J. Astley, G.J. Macaulay and J.P. Coyette, Mapped wave envelope elements for acoustical radiation and scattering, J. Sound Vib., 7, 97 8 (994). [] L. Cremers, K.R. Fyfe and J.P. Coyette, A variable order infinite acoustic wave envelope element, J. Sound Vib., 7, (994). [] R.J. Astley, G.J. Macaulay, J.P. Coyette and L. Cremers, Three-dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part I. Formulation in the frequency domain, J. Acoust. Soc. Am., 3, (998). [] J.P. Coyette and B.V.d. Nieuwenhof, A conjugated infinite element method for half-space acoustic problems, J. Acoust. Soc. Am., 8, (). [3] T.H.H. Pian, Derivation of element stiffness matrices by assumed stress distributions, AIAA J.,, (964). [4] P. Tong, New displacement hybrid finite element models for solid continua, Int. J. Numer. Methods Eng.,, (97). [5],, 3, pp (98). [6],,,,,, J65-A, 6 69 (98). [7],,,, A,, 7 34 (99). [8],, C, 5, (984). [9],,,,, J68-A, (985). [],,,, pp (4.3). [], /,, 98, 7-. [],,,,, I,, 98, [3],,, III,, 96, pp [4],, II,, 99, pp [5] W.J. Thompson, Atlas for Computing Mathematical Functions (John Wiley & Sons, New York, 997), Chap ,,

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