Vol.1( ) No JASCOME Trefftz ( ) SIMULATION OF SLOSHING PHENOMENON BY INDIRECT TREFFTZ METHOD (EXTENSION OF SIMULATION SCHEME) 1), 2),
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1 Vol.1( ) No JASCOME Trefftz ( ) SIMULATION OF SLOSHING PHENOMENON BY INDIRECT TREFFTZ METHOD (EXTENSION OF SIMULATION SCHEME) 1), 2), 3) 4), Yoichi IKEDA, Jun ichi KATSURAGAWA, Eisuke KITA and Norio KAMIYA 1) ( , ikeda@daidoh-it.ac.jp) 2) ( , jk@mech.nagoya-u.ac.jp 3) ( , kita@info.human.nagoya-u.ac.jp) 4) ( , b41861a@nucc.cc.nagoya-u.ac.jp) This paper describes the application of the Trefftz-type boundary element method to the simulation of the sloshing phenomenon. Assuming that the fluid could be the perfect one, the phenomenon can be modeled as the initial and the boundary values problem of the Laplace equation with respect to the velocity potential. The governing equation is firstly solved with the adequate boundary conditions to determine the components of velocity and acceleration on the fluid surface. In order to solve the initial value problem, we will compare the simple Euler scheme using velocity vector on the free surface alone and the extended scheme using both velocity and acceleration components. The Trefftztype boundary element method is applied for estimating the velocity and acceleration components. Finally, the present scheme is applied to the simulation of the sloshing phenomenon on the fluid in a rectangular vessel. Key Words : Indirect Trefftz Method, Sloshing Phenomenon, Boundary Derivatives, T-complete Functions 1. (1,2,3,4,5,6,7,8,9) Trefftz Trefftz (10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25)
2 u n 2 u = 0 (in Ω) u = ū (on Γ 1 ) = q =0 (onγ2) (5) Du Fig. 1 Problem Statement = 1 u u gξ + A(t)η 2 Dξ = u (6) x Dη = u y T-complete 2.3. (5) Trefftz Trefftz (1) T-complete 2 T-complete u j (15) Trefftz 1 Euler u = {u 1,,u 2µ,u 2µ+1, } T = {1,, <[r µ e jµθ ], =[r µ e jµθ ], } T (7) (r, θ) µ j = 2. 1 <, = 2.1. (7) u u ' ũ = a 1u 1 + a 2u a N u N ξ η ( 1) Ω Γ 1 Γ 2 = a T u (8) u(x, y, t) u, a T-complete N (8) (3, 4) q ' q ũ n = a 1q1 + a 2 q2 + + a N qn 2 u =0 (inω) (1) = a T q (9) Du Dξ Dη q u n =0 (onγ 2) (2) = 1 u u gξ + A(t)η 2 = u x = v x = u y = vy (3) q T-cocmplete (8) (9) ) R 1 ũ ū = a T u ū 6= 0 onγ 1 (10) R 2 q q = a T q q 6= 0 onγ 2 (10) g, A(t) v x,v y P i (10) 0 ) a T u (P i ) = ū(p i ) (P i Γ 1 ) (11) a T q (P i ) = q(p i ) (P i Γ 2 ) t =0 Γ 1 Ka = f (12) ξ = ξ, η =0,u=0 (onγ 1) (4)
3 (1) Euler Euler t u k+1 = u k + t Du ξ k+1 = ξ k + t Dξ η k+1 = η k + t Dη Du/, Dξ/, Dη/ (3) (3) u/ x, u/ y (8) x, y (13) u,x = a T u,x,u,y = a T u,y (14) (),x / x, (),y / y Fig. 2 Object under consideration (1) (2) t + t u(t + t), ξ(t + t), η(t + t) 2 u k+1 ξ k+1 η k+1 = u k + t Du ( t)2 D 2 u 2 (15) = ξ k + t Dξ ( t)2 D 2 ξ 2 (16) = η k + t Dη ( t)2 D 2 η 2 (17) 1 Du/, Dξ/, Dη/ (6) 2 Fig. 3 Initial placement of collocation points D 2 u/ 2 D 2 u = D µ 1 u u gξ + A(t)η 2 2 D 2 ξ = v x + D 2 η 2 vy + D [A(t)ξ] 2 gvy (18) D 2 ξ/ 2,D 2 η/ 2 D 2 ξ/ 2,D 2 η/ 2 D 2 ξ = Dv µ x u = 2 t D 2 µ η = Dvy u = 2 t,x,y D 2 u/ 2 A(t) + u,x u,xx + u,y u,xy + u,x u,yx + u,y u,yy u,x,u,y,u,xx,u,xy,u,yy (8) 1 (1) u/ t 2 u t = 0 (in Ω) u t u n t = q =0 (onγ2 ) = 1 u u gξ + A(t)η 2 (on Γ 1) Trefftz u/ t T-complete x, y ( u/ t),x, ( u/ t),y 3. (19) 2 L=0.9( ) 2 H=0.6( ) d ω 2 A(t) =dω 2 sin(ωt) (t 0) (20) d = (m), ω =5.5(rad/sec) Fig. 4 Elevation of free surface (Euler scheme)
4 Fig. 5 Comparison with finite element solutions 10cm 4. 2 Trefftz Trefftz 2 ABM Fig. 6 Elevation of free surface (Extended scheme) T-complete (1).., (2) J. W. Dold and D. H. Peregrine. Steep unsteady water waves - an efficient computational scheme. In Proc th Coastal Eng. Conf., Vol. 1, pp , (3) T. Nakayama. A computational method for simulating transient motions of an incompressible inviscid fluid with a free-surface. International Journal of Numerical Methods in Fluid, Vol. 10, pp , (4),,.., Euler Vol. 6, pp , t =0.01, 0.001, (5) N. Tosaka and R. Sugino. Boundary Element Analysis t =9.5(s) of Non-linear Liquid Motion in Two-dimensional Containers, pp Springer Verlag, t = (6) K.Wasizu,T.Nakayama,M.Ikegawa,Y.Tanaka,and (6) 5 T. Adachi. Some Finite Element for Techniques Analysis of Nonlinear Sloshing Problem, chapter 5, pp John Wile & Sons Ltd., (7),. t =., Vol. 45, pp , 0.01, 0.001, t =9.5(s) (8),. 6., Vol. 45, pp , t = (9) M.A.Hamzah,,. T =0.54, 1.12, 1.68, 2.24, 2.81, 3.37, 5.05, 6.75(s). 7 8, Vol. 45, pp , 1998.
5 (a) T=0.54 (s) (e) T=2.81 (s) (b) T=1.12 (s) (f) T=3.37 (s) (c) T=1.68 (s) (g) T=5.05 (s) (d) T=2.24 (s) (h) T=6.75 (s) Fig. 7 Wave profiles Fig. 8 Wave profiles (Cont.) (10) N. Kamiya and E. Kita. Advances in Engineering Software: Special Issue on Trefftz Method 70 Years, Vol. 24. Elsevier Science Pub., (11) H. Antes. On a regular boundary integral equation and amodified Trefftz method in Reissner s plate theory. Engineering Analysis, Vol. 1, No. 3, pp , (12) Y. K. Cheung, W. G. Jin, and O. C. Zienkiewicz. Direct solution procedure for solution of harmonic problems using complete, non-singular, Trefftz functions. Communications in Applied Numerical Methods, Vol.5,pp , (13) W. G. Jin, Y. K. Cheung, and O. C. Zienkiewicz. Application of the Trefftz method in plane elasticity problems. International Journal for Numerical Methods in Engineering, Vol. 30, pp , (14) Y. K. Cheung, W. G. Jin, and O. C. Zienkiewicz. Solution of Helmholtz equation by Trefftz method. International Journal for Numerical Methods in Engineering, Vol. 32, pp , (15) I. Herrera. Boundary Methods : An Algebraic Theory. Pitman, (16) Ismael Herrera. Trefftz-Hrrera domain decomposition.
6 Advances in Engineering Software, Vol. 24, No. 1-3, pp , (17) Ch. Hochard and L. Proslier. A simplified analysis of plate structures using Trefftz functions. International Journal for Numerical Methods in Engineering, Vol. 34, pp , (18) J. Jirousek and Lan Guex. The hybrid-trefftz finite element model and its application to plate bending. International Journal for Numerical Methods in Engineering, Vol. 23, pp , (19) J. Jirousek and A. Venkatesh. Generation of optimal assumed stress expansions for hybrid-stress elements. Computers & Structures, Vol. 32, No. 6, pp , (20) N. Kamiya and S. T. Wu. Generalized eigenvalue formulation of the Helmholtz equation by the Trefftz method. Engineering Computations, Vol. 11, pp , (21) E. Kita, N. Kamiya, and Y. Ikeda. An application of Trefftz method to the sensitivity analysis of twodimensional potential problem. International Journal for Numerical Methods in Engineering, Vol. 38, No. 13, pp , (22),,. Trefftz.,Vol.5,pp , (23) E. Trefftz. Ein Gegenstück zum ritzschen Verfahren. Proc. 2nd Int. Cong. Appl. Mech., Zurich, pp , (24) B. Szybinski and A. P. Zielinski. Alternative T- complete systems of shape functions applied in analytical Trefftz finite elements. Numerical Methods for Partial Differential Equations, Vol. 11, pp , (25) O. C. Zienkiewicz, D. W. Kelly, and P. Bettess. Marriage álamode thebestofbothworlds(finite elements and boundary integrals). In R. Glowinski, E. Y. Rodin, ando.c.zienkiewicz,editors,energy Methods in Finite Element Analysis, pp John Willy & Sons, 1979.
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,, Department of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyo-ku, Tokyo 112 8551, JAPAN E-mail : atsu1005@kc.chuo-u.ac.jp E-mail : kawa@civil.chuo-u.ac.jp SATO KOGYO CO., LTD. 12-20, Nihonbashi-Honcho
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