oundary Elements and Other Mesh Reduction Methods XXV 205 terative coupling in fluid-structure interaction: a EM-EM based approach D. Soares Jr 1,2, W. J. Mansur 1 & O. von Estorff 3 1 Department of Civil Engineering, COPPE - ederal University of Rio de Janeiro, Rio de Janeiro, razil 2 Structural Engineering Department, ederal University of Juiz de ora, MG, razil 3 Modelling and Computation, Hamburg University of Technology, Hamburg, Germany Abstract An iterative coupling of finite element and boundary element methods for the investigation of coupled fluid-solid systems is presented. While finite elements are used to model the solid, the adjacent fluid is represented by boundary elements. n order to perform the coupling of the two numerical methods, a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved. Keywords: iterative coupling, solid-fluid interaction, EM, EM, adjustable time-steps, nonlinear analysis. 1 ntroduction Most of the EM/EM coupling algorithms [1-3] are formulated in a way that, first, a coupled system of equation is established, which afterwards has to be solved using a standard direct solution scheme. Such a procedure leads to several problems with respect to accuracy and efficiency. irst, the coupled system of equation has a banded structure only in the E part, while in the E part it is fully populated. Consequently, for its solution the optimized solvers usually used in the EM cannot be employed anymore, which leads to rather expensive calculations with respect to computer time. Second, the duration of a time step needs to be the same in all subsystems. n general, however, the velocities of the WT Transactions on Modelling and Simulation, Vol 42, 2006 WT Press www.witpress.com, SSN 1743-355X (on-line) doi:10.2495/e06021
206 oundary Elements and Other Mesh Reduction Methods XXV propagating waves in the solid and the fluid are quite different, such that a unified time step may cause serious problems in the numerical solution algorithms (instabilities, lack of accuracy etc.). Third, in the case of taking into account some nonlinearity within the E sub-region, the rather big coupled system of equations needs to be solved in each step of the iteration process, i.e., a few times within each time step. This is very computer time consuming. 2 EM/EM coupling Considering the coupling conditions at each node i of the EM/EM interface (superscript ), the following equations must hold ( ρˆ is the fluid mass density): T ( ) = 0 (1) i N ( ) = ( P ) (2) i i N( U ) = (1/ ˆ ρ) ( Q) (3) i i where - in order to obtain consistency between the E (subscript ) and the E formulation (subscript ) - P represents the resultant nodal hydrodynamic pressure force, which is obtained from the potential distributions P. denotes the E nodal forces; Q is the fluid flux and U is the solid acceleration. The functions N (.) i and T (.) i lead to the normal and the tangential component of their arguments, respectively. 2.1 terative coupling n the iterative EM/EM coupling, first the E problem is solved and the t accelerations U ( k+ α ) are obtained. Then a relaxation parameter α is introduced, according to equation (4), in order to ensure and/or to speed up convergence: U t ( U + ( k ) (4) t t ( k 1) = α U ( k + α ) + 1 α) Once the EM accelerations at the interface are computed, equation (3) can be t used to obtain the EM flux Q ( k+ 1). n the present formulation different timestep durations in each subdomain can be taken into account by means of extrapolations and interpolations (with respect to time) of the variables at the interface. These interpolations and extrapolations are done according to the time interpolation functions adopted by the E formulation, as indicated by equations (5-6) (piecewise constant interpolation for the flux and linear interpolation for the potential). WT Transactions on Modelling and Simulation, Vol 42, www.witpress.com, SSN 1743-355X (on-line) 2006 WT Press
oundary Elements and Other Mesh Reduction Methods XXV 207 t Q = Q ( k+ 1) (5) t ( k+ 1) ( ) ( ) P = t t / t P + 1 t t / t P (6) t t t t ( k+ 1) ( k+ 1) ( k+ 1) t Once Q ( k+ 1) is obtained (equation (5), the E subdomain can be solved, having t prescribed flux values at the interface. As the result, P ( k+ 1) is obtained and it needs to be interpolated as well, in order to be used by the EM. y means of t equations (1-2) and P ( k+ 1) (equation (6)) one finally obtains the new EM nodal t forces ( k+ 1), which are needed to solve the E problem once more. The iterative loop goes on until convergence is achieved. A sketch of the iterative coupling is shown in igure 1. f t > t t= t+ t t= t+ t Adoption of parameter α Equation (4) NPUT OUTPUT Solve EM t Obtain: U rom rom U obtain t Equation (3) EM/EM TERATVE COUPLNG P obtain t Equations (1-2) t Q t Time extrapolation t t Q Q Equation (5) Solve EM t Obtain: P Time interpolation t P P t Equation (6) f t+ t > t EM results actualization EM results actualization igure 1: terative EM/EM coupling scheme. 2.2 Numerical example n this example, a dam-reservoir system, as depicted in igure 2, is analyzed. The structure is subjected to a sinusoidal, distributed vertical load on its crest, acting with an angular frequency ω = 18 rad/s. The material properties of the dam are: Poisson s ratio ν = 0.25; Young s modulus E = 3.437 10 6 N/m 2 ; mass density ρ = 2.00 Ns 2 /m 4. A perfectly plastic material obeying the Drucker-Prager yield criterion is assumed: cohesion c 0 = 0.15 N/m 2 ; internal friction angle φ = WT Transactions on Modelling and Simulation, Vol 42, www.witpress.com, SSN 1743-355X (on-line) 2006 WT Press
208 oundary Elements and Other Mesh Reduction Methods XXV 20 o. The adjacent water is characterized by a mass density ρˆ = 1.00 Ns 2 /m 4 and a wave velocity c = 1436 m/s. The time-step duration adopted for the EM and EM are t = 0.00350 s and t = 0.00175 s, respectively. The transient behaviour of the vertical displacement at point A is shown in igure 3(a). Results of linear as well as nonlinear analyses are given and two different water levels, namely h = 50 m and h = 35 m, and their influence on the dam are investigated. n igure 3(b) the transient hydrodynamic pressure at point is depicted. A comparison of the results obtained with the iterative coupling procedure with those from the standard coupling scheme used in von Estorff and Antes [1] shows good agreement. The average number of iterations per time step (it was adopted α = 0.5), for the current linear model, was 2.4 (h = 50 m) and 2.2 (h = 35 m). or the nonlinear analysis, the average number of iterations per time step was 2.6 (h = 50 m) and 2.4 (h = 35 m). As one can observe, the convergence is quite fast: the iterative coupling can be regarded as a very attractive tool to deal with high scale linear or (especially) nonlinear models. or more details on the iterative EM/EM coupling taking into account solid-solid interactions, one is referred to Soares Jr et al. [4]. 10 Dam (93 E) P P P(t) = sin(wt) (ω t ) A t Storage-lake 60 5 = 1 E t = 0.00350s t = 0.00175s h 35 75 igure 2: Dam with storage-lake. 3 Conclusions n the present paper, an iterative coupling scheme for the investigation of a continuous solid coupled to an adjacent fluid was presented. The major advantage of such a procedure can be seen in the fact, that the E and E subsystems can be solved separately using optimised solution algorithms according to the special features of the respective system of equations, which are a further advantage much smaller than the coupled matrices resulting from a WT Transactions on Modelling and Simulation, Vol 42, www.witpress.com, SSN 1743-355X (on-line) 2006 WT Press
oundary Elements and Other Mesh Reduction Methods XXV 209 standard coupling approach. n addition, the iterative coupling offers two advantages: t is straightforward to use different time steps in each subdomain; and, moreover, to take into account nonlinearities (within the E subdomain) in the same iteration loop that is needed for the coupling. 3 (a) Vertical displacement at point A (10-6 m) 2 1 0-1 -2-3 h = 50 h = 35 Standard - linear terative - linear terative - nonlinear -4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (s) 0.025 (b) igure 3: Hydrodynamic pressure at point (N/m) 0.020 0.015 0.010 0.005 0.000-0.005-0.010-0.015-0.020-0.025 Standard - Linear terative - Linear h = 35 h = 50-0.030 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Time (s) Results for the standard [1] and the iterative EM/EM coupling: (a) vertical displacements at point A; (b) hydrodynamic pressure at point. WT Transactions on Modelling and Simulation, Vol 42, www.witpress.com, SSN 1743-355X (on-line) 2006 WT Press
210 oundary Elements and Other Mesh Reduction Methods XXV References [1] von Estorff, O. & Antes, H., On EM-EM coupling for fluid-structure interaction analysis in the time domain. nternational Journal of Numerical Methods in Engineering, 31, pp. 1151-1168, 1991. [2] Czygan, O. & von Estorff, O., luid-structure interaction by coupling EM and nonlinear EM. Engineering Analysis with oundary Elements, 26, pp. 773-779, 2002. [3] Yu, G.Y., Lie, S.T. & an, S.C., Stable boundary element method/finite element method procedure for dynamic fluid structure interactions. Journal of Engineering Mechanics, 128, pp. 909-915, 2002. [4] Soares Jr, D., von Estorff, O. & Mansur, W.J., terative coupling of EM and EM for nonlinear dynamic analyses. Computational Mechanics, 34, pp. 67-73, 2004. WT Transactions on Modelling and Simulation, Vol 42, www.witpress.com, SSN 1743-355X (on-line) 2006 WT Press