Hosseini Zahraei, S. M.

Seubers, Jan

van der Plas, Peter

Veldman, Arthur

Wellens, P.R.

English

Dissertations of the University of Groningen

   University of GroningenStrong quasi-simultaneous coupling of fluid-structure interaction in offshore applicationsVeldman, Arthur; Seubers, Jan; Hosseini Zahraei, S. M.; van der Plas, Peter; Wellens, P.R.IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Publication date:2018Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Veldman, A., Seubers, J., Hosseini Zahraei, S. M., van der Plas, P., & Wellens, P. R. (2018). Strong quasi-simultaneous coupling of fluid-structure interaction in offshore applications. Paper presented at NumericalTowing Tank Symposium , Cortona, Italy.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 04-06-2022Strong quasi-simultaneous coupling for fluid-structure interactionin offshore applicationsArthur E.P. Veldmana∗, Henk Seubersa, Matin Hosseini Zahraeib,Peter van der Plasa, Peter Wellensba Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, TheNetherlandsb Department of Ship Hydrodynamics, Delft University of Technology, The Netherlands∗ Corresponding author: a.e.p.veldman@rug.nl1 IntroductionSimulating the hydrodynamics of deformable, floating structures using a partitioned strategy poses amajor challenge when the physical coupling between the fluid and the structure is strong and the ratioof the added mass to the structural mass is considerate. In numerical simulations of such systems withtwo-way interaction numerical instabilities can occur, which do not occur in systems with inherently one-way interaction. Existing computational procedures for fluid-structure interaction become less efficientor even unstable. In these situations, it is advisable to modify the coupling to allow the fluid to respondbetter to the body motions. A simultaneous solution of the equations governing fluid and (solid or elastic)body would be a stable choice, but is often not feasible. Usually the numerical problems are taken careof with sub-iterations between fluid and structure, but their convergence can be slow; e.g. [Causin et al.,2005, Forster et al., 2006].In this paper we present a more powerful, quasi-simultaneous approach, which tries to mimic a fullysimultaneous coupling in an affordable way as introduced by Veldman for aerodynamic viscous-inviscidboundary-layer interaction [Veldman, 1981, Veldman, 2009]. In our application, the quasi-simultaneousapproach makes use of a simple approximation (termed interaction law) of the elastic-body dynamics,based on the (6 DOF) solid-body modes and the main elastic modes of the structure.The fluid solver in this study is the symmetry preserving finite-volume VOF method ComFLOW[Kleefsman et al., 2005, Veldman et al., 2007, Wemmenhove et al., 2015]. A finite element method isused to solve the elastic structure response based on a Euler-Bernoulli beam. Kinematic and dynamicrelations couple fluid dynamics to the structural dynamics.2 Mathematical modelFlow model Incompressible, turbulent fluid flow can be modeled by means of the Navier–Stokes equa-tionsMu = 0,∂u∂t+ C(u)u + Gp − Vu = f. (1)Here M is the divergence operator, which describes conservation of mass. Conservation of momentumis based on the convection operator C(u)v ≡ ∇(u ⊗ v), the pressure gradient operator G = ∇, the viscousdiffusion operator V(u) ≡ ∇ · ν∇u and a forcing term f. The kinematic viscosity is denoted by ν. Theflow equations are discretized on a staggered grid with a finite-volume method. The convective termsare discretized in a skew-symmetric energy-preserving way [Verstappen and Veldman, 2003]. Also thediscrete pressure gradient G is the negative transpose of the discrete divergence operator M, such thatalso the pressure does not influence the energy balance.The free-surface location is indicated by a Volume-of-Fluid [Hirt and Nichols, 1981] function φ, andreconstructed by means of Youngs’ PLIC method [Youngs, 1987, Düz, 2015].In this exposition, for simplicity reasons the first-order forward Euler time integration will be used.In the actual calculations, a second-order Adams–Bashforth method is being applied.The solution of the discrete Navier–Stokes equations is split into two steps. Firstly, an auxiliaryvariable ũ is introducedΩũ − unδt= −C(un) un + Vun + f, such that un+1 = ũ − δt Ω−1Gpn+1. (2)Secondly, by imposing discrete mass conservation at the new time level, substitution of (2) results in adiscrete Poisson equation for the pressure:δt M0Ω−1Gpn+1 = M0ũ + MΓun+1Γ , (3)where M0 and MΓ represent the discrete continuity operator in the interior and along the boundary, re-spectively.Structural model For simplicity, in this study the structure is selected to be a one dimensional Euler–Bernoulli beam. Assuming a constant cross section A for the beam, its equation of motion isρsA∂2d∂t2+ EI∂4d∂s4= f , (4)with appropriate initial and boundary conditions. Here, s denotes a coordinate along the beam, d thebeam deformation, ρs the beam density, E Young’s modulus, I the second moment of inertia and f theload per unit length of the beam.The structural response of the elastic body (eb) is modeled with a finite element method. Omittingthe technical details, the resulting discrete set of equations can be written in the formMebd̈ + Kebd = f, (5)where Meb is the discrete mass operator and Keb the discrete stiffness operator. These matrices can besimultaneously diagonalized as QT MebQ = I and QT KebQ = Λ, where Q contains the normalized elasticbody eigenvectors (modes) with eigenvaluesΛ. In this way, the elastic body dynamics (5) can be rewrittenasQ−1d̈ + ΛQ−1d = QT f. (6)The temporal integration of the structure equations is performed by means of the generalized-αmethod [Chung and Hulbert, 1992].Fluid-solid interface The coupling relations along the fluid-solid interface ΓFS consist of two rela-tions: the kinematic and the dynamic conditions. The kinematic condition states that the motion of theinterface on both sides matches. The dynamic condition expresses equilibrium of stresses at the fluid-solid interface.3 Numerical couplingTime marching with sub-iterations Information has to be exchanged between the two subsystems: thefluid solver provides the loads to the structural solver, whereas in return the latter provides the structuralmotion to the flow solver. In an explicit, weak coupling this exchange only happens once per time step,but this process is unstable for larger mass ratios. Therefore often per time step sub-iterations with severeunder-relaxation (e.g. Aitken [Forster et al., 2006]) are introduced to achieve a strong coupling betweenthe subsystems; Fig. 1(left). The convergence of this process can be rather slow, and each iteration asolid-fluid solve has to be performed. This makes this coupling method inefficient. In the next sectionwe will present a method that shares the strong coupling with the above sub-iterations, but at a limitedadditional computational effort.Quasi-simultaneous coupling For two-way coupled problems, a monolithic or simultaneous approachis not always possible, as the sub-domain solvers have to be coupled at a deep iterative level. In this sec-tion we will describe an approach that tries to combine the simplicity of a hierarchical coupling approachwith the iterative power of a monolithic approach. In the quasi-simultaneous method, an approximationof the body dynamics is solved simultaneously with the fluid. This interaction law anticipates the bodyresponse in advance of the actual solid dynamics computation. As such, the interaction law (Fig. 1(right)).Fig. 1: (Left) Sub-iterative loop of FSI with relaxation within one time step; (right) adding an approxi-mate interaction law. Force f, displacement d, whereas k is iteration level.To understand the coupling stability, it is useful to formulate the coupling problem in terms of in-terface variables only: the velocity along the interface uΓ (related to the displacement d, and the loadexerted by the fluid to the structure fΓ (for an elastic body found from the local stresses, for a solid bodyfound from their integration along the interface).Elastic body couplingBecause the structural equation (6) contains both d̈ and d, first a discrete time integration is carriedout. Thereafter, the discrete version of the hierarchically coupled problem at the new time level can bedenoted asElastic body Q−T(Iδt2+ Λ)Q−1dk+1Γ = fkΓ, Fluid fk+1Γ = −Madδt2dk+1Γ . (7)The contribution from the previous time steps is omitted in view of clarity; it is just an inhomogeneousterm in the right-hand side, which is not relevant for the convergence of the sub-iterations per time step.The displacement dk+1 can be eliminated from the system of equations (7), after which the iterativeprocess can be written asfk+1 = −MadQ(I + δt2Λ)−1QT fk. (8)For small enough δt, the amplification factor simplifies to −MadQQT , where QQT has the dimension of1/kg. Thus we recognize again an added-mass ratio, leading to divergence when the added mass of thefluid is too large.In the quasi-simultaneous treatment we approximate the behavior of the elastic body by a simplereduced-order model consisting of a limited number of modes, including the solid-body modes. Letthese be collected in the matrix Q̃. Thus, the proposed interaction law readsdk+1 − Q̃(1δt2+ Λ̃)−1Q̃T fk+1 =Q(1δt2+ Λ)−1QT − Q̃(1δt2+ Λ̃)−1Q̃T fk. (9)Letting δt → 0, i.e. studying zero-stability, and combining with the fluid-flow model (7), the quasi-simultaneous iterative process reduces tofk+1 = −(M−1ad + Q̃Q̃T )−1(QQT − Q̃Q̃T )fk. (10)Clearly, if all of the modes are incorporated into the interaction law, the spectral radius will become zeroand the method simultaneously solves the fluid with the ‘exact’ body.ImplementationThe interaction law is a relation between the pressure and the local velocity of the body surface. Thisrelation can be substituted in the right-hand side of the discrete Poisson equation (3) and thus can be con-sidered as a boundary condition. It can be shown that the latter retains its favorable numerical properties(symmetric, negative definite), such that its iterative solution can proceed as before.Fig. 2: Schematic of the first test case; the domain with free-surface flow on top and flexible beam at thebottom4 ExamplesElastic body: tank with membrane bottomIn order to assess the performance of the quasi-simultaneous approach for different mass ratios, a testcase has been designed in which this ratio can be varied.At the bottom of a rectangular container (1.0 × 0.1 × 0.5 m3) filled with 50 kg of water, a flexiblebeam is placed as illustrated in Fig.2. The mass of the beam is varied between 1 kg and 50 kg; its moduleof elasticity is 1 MPa. The interaction law is made out of truncated structural modes; the number ofmodes dictates its accuracy. The relaxation parameter is adjusted by Aitken’s method. Five cases withdifferent mass ratios are solved with and without the interaction law. Also, the effect of the accuracy ofthe interaction law on the performance is studied.First, the effect of the added mass ratio is studied. The interaction law is constructed out of the first10 modes, while the symmetry of the problem cancels out the effect of odd modes. For all the casesthe initial relaxation parameter is set to 1, while from the third FSI iteration this value is adjusted byAitken’s method. Fig. 3 shows the convergence of the FSI iterations during the first time step. As shownin Fig. 3(left), higher mass ratios require more computational effort. Plugging in the interaction law,Fig. 3(right) reproduces the same solution while the computational effort is less. It can be observed thatthe difficult cases with higher mass ratios speed-up a lot more from the quasi-simultaneous approachthan the easier cases with modest mass ratios. The difference in convergence rates is clearly visible, withthe quasi-simultaneous method hardly needing any under-relaxation.Fig. 3: Convergence history for the first time step for different added mass ratios: (left) without interactionlaw, (right) with interaction law.Next, the effect of the number of modes in the interaction law is shown in Fig. 4. Obviously, employ-ing more modes improves the structure response estimation in the interaction law. But there is a tradeoff, as the gain goes down while making the interaction law itself more expensive. When the mass ratiois low, the number of effective modes is lower than the case when mass ratio is 50.Rubber gateThe validation test case is an elastic rubber gate placed in front of a bulk of water. This experimenthas been performed by Antoci [Antoci et al., 2007]. The gate separating the fluid from the air is partlyrigid, but the lower part is elastic. The rubber gate is clamped on the top side and free on the otherside. The density of the beam is ρs = 1100 kg/m3 and the Young modulus for elasticity is E = 10Fig. 4: Convergence history for the first time step for varying accuracy (number of included modes) ofthe interaction law: (left) mass ratio 1; (right) mass ratio 50.MPa. According to [Antoci et al., 2007], the problem is mostly two dimensional, so in the current studynumerical simulations are performed in 2D. A grid of 250 × 2 × 50 is used. The initial time step is set tobe 1 × 10−4; this value is automatically adjusted during the simulation based on the Courant condition.Fig. 5: Snapshots of the simulation past a rubber gate, compared with the experiment of [Antoci et al.,2007]The results after 0.12 s, with the snapshots from experiment and simulation at time intervals of 0.4 s,are compared in Fig. 5.5 ConclusionIt has been demonstrated that the ratio of the fluid added mass versus the solid-body mass is an impor-tant parameter controlling the (in)stability of a fluid-structure system. Whereas traditional (sub-)iterationmethods require severe under-relaxation, a quasi-simultaneous method can handle large added mass ra-tios more efficiently. An interaction law approximating the structural dynamics is its key ingredient. Itis shown that only a couple of dominant modes are needed to achieve a stable method. The stabilityand resulting efficiency has been demonstrated by comparing the computational effort with and withoutinteraction law on a number of offshore-related applications.6 AcknowledgmentsThis work is part of the research programme Maritime2013 with project number 13267 which is (partly)financed by the Netherlands Organization for Scientific Research (NWO).ReferencesAntoci, C., Gallati, M., and Sibilla, S. (2007). Numerical simulation of fluidstructure interaction by SPH. J.Comp. Struct., 85:879–890.Causin, P., Gerbau, J. F., and Nobile, F. (2005). Added-mass effect in the design of partitioned algorithms forfluid-structure problems. Comp. Meth. Appl. Mech. Eng., 194:4506–4527.Chung, J. and Hulbert, G. M. (1992). A time integration algorithm for structural dynamics with improved numer-ical dissipation: the generalized-α method. J. Appl. Mech., 60(2):371–375.Düz, B. (2015). Wave generation, propagation and absorption in CFD simulations of free surface flows. PhDthesis, Technical University Delft.Forster, C., Wall, W. A., and Ramm, E. (2006). Artificial added mass instabilities in sequential staggered couplingof nonlinear structures and incompressible viscous flows. Comp. Meth. Appl. Mech. Eng., 196:1278–1293.Hirt, C. W. and Nichols, B. D. (1981). Volume of fluid (VOF) method for the dynamics of free boundaries.J. Comput. Phys., 39:201–25.Kleefsman, K. M. T., Fekken, G., Veldman, A. E. P., Iwanowski, B., and Buchner, B. (2005). A Volume-of-Fluidbased simulation method for wave impact problems. J. Comput. Phys., 206:363–393.Veldman, A. E. P. (1981). New, quasi-simultaneous method to calculate interacting boundary layers. AIAA J.,19:79–85.Veldman, A. E. P. (2009). A simple interaction law for viscous-inviscid interaction. J. Eng. Math., 65:367–383.Veldman, A. E. P., Gerrits, J., Luppes, R., Helder, J. A., and Vreeburg, J. P. B. (2007). The numerical simulationof liquid sloshing on board spacecraft. J. Comput. Phys., 224:82–99.Verstappen, R. W. C. P. and Veldman, A. E. P. (2003). Symmetry-preserving discretization of turbulent flow.J. Comput. Phys., 187:343–368.Wemmenhove, R., Luppes, R., Veldman, A. E. P., and Bunnik, T. (2015). Numerical simulation of hydrodynamicwave loading by a compressible two-phase flow method. Computers & Fluids, 114:218–231.Youngs, D. L. (1987). An interface tracking method for a 3d Eulerian hydrodynamics code. Technical ReportAWRE/44/92/35, Atomic Weapons Research Establishment.

Strong quasi-simultaneous coupling of fluid-structure interaction in offshore applications

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Strong quasi-simultaneous coupling of fluid-structure interaction in offshore applications

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