In linear vibration analysis of partially ﬁlled elastic tanks [1], even if the the structure is submitted by a gaz or a liquid pressure, the reference conﬁguration is generally used without the eﬀect of static loads. In the case of very thin structures or soft material, the static state is considered as prestressed, due to geometrical nonlinearities of the deformed tank. The global stiﬀness of the structure may change in function of the ﬂuid volume amount [2, 3, 4]. The aim of the paper is to quantify the prestressed eﬀets on the linearized dynamic behavior of the ﬂuid-structure system. The chosen methodology is the following: (i) A quasi-static solution is computed from an empty to a fully ﬁlled state of the tank, by considering geometrical nonlinearities and hydrostatic follower forces [5] (no volumetric mesh of the ﬂuid is needed for this step); (ii) after a volumetric remeshing of the ﬂuid at each states, a linearized hydroelastic displacement-pressure formulation around the prestressed state, without gravity eﬀects, is established; (iii) a reduced basis of the hydroelastic problem is generated by using prestressed dry modes to minimize the computation of the added mass matrix. Numerical examples are given to illustrate the proposed approaches

Deü, J.-F.

Hoareau, C.

Ohayon, R.

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Prestressed vibrations of partially filled tanks containing a free-surface fluid: finite element and reduced order models

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Prestressed Vibrations of Partially Filled Tanks Containing a Free-Surface Fluid: Finite Element and Reduced Order ModelsC. Hoareau, J.-F. Deü and R. Oha onVIII International Conference on Computational Methods for Coupled Problems in Science and EngineeringCOUPLED PROBLEMS 2019E. On˜ate, M. Papadrakakis and B. Schrefler (Eds)PRESTRESSED VIBRATIONS OF PARTIALLY FILLEDTANKS CONTAINING A FREE-SURFACE FLUID:FINITE ELEMENT AND REDUCED ORDER MODELSC. HOAREAU∗, J.-F. DEU¨∗ AND R. OHAYON∗∗Laboratoire de Me´canique des Structures et des Syste`mes Couple´es (LMSSC)Conservatoire national des arts et me´tiers (Cnam), 292 Rue Saint-Martin, Paris 75003, Francee-mail: {christophe.hoareau, jean-francois.deu, roger.ohayon}@lecnam.net, web page:http://www.lmssc.cnam.fr/enKey words: reduced order model, hydroelasticity, geometrical nonlinearity, prestressedvibrations, finite elementsAbstract. In linear vibration analysis of partially filled elastic tanks [1], even if thethe structure is submitted by a gaz or a liquid pressure, the reference configuration isgenerally used without the effect of static loads. In the case of very thin structures or softmaterial, the static state is considered as prestressed, due to geometrical nonlinearities ofthe deformed tank. The global stiffness of the structure may change in function of thefluid volume amount [2, 3, 4]. The aim of the paper is to quantify the prestressed effets onthe linearized dynamic behavior of the fluid-structure system. The chosen methodology isthe following: (i) A quasi-static solution is computed from an empty to a fully filled stateof the tank, by considering geometrical nonlinearities and hydrostatic follower forces [5](no volumetric mesh of the fluid is needed for this step); (ii) after a volumetric remeshingof the fluid at each states, a linearized hydroelastic displacement-pressure formulationaround the prestressed state, without gravity effects, is established; (iii) a reduced basisof the hydroelastic problem is generated by using prestressed dry modes to minimize thecomputation of the added mass matrix. Numerical examples are given to illustrate theproposed approaches.1330C. Hoareau, J.-F. Deu¨ and R. Ohayon1 INTRODUCTIONFor geometrical simple fluid-structure systems such as cylinders, relevant of semi-analytical analysis, the problem has been solved by using, as a projection basis, thestructural dry modes (i.e. modes in vacuo).For complex structures, it is known that the presence of an added mass operator maylead to prohibitive computational times and memory (computation of a Schur complementmatrix). An alternative formulation, for partially filled structures, using the computationof the added mass for a limited number of filled structures, has been presented in [1]. Inthe paper, the aim is to evaluate here the approach using only dry-structural modes, butprestressed, in order to reduce future computational times.1.1 Definition of the emptying rateWe consider an elastic tank partially filled by an internal fluid. We denote by Vt theinitial volume tank and Vf the current fluid volume corresponding to various level of liquidin the tank. We define the emptying rate τ such thatτ =Vt − VfVt(1)The main objective of the study is to estimate the influence of this parameter on theprestressed dynamic behavior of the fluid-structure system. We make the assumptionthat the frequency range of interest is high enough (f > 5 Hz) compared to the evolutionof the emptying rate, so the evolution of the fluid volume is supposed to be quasi-staticin our study. Thereby, at each emptying rate the fluid-structure system vibrations arecomputed around a quasi-static equilibrium state.1.2 Fluid-structure-hypothesesStructure - In the numerical examples, the structure is supposed to be isotropic, homo-geneous, elastic and prestressed. At the equilibrium, the displacements of the structureare high enough to take into account the prestressed effect on the dynamic behavior dueto an hydrostatic or a gas pressurization. The linear vibrations are computed around eachprestressed configurations. Three states of the structural domain, described in Fig. 1, areconsidered. At first, the reference configuration Ω0 is the domain occupied by the solidwithout any external forces. The associated position vector is notedX0. Then, the current”quasi-static” Ωq is the domain occupied by the structure considering finite deformations.The position of the prestressed state is associated with xq = X0 + uq. Finally, we denoteΩs the current ”dynamic” configuration under the assumption of harmonic excitationsand the position vector xs = xq+us. In the following, Ωs and Ωq coincide because ‖us‖ isvery small compared the thickness t of the elastic tank. All the domains are bounded of R3.2331C. Hoareau, J.-F. Deu¨ and R. OhayonFigure 1: (a) Reference configuration for a dry structure; (b) Current ”quasi-static” configuration infinite deformation; (c) Linearized ”structural” dynamic configuration around the prestressed state.Free-surface fluid hypotheses - At the equilibrium, the horizontal free-surface fluid issupposed to be at rest, homogeneous, inviscid, incompressible and without surface tension.We neglect, as usually done in hydroelastic linear vibrations, the gravity effect (sloshing).Consequently, the pressure fluctuation is null on the free-surface Γ and written asp = 0 on Γ (2)Finally, we denote by Σ the fluid-structure interface on the current configuration. TheFig. 2 recapitulates the methodology such that:• Phase 1: Computation of the quasi-static deformation in finite deformation;• Phase 2: Coincident re-meshing of the fluid and the structure around the prestressedstate;• Phase 3: Prestressed hydroelastic vibrationsOnly Phase 3 will be presented in the current paper.Figure 2: (a) Phase 1: Prestressed state computed with follower forces (due to a gas or liquide pressure);(b) Phase 2: Current configuration of the structure Ωq around a quasi-static equilibrium and the fluiddomains Ωf at rest; (c) Phase 3: Hydroelastic vibrations around the prestressed state with an excitationf and a pressure fluctuation p = 0 on the free surface Γ3332C. Hoareau, J.-F. Deu¨ and R. Ohayon2 Discretized added mass prestressed hydroelastic formulationIn this section, the nodal finite element displacementUq of a deformed elastic structure,related to the quasi-static finite deformation, is supposed to be known. We denote byUs the fluctuation of displacement vector around the prestressed state. The structure issupposed to be clamped at ∂uΩ0 = ∂uΩs.2.1 Symmetric eigenvalue problem with added mass matrixWe will use an intermediate discrete variable called the fluid displacement potential,denoted by ϕ, related to the pressure fluctuation vector P by P = ρfω2ϕ (with ϕ = 0on Γ). Using the condition of P or ϕ, the discretized variational approach of the FSIproblem leads to a pure structural eigenvalue problem written as[Ktan − ω2(M+Ma)]Us = 0 (3)Ma = ρfC∗H−1∗ CT∗ (4)where ρf is the fluid density, ∗ corresponds to the matrices which take into account ϕ = 0on Γ and:• Ktan is the tangent stiffness matrix which is symmetric and supposed to be positive-definite. The tangent matrix is a contribution of the geometrical, material and loadcontribution to the stiffness of the prestressed structure, respectively denoted byKg, Km and Kf, such that:Ktan = Kg +Km +Kf (5)• M is the positive-definite mass matrix of the structure;• Ma is the added mass matrix which is positive-definite, symmetric and full in thecolumns and the lines related to the fluid-structure interface displacements degreesof freedom;• H is a fluid operator related to the kinetic energy of the displaced fluid induced bythe displacement of the fluid-structure interface;• C is the coupling matrix at the fluid-structure interface.We shall note thatKtan,M, C andH depend on the known quasi-static displacement Uq.The eigenvalues and eigenvectors of the system Eq. (3) are respectively the prestressedhydroelastic natural angular frequencies ωα and the structural part of the hydroelasticmodesUα. Due to the fluid incompressibility and geometric considerations from the fluid-structure interface displacements, the potential of displacement part is then deduced, foreach hydroelastic modes, with the following equations:H∗ϕα = −CT∗Uα (6)4333C. Hoareau, J.-F. Deu¨ and R. OhayonDue to the matrix inverse computation H−1∗ in Eq. (4), the computational time mightbecome prohibitive.3 PROJECTION ON PRESTRESSED DRY MODESThe projection of the coupled problem on the dry modes is presented in three steps :Step 1: We compute N eigenvectors by solving the linearized eigenvalue problem of thestructure in vacuo:[Ktan − ω2M]Us = 0 (7)Step 2: For each deformed shapes Uβ, we solve a linear system to compute the associ-ated fluid potential displacement [1] expressed as:H∗ϕβ = −CT∗Uβ (8)where ϕβ is the associated fluid potential displacement (ϕβ = 0 on the discretized freesurface Γ). We introduce the following notations:B =......U1 . . . UN...... , Φ =......ϕ1 . . . ϕN...... and κ =κ1...κN (9)where B and Φ are the rectangular matrices containing respectively each structural eigen-vectors (normalized by the mass matrixM) and the associated potential of displacementson the fluid for each mode. κ is the generalized unknown coordinates vector. We supposethat N is high enough such that Us  Bκ.Step 3: The induced pressure associated with the βth dry mode is written in the fol-lowing form:Pβ = −ω2ρfϕβκβ (10)We take into account each pressure contribution of the fluid, induced by the structuraldry modes displacements such that an external load appear in the right side of Eq. (7)written as:[Ktan − ω2M]Bκ = ω2ρfCΦκ (11)Finally by multiplying Eq. (11) by BT in the left, we obtain the following reduced eigen-value equation:[Ω− ω2(I+ ra)]κ = 0 (12)where Ω = BTKtantB, I = BTMB and ra = ρfBTCTΦ. We shall denote the followingremarks:5334C. Hoareau, J.-F. Deu¨ and R. Ohayon• The resulting reduced matrix problem is very small due to the fact that N is sup-posed to be very small compared to the displacement structural degree of freedom;• All the matrices are symmetric, according to Eq. (8), ra = ρfBTCH−1CTB, exceptthat no matrix inversion have been computed for ra but only N linear system ofequations;• If N is high enough, the K first eigenvalues of this reduced problem (with K ≤ N)converge to the K first one of the hydroelastic with the full added mass matrix;• The N linear system of equations can be done in parallel.4 Numerical examplesThe objective of the numerical examples is to compare the hydroelastic dynamic be-havior computed with (i) classical added mass formulation and (ii) the coupled problemprojected on the dry mode. Two cases are presented, one with prestressed effects and onewithout.4.1 Example without prestressed effects of a partially filled cylinderThe first case concerns the computation of a partially filled elastic clamped cylinderwith a rigid bottom (see Fig. 3 for the geometry and the mesh parameterization). Thematerial properties are the Young modulus E = 200× 109 Pa, the Poisson ratio ν = 0.33,the structural mass density ρs = 7.8× 103 kg/m3 and ρf = 1.0× 103 kg/m3.Figure 3: (a) Geometry of the partially filled elastic clamped cylinder with rigid bottom with t = 6 mm,L = 6 m and R = 2 m; (b) Mesh parameterization, with 20 hexaedral elements, where nt = 1 and nl = 4are fixed.The objective of this example consist to show the convergence of the problem projectedon the dry modes toward the classic added mass formulation involving the computationof an inverse matrix. Three cases are analyzed :6335C. Hoareau, J.-F. Deu¨ and R. Ohayon• An empty cylinder τ = 0 with nθ = 10, nz = 10, nf = 0;• A half filled cylinder τ = 0.5 with nθ = 10, nz = 5, nf = 5;• A fully filled cylinder τ = 1 with nθ = 10, nz = 0, nf = 10.The hydroelastic analysis, done with the added mass formulation are given in Fig. 4.As expected, the natural hydroelastic frequencies rises when τ decreased.0 10 200501001502002503003504004500 10 200501001502002503003504004500 10 20050100150200250300350400450Figure 4: Representation of the circumferential and the longitudinal wave numbers n and m. (a)Hydroelastic frequencies with τ = 1; ; (b) Hydroelastic frequencies with τ = 0.5; (c) Natural frequenciesin vacuoA convergence analysis of the K = 20 first hydroelastic natural frequencies is presentedin Fig. 5 and Fig. 6 for τ = 1 and τ = 0.5. For both cases the hydroelastic naturalfrequencies computed with the projection on N dry modes are done with N = 20, N = 50,N = 100 and N = 150:• the proposed method based on the projection of the coupled problem on N drymodes converge toward the classic added mass formulation when N increases;7336C. Hoareau, J.-F. Deu¨ and R. Ohayon• the maximum error within the K = 20 hydroelastic frequencies, decreases, but notcontinuously. In fact, it depends on the contribution of a specifics number of modes.It means that the number of linear system of equation could be reduced if the modalcontribution could be known a priori.5 10 15 2001020304050607080905 10 15 2030405060708090Figure 5: Value natural hydroelastic frequencies for the 20 first K modes in dashed blue (added massformulation) and hydroelastic frequencies computed with the projection on N dry modes in red (N = 20,N = 50, N = 100 and N = 150); (a) τ = 1; (b) τ = 0.5.50 100 150010203050 100 1500102030Figure 6: Evolution of the maximum error on the K = 20 first hydroelastic modes between the twoapproaches in function of the number of dry modes. The added mass formulation with the full matrix isthe reference.A comparison between computational times is done for the fully filled cylinder τ , con-sidering various mesh parameterizations which increase the number of fluid degrees offreedom (see Tab. 1). Two computational times are considered:8337C. Hoareau, J.-F. Deu¨ and R. Ohayon• the time needed to compute H−1∗ ;• the time needed to solve N linear system form Eq. (8).(nθ,nf) (5, 5) (10, 10) (15, 15)nelem-s (structure) 100 400 900nelem-f (fluid) 500 2600 6975ndof-s (structure) 2100 8400 18900ndof-f (fluid) 2260 11020 29280Table 1: Number of elements and number of degrees of freedom for the structure and the fluid in functionof the mesh parameterizationIn Fig. 7, two graphs are proposed (one for N = 20 and the other for N = 150) inwhich the computational time is plotted in function of the number of degrees of freedomon the fluid.0 0.5 1 1.5 2 2.5 310 410 -110 010 110 210 310 40 0.5 1 1.5 2 2.5 310 410 010 110 210 310 4Figure 7: Computational time in function of the number of pressure degrees of freedom (in red a matrixinversion is computetd, in blue N linear system is computed one by one and in dashed blue line N linearsystem is done in parallel with 12 cores).• As expected, if the number of degrees of freedom is to high, the computation timeneeded to inverse the matrix is prohibitive (more than 2000 second for 29280 fluiddegrees of freedom);9338C. Hoareau, J.-F. Deu¨ and R. Ohayon• In the proposed approach with the projection on dry modes, N linear system ofequations are computed. Except for very coarse systems, the computational timeof the projection on dry modes is lower than the matrix inversion (it depend on thefluid degree of freedoms and the number N of linear system). Even if those systemscan be solved in parallel, among the N contributions, it seems that only a few ofthem contributes to the convergence. The number of linear systems could be evenmore reduced.4.2 Validation with prestressed effectsThe projection on prestressed dry mode have been performed in a clamped elasticbottom based on an experiment done in [2]. The problem consists in computing thehydroelastic angular frequencies of the prestressed elastic circular plate in function of thefluid height. Geometrical nonlinearities are taken into account and have been computedbefore the dynamic analysis. The geometrical and mesh parameters are exposed in Fig.8. The material parameters are E = 5.47 × 109, ν = 0.38, ρs = 1.4 × 103 kg/m3 andρf = 1.0× 103 kg/m3.Figure 8: (a) Geometrical parameterization of the elastic bottom submetted by an hydrostatic pressureof a fluid column of height h with Rint = 0.144 m and t = 0.35 mm; (b) Mesh parameterization, with20 hexaedral elements, used for the computation of the hydroelastic angular frequencies with nr = 10,nf = 10, nθ = 10 nt = 2.In Fig. 9, the evolution of natural angular hydroelastic frequencies are plotted between0 and 80 Hz in function of the fluid height h ∈ [0, 250] mm and dh = 5.• 50 hydroelastic reduced eigenvalues problems have been computed by using theprojection of prestressed dry modes in 1 hour;• For each fluid height, the first N = 25 dry modes have been selected;10339C. Hoareau, J.-F. Deu¨ and R. Ohayon• Except for the first modes, the angular natural frequencies decrease, then increasewhen the fluid height increase in the given fluid height range;• Very good agreements with the experiment is observed, compared to the literature;• One computation of the hydroelastic angular frequencies, for a given fluide height,with the full inversion of the added mass matrix takes more than 3h for the samemesh parameterization.Figure 9: Evolution of the angular natural hydroelastic frequencies in function of the fluid height withh ∈ [0, 250] mm and dh = 5.5 CONCLUSIONS AND OUTLOOKSIn this paper, the computation of the linearized prestressed hydroelastic behaviour ofpartially filled elastic tanks have been proposed. The numerical estimation, done with thefinite element method, of the prestressed effects on the fluid-structure dynamic analysishas been adressed. A methodology based on the projection of the fluid-structure problemon the structural prestressed dry modes is detailed in a discrete framework. It is shownthat for a given number of hydroelastic modes, the proposed approach converges to a full11340C. Hoareau, J.-F. Deu¨ and R. Ohayonhydroelastic formulation, without computing any full matrix inverse. Consequently, thecomputational time is reduced. Two numerical examples are treated, one with prestressedeffects and one without. Both results shows the effectiveness of the projection on a drymode basis. Further developments are expected in the future such as the use of theapproach on much more complex structures and the selection of a minimum number ofdry modes needed to minimize the number of linear system that has to be solved.REFERENCES[1] Morand, H.-J.P. and Ohayon, R. Fluid Structure Interaction, John Wiley & Son,1995.[2] Chiba, M. Nonlinear hydroelastic vibration of a cylindrical tank with an elastic bot-tom, containing liquid. Part I: Experiment, J. Fluids Struct. (1992) 6:181–206.[3] Schotte´, J.-S. and Ohayon, R. Various modelling levels to represent internal liquidbehaviour in the vibration analysis of complex structures Comput. Methods Appli.Mech. Engrg. (2009) 198:1313–1925.[4] Hoareau, C. and Deu¨, J.-F. A finite element approach for hydroelastic vibrations ofpartially filled prestressed elastic tanks in Proceeding of the 6th European Conferenceon Computational Mechanic. (2019), ECCM6, Glasgow, UK.[5] Haßler, M. and Schweizerhof, K. On the static interaction of fluid and gas loadedmulti-chamber systems in large deformation finite element analysis Comput. MethodsAppli. Mech. Engrg. (2009) 197:1725–1749.12341

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Prestressed vibrations of partially filled tanks containing a free-surface fluid: finite element and reduced order models

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