Rumpel, Thorsten

Schweizerhof, Karl

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Rumpel

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Finite element analysis of liquid filled structures under gravity loading

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Rumpel, T.; Schweizerhof, K.Finite element analysis of liquid filled structures under gravity loadingThe large deformation analysis of membrane or shell structures loaded and/or supported by liquid can be based on afinite element description for the structure only. Then in statics the effects in the liquid have to be considered by usingthe equations of state for the liquid, the information about the current volume and the current shape of the structure.The interaction with the structure is then modeled by a pressure resulting from the liquid always acting normal tothe current wetted structural part. This description can be easily used to model the filling process without all thedifficulties involved with standard discretization procedures. In addition the consistent derivation of the nonlinearformulation and the linearization for a Newton type scheme results in a particular formulation which can be castinto a very efficient solution procedure based on a sequential application of the Sherman-Morrison formula.1. Governing equationsThe mathematical description of static liquid structure interaction can be based on the principle of stationarity forthe total potential energy δW of a liquid in an elastic structure and additional equations describing the physicalbehaviour of the liquid. The variation of the elastic potential of the structure is specified by δelV ; δV denotes thevariation of a compressible gravity potential which interferes between structure and fluid; δexΠ is the virtual workof external forces acting on the structure, see also [1].δW = δelV + δV − δexΠ = 0 (1)The interaction term between liquid and structure is described by a body fixed pressure force p ∗n, with a non-normalized normal vector ∗n = x,ξ ×x,η and the pressure level p, see equation (2). x,ξ, x,η denote covariant unitvectors on the wetted surface of the structure. The pressure acts normal to the covariant surface element dξdη alongthe virtual displacement δu. Therefore a variational expression of a follower force is given.δV =∫η∫ξp ∗n · δu dξdη (2)The hydrostatic pressure law p for compressible liquids can be derived from a variational analysis of the gravitypotential and the virtual work expression of the pressure resulting from Hooke’s law hp = −K v−VV by respecting massconservation for the liquid. For technical applications an uniform density and compression distribution throughoutthe volume v of the liquid can be assumed. K denotes the bulk modulus and V the undeformed volume of the liquid.p = cp − xp − hp = ρ(v)g · (c − x) − hp (3)With g as gravity and ρ as density, the pressure at the center c of the liquid is given by: cp = ρ(v)g ·c and resp. thepressure at an arbitrary point x on the wetted structure by: xp = ρ(v)g ·x. In the view of a mesh-free representationof the liquid, the constitutive equations are described by the shape and the volume of the structure via a boundaryintegral representation. The volume v and the center c of the liquid can be computed via:v =13∫η∫ξx ·∗ n dξdη and c = 14v∫η∫ξx x ·∗ n dξdη. (4)2. Linearization, discretization and solution algorithmA large deformation analysis of the structure including the liquid can be performed using a Newton type scheme byapplying a Taylor series expansion on the total potential energy and the constraint equations. The discretizationwith standard FE-shell elements results in a symmetric displacement formulation. This implies that the proposedmodel is conservative. The system of equations with the elastic and load stiffness matrix el,K, the residual ofinternal, external and interaction forces F, the nodal displacement vector d, a volume pressure gradient α and tworank-one vectors a and b results in:[el,K + αa ⊗ a + b⊗ a + a ⊗ b]d = F. (5)This can be interpreted as a symmetric rank-three update of the matrix el,K coupling all wetted degrees of freedomtogether. Applying the Sherman - Morrison formula an efficient solution can be computed by two additional forward-PAMM · Proc. Appl. Math. Mech. 3, 312–313 (2003) / DOI 10.1002/pamm.200310428  First published in: EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository) http://digbib.ubka.uni-karlsruhe.de/volltexte/1000000546 backward substitutions:d1 =el, K−1 F, d2 = el,K−1 a, d3 =el, K−1 b. (6)The nodal displacement vector for one iteration step is given by a linear combination of the three auxiliary solutionvectors: d = d(d1,d2,d3).3. Numerical exampleAn elastic cylindrical vessel (weightless, elastic modulus E = 21 · 1010 Nm2 , Poisson’s ratio ν = 0.3) with a very thinwall - close to a membrane - is completely filled with water (density ρ = 1000 kgm3 , bulk modulus K = 0.5 · 109 Nm2 ).In a first load step the vessel is pressurized by 1 bar at the top of the vessel indicating the weight of the plate. In asecond step the structure is loaded by a given displacement uext of the loading plate, see figure (1b).a) c)b)hduext0 1 2 3 40.90.9511.051.121002200230024002500260000.511.522.533.544.50 1 2 3 42.533.544.55uext [m] v [m3] uext [m]ρρ0vv0mh,gp[bar]liquidgas·103c[m]d) e) f)Figure 1: a) radial displacement vectors - first load step; b) height h = 10m, diameter d = 10m, piston diplacementuext = 4m; c) radial displacement vectors - final load step; d) mass conservation vs. piston displacement uext; e)liquid pressure hp / gas pressure gp vs. fluid volume v; f) location of center c of volume vs. piston displacement uextDue to the large deformation with a maximum of uext = 4m density and volume change according to the conservationof mass, see figure (1) d). The decrease of the volume implicates an increase in the pressure level hp in the liquid.For comparison only a gas filling is considered too, see figure (1) e). A provisional result is the position of the centerc of the volume, which changes with the displacement of the piston, see figure (1) f).4. ConclusionThe proposed approach to describe the interaction of liquids with strongly deforming structures by state equationshas several advantages. First, the mesh-free analysis of the liquid avoids remeshing procedures in large deformationanalysis. Second, no contact models between liquid and structure have to be considered. Third, stability inves-tigations can be carried out taking the specific decomposition of the stiffness matrix of the coupled problem intoaccount. Fourth, the solution of the coupled equation can be efficiently performed based on the subsequent use ofthe Sherman - Morrison formula involving only the triangular decomposition of the structural matrix. Summarizingall, the computational effort is significant lower and better adjusted than in conventional fully discretized methods.5. References1 T. Rumpel and K. Schweizerhof.: On an efficient model for gas and fluid supported membranes and shell structures,In E. Oñate and B. Kröplin (Eds), Textile Composites and Inflatable Structures, CIMNE, Barcelona (2003)T. Rumpel, Institut für Technische Mechanik, Universität Karlsruhe (TH), D-76128 KarlsruheK. Schweizerhof, Institut für Mechanik, Universität Karlsruhe (TH), D-76128 Karlsruhe   Section 8: Computational mechanics 313  

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Finite element analysis of liquid filled structures under gravity loading

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