Efficient Finite Element Modelling and Simulation of Gas and Fluid Supported Membrane and Shell Structures T.Rumpel,K.Schweizerhof and M.HaBler1 Institute of Mechanics,University Karlsruhe (TH),Kaiserstr.12,76128 Karlsruhe, Germany,Karl.Schweizerhof@bau-verm.uni-karlsruhe.de Summary.In statics,the large deformation analysis of membrane or shell struc- tures loaded and/or supported by gas or fluid can be based on a finite element description for the structure only.Then in statics the effects in the gas or the fluid have to be considered by using the equations of state for the gas or the fluid,the information about the current volume and the current shape of the structure.The interaction of the gas/fluid with the structure,which can be also otherwise loaded,is then modelled by a pressure resulting from the gas/fluid always acting normal to the current wetted structural part.This description can be also directly used to model slow filling processes without all the difficulties involved with standard discretization procedures.In addition the consistent derivation of the nonlinear formulation and the linearization for a Newton type scheme results in a particular formulation which can be cast into a very efficient solution procedure based on a sequential applica- tion of the Sherman-Morrison formula.The numerical examples show the efficiency and the effects of the developed algorithms which are particularly important for structures,where the volume of the gas of fluid has to be considered. Key words:Pressure loading,hydrostatics,large deformations,finite ele- ments,deformation dependent loading,membranes,shell structures 1 Introduction The simulation of the inflation resp.filling and the support of thin membrane or shell type structures by gas or fluids can be usually performed in an effi- cient way by assuming an internal pressure in the structure which acts normal to the inner surface [2],[3],[12]besides any other loading.The restriction of this model is that it does not take into account the change of the volume of the gas or fluid due to the deformation of the structure even if there is no further inflation or filling.Also the pressure may change due to temperature modifications of the gas/fluid.In both cases the volume of the gas resp.the mass conservation of the fluid has to be considered in the model [1],[7];this 153 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,153-172. C 2005 Springer.Printed in the Netherlands
Efficient Finite Element Modelling and Simulation of Gas and Fluid Supported Membrane and Shell Structures T. Rumpel, K. Schweizerhof and M. Haßler1 Institute of Mechanics, University Karlsruhe (TH), Kaiserstr. 12, 76128 Karlsruhe, Germany, Karl.Schweizerhof@bau-verm.uni-karlsruhe.de Summary. In statics, the large deformation analysis of membrane or shell structures loaded and/or supported by gas or fluid can be based on a finite element description for the structure only. Then in statics the effects in the gas or the fluid have to be considered by using the equations of state for the gas or the fluid, the information about the current volume and the current shape of the structure. The interaction of the gas/fluid with the structure, which can be also otherwise loaded, is then modelled by a pressure resulting from the gas/fluid always acting normal to the current wetted structural part. This description can be also directly used to model slow filling processes without all the difficulties involved with standard discretization procedures. In addition the consistent derivation of the nonlinear formulation and the linearization for a Newton type scheme results in a particular formulation which can be cast into a very efficient solution procedure based on a sequential application of the Sherman-Morrison formula. The numerical examples show the efficiency and the effects of the developed algorithms which are particularly important for structures, where the volume of the gas of fluid has to be considered. Key words: Pressure loading, hydrostatics, large deformations, finite elements, deformation dependent loading, membranes, shell structures 1 Introduction The simulation of the inflation resp. filling and the support of thin membrane or shell type structures by gas or fluids can be usually performed in an effi- cient way by assuming an internal pressure in the structure which acts normal to the inner surface [2], [3], [12] besides any other loading. The restriction of this model is that it does not take into account the change of the volume of the gas or fluid due to the deformation of the structure even if there is no further inflation or filling. Also the pressure may change due to temperature modifications of the gas/fluid. In both cases the volume of the gas resp. the mass conservation of the fluid has to be considered in the model [1], [7]; this 153 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 153–172. © 2005 Springer. Printed in the Netherlands
154 T.Rumpel,K.Schweizerhof and M.HaBler is also important for stability considerations of the gas/fluid filled structure under any other external loading.Then in the case of gas filling the inter- nal gas pressure formally provides an additional rank-one update of the FE stiffness matrix which stabilizes from an engineering point of view an almost completely flexible structure.Hydrostatics with a free fluid surface,also leads to an additional rank-one update of the FE stiffness matrix,whereas com- pressible,heavy fluids lead to a rank-two update 5. Thus in the case of fluid filling [6],[8],[10]or a mixture of gas and fluid [9] the filling of membrane-like structures can be performed without a separate discretization of the fluid with e.g.FE or Finite Volumes or similar.Also fully fluid filled structures can be analyzed without separate discretization of the fluid [9].Several cases have to be distinguished,fluid with a free surface [8], [10],structure under overpressure of fluid [9]and fluid with free surface but gas overpressure.The contribution shows the derivation of the variational formu- lation and the corresponding Finite Element discretization for compressible fluids under gravity loading.A particular focus is on the consistent lineariza- tion of the nonlinear equations and the accompanying constraint equations. Also the specific solution of the linearized equation system based on a sequen- tial application of the Sherman-Morrison formula is presented.The numerical examples show large deformation analyses of gas and fuid filled shell struc- tures with rather thin flexible walls under various conditions,such as filling and loading. 2 Governing Equations The mathematical description of static fluid structure interaction can be based on the principle of stationarity for the total potential energy oW of a fluid in an elastic structure and additional equations describing the physical behavior of different fluids or gases. 2.1 Virtual Work Expression The variation of the elastic potential of the structure is specified by oev,6'IT denotes the virtual work of the pressure loading which acts between the fluid i and the structure,ger II is the virtual work of other external forces acting on the structure 6W=6elV+62Ⅱ-6ezⅡ=0 (1) The interaction term between fluid and structure is described by a body fixed pressure force 'p'n,with a non-normalized normal vector 'n =es x en and the pressure level p,see equation (2).e=,en=denote covariant non-normalized vectors on the wetted surface of the structure (2)
154 T. Rumpel, K. Schweizerhof and M. Haßler is also important for stability considerations of the gas/fluid filled structure under any other external loading. Then in the case of gas filling the internal gas pressure formally provides an additional rank-one update of the FE stiffness matrix which stabilizes from an engineering point of view an almost completely flexible structure. Hydrostatics with a free fluid surface, also leads to an additional rank-one update of the FE stiffness matrix, whereas compressible, heavy fluids lead to a rank-two update [5]. Thus in the case of fluid filling [6], [8], [10] or a mixture of gas and fluid [9] the filling of membrane-like structures can be performed without a separate discretization of the fluid with e.g. FE or Finite Volumes or similar. Also fully fluid filled structures can be analyzed without separate discretization of the fluid [9]. Several cases have to be distinguished, fluid with a free surface [8], [10], structure under overpressure of fluid [9] and fluid with free surface but gas overpressure. The contribution shows the derivation of the variational formulation and the corresponding Finite Element discretization for compressible fluids under gravity loading. A particular focus is on the consistent linearization of the nonlinear equations and the accompanying constraint equations. Also the specific solution of the linearized equation system based on a sequential application of the Sherman-Morrison formula is presented. The numerical examples show large deformation analyses of gas and fluid filled shell structures with rather thin flexible walls under various conditions, such as filling and loading. 2 Governing Equations The mathematical description of static fluid structure interaction can be based on the principle of stationarity for the total potential energy δW of a fluid in an elastic structure and additional equations describing the physical behavior of different fluids or gases. 2.1 Virtual Work Expression The variation of the elastic potential of the structure is specified by δelV , δi Π denotes the virtual work of the pressure loading which acts between the fluid i and the structure, δexΠ is the virtual work of other external forces acting on the structure δW = δelV + δi Π − δexΠ = 0 (1) The interaction term between fluid and structure is described by a body fixed pressure force i p ∗n, with a non-normalized normal vector ∗n = eξ × eη and the pressure level i p, see equation (2). eξ = ∂x ∂ξ , eη = ∂x ∂η denote covariant non-normalized vectors on the wetted surface of the structure η δi Π = η ξ i p ∗n · δu dξdη (2)
FE Modelling and Simulation of Gas and Fluid Supported Structures 155 The pressure acts normal to the surface element dedn along the virtual dis- placement Ou.Therefore a virtual work expression of a follower force is given. Possible physical properties of the fluid i are summarized in the following paragraphs: 2.2 Compressible Fluids If the dead weight of a fluid is neglected,we can distinguish between a pneu- matic model,see [1,[7 and a hydraulic description.The corresponding con- stitutive equations are the Poisson's law for a pneumatic (i=p)and the Hooke's law (i=h)for a hydraulic model. Pneumatic Model In realistic physical situations the investigations can be restricted to conser- vative models,which entails the application of the adiabatic state equation Pp uh PP V*const. (3) Pp,v are the state variables (pressure and volume)of the gas in the deformed state,capital letters denote the initial state and k the isentropy constant. Hydraulic Model For an analysis of hydraulic systems the fluid pressure is given by Hooke's law."p is the mean pressure in the fluid determined by the bulk modulus K and the relative volume change of the fluid with V as initial volume @=-' (4) 2.3 Hydrostatic Loading-Incompressible Fluids under Gravity Loading For partially filled structures the liquid can be treated as incompressible,see [6],[8],[10.The pressure distribution is given by the hydrostatic pressure law,with p as the constant density,g as the gravity and with the difference of the upper liquid level x and an arbitrary point x on the wetted structure.A conservative description is achieved,if the volume conservation of the liquid is taken into account during the deformation of liquid and structure,too 9p=Pg·(°x-x) (5) and v=const. (6)
FE Modelling and Simulation of Gas and Fluid Supported Structures 155 The pressure acts normal to the surface element dξdη along the virtual displacement δu. Therefore a virtual work expression of a follower force is given. Possible physical properties of the fluid i are summarized in the following paragraphs: 2.2 Compressible Fluids If the dead weight of a fluid is neglected, we can distinguish between a pneumatic model, see [1], [7] and a hydraulic description. The corresponding constitutive equations are the Poisson’s law for a pneumatic (i = p) and the Hooke’s law (i = h) for a hydraulic model. Pneumatic Model In realistic physical situations the investigations can be restricted to conservative models, which entails the application of the adiabatic state equation pp vκ = pP V κ = const. (3) pp, v are the state variables (pressure and volume) of the gas in the deformed state, capital letters denote the initial state and κ the isentropy constant. Hydraulic Model For an analysis of hydraulic systems the fluid pressure is given by Hooke’s law. hp is the mean pressure in the fluid determined by the bulk modulus K and the relative volume change of the fluid with V as initial volume hp(v) = −K v − V V (4) 2.3 Hydrostatic Loading – Incompressible Fluids under Gravity Loading For partially filled structures the liquid can be treated as incompressible, see [6], [8], [10]. The pressure distribution is given by the hydrostatic pressure law, with ρ as the constant density, g as the gravity and with the difference of the upper liquid level ox and an arbitrary point x on the wetted structure. A conservative description is achieved, if the volume conservation of the liquid is taken into account during the deformation of liquid and structure, too gp = ρg · ( ox − x) (5) and v = const. (6)
156 T.Rumpel,K.Schweizerhof and M.HaBler 2.4 Compressible Hydrostatic Loading-Compressible Fluids Under Gravity Loading A further important case is the composition of dead weight and compressibility of fluid (i=hg),see [5],9].The corresponding pressure law for technical applications can be found by combining Hooke's law and mass conservation with the assumption of an uniform density distribution throughout the fluid. The hydrostatic pressure law for compressible fluids can be derived from a variational analysis of the gravity potential and the virtual work expression of the pressure resulting from Hooke's law hop=p-ip-hp (7)） =p(u)g·(c-x)-hp (8) with p(v)v=const. (9) p =p(v)g.c is the pressure at the center c of volume,p =p(v)g.x denotes the pressure at an arbitrary point x on the wetted structure.In the view of a mesh-free representation of the fluid,the constitutive equations are dependent on the shape and on the volume of the gas or fluid enclosed by the structure or by parts of the structure.It must be noted that the term p is responsible for the compression of the fluid due to its own dead weight. 2.5 Boundary Integral Representation of Volume and Center of Volume The goal of this approach is that all necessary quantities can be expressed by a boundary integral representation.This allows to formulate all state variables via an integration of the surrounding wetted surface.The fluid volume v and the center c of the volume can be computed via: 3n .*ndξdn (10) and xx.*n dedn (11) A large deformation analysis of the structure including the fluid can be per- formed using a Newton type scheme for the solution by applying a Taylor series expansion on the governing equations.The following linearization is shown in short for all four cases discussed above.For details we refer to [1], [61,[7]. 3 Linearization of the Volume Contribution for Gas and Fluid Models Within the Newton scheme the deformed state is computed iteratively.Both, the virtual expression and the different additional constraint equations have
156 T. Rumpel, K. Schweizerhof and M. Haßler 2.4 Compressible Hydrostatic Loading – Compressible Fluids Under Gravity Loading A further important case is the composition of dead weight and compressibility of fluid (i = hg), see [5],[9]. The corresponding pressure law for technical applications can be found by combining Hooke’s law and mass conservation with the assumption of an uniform density distribution throughout the fluid. The hydrostatic pressure law for compressible fluids can be derived from a variational analysis of the gravity potential and the virtual work expression of the pressure resulting from Hooke’s law hgp = c p − xp − hp (7) = ρ(v)g · (c − x) − hp (8) with ρ(v)v = const. (9) cp = ρ(v)g · c is the pressure at the center c of volume, xp = ρ(v)g · x denotes the pressure at an arbitrary point x on the wetted structure. In the view of a mesh-free representation of the fluid, the constitutive equations are dependent on the shape and on the volume of the gas or fluid enclosed by the structure or by parts of the structure. It must be noted that the term cp is responsible for the compression of the fluid due to its own dead weight. 2.5 Boundary Integral Representation of Volume and Center of Volume The goal of this approach is that all necessary quantities can be expressed by a boundary integral representation. This allows to formulate all state variables via an integration of the surrounding wetted surface. The fluid volume v and the center c of the volume can be computed via: v = 1 3 η ξ x · ∗ n dξdη (10) and c = 1 4v η ξ x x · ∗ n dξdη (11) A large deformation analysis of the structure including the fluid can be performed using a Newton type scheme for the solution by applying a Taylor series expansion on the governing equations. The following linearization is shown in short for all four cases discussed above. For details we refer to [1], [6], [7]. 3 Linearization of the Volume Contribution for Gas and Fluid Models Within the Newton scheme the deformed state is computed iteratively. Both, the virtual expression and the different additional constraint equations have
FE Modelling and Simulation of Gas and Fluid Supported Structures 157 to be consistently linearized.The linearization of the virtual work expression leads always to three parts,the residual part ophgllt,the follower force part phgⅡAn and the pressure level part6phgⅡAp 6hgin=6hgⅡ：+sphg II An+phgⅡAp (12) /e9p:*nu+hu△'n+Ahop'n）·idd- (13) 3.1 Pneumatic and Hydraulic Model The follower force part is dependent on the structural displacements Au re- spectively the change of the non-normalized normal*n,and thus indirectly on the size of the wetted surface with △*n=△u,e×xtn+xt×△u,n (14) The pressure change differs only slightly for both models and is only depen- dent on the volume change pneumatic model:Ap=-kP *nt·△dfdn (15) rmod4p=-长/人aaa5ihV=wa间 Introducing both into (13)and integrating by parts,we obtain a field and boundary valued problem.The boundary value part vanishes completely for closed structures respectively the parts enclosing the gas/fluid volume.Thus the linearized virtual expression reads: 8p.Itin =op.hI: 6u*ntdξdn *nt·△udξdn pneumatic - 6u.*ntd延dm *nt·△udξdn hydraulic 0 h D △u Ou.E u.E d延dn(17) 2 0 △u,n with the skew symmetric tensors Wf=*nt⑧e5-ef⑧*nW”=*nt⑧e”-e”⑧*nt: (18) Obviously the final linearized expression is a symmetric displacement formu- lation indicating that the proposed model is conservative as expected
FE Modelling and Simulation of Gas and Fluid Supported Structures 157 to be consistently linearized. The linearization of the virtual work expression leads always to three parts, the residual part δphgΠt, the follower force part δphgΠ∆n and the pressure level part δphgΠ∆p δphgΠlin = δphgΠt + δphgΠ∆n + δphgΠ∆p (12) = η ξ ( phgpt ∗nt +phg pt∆∗n + ∆phgp∗nt) · δu dξdη. (13) 3.1 Pneumatic and Hydraulic Model The follower force part is dependent on the structural displacements ∆u respectively the change of the non-normalized normal ∗n, and thus indirectly on the size of the wetted surface with ∆∗n = ∆u,ξ ×xt,η +xt,ξ ×∆u,η . (14) The pressure change differs only slightly for both models and is only dependent on the volume change pneumatic model: ∆pp = −κpt vt η ξ ∗nt · ∆u dξdη (15) hydraulic model: ∆hp = −K V η ξ ∗nt · ∆u dξdη with V ≡ v0 (16) Introducing both into (13) and integrating by parts, we obtain a field and boundary valued problem. The boundary value part vanishes completely for closed structures respectively the parts enclosing the gas/fluid volume. Thus the linearized virtual expression reads: δp,hΠlin = δp,hΠt − κpt vt η ξ δu · ∗ nt dξdη η ξ ∗nt · ∆u dξdη pneumatic − K V η ξ δu · ∗ nt dξdη η ξ ∗nt · ∆u dξdη hydraulic + p,hpt 2 η ξ ⎛ ⎝ δu δu,ξ δu,η ⎞ ⎠ · ⎛ ⎝ 0 Wξ Wη WξT 0 0 WηT 0 0 ⎞ ⎠ ⎛ ⎝ ∆u ∆u,ξ ∆u,η ⎞ ⎠ dξdη (17) with the skew symmetric tensors Wξ =∗ nt ⊗ eξ − eξ ⊗∗ ntWη =∗ nt ⊗ eη − eη ⊗∗ nt. (18) Obviously the final linearized expression is a symmetric displacement formulation indicating that the proposed model is conservative as expected
158 T.Rumpel,K.Schweizerhof and M.HaBler 3.2 Hydrostatic Loading-Incompressible Fluids Under Gravity Loading The follower force part depends as in 3.1 from the change in the normal and of the gradient of the fluid under gravity loading.The latter comes into the formulation after partial integration resulting in 哈= u·[g:eeW+genW"y]△dEdn ou 0 w:w 6u, 0 dedn.(19) 0 △1n It is obvious that the first part is non-symmetric and disappears if g is set to zero.The interesting part is the volume conservation and its influence on the pressure in the linearized form.The linearized pressure is a function of the variation of the fluid level A u and the local structural deformation Au △9p=pg·(△°u-△u) (20) The volume change is zero thus the linearization is zero as well: ont·△°uddn=0. (21) Focusing on the fluid load part-the second part in(21)-we obtain based on the direction of the normal on the fluid level,which is identical to the direction of gravity,the components of the free fluid surface and the corresponding displacement ont =ont. g gl' (22) △°u=△u· (23) Thus the volume change due to the change in the fluid level can be written as g△°u· g dogdon (24) Obviously both quantities in the integral are scalars;in addition the fluid level displacement is uniform,thus we obtain (25)
158 T. Rumpel, K. Schweizerhof and M. Haßler 3.2 Hydrostatic Loading – Incompressible Fluids Under Gravity Loading The follower force part depends as in 3.1 from the change in the normal and of the gradient of the fluid under gravity loading. The latter comes into the formulation after partial integration resulting in δgΠ∆n lin = ρ 2 η ξ δu · [g · eξWξ + geηWη] ∆u dξdη + η ξ gpt 2 ⎛ ⎝ δu δu,ξ δu,η ⎞ ⎠ · ⎛ ⎝ 0 Wξ Wη WξT 0 0 WηT 0 0 ⎞ ⎠ ⎛ ⎝ ∆u ∆u,ξ ∆u,η ⎞ ⎠ dξdη. (19) It is obvious that the first part is non-symmetric and disappears if g is set to zero. The interesting part is the volume conservation and its influence on the pressure in the linearized form. The linearized pressure is a function of the variation of the fluid level ∆0u and the local structural deformation ∆u ∆gp = ρg · (∆ou − ∆u). (20) The volume change is zero thus the linearization is zero as well: ∆gv = η ξ ∗nt · ∆u dξdη + η ξ ont · ∆ou doξdoη = 0. (21) Focusing on the fluid load part – the second part in (21) – we obtain based on the direction of the normal on the fluid level, which is identical to the direction of gravity, the components of the free fluid surface and the corresponding displacement ont = ont · g |g| , (22) ∆ou = ∆o u · g |g| . (23) Thus the volume change due to the change in the fluid level can be written as ∆ov = η ξ ont · g |g| ∆ou · g |g| doξdoη. (24) Obviously both quantities in the integral are scalars; in addition the fluid level displacement is uniform, thus we obtain ∆ov = ∆ou η ξ ont · g |g| doξdoη = ∆ouSt. (25)
FE Modelling and Simulation of Gas and Fluid Supported Structures 159 S,is the size of the water surface,which can also be computed via a boundary integral over the enclosure of the fluid volume projected onto the direction of gravity nt' g dEdn. (26) Thus the change in the water level height can be written as A'u-5 1 St JoJ *nt·△udξdn (27） and the corresponding pressure change becomes △p=pg·△°u-pg·△u *nt·△udξdm-pg·△u (28) The linearized variational form of the gravity potential depending on the fluid level is then obtained as △pnt·6ud'dm *nt·△uddn u*ntg·△dedn. (29) Obviously the second part of this equation is a non-symmetric term.However, combining both non-symmetric parts ofanda symmetric ex- pression results for the complete sum 9nm=9+69+9n: =69Ⅱ：+ + u*ndd n·△udd Term I 6u·('nt图g+g8*nt)△ud'dm Term II ou 0 w:w Au 0 0 △u,e dfdn. Su:n 0 △un Term III (30)
FE Modelling and Simulation of Gas and Fluid Supported Structures 159 St is the size of the water surface, which can also be computed via a boundary integral over the enclosure of the fluid volume projected onto the direction of gravity St = η ξ ∗nt · g |g| dξdη. (26) Thus the change in the water level height can be written as ∆ou = ∆ov St = 1 St η ξ ∗nt · ∆u dξdη (27) and the corresponding pressure change becomes ∆p = ρg · ∆ou − ρg · ∆u = ρ |g| St η ξ ∗nt · ∆u dξdη − ρg · ∆u. (28) The linearized variational form of the gravity potential depending on the fluid level is then obtained as δgΠ∆p lin = η ξ ∆p∗nt · δu dξdη = ρ |g| St η ξ δu · ∗ nt dξdη η ξ ∗nt · ∆u dξdη −ρ η ξ δu · ∗ nt g · ∆u dξdη. (29) Obviously the second part of this equation is a non-symmetric term. However, combining both non-symmetric parts of δgΠ∆n lin and δgΠ∆p lin , a symmetric expression results for the complete sum δgΠlin = δgΠ∆n lin + δgΠ∆p lin + δgΠt = δgΠt + + ρ |g| St η ξ δu · ∗nt dξdη η ξ ∗nt · ∆u dξdη Term I − ρ 2 η ξ δu · ( ∗nt ⊗ g + g ⊗ ∗nt)∆u dξdη Term II + η ξ gpt 2 ⎛ ⎝ δu δu,ξ δu,η ⎞ ⎠ · ⎛ ⎝ 0 Wξ Wη WξT 0 0 WηT 0 0 ⎞ ⎠ ⎛ ⎝ ∆u ∆u,ξ ∆u,η ⎞ ⎠ dξdη. Term III (30)
160 T.Rumpel,K.Schweizerhof and M.HaBler 3.3 Compressible Fluids Under Gravity Considering only the gravity potential of a compressible fluid,we have to integrate over the total volume of the fluid =- p(x)g·xdu+const. (31) In general the density p(x)is dependent on the height of the fluid,however, for technical applications with standard heights the density can be assumed to be given by the law of mass conservation.Then for compressible and in- compressible fluids the potential is only a function of the form and the volume of the enclosed fluid hgⅡ=-p(v) g·xdv+const. (32) This can be written as a surface integral wn-) g·xx.*ndξdm+const. =-p(v)g·s, s:1.order volume moment (33) The corresponding linearized functional contains two major parts: 6g亚im=-6p(u)g·s-p(v)g·s. (34) From mass conservation p(v)v=PoV with po,V as reference values,we obtain 6p(u)-p6odse V。 (35) with *n·ud诞dn (36) The variation of the second part is identical to the variation shown in the previous paragraph.After defining the location of the center of gravity of the fuid c= (37) the variation follows as n=pe-刘nmds (38) Introducing the compressibility of the fluid in an identical fashion as in 3.1 with Hooke's law =-' (39)
160 T. Rumpel, K. Schweizerhof and M. Haßler 3.3 Compressible Fluids Under Gravity Considering only the gravity potential of a compressible fluid, we have to integrate over the total volume of the fluid hgΠ = − v ρ(x)g · xdv + const. (31) In general the density ρ(x) is dependent on the height of the fluid, however, for technical applications with standard heights the density can be assumed to be given by the law of mass conservation. Then for compressible and incompressible fluids the potential is only a function of the form and the volume of the enclosed fluid hgΠ = −ρ(v) v g · xdv + const. (32) This can be written as a surface integral hgΠ = −ρ(v) η ξ g · x x · ∗ n dξdη + const. = −ρ(v)g · s , s: 1.order volume moment (33) The corresponding linearized functional contains two major parts: δhgΠlin = −δρ(v)g · s − ρ(v)g · δs. (34) From mass conservation ρ(v)v = ρoV with ρo, V as reference values, we obtain δρ(v) = −ρo V v2 δv = −ρ(v) v δv (35) with δv = η ξ ∗n · δu dξdη. (36) The variation of the second part is identical to the variation shown in the previous paragraph. After defining the location of the center of gravity of the fluid c = s v , (37) the variation follows as δhgΠ = ρ(v) η ξ g · (c − x) ∗n · δu dξdη. (38) Introducing the compressibility of the fluid in an identical fashion as in 3.1 with Hooke’s law hp(v) = −K v − V V , (39)
FE Modelling and Simulation of Gas and Fluid Supported Structures 161 the final term of a compressible fluid is given as Shgw =hop(v) *n·ud延dm (40) It is of some help to subdivide the pressure into three parts,p(v)as above and 严p=p(v)g·x (41) cp=p(w)g·c (42) Then the form shown in(7)is given and linearization is a straightforward process with h9p=△p-△p-△p (43) 4p=-2 *nt'△a dEdn+ *nt△udd (44) Ut Ut nt·△uddm+ptg·△u (45) =- *nt·△uddn (46) The summary of the pressure changes introduced into the linearized virtual work expression results in a symmetric displacement formulation.This implies that the proposed model is conservative gⅡn=gⅡ + -2 *nt·6ud延d而 part I 人maus part II 6u.(*nt⑧g+g图*nt)△uddn part III u 0 w:w △u Su.E △u,E dcdm.(47） Su.n 00 part IV The different linearized parts can be interpreted as follows: I The multiplication of the two surface integrals indicates the volume de- pendence of the compression level and of the pressure at the center of the fluid volume
FE Modelling and Simulation of Gas and Fluid Supported Structures 161 the final term of a compressible fluid is given as δhgW = hgp(v) η ξ ∗n · δu dξdη. (40) It is of some help to subdivide the pressure into three parts, hp(v) as above and xp = ρ(v)g · x, (41) cp = ρ(v)g · c. (42) Then the form shown in (7) is given and linearization is a straightforward process with ∆hgp = ∆c p − ∆xp − ∆hp (43) ∆c p = −2 cpt vt η ξ ∗nt · ∆u dξdη + η ξ xpt vt ∗nt · ∆u dξdη (44) ∆xp = − xpt vt η ξ ∗nt · ∆u dξdη + ρtg · ∆u (45) ∆hp = −K V η ξ ∗nt · ∆u dξdη (46) The summary of the pressure changes introduced into the linearized virtual work expression results in a symmetric displacement formulation. This implies that the proposed model is conservative δhgΠlin = δhgΠt + (K V − 2 cpt vt ) η ξ ∆u · ∗nt dξdη η ξ ∗nt · δu dξdη part I + η ξ xpt vt ∗nt · ∆u dξdη η ξ ∗nt · δu dξdη + η ξ ∗nt · ∆u dξdη η ξ xpt vt ∗nt · δu dξdη part II − ρt 2 η ξ δu · ( ∗nt ⊗ g + g ⊗ ∗nt)∆u dξdη part III + 1 2 η ξ hgpt ⎛ ⎝ δu δu,ξ δu,η ⎞ ⎠ · ⎛ ⎝ 0 Wξ Wη WξT 0 WηT 0 0 ⎞ ⎠ ⎛ ⎝ ∆u ∆u,ξ ∆u,η ⎞ ⎠ dξdη. (47) part IV The different linearized parts can be interpreted as follows: I The multiplication of the two surface integrals indicates the volume dependence of the compression level and of the pressure at the center of the fluid volume
162 T.Rumpel,K.Schweizerhof and M.HaBler II The change of the local pressure is influenced by changes of the total volume and changes of the location of the center of the volume. III A hydrostatic pressure generates a nonuniform pressure field,which is rep- resented by a symmetric field equation under realistic boundary conditions, see[11],[12],[2. IV Follower forces create a symmetric field equation too,considering realistic boundary conditions,see [11],[12],[2],[4],[13]. 4 FE-Discretization and Solution Algorithm The virtual work expression 6W=6elV+6phgⅡ-6exⅡ=0 (48) followed by the linearization process as shown in chapter 3 leads to a residual and a linear term,depending only on the surfaces of the wetted resp.closed volumes.These terms have to be discretized with standard FE shell,mem- brane or continuum elements.Thus the boundary description is based on the surfaces of the FE elements wetted by the fluid or gas.Further,the discretized and linearized constraint equations as Hooke's law,the mass conservation of the fluid and the computation of the pressure at the center of the structure are included,resulting in general in a hybrid symmetric system of equations for the coupled problem: el.phgK -a -b d phgF -ar 0 0 △kp -bT (49) 0 U 0 △cD This symmetric system can be reduced to a conventional symmetric displace- ment representation with the elastic and load stiffness matrix el.phgK =el K+phgK,the residual ph9F of internal elf,external erf and interaction forces phaf,the nodal displacement vector d,a volume pressure gradient ph9a;and two rank-one vectors a and b [e,phgK+b⑧a+a⑧b]d=hgF exf-phg f_el f (50) This can be interpreted as a symmetric rank-two update of the matrix el.phgK coupling all wetted degrees of freedom together.Applying the Sherman- Morrison formula an efficient solution can be computed by two additional forward-backward substitutions: d1 =el.phg K-1 ph9F,d2=el.phgK-1 a,d3 =el.phg K-1 b.(51) The load stiffness matrix and the other pressure related terms are:
162 T. Rumpel, K. Schweizerhof and M. Haßler II The change of the local pressure is influenced by changes of the total volume and changes of the location of the center of the volume. III A hydrostatic pressure generates a nonuniform pressure field, which is represented by a symmetric field equation under realistic boundary conditions, see [11], [12], [2]. IV Follower forces create a symmetric field equation too, considering realistic boundary conditions, see [11], [12], [2], [4], [13]. 4 FE-Discretization and Solution Algorithm The virtual work expression δW = δelV + δphgΠ − δexΠ = 0 (48) followed by the linearization process as shown in chapter 3 leads to a residual and a linear term, depending only on the surfaces of the wetted resp. closed volumes. These terms have to be discretized with standard FE shell, membrane or continuum elements. Thus the boundary description is based on the surfaces of the FE elements wetted by the fluid or gas. Further, the discretized and linearized constraint equations as Hooke’s law, the mass conservation of the fluid and the computation of the pressure at the center of the structure are included, resulting in general in a hybrid symmetric system of equations for the coupled problem: ⎡ ⎢ ⎢ ⎣ el,phgK −a −b a −aT − K V 0 0 −bT 0 −2 cpv v aT 0 v 0 ⎤ ⎥ ⎥ ⎦ ⎛ ⎜⎜⎝ d ∆kp ∆ρ ρ ∆cp ⎞ ⎟⎟⎠ = ⎛ ⎜⎜⎝ phgF 0 0 0 ⎞ ⎟⎟⎠ . (49) This symmetric system can be reduced to a conventional symmetric displacement representation with the elastic and load stiffness matrix el,phgK =el K+phg K, the residual phgF of internal elf, external exf and interaction forces phgf, the nodal displacement vector d, a volume pressure gradient phgαt and two rank-one vectors a and b [ el,phgK + b ⊗ a + a ⊗ b]d = phgF = exf −phg f −el f (50) This can be interpreted as a symmetric rank-two update of the matrix el,phgK coupling all wetted degrees of freedom together. Applying the ShermanMorrison formula an efficient solution can be computed by two additional forward-backward substitutions: d1 =el,phg K−1 phgF, d2 = el,phgK−1 a, d3 =el,phg K−1 b. (51) The load stiffness matrix and the other pressure related terms are: