FE Analysis of Membrane Systems Including Wrinkling and Coupling Riccardo Rossi,Vitaliani Renatol,and Eugenio Onate2 1 Universita di Padova ricrossi@caronte.dic.unipd.it 2 CIMNE,UPC,Barcelona onate cimne.upc.es 1 Abstract Current work summarizes the experience of the writer in the modeling of membrane systems.A first subsection describes an efficient membrane model, together with a reliable solution procedure.The following section addresses the simulation of the wrinkling phenomena providing details of a new solution procedure.The last one proposes an efficient technique to obtain the solution of the fluid structural interaction problem. 2 The Membrane Model A membrane is basically a 2D solid which "lives"in a 3D environment.Given the lack of flexural stiffness,membranes can react to applied load only by using their in-plane resistance "choosing"the spatial disposition that is best suited to resist to the external forces.The consequence is that membrane structures tend naturally to find the optimal shape(compatible with the ap- plied constraints)for any given load.In this "shape research",they typically undergo large displacements and rotations. From a numerical point of view,this reflects an intrinsic geometrical non- linearity that has to be taken in account in the formulation of the finite element model.In particular,an efficient Membrane Element should be able to represent correctly arbitrary rotations both of the element as a whole and internally to each element.The possibility of unrestricted rigid body motions constitutes a source of ill-conditioning or even of singularity of the tangent stiffness matrix introducing the need of carefully designed solution procedures. 2.1 Finite Element Model Current section describes a finite element model that meets all of the require- ments for the correct simulation of general membrane systems.The derivation 89 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,89-108. C 2005 Springer.Printed in the Netherlands
FE Analysis of Membrane Systems Including Wrinkling and Coupling Riccardo Rossi 1, Vitaliani Renato1, and Eugenio Onate2 1 Universit`a di Padova ricrossi@caronte.dic.unipd.it 2 CIMNE, UPC, Barcelona onate@cimne.upc.es 1 Abstract Current work summarizes the experience of the writer in the modeling of membrane systems. A first subsection describes an efficient membrane model, together with a reliable solution procedure. The following section addresses the simulation of the wrinkling phenomena providing details of a new solution procedure. The last one proposes an efficient technique to obtain the solution of the fluid structural interaction problem. 2 The Membrane Model A membrane is basically a 2D solid which “lives” in a 3D environment. Given the lack of flexural stiffness, membranes can react to applied load only by using their in–plane resistance “choosing” the spatial disposition that is best suited to resist to the external forces. The consequence is that membrane structures tend naturally to find the optimal shape (compatible with the applied constraints) for any given load. In this “shape research”, they typically undergo large displacements and rotations. From a numerical point of view, this reflects an intrinsic geometrical non– linearity that has to be taken in account in the formulation of the finite element model. In particular, an efficient Membrane Element should be able to represent correctly arbitrary rotations both of the element as a whole and internally to each element. The possibility of unrestricted rigid body motions constitutes a source of ill-conditioning or even of singularity of the tangent stiffness matrix introducing the need of carefully designed solution procedures. 2.1 Finite Element Model Current section describes a finite element model that meets all of the requirements for the correct simulation of general membrane systems. The derivation 89 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 89–108. © 2005 Springer. Printed in the Netherlands
90 Riccardo Rossi,Vitaliani Renato,and Eugenio Onate makes use exclusively of orthogonal bases simplifying the calculations and al- lowing to express all the terms directly in Voigt Notation,which eases the implementation. Einstein's sum on repeated indices is assumed unless specified otherwise .xI=,is the position vector of the I-th node in the cartesian space(3D space) .={describes the position of a point in the local system of coordi- nates Capital letter are used for addressing to the reference configuration Nr(g)is the value of the shape function centered on node I on the point of local coordinatesξ The use of the standard iso-parametric approach allows to express the position of any point as x()=NI()xI. In the usual assumptions of the continuum mechanics it is always possible to define the transformation between the local system of coordinates and the cartesian system as 长+，P-长，P-0x5=gG ∂灰 (1) {长，7+dr-长，nr-05l=g, (2) in which we introduced the symbols gE= aN,x灯 (3) DE gn= aN,dx灯 8n (4) the vectors ge and gn of the 3D space,can be considered linearly independent (otherwise compenetration or self contact would manifest)follows immedi- ately that they can be used in the construction of a base of the 3D space.In particular an orthogonal base can be defined as v1二ge gE (5) n= g:X gn →V2=nXV1 (6) lge×gml V3=g X gn (7) Vectors vI and v2 describe the local tangent plane to the membrane while the third base vector is always orthogonal.Follows the possibility of defining a local transformation rule that links the local coordinates and the coordinates
90 Riccardo Rossi, Vitaliani Renato, and Eugenio Onate makes use exclusively of orthogonal bases simplifying the calculations and allowing to express all the terms directly in Voigt Notation, which eases the implementation. Einstein’s sum on repeated indices is assumed unless specified otherwise • xI = {xI , yI , zI } T is the position vector of the I–th node in the cartesian space (3D space) • ξ = {ξ, η} T describes the position of a point in the local system of coordinates • Capital letter are used for addressing to the reference configuration • NI (ξ) is the value of the shape function centered on node I on the point of local coordinates ξ The use of the standard iso-parametric approach allows to express the position of any point as x(ξ) = NI (ξ)xI. In the usual assumptions of the continuum mechanics it is always possible to define the transformation between the local system of coordinates and the cartesian system as {ξ + dξ, η} T − {ξ, η} T → ∂x(ξ,η) ∂ξ dξ = gξdξ (1) {ξ, η + dη}T − {ξ, η} T → ∂x(ξ, η) ∂η dη = gηdη (2) in which we introduced the symbols gξ = ∂NI (ξ, η) ∂ξ xI (3) gη = ∂NI (ξ, η) ∂η xI (4) the vectors gξ and gη of the 3D space, can be considered linearly independent (otherwise compenetration or self contact would manifest) follows immediately that they can be used in the construction of a base of the 3D space. In particular an orthogonal base can be defined as v1 = gξ gξ (5) n = gξ × gη gξ × gη → v2 = n × v1 (6) v3 = gξ × gη (7) Vectors v1 and v2 describe the local tangent plane to the membrane while the third base vector is always orthogonal. Follows the possibility of defining a local transformation rule that links the local coordinates ξ and the coordinates
FE Analysis of Membrane Systems Including Wrinkling and Coupling 91 x in the local tangent plane base.This can be achieved by considering that an increment fde,0}maps to the new base as ge·V1 gm·V1 dn (8) 0 d切 gV2 this is sinthetized by the definition of the linear application d⑦ ge·V1gm●V1 d在2 (9) ge●V2gm●V2 dn →d=jd延 it should be noted that ge.v3 and gn.v3 are identically zero consequently no components are left apart in representing the membrane in the new co- ordinates system.Taking into account the definition of the base vectors the tensor j becomes (after some calculations) (10) and its determinant det)=lv3l=lgs×gl (11) As the interest is focused on the purely membranal behavior,it is not needed to take in account the deformation of the structure over the thickness,as this can be calculated "a posteriori"once known the deformation of the mid-plane. On the base of this observation,the deformation gradient which describes the deformation of the membrane as a 3D body dx F3x3=8区 (12) can be replaced by 宜2x2= 0胶0胶05 胶=c胶 =j订-1 (13) taking in account the behavior over the thickness in the definition of the (two dimensional)constitutive model to be used (for example making the assumption of plane stress).The symbol J is used here and in the following to indicate j calculated in the reference position. under this considerations the subsequent development of the finite element follows closely the standard procedure for a non-linear 2D finite element,the only difference being that the local base changes on all the domain,which makes the linearization slightly more involved. To proceed further we need therefore to write the Right Cauchy Green Strain tensor C=FTF which takes the form C=(J-TjTjJ-1)=(GTgG) (14)
FE Analysis of Membrane Systems Including Wrinkling and Coupling 91 x, in the local tangent plane base. This can be achieved by considering that an increment {dξ, 0} maps to the new base as dξ 0 → gξ • v1 gξ • v2 dξ ; 0 dη → gη • v1 gη • v2 dη (8) this is sinthetized by the definition of the linear application dx,1 dx,2 = gξ • v1 gη • v1 gξ • v2 gη • v2 dξ dη → dx, = jdξ (9) it should be noted that gξ • v3 and gη • v3 are identically zero consequently no components are left apart in representing the membrane in the new coordinates system. Taking into account the definition of the base vectors the tensor j becomes (after some calculations) j = gξ gξ•gη gξ 0 v3 gξ (10) and its determinant det(j) = v3 = gξ × gη (11) As the interest is focused on the purely membranal behavior, it is not needed to take in account the deformation of the structure over the thickness, as this can be calculated “a posteriori” once known the deformation of the mid–plane. On the base of this observation, the deformation gradient which describes the deformation of the membrane as a 3D body F3×3 = ∂x ∂X (12) can be replaced by F,2×2 = ∂x, ∂X, = ∂x, ∂ξ ∂ξ ∂X, = jJ−1 (13) taking in account the behavior over the thickness in the definition of the (two dimensional) constitutive model to be used (for example making the assumption of plane stress). The symbol J is used here and in the following to indicate j calculated in the reference position. under this considerations the subsequent development of the finite element follows closely the standard procedure for a non–linear 2D finite element, the only difference being that the local base changes on all the domain, which makes the linearization slightly more involved. To proceed further we need therefore to write the Right Cauchy Green Strain tensor C = FT F which takes the form C = J−T j T jJ−1 = GT gG (14)
92 Riccardo Rossi,Vitaliani Renato,and Eugenio Onate where we introduced the symbols g=jTj and G=J-1.Operator g takes, after some calculations,the simple form gE●gEgm·g g= (15) ge●gngm●gm From the definition of he Green Lagrange strain tensor E =(C-I) we obtain immediately E =6C.This allows to write the equation of vir- tual works in compact form as (taking in consideration only body forces and pressure forces) SWint 6Wert 6Wpress (16) 、 6C:S=ho x●b+ p6x●n (17) Internal Work The termC:S describes the work of internal forces during the defor- mation process.Operator G=J-1 is referred to the reference configuration and is therefore strictly constant,follows immediately that C=GTogG (18) The term 6C:S becomes in Einstein's notation 3iC:8-i0u51=06m5m 1 (19) 2 introducing the symbols 6g11 S11 29则→26g}= 6g22 SJ→{S}= S22 (20) 2ǒg12 S12 (G11)2(G12)22G11G12 GuGJ→[Q1T 0 (G22)2 0 (21) 0 G12G22G11G22 it is possible to express the (19)in Voigt form as ic:5-0g1rQrs)-gr母：o=Qrs (22) considering the definition(15),introducing the symbol x)(x)..x and taking in account the isoparametric approximation one obtains Nx·g影=6x {g 2g11= (23) {g}/
92 Riccardo Rossi, Vitaliani Renato, and Eugenio Onate where we introduced the symbols g = jT j and G = J−1. Operator g takes, after some calculations, the simple form g = gξ • gξ gη • gξ gξ • gη gη • gη (15) From the definition of he Green Lagrange strain tensor E = 1 2 (C − I) we obtain immediately δE = 1 2 δC. This allows to write the equation of virtual works in compact form as (taking in consideration only body forces and pressure forces) δWint = δWext + δWpress (16) h0 2 Ω δC : S = h0 Ω δx • b + ω pδx • n (17) Internal Work The term h0 2 ) Ω δC : S describes the work of internal forces during the deformation process. Operator G = J−1 is referred to the reference configuration and is therefore strictly constant, follows immediately that δC = GT δgG (18) The term δC : S becomes in Einstein’s notation 1 2 δC : S = 1 2 δCIJ SIJ = 1 2 δgijGiIGjJ SIJ (19) introducing the symbols 1 2 δgij → 1 2 δ {g} = 1 2 ⎛ ⎝ δg11 δg22 2δg12 ⎞ ⎠ ; SIJ → {S} = ⎛ ⎝ S11 S22 S12 ⎞ ⎠ (20) GiIGjJ → [Q] T = ⎛ ⎝ (G11)2 (G12)2 2G11G12 0 (G22)2 0 0 G12G22 G11G22 ⎞ ⎠ (21) it is possible to express the (19) in Voigt form as 1 2 δC : S = 1 2 {δg}T [Q] T {S} = 1 2 {δg}T {s} ; {s} = [Q] T {S} (22) considering the definition(15), introducing the symbol {δx}T= {δx1} T . . . {δxk} T and taking in account the isoparametric approximation one obtains 1 2 δg11 = ∂NI ∂ξ δxI • gξ = {δx}T ⎛ ⎝ ∂N1 ∂ξ {gξ} ... ∂Nk ∂ξ {gξ} ⎞ ⎠ (23)
FE Analysis of Membrane Systems Including Wrinkling and Coupling 93 ga-21·8=刘 {g} (24) ON (gn) 202g12= Nx灯·g<+0c Nx·g={6x {g}+盼{ge) (25) on 股{g}+册{8<} by defining the matrix /器{g}器{g}器{g}+器{g} [b)T (26) 驶{g<}{g}{g}+{g} it is then possible to write acy7s)=EΨs=7rQrs (27) Defining the symbol B] [B]=[Q][b] (28) we finally obtain (fint}=ho [B]T (S)do (29) 6Wint={ox){fint} (30) External Work Derivation of the work of external conservative forces follows the standard procedure and can be found on any book on nonlinear finite elements.The expression of the work of follower forces (body forces)is on the other hand a little more involved.In the following the pressure is considered constant,the non linearity being introduced by the change of direction of the normal.For the derivation of the pressure contributions it is much easier to perform the integration over the actual domain then over the reference one. 6Wor p6x.nd p6x.ndet(j)dedn (31) taking in account the definition of the base vectors (6)(7),and considering (11)we obtain immediately }= Ni(,m)p(ξ，jv3(5,n)dd (32) }=（《6}T.…{.}T） (33) 6Wpr ={6x}(fpr} (34)
FE Analysis of Membrane Systems Including Wrinkling and Coupling 93 1 2 δg22 = ∂NI ∂ξ δxI • gη = {δx} T ⎛ ⎝ ∂N1 ∂η {gη} ... ∂Nk ∂η {gη} ⎞ ⎠ (24) 1 2 δ2g12 = ∂NI ∂η δxI • gξ + ∂NI ∂ξ δxI • gη = {δx}T ⎛ ⎝ ∂N1 ∂ξ {gη} + ∂N1 ∂η {gξ} ... ∂Nk ∂ξ {gη} + ∂Nk ∂η {gξ} ⎞ ⎠ (25) by defining the matrix [b] T = ⎛ ⎝ ∂N1 ∂ξ {gξ} ∂N1 ∂η {gη} ∂N1 ∂ξ {gη} + ∂N1 ∂η {gξ} ... ... ... ∂Nk ∂ξ {gξ} ∂Nk ∂η {gη} ∂Nk ∂ξ {gη} + ∂Nk ∂η {gξ} ⎞ ⎠ (26) it is then possible to write 1 2 {δC}T {S} = {δE}T {S} = {δx}T [b] T [Q] T {S} (27) Defining the symbol [B] [B] = [Q] [b] (28) we finally obtain {fint} = Ω h0 [B] T {S} dΩ (29) δWint = {δx} T {fint} (30) External Work Derivation of the work of external conservative forces follows the standard procedure and can be found on any book on nonlinear finite elements. The expression of the work of follower forces (body forces) is on the other hand a little more involved. In the following the pressure is considered constant, the non linearity being introduced by the change of direction of the normal. For the derivation of the pressure contributions it is much easier to perform the integration over the actual domain then over the reference one. δWpr = ω pδx • ndω = ξ,η pδx • ndet(j)dξdη (31) taking in account the definition of the base vectors (6)(7), and considering (11) we obtain immediately {fI} = ξ,η NI (ξ, η)p(ξ, η)v3(ξ, η)dξdη (32) {fpr} = {f1}T . . . {fk}T T (33) δWpr = {δx} T {fpr} (34)