《纺织复合材料》课程参考文献（Textile Composites and Inflatable Structures）09 Equilibrium Consistent Anisotropic Stress Fields in Membrane Design

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design Kai-Uwe Bletzingerl,Roland Wiichner2 and Fernass Daoud3 1 Lehrstuhl fuir Statik Technische Universitat Miinchen 80290 Miinchen,Germany kub@bv.tum.de 2 Lehrstuhl fiir Statik Technische Universitat Miinchen 80290 Miinchen,Germany wuechner@bv.tum.de 3 Lehrstuhl fuir Statik Technische Universitat Miinchen 80290 Miinchen,Germany daoud@bv.tum.de Key words:form finding,anisotropic surface stress,minimal surface,mesh control. Abstract.This paper deals with the control of mesh distortions which may appear during the form finding procedure of membrane design.The reason is the unbalance of surface stresses either due to the interaction of edge and sur- face or incompatibilities along the sewing lines of adjacent membrane patches. An approach is presented which is based on a rational modification of the surface tension field.The criterion is based on the control of the element dis- tortion and derived from differential geometry.Several examples demonstrate the success of the method. 1 Introduction Two different lines of research have developed which deal with the generation of structural shapes:the fields of "form finding"and "structural optimiza- tion",respectively.The methods of form finding are usually restricted to tensile structures(cables and membranes)whereas the methods of structural optimization are far more general and can usually be applied to any kind of structure.So far,The differences of the two approaches are not only the level 143 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,143-151. C 2005 Springer.Printed in the Netherlands

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design Kai-Uwe Bletzinger1, Roland W¨uchner ¨ 2 and Fernass Daoud3 1 Lehrstuhl fur Statik ¨ Technische Universit¨at M¨unchen ¨ 80290 Munchen, Germany ¨ kub@bv.tum.de 2 Lehrstuhl fur Statik ¨ Technische Universit¨at M¨unchen ¨ 80290 Munchen, Germany ¨ wuechner@bv.tum.de 3 Lehrstuhl fur Statik ¨ Technische Universit¨at M¨unchen ¨ 80290 Munchen, Germany ¨ daoud@bv.tum.de Key words: form finding, anisotropic surface stress, minimal surface, mesh control. Abstract. This paper deals with the control of mesh distortions which may appear during the form finding procedure of membrane design. The reason is the unbalance of surface stresses either due to the interaction of edge and sur￾face or incompatibilities along the sewing lines of adjacent membrane patches. An approach is presented which is based on a rational modification of the surface tension field. The criterion is based on the control of the element dis￾tortion and derived from differential geometry. Several examples demonstrate the success of the method. 1 Introduction Two different lines of research have developed which deal with the generation of structural shapes: the fields of ”form finding” and ”structural optimiza￾tion” , respectively. The methods of form finding are usually restricted to tensile structures (cables and membranes) whereas the methods of structural optimization are far more general and can usually be applied to any kind of structure. So far, The differences of the two approaches are not only the level 143 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 143–151. © 2005 Springer. Printed in the Netherlands

144 Kai-Uwe Bletzinger,Roland Wiichner and Fernass Daoud of specialization but also their aims.Form finding methods are designed to determine structural shapes from an inverse formulation of equilibrium and are derived from the simulation of physical phenomena of soap films and hang- ing models.In the case of soap films the structural shape is defined by the equilibrium geometry of a prescribed field of tensile surface stresses.It is well known that the shapes related to isotropic surface stresses are minimal sur- faces which have minimal surface area content within given edges.Minimal surfaces have the additional property of zero mean curvature,or,with respect to pneumatically loaded surfaces of constant mean curvature. Using a variational approach for the solution we realize that only those shape variations are meaningful which result in a variation of the area content. In other words,the variation of the position of any point on the surface must have a component normal to the surface.A variation of the position along the surface will not alter the area content.That means.that if a finite element method is used to solve the problem and the surface is discretized by a mesh of elements and nodes the stiffness with respect to a movement of the nodes tangential to the surface vanishes.This problem is well known since long from shape optimal design also where the design parameters must be chosen such that their modification must have an effect on the structural shape.Shape optimal design is controlled by the modification of the boundary. There exist several remedies.Two techniques have been accepted as state of the art in shape optimal design:(i)linking the movement of internal nodes to key nodes using mapping techniques from CAGD,the so called design ele- ment technique,and,(ii)defining move directions for nodes on the boundary to guarantee relevant shape modifications.The positive side effect is that the number of optimization variables can drastically be reduced by this approach which is very attractive for optimal design.On the other hand,however,the space of possible shapes is also reduced.That is unacceptable if a high vari- ability of shape modification is needed,as e.g.for the shape design of tensile structures or for the problem to find shapes of equilibrium of tension fields at the surface of liquids or related fields.Then methods are needed which are able to stabilize the nodal movement such that all three spatial coordinates of any finite element node may be variables in the shape modification process. For the special case of form finding of tensile structures the updated reference strategy (URS)is designed to find the shape of equilibrium of pre-stressed membranes.It is a general approach which can be applied to any kind of spe- cial element formulation (membranes or cables).A stabilization term is used which fades out as the procedure converges to the solution.The method is based on the specific relations of Cauchy and 2nd Piola-Kirchhoff stress ten- sors which appear to be identical at the converged solution [5],[8],[6].Other alternative stabilization approaches have the same intention but used other methodologies,one may find many references in [4],[7],[3],[1]. All methods,however,will have problems or even fail if they are applied to physically meaningless situations without solution.E.g.it is not sure if a minimal surface exists for a given edge,or,a practical question from tent

144 Kai-Uwe Bletzinger, Roland W¨uchner and Fernass Daoud ¨ of specialization but also their aims. Form finding methods are designed to determine structural shapes from an inverse formulation of equilibrium and are derived from the simulation of physical phenomena of soap films and hang￾ing models. In the case of soap films the structural shape is defined by the equilibrium geometry of a prescribed field of tensile surface stresses. It is well known that the shapes related to isotropic surface stresses are minimal sur￾faces which have minimal surface area content within given edges. Minimal surfaces have the additional property of zero mean curvature, or, with respect to pneumatically loaded surfaces of constant mean curvature. Using a variational approach for the solution we realize that only those shape variations are meaningful which result in a variation of the area content. In other words, the variation of the position of any point on the surface must have a component normal to the surface. A variation of the position along the surface will not alter the area content. That means, that if a finite element method is used to solve the problem and the surface is discretized by a mesh of elements and nodes the stiffness with respect to a movement of the nodes tangential to the surface vanishes. This problem is well known since long from shape optimal design also where the design parameters must be chosen such that their modification must have an effect on the structural shape. Shape optimal design is controlled by the modification of the boundary. There exist several remedies. Two techniques have been accepted as state of the art in shape optimal design: (i) linking the movement of internal nodes to key nodes using mapping techniques from CAGD, the so called design ele￾ment technique, and, (ii) defining move directions for nodes on the boundary to guarantee relevant shape modifications. The positive side effect is that the number of optimization variables can drastically be reduced by this approach which is very attractive for optimal design. On the other hand, however, the space of possible shapes is also reduced. That is unacceptable if a high vari￾ability of shape modification is needed, as e.g. for the shape design of tensile structures or for the problem to find shapes of equilibrium of tension fields at the surface of liquids or related fields. Then methods are needed which are able to stabilize the nodal movement such that all three spatial coordinates of any finite element node may be variables in the shape modification process. For the special case of form finding of tensile structures the updated reference strategy (URS) is designed to find the shape of equilibrium of pre-stressed membranes. It is a general approach which can be applied to any kind of spe￾cial element formulation (membranes or cables). A stabilization term is used which fades out as the procedure converges to the solution. The method is based on the specific relations of Cauchy and 2nd Piola-Kirchhoff stress ten￾sors which appear to be identical at the converged solution [5], [8], [6]. Other alternative stabilization approaches have the same intention but used other methodologies, one may find many references in [4], [7], [3], [1]. All methods, however, will have problems or even fail if they are applied to physically meaningless situations without solution. E.g. it is not sure if a minimal surface exists for a given edge, or, a practical question from tent

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design 145 design,it is practically impossible to a priori satisfy equilibrium along the common edge of membrane strips which are anisotropically pre-stressed in different directions.The unbalance of stresses can be detected by a cumula- tive distortion of the FE-mesh during iteration.Methods are needed to control the mesh distortion by adapting the stress distribution.The present approach defines a local criterion to modify the pre-stress such that the element distor- tion is controlled.Put into the context of the updated reference strategy it appears to converge to homogeneous meshes during the regular time of itera- tion needed by the form finding procedure.The additional numerical effort is negligible.The results are surfaces of balanced shape representing equilibrium as close as possible at the desired distribution of stresses.The approach will be extended to other situations where mesh control might be necessary,e.g. in general shape optimal design or other surface tension problems [2. 2 The Updated Reference Strategy The basic idea of URS will be briefly shown to understand the following chapters.A detailed description is given in [5].Suppose one wants to find the equilibrium shape of a given surface tension field o.The solution is defined by the stationary condition 6w= ux da= detF(o.F-T):6F dA=0 (1) which represents the vanishing virtual work of the Cauchy stresses o and u.x is the spatial derivative of the virtual displacements,F is the deformation gradient,a and A are the surface area of the actual and the reference configu- ration,respectively.As mentioned in the introduction the direct discretization of(1)w.r.t.all spatial coordinates will give a singular system of equations. The problem is stabilized by blending with the alternative formulation of(1) in terms of 2nd Piola-Kirchhoff stresses S: dw=入detF(o·F-T):6FdA+(1-) (F.S):6Fd4=0 (2) 4 where the coefficients of S are assumed to be constant and identical to those of o.The continuation parameter A must be chosen small enough,even zero is possible.As the solution of (2)is used as the reference configuration for a following analysis the procedure converges to the solution of(1).Then F becomes the identity and both stress tensors are identical.The stabilization fades out. 3 Equilibrium of Surface Stresses As also mentioned in the introduction it is not always possible to find a shape of equilibrium for each combination of edge geometry and surface stress dis- tribution,isotropic or anisotropic.In these cases the size of some elements

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design 145 design, it is practically impossible to a priori satisfy equilibrium along the common edge of membrane strips which are anisotropically pre-stressed in different directions. The unbalance of stresses can be detected by a cumula￾tive distortion of the FE-mesh during iteration. Methods are needed to control the mesh distortion by adapting the stress distribution. The present approach defines a local criterion to modify the pre-stress such that the element distor￾tion is controlled. Put into the context of the updated reference strategy it appears to converge to homogeneous meshes during the regular time of itera￾tion needed by the form finding procedure. The additional numerical effort is negligible. The results are surfaces of balanced shape representing equilibrium as close as possible at the desired distribution of stresses. The approach will be extended to other situations where mesh control might be necessary, e.g. in general shape optimal design or other surface tension problems [2]. 2 The Updated Reference Strategy The basic idea of URS will be briefly shown to understand the following chapters. A detailed description is given in [5]. Suppose one wants to find the equilibrium shape of a given surface tension field σ. The solution is defined by the stationary condition δw =  a σ : δu,x da =  A detF(σ · F−T ) : δF dA = 0 (1) which represents the vanishing virtual work of the Cauchy stresses σ and δu,x is the spatial derivative of the virtual displacements, F is the deformation gradient, a and A are the surface area of the actual and the reference configu￾ration, respectively. As mentioned in the introduction the direct discretization of (1) w.r.t. all spatial coordinates will give a singular system of equations. The problem is stabilized by blending with the alternative formulation of (1) in terms of 2nd Piola-Kirchhoff stresses S: δw = λ  A detF(σ · F−T ) : δF dA + (1 − λ)  A (F · S) : δF dA = 0 (2) where the coefficients of S are assumed to be constant and identical to those of σ. The continuation parameter λ must be chosen small enough, even zero is possible. As the solution of (2) is used as the reference configuration for a following analysis the procedure converges to the solution of (1). Then F becomes the identity and both stress tensors are identical. The stabilization fades out. 3 Equilibrium of Surface Stresses As also mentioned in the introduction it is not always possible to find a shape of equilibrium for each combination of edge geometry and surface stress dis￾tribution, isotropic or anisotropic. In these cases the size of some elements

146 Kai-Uwe Bletzinger,Roland Wiichner and Fernass Daoud will steadily increase during iteration although URS is designed to control the mesh quality.That is because it is not possible to satisfy equilibrium of the given surface stresses at the location of those elements.For example consider a catenoid,Fig.1.If the height exceeds the critical value the surface will col- lapse as shown.During iteration that is indicated by the increasing length of the related elements.To avoid the collapse the meridian stresses should be increased. Now,consider the catenoid with an constant anisotropic stress field with larger meridian stresses,Fig.2.The surface does not collapse anymore,but still the elements at the top get out of control as indicated by the tremendous increase of size.Again,the physical explanation is that equilibrium cannot be found with a constant distribution of meridian stresses over the height. Fig.1.Collapsed catenoid with different end rings. Fig.2.Catenoid with anisotropic pre-stress (meridian/ring=3/1). A similar situation arises in membrane design.Typically those structures are sewed together by several strips,each of them made of anisotropic ma- terial and anisotropically pre-stressed.Along the common line there exists also an intrinsic unbalance of equilibrium of stresses which must be handled with during the form finding procedure.All cases demand for automatically adjusted stresses. 4 Element Size Control The element size is used as indicator to adjust the surface stress field.At each Gauss point of the element discretization an additional constraint is intro-

146 Kai-Uwe Bletzinger, Roland W¨uchner and Fernass Daoud ¨ will steadily increase during iteration although URS is designed to control the mesh quality. That is because it is not possible to satisfy equilibrium of the given surface stresses at the location of those elements. For example consider a catenoid, Fig. 1. If the height exceeds the critical value the surface will col￾lapse as shown. During iteration that is indicated by the increasing length of the related elements. To avoid the collapse the meridian stresses should be increased. Now, consider the catenoid with an constant anisotropic stress field with larger meridian stresses, Fig. 2. The surface does not collapse anymore, but still the elements at the top get out of control as indicated by the tremendous increase of size. Again, the physical explanation is that equilibrium cannot be found with a constant distribution of meridian stresses over the height. Fig. 1. Collapsed catenoid with different end rings. Fig. 2. Catenoid with anisotropic pre-stress (meridian/ring=3/1). A similar situation arises in membrane design. Typically those structures are sewed together by several strips, each of them made of anisotropic ma￾terial and anisotropically pre-stressed. Along the common line there exists also an intrinsic unbalance of equilibrium of stresses which must be handled with during the form finding procedure. All cases demand for automatically adjusted stresses. 4 Element Size Control The element size is used as indicator to adjust the surface stress field. At each Gauss point of the element discretization an additional constraint is intro-

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design 147 F reterence G G actual initial Bmax 2 F max.allowable gmar! Fig.3.Configurations during form finding. duced to check for the length of base vectors against the allowable maximal deformation.The standard optimization approach by the Lagrangian multi- plier technique is prohibitive because of the size of the problem.Therefore, a local criterion is defined which can be solved at each Gauss point indepen- dently.The approach is analogous to what is done in elastic-plastic analysis. In contrast to that,now,the surface stresses are constant until the critical deformation is reached and will then be adjusted to the further change of deformation. Consider the following configurations and the related definition of base vectors:the initial configuration,Go,where the form finding was started,the reference configuration,G,defined by the update procedure of URS,and the actual configuration,g,as the state of equilibrium of the current time step.An additional configuration,gmaz is defined which states the maximum allowable element size.There are several deformation gradients F defined which map differential entities between the configurations,Fig.3. The deformation gradient Fin time stepk which describes the total deformation is multiplicatively created by F)=F.F)where F=g:⑧G(andF=g)®Gi If the actual deformation exceeds the allowable limit,i.e. F)F,a modified surface stress tensor mod is generated by the following rule of nested pull back and push forward operations: 1.apply the surface Cauchy stress o to the"max.allowable"config- uration 2.determine the related 2nd Piola-Kirchhoff stress So w.r.t.to the initial configuration by pull back using Fma 3.push So forward to the actual configuration applying Ft The resulting operation is: Omod= dtEF~Fo,E·野 detF (3)

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design 147 Fig. 3. Configurations during form finding. duced to check for the length of base vectors against the allowable maximal deformation. The standard optimization approach by the Lagrangian multi￾plier technique is prohibitive because of the size of the problem. Therefore, a local criterion is defined which can be solved at each Gauss point indepen￾dently. The approach is analogous to what is done in elastic-plastic analysis. In contrast to that, now, the surface stresses are constant until the critical deformation is reached and will then be adjusted to the further change of deformation. Consider the following configurations and the related definition of base vectors: the initial configuration, G0, where the form finding was started, the reference configuration, G, defined by the update procedure of URS, and the actual configuration, g, as the state of equilibrium of the current time step. An additional configuration, gmax is defined which states the maximum allowable element size. There are several deformation gradients F defined which map differential entities between the configurations, Fig. 3. The deformation gradient F(k) t in time step k which describes the total deformation is multiplicatively created by F(k) t = F · F(k−1) t where F = gi ⊗ Gi(k) and F(k) t = g(k) i ⊗ Gi 0. If the actual deformation exceeds the allowable limit, i.e. F(k) t > Fmax , a modified surface stress tensor σmod is generated by the following rule of nested pull back and push forward operations: 1. apply the surface Cauchy stress σ to the ”max. allowable” config￾uration 2. determine the related 2nd Piola-Kirchhoff stress S0 w.r.t. to the initial configuration by pull back using Fmax 3. push S0 forward to the actual configuration applying Ft The resulting operation is: σmod = det Fmax det Ft Ft · F−1 max · σ · F−T max · FT t (3)

148 Kai-Uwe Bletzinger,Roland Wuichner and Fernass Daoud and,considering the 2nd Piola-Kirchhoff stress tensor S(used in the stabilizing term of URS): tEnF,EmF.F明 omod=detFdetF (4) The "max.allowable"configuration is defined by multiples of the base vectors of the actual configuration: gmar i=B(i)gi (5) Then,we can determine the related modified deformation gradient Fmod Fmod=EFd·F=1 gi图G (6) The change of element size is determined by tracing the length change of base vectors during the deformation process.Introducing the ratio a=‖g(a‖/|G(o‖(no summation)the total change of geometry in time stepidefined If the total change of geometrys larger than the allowable maximum mai then the constraint is active.The factor Bi which is used to define the "max.allowable"configuration can now be determined as: ifa>amar间then:amar间=a喝属 ..32= (mar(i) 喝 (7) Considering both surface base vectors gi and g2 the determinant of Fmod is now det Fmod =(B1B2)(-1)det F (8) If constraints are active the state of stresses must change with time and the modified Cauchyeten)for the next time step1 which is acting in the "max.allowed"configuration follows as: o(k1) mod mod的.F职od det F (9) In the contet of URS the components ofwill also be usdas the components of S(k+1)in the next time step.They are: o2经+)= 32_S11() B detF 22(k+1)= Omod S22() B2 detF 12(k+1)= Omod deiF) 1 (10) The modification rule (10)simply says that the surface stress of an element which became too long must be increased to shorten it and vice versa

148 Kai-Uwe Bletzinger, Roland W¨uchner and Fernass Daoud ¨ and, considering the 2nd Piola-Kirchhoff stress tensor S (used in the stabilizing term of URS): σmod = det Fmax det F det Ft Ft · F−1 max · F · S · FT · F−T max · FT t (4) The ”max. allowable” configuration is defined by multiples of the base vectors of the actual configuration: gmax i = β(i)gi (5) Then, we can determine the related modified deformation gradient Fmod Fmod = Ft · F−1 max · F = 1 β(i) gi ⊗ Gi (6) The change of element size is determined by tracing the length change of base vectors during the deformation process. Introducing the ratio αi = g(i) / G(i) (no summation) the total change of geometry in time step k is defined as: α(k) ti = α(i)α(k−1) t(i) . If the total change of geometry is larger than the allowable maximum αmax i then the constraint is active. The factor βi which is used to define the ”max. allowable” configuration can now be determined as: if α(k) ti > αmax (i) then : αmax (i) = α(k) t(i) βi ∴ βi = αmax(i) α(k) t(i) (7) Considering both surface base vectors g1 and g2 the determinant of Fmod is now det Fmod = (β1β2) (−1) det F (8) If constraints are active the state of stresses must change with time and the modified Cauchy stress tensor σ(k+1) mod for the next time step k + 1 which is acting in the ”max. allowed” configuration follows as: σ(k+1) mod = β1β2 det F Fmod · S(k) · FT mod (9) In the context of URS the components of σ(k+1) mod will also be used as the components of S(k+1) in the next time step. They are: σ11(k+1) mod = β2 β1 det F S11(k) σ22(k+1) mod = β1 β2 det F S22(k) σ12(k+1) mod = 1 det F S12(k) (10) The modification rule (10) simply says that the surface stress of an element which became too long must be increased to shorten it and vice versa.

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design 149 5 Examples 5.1 Catenoid This example shows the effect of different values of omaz on the final shape of a catenoid which was initially isotropically pre-stressed.The choice of omar controls the maximal deformation of the elements.A value of 1.0 means almost no element deformation.Compare with Fig.1 how collapse could be omitted. (a)amaz =1.0 (b)amaz =1.2 (c)amar =1.5 (d)amar=2.0 Fig.4.Stabilized catenoid of initially isotropic surface stress. (a)no constraint (b)amaz =1.2 (c）amar=2.0 Fig.5.Initially anisotropically pre-stressed catenoid (meridian/ring=1.2/1). 5.2 Tent Hiifingen This tent is composed by 5 membrane patches which are anisotropically pre- stressed towards the center of the tent.A top view shows the consequences of unbalanced stresses at the common edges of adjacent patches where the mesh is distorted,Fig.6,left.The mesh quality is maintained using a value of amar =1.0,Fig.6,right.Fig.7 shows the generated shapes using different values for the mesh control factor

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design 149 5 5.1 Catenoid This example shows the effect of different values of αmax on the final shape of a catenoid which was initially isotropically pre-stressed. The choice of αmax controls the maximal deformation of the elements. A value of 1.0 means almost no element deformation. Compare with Fig. 1 how collapse could be omitted. Fig. 4. Stabilized catenoid of initially isotropic surface stress. Fig. 5. Initially anisotropically pre-stressed catenoid (meridian/ring=1.2/1). 5.2 Tent Hufingen ¨ This tent is composed by 5 membrane patches which are anisotropically pre￾stressed towards the center of the tent. A top view shows the consequences of unbalanced stresses at the common edges of adjacent patches where the mesh is distorted, Fig. 6, left. The mesh quality is maintained using a value of αmax = 1.0, Fig. 6, right. Fig. 7 shows the generated shapes using different values for the mesh control factor. Examples (a) αmax = 1.0 (b) αmax = 1.2 (c) αmax = 1.5 (d) αmax = 2.0 (a) no constraint (b) αmax = 1.2 (c) αmax = 2.0

150 Kai-Uwe Bletzinger,Roland Wiichner and Fernass Daoud Fig.6.Top view after form finding,without or with mesh control (left,right). (a)no mesh control (b)amaz =1.0 (c)amaz =1.5 Fig.7.Effects of varied mesh control 6 Conclusion Motivated by the specific problems of form finding an approach was developed to control the mesh quality of the finite element discretization.The idea of the method is to adjust the value of the applied surface stresses by a simple scaling of the stress components which is derived from geometric considera- tions.The method is very effective without additional effort.Several examples demonstrate the success.Further developments are directed towards the gen- eralization of the approach for general problems of shape optimal design where similar situations occur

150 Kai-Uwe Bletzinger, Roland W¨uchner and Fernass Daoud ¨ Fig. 6. Top view after form finding, without or with mesh control (left, right). Fig. 7. Effects of varied mesh control 6 Conclusion Motivated by the specific problems of form finding an approach was developed to control the mesh quality of the finite element discretization. The idea of the method is to adjust the value of the applied surface stresses by a simple scaling of the stress components which is derived from geometric considera￾tions. The method is very effective without additional effort. Several examples demonstrate the success. Further developments are directed towards the gen￾eralization of the approach for general problems of shape optimal design where similar situations occur. (a) no mesh control (b) αmax = 1.0 (c) αmax = 1.5

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design 151 References 1.B.Maurin and R.Motro.Cutting pattern of fabric membranes with the stress composition method.Int.Journal of space structures,14:121-130,1999. 2.D.Peric and P.H.Saksono.On finite element modeling of surface tension: variational formulation and applications.In W.Wall,K.-U.Bletzinger,and K. Schweizerhof,editors,Trends in Computational Structural Mechanics,pages 731- 740.CIMNE,Barcelona,2001. 3.K.Ishii.Form finding analysis in consideration of cutting patterns of membrane structures.Int.Journal of space structures,14:105-120,1999. 4.K.Linkwitz.Formfinding by the Direct Approach and pertinent strategies for the conceptual design of prepressed and hanging structures.Int.Journal of space structures,14:73-88,1999. 5.K.-U.Bletzinger and E.Ramm.A general finite element approch to the form finding of tensile structures by the updated reference strategy.Int.Journal of space structures,14:131-146,1999. 6.K.-U.Bletzinger and R.Wiichner.Formfinding of anisotropic pre-stressed mem- brane structures.In W.Wall,K.-U.Bletzinger,and K.Schweizerhof,edi- tors,Trends in Computational Structural Mechanics,pages 595-603.CIMNE, Barcelona,2001. 7.M.Barnes.Form finding and analysis of tension structures by dynamic relaxation. Int.Journal of space structures,14:89-104,1999. 8.R.Wiichner.Formfindung von Membrantragwerken unter Beriicksichtigung anisotroper Vorspannung.Master's thesis,Lehrstuhl fiir Statik,TU Miinchen, 2000

Equilibrium Consistent Anisotropic Stress Fields in Membrane Design 151 References 1. B. Maurin and R. Motro. Cutting pattern of fabric membranes with the stress composition method. Int. Journal of space structures, 14:121–130, 1999. 2. D. Peric and P. H. Saksono. On finite element modeling of surface tension: variational formulation and applications. In W. Wall, K.-U. Bletzinger, and K. Schweizerhof, editors, Trends in Computational Structural Mechanics, pages 731– 740. CIMNE, Barcelona, 2001. 3. K. Ishii. Form finding analysis in consideration of cutting patterns of membrane structures. Int. Journal of space structures, 14:105–120, 1999. 4. K. Linkwitz. Formfinding by the Direct Approach and pertinent strategies for the conceptual design of prepressed and hanging structures. Int. Journal of space structures, 14:73–88, 1999. 5. K.-U. Bletzinger and E. Ramm. A general finite element approch to the form finding of tensile structures by the updated reference strategy. Int. Journal of space structures, 14:131–146, 1999. 6. K.-U. Bletzinger and R. W¨uchner. Formfinding of anisotropic pre-stressed mem- ¨ brane structures. In W. Wall, K.-U. Bletzinger, and K. Schweizerhof, edi￾tors, Trends in Computational Structural Mechanics, pages 595–603. CIMNE, Barcelona, 2001. 7. M. Barnes. Form finding and analysis of tension structures by dynamic relaxation. Int. Journal of space structures, 14:89–104, 1999. 8. R. Wuchner. Formfindung von Membrantragwerken unter Ber¨ ¨ ucksichtigung anisotroper Vorspannung. Master’s thesis, Lehrstuhl f¨ur Statik, TU M¨unchen, 2000