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 surface 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 distortion 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 optimization” , 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 hanging 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 surfaces 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 element 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 variability 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 special 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 tensors 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 cumulative 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 distortion is controlled. Put into the context of the updated reference strategy it appears to converge to homogeneous meshes during the regular time of iteration 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 configuration, 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 distribution, 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 collapse 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 material 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 multiplier technique is prohibitive because of the size of the problem. Therefore, a local criterion is defined which can be solved at each Gauss point independently. 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” configuration 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)