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 tent144 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