* Corresponding author. Fax: +49 7031 874521.

E-mail address: kub@bv.tum.de (K.-U. Bletzinger).

Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452

www.elsevier.com/locate/cma0045-7825/$ - see front matter 2005 Published by Elsevier B.V.1. Introduction

Lightweight structures as shells and membranes are dened by the optimal use of material to carry exter-

nal loads or pre-stress. Material is used optimally within a structural member if the member is subjected to

membrane forces rather than bending. An important objective of a procedure to determine layout and

shape of a lightweight structure is, therefore, to minimize bending or more general, to minimize the strain

energy rather than structural weight as the term lightweight may imply [1].

Two dierent lines of research have developed which deal with the generation of structural shapes: theelds of form nding and structural optimization, respectively. The methods of form nding areAbstract

Free form shells optimized for stiness under given loading and membrane structures act in a pure membrane state

of stresses, either because bending is minimized or not even present by denition. Physical experiments as soap lms and

hanging models have been used since centuries to generate optimal shapes of membranes in tension and shells in com-

pression. The paper presents numerical methods to simulate the physical experiments as well as how they can be merged

among each other and with the most general technology of structural optimization. The combined approach represents

the combined power of each technique. Several examples illustrate the methods and typical applications.

2005 Published by Elsevier B.V.

Keywords: Form nding; Shape optimization; Free form shell; Hanging model; Minimal surfaceComputational methods for form nding and optimizationof shells and membranes

Kai-Uwe Bletzinger *, Roland Wuchner, Ferna Daoud, Natalia Camprub

Lehrstuhl fur Statik, Technische Universitat Munchen, Arcisstr. 21, D-80290 Munchen, Germany

Received 20 January 2004; received in revised form 23 August 2004; accepted 6 December 2004doi:10.1016/j.cma.2004.12.026

usually restricted to tensile structures (cables and membranes) whereas the methods of structural optimiza-

tion are far more general and can be applied to any kind of structure [24]. The dierences of the two

approaches, however, are not only the level of specialization but also their aims. Form nding methods

are designed to determine structural shape from an inverse formulation of equilibrium. Again, two dier-

ent methods are known: the soap lm analogy for the generation of structures acting in pure tensionwhich are related to minimal surfaces, and hanging models to generate structures in compression by inver-

sion of tensile structures. These methods are well known from tradition and are established since centu-

ries by physical experiments. Today, these methods can be eectively simulated by nite element

methods as demonstrated in the sequel. In the context of structural optimization other criteria or even a

combination of several criteria can be chosen to dene the shape with respect to the special problem

under consideration. The most general approach of structural optimization, however, has to be paid by

numerical expense and is therefore often restricted to rather few shape degrees of freedom. On the other

hand, numerical simulations of form nding methods are comparatively robust, are suited for complex

2.2. Virtual work of a surface stress eld

K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452 3439Consider a stress eld r which acts tangential to a surface and which is assumed to be in self-equilibrium

(Fig. 1). The governing equation is the principle of virtual work

y

z

x shapes but are designed for very special applications. The paper will give an overview on the both

approaches and will show a strategy about how to combine them with respect to eectiveness and mechan-

ical relevance.

2. Form nding methods

2.1. Pre-stressed membranessoap lm analogy

Pre-stressed membranes in tension are optimal in two respects. First, in the case of uniform surface stres-

ses they reect surfaces of minimal area or minimal weight, e.g. minimal surfaces [5], and, secondly, struc-

tures of optimal material usage since bending is omitted a priori by denition. Experimentally minimal

surfaces can be realized by soap lms (soap lm analogy) [1]. Numerical simulation results in a compli-cated inverse problem why there exist many dierent solution approaches, e.g. [615], further references

can be found in [13].Fig. 1. Tangential surface stress eld.

With

S is tIt

The m

At th

the st

a give

unles

norm

3440 K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452du;x odu odx oX d ox

F1 dF F1: 2a ox a

which is used to nd the unknown shape x of equilibrium. r is the prescribed Cauchy stress tensor whichacts on the surface, du,x is the derivative of the virtual displacement with respect to the geometry x of thesurface in equilibrium. The thickness of the membrane is denoted by t. It is comparatively thin and assumed

to be constant during deformation.

The standard notation of continuum mechanics to describe the deformation of bodies denes the dis-

placement eld u as the dierence between the surface in equilibrium x and any reference surface X,u = x X, Fig. 2. Following the chain rule of dierentiation the derivative of the virtual displacementdu with respect to x can be explained by use of the real and virtual deformation gradients F and dFdw tZ

r :odu

da tZ

r : du;x da 0 1Fig. 2. Deformation of surface.Y, y

X, x

Z, z

1

x(1,2)P g1

actual configuration X( 1,2) u(1,2) P 2 1 g2configuration

2

2 1

G G

deformation

reference ox oX ox oX

(2) the virtual work equation (1) can be written as

dw tZar : du;x da t

ZAdetFr FT : dFdA t

ZAF S : dF dA 0; 3

he second KirchhoPiola stress tensor.

is related to r by

S detFF1 r FT: 4apping of the area dA from the reference to the equilibrium state da is dened by

da detF dA: 5is point (3) reects alternative formulations of the state of equilibrium. By the special relation (4) of

ress tensors, r reects the deformation from the reference conguration to the state of equilibrium and

n tensor S or vice versa. Or, in other terms, if r denes a uniform surface stress then S is non-uniforms the reference geometry X coincides with the equilibrium geometry x. Note that all stress componentsal to the surface are zero, i.e.

ri3 r3i Si3 S3i 0: 6

In the case of a minimal surface the Cauchy stress tensor is a scalar multiple of the unit tensor I, r = sI and(3) reduces to

dw stZaI : du;x da st

ZAdetFFT : dF dA 0 7

which is the governing equation of a surface of minimal area content within a given boundary [5]. The ana-

logy between this geometrical explanation and the initial mechanical statement of equilibrium is the well

known soap lm analogy.

From the geometrical analogy it is obvious, that the variation of shape must have at least a normal com-

ponent to the surface. Otherwise, the shape maps to itself and the area content does not change. In terms of

a nite element discretization tangential displacement degrees of freedom are irrelevant and lead to singular

structural response.

2.3. Regularization

The conclusion of the previous section implies that a solution of (1) by a nite element technique allows

for only one displacement degree of freedom at each node which remains normal to the surface during the

form nding procedure, Fig. 3.

However, for application to tents the boundaries are exible cables and the surface nodes must move

tangentially also to ensure a proper mesh, Fig. 4.

That means, a generally applicable discretization technique must use three displacement degrees of free-dom at each node and, as a consequence, must be enhanced by some regularization methodology to circum-

K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452 3441directions to guarantee mesh qualityNodes must move in all three spatial final shapeinitial shapevent the above mentioned singularity problems.

There exist several regularization techniques. The method of dynamic relaxation adds articial inertia

terms to the governing equations (3). Other ideas suggest neglecting certain terms after linearization to

move directions of nodes normal to surface

diagonal section

Fig. 3. Form nding of a HP-like surface with rigid boundaries.Fig. 4. Form nding of a HP-like surface with exible boundaries, top view.

prevent the stiness matrix from rank deciency. A further alternative is based on an intelligent mixture of

congurations.

Starting from (3) one can introduce a continuation factor k and write

dw ktZAdetFr FT : dF dA 1 kt

ZAF S : dF dA 0: 8

Nothing is changed so far.

Next, we assume that instead of applying (4) the second PiolaKircho stresses S are dened identicallyas the Cauchy stresses r, e.g. in the case of uniform surface stresses

r S sI : 9Now, (8) is modied to be

dwk stZAk detFFT 1 kF : dF dA 0; 1 < k 6 0: 10

If k is chosen properly the second term stabilizes the original expression and allows for standard nite ele-

3442 K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452eventually also of continuation factor Solve (10) Inner loop, Iterative solution of (10) one linear step if = 0external loop, update of

reference geometry, Choose ment discretization and solution. The modication has the convincing property that it disappears at the

solution of a minimal surface. Then the deformation gradient reduces to be the identity tensor. It should

be mentioned that this regularization technique can be generalized for other situations where tangential

movements of nodes must be stabilized.

2.4. The updated reference strategy

The modied and stabilized expression (10) is non-linear with respect to the nal geometry x and must besolved iteratively applying the NewtonRaphson method. The result deviates from the true solution

depending on the choice for k. The procedure might be repeated with a modied choice of k closer toone until one gets close enough to the solution [15].

Alternatively, k can be chosen as a xed number. Then the resulting geometry ~x of the modied system(10) can be used to update the reference geometry X for the next iterations. By repeatedly updating the ref-erence geometry the procedure safely and robustly converges to the nal solution, Fig. 5. Even k = 0 mightbe chosen. Then the problem reduces to be linear and can be solved within one iteration. However, moreFig. 5. Flow scheme of the updated reference strategy.

K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452 3443external loops are necessary to update the reference geometry. The method converges very fast and robust,

Fig. 6. Minimal surfaces can be found in a very reliable manner, Fig. 7.

2.5. Anisotropic pre-stress

There exist several situations where ideal uniform surface stresses are not possible if the structure is real-ized by anisotropic textile material or is loaded by additional load cases as wind and or snow. Then aniso-

tropic pre-stresses must be considered which result in shapes which dier from the ideal minimal surface.

The updated reference method is readily modied for this new situation if the stress tensors r and S in thegoverning equation (8) are formulated accordingly [16]. With respect to the production of membrane struc-

tures by the composition of initially at strips we dene the stress eld in terms of an additional, develop-

able surface, denoted as reference strip. For simplicity of presentation we take a plane assuming rather

Fig. 6. Updating reference strategy, adaptation of k: Steps 0 to 3; updating geometry: Step 4.

Fig. 7. Membrane structure of constant surface stress (minimal surface).

Gat shapes as solution. Fig. 8 shows part of the reference membrane surface and below the additional ref-

erence strip. Vectors f1 and f2 dene the directions of prescribed pre-stress on the strip which are initiallydened as problem input. The direction T1 of pre-stress on the reference surface is dened by the intersec-tion line of tangential and projection plane (Fig. 8). Direction T2 is determined to be orthogonal to T1 andthe surface normal G3.

x

Fig. 8. Projection of pre-stress on membrane surface.Th

Pre-s

Pre-s

wher

The p

den

at the

as

The n

choic

Prthe c

can b1x2e1

e2 reference strip

e3 F f1f2f 3 x3

1

projection raysplane ofT1

1 G2

tangent plane

T3=G3 2

reference surface coordinate lines on

3444 K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452e following chain of stress transformation is dened:

tress on the strip:

r rabf a f b: 11tress on reference surface:

S rabTa Tb SabGa Gb; 12e the components of S appear to be

Sab rcdGa TcTd Gb: 13re-stress tensor r acting on the equilibrium surface can be determined by an analogous procedure,

ing vectors (t1, t2) dened by the intersection of the projection plane and the tangential plane, now,equilibrium surface. The Cauchy stress components rab with respect to the basis (g1,g2) are dened

rab rcdga tctd gb: 14umerical eort of consistent linearization with respect to the discretization parameters for a general

e of k can be avoided if the URS scheme with continuation factor k = 0 is chosen.e-stress of dierent magnitudes and orientations in adjacent strips are usually not in equilibrium alongommon seam. In these cases the magnitude of pre-stress must be corrected during form nding. This

e done without altering the general layout of the method.

K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452 3445A typical application is shown in Fig. 9 where an anisotropic pre-stress is used to modify the shape of

tents slightly to prevent water ponds under additional snow loading.

2.6. Hanging models

The hanging chain and its inverse is one of the oldest methods which are known to generate the shape of

an arch which is free of bending, subjected only to compressive axial force. The method has been used

intensively during the centuries, e.g. by Antoni Gaud, to mention one well known name among all the

others. Extended to two directions to dene the shape of shells the hanging model concept has been brought

to perfection by Isler [17].

The goal of hanging models is to perform the transition from a bending structure to a membrane struc-ture thus minimizing the bending part of strain energy. The optimal shape generated by hanging models isthe result of a mechanical deformation for one load case. It is dominated by the size of the undeformed

original piece of material (chain, cloth) which has been used for the experiment. In two dimensions for

the generation of shells the choice of the initial shape is also critical with respect to wrinkles and folds which

may develop during the deformation. The choice of material, isotropic or anisotropic, also aects the result.

The implicit interaction of initial and optimal shape through the mechanical deformation yields that the

design variables are not at all obvious. It might be very complex to identify and to vary them. The variety

of possible solutions is almost innite, because a further classication of structural quality besides the ab-

sence of bending is not part of the method. And, stability eects cannot be considered by hanging models.The numerical model is derived from the virtual work equation (3), again allowing for large displace-

ments and considering an external load p

Fig. 9. Snow loaded 5-point tent.

which denes an elastic relation between the GreenLagrange strain tensor E and S where k and l are the

3. Str

[18] T

metri

theyprobl

The oshape

as ten

shell

Th

weigh

by thhey combine highly specialized methods from dierent disciplines as there are computer aided geo-

cal design, computational mechanics, and non-linear mathematical programming, Fig. 11. TogetherThe methods of structural optimization are the most general optimization tools, for an overview see e.g.uctural optimizationLame constants. The kinematic assumptions for thin membranes are applied to derive a standard nite ele-

ment discretization.

The hanging model experiment is done numerically starting from an undeformed reference structure, e.g.

a plane membrane, applying a shape generating load case, usually a distributed surface load like self weight

or snow. In general, the equilibrium shape is found after large displacements and large strains as can be

seen by the following examples. Nevertheless, the St. VenantKirchho law can be used which is restricted

to small strains. The hanging model is used only for shape generation and not for the validation of the nal

structure after inversion and application of additional loading, Fig. 10.dw tZAF S : dF pdudA t

ZAS : dE pdudA 0: 15

The isotropic St. VenantKirchho material law is used

S k trEI 2lE 16

Fig. 10. Hanging models and inverted shapes.3446 K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452dene a modular tool box which can be applied for the denition of a very general optimizationem:

f s; u ! mins

;

his; u 0; i 1; . . . ; no: of equality constraints;gks; u 6 0; k 1; . . . ; no: of inequality constraints:

17

ptimization variables s are the spatial control node positions of the design model which in turn deneand FE mesh, Fig. 12. In terms of Computer Aided Geometrical Design the design model is dened

sor product surfaces, e.g. Bezier-spline and B-spline patches. The example shows an initially spherical

which is approximated by 6 Bezier patches [3,4].

e analysis model denes type of structure, material and loading conditions, e.g. shell, concrete, self

t, respectively. The structural behavior is a function of the state variables u. The optimum is dened

e objective function f and the constraints g and h which are functions of the design s and the state u. As

K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452 3447objective and constraints all structural properties and response quantities can be used which dene the

considered optimization problem, e.g. weight, mass, strain energy, absolute stresses, stress dierences,

Optimization Model

DesignModel

Analysis Model

CARAT

Fig. 11. The models of structural optimization.

Fig. 12. Shape optimization of a shell by means of structural optimization.

strate

Ta

tical

is possible because doubly curved shells are able to carry dierent loads by pure membrane action. Shape

generating and design load cases can be dierent. The governing equations are the sum of (3) and (15)dw tZAF S Spre : dF pdudA t

ZAS Spre : dE pdudA 0: 18

The pre-stresses belonging to the soap lm part are nominated by Spre and S are related to the hanging4. Merging form nding and hanging models

Soap lm analogy and hanging model experiment can be combined to nd structures of maximum sti-

ness in situations which are characterized by a combination of distributed surfaces loads and pre-stress. Thismode

by us

Th

had texper

The r

shape

can b

ment

result

Doub

5. M

Th

its wegies can be used, e.g. mathematical programming or optimality criteria methods.

king minimum strain energy as objective whilst restricting mass the optimal shape is in principle iden-

with the related hanging model and acts mainly as a membrane and is almost free of bending.displacements, eigenfrequencies, etc. Finally, the procedure is controlled by the optimizer, where several

Fig. 13. Periodically arranged concrete pillars.3448 K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452l and reect elastic deformations. Depending on the relation between both parts in (18) regularization

e of the updated reference strategy may be necessary.

e approach was successfully applied for a real application which is shown in Fig. 13. Concrete pillars

o be designed to carry heavy loads. The original design was motivated by the inversion of a principaliment shown at top of the gure. A closed thread is pulled out of a soap lm to form a singular vertex.

ow below shows on the left side the application of this idea to form periodically arranged pillars. This

modeled as a shell and loaded by distributed surface load instead of the shape generating load case

e further improved by simultaneously considering the soap lm analogy and the hanging model experi-

. The critical part of the design is the edge of the inner eyes as can be seen by comparison of the two

s. Here the structure reacts very sensitive to modied shape generating loads by bending response.

ly curved inner regions are able to carry several load cases by membrane action.

erging hanging models and structural optimization

e general denition of structural optimization is the reason for the power of the approach but also for

akness if applied to special tasks. Since the structural geometry is dened in mathematical terms it

K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452 3449Fig. 14. Hanging model experiment as shape generator in a structural optimization loop.canno

where

mode

insteathe cl

looki

hangi

addit

or for

of the

The l

given

repor

conteFig

sical

the co

eleme

are th

modet reect anything of the mechanical behavior. This is the case for form nding of free form shells

it is initially known that the optimum shape will be free of bending. The idea of merging hanging

ls with structural optimization is to use a numerical hanging model experiment as shape generator

d of a CAGD design model, Fig. 14 [19,20]. Doing so, we implicitly search for the optimum withinass of membrane shapes. We add to the hanging model experiment an additional optimization loop

ng for the best available hanging model. Optimization variables can be all parameters which dene a

ng model experiment, e.g. the intensity and distribution of the shape generating loading case p. Inion, we can directly test the quality of a hanging model shape with respect to design load cases qstability. Dening the structural shape x(p) as the deection of the hanging model, i.e. as a functionload generating load parameters p, the optimization problem states as

f xp; u ! min;gjxp; u 6 0; j 1 . . . no: of ineq: constr:;hjxp; u 0; j 1 . . . no: of eq: constr:

19

oad parameters p are the optimization variables. For the following examples we took the intensity of a

load distribution, e.g. the number of optimization variables was reduced to one only. Chargin et al.

ted about a similar method taking also the deformation of statical load cases as shape vectors in the

xt of structural optimization [21].. 15 shows the results of a simple, principal example and compares the mentioned approach with clas-

shape optimization. The shape of a shell is to be optimized with respect to minimal strain energy. In

ntext of classical structural optimization the design model is dened as a patch of four Bezier-

nts. Considering geometrical continuity and symmetry ve independent shape variables remain which

e vertical coordinates of design nodes as indicated in the gure. In the middle of the gure the hanging

l of a vertically loaded membrane as alternative design model is shown. The independent shape

3450 K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452Fig. 15. Comparison of conventional structural optimization and the merged approach.variable is the intensity of the shape generating load p. The structural load case q is chosen to be the neg-ative of p. The thickness of the structure is constant and the same for both approaches. The bottom of thegure gives the optimal results for both alternatives. Both structures act in a membrane state of stresses andhave negative curvature at the free edges. The classical optimum gives the better optimal value of the

objective strain energy p. That reects the fact that the classical approach makes use of the larger numberof degrees of freedom. However, the merged result appears to be more pleasing and natural because the

shape has higher order continuity properties than the alternative. The classical optimum is the result

of minimizing bending whereas the merged optimum reects the best available shape which is generated

load path along edge beamconventional result:

load path along diagonal arches result of merged method:

Fig. 16. Load carrying behavior of dierent membrane solutions.

strain energy of normalized value 75,084 both alternative results are considerably better solutions.

The edges remain to be soft. Loads are, therefore, transferred along an internal diagonal pair of arches

they are combined or even merged with the most general method of mathematically based structural opti-mization procedures are dened which combine the power of each single technique. Various examples show

the success. Actual research deals with the generalization of the demonstrated techniques for other appli-cations as shown here.

Acknowledgement

Financial support by the German Ministry for Education and Science is gratefully acknowledged.

Referencesdown to the supports. Interestingly, both structural alternatives had been realized by engineers. They rep-

resent two local optima of shells which are optimized for stiness. Fig. 17 shows two other examples where

the merged methods had been applied.

6. Conclusions

Free form shells and membranes are considered to be optimal with regard to structural behavior if they

act in a pure state of membrane internal forces. This mechanical property is exploited by traditional physi-

cal experiments as the hanging model and the soap lm analogy. Very powerful numerical techniques sim-

ulate these experiments and allow generating even complex shapes by a simple and robust methodology. IfThe optimal structures dier considerably in the load carrying behavior, Fig. 16. Because of the high

curvature along the edges of the conventionally optimized structure the edges act like sti beams. As

a consequence, loads are transferred from the inner domain rst to the edge and then down to the support.An orthogonal grid of arches is formed. The resulting structure of the merged approach behaves dierent.by p and loaded by q. A more shallow shape would exhibit higher stresses, a steeper shape would have alarger volume. Both would give larger values of strain energy. If compared to a plane plate pure bending

Fig. 17. Combining hanging models and structural optimization: two further examples.

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Copenhagen, 1991, pp. 5967.

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[4] K.-U. Bletzinger, E. Ramm, Structural optimization and form nding of light weight structures, Comput. Struct. 79 (2001) 2053

2062.

[5] S. Hildebrandt, A. Tromba, Mathematics and Optimal Form, Scientic American Library, 1985.

[6] T. Suzuki, Y. Hangai, Shape analysis of minimal surface by the nite element method, in: International IASS Symposium on

Spatial Structures at the Turn of the Millenium, Copenhagen, 1991, pp. 103110.

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at Berkeley, 1972.

[9] R. Haber, J. Abel, Initial equilibrium solution methods for cable reinforced membranes, Comput. Methods Appl. Mech. Engrg.

30 (1982) 263284.

[10] M. Barnes, Form and stress engineering of tension structures, Struct. Eng. Rev. 6 (1994) 175202.

[11] W.J. Lewis, Tension Structuresform and Behaviour, Thomas Telford Ltd, London, 2003.

[12] B. Maurin, R. Motro, Surface stress density method as a form nding tool for tensile membranes, Engrg. Struct. 20 (1998) 712

719.

[13] R. Motro (Ed.), Int. J. Space Struct., Tensile structures vol. 14 (1999) (special issue).

[14] K. Ishii, Membrane Structures in Japan, SPS Publ. Co., Tokyo, 1995.

[15] K.-U. Bletzinger, E. Ramm, A general nite element approach to the form nding of tensile structures by the updated reference

strategy, Int. J. Space Struct. 14 (1999) 131146.

[16] K.-U. Bletzinger, R. Wuchner, Form nding of anisotropic pre-stressed membrane structures, in: Wall, Bletzinger, Schweizerhof

(Eds.), Trends in Computational Structural Mechanics, CIMNE, Barcelona, 2001, pp. 595603.

[17] H. Isler, Concrete shells derived from experimental shapes, Struct. Engrg. Int. 3/94 (1994) 142147.

[18] R.T. Haftka, Z. Gurdal, Elements of Structural Optimization, third ed., Kluwer, 1991.

[19] K.-U. Bletzinger, Form nding and optimization of tensile structures, in: International IASS-IACM Conference on

Computational Methods for Shell and Spatial Structures, Chania, Greece, 2000.

[20] L. Schiemann, Formndung und Formoptimierung von Schalen mit numerischen Hangemodellen und mathematischer

Programmierung, Diplomarbeit, Uni Karlsruhe, TU Munchen, 2000.

[21] M.K. Chargin, I. Raasch, R. Bruns, D. Deuermeyer, General shape optimization capability, Finite Elements Anal. Des. 7 (4)

(1991) 343354.

3452 K.-U. Bletzinger et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 34383452

Computational methods for form finding and optimization of shells and membranesIntroductionForm finding methodsPre-stressed membranes mdash soap film analogyVirtual work of a surface stress fieldRegularizationThe updated reference strategyAnisotropic pre-stressHanging models

Structural optimizationMerging form finding and hanging modelsMerging hanging models and structural optimizationConclusionsAcknowledgementReferences