A General Finite Element Approach to the Form Finding of Tensile Structures by the Updated Reference Strategy

1 INTRODUCTIONMembrane structures are very attractive alternatives tospan large distances They are very light elegant andeffective [1-3] The material is optimally used sincethe structures are subjected only to membrane tensionstresses The art of form finding means to find theoptimal deflected and finally visual shape due to agiven stress distribution acting on the deformedstructure This fact should be clearly pointed out sinceit is different compared to conventional structuraloptimization where the undeformed shape isoptimized with respect to displacements due to givenload cases

The problem is very closely related to thedetermination of minimal surfaces Since centuriesthey are one of the oldest toys of mathematiciansThere exists an enormous amount of knowledge ofhow to describe and to generate them [4] From themechanical point of view minimal surfaces aredetermined by an isotropic stress field that can beexperimentally simulated by the soap film analogyFor the engineering application ldquoform finding oftensile membrane structuresrdquo the later approach ismost promising since it is directly formulated withrespect to stresses which restrict and dominate thefinal design because of strength resistance and the

International Journal of Space Structures Vol 14 No 2 1999 131

A General Finite ElementApproach to the Form Finding

of Tensile Structures by the Updated Reference

StrategyKai-Uwe Bletzinger and Ekkehard Ramm

Institute of Structural Analysis University of KarlsruheKaiserstr 12 76131 Karlsruhe Germanye-mail kubbau-vermuni-karlsruhede

web page httpwwwuni-karlsruhede~baustatikMitarbeiterkubhtml

Institute of Structural Mechanics University of Stuttgart70550 Stuttgart Germany

e-mail erammstatikuni-stuttgartdeweb page httpwwwuni-stuttgartdeibsrammhtml

(Received 6th November 1998 revised version received 28th April 1999)

ABSTRACT Starting from a rigorous mechanical formulation a numericalprocedure is developed for the form finding of minimal surfaces and pre-stressed membrane structures The method is based on an iteratively adaptedregularisation of the stiffness matrix which gives the method its name theupdated reference strategy The method can be applied to any finite elementdiscretization of cable and membrane structures subjected to pre-stress as wellas lateral pressure or other external loading It is very robust and reliable as isshown by many illustrative examples Further analysis states that the wellknown force density method is the special case of applying the updatedreference strategy to cable nets

anisotropic material behavior in the directions of weftand warp [5] Furthermore an unlimited variation ofshapes is generated by prescribing anisotropic stressdistributions

Inspired eg by the pioneer work of Frei Otto [6]many technical procedures and algorithms had beendeveloped which many of them are based on dynamicrelaxation [78] Others like the force density methodare based on special discretization and linearizationtechniques Originally it was developed for the formfinding of cable structures [9-11] Recently themethod was extended to triangular membraneelements [1213] The existence of all the differentmethods is explained by the mathematical problemwhich arises by solving an inverse mechanicalproblem It is defined by the prescribed stressdistribution as the driving degree of freedom in thedesign process This is inverse to standard mechanicswhere stresses are the structural response to thedeformation of material As a consequence the relatednumerical solution methods are faced to mathematicalsingularities which are overcome by severaltechniques eg by a modified Newton-Raphsoniteration [14-16] The updated reference strategywhich will be presented in the following is a furtheralternative [17-19] It is consistently derived fromcontinuum mechanics of elastic bodies with respect tolarge deflections and small strains The numericalsolution follows standard finite element discretizationprocedures which means that the method can beapplied to any triangular or quadrilateral finiteelement formulation The above mentionedsingularities are regularized by a homotopy mapping[20] which is based on approximations for Cauchy andPiola-Kirchhoff stresses It will be shown that thisapproach leads to the force density method if it isapplied to one dimensional elastic bodies eg cablesVice versa the updated reference strategy is theconsistent generalization of the force density methodapplicable for any structure It can also be used as anapproximation technique in structural shapeoptimization [19]

2 THE GOVERNING EQUATION OFFORM FINDING

21 DIRECT GEOMETRICAL APPROACH TOMINIMAL SURFACESMinimal surfaces are defined as surfaces of minimalarea which are enclosed by a given boundaryConsider the procedure to determine a minimalsurface as an iterative process (Fig 1) The surface of

the intermediate state or actual configuration withactual area a is understood as to be generated bydeformation of an initial state or referenceconfiguration of area A A point P on the actual surfacewith position vector x (u 1u 2 ) had originally theposition X (u 1u 2) on the reference surface The pointis identified by its surface coordinates u 1u 2 whichstay invariant during deformation The displacement uof point P is defined as

u( u 1u 2)=x ( u 1u 2)ndashX( u 1 u 2) (1)

The surface area a of the actual configuration can be written with respect to the referenceconfiguration as

a = Ea

da = EA

detFdA (2)

where detF is the determinant of the deformationgradient F Refer to the appendix for further details ondifferential geometry

Figure 1 Deformation of surface

The minimum of (2) is determined by the vanishingvariation d a of the surface area with respect tovariation d x of the shape This states the governingequation

d a= d 1 E ada 2 = EAd (detF)dA=E

AdetFFndashT d FdA=0 (3)

where

d F= d giAuml G i= (4)

The shape variation cannot be chosen arbitrarily It

para( d x)

paraX

A General Finite Element Approach to the Form Finding of Tensile Structures by the Updated Reference Strategy

132 International Journal of Space Structures Vol 14 No 2 1999

must have a normal component at any point on thesurface ie the condition d xmiddotg3plusmn 0 must be fulfilledeverywhere If the variation d x would map the surfaceback on itself the area content would not change evenfor a non-optimal geometry

22 THE MECHANICAL APPROACH ndashVIRTUAL WORK OF A SURFACE STRESSFIELDConsider a stress field which acts tangential to asurface and which is assumed to be in self-equilibrium(Fig 2) eg one can imagine the stress field as theresulting stresses caused by initial pre-stress andstresses due to deformation of a textile membranewithin a rigid boundary However at this point it is notnecessary to specify any special material since weconsider the stress field as given no matter how it wasgenerated The problem is to find the geometry of therelated surface which allows the stress field to be inequilibrium The governing equation is the principleof virtual work

Figure 2 Tangential surface stress field

d w=tEas da=tE

as d uxda=0 (5)

which states that the virtual work of a stress field inequilibrium vanishes s is the prescribed Cauchystress tensor which acts on the surface in equilibriumd ux is the derivative of the virtual displacement withrespect to the geometry of the actual surface Thethickness of the membrane is denoted by t It iscomparatively thin and assumed to be constant duringdeformation ie the Poisson effect in thicknessdirection is neglected This is in accordance with thebehavior of available membrane material

d ux can be expressed in terms of the deformationgradient From (1) it follows that d u = d x and

d ux= = =para 1 2 Fndash1= d paraFFndash1 (6)

(6) inserted into (5) and integration over thereference area A

d w=tEa

s d ux da=tEA

s d uxdetFdA

=t EA

s ( d FFndash1)detFdA=0 (7)

Rearrangement of tensors gives the alternativeequivalent formulation

d xw=t EA

s ( d FmiddotFndash1)detFdA=0(8)

=t EAdetF(d middotFndashT) d FdA=0

A further rearrangement of (8) leads to

d w=t EA

detF(FmiddotFndash1 s middotF-T) d xFdA(9)

=t EA(Fmiddot(detFF-1middots middotF-T))x d FdA=tE

A(FmiddotS)d FdA=0

where S is the 2nd Piola-Kirchhoff stress tensor S isrelated to s by

s = s ab ga Auml g b (10)

S=detFF-1middot s middotF-T =detFs ab GaAuml Gb =Sa b GaAuml Gb

The components S a b of the 2nd Piola-Kirchhoffstress tensor are identical to the components s a b ofthe Cauchy stress tensor multiplied by detF Howeverthey are related to the base vectors of the referenceconfiguration whereas the Cauchy stress componentsare related to the actual configuration In general if sis homogeneous than S is not a vice versa S and sare identical if actual and reference configuration areidentical Note that all stress components normal tothe surface area are zero ie

s i3=s 3i=Si3=S3i=0 (11)

For further reading on the mechanics of tensionstructures see eg Leonard [22]

23 SOAP FILM ANALOGY ndash THEEQUIVALENCE OF GEOMETRICAL ANDMECHANICAL APPROACHUp to here we developed two different equations todefine (i) minimal surfaces

paraxparaX

paraXparax

para(d x)

paraX

para( d u)

parax

para(d u)

d x

Kai-Uwe Bletzinger and Ekkehard Ramm


d a= EA

detFF-T d FdA=0 (3)

and (ii) the shape of equilibrium of a given stress field

d w=t EAdetF( s middotFndashT) d FdA=0 (8)

If an isotropic homogeneous stress field isprescribed ie s = sI is a scalar multiple of the unittensor I (8) reduces to (3) and the shape ofequilibrium is a minimal surface This well known factis called the soap film analogy which reflects theexperiment to produce minimal surfaces of thin soapfilms within a rigid wire

(8) is the theoretical basis of the form findingprocedure of membrane structures The manipulationof s allows for a broad range of structural shapesother than minimal surfaces It is the governing degreeof freedom in the design procedure On the other handit reflects the fact that shapes of isotropichomogeneous stress distributions often cannot bebuild because of technical reasons of manufacturing orthat minimal surfaces may be bad alternatives to carryadditional load as eg wind and snow

3 DISCRETIZATION OF THE GOVERNINGEQUATIONSThe equations (3) and (8) are solved numerically bythe finite element method Geometry anddisplacements are discretized by standarddisplacement elements The surface geometry and the

displacement field are piecewise approximated by theinterpolation of nodal coordinates or displacementsrespectively (Fig 3)

X= ^nel

k=1Nk ( u 1u 2)Xw k u= ^

nel

k=1Nk ( u 1u 2 )uw k

(12)

x= ^nel

k=lNk ( u 1u 2 ) (Xw k +uw k )

The upper bar denotes nodal values egndash Xndashw k is thereference position vector of the k-th element node Nk

are the standard Co-continuous shape functions andnel is the number of nodes of one finite element Forthe sake of simplicity no difference in notationbetween real and approximated fields is made

The shape functions of a three node element are

X1 Xw 1k X2 = ^

3

k=1Nk Xw 2

k (13)

X3 Xw 3k

N1=1ndashu 1ndash u 2 paraN1 para u 1=ndash1 paraN1 parau 2=ndash1

N2= u 1 paraN2parau 1=1 paraN2para u 2=0


and of a four node element (Fig 3)

N1= (1+ u 1)middot(1+ u 2) N2= (1ndash u 1)middot(1+ u 2)(14)

N3= (1ndash u 1)middot(1ndash u 2) N4= (1+ u 1)middot(1ndash u 2)14

14

14

14



Figure 3 Finite element surface discretization

The covariant base vectors are determined straightforward to be

Ga= =Xa= ^nel

k=1Nka Xw k

(15)

ga= =xa= ^nel

k=1Nka (Xw k +uw k )

which gives for the three node triangle

G1 =Xw 2 ndashXw 1 g1 =(Xw 2 +uw 2 ) ndash (Xw 1 +uw 1 )

G2 =Xw 3 ndashXw 1 g2 =(Xw 3 +uw 2 ) ndash (Xw 1 +uw 1 )(16)

The nodal displacement components are defined tobe the free discrete parameters of form finding Theyare arranged in a column matrix b of dimension ndof(number degrees of freedom) Variation of any entityeg the deformation gradient F means now variationwith respect to the free parameters By use of the chainrule of differentiation we get

(17)

d F= d br =d giAuml Gi= d br Auml Gi r=1ndorf

and

paraga=paraxa = parabr= ^nen

k=1Nka parabr r=lndorf

(18)

where br is the r-th component of b ie the r-th degreeof freedom of the discretized problem

Discretization of (8) yields

paraw=parabr tEAdetF( s middotFndashT) dA=0 (19)

which has to be fulfilled for any choice of d b andfinally we arrive at the non-linear system of ndofequations

(20)=t E

AdetF(s middotFndashT) dA=0

4 LINEARIZATION

41 MEMBRANE ELEMENTThe governing discrete system of equations (20) isnon-linear in terms of the discretization parameters brIt is solved iteratively by subsequent linearizationusing the Newton-Raphson method Linearization of

(20) yields

LIN( )=t EA

detF(s middotFndashT) dA+D bs t EA

(21)1 detF(s middotFndashT) 2 dA=0 rs=1ndof

Using the standard vocabulary of the finite elementmethod

(221)R=t E

AdetF(s middotFndashT) dA

is the vector of unbalanced forces

and K=t EA 1 detF(s middotFndashT) 2 dA (222)

the stiffness matrix

As a consequence of discretization the covariantbase vectors gi are only linearly dependent on theparameters br and therefore the second orderderivative

= f G1 is equal to zero

The stiffness matrix of the membrane finite elementcan now be written as

Krs =tEA

(detF(s xmiddotFndashT)) dA (23)

or in components

(24)Rr =t EA

detFs ab (gar middotg b ) dA

Krs =t EA3 detFs ab (garmiddotg b s )+(detFs ab )s (garmiddotgb ) 4 dA

( )r = and ( )s =

Note that even for constant components of stresstensor s the derivative s a b s does not vanish sincethe related covariant base vectors ga are functions ofthe discretization parameters

For the case of isotropic stress fields ( s = s ^I ) orminimal surfaces the expressions reduce to

Rr = =ts ^ EA

detFgar middotgadA (25)

Krs= =ts ^ EAdetF [( gar middotg a )(g b s middotg b )ndash (g armiddotg b )(g b smiddotg

a)]dApara2a

parabr parabs

paraaparabr

paraparabs

paraparabr

paraFparabr

paraparabs

para2giparabr parabs

para2 Fparabr parabs

paraFparabr

paraparabs

paraFparabr

paraFparabr

paraparabs

paraFparabr

parawparabr

paraFparabr

parawparabr

paraFparabr

parauw kparabr

paragaparabr

paragiparabr

paraFparabr

paraxparau a

paraXpara u a



and Krs is identified as the Hessian matrix of thesurface area ie the matrix of second order derivativeswith respect to the discretization parameters

42 CABLE ELEMENTMembrane structures such as tents have usuallyflexible edges which are reinforced by cables Theinteraction of these cables with the attachedmembrane and as a result their shape are determinedby the forces which act in the cables During the formfinding procedure these forces are assumed to begiven and the same principle ideas as above apply todetermine the unbalanced force vector and thestiffness matrix of a cable element First we take (8)and reduce it to one dimension The remainingrelevant covariant base vector g1 is tangential to thecable The other two base vectors g2 and g3 areorthogonal to g1 and of unit length As a consequencedet F reduces to be det F = ||g1|| ||G1|| The remainingstress component acts in direction of g1 along thecable axis Ie the stress tensor can be set to be amultiple of the unit tensor s I = s ^I

Now the virtual work displays as

d w=Ac ES

detF(s middotFndashT)d FdS=s X Ac ES

g11(d g1middotg1 )dS

(26)

where integration is performed along the arc length Sof the reference configuration and the cable crosssection area Ac is assumed to be constant duringdeformation ie the Poisson effect is neglected

Figure 4 2-node cable finite element

We further discretize the cable by simple two-nodeelements (Fig 4) By this assumption we can set i g1i= ` i G1i = L g11 = 1`2 the element length beforeL and after ` deformation Now the integration can

be performed explicitly

d w= s X Ac (girmiddotg1)L d br= (girmiddotg1) d xbr

(27)

= (girmiddotgl) d br

where the last expression of (27) reflects thealternative formulation using the 2nd Piola-Kirchhoffstress SX

(28)S=SX I=detFFndash1middot sX ImiddotFndashT= sX I

The unbalanced force vector and the stiffnessmatrix display as

(29)

Rr = (glrmiddotg1)

Krs= 3 (glrmiddotgls)ndash (glrmiddotg1)(glsmiddotg1) 4with ` = Iuml gw 1middotw gw 1

w and ` s=

5 REGULARIZATION

51 NUMERICAL CONTINUATIONSo far the procedure is straight forward However aclose inspection shows that the stiffness matrix issingular with respect to tangential shape variationsThe total area content of the structural surface or theoverall length of the cable respectively are not alteredby these shape modification This reflects the generalrestriction for shape variations already mentionedabove The deficiency can be overcome simply byrestricting the shape variations to directions whichhave a significant normal component to the actualshape That means we link the degrees of freedom atthe finite element nodes by a prescribed movedirection which is at least not tangential to the actualshape The initially three degrees of freedom at a nodein space are reduced to one Tracing the form findingprocedure the move directions in general describecurves in space From a technical point of view itmight be difficult to define such a curve in advance inparticular if the surface undergoes large changes inshape

More severe however are the problems which areinduced by the flexible cable reinforced edges of

g1middotgls`

1` 2

s Ac`

s Ac`

L`

SAcL

s xAcl

l 1L l2

||g1 ||||G1 ||



X

X

X

X

membrane structures In general the edges are movingtangentially inwards during the form findingprocedure forcing a tangential adjustment of thesurface finite element mesh It is practicallyimpossible to consider this effect by prescribed movedirections We have to find other methods to regularizethe problem

General mathematical methods to approach asolution of a singular problem are the methods ofnumerical continuation [20] also called homotopymethods or path following methods The idea is tomodify the original problem by a related one whichfades out as we approach the solution The methodsare used eg to determine the non-linear loaddeflection relations in structural limit state analysis[23] or structural optimization [24]

Eg consider the problem f(x) Otilde min Theobjective f(x) is assumed to be singular at the solutionthe standard methods of optimization fail However arelated problem f rsquo(x) Otilde min min is known which is notsingular The function f rsquo(x) can be used to regularizethe original problem by mapping with the continuationfactor l

(30)min fl (x)=min[ l f (x)+(1ndashl )f rsquo(x)]

The solution of (30) approaches the solution of theoriginal problem as we trace l from 0 to 1 The factorl is identified as the step length parameter of a curvein the extended parameter space (x l ) The method isthe more successful as the related function f rsquo(x) iscloser to the original function f (x)

Applied to the problem of form finding ofmembranes and minimal surfaces we can use thealternative formulation of the virtual work (9) by useof the 2nd Piola-Kirchhoff stress tensor S as a relatedproblem The difference to the original problem is thatwe now assume the 2nd Piola-Kirchhoff stresses to begiven instead of the Cauchy stresses s The modifiedstationary condition states as

d wl = l tEAdetF( s middotFndashT) d FdA

(31)+(lndash l )t E

A(FmiddotS) d FdA=0

For the modified problem Cauchy as well as 2nd

Piola-Kirchhoff stresses are prescribedAs the 2nd Piola-Kirchhoff stresses are defined in

terms of the fixed reference configuration S remainsconstant if the shape is modified the derivatives withrespect to the discretization parameters vanish Sr = 0

Thus the Hessian of the modified problem has theform

K(l )rs= l t EA(detFs middotFndashT)sFrdA

(32)+(lndash l )t E

A(FsmiddotS)FrdA

where the regularization term

(33)t EA(FsmiddotS)FrdA=tE

ASab garmiddotgb sdA

is positive definite if S is positive definite which is thecase for tension stresses As a consequence the totalstiffness matrix is also positive definite if l is smallenough The quality of the approximation improves asthe reference configuration gets closer to the optimalshape Note that for given S (33) is a constantexpression since the covariant base vectors ga are onlylinear dependent on the discretization parameters br Innon-linear structural analysis (33) is also known asgeometrical stiffness or initial stress matrix

For an isotropic stress distribution (S = SXI) orminimal surfaces (33) reduces to

t EA(FsmiddotS)FrdA=S

Xt E

AGab garmiddotg b sdA (34)

The modified unbalanced force vector is

R(l )r= l t EAdetF( s middotFndashT)FrdA

(35)

+(1ndashl )tEA

(FmiddotS)FrdA

52 APPROACHING THE OPTIMUM ndashTHE UPDATED REFERENCE STRATEGY(URS)Several strategies are suggested to trace thecontinuation factor l towards the optimal solution

(i) Adaptation of l by trial and error

The continuation factor l is set constant and smallenough to stabilize the problem Trace lexperimentally from 0 to 1 in series of analyses untilthe solution fails Take the last solution of themodified problem (31) as optimal solution Thisprocedure is simple however not very satisfactory inparticular for large problems



(ii) Optimization of l by extended systemmethods

A solution of the singular problem can directly bedetermined by so called extended system methodsThese methods have recently been used to determinebuckling or limit loads in the context of non-linearstructural analysis [24-26] A detailed list of relatedliterature is eg given by Reitinger [24] Usually themethods consider critical points with one unique zeroeigenmode The basic idea is to extend the singularproblem by additional equations which state thesingularity condition and a normalization of therelated eigenvector After discretization the extendedsystem of equations displays as

R(l )r(bl ) =0 stationary condition

K( l )rs(bl )fs =0 singularity condition (36)

l(f ) =0 normalization of eigenvectorf

The singularity condition is enhanced by an additionalpenalty term to be able to solve (36) Several exact andapproximate penalty formulations are known inliterature Exact formulations fail because the originalproblem is multiple singular An approximateformulation given by Reitinger might instead beapplied which underestimates the correct solution andgives save approximations

R(l )r(bl ) =0

[K(l )rs(bl )+cX ereTs ]f sndashcX [esf sndashfX ]er =0 (37)

eTr f rndashfX =0

cX penalty parameter er =lr|max(f r)r=1 ndorf0otherwise

Although theoretical fascinating the extendedsystem methods may appear in this context as notstable enough for routine usage

(iii) The Updated Reference Strategy (URS)

The solution of the modified problem (31) with anysuitable choice of l is used as a new improvedreference state for the next approximation Thereference configuration is iteratively adapted towardsthe optimal solution If the differences of subsequentsolutions are small enough to be accepted the series ofapproximations is terminated The strategy convergesto the optimum for any value of the continuation factorl which is small enough to ensure the regularity of theHessian matrix For the special case l = 0 (31) is even

linear and is solved within one step for the cost ofadditional approximation steps In practice one willstart with a sufficiently small value eg l pound05 andwill enlarge l for the next steps to improveconvergence The procedure is absolutely robust andgives reliable results very fast The amount of trainingis negligible

53 THE FORCE DENSITY METHOD AS ASPECIAL CASE OF URSAs stated above URS converges for almost any valuefor the continuation factor l If l is set to 0 and URSapplied to 1D structures (spatial cables and struts) itcan be further specialized Consider an element of acable net It has the length L in the undeformedreference configuration and the length ` in the actualconfiguration Yet the deformation gradient F isdetermined to be F = ` L = (L+u)L where u is thediscretization parameter and describes the longitudinaldeflection The derivation of F wrt u yields Fu = 1LAgain we assume Ac to be constant throughout theentire form finding process This means that theCauchy stress in the cable is expressed by s = nAcwhere n is the prescribed tension force Putting alltogether we get

S=SX I=detFFndash1middots middotFndashT= I (38)

And for the stiffness with respect to thediscretization parameter u

K=AcEL(FumiddotS)Fuds=AcE

L 1 2 ds= =q (39)

Finally we arrive at the expression q which is wellknown as ldquoforce densityrdquo and relates the prescribedcable force n to the deformed length ` of the memberThe force density q may be interpreted as a ldquo2nd Piola-Kirchhoff forcerdquo It is usually set constant duringcalculation which yields linear systems of equationsFinally the force density method appears to be aconsistent part of the more general theory of theupdated reference strategy

6 SURFACE PRESSURE LOADTension structures may be classified by the way howthey are prestressed either ldquomechanicallyrdquo bystretching the edges or ldquopneumaticallyrdquo by a surfacepressure load Pressure is characterized by constantmagnitude while the direction is always normal to the

n`

lL

L n` Ac

lL

I=Ln` Ac

` LnLL` Ac`



changing surface during all stages of deformationLoad of that type is therefore often called followerload Because of that follower loads contribute to thestiffness matrix It depends on the boundaryconditions whether this load is conservative or not[27] ie the stiffness matrix is symmetrical or notrespectively

The equation of virtual work (5) is enhanced by thepressure term

d w=tEas d uxdandashE

apmiddot d u da=0 (40)

The geometrical equivalence are surfaces ofconstant positive mean curvature as a generalizationof minimal surfaces which are defined by zero meancurvature

Figure 5 Surface normal pressure

Given the pressure magnitude pX the pressure vectoracts in the direction of the surface normal (Fig 5)

p=pX g3g3= = (41)

The virtual work of the pressure load can now bewritten as

Ea

pmiddotd uda=Eu 1 E u 2

jpmiddotd udu 1d u 2

(42)

= Eu 1 E u 2

pX (g13 g2) d ud u 1du 2

Considering the discretization of the displacementu in terms of the parameters br and the definition (49)of the base vectors ga ie

parau=parax= d br=xrd br and ga= =xa

the virtual work expression is further modified to be

Ea

pmiddotd uda=d br Eu 1E u 2

pX (x13 x2)middotxr du 1du 2 (43)

Linearization gives

lin(Ea

pmiddot d uda)=d brEu 1E u 2

pX (x1 3 x2)middotxr du 1du 2

(44)

+d br D bsEu 1E u 2

pX [(x13 x2)smiddotxr+(x13 x2)middotxrs]du 1du 2

and we can identify the additional contributions R(p)r andK(p)rs to unbalanced force vector and stiffness matrix

R(p)r=ndash Eu 1E u 2

pX (x13 x2)middotxr du 1du 2

K(p)rs=ndashEu 1E u 2


ndashEu 1E u 2

pX (x13 x2)smiddotxrdu1du 2 (45)

which have to be added to (24) to yield the totalexpressions In the case of standard displacementelements the second term of K(p)rs vanishes This isbecause the geometry x is only linearly dependent onthe discretization parameters br and the secondderivatives of x with respect to br are therefore zero

Making use of (a 3 b) c = ndash(b 3 a) c we integrate(45) by parts and get after some rearrangement

K(p)rs=ndash pX Eu 1

Eu 2

(x13 x2)smiddotxr du 1du 2

=ndash pX Eu 1

(x13 xs)middot xrdu 1ndashpX Eu 2

(xs3 x2)middotxrdu 2

ndash pX Eu 1E u 2

(x13 x2)rmiddotxs du 1du 2 (46)

Adding K(p)rs = ndash pX e u 1 e u 2 (x13 x2)smiddotxrdu1 du 2 on

both sides and dividing by 2 gives finally

(47)

Obviously the domain integral of (47) issymmetrical The boundary integrals vanish at finiteelement interfaces if adjacent elements share common

paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||



tangent base vectors and common displacementderivatives ur That is the case for conformingelements as eg for the chosen displacement elementsThus the boundary conditions decide on thesymmetry of the overall stiffness matrix and theconservative or non-conservative character of thepressure loading The boundary integrals vanish if thedisplacement is prescribed at the structures boundary(ur 0) or if (g1 3 us)middotur 0 or (us3 g2)middotur 0 atthe supported edge The latter conditions reflect thesituation that at every point on the boundary thedisplacement derivatives ur and the tangent basevector share a common plane eg support in normaldirection The typical example of an inflated bubblewhich is fixed around a hole where the filling mediacomes through is a conservative problem withsymmetrical stiffness matrix

7 EXAMPLES

71 CATENOID

The shape of a catenoid is determined starting from acylindrical initial shape of diameter 2 and height 12(Fig 6) All three solution strategies have been appliedand the numerical solutions are compared with theanalytical result One eighth of the shape wasdiscretized by 144 4-noded isoparametric elementsThe best approximation of shape was achieved by theupdated reference strategy The comparison of surfacearea is misleading because of the discretization errorEven for strategy ldquotrial and errorrdquo the principalstresses are almost homogeneous Other extendedsystem methods than Reitingerrsquos failed

no ofstrategy Surface area time steps l

analytical 70Trial and error 6998 1 085

URS 6993 5 08ext system 6993 1 + 08

ext system 0979



Figure 6 Catenoid Form finding results

Figure 7 Scherkrsquos surface Scherkrsquos tower Ennepperrsquos surface

72 OTHER MINIMAL SURFACESThe success of URS is demonstrated by the generationof well known mathematical examples as there areScherkrsquos or Enneperrsquos surfaces (Fig 7)

73 INFLATED BUBBLEThe shape of a bubble is determined where themagnitude of the principal stresses in the skin oughtto be 10 Fig 8 The initial shape is a flat circularplate of diameter 20 As the internal pressure isincreased the bubble grows and exhibits the ideal

shape of a discretized sphere The problem ismodelled by 80 4-node iso-parametric elementsThe continuation factor is 095 throughout thewhole calculation

74 TENTSSeveral student projects [28-30] of membranes withcable reinforced free edges are shown to give animpression of the variety of applications Fig 9 Someof them are related to existing structures as eg theldquoTanzbrunnenrdquo by Frei Otto



Figure 8 Inflated bubble

Figure 9 Student projects

75 RELATIONS TO CADThis example illustrates some procedural aspects ofform finding Fig 11 shows the stress distribution of atent after form finding with URS The stresses arehomogeneously distributed indicating a minimalsurface The results are transmitted to a CAD toolwhich allows architectural judgment Fig 10 andfurther modifications of the geometry Thechallenging aspect of this project which was proposedby the architect Siegfried Gass is the free central massof the tent It is supported by the membrane itself andonly two edge cables Considering practical loading aswind and snow the spatial support of the mast top isvery sensitive with respect to the overall geometry ofthe structure

Figure 10 CAD realization



Figure 11 Distribution of principal stresses

Figure 12 URS interation

76 URS CONVERGENCE BEHAVIORThe convergence behavior of the updated referencestrategy is demonstrated by this example The problemis to find the minimal surface of the displayedstructure (Fig 12) Surface stress and edge cable forceare set to 1 The significant displacement of a selectedfinite element node is traced versus the continuationfactor for iteration steps 1 to 4 and the final valuelabeled as 5 Fig 13) The rapid convergence isobvious

8 CONCLUSIONSForm finding ie the problem to determine the shapeof equilibrium of a stress field with given intensity butunknown orientation is a challenging esthetical aswell as analytical and numerical task Classicalmethods of consistent linearization fail because theproblem is singular at the solution The updatedreference method overcomes the difficulty by arigorous algorithm which is based on differentialgeometry continuum mechanics and numericalcontinuation Internal pressure is implementedconsidering the interaction of shape and load directionand the consequence for the non-linear characteristicsof form finding The method can be applied to theform finding of membranes and cable structures usingany finite element discretization scheme eg two-three- or four-node elements or even elements ofhigher order It is shown that the force density methodfor cable nets is a special case of the presentedmethod Several typical examples illustrate accuracybehavior and applicability of the method

9 REFERENCES1 Otto F and Rasch B Finding Form Deutscher

Werkbund Bayern Edition A Menges 1995

2 Schock H-J Segel Folien und Membrane BirkhaumluserBasel 1997

3 Berger H Light structures ndash structures of lightBirkhaumluser Basel 1996

4 Dierkes U Hildebrandt S Kuumlster A Wohlrab OMinimal surfaces I and II Grundlehren dermathematischen Wissenschaften 295-296 SpringerBerlin Heidelberg 1992

5 Suzuki T Hangai Y Shape analysis of differently stressedsurfaces by the finite element method in Srivastava N KSherbourne A N and Rooda J eds Innovative LargeSpan Structures (vol 2) Proc IASS-CSCS InternationalCongress Toronto Canada 1991 400-411

6 Otto F et al Zugbeanspruchte Konstruktionen Vols Iand II Ullstein Verlag Franfurt Berlin 1962 1966

7 Barnes M Form and stress engineering of tensionstructures Structural Engineering Review 6 1994 175-202

8 Lewis W J Lewis T S Application of Formian anddynamic relaxation to the form finding of minimalsurfaces Journal of the IASS 37(3) 1996 165-186

9 Linkwitz K and Schek H-J Einige Bemerkungen zurBerechnung von vorgespannten SeilnetzkonstruktionenIngenieur-Archiv 40 1971 145-158

10 Schek H-J The force density method for form findingand computations of general networks ComputerMethods in Applied Mechanics and Engineering 31974 115-134



Figure 13 URS onvergence behaviour

11 Linkwitz K Least squares methods in non-linear formfinding and analysis of pre-stressed and hanging nets inNon-Linear Analysis and Design for Shell and SpatialStructures Proc SEIKEN-IASS Symposium Tokio1993 601-610

12 Singer P Die Berechnung von MinimalflaumlchenSeifenblasen Membrane und Pneus aus geodaumltischerSicht PhD Thesis University of Stuttgart 1995

13 Maurin B and Motro R The surface stress densitymethod as a form-finding tool for tensile membranesEngineering Structures 20(8) 1998 712-719

14 Haug E and Powell G H Finite element analysis ofnonlinear membrane structures Report UCSESM 72-7University of California at Berkeley 1972

15 Haug E and Powell G H Finite element analysis ofnonlinear membrane structures in Tension and spacestructures (vol2) Proc 1971 IASS Pacific SymposiumTokio and Kyoto 1972 165-175

16 Haber R and Abel J Initial equilibrium solutionmethods for cable reinforced membranes ComputerMethods in Applied Mechanics and Engineering 301982 263-284

17 Bletzinger K-U Form finding of membrane structuresand minimal surfaces in Rozvany G and Ohlhoff Neds Proc 1st World Congress of Structural andMultidisciplinary Optimization Goslar Germany 1995563-568

18 Bletzinger K-U Form finding of tensile structures bythe updated reference strategy in Chilton J Lewis Wet al eds Structural Morphology ndash Towards the NewMillennium Proc IASS International ColloquiumAugust 15-17 1997 University of Nottingham UK1997

19 Bletzinger K-U Shape optimization by homotopymethods with special application to membranestructures in Proc 6th AIAANASAISSMO Symposiumon Multidisciplinary Analysis and Optimization Bellevue Washington USA 1996 122-129

20 Allgower E L and George K Numerical ContinuationMethods Springer Series in ComputationalMathematics Springer Berlin Heidelberg New York1990

21 Kreyszig E Differentialgeometrie AkademischeVerlagsgesellschaft Leipzig 1968

22 Leonard J W Tension Structures McGraw-Hill NewYork 1988

23 Ramm E The RiksWempner approach ndash An extensionof the displacement control method in non-linearanalysis in Hinton E Owen D R J and Taylor Ceds recent advances in non-linear computationalmechanis Pineridge Press UK 1982 63-86

24 Reitinger R Bletzinger K-U and Ramm E Shapeoptimization of buckling sensitive structures ComputingSystems in Engineering 5 65-75 1993

25 Wagner W Zur Behandlung von Stabilitaumltsproblemender Elastostatik mit der Methode der finiten ElementeHabilitationsschrift Bericht-Nr F 911 Institut fuumlrBaumechanik und Numerische Mechanik UniversitaumltHannover 1991

26 Wriggers P and Simo J C A general procedure for thedirect computation of turning and bifurcation pointsInternational Journal Numerical Methods inEngineering 30 1990 155-176

27 Schweizerhof K Nichtlineare Berechnung vonTragwerken unter verformungsabhaumlngiger Belastung mitfiniten Elementen PhD Thesis University of Stuttgart1982

28 Wahl C Numerische Formfindung von vorgespanntenSeil- und Flaumlchentragwerken Diploma Thesis Institutfuumlr Baustatik University of Stuttgart 1995

29 Krapf A Formfindung und Statik vorgespannterMembrantragwerke Diploma Thesis Institut fuumlrBaustatik University of Stuttgart 1996

30 Seebacher F Formfindung Zuschnitt und Statikvorgespannter textiler MembrankonstruktionenDiploma Thesis Institut fuumlr Baustatik University ofKarlsruhe 1998

APPENDIXThe basis of differential geometry will be brieflydisplayed [21]

The position vectors x and X to a point P on aspatially curved surface in the actual and referenceconfiguration respectively are vector functions of thesurface parameters u 1 and u 2

x=x( u 1u 2) X=X( u 1u 2) (48)

The covariant vectors g1 g2 and G1 G2 are definedas

g1= g2= and G1 =G2 (49)

The covariant base vectors are tangential to thecorresponding coordinate lines eg g1 is tangential tothe coordinate line u 1 = const in the actualconfiguration Covariant base vectors are in generalnot orthogonal and are not of unit length Their scalar

paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1



products gij ndash the components of the covariant metrictensor ndash reflect the metric of the surface ie the lengthof the covariant base vectors and the angle betweenthem

g11=g1middotg1sup1 1 g22=g2middotg2sup1 1 g12=g21=g1middotg2sup1 0 (50)

The surface normal is determined by g3 which isdefined by the normalized cross product of g1 and g2

g3= ||g3||=1 and G3= ||G3||=1

(51)

Th contravariant base vectors g i are the dual of thecovariant base vectors g j They are defined by the rule

(52)gimiddotgj= d ij and GimiddotGj= d i

j

where d ij is the Kronecker delta

Any vector or tensor can be written with respect to co-or covariant base vectors gi Gi or gi Gi

a=aigi=atilde iGi=aigi=atilde iG

i(53)

T=T ijgiAuml gj=T~ijGiAuml Gj=Tij g

iAuml g j=T~

ij GiAuml Gj

The unit tensor I is defined in terms of the co- orcovariant base vectors as

(54)I=GijG

iAuml G j=G ijGiAuml Gj=gijgiAuml g j=g ijgiAuml gj

Auml means the tensor product and we make use ofEinsteinrsquos summation convention

(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga

Note that Latin dummy indices run from 1 to 3 andGreek from 1 to 2 Note also that covariantcomponents eg ai go together with contravariantbase vectors gi and vice versa

The differential piece of area da is defined as the

vector parallelogram which is given by the covariantbase vectors g1 and g2 The area content is determined as

da=||g1 3 g2||=(g13 g2)middotg3=det(gij)=j and(56)

dA=||G13 G2||=(G13 G2)middotG3=det(Gij)=J and

The total area a of a surface in the actualconfiguration can be expressed in terms of the surfacecoordinates as

(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2

The reference configuration is transformed intothe actual configuration by the deformationgradient F

F= =giAuml Gi FT=G iAuml gi

(58)

Fndash1= =GiAuml g i FndashT=giAuml Gi

The deformation gradient F transforms basevectors by

gi=FmiddotGi=gjAuml GjmiddotGi= d ji gj=gi

Gi=FndashTmiddotgi=GjAuml g jmiddotgi= d ji Gj=G (59)

gi=FndashTmiddotGi=g jAuml GjmiddotGi= d i

j g j=gi

Gi=F Tmiddotg i=G jAuml gjmiddotgi= d j

i Gj=Gi

and the differential area by

da=j=||g13 g2||=detFdA=detFJ=detF||G13 G2||

=detF= = (60)

The surface area of the actual configuration canbe written with respect to the referenceconfiguration

||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||



anisotropic material behavior in the directions of weftand warp [5] Furthermore an unlimited variation ofshapes is generated by prescribing anisotropic stressdistributions

Inspired eg by the pioneer work of Frei Otto [6]many technical procedures and algorithms had beendeveloped which many of them are based on dynamicrelaxation [78] Others like the force density methodare based on special discretization and linearizationtechniques Originally it was developed for the formfinding of cable structures [9-11] Recently themethod was extended to triangular membraneelements [1213] The existence of all the differentmethods is explained by the mathematical problemwhich arises by solving an inverse mechanicalproblem It is defined by the prescribed stressdistribution as the driving degree of freedom in thedesign process This is inverse to standard mechanicswhere stresses are the structural response to thedeformation of material As a consequence the relatednumerical solution methods are faced to mathematicalsingularities which are overcome by severaltechniques eg by a modified Newton-Raphsoniteration [14-16] The updated reference strategywhich will be presented in the following is a furtheralternative [17-19] It is consistently derived fromcontinuum mechanics of elastic bodies with respect tolarge deflections and small strains The numericalsolution follows standard finite element discretizationprocedures which means that the method can beapplied to any triangular or quadrilateral finiteelement formulation The above mentionedsingularities are regularized by a homotopy mapping[20] which is based on approximations for Cauchy andPiola-Kirchhoff stresses It will be shown that thisapproach leads to the force density method if it isapplied to one dimensional elastic bodies eg cablesVice versa the updated reference strategy is theconsistent generalization of the force density methodapplicable for any structure It can also be used as anapproximation technique in structural shapeoptimization [19]

2 THE GOVERNING EQUATION OFFORM FINDING

21 DIRECT GEOMETRICAL APPROACH TOMINIMAL SURFACESMinimal surfaces are defined as surfaces of minimalarea which are enclosed by a given boundaryConsider the procedure to determine a minimalsurface as an iterative process (Fig 1) The surface of

the intermediate state or actual configuration withactual area a is understood as to be generated bydeformation of an initial state or referenceconfiguration of area A A point P on the actual surfacewith position vector x (u 1u 2 ) had originally theposition X (u 1u 2) on the reference surface The pointis identified by its surface coordinates u 1u 2 whichstay invariant during deformation The displacement uof point P is defined as

u( u 1u 2)=x ( u 1u 2)ndashX( u 1 u 2) (1)

The surface area a of the actual configuration can be written with respect to the referenceconfiguration as

a = Ea

da = EA

detFdA (2)

where detF is the determinant of the deformationgradient F Refer to the appendix for further details ondifferential geometry

Figure 1 Deformation of surface

The minimum of (2) is determined by the vanishingvariation d a of the surface area with respect tovariation d x of the shape This states the governingequation

d a= d 1 E ada 2 = EAd (detF)dA=E

AdetFFndashT d FdA=0 (3)

where

d F= d giAuml G i= (4)

The shape variation cannot be chosen arbitrarily It

para( d x)

paraX






d w=tEas da=tE

as d uxda=0 (5)





d w=tEa

s d ux da=tEA

s d uxdetFdA

=t EA



d xw=t EA




d w=t EA



A(FmiddotS)d FdA=0





s i3=s 3i=Si3=S3i=0 (11)



paraxparaX

paraXparax

para(d x)

paraX

para( d u)

parax

para(d u)

d x



d a= EA

detFF-T d FdA=0 (3)







X= ^nel

k=1Nk ( u 1u 2)Xw k u= ^

nel


(12)

x= ^nel





X1 Xw 1k X2 = ^

3

k=1Nk Xw 2

k (13)

X3 Xw 3k







14

14

14





Ga= =Xa= ^nel

k=1Nka Xw k

(15)

ga= =xa= ^nel






(17)


and



(18)





(20)=t E


4 LINEARIZATION


(20) yields

LIN( )=t EA




(221)R=t E








Krs =tEA


or in components

(24)Rr =t EA



( )r = and ( )s =



Rr = =ts ^ EA



a)]dApara2a

parabr parabs

paraaparabr

paraparabs

paraparabr

paraFparabr

paraparabs



paraFparabr

paraparabs

paraFparabr

paraFparabr

paraparabs

paraFparabr

parawparabr

paraFparabr

parawparabr

paraFparabr

parauw kparabr

paragaparabr

paragiparabr

paraFparabr

paraxparau a

paraXpara u a






d w=Ac ES



(26)






(27)





(29)

Rr = (glrmiddotg1)


w and ` s=

5 REGULARIZATION



g1middotgls`

1` 2

s Ac`

s Ac`

L`

SAcL

s xAcl

l 1L l2

||g1 ||||G1 ||



X

X

X

X








(31)+(lndash l )t E

A(FmiddotS) d FdA=0






(32)+(lndash l )t E

A(FsmiddotS)FrdA







Xt E




(35)

+(1ndashl )tEA

(FmiddotS)FrdA












R(l )r(bl ) =0


eTr f rndashfX =0










L 1 2 ds= =q (39)



n`

lL

L n` Ac

lL

I=Ln` Ac

` LnLL` Ac`










p=pX g3g3= = (41)


Ea



(42)

= Eu 1 E u 2





Ea



Linearization gives

lin(Ea



(44)

+d br D bsEu 1E u 2







ndashEu 1E u 2





Eu 2


=ndash pX Eu 1



ndash pX Eu 1E u 2




(47)


paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||






d w=tEas da=tE

as d uxda=0 (5)





d w=tEa

s d ux da=tEA

s d uxdetFdA

=t EA



d xw=t EA




d w=t EA



A(FmiddotS)d FdA=0





s i3=s 3i=Si3=S3i=0 (11)



paraxparaX

paraXparax

para(d x)

paraX

para( d u)

parax

para(d u)

d x



d a= EA

detFF-T d FdA=0 (3)







X= ^nel

k=1Nk ( u 1u 2)Xw k u= ^

nel


(12)

x= ^nel





X1 Xw 1k X2 = ^

3

k=1Nk Xw 2

k (13)

X3 Xw 3k







14

14

14





Ga= =Xa= ^nel

k=1Nka Xw k

(15)

ga= =xa= ^nel






(17)


and



(18)





(20)=t E


4 LINEARIZATION


(20) yields

LIN( )=t EA




(221)R=t E








Krs =tEA


or in components

(24)Rr =t EA



( )r = and ( )s =



Rr = =ts ^ EA



a)]dApara2a

parabr parabs

paraaparabr

paraparabs

paraparabr

paraFparabr

paraparabs



paraFparabr

paraparabs

paraFparabr

paraFparabr

paraparabs

paraFparabr

parawparabr

paraFparabr

parawparabr

paraFparabr

parauw kparabr

paragaparabr

paragiparabr

paraFparabr

paraxparau a

paraXpara u a






d w=Ac ES



(26)






(27)





(29)

Rr = (glrmiddotg1)


w and ` s=

5 REGULARIZATION



g1middotgls`

1` 2

s Ac`

s Ac`

L`

SAcL

s xAcl

l 1L l2

||g1 ||||G1 ||



X

X

X

X








(31)+(lndash l )t E

A(FmiddotS) d FdA=0






(32)+(lndash l )t E

A(FsmiddotS)FrdA







Xt E




(35)

+(1ndashl )tEA

(FmiddotS)FrdA












R(l )r(bl ) =0


eTr f rndashfX =0










L 1 2 ds= =q (39)



n`

lL

L n` Ac

lL

I=Ln` Ac

` LnLL` Ac`










p=pX g3g3= = (41)


Ea



(42)

= Eu 1 E u 2





Ea



Linearization gives

lin(Ea



(44)

+d br D bsEu 1E u 2







ndashEu 1E u 2





Eu 2


=ndash pX Eu 1



ndash pX Eu 1E u 2




(47)


paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||



d a= EA

detFF-T d FdA=0 (3)







X= ^nel

k=1Nk ( u 1u 2)Xw k u= ^

nel


(12)

x= ^nel





X1 Xw 1k X2 = ^

3

k=1Nk Xw 2

k (13)

X3 Xw 3k







14

14

14





Ga= =Xa= ^nel

k=1Nka Xw k

(15)

ga= =xa= ^nel






(17)


and



(18)





(20)=t E


4 LINEARIZATION


(20) yields

LIN( )=t EA




(221)R=t E








Krs =tEA


or in components

(24)Rr =t EA



( )r = and ( )s =



Rr = =ts ^ EA



a)]dApara2a

parabr parabs

paraaparabr

paraparabs

paraparabr

paraFparabr

paraparabs



paraFparabr

paraparabs

paraFparabr

paraFparabr

paraparabs

paraFparabr

parawparabr

paraFparabr

parawparabr

paraFparabr

parauw kparabr

paragaparabr

paragiparabr

paraFparabr

paraxparau a

paraXpara u a






d w=Ac ES



(26)






(27)





(29)

Rr = (glrmiddotg1)


w and ` s=

5 REGULARIZATION



g1middotgls`

1` 2

s Ac`

s Ac`

L`

SAcL

s xAcl

l 1L l2

||g1 ||||G1 ||



X

X

X

X








(31)+(lndash l )t E

A(FmiddotS) d FdA=0






(32)+(lndash l )t E

A(FsmiddotS)FrdA







Xt E




(35)

+(1ndashl )tEA

(FmiddotS)FrdA












R(l )r(bl ) =0


eTr f rndashfX =0










L 1 2 ds= =q (39)



n`

lL

L n` Ac

lL

I=Ln` Ac

` LnLL` Ac`










p=pX g3g3= = (41)


Ea



(42)

= Eu 1 E u 2





Ea



Linearization gives

lin(Ea



(44)

+d br D bsEu 1E u 2







ndashEu 1E u 2





Eu 2


=ndash pX Eu 1



ndash pX Eu 1E u 2




(47)


paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||




Ga= =Xa= ^nel

k=1Nka Xw k

(15)

ga= =xa= ^nel






(17)


and



(18)





(20)=t E


4 LINEARIZATION


(20) yields

LIN( )=t EA




(221)R=t E








Krs =tEA


or in components

(24)Rr =t EA



( )r = and ( )s =



Rr = =ts ^ EA



a)]dApara2a

parabr parabs

paraaparabr

paraparabs

paraparabr

paraFparabr

paraparabs



paraFparabr

paraparabs

paraFparabr

paraFparabr

paraparabs

paraFparabr

parawparabr

paraFparabr

parawparabr

paraFparabr

parauw kparabr

paragaparabr

paragiparabr

paraFparabr

paraxparau a

paraXpara u a






d w=Ac ES



(26)






(27)





(29)

Rr = (glrmiddotg1)


w and ` s=

5 REGULARIZATION



g1middotgls`

1` 2

s Ac`

s Ac`

L`

SAcL

s xAcl

l 1L l2

||g1 ||||G1 ||



X

X

X

X








(31)+(lndash l )t E

A(FmiddotS) d FdA=0






(32)+(lndash l )t E

A(FsmiddotS)FrdA







Xt E




(35)

+(1ndashl )tEA

(FmiddotS)FrdA












R(l )r(bl ) =0


eTr f rndashfX =0










L 1 2 ds= =q (39)



n`

lL

L n` Ac

lL

I=Ln` Ac

` LnLL` Ac`










p=pX g3g3= = (41)


Ea



(42)

= Eu 1 E u 2





Ea



Linearization gives

lin(Ea



(44)

+d br D bsEu 1E u 2







ndashEu 1E u 2





Eu 2


=ndash pX Eu 1



ndash pX Eu 1E u 2




(47)


paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||






d w=Ac ES



(26)






(27)





(29)

Rr = (glrmiddotg1)


w and ` s=

5 REGULARIZATION



g1middotgls`

1` 2

s Ac`

s Ac`

L`

SAcL

s xAcl

l 1L l2

||g1 ||||G1 ||



X

X

X

X








(31)+(lndash l )t E

A(FmiddotS) d FdA=0






(32)+(lndash l )t E

A(FsmiddotS)FrdA







Xt E




(35)

+(1ndashl )tEA

(FmiddotS)FrdA












R(l )r(bl ) =0


eTr f rndashfX =0










L 1 2 ds= =q (39)



n`

lL

L n` Ac

lL

I=Ln` Ac

` LnLL` Ac`










p=pX g3g3= = (41)


Ea



(42)

= Eu 1 E u 2





Ea



Linearization gives

lin(Ea



(44)

+d br D bsEu 1E u 2







ndashEu 1E u 2





Eu 2


=ndash pX Eu 1



ndash pX Eu 1E u 2




(47)


paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||










(31)+(lndash l )t E

A(FmiddotS) d FdA=0






(32)+(lndash l )t E

A(FsmiddotS)FrdA







Xt E




(35)

+(1ndashl )tEA

(FmiddotS)FrdA












R(l )r(bl ) =0


eTr f rndashfX =0










L 1 2 ds= =q (39)



n`

lL

L n` Ac

lL

I=Ln` Ac

` LnLL` Ac`










p=pX g3g3= = (41)


Ea



(42)

= Eu 1 E u 2





Ea



Linearization gives

lin(Ea



(44)

+d br D bsEu 1E u 2







ndashEu 1E u 2





Eu 2


=ndash pX Eu 1



ndash pX Eu 1E u 2




(47)


paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||









R(l )r(bl ) =0


eTr f rndashfX =0










L 1 2 ds= =q (39)



n`

lL

L n` Ac

lL

I=Ln` Ac

` LnLL` Ac`










p=pX g3g3= = (41)


Ea



(42)

= Eu 1 E u 2





Ea



Linearization gives

lin(Ea



(44)

+d br D bsEu 1E u 2







ndashEu 1E u 2





Eu 2


=ndash pX Eu 1



ndash pX Eu 1E u 2




(47)


paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||










p=pX g3g3= = (41)


Ea



(42)

= Eu 1 E u 2





Ea



Linearization gives

lin(Ea



(44)

+d br D bsEu 1E u 2







ndashEu 1E u 2





Eu 2


=ndash pX Eu 1



ndash pX Eu 1E u 2




(47)


paraxpara u a

paraxparabr

g13 g2j

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||




7 EXAMPLES

71 CATENOID




URS 6993 5 08ext system 6993 1 + 08

ext system 0979

























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||























































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||















































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||









































x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||

























x=x( u 1u 2) X=X( u 1u 2) (48)


g1= g2= and G1 =G2 (49)


paraXpara u 2

paraXpara u 1

paraxpara u 2

paraxparau 1






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||






g3= ||g3||=1 and G3= ||G3||=1

(51)



j




i(53)


iAuml g j=T~

ij GiAuml Gj


(54)I=GijG



(55)a=a igi= ^

3

i=1aigi b=baga= ^

2

a=1baga







(57)a=Eu 1E u 2

jdu 1du 2=Eu 1E u 2

||g1 3 g2||du 1du 2



(58)






j g j=gi


i Gj=Gi



=detF= = (60)


||g13 g2|| ||G13 G2||

jJ

dadA

paraXparax

paraxparaX

G13 G2 ||G13 G2||

g13 g2||g13 g2||



Documents

A General Finite Element Approach to the Form Finding of Tensile Structures by the Updated Reference Strategy