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9th World Congress on Structural and Multidisciplinary Optimization

June 13 -17, 2011, Shizuoka, Japan

FUNDAMENTAL STUDY OF DIRECT MINIMIZATION ON STATIC PROBLEMS OF

MEMBRANES

Masaaki Miki1, Kenichi Kawaguchi2

1 The University of Tokyo, Tokyo, Japan, [email protected] The University of Tokyo, Tokyo, Japan, [email protected]

1. Abstract

Within this work, direct minimization approaches on static problems of membranes are discussed. First, standardminimization approaches are described in the framework of 2-term method and 3-term method. It is also

discussed minimization problems under constraint conditions. Then, the scope of the direct minimizationapproaches is examined. We start with the principle of virtual work for general self-equilibrium membranes, and

then, it is extended forN-dimensional Riemannian manifolds. Finally, some principle of virtual works that can besolved by direct minimization approaches are illustrated.

2. Keywords: Principle of Virtual Work, Three Term Method, Membrane, Continuum Mechanics, Minimization

3. Introduction

For form-finding of tension structures such as cable-net structures and membrane structures, direct minimizationapproaches are sometimes very effective [1]. In the direct minimization procedures, the assigned parameters can

be varied directly at any moment so that the forms can be varied freely.On the other hand, to solve static problems of continuum bodies, non-linear Finite Element Method, or just the

Newton-Raphson method family, has been widely used. However, for some special cases that can not be solvedby the Newton-Raphson method, e.g. the minimal surface problem, it has been reported that the Dynamic

Relaxation Method, a family method of direct minimization approaches, can solve them [2].This work contributes to the scope of direct minimization approaches on various types of static problems of

deformable bodies.

4. Direct Minimization Approaches

4.1. Three Term Method without Constraint Conditions

Let us consider form-finding of cable-net structures. As an example, the following stationary problem can beused:

stationary)()(1

2=

=

m

j

jjLw xx , (1)

where jj Lw , are the weight coefficient and the length of the j-th cable respectively. In addition, xis a column

vector containing unknown variables. Let us define x and corresponding gradient operator by

[ ]Tn

xx 1

x , and

nx

f

x

ff

1

. (2)

It has been reported by the authors [1] that Eq. (1) simply represents the Force Density Method [3] which wasfirst proposed by H. J. Schek and K. Linkwitz in 1973. While inverse matrix is always used to solve Eq. (1) in

the original one, this work puts a focus on the fact that the same equation can be solved by general direct

minimization approaches.

In the following discussion, suppose that },,{ 1 nxx represents the Cartesian coordinates of the free nodes and

remark that those of the fixed nodes were eliminated beforehand and directly substituted into each )(xjL .

The stationary condition of Eq. (1) is as follows:

0== j

jjjLLw2

. (3)

It seems obvious that Eq. (3) can be easily solved by direct minimization approaches, in which , thegradient, is used as the standard search direction. The simplest direct minimization approach is represented as

follows:

1


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,

),(

CurrentCurrentNext

Current

T

Current

rxx

xxr

=

=

=

(4)

which is usually called the steepest decent method; however, in the relation with the following discussion, we

shall call Eq. (4) as 2-term method. In Eq. (4), is normalized because too large or too small

sometimes causes a trouble.Eq. (4) contains as an only unique parameter which can be adjusted manually for controlling the step-size.By the way, the following remedy sometimes provides a remarkable improvement of global convergenceefficiency:

,

,98.0

),(

NextCurrentNext

CurrentCurrentNext

Current

T

Current

qxx

rqq

xxr

+=

=

=

=

(5)

which represents the simplest 3-term method.Fig. 1(a) shows a numerical example for verification that can be solved by both the 2-term and 3-term method,

which consists of 220 cables and 5 fixed nodes. The initial sets of },,{ 1 nxx which was tested by the first

author is given by random numbers ranging from -2.5 to 2.5 as shown in Fig. 1(b). Then, when }{ 1 mww

wasprescribed as }11{ , Fig.1 (c) was obtained. The corresponding minimum number of the objective function was

160.214. Another result, which was obtained when 4 times greater weight coefficients were assigned onto the

boundary cables, is shown by Fig. 1 (d). The corresponding minimum number was 188.09. For both 2-term and3-term method, was always set as 0.2.The history of the objective function obtained by 3-term method which corresponds to Fig. 1 (c) is shown byFig. 2. In the 3-term method, by keeping as 0.2, the form was able to be varied smoothly at any moment by

varying }{ 1 mww . When precise solution was needed, was gradually decreased manually so that

decreased gradually as shown in Fig.3.Here, it must be noted that Eq. (5) has a close relation with the family methods with three term recursion

formulae. In 1982, M. Papadrakakis stated that two popular minimization approaches, the Dynamic Relaxationmethod and the Conjugate Gradient method, can be classified under family methods with three term recursion

formulae [4], which was first proposed by M. Engeli, H. Rutishauser et. al [3] in 1959.-10,-10,0) (10,-10,0)

(0,0,10)

(10,10,0)(-10,10,0)

(a) Connections (b) Initial Step (c) min2 L (d) min2 wL

Figure 1: Form-finding of Cable-net Structure

0

100

200

300

400

500

600

700

100 200 300 400 500 600

2=j

jL

Step #

=633Step1:

=160St

Figure 2: History of ( =0.2) (by 3-term method)

2


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1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

100 200 300 400 500 600

1.0E- 05

1.0E- 04

1.0E-03

1.0E- 02

1.0E- 01

1.0E+00

Step #

Figure 3: History of (by 3-term method)

By using either 2-terrm method or 3-term method, both Fig. 1(c) and (d) were able to be easily obtained.However, the behavior of the 2-term method was somehow awkward, whereas, the behavior of the 3-term

method was always very smooth and seems to be the best method for interactive design process. Therefore, thescope of the 3-term method must be examined.

4.2. Three Term Method with Constraint Conditions

(a) Connections (b) Result ( 180004= jL )

Figure 4: Form-Finding ofSimplexTensegrity

Let us consider a simple minimization problem with constraint conditions such as:

=

=

= =

0))((

0))((

t.s.

min,)()(

1212

1010

9

1

4

LL

LL

Lwj

jjw

x

x

xx

, (6)

which gives the forms of Simplex Tensegrities that consist of 3 struts and 9 cables. The connections over the

members are shown by Fig. 4(a). When { } { }1,,1,, 91 =ww and { } { }10,,10,, 1210 =LL , Fig. 2 (b) wasobtained and the corresponding minimum number was 18000. The methods for this type of problem is describedbelow.

Applying theLagrange multiplier method, Eq. (6) reduces to:

stationary))(()(),(11

4+=

=++

=

r

k

kmkmk

m

j

jjLLLw xxx , (7)

where the first sum is taken for all the cables and the second sum is taken for all the struts. In addition, is a

column vector containing the Lagrange multipliers. Let us define x, and the corresponding differential

operators by:

[ ]Tn

xx 1

x ,

nx

f

x

ff

1

)(

x

x, (8)

[ ]r

1

,

T

r

fff

1

)(

. (9)

Additionally, letx

x

)(ff . The stationary condition of Eq. (7) is represented by

0

0 =

= & . (10)

Let us firstly discuss the first stationary condition, namely

0=+= k

kk

j

jjjLLLw

34 . (11)

Due to the unknownLagrange multipliers { }r1 , can not be determined uniquely when only x is given.

The simplest idea to overcome this difficulty is to use general inverse matrix. Let us rewrite Eq. (11) into the

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following form:

0Jx =+=

)(w (12)

w=

J , where =

j

jjjwLLw

34 and

=

+

+

rm

m

L

L

1

J . (13)

Eq. (12) has a solution only if xrepresents a solution. Anyway, by using the Moore-Penrose type pseudo

inverse matrix+

J , can be determined uniquely by

+= Jx )(w , (14)

which basically provides a least squared solution. This strategy seems rather rough but it works fine because

when Eq. (14) turns to a least norm solution, Eq. (12) is simultaneously satisfied and vice versa.

Within the examples discussed in this work, J is always a full-rank matrix and r


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r:constantw

J

(a) Composition of forces (b) Two types of search directions

Figure 5: Overview of Minimization under Constraint Conditions

Here is another problem, which can be used for form-finding of tension structures that consist of cables,membranes and struts:

stationary))(()()(),(

24++=

llll

kkk

jjj LLSwLw xxxx , (22)

where the first sum is taken for all the linear elements, the second is for all the triangular elements, and the third

is for all the struts. In addition, jL and kS are respectively functions to give the length ofj-th linear element and

the area ofk-th triangular element. Fig. 6(a) shows the initial set of }{ 1 nxx , which was given by random

numbers. By using the 3-term method, the form was able to be varied smoothly at any moment by varying

lkjLww ,, , as shown by Fig.6 (b) and (c) (see Ref. [1]).

Pseudo codes for both 2-term and 3-term method under constraint conditions are presented in Appendix A.

(a) Initial Step (b) Variation 1 (c) Final decision

Figure 6: Form-Finding of Tanzbrunnen Koln (F. Otto, 1959)4.2. Gradients

For verification kj SL , can be calculated by referring to the following calculations.

( ) ( )22),,,,,(zzxxzuxzux

qpqpqqqpppL ++ ,

zyxzyxq

L

q

L

q

L

p

L

p

L

p

LL (23)

=

L

pq

L

qp

L

qp

L

qpL zzzz

yyxx (24)

N

NnprpqNNNrqp = ),()(,

2

1),,(S ,

zyzyxr

S

r

S

p

S

p

S

p

SS (25)

=

1

0

0

,

0

1

0

,

0

0

1

)(

1

0

0

,

0

1

0

,

0

0

1

)(

1

0

0

,

0

1

0

,

0

0

1

)(2

1 pqrpqrnS (26)

5. Principle of Virtual Work

5.1. Principle of Virtual Work for Minimal Surfaces and Self-equilibrium Membranes

Einstein summation convention is used in this section. Let us discuss the surface area of a surface, namely

0da == aa , )2,1(detda 21 jiddgij , (27)where

21,, ij

g represents theRiemannian metric and the local coordinates defined on a surface respectively.

Using gggg ijij

ij detdet = , where ( )1

ijij

gg , the variation of the surface area is given by

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0da2

1== a ij

ij gga . (28)

By the way, let us discuss a self-equilibrium membrane whose boundary is fixed. The principle of virtual workfor such a membrane is expressed as

)2,1(0da2

1== jigtw a ij

ij , (29)

where t, ij represents the thickness and the Cauchy stress tensor of the membrane respectively. Here, ijg is

used instead of the variation of strain tensor ije , due to the essential identity between them.

Using tensor component transformation lawkj

kiij g= , the principle of virtual workfor such a membrane is

transformed into

)2,1(0da2

1== jiggtw a ij

kjk

i . (30)

When a new stress tensori

kT is defined by k

ik

i tT , Eq. (30) can be rewritten as

)2,,1(0da2

1== kjiggTw a ij

kjk

i . (31)

On the other hand, Eq. (28) can be rewritten as

)2,,1(0da21 == kjigga a ij

kjk

i , (32)

which can be a simple demonstration of the essential identity between minimal surface and uniform stresssurface.

5.2. Principle of Virtual Work for N-Dimensional Riemannian Manifolds

The length of a curve, the area of a surface, and the volume of a body are respectively defined by

= ll dl , = aa da , = vv dv , where (33))1,(detdl 1 = jidg

ij , )2,1(detda

21 jiddgij

, )3,1(detdv321 jidddg

ij . (34)

By using the concept ofN-dimensional Riemannian manifold, they can be unified. The volume element, the

volume, and the variation of the volume of anN-dimensional Riemannian manifoldMcan be given by

N

ij

N ddg 1detdv , MNNv dv , and ),,1(dv

21 Nkjiggv

M

N

ij

ijN = . (35)Hence, 321 ,,,. vvvval = . Then, the variational problem of the volume ofMcan be defined as the standardvariational problem ofM, namely:

),,1(0dv2

1Nkjiggv

M

N

ij

ijN == . (36)By the way, the principle of virtual work for self-equilibrium cables, membranes, deformable bodies arerepresented as follows:

)1,,(0dl2

11 === kjiggAw l ijkj

ki , (37)

)2,,1(0da2

12 == kjiggtw a ijkj

ki , (38)

)3,,1(0dv2

13 == kjiggw v ijkj

ki , (39)

where tA, respectively denote the sectional area of a cable and the thickness of a membrane. Then, when a new

stress tensor kiT is separately defined for each dimensionalRiemannian manifold by

)1( = NAT kiki , )2( = NtT kiki , and )3( = NT kiki , (40)Eq. (37), (38), (39) can be unified by

),,1(0dv2

1NkjiggTw

M

N

ij

kjk

iN == , (41)which is the principle of virtual workfor a self-equilibrium N-dimensional Riemannian manifold M. This can

be naturally read thatkj

kiij gTT = is a general force which act withinMand tend to produce small change of ijg

[6].

Recalling Eq. (36), it can be transformed into

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),,1(0dv2

1Nkjiggv

M

N

ij

kjk

iN == , (42)therefore, the standard variational problems are special cases of the principle of virtual works. In turn, theprinciple of virtual works are one of natural generalizations of the standard variational problems.

6. Galerkin Method applied to the Principle of Virtual Work for N-dimensional Riemannian Manifold

6.1. Galerkin MethodAs a result of previous section, it would be reasonable discussing the approximation of

0dv2

1== M

N

ij

kjk

iN ggTw (43)

in advance. When the form of a curve, a surface, or a body is represented by n independent parameters such as

[ ]Tn

xx 1

=x , (44)

the gradient vector of a functionfwith respect to x is defined by

nx

f

x

ff

1

. (45)

In such cases, at most n independent ijg can satisfy Eq. (43). Such n independent ijg are provided by weighted

residual method family. The Galerkin method is one of them. Using the Galerkin method, ijg is altered into

x =ijij

gg , (46)

where x is the variation of x, or just an arbitrary column vector, i.e. [ ]Tn

xx 1

x .

Then the discrete principle of virtual workdeduces as

== 0dv21

M

N

ij

kjk

iN ggTw x 0dv2

1=

= x M

N

ij

kjk

iN ggTw , then, (47)

0 == MN

ij

kjk

i ggT dv2

1, (48)

which is the discrete stationary condition for self-equilibrium problem ofM. For general problems of statics,

the following discrete stationary condition can be used:

0pxpx ===

MN

ij

kjk

i

M

N

ij

kjk

i ggTggT dv

2

1dv

2

1 , (49)

where p represents the nodal forces.

6.2. N-dimensional Simplex Elements

When the form of a structure is represented by m independent elements such as linear elements (N=1), triangular

elements (N=2), and tetrahedron elements (N=3), the discrete stationary condition deduces to

0 == j

j

NggT dv2

1

, (50)

where the sum is taken for all the elements and the integrals are performed separately within each j-th element.

Let us define

jNN

jggTT dv

2

1)(

, (51)

so that the discrete stationary condition can be simply represented as( ) 0 ==

j

N

jT

or ( ) 0p == Tj

N

j . (52)

The simplest idea to perform the integral in Eq. (51) is to use N-dimensional Simplex elements which are shown

by Fig. 7. As shown in Fig. 7, the covariant bases Ngg1 are constant within each element and they are given

by

)1(1

Niiii

= +ppg . (53)where ip represents the global Cartesian coordinate of i-th node. Then, Riemannian metric is also constant

within each element and is given by

jiijg gg = . (54)

In almost cases, kiT is only dependent on

ij

ijgg and , then k

iT is also constant within each element. Therefore,

7


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j

jjggTLT

11

111

11

2

1)( = , ( )

j

jjgg

gg

gg

gg

TT

TTST

=

2221

1211

2221

1211

22

12

21

11

2:

2

1

,

( )

j

jj

ggg

ggg

ggg

ggg

ggg

ggg

TTT

TTT

TTT

VT

=

333231

232221

131211

333231

232221

131211

332313

32

22

12

31

21

11

3:

2

1

, (55)

where jjj VSL ,, respectively denote the length, the area, and the volume of each element, namely

jj

jj

jj

gVgSgL

det6

1,det

2

1,det === . (56)

Here, note that : symbol represents the inner product between two matrices defined by ijijij BABA : .

Figure 7: Simplex Elements

Interestingly and amazingly,

jjL= )(

1

, jj S= )(

2

, and jj V= )(

3

. (57)

Therefore, in the simple cases that kiT is prescribed as just a scalar multiple of k

i , the discrete stationary

condition reduces to a simple stationary problem. Eq. (1), (7) and (22) are one of such cases. For example,

statinoary2)2(21 ===

j

jj

j

jjj

j

jjj LwLLwLw 00 . (58)

Then, it seems obvious that such simple cases can be solved by direct minimization approaches. In the next

section, other cases in which T is not a scalar multiple of are discussed. Since is jsut a mixture of

gradients, it may be possible to alter into in the direct minimization approaches.For calculating ijg , the following equations are helpful. First,

)()()()(1111 jjiijjiiij

dddg pppppppp += ++++ . (59)

Second, when [ ]Tiiii

zyx=p represents the Cartesian coordinates ofi-th node, Eq. (55) can be expanded as:

).()()()(

)()()()(

)()()()(

111111

111111

111111

iijiijjjijji

iijiijjjijji

iijiijjjijjiij

zzdzzzzdzzzdzzdz

yydyyyydyyydyydy

xxdxxxxdxxdxxxxddg

++

++

+=

++++++

++++++

++++++

(60)

Finally, comparing the following general relation with Eq. (60), ijg can be obtained.

=

=

n

ij

n

n

ijij

ij

dx

dx

g

dx

dx

x

g

x

gdg

11

1

. (61)

7. Numerical examples

To solve the discrete principle of virtual workby direct minimization approaches, an explicit expression ofi

kT

that provides 1 to 1 mapping between ijg andi

kT must be prescribed. In this work, only

)(symmetric)(),(kj

lklk

ilkji

klklk

ili

kgggEggTggEgT == (62)

has been already tested, which is a simplest constitutive law (Poisson ratio is 0). Here, note that the geometric

property of undeformed state is given by lkg , which is measured on the unreformed shape.

Fig. 8 and 9 show natural forms of a handkerchief under gravity whose dimension is 8.0-8.0. For both results,

( ) 0p == j

i

kjT

2

(63)

was solved by the 3-term method (see Eq. (5)), in which is altered into . The parameters were as

follows:E=50, [ ]1.001.0001.000 = p , and = 0.2.

8


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When Eq. (62) is used, lkg , the Riemannian metric on the undeformed shape, must be calculated in advance.

Then, in each initial step, the initial set of { }nxx 1 was given by those of the undeformed shape, and wasnot given by random numbers.

(a) Undeformed shape (b) ResultFigure 8: Natural Forms of Handkerchief under Gravity Hanged by 1-Point

(a) Undeformed shape (b) Result1 (c) Result2

Figure 9: Natural Forms of Handkerchief under Gravity Hanged by 2-PointsFig. 10 shows a large deformation analysis of a cantilever whose dimension is 2.0-2.0-12.0. Fig. 11 also shows a

large deformation of the same deformable body after bucking. For obtaining both the results,

( ) 0p == j

i

kjT

3

(64)

was solved by the 3-term method ( 2.0and50 == E ). In Fig. 10, p represents the value which was set to the z-

components of all the nodal loads. In Fig 11, p represents the value which was set to only the z-components of 9nodal loads, which act on 9 nodes on top of the body.

For the deformable body shown by Fig. 11, theEulerbuckling load is 14.1=crp , then its division by 9 is 0.126,which places between Fig. 11 (a) and (b).

(a) p=0.00 (b) p=0.05 (c) p=0.1Figure 10: Large Deformation of Cantilever under Gravity

(a) p=0.10 (b) p=0.20 (c) p=1.0

Figure 11: Large Deformation after Buckling

8. Conclusions

2-term method and 3-term method were described for not only simple minimization problems but also forminimization problems under constraint conditions. The principle of virtual work for N-dimensional

9


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Riemannian manifolds was also formulated. Finally, some principle of virtual works that can be solved by 3-term method were illustrated, which imply the potential ability of 3-term method on various types of static

problems.

5. Acknowledgements

This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology,

Grant-in-Aid for JSPS Fellows, 10J09407, 2010

6. References[1] M. Miki, K. Kawaguchi, Extended force density method for form finding of tension structures, Journal of the

International Association for Shell and Spatial Structures. Vol. 51, No.3 (2010) 291-303.[2] R. M. O. Pauletti, D. M. Guirardi, Direct area minimization through dynamic relaxation, Proceedings of the International

Association for Shell and Spatial Strucutres (IASS) Symposium, (2010), 1222-1234.

[3] H.J. Schek, The force density method for form finding and computation of general networks, Computer Methods Inapplied Mechanics and Engineering. 3 (1974) 115134.

[4] M. Engeli, T. Ginsburg, R. Rutishauser, E. Stiefel, Refined Iterative Methods for Computation of the Solution and theEigenvalues of Self-Adjoint Boundary Value Problems, Basel/Stuttgart, Birkhauser Verlag, 1959.

[5] M. Papadrakakis, A family of methods with three-term recursion formulae, International Journal for Numerical MethodsIn Engineering. 18 (1982) 17851799.

[6] J. L. Lagrange (author), Analytical mechanics, A. C. Boissonnade and V. N. Vagliente (translator), Kluwer, (1997).

[7] I. I. Dikin, Iterative solution of problems of linear and quadratic programming, Soviet Mathematics Doklady. 8 (1967)674-675

10


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Appendix A. Pseudo Codes

(a) 3-term method (b) 3-term method under constraint conditions

}

}

);visualize(call

;1;1

;:

;98.0:

;:

usingby;2:

GUI;from},,{,read

{

)while(

;2:;1:

;:;:numbers;random:

();initGUI

{

()solve

stationary)(

1

111

2

Current

NextCurrentNext

CurrentCurrentNexxt

T

Current

Current

j

jjj

m

j

jj

NextNextCurrentCurrent

LLw

ww

true

NextCurrent

Lw

x

qxx

pqq

p

x

0q0px

x

+=+=

+=

=

=

=

==

===

}

}

);visualize(call

;1;1

;:

;98.0:

;:

;:

;:

usingby;:

usingby;4:

);5.0(byupdate

;

;

)(

)(

:

usingby;:

GUI;from},,{},,,{,read

{

)while(

;2:;1:

;:;:numbers;random:

();initGUI

{

solve()

.stationary))(()(

1

1

3

11

1

11

111

11

4

Current

NextCurrentNext

CurrentCurrentNexxt

T

Current

w

w

Current

rm

m

Current

m

j

jjjw

CurrentCurrent

rmCurrentrm

mCurrentm

Current

rm

m

rm

r

k

mkmkk

m

j

jj

NextNextCurrentCurrent

L

L

LLw

LL

LL

L

L

LLww

true

NextCurrent

LLLw

x

qxx

pqq

p

J

J

xJ

x

xxx

rJx

x

x

r

xJ

0q0px

xx

+=+=

+=

=

=

+=

=

=

=

+

=

=

=

==

===

+

++

+

=

+

++

++

+

+

=++

=

11

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