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Three-term Method and Dual Estimate for Search of Shapesin Equilibrium

Masaaki MIKI and Kenichi KAWAGUCHI2

1 Department of Engineering, the University of Tokyo, Tokyo, Japan, mikity@iis.u-tokyo.ac.jp

2 Institute of Industrial Science, the University of Tokyo, Tokyo, Japan, kawaken@iis.u-tokyo.ac.jp

Summary

This paper contributes to standards of direct minimization approaches for use in searches ofequilibrium shapes in structures. Two-term and three-term methods are presented as standards ofdirect minimization approaches. It is indicated that constrained conditions can be easily and naturallyinvolved into two-term and three-term methods by using dual estimate. Moreover, some basicgradient vectors are extended in a natural way and then the scope of the direct minimization

approaches are extended to generalized minimization problems, in which the objective functions cannot be generally found. The simultaneous non-linear equations that are solved in the finite elementmethods are typical examples of such generalized problems.

Keywords: direct minimization approach; tension structures; principle of virtual work; three-termmethod; dual estimate; continuum mechanics; multiplier method; finite element method

1. Introduction

Minimization problems or stationary problems of objective functions typically appear in structuralengineering. A family of methods that are based on a simple idea, tracking of gradient vectors ofobjective functions, is often called the direct minimization approaches.

By using the direct minimization approaches, without terminating computation, we can explorevarious equilibrium shapes by alternating some parameters. The decision of changing parameters can

be made before the calculation converges to an optimum. Hence, direct minimization approaches arepotentially applicable for constructing interactive interface for structural design.

This paper describes two-term and three-term methods as standards for such direct minimizationapproaches. The dual estimate is also presented as a powerful strategy to involve constraintconditions into them. Moreover, some basic gradient vectors are extended in a natural way and thenthe scope of the direct minimization approaches are extended to generalized stationary problems, theobjective functions of which are not generally found. The simultaneous equations that are solved inthe non-linear finite element methods are typical examples of such generalized problems.

The authors have already presented a method for form-finding problems in light-weight tensionstructures [1]. However, only choices of objective functions of the problems are discussed in it. Theprimal aim of this paper is to explain the numerical strategies that have been used by the first author.

2. Two-term and three-term methods

2.1 Direct Minimization Approaches without Constraint Conditions

Let us consider the form-finding problem of cable-net structures. For example, the followingstationary problem can be used:

stationary)()(1

2=

=

m

j

jjLw xx , (1)

where jj Lw , are the weight coefficient and the length ofj-th cable respectively. In addition, x is acolumn vector containing n independent parameters. Let us define x and corresponding gradient by

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

(0,0,10)

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

(a) Connections (b) Initial Step (c) min2

LFigure 1: Form-finding of Cable-net Structure

[ ]Tn

xx 1

x , and

nx

f

x

ff

1

, (2)

where f is a real valued function ofx. In the following, suppose },,{ 1 nxx represents the Cartesiancoordinates of the free nodes and remark that those of the fixed nodes are eliminated beforehand anddirectly substituted into each )(xjL . The stationary condition of Eq. (1) is as follows:

( ) 0x == j

jjjLLw2

. (3)

To enable the direct minimization approaches, let us define

=

T

)(xr , where T= , (4)

as the standard search direction, which is the solution of the following optimization problem:

.1s.t.

max,)(

=

=

rr

rrT (5)

The simplest direct minimization approach is given by

.

),(

CurrentCurrentNext

CurrentCurrent

rxx

xxr

=

=

=

T

(6)

We shall call the iteration based on Eq. (6) as two-term method. While the two-term method isalmost identical with the steepest decent method, they are different in two aspects: The gradientvector is always normalized and the step-size factor is treated as a constant and to be adjusted by amanual operation. Similar to the steepest decent method, the global convergence efficiency of thetwo-term method is very poor. Then, the following remedy sometimes provides a remarkableimprovement of global convergence efficiency:

.

,98.0

),(

NextCurrentNext

CurrentCurrentNext

Current

T

Current

qxx

rqq

xxr

+=

=

=

=

(7)

The iteration based on Eq. (7) is called the three-term method by the authors. Because the additionalvariable q can be considered as velocity, Eq. (7) can be considered as a Newtons equation of motionwith a damping term. Therefore, the basic idea of the three-term method is almost identical with theDynamic Relaxation Method (see, e.g. Ref. [2]), but it is highly simplified and standardized. Thethree-term method can be also considered as the simplest method based on the three-term recursionformulae (see Ref. [4] and [5]).

Fig. 1(a) shows a numerical example that can be solved by both the two-term and three-termmethods, which consists of 220 cables and 5 fixed nodes. The first author obtained the results bysetting initial set of },,{ 1 nxx random numbers ranging from -2.5 to 2.5 as shown in Fig. 1(b). Fig. 1(c) shows the shape that the sum of squared length is minimum value, 640.9.

In the following, only the detail of the analysis by using the three-term method is reported becauseof its good global convergence efficiency. First, step-size factor was fixed at 0.2 and shortly theobjective function converged and vibrated around 640. At this time, was around 0.27, whichmay not be considered sufficiently small. However, even in such cases, as shown in Fig. 2(b),

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can be gradually decreased to a sufficiently small value, by decreasing by a manual operation. Inthe following numerical examples, the step-size factor was always fixed at 0.2 because very high

precision is not required for form-finding problems in general.

=0.2

0

500

1000

1500

2000

0 2 00 4 00 6 00 8 00 10 001.0E-01

1.0E+00

1.0E+01

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

0 1000 2000 3000 4000 50001.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

(a) History of and (b) History of and

Figure 2: History three-term method

2.2 Dual Estimate for Involving Constraint Conditions into Direct Minimization

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

Figure 3: Form-Finding of Simplex Tensegrity

A tensegrity is a self-equilibrium structure that consists of both tension members and compressionmembers. Let us consider a form-finding problem ofSimplex tensegrity which is shown by Fig. 4. Forexample, the following stationary problem that can be obtained by applying the Lagrange multipliermethod to a minimization problem with constraint conditions can be used:

stationary))(()(),(11

4+=

=++

=

r

k

kmkmk

m

j

jjLLLw xxx , (8)

where the first sum is taken for all the tension members and the second, for all the compressionmembers. In addition, is a row vector that contains theLagrange multipliers. Let us define x, and the corresponding differential operators by:

[ ]Tn

xx 1

x ,

nx

f

x

ff

1

)(

x

x,

x

x

)(ff , (9)

[ ]r

1

,

T

r

fff

1

)(

, (10)

wherefis a real valued function ofxand . The stationary condition of Eq. (8) is as follows:

0

0 =

= & . (11)

Let us examine the first stationary condition (with respect to x). It is expressed as follows:

0=+= =

+=

r

k

kmk

m

j

jjjLLLw

11

34 . (12)

Due to the unknown Lagrange multipliers { }r1 , can not be determined uniquely when x isgiven. The simplest idea to determine is making use of general inverse matrix. Let us rewrite Eq.

(12) into the following form:

0Jx =+=)(w , where =

j

jjjwLLw

34 and

=

+

+

rm

m

L

L

1

J . (13)

Eq. (13) is an over-conditioned problem, because n>rin usual. Then, the solution can not be basically

found; However, by using the Moore-Penrose type pseudo inverse matrix +J , can be determined

uniquely by

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+=

Jx )(

w, (14)

which basically gives a least squared solution. This strategy can be used because when an exact

solution is exists in Eq. (13), Eq. (14) returns an exact solution. Within the examples reported in

this paper, J is always a full-rank matrix and r

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element and the area of k-th triangle element. By using the three-term method, the form was able to

be varied by varying lkj Lww ,, , without terminating the computation, as shown in Fig.5. Contrary, the

two-term method did not prove useful. Then, the first author strongly interested in the applicability

of the three-term method.

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

Figure 5: Form-Finding of Tanzbrunnen Koln (F. Otto, 1959)

3. Generalized Stationary Problems

Let jj SL and be the functions of the length ofj-th line element and the area ofk-th triangle element.

The explicit representations of jj SL , are provided in the following:

( ) ( )

22

),,,,,( zzxxzuxzux qpqpqqqpppL++

,

zyxzyx q

L

q

L

q

L

p

L

p

L

p

LL

(21)

=

L

pq

L

qp

L

qp

L

qpL zzzz

yyxx (22)

N

NnprpqNNNrqp = ),()(,

2

1),,(S ,

zyzyxr

S

r

S

p

S

p

S

p

SS (23)

=

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 (24)

Because jj SL and look very different, when the volume of a tetrahedron element jV is considered,

it seems that we must calculate jV again. However, a general form of jjj VSL ,, exists, then we

can simplify the calculations of jjj VSL ,, .

3.1 Principle of Virtual Work forN-DimensionalRiemannian Manifolds

In the following, the Einsteins summation convention and standard tensor notations are used. Thevolume element and the volume of anN-dimensionalRiemannian manifoldMis given by

N

ij

N ddg 1detdv , MNNv dv . (25)

AnN-dimensionalRiemannian manifolds is respectively a curve, a surface, or a body whenN=1, 2, or3. Similarly, the volume is respectively the length, the area, or the volume. In addition, ( )N ,,1 represents a local coordinate defined on each point ofMbut is treated like a global coordinate. ijg

represents aRiemannian metric defined on each point ofM. Then, suppose the boundary ofMis fixedand let us discuss the variation of the volume which is given by

),1(det 1 NjiddgvM

N

ij

N = . (26)

By using a well-known relation, gggg ijij

ijdet

2

1det = , we have

),1(dv2

1Njiggv

M

N

ij

ijN = , (27)

where,ijg is the inverse matrix of ijg . Suppose, ijg is restricted to small change of ijg when the

shape ofM is varied arbitrarily by keeping its boundary fixed. Such a ijg is called kinematicallyadmissible. Then, the minimum volume problem ofMcan be expressed as

),1(0dv2

1Njiggv

M

N

ij

ijN == . (28)

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Unlink one of the dummy indices and factor out theKroneckers delta symbol ki , we have

),,1(0dv2

1Nkjiggv

M

N

ij

kjk

iN == . (29)

Because theKroneckers delta symbol simultaneously represents a unit matrix, it is natural to considera naturally generalized variational problem expressed in the following form:

),,1(0dv21 NkjiggTw

M

Nij

kjk

iN== , (30)

where kiT is a general NxN real valued matrix. We shall call Eq. (30) as the principle of virtualwork ofM, and let us consider only Eq. (32) in the following, because we can make Eq. (30)equivalent to the principle of virtual works for the structures when the relation between kiT andCauchy stress tensor ki is moderately defined. Particularly, when

)1( = NAT kiki , )2( = NtT kiki , or )3( = NT kiki , (31)

whereA and t represent the sectional area of a cable and the thickness of a membrane respectively,Eq. (31) becomes respectively equivalent to the principle of virtual workfor self-equilibrium cables,membranes or three-dimensional bodies (Suppose that the boundaries of them are fixed).

3.2 Galerkin MethodThe principle of virtual work (Eq. (30)) is a field equation and the degree of freedom ofgij isinfinite; hence it is obvious that the three-term method can not solve Eq. (30). However, when thedegree of freedom is reduced to a finite number, thee-term method can be performed. First, supposethe shape of geometry is explicitly represented by n independent parameters },,{ 1 nxx . Then, ,, xx can be defined by the same manner in section 2. Because the degree of freedom of the shape is n, onlyn independent gij can satisfy Eq. (30). Then, any set of },,{ 1 nxx that n independent gij can satisfyEq. (30) is often adopted as an approximated solution. One of the natural ways of giving such nindependent gij is an substitution of

x =ijij

gg~ (32)

into gij. The function ijg appeared in the right-hand-side is a composite function of the original

metric, ),,(1 N

ijg

, with the explicit representation of the shape by },,{ 1 nxx

. Therefore, this is theGalerkins method and the following formulations are the simultaneous equations that are typicallysolved in the finite element methods. Note that ijg

~ automatically becomes kinematically admissible.By substituting gij. by ijg

~ , the discrete principle of virtual work and stationary condition areobtained as

0dv~

2

1~ == MN

ij

kjk

iN ggTw , ( ) 0x == MN

ij

kjk

i ggT dv2

1. (33)

When external forces are considered, we can use

( ) 0px == MN

ij

kjk

iggT dv

2

1, (34)

where the elements of p are thex,y,zcomponents of the nodal forces.

3.3 N-dimensionalSimplexelement

Suppose the integral domain ofMis subdivided into m elements and let us denote each element byj.When the element integral within each elementj is defined by

jNN

jggTT dv

2

1)(

, (35)

( )x can be simply expressed by

( ) =j

N

jT )(

x or ( ) px =

j

N

jT )(

. (36)

The simplest idea to calculate )(

TN

j is to make the integrated function constant within each

element. Let us introduce N-dimensional Simplex element that is shown in Fig. 6. As shown in the

figure, anN-dimensional element is just a line, triangle, or tetrahedron element and they always haveN+1 nodes. Let us denote the position vectors of the nodes by { }11 ,, +Npp . In this paper, suppose each

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position vector represents the Cartesian coordinated of each node and simple local coordinate( )N ,,1 is defined within each element. The range of each i is from 0 to 1. The position of a pointin an element is given by the following interpolating function:

( ) ( ) ( )1121

11 ,, ++ +++= NNNNN

pppppr . (37)

Then, the base vectors ig andRiemannian metric ijg can be calculated as

ii rg )1(1 Niiii = +ppg , and jiijg gg = . (38)

Then, each of them is constant within each element. When only elastic bodies are discussed,

T canbe regarded as function of ijg , then

T is also constant within each element. After a manipulation, thefollowing formulations are obtained

[ ] ( )1,,2

1)(

1=

jjjggTLT , (39)

[ ] ( )2,,12

1)(

2

jjjggTST , (40)

[ ] ( )3,,12

1)(

3

jjjggTVT , (41)

where, jjj VSL ,, are respectively the length, area and the volume of each element.

Lj

(a) N=1

(0,0,0)(0,0,1)

(0,1,1)(1,1,1) g1

g2

g3

Vj

(c) N=3

1) (0)g1

(1,1) (0,1)

(0,0)(1,0)

g2

g1

(b) N=2

1 2

3

Sj

1 2

3

1

Figure 6: N-dimensional Simplex elements

3.4 General form of gradient vectors

When

=T is substituted into )(

TN

j , the following exact relations are formed:

jjL= )(

1

, jj S= )(

2

, jj V= )(

3

. (42)

Eq. (42) can be demonstrated by replacing symbols by symbol in Eq. (29). Eq. (42) indicatesthat )(

TN

j is the general form of jjj VSL ,, . Hence, we can use )(

TN

j instead of explicit

representations of jjj VSL ,, . Even more, we can now substitute any constitutive law, )( ijgTT

= ,into )(

TN

j . Additionally, because )(

TN

j is a mixture of the gradient vectors ijg , ( )x is a row

vector that highly resembles the gradient vectors, even though a function )(x that satisfy ( ) =x is not generally found. The stationary problems that are discussed in section 2 are special cases thatsuch functions )(x can be found. Then, it is expected to solve Eq. (33) or Eq. (34) by using the two-term method or the three-term method by just replacing by ( )x .

4. Numerical examples

To solve the discrete principle of virtual workby the direct minimization approaches, a constitutive

law, )( ijgTT

= , must be prescribed in advance. In this work, only

)(symmetric)(),(

gggEgTggEgT == (43)

is considered, where g is theRiemannian metric measured on the undeformed shape. Suppose thelocal coordinate ( )N ,,1 is embedded within each element.

Fig. 7 shows natural forms of a handkerchief under gravity. In the analysis

( ) 0p == j

i

kjT

2

(44)

was solved by the three-term method (see Eq. (7)), in which is replaced by . The parameterswere as follows:E=50, [ ]1.001.0001.000 = p , and = 0.2.

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In an initial step, the initial set of nxx 1 was given by those of the undeformed shape, and wasnot given by random numbers. Then, g was measured on the undeformed shape in the initial step.

Fig. 8 shows a large deformation analysis of a cantilever. In the analysis,

( ) 0p == j

i

kjT

3

(45)

was solved by the three-term method ( 2.0and50 == E [ ]pp = 0000 p ).

(a) Initial shapes (b) Results

Figure 7: Natural Forms of Handkerchief under Gravity Hanged by 1-Point

(a) p=0.0 (b) p=0.05 (c) p=0.1

Figure 8: Large Deformation of Cantilever under Gravity

5. Conclusions

The two-term and three-term methods were described. It was indicated that constraint conditionscan be easily involved into them by using the dual estimate. The general form of some gradientvectors was derived and the scope of the direct minimization approaches was extended to generalizedstationary problems. The simultaneous equations that are typically solved in the finite element

methods are typical examples of such generalized problems.

6. Acknowledgements

This research was partially supported by the Ministry of Education, Culture, Sports, Science andTechnology, Grant-in-Aid for JSPS Fellows, 10J09407, 2010

7. References

[1] M. Miki, K. Kawaguchi,Extended force density method for form finding of tension structures,J. IASS, 51(3), 2010, pp. 291-303.

[2] R. M. O. Pauletti, D. M. Guirardi, Direct area minimization through dynamic relaxation,

Proceedings of IASS Symposium (Shanghai), 2010, pp. 1222-1234.[3] H.J. Schek, The force density method for form finding and computation of general networks ,

Comput. Meth. Appl. Mech. Eng, 3, 1974, pp. 115134.

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

[5] M. Papadrakakis, A family of methods with three-term recursion formulae, Int. J. Numer.Meth. Eng., 18, 1982, pp. 17851799.

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