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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng 0000; 00:128

Published online in Wiley InterS ien e (www.inters ien e.wiley. om). DOI: 10.1002/nme

The Geodesi Dynami Relaxation Method for Problems of

Equilibrium with Equality Constraint Conditions

Masaaki Miki

1,3†∗Sigrid Adriaenssens

2∗Takeo Igarashi

3,4∗Ken'i hi Kawagu hi

5

1Department of Ar hitre ture, The University of Tokyo, Japan

2Department of Civil and Environmental Engineering, Prin eton University

3Japan S ien e and Te hnology Agen y/ERATO

4Department of Computer S ien e, The University of Tokyo, Japan

5Institute of Industrial S ien e, The University of Tokyo, Japan

SUMMARY

This paper presents an extension to the existing Dynami Relaxation method to in lude equality onstraint

onditions in the pro ess. The existing Dynami Relaxation method is presented as a general, gradient-

based, minimization te hnique. This representation allows for the introdu tion of the proje ted gradient,

dis rete parallel transportation and pull ba k operators that enable the formulation of the Geodesi Dynami

Relaxation method, a method whi h a ounts for equality onstraint onditions. The hara teristi s of

both the existing and the Geodesi Dynami Relaxation methods are dis ussed in terms of the system ’s onservation of energy, damping (vis ous, kineti and drift) and geometry generation. Parti ular attention

is drawn to the introdu tion of a novel damping approa h named drift damping. This te hnique is essentially

a ombination of vis ous and kineti damping. It allows for a smooth and fast onvergen e rate in both

the existing and the Geodesi Dynami Relaxation pro esses. The ase study was performed on the form-

nding of an i oni , ridge-and-valley, pre-stressed membrane system, whi h is supported by masts. The

study shows the potential of the proposed method to a ount for spe ied (total) length requirements.

The Geodesi Dynami Relaxation te hnique is widely appli able to the form-nding of for e-modelled

systems (in luding me hani ally and pressurized pre-stressed membranes) where equality onstraint ontrol

is desired. Copyright © 0000 John Wiley & Sons, Ltd.

Re eived . . .

KEYWORDS: Pseudo Inverse Matrix; Dynami Relaxation Method; Geodesi s; Constraint Conditions;

Form-nding; Tension Stru tures

1. INTRODUCTION, CONTEXT AND RESEARCH QUESTION

The on ept of Dynami Relaxation (DR) was rst introdu ed by Day [1 as an expli it solution

method for the stati behavior of stru tures from an analogy with tidal ow omputations. Brew

and Brotton [2 proposed a DR expression that deta hed the equations of equilibrium from those of

ompatibility and, thus, avoided the formulation of a stiffness matrix. This DR formulation has

been extensively employed, spe i ally for highly non-linear stru tures, and is implemented in

this paper. During the 1970 ’s, Barnes, Papadrakakis and Wakeeld extensively developed DR

algorithms for the form-nding and analysis of a wide range of tension stru tures in luding able

networks, membranes and pneumati stru tures [310. More re ently, new DR developments have

been proposed in the works of Wood [11 and Han and Lee [12, 13. To allow for the form-

nding and analysis of a wider range of for e-modelled systems, more re ent element formulations

†E-mail: masaakima m.org

∗Corresponden e to: Masaaki MIKI, Department of Ar hite ture, The University of Tokyo, Japan

Copyright © 0000 John Wiley & Sons, Ltd.

Prepared using nmeauth. ls [Version: 2010/05/13 v3.00

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have in luded alternative membrane (Gosling and Lewis [14, Hegyi et al. [15) and pneumati

(Rodriguez et al. [16) elements, non-regular tensegrity modules (Zhang et al. [17, Bel Hadj Ali et

al. [18), re ipro al frames links (Douthe and Baverel [19), pulley elements with fri tion (Hin z

[20) and beam and torsion elements (Adriaenssens and Barnes [20,21).

The method is based on Newton’s se ond law of motion and follows the movement of ea h

node of a stru ture, for small time intervals, from its initial position until all vibrations have be ome

negligible due to arti ial damping. Although the name DR ontains the term dynami , it is most

widely used as a omputational method to solve stati problems. Due to the law of inertia, it

solves equilibrium problems more ef iently than the steepest des ent method, whi h is a standard,

gradient-based, minimization method. Similar to the steepest des ent method, the DR method also

evaluates the gradient of energy fun tion only. Therefore, it has potential to be a powerful alternative

of steepest des ent method.

Little resear h has been performed on the in lusion of equality onditions onstraints into this DR

pro ess. Su h onstraints are mostly attributed to physi al and geometri limitations of the te hni al

nature of a proje t. To in lude these onstraints into the DR method, developments have in luded

formulations for uniform net meshes [22, element distortion ontrol [11, nodal planarity [23 and

length pres ription [24.

The equality ondition onstraints dis ussed in this paper are different from those based

on geometri onstru tion limitations; they provide an alternative way to model and ontrol

key stru tural elements in a for e-modelled system. For example, ompression struts in a pre-

stressed membrane roof stru ture ould be modelled as elements with a spe ied elasti stiffness.

Alternatively, it might be reasonable to model these struts with length onstraints while the other

omponents (su h as the pre-stressed ables or membranes) are treated as omponents with spe ied

elasti stiffnesses. Similarly, it might be bene ial to model spe i pre-stressed ables with total

length onstraints. Another lear example for desirable in lusion of equality onstraint onditions

in the DR pro ess relates to the modeling of air-supported, pneumati stru tures, in whi h the air is

treated as a volume onstraint. From an engineering viewpoint, the axial for es, in length- ontrolled

struts or ables, and the pressure a ting on the membrane, in a pneumati with a onstant air volume,

should be taken into onsideration in the DR pro ess. These for es an be onsidered rea tion for es

produ ed by the equality onstraint onditions.

The arising resear h question be omes: how an equality onstraint onditions be in orporated in

the DR pro ess while appropriately a ounting for the rea tion for es produ ed by these onditions?

The remainder of the paper is organized as follows. Se tion 2 des ribes the existing DR pro ess as

a general, gradient-based, minimization te hnique and dis usses its hara teristi s. In se tion 3, the

Geodesi Dynami Relaxation method is presented. This te hnique allows for the introdu tion of

equality onstraint onditions. Its features are dis ussed in se tion 4. Se tion 5 shows the a ura y

and validity of the presented developments based on the ase study of an existing, pre-stressed,

membrane system. In se tion 6, on lusions and a summary of this paper are given.

2. DYNAMIC RELAXATION PROCESS

The DRmethod ould be onsidered as a general, gradient-based, minimization approa h that solves

findx ∈ Rn | f(x)→min, (1)

where x represents unknown variables, f is a real-valued fun tion of x, Rnis the set of n-

dimensional real ve tors, and n denotes the total number of unknown variables. In problems of

equilibrium, f might represent the sum of elasti energies plus additional potential fun tions. In this

paper, we use an n-dimensional olumn ve tor to represent x , su h as x =[

x1 · · · xn]T.

We only onsider problems of equilibrium. As ea h lo al minimum represents an equilibrium state

of the system, we study lo al, not global, minima of the fun tion f . A lo al minimum is a point

x ∈ Rnthat satises the stationary ondition of the fun tion or

∇f = 0, (2)

Copyright © 0000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (0000)

Prepared using nmeauth. ls DOI: 10.1002/nme

3

where ∇f is the gradient of f dened as

∇f =[

∂f∂x1

· · · ∂f∂xn

]

(3)

and 0 is an n-dimensional row ve tor of whi h all the omponents are set to 0.In this paper we use an n-dimensional row ve tor to represent gradient ve tors. In order to

emphasize the fa t that ∇f is a fun tion of x, we write

ω (x) = ∇f. (4)

Note that, in problems of equilibrium, ω (x) represents external for e a ting on a single parti le

whose position is denoted by x. In typi al problems of equilibrium that are solved by DR, x is the

set of x, y and z oordinates of all the free nodes. In those ases, ω (x) an be de omposed to nodal

for es whi h a t on individual nodes. As a result, the problem of equilibrium dis ussed in this paper

takes the form

findx ∈ Rn | ω(x) = 0. (5)

To further develop these expressions for DR, we initially onsider the Steepest Des ent Method

(SDM), whi h is the rst standard, gradient-based, minimization approa h. For a given initial

onguration x0, SDM iteratively generates a point series x0, · · · ,xt, · · · based on

rt = −ω (xt)T

xt+1 = xt + αrt

, (6)

where α is a step-size fa tor, t is the step number, and rt is the standard sear h dire tion. By keeping

the step-size α onstant, suf ient onvergen e ef ien y annot be a hieved. Therefore, α is often

determined by a line sear h algorithm (e.g. see Flet her [25, se tion 2.6). However, a typi al line

sear h algorithm alls for more than one iteration in ea h line sear h. As a result, not only ω (x),but also f (x), are typi ally evaluated and taken into a ount. Be ause the original SDM, without

line sear h, only assesses ω (x), it would be preferable to onsider the integration of SDM with a

typi al line sear h algorithm, whi h only evaluates ω (x) and not f (x).The DR method an be introdu ed as a te hnique for this natural integration. In DR, instead of

performing a line sear h, the point series x0, · · · ,xt, · · · tends to move along a straight line by

introdu ing a velo ity parameter, denoted by q, and represented in an n-dimensional olumn ve tor.

For a given initial onguration, x0, q0, DR generates a point and ve tor series

x0, q0 , · · · , xt, qt , · · · based on

rt = −ω (xt)T

qt+1 = γqt + βrt

xt+1 = xt + βqt+1

, (7)

where β is a step-size fa tor, and γ is a damping oef ient. The DR pro ess an be intuitively

understood as a dis rete Newton's equation of motion by onsidering ω, rt, qt, and xt as the for e,

a eleration, velo ity, and position of a parti le moving in an n-dimensional Eu lidean spa e. It

should be noted that the name of this method differs between ommunities to ommunity and has

been named Verlet's s heme [26 and Symple ti Euler method [2731.

2.1. Conservation of energy

Empiri ally, the energy onservation law is observed in DR. We demonstrate this phenomenon with

the fun tion

f(x, y) = (10 + r (cos (8θ + 30r)))2 , (8)

where (r, θ) is a polar oordinate whi h an be onverted from (x, y) as follows:

r =(

x2 + y2)

,

θ =

π (x = 0)

atan (y/x) otherwise.

(9)



4

To ex lude the possibility that the observed energy onservation law o urs as a oin iden e, we

reated this fun tion to be suf iently omplex. Following the general manner des ribed above, x

and y are pa ked together as x =[

x y]T.

Starting with x0 =[

1 0]T

and q0 =[

0 0]T, we minimize Equation (8) using DR and

generate a point series x0, · · · ,xt, · · · . We use β = 0.005 for the step-size. By setting the damping

oef ient γ to 1.0, the point series does not onverge nor diverge as shown in Figure 1 (a). This

observation suggests that ertain quantities are onserved. We dene the kineti energy,Kt, and the

total energy, Et, at step t by

Kt =1

2qTt qt, Et = Kt + f (xt) . (10)

As shown in Figure 1 (b), the value Et is roughly kept onstant throughout the omputational

pro ess, and it does not seem at in a stri t sense. However, tra ing the traje tories of f andK , in the

same plot, shows a path with larger magnitudes whi h an el ea h other out. Similar observations

were reported and explained in studies on the Symple ti Euler method [2731. When γ = 0.0, theDR be omes an exa t SDM, in whi h the step size of SDM is given by α = β2

. The DR obeys the

law of onservation of energy, when γ = 1.0.

(a) Trajectory (b) History of energy

E=f+K

K

f

Step#

Ener

gy

y

Total Energy E=f+K

Kinetic energy K

x

Figure 1. The energy onservation law, observed in the DR pro ess: (a) plot of the generated point series

superimposed on a ontour of the fun tion and (b) plot of time steps versus total energy E and kineti energy

K.

2.2. Damping oef ient

The basis of the DR method is to follow the movement of ea h node of the stru ture, for small time

intervals, from its initial position until all vibrations have died out due to arti ial damping.

Usually, a nite element form of DR is des ribed as being the means by whi h any unstable

dis retized system might be brought to rest through the appli ation of vis ous damping, applied

through the damping oef ient γ, of the nodal movements. In order to a hieve the most rapid

onvergen e, the lowest mode of vibration of the stru ture is riti ally damped and the titious

nodal mass omponents are adjusted to be proportional to the orresponding dire t stiffness

omponents. In some ases, the riti al damping oef ient, γ, might be dif ult to estimate.

Starting from a rather ina urate initial position, ertain elasti members are grossly deformed

in the initial DR stages and indu e lo ally unbalan ed for es and related high frequen y modes

[3. Therefore, additional ontrol measures su h as varying mass omponents, lo ally varying

damping onstants and titious member stiffnesses are needed in different stages of the analysis

to obtain onvergen e [32. Frieze, Hobbs and Dowling [33 used similar variable ontrols for



5

the investigation of plate bu kling; signi antly different levels of damping were needed for

varying levels of applied loading. Zhang and Yu [34 have presented a modied adaptive Dynami

Relaxation Methods based on the vis ous damping approa h. In this ase, the damping oef ient

is based on a fun tion of the urrent system onguration, the internal element for e and the mass

matrix. Re ently, Rezaiee-pajanda et al. [35 proposed a method that minimizes errors between two

su essive iterations to obtain optimum titious mass and vis ous damping with the aid of the

Stodola iterative pro ess.

While working on unstable, geo-me hani al problems, Cundall [36 rst suggested using kineti

damping, whi h proved to be entirely stable and rapidly onverging when dealing with large

unbalan ed for es [3. Sin e vis ous damping is negle ted, there is no need for prior determination

of the damping onstant. The underlying basis of kineti damping is that as an os illating body

passes through a minimum potential energy state, its total kineti energyKt rea hes a lo al

maximum. Upon dete tion of this lo al peak, all urrent nodal velo ities are set to zero. The

pro ess is then restarted from the urrent geometry and ontinued through generally de reasing

peaks until all energy of all modes of vibration has been dissipated and the stru ture rea hes its

stati equilibrium state. For a omprehensive overview of kineti damping in DR, the reader is

referred to Shugar [37.

In this se tion, we present a framework that expresses both vis ous and kineti damping as similar

parameters. First, we dene θt, an entity between the a eleration and the velo ity at step t by

θt =qTt rt

|qt| |rt|, (11)

where |·| is the standard Eu lidean norm. Note that θt = 1.0 when the a eleration and the velo itypoint in the same dire tion, θt = 0.0 when one is perpendi ular to the other, and θt = −1.0 when

they point in opposite dire tions. Se ond, we assume that the damping oef ient at ea h step is

given by a fun tion of θt, i.e.,γt = γ (θt) (12)

This fun tion gives a hara teristi urve between θ and γ, and determines the behavior of DR. By

optimizing this fun tion, the performan e of the DR methods an, thus, further be improved.

In this ontext, vis ous and kineti damping an be des ribed as follows. Vis ous damping is

hara terized by

γ (θ) = const. (13)

The traje tory and the history of the energy in DR with vis ous damping (γ = 0.98) applied to

Equation (8) is shown in Figure 2.

On the other hand, kineti damping is hara terized by

γ (θ) =

1.0 (1 ≥ θ > 0)

0.0 (0 ≥ θ ≥ −1). (14)

These equations an be explained as follows. When kineti energyKt rea hes a lo al maximum, the

a eleration rt be omes orthogonal to the velo ity qt, i.e., θ = 0. Prior to that instan e, θ is greaterthan 0 be ause the kineti energy in reases. Hen e, setting γ to 0, when θ ≤ 0, des ribes kineti damping. In other words, the system a elerates with no damping until a eleration and velo ity

o ur perpendi ular to ea h other. On e they are perpendi ular to ea h other, the system de elerates

instantly.

The traje tory and the history of energies in the DR pro ess, with kineti damping applied to

Equation (8), are shown in Figure 3. Although the approa h with kineti damping proves to be the

most advantageous in the majority of numeri al examples, this example shows that adopting kineti

damping in the DR pro ess might bring about an inef ient traje tory due to the dis ontinuity in the

hara teristi urve.

In addition to vis ous and kineti damping, we introdu e the on ept of drift damping, whi h is

dened by

γ (θ) = 0.95 + θ/20 (15)



6

This fun tion is developed to solve the dis ontinuity in kineti damping. The traje tory and the

history of the energies in the DR pro ess, with drift damping applied to Equation (8) are shown

in Figure 4. By using drift damping, the phase of DR pro ess is dynami ally adjusted between

a eleration and de eleration as well as kineti damping. Be ause Equation (15) adjusts γ from 1.0to 0.90 smoothly, as opposed to kineti damping, the phase transition in drift damping is gentle rather

than instant. Note that, be ause the effe t of damping works exponentially, the damping effe t with

γ = 0.92, 0.95 and 0.98 works in very different ways. Table I shows that, after 100 step iterations, apulse for e at the rst step almost vanishes if the damping oef ient is 0.92 or 0.95. However, 10%of that pulse for e is still present after 100 iterations, when the damping oef ient has a value of

0.98.Based on trial runs we pose that a desirable duration of pulse for e is around 100 iterations. The

bias degree, 1/20, in Equation (15) is adjusted to a hieve this duration. In se tions 4 and 5, γ = 0.98is used for vis ous damping. This parti ular number 0.98 is also adjusted to ensure that the initial

pulse for e is dissipated after 100 iterations. Therefore, if a different expe ted duration of pulse

for e is hosen, Equation (15) or the onstant value in vis ous damping may be hanged.

In Figure 5 the three γ − θ relation urves for kineti , vis ous and drift damping are plotted. The

ow hart for the DR method ombined with the proposed exible framework of damping is shown

in Figure 6. The DR pro ess hanges its performan e a ording to different parameter hoi es in

Equations (13) and (15). A detailed explanation of their respe tive performan es is provided in

Appendix A for a ben hmark test. All three damping approa hes are applied to the geodesi DR

framework presented in se tion 3.

E=f+K

Step#

Ener

gy

y

K


Total Energy E=f+K

Kinetic energy K

x

Figure 2. Vis ous Damping (γ = 0.98): (a) plot of the generated point series superimposed on a ontour of

the fun tion and (b) plot of time steps versus total energy E and kineti energy K.

γ 0.92 0.95 0.98

γ100 2.39E-04 0.592E-03 1.32E-01

t that gives γt = 0.1 27 44 113

Table I. For three different damping oef ients γ, the strength of a pulse for e present after 100 iterations

γ100 and number of iterations t at whi h the pulse for e is redu ed to 10% of its initial value are given.

2.3. Straight line geometry generation

By setting r and γ to 0 and 1.0 respe tively in DR, i.e.,

rt = 0

qt+1 = qt + βrt

xt+1 = xt + βqt+1

, (16)



7


E=f+K

K

Step#

Ener

gy

x

y

Total Energy E=f+K

Kinetic energy K

Figure 3. Kineti Damping: (a) plot of the generated point series superimposed on a ontour of the fun tion

and (b) plot of time steps versus total energy E and kineti energy K.

(a) Trajectory (b) History of energyx

y

E=f+K

K

Step#

Ener

gy

Total Energy E=f+K

Kinetic energy K

Figure 4. Drift Damping: (a) plot of the generated point series superimposed on a ontour of the fun tion

and (b) plot of time steps versus total energy E and kineti energy K.

1.0

1.0

0.0

-1.0-0.0

γ

θ

1.0

1.0

0.0

-1.0-0.0

γ

θ

1.0

1.0

0.0

-1.0-0.0

γ

θ

(a) Viscous Damping (b) Kinetic Damping (c) Drift Damping

Figure 5. Plots of θ versus γ for vis ous (a), kineti (b) and drift ( ) damping.

an important feature of the DR pro ess is revealed. Starting from q0, whi h does not equal 0, and

an arbitrary x0, the DR pro ess generates a straight line on sear h spa e Rn, i.e., xt = x0 + βtq0.

In terms of Newtonian me hani s, this is alled the law of inertia. Note that this straight line does

not orrespond to a line drawn on a physi al stru ture su h as a membrane roof, but a point series

dened on Rn. Hen e, before the for e is applied, the DR method keeps generating a straight line.



8

Inputs: ,

Entity between

and

Damping

Acceleration

Velocity

Position

,

Figure 6. Flow hart 1: Dynami Relaxation method ombined with a exible framework of damping.

When a for e ω (x) is applied to the system, su h a straight line is gradually altered based on ω (x)step by step.

This preliminary dis ussion of the DR pro ess (se tion 2) highlights parti ular hara teristi s

that are worth dis ussing when handling equality onstraint onditions, whi h are addressed in the

se tion 3. These features an be summarized as (i) the DR algorithm only evaluates ω (x), and notf (x), (ii) with no damping applied, the total energy of the for e-modelled system is onserved, (iii)

vis ous, kineti and drift damping an su essfully be applied to the DR pro ess and (iv) the DR

pro ess simply generates a straight line when no for e or strain energy is applied to the system.

3. GEODESIC DYNAMIC RELAXATION METHOD

In this paper, we present an extension to the DR method to allow for equality onstraint onditions

to be in orporated into the pro ess. Problems of equilibrium with equality onstraint onditions are

typi ally dened as

findx ∈ Rn | f (x) → min, (17)

s.t.

g1 (x) = g1.

.

.

gm (x) = gm

,

where m is the total number of onstraint onditions, g1 (x) , · · · , gm (x) are quantities to be

onstrained, and g1, · · · , gm are the values to whi h the quantities are onstrained. In this paper,

we assumem < n and that g1 (x) , · · · , gm (x) have gradients ∇g1, · · · ,∇gm.

As we onsider problems of equilibrium, we only have to onsider a lo al minimum whi h

satises the stationary ondition of a Lagrangian omposed of this problem. The Lagrangian of

this problem is given by

L (x,λ) = f (x) +

m∑

j=1

λj (gj (x)− gj) , (18)



9

where λ =[

λ1 · · · λm]

represents the Lagrange multipliers. Note that we use an m-

dimensional row ve tor to represent the Lagrange multipliers. The stationary ondition of L (x,λ)is given by

∂L∂x

= 0 and∂L∂λ

= 0. (19)

For fun tions of both x and λ, we dene ∇f (x,λ) ≡ ∂f∂x

and ex lude the partial derivatives with

respe t to the Lagrange multipliers from ∇f . Then, the rst and se ond stationary onditions are

respe tively expanded as

∇L = ∇f +

m∑

j=1

λj∇gj = 0 and

g1 (x) = g1.

.

.

gm (x) = gm

. (20)

The rst ondition represents an equilibrium state between ∇f and the rea tion for es supplied by

the onstrained quantities. The se ond ondition is the given set of equality onstraint onditions.

For example, if gj (x) represents a length of a strut, volume of air, or angle of a hinge joint, λjrepresents axial for e, pressure or moment a ting on those stru tural omponents, respe tively.

We an regard Rnas an n-dimensional, Eu lidean spa e by onsidering the elements x ∈ R

nas

points of whi h oordinates are (x1, · · · , xn). By olle ting all the points in Rnthat satisfy all the

onstraint onditions, we dene

S ≡ X ∈ Rn | gj (X) = gj, ∀j = 1, · · · ,m , (21)

whi h forms an isomanifold in Rn. We all the S onstraint isomanifold. In order to emphasize the

fa t that ∇L is a fun tion of both x and λ, we write

τ (x,λ) = ∇L = ω (x) + λJ (x) , (22)

where

ω (x) = ∇f andJ (x) =

∇g1.

.

.

∇gm

. (23)

In problems of equilibrium with equality onstraint onditions, three different for es are taken into

a ount: ω (x) as an external, λJ (x) as rea tion, and τ (x,λ) as resultant for e. Note that J is an

m× n matrix and is often alled a Ja obian matrix. In this paper, we assume that J is a full-rank

matrix. Hen e, as we assumed thatm < n, the rank of J should bem.

Consequently, the problems of equilibrium with equality onstraint onditions typi ally take the

form

find

x ∈ S ⊂ Rn, λ ∈ R

m

| τ (x,λ) = 0. (24)

UnlikeRn, the onstraint isomanifold S is assumed to be a urved subspa e ofR

n. On su h a urved

subspa e, straight lines do not exist, but geodesi s an be used as an alternative to straight lines.

Therefore, this extension to the existing DR method is named the Geodesi Dynami Relaxation

method.

In the geodesi DR, ea h iteration is extended to

rt = −φ (ω (xt))T

qt+1 = γϕ (qt) + βrt

xt+1 = ψ(

xt + βqt+1

)

, (25)

where φ (ω), ϕ (q) and ψ (x) are proje tion operators to proje t for e, velo ity and position ve torsto appropriate subspa es. More spe i ally, these terms are alled proje ted gradient, dis rete

parallel transportation, and pull ba k, respe tively.



10

The relationship between the geodesi and existing DR method is very similar to the one between

the proje ted gradient method [25, 38, 39 and SDM. The proje ted gradient method iteratively

generates a point series x0, · · · ,xt, · · · based on

rt = −φ (ω (xt))T

xt+1 = ψ (xt + αrt)(26)

where φ (ω) and ψ (x) are exa tly the same operators as the proje ted gradient and pull ba k

operators in the geodesi DR iteration. In addition, the similarity between the fun tion of pull ba k

operator and the fast proje tion method [40 is noted.

Therefore, the key ontribution in this paper is to present the dis rete parallel transportation ϕ (q).This guarantees the onservation of the total energy of the system during the DR pro ess when the

damping oef ient γ is set to 1.0.Before des ribing these three proje tion operators, we introdu e a pseudo inverse matrix of

J [41, whi h will be denoted by J+, and larify the geometry underlying the problem dened

in Equation (17).

3.1. Underlying geometry and pseudo inverse matrix

By varying g1, · · · , gm arbitrarily, the isomanifold S an be moved in Rn, and all possible

isomanifolds an be olle ted. On e empty manifolds are deleted, the remaining isomanifolds over

whole Rnwith no overlap. Hen e, any point, x ∈ R

n, belongs to one of those isomanifolds. Su h

an isomanifold spe ied by a given x ∈ Rnis identied by

S (x) ≡ X ∈ Rn | gj (X) = gj (x) , ∀j = 1, · · · ,m . (27)

This is a set of all the points that give the same values of g1 (x) , · · · , gm (x).We use a pseudo inverse of J in order to represent the tangent spa e and its orthogonal

omplement of S (x) at x. Although the pseudo inverse matrix is dened in a more general way, in

this work, we ompute the pseudo inverse matrix of J by

J+ = JT(

JJT)−1

. (28)

This operation is allowed only whenm < n and J is a full-rank matrix. It is likely that the geodesi

DR an also be performed in more general ases, in whi h J+is omputed in a more general way.

However, the dis ussion of more general ases is outside the s ope of this paper. In the presented

ase, it is obvious that JJ+ = Im, where Im is an m×m unit matrix. However, ontrary to

expe tations, J+J 6= In, where Im is an n× n unit matrix. This means that J+is a right inverse

and not a perfe t inverse.

On S (x), we dene the following four ve tor spa es at x:

TS (x) =

α ∈ Rncol | α =

(

In − J+J)

β, ∃β ∈ Rncol

, (29)

T ∗S (x) =

α ∈ Rnrow | α = β

(

In − J+J)

, ∃β ∈ Rnrow

, (30)

OS (x) =

α ∈ Rncol | α = J+Jβ, ∃β ∈ R

ncol

, (31)

O∗S (x) =

α ∈ Rnrow | α = βJ+J , ∃β ∈ R

nrow

, (32)

where Rncol

and Rnrow are the sets of n-dimensional row and olumn ve tors respe tively. From the

viewpoint of differential geometry, both TS (x) and T ∗S (x) represent a tangent spa e of S (x)at x. If T ∗S (x) is differentiated from TS (x), it would be alled a otangent spa e. Additionally,

OS (x) and O∗S (x) are the orthogonal omplements of them. Note that TS (x)⊕OS (x) = Rncol

and T ∗S (x)⊕O∗S (x) = Rnrow be ause

(

In − J+J)

+ J+J = In. The tangent spa es are the

sets of dire tions that do not hange the values of g1 (x) , · · · , gm (x). The orthogonal omplements

are the sets of dire tions that hange the values of g1 (x) , · · · , gm (x) most effe tively.



11

ωϕ(ω)x

S(x)

S

J

Figure 7. φ (ω): Proje ted gradient omputes a omposition of external for e ω and rea tion for e λJ .

Adopting the same approa h, T S, T ∗S, OS, and O∗S are also dened on S. However, we shouldnot assume that xt belongs to S at ea h step. Instead, we dene the three proje tion operators on

any S (x) and later employ an iterative strategy (i.e., pull-ba k) to superimpose S (x) onto S.

3.2. Proje ted gradient operator

As shown in Figure 7, the proje ted gradient operator φ (ω) proje ts external for e ω to T ∗S (x).This proje ted gradient is dened by

φ (ω) = ω(

In − J+J)

. (33)

As the result, φ (ω) always points in the dire tion that does not hange the values of

g1 (x) , · · · , gm (x). In other words, the orthogonal omponent of external for e to the tangent spa e

is ne essarily eliminated. As the result, a eleration rt in the geodesi DR method also points in the

dire tion that does not hange the values of g1 (x) , · · · , gm (x).In identally, the proje ted gradient method an be understood as a omposite fun tion of τ (x,λ)

with a multiplier estimate given by

λ (x) =(

−ωJ+)

. (34)

By dire tly substituting Equation (34) into Equation (22), the proje ted gradient φ (ω) is obtained.As shown by Figure 7, the proje ted gradient φ (ω) an be de omposed as two for es, ω and

orthogonal rea tion for e λJ . Due to the multiplier estimate, the rea tion for e λJ is always an

element of orthogonal omplement O∗S (x).The multiplier estimate annot be derived from any prin iples, but its use is bene ial be ause

the right-hand side of Equation (34) depends only on x. Hen e, the multiplier estimate gives a one

to one mapping from x to λ, and the multipliers an be eliminated from the problem. Additionally,

the multipliers are arried over between the different analyses and follow the stress analysis stages

smoothly. This o urs be ause, when the DR pro ess is judged to onverge, the rea tion for es an

be omputed by Equation (34).

Due to the multiplier estimate, only the initial ongurations of position and velo ity ( i.e., x and

q) are given and updated in the geodesi DR pro ess as well as in the existing DR method.

3.3. Dis rete parallel transportation operator

As shown in Figure 8, the dis rete parallel transportation operator ϕ (q) proje ts q to TS (x) and isgiven by

ϕ (q) =|q|

∣

∣

(

In − J+J)

q∣

∣

(

In − J+J)

q. (35)

This operator is similar to the proje ted gradient, but preserves the norm of the given ve tor. This

hara teristi ontributes to the energy onservation law in the geodesi DR. In order to a hieve

energy onservation, the kineti energy must, at least, be onserved when no damping nor external

for es are applied. If the norm of the velo ity is not preserved during the proje tion, the kineti



12

x

q

φ(q)S(x)

S

Figure 8. ϕ(q): Dis rete parallel transportation preserves the norm of the velo ity while proje ting it to the

tangent spa e of S (x).

energy de reases, even if no damping is applied. This means that, if the same operator as the

proje ted gradient is applied to the velo ity, unexpe ted damping effe ts might arise.

Similarly to the proje ted gradient, the dis rete parallel transportation operator guarantees that

the velo ity always points in the dire tion that does not hange the values of g1 (x) , · · · , gr (x).In the geodesi DR, ϕ (qt) should be used for omputation of the entity θ between the velo ity

and the a eleration, i.e.,

θt =ϕ (qt)

Trt

|ϕ (qt)| |rt|(36)

When the damping oef ient γ is given by a damping fun tion γ (θ), the se ond line of the geodesi DR (Equation (25)) is repla ed with

qt+1 = γ (θt)ϕ (qt) + βrt. (37)

3.4. Pull ba k operator

Figure 9 shows that pull ba k operator, ψ (x), tries to pull x ba k onto S. Unlike the proje ted

gradient and the dis rete parallel transportation operators, ψ (x) is not a simple operator. Instead, it

an be dened by the iterative algorithm given in ow hart 2 shown in Figure 10.

In the ow hart, b represents the residual of onstraint onditions dened by

b (x) =

g1 (x)− g1.

.

.

gr (x)− gr

. (38)

Also, ∆x = −J+b is a minimum norm solution for a system of linear equations

J (x)∆x = −b (x) , (39)

whi h is a linear approximation of b (x+∆x) = 0. Be ause Equation (39) is a linear

approximation, ideally, x should ideally be lose to S in ea h step. It should be noted that

∆x = −J+b is an element of the orthogonal omplement OS (x).The ow hart in Figure 10 further shows that s, a ounter for iteration y les, and |b|, the norm of

residual of onstraint onditions, are updated in every step and used to evaluate whether the y les

are ontinued or not. The threshold values for both the ounter, s, and the norm, |b|, denoted by Nand ξ, respe tively, in the ow hart, are given. The iteration y les are terminated when the iteration

ount s rea hes N or when |b| be omes less than ξ.Even if the proje ted gradient and dis rete parallel transportation operators make the a eleration

rt and velo ity qt always point toward the dire tion that does not hange the values of

g1 (x) , · · · , gr (x), small errors in onstraint onditions a umulate be ause an isomanifold S (x)is generally not at. Hen e, they are not suf ient to keep the values of g1 (x) , · · · , gr (x) onstant.This phenomenon means that, even if x truly belongs to the onstraint isomanifold S at step t,



13

S(x)x

(x)S

Figure 9. ψ(x): Pull ba k operator proje ts x onto S by iteratively solving b (x+∆x) = 0.

Input:

Output:

No

Yes

-

Figure 10. Flow hart 2: Pull ba k operator.

x gradually deta hes itself from S in later steps, unless at least one iteration of the pull ba k is

performed in ea h step. However, if S (x) is suf iently lose to S, i.e., |b| is suf iently small, only

single iteration is suf ient for the pull ba k be ause Equation (39) is an appropriate approximation

of b (x+∆x) = 0. Therefore, the basi number of iterations in the pull ba k is one. More than one

iterations is alled when |b| is greater than ξ.Having dened the three operators (proje ted gradient, dis rete parallel transportation and pull

ba k) in the se tions 3.2-3.4, the geodesi DR an be summarized. In ea h step, the external for e

ω and velo ity q are proje ted to tangent spa es, T ∗S (x) and TS (x). The norm of q is preserved

before and after the proje tion, while a simple orthographi proje tion is applied to external for e

ω. By using the proje ted ve tors, a single iteration, whi h is exa tly the same operation as the

existing DR, is performed. After this operation, if the new x is suf iently lose to the onstraint

isomanifold S, the single step pull ba k operation is performed. If not, the iterative pull ba k

operation is performed until the iteration y le ount rea hes a pres ribed maximum number or



14

Inputs: ,

:Discrete parallel

transportation

Entity between

and

Damping

Acceleration

Velocity

Position

,

:Pull-back

:Projected

Gradient

Figure 11. Flow hart 3: geodesi Dynami Relaxation Method

the norm of onstraint ondition residual be omes less than a pres ribed threshold. This approa h

is further illustrated in ow hart 3 (Figure 11).

Eventually, the onditions φ (ω) = 0 and S (x) = S are satised with an a eptable toleran e

and the geodesi DR method onverges. Due to the denition of the proje ted gradient, when

the proje ted gradient vanishes, a relation of ω = ωJ+J is established. Choosing the multipliers

as λ = −ωJ+, we have ω = −λJ and, hen e, obtain ω + λJ = 0, whi h is the rst stationary

ondition. This ondition an be expressed as τ (x,λ) = 0. It is lear that S (x) = S ensures the

se ond stationary ondition. As shown in Figure 12, both ω and λJ belong to O∗S (x). On the

other hand, φ (ω), whi h belongs to T ∗S (x), disappears. Figure 12 also indi ates that the externalfor e ω and the rea tion for e λJ an el ea h other out. Therefore, their de omposition disappears,

whi h is a reinterpretation of φ (ω) = 0 in terms of stati s.

In ontrast with the existing DR method, the geodesi DR te hnique evaluates ω (x) and J (x).However, J (x) ontains gradients of the onstrained quantities, and the geodesi DR only assesses

the rst derivatives of the fun tions. Therefore, neither the existing nor the presented geodesi DR

methods require the omputation of the se ond derivatives of the fun tion. As a tradeoff between

avoiding omputation or estimation of se ond derivatives, the DR method is might be onsidered

slow when ompared with the Newton methods family (e.g. Newton-Raphson and quasi-Newton

methods). Nevertheless, the extension presented in this paper has potentially large impa t as it

is independent of the omputation of se ond derivatives. The presented extension improves the

performan e of the existing DR method, whi h has been widely adopted by a ademia and industry

for the form-nding and nonlinear analysis of pre-stressed stru tural systems.

4. DISCUSSION OF THE GEODESIC DYNAMIC RELAXATION METHOD

In parallel with se tion 2, the major hara teristi s ( onservation of energy, damping oef ient and

geodesi generation) of the geodesi DR method are dis ussed.



15

ω

x

S(x)=SJ

T S(x)∗

O S(x)∗

Figure 12. Equilibrium established in a problem of equilibrium with equality onstraint onditions: External

for e ω and rea tion for e λJ an el ea h other out at x.

4.1. Conservation of Energy

First, the effe t of applying the dis rete parallel transportation operator to the velo ity in the

geodesi DR method on the energy onservation law is investigated. We apply the geodesi DR

method to the following numeri al example:

f (x, y, z) = 0.1(z + 5) → min, (40)

s.t.

(

x− RA√x2 + z2

x

)2

+ y2 +

(

z − RA√x2 + z2

z

)2

= R2B,

where the fun tion f represents a gravity potential, and the onstraint ondition is an impli it

representation of a torus with RA as the major and RB as the minor radii. The initial ongurations

are

x0 =[

RA +RB sin v RB cos v 0]

, q0 =[

0 0 1]

, (41)

where ν is a parameter to movex0 along a meridian. It should be noted thatx0 satises the onstraint

ondition and, hen e, S (x) = S is satised in the initial step.

We use the following values: RA = 3, RB = 2, and v = 2.34. For the time step, we set β = 0.1,and N = 10 and ξ = 0.01 for the pull ba k. The traje tory generated by the geodesi DR method

with no damping (γ = 1.0) is shown in Figure 13 (a). This gure shows that the traje tory does not onverge nor diverge. The history of the total and kineti energies versus the number of iterations in

the geodesi DR method is plotted in Figure 13 (b). This gure illustrates that the total energy of the

system is roughly onserved throughout the omputation. If the proje ted gradient is applied to the

velo ity, instead of the dis rete parallel transportation, the total energy gradually de reases be ause

the proje ted gradient always de reases the norm of the velo ity without amplifying it.

4.2. Damping oef ient

The effe t of adopting different damping fun tions (vis ous, kineti and drift) on the onvergen e of

the geodesi DR method is investigated in the ase study presented in se tion 4.1. The traje tories

and histories of the energies versus number of time steps are plotted in Figure 14. These plots

illustrate the effe ts of the three damping approa hes well. When kineti damping is solely applied,

the kineti energy is not absorbed until the kineti energy rea hes a lo al maximum. On the other

hand, when vis ous damping is solely applied, a small amount of the kineti energy is ontinuously

dissipated along the pro ess. The drift damping approa h exhibits the bene ial hara teristi s

of both the kineti and the vis ous damping methods. Throughout the geodesi DR pro ess, it

ontinuously dissipates kineti energy ( fr., vis ous damping). However, the rate of dissipation of

kineti energy is adjusted dynami ally, and it rapidly in reases on e the pro ess hits a lo al peak in

kineti energy ( fr., kineti damping).



16

(a) Trajectory (b) History of Energy

E=f+K

K

f

K

Step#

Ener

gy

Initial velocity

Trajectory

Gravity

Initial position

Figure 13. Energy onservation law in the geodesi DR (β = 0.1,γ = 1.0,v = 2.34).

Step#

Ener

gy

(a) Viscous damping

(a) Viscous damping (b) Kinetic damping (c) Drift damping

Step#

Ener

gy

(c) Drift damping

Step#

Ener

gy

(b) Kinetic damping

E=f+K

K

Initial velocity

Trajectory

Gravity

Initial position

Converged position

Figure 14. Top: Traje tory plots of geodesi DR with (a) vis ous (γ = 0.98), (b) kineti , and ( ) drift

damping. All three approa hes lead a point on the torus onverge to a position where its gravity potential is

minimum. Bottom: Total energy plots versus number of iterations for vis ous, kineti and drift damping.

4.3. Geodesi generation

In the geodesi DR method, the law of inertia on a urved spa e an be reprodu ed by setting the

external for e, ω, to 0 and damping oef ient, γ, to 1.0. In other words, we an draw a geodesi on

an impli itly represented surfa e (or on a higher dimensional manifold) by using the geodesi DR



17

Inputs ,

Figure 15. Flow hart 4: Geodesi s generation: Repetition of dis rete parallel transportation, position update,

and pull ba k.

method. Be ause both the external for e and damping are not given, the geodesi DR method an

be simplied for this spe i purpose, as illustrated in ow hart 4 (Figure 15).

Using the algorithm presented in Figure 15, geodesi s are generated on the torus dened in se tion

4.1. The same initial onguration as Equation (41) is used but the value of v is varied from 0 to 2π.The other parameters are kept onstant (i.e. for the time step, β = 0.1, the pull-ba k, N = 10 and

ξ = 0.01 , the radii of torus, RA = 3, and RB = 2).In Figure 16, the top row gures show the traje tories on the torus generated by geodesi DR

approa h while the bottom row gures show geodesi s on the torus given by a program developed

with ommer ially available software, Mathemati a® [42. The program algorithm solves an

ordinary differential equation that gives geodesi s on a parametri surfa e, given by an expli it

representation.

Even though our algorithm is te hni ally very different from the Mathemati a®program

algorithm, both routines visually give the same results with the ex eption of one part indi ated

by ellipses in the gure. Hen e, we laim that our method an generate a geodesi s on an impli it

surfa e or, more broadly, that the presented geodesi DR method generates geodesi s when no for e

is applied.

The dis ussion of the hara teristi s of the geodesi DR method suggests a number of similarities

and differen es with the existing DR method. Both methods preserve the total energy of the system

when no damping is applied and work with different damping approa hes (vis ous, kineti and drift

damping). However, unlike the DR method, the geodesi DR produ es geodesi s as opposed to

generating straight lines. From a differential geometry perspe tive, geodesi s are one of the natural

generalizations of straight lines to urved spa es. Hen e, the geodesi DR an be thought as a natural

extension of the existing DR method.

5. CASE STUDY

To show the validity and a ura y of the geodesi DR method, we arried out the form nding of

an existing, pre-stressed membrane stru ture, Tanzbrunnen (1957, Cologne, Germany, Frei Otto).

The original design of Tanzbrunnen is a minimal surfa e stru ture based upon the geometry of a

physi al soap lm model. This system onsists of a radially, pre-stressed, ridge-valley membrane

with an o ulus inner ring. The membrane is supported by six masts, equally spa ed along the outer

tension able ring of the membrane. The ridge-valley onguration is a hieved by 12 pre-stressed

ables, radially positioned in the membrane. These ables run from the inner o ulus tension ring

either (i) over the tops of the masts or (ii) to a point, positioned on the membrane's outer tension

ring between two mast heads. The masts' footings themselves are pinned to the foundation and

the mast heads are further stabilized with stay ables. The points on the membrane outer tension



18

(a-1) v=0.0 (a-2) v=1.04 (a-3) v=2.34

(b-1) v=0.0 (b-2) v=1.04 (b-3) v=2.34

(a-4) v=3.54 (a-5) v=4.44 (a-6) v=5.62

(b-4) v=3.54 (b-5) v=4.44 (b-6) v=5.62

A part that two curves

do not match.

Initial velocity

Trajectory

Initial position

Figure 16. Geodesi s on a torus (a): Geodesi s omputed by our method for different values of ν. (b): Curvesgenerated by a Mathemati a®program [42. The arrows show velo ity at the initial point. Ex ept one part

indi ated by the ellipses, they visually mat h well.

ring are also stabilized by one stay able going to the foundation. The membrane and the ables on

the inner and outer tension rings are pre-stressed. The length of the masts and the stay ables are

onstrained. The total lengths of the 12 pre-stressed ables, positioned in the membrane surfa e, are

also onstrained to a xed length.



19

5.1. Form nding pro ess

The initial onguration, used for x0 in the form nding pro ess, is given in Figure 17 (a). Figure

17 (b) shows whi h nodes are xed, whi h elements are pre-stressed and whi h elements have been

assigned onstraint onditions. L and S, shown in Figure 17 (b), represent the length between twonodes of a linear element, and the area of a triangular element dened by three nodes, respe tively.

Additionally, w is a weight oef ient that is assigned and used to make the elements larger or

smaller. In this ase study, we only onsider onstraints related to length.

The energy fun tion to be minimized by the geodesi DR method is expressed as follows:

f(x) = ws

∑

j∈D

Ssj + w1

∑

j∈E

Ltj + w2

∑

j∈F

Ltj , (42)

where ws, w1, and w2 are weight oef ients for membrane and the outer and inner tension rings,

respe tively, and s and t are exponents of S and L, respe tively. Typi ally, these exponents have

values of 1 or 2 in form nding problems. The hoi e of these exponents is dis ussed further in this

se tion. Additionally, D, E and F are the sets of (triangular or linear) elements in the membrane,

outer and inner ring, respe tively.

In the model, there are 504 free nodes and 18 nodes (i.e. 6 mast bottoms and 12 onne tions of

the stay ables to the foundation) pinned in all three dire tions. Therefore, the degree of freedom,

i.e. the dimension of x, in this problem is 1512. As shown by Figure 17 ( ), for the pre-stressed

elements (membrane and tension ring ables) the weight oef ients ws, w1, and w2 are treated as

design parameters to ontrol and study the shape of the membrane stru ture. On the other hand, for

the elements for whi h the lengths are onstrained (masts, stay ables and radial ables), the lengths

themselves are treated as design parameters.

Two different types of onstraint onditions are in orporated into the design problem. The rst

type is a simple length onstraint ondition of a linear element and the se ond type is a total length

onstraint ondition of a set of linear elements. There are 18 onstraint onditions of the rst type

and 12 onstraint onditions of the se ond. Hen e, there are 30 multipliers pa ked together in λ.

In order to attain radial symmetry in the stru ture, the onstraint onditions are assembled to

one group per 6 onditions, and hen e there are 5 groups of onstraint onditions. In ea h group, a

ommon onstraint value is used for six onstraint onditions, su h as L1, · · · , or L5.

For example, the rst group ontains the six masts with onditions des ribed as

L145 (x) = L1

.

.

.

L150 (x) = L1

. (43)

Similarly, the se ond and third groups ontain onstraint onditions of the same type for stay

ables, in whi h L2 and L3 are used for onstraint values. The fourth and fth groups ontain six

onditions for six pre-stressed ables, in whi h L4 and L5 are used to pres ribe total lengths of

ables, su h as

168∑

j=163

Lj (x) = L4

.

.

.

198∑

j=193

Lj (x) = L4

. (44)

We arried out the geodesi DR method with spe i design parameters: ws = 0.8, w1 = 5,w2 = 12 and L1 = 4.69, L2 = 1.764, L3 = 5.176, L4 = 5.357, L5 = 5.092 with exponents

s = 1, t = 1. We used β = 0.05 for the time step and N = 50 and ξ = 0.0001 for pull ba k.

However, an unexpe ted result was obtained, as shown by Figure 17 (d). This result an be attributed



20

to the exponents hosen. We tested another hoi e of exponents s = 1, t = 2 while holding the

other parameters to the same values. Although improvement was observed, very small and distorted

triangular elements have been noted in the onverged solution model, as shown by Figure 17 (e).

This o urren e is due to the singularity problem, whi h has been reported in the ontext of minimal

surfa e problems [43. When the Newton method family is used to solve minimal surfa e problems,

this singularity prevents the pro ess from onverging be ause the stiffness matrix be omes harder

to invert as the method iterates. On the ontrary, the DR method an dire tly minimize the sum of

element areas without onfronting this singularity problem. However, as one an observe in Figure

17 (d) and (e), this singularity still auses unexpe ted results, even if the DR method is employed

for minimization. Therefore, regardless of the DR method’s ability to solve the problem, we should

not ignore the singularity problem and adopt a strategy to avoid it.

Wu hner and Bletzinger ta kled this singularity problem [43 in the minimal surfa e problem.

Their approa h was to modify the minimal surfa e problem arefully to make the modied problem

as similar to the original as possible, but the singularity no longer exists in the modied problem.

Be ause further dis ussion of minimal surfa e problem is outside the s ope of this paper, instead,

we modied the original problem largely and simply hose s = 2, t = 2 as exponents. This hoi eis equivalent to giving surfa e stress density [44 to the triangular elements and for e density to the

linear elements [45.

Thus, a stable shape for Tanzbrunnen stru ture was obtained, as shown by Figure 17 (f) with

spe i design parameters: s = 2, t = 2, ws = 0.8, w1 = 1, w2 = 12 and L1 = 4.69, L2 =1.764, L3 = 5.176, L4 = 5.357, L5 = 5.092. Note that we hanged w1 as well as the exponents to

obtain appropriate proportions for the stru ture. For form-nding problems, espe ially for problems

with equality onstraint onditions, a higher exponent for s and t sometimes provides ner result.

For further dis ussion, the interested reader is referred to [46. With higher exponents, the physi al

meaning of energy fun tion be omes rather un lear. However, by following a systemati stati

approa h of stati s, the onverged stress state under stati equilibrium is further laried, as

des ribed in se tion 5.2.

With the spe i design parameters used for Figure 17 (f), we tested different damping

approa hes. When no damping is applied, (i.e. γ = 1.0), onservation of energy is observed, as

shown in Figure. 18. When using vis ous (γ = 0.98), kineti , or drift damping, the geodesi DR

method onverged to a stable equilibrium state after 5000 iterations. This took 3 minutes and

30 se onds with a Core i5 2.56 GHz. Further plots of history of energies and norm of proje ted

gradient, φ (ω), are given in Figure 19. The plots in Figure 19 learly show that the shapes of

history urves of the geodesi DR pro ess, with drift damping, are smooth and approximate those

of vis ous damping. However, the onvergen e rate is more omparable to that of kineti damping.

Additional plots about pull ba k operator in the geodesi DR pro ess are given in Figure 20. Figure

20 further shows that the residual of the onstraint onditions redu es to a suf iently small amount

shortly after the geodesi DR pro ess starts. Ex ept for the initial phases of the onvergen e pro ess,

only one single iteration is needed for pull ba k in the later phases.

5.2. Stress analysis

When the geodesi DR pro ess onverges, a stable equilibrium state is a hieved. The stress state

in this parti ular equilibrium ondition an be further analyzed. The obtained stress state only

implies a ratio of for es in a onstru ted (pre-tensioned) stru ture, but it does not dis lose the

absolute magnitude of those for es. Moreover, the ratios between the for es in the onstru ted

(pre-tensioned) stru ture might be different from those in the form-found system, where titious

material properties are used. In general, the pre-stress in a real tensile surfa e realisti ally depends

on a number of parameters su h as the stiffnesses of the hosen te hni al textile, the method of

pre-stressing the membrane (whi h an in lude more than one pre-stressing devi e), and the way

the different stru tural omponents are onne ted. Nevertheless, we demonstrate the stress analysis

for the problem dis ussed in se tion 5.1 be ause of its dida ti value. The for e ratios and stress

eld obtained by the following stress analysis would be, at least, onsulted in the later engineering

stages.



21

(a) Initial configuration used for x0. (b) Numerical modling of Tanzbrunnen.

Strain energy is given.

Total length is constrained.

Length is constrained.

(c) Design parameters.

(f) Numerical result with (s=2, t=2, ws=0.8, w

1=1, w

2=12,

L1=4.69, L

2=1.764, L

3=5.176, L

4=5.357, L

5=5.092).

(d) Numerical result with (s=1, t=1, ws=0.8, w

1=5, w

2=12,

L1=4.69, L

2=1.764, L

3=5.176, L

4=5.357, L

5=5.092).

(e) Numerical result with (s=1, t=2, ws=0.8, w

1=5, w

2=12,

L1=4.69, L

2=1.764, L

3=5.176, L

4=5.357, L

5=5.092).

1

2

2

34

5

1

ss

A node that is pinned

in all three directions.mast

stay cable 1

staycable 2

9[m]4[m]

5[m]

Figure 17. Numeri al model of Tanzbrunnen (F. Otto, 1956, Kologne, Germany): (a) an initial onguration

dimensions are expressed in [m, (b) strain energies and onstraint onditions given to the elements, ( )

design parameters to ontrol the size of the elements, (d) a numeri al result with a spe i ombination of

design parameters.



22

Total Energy E=f+K

Kinetic energy K

E

K

f

Time step#

Energy

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000

Figure 18. Energy onservation law observed in the form nding pro ess (β = 0.001, γ = 1.0, N = 50,ξ = 0.0001).

(a) Viscous damping (γ=0.98) (b) Kinetic damping (c) Drift damping


Time step#

Energy Energy Energy

Time step# Time step#

Time step# Time step# Time step#

Total Energy E=f+K

Kinetic energy K

E E E

K K K

Norm ofprojected gradient



0

50

100

150

200

0

1000

2000

3000

4000

5000

0

50

100

150

200

0

1000

2000

3000

4000

5000

0

50

100

150

200

0

1000

2000

3000

4000

5000

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

0

1000

2000

3000

4000

5000

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

0

1000

2000

3000

4000

5000

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

0

1000

2000

3000

4000

5000

Figure 19. Plots of the form nding pro ess using geodesi DR method with (a) vis ous (γ = 0.98), (b)kineti , and ( ) drift damping. Top: Plots of energies versus number of iterations. The shape of the total

energy urve in ( ) is smooth and lose to (a), but its onvergen e rate is rather loser to (b). Bottom: Plots

of the norm of proje ted gradient φ (ω) versus number of iterations (0 is optimum). Drift damping exhibits

equal ef ien y to kineti damping.

The variation of Lagrangian omposed of the problem takes the form

δL =∑

j∈D

σjδSj +∑

j∈E,F,G

njδLj = 0, (45)



23





Number of iterationsin pull back

Norm ofconstraint residual

Norm ofconstraint residual

Norm of

constraint residual



0

5

10

15

20

25

30

35

40

0

1000

2000

3000

4000

5000

0

5

10

15

20

25

30

35

40

0

1000

2000

3000

4000

5000

0

5

10

15

20

25

30

35

40

0

1000

2000

3000

4000

5000

1E-15

1E-13

1E-11

1E-09

1E-07

1E-05

0.001

0.1

10

0

1000

2000

3000

4000

5000

1E-15

1E-13

1E-11

1E-09

1E-07

1E-05

0.001

0.1

10

0

1000

2000

3000

4000

5000

1E-15

1E-13

1E-11

1E-09

1E-07

1E-05

0.001

0.1

10

0

1000

2000

3000

4000

5000

Figure 20. Plots about pull ba k operator in the same analysis as Figure 19. Top: number of iterations

performed in pull ba k operator in ea h step of geodesi DR pro ess with (a) vis ous (γ = 0.98), (b) kineti ,and ( ) drift damping. Ex ept for the rst step, only one single iteration is required in most of later steps.

Bottom: Norm of residual of onstraint onditions (0 is optimum). With any of three damping approa hes,

the satisfa tion of onstraint onditions is guaranteed shortly after omputation starts.

where δ is a variational operator with respe t to x, and G is a set of linear elements in the onstraint

onditions. This means that the variations of the terms with respe t to onstraint onditions, whi h

have the form of either λkδLk or λk∑

f δLf , is merged with the se ond term be ause they have the

same form as the se ond term. In terms of stati me hani s, Equation (45) is alled the prin iple of

virtual work. In Equation (45), it an be observed that σj represents the magnitude of the surfa e

stress a ting on the j-th triangular element, and nj , the magnitude of the axial for e a ting on the

j-th linear element.

Comparing Equation (45) with the variation of ws

∑

j∈D S2j that was used in the form-nding

analysis, we obtain

σj = 2wsSj . (46)

This expression means that the surfa e stress is proportional to element area and ws =1

2

σj

Sj, whi h

is a half of the surfa e stress density dened in [44. Similarly, omparing Equation (45) with the

variations of w1

∑

j∈E L2j and w2

∑

j∈F L2j , we obtain

nj = 2w1Lj for j ∈ E, and nj = 2w2Lj for j ∈ F, (47)

whi h are the axial for es a ting in the linear elements on the outer and inner rings, respe tively.

Therefore, the axial for e is proportional to element length and w = 1

2

nj

Lj, whi h is half of the for e

density dened in [45. For the linear elements in the onstraint onditions, in whi h either ea h

length of the linear element or total length of a set of linear elements is onstrained to a pres ribed

value, we obtain

nj = λ1, · · · , ornj = λ5 (48)



24

whi h also represent axial for es. As su h, the multipliers generally represent the magnitudes of

rea tion for es produ ed by stru tural omponents whi h are modeled using equality onstraint

onditions. The multipliers have a negative sign for the ompression mast and a positive sign for

the tension elements. This sign onvention is not applied in any stage of the form-nding pro ess

and should be he ked after the geodesi DR pro ess has onverged. On the ontrary, when values

greater than 0 are given to the weight oef ients, Equations (46) and (47) are guaranteed to have

the appropriate sign, whi h indi ates that the elements are in a tension state.

The stress tensor eld in the elements an be further analyzed as follows. We write Aij for thematrix whose elements are denoted by Aij . A matrix with lower indi es, su h as Aij, is used to

represent the ovariant omponents of a 2nd-order tensor. On the other hand, a matrix with upper

indi es, su h as

Bij

, is used to represent ontra-variant omponents of a tensor. Additionally,

we employ alternative notations with Einstein summation onvention [47,48, in whi h summation

symbols are omitted be ause double indi es usually implies summation.

First, we assume that a natural oordinate system

(

θ1)

or

(

θ1, θ2)

[49 is adequately setup on a

linear or triangular element and denote the domain of

(

θ1)

or

(

θ1, θ2)

on ea h element Ω. Then, forea h linear or triangular element, we dene the metri tensors (or rst fundamental forms) gij by

gij = [g11] or gij =

[

g11 g12g21 g22

]

, (49)

where gij = gi · gj and gi represent natural ( ovariant) basis on the elements. For ea h linear or

triangular element, the length and the area of the elements are dened by

L =

∫

Ω

dv1, S =

∫

Ω

dv2, (50)

where dv1 and dv2 are dened in a ommon form as

dvN =√gdθ1 · · · dθN (51)

and g is the determinant of gij.In addition, we dene inverse matri es of gij,

gij

=

[

1

g11

]

or

gij

=1

g

[

g22 −g12−g21 g11

]

. (52)

In order to al ulate the variations of L and S systemati ally, we use the relation δdvN =1

2

gαβ

: δgαβ dvN , where we dene the inner produ t between two matri es by

Aαβ

:

Bαβ =∑

αβAαβBαβ . It should be noted that the ranges of indi es hange a ording to the

dimensions of the element, i.e. α, β ∈ 1 for linear elements and α, β ∈ 1, 2 for triangular

elements. When the Einstein summation onvention is used, this is simply written as δdvN =1

2gαβδgαβdv

N. As a result, we obtain

δL =1

2

∫

Ω

gαβ

: δgαβ dv1, δS =1

2

∫

Ω

gαβ

: δgαβ dv2. (53)

The virtual work done by Cau hy stress tensor takes the form

δw =1

2

∫

Ω

Tαβ

: δgαβ dvN , (54)

where Tαβis the ontra-variant omponent of the Cau hy stress tensor denoted by T =

∑

α

∑

β Tαβgα ⊗ gβ , where ⊗ represents a dyadi (or tensor) produ t of the base ve tors. When

the Einstein summation onvention is used, it is simply denoted by T = Tαβgα ⊗ gβ .

Comparing Equations (53) and (54) with the virtual works, that were denoted by δwj = njδLj or

δwj = σjδSj ignoring the summation onvention, we obtain

T = njI or T = σjI. (55)



25

where I =∑

α

∑

β gαβgα ⊗ gβ is a unit tensor dened on ea h element. When the Einstein

summation onvention is used, it is denoted by I = gαβgα ⊗ gβ .

If all the triangle elements take the same value for σj at the same time, the global membrane stress

eld, over all the triangle elements, an be des ribed by T = σI with a unique ommon s alar σ.This representation is a uniformly distributed, isotropi stress eld, whi h is preferred in the early

stage of design pro ess of membrane stru tures [50. This spe ial stress eld is also known as the

minimal surfa e stress eld, observed in physi al soap lm experiments. The obtained membrane

stress σj = 2wsSj indi ates that elements that have different element areas take different values

for σj . However, doe to the minimization pro ess of

∑

j∈D S2j , the deviation of element areas in

the obtained result is very small, and, hen e, the obtained shape is expe ted to be lose to the one

obtained by a physi al minimal surfa e soap lm models.

6. CONCLUSIONS AND AVENUES FOR FURTHER RESEARCH

In this paper, the geodesi DR method was formulated and dis ussed. This approa h is a novel

extension of the existing DR method be ause it allows us to in orporate equality onstraint

onditions in for e-modelled systems. In both the existing and geodesi DR methods, the total

energy is preserved when no damping is given.

Drift damping was also introdu ed as a new damping approa h. Drift damping is essentially

a ombination of two existing typi al damping te hniques, i.e., vis ous and kineti damping. This

approa h represents a novel way to dynami ally adjust the damping oef ient between a eleration

and de eleration. The drift damping te hnique inherits the smoothness from the vis ous damping

and dynami adjustment feature from the kineti damping. Hen e, the drift damping is as robust as

vis ous damping and as ef ient as kineti damping. In both existing and geodesi DR methods,

vis ous, kineti and drift damping an be used.

The existing DR generates straight lines by setting the external for e to zero and damping

oef ient to one (i.e., no damping is given). In ontrast, the geodesi DR generates geodesi s,

whi h are a natural extension of straight lines to urved spa es. As an interesting byprodu t, the

geodesi DR method an be used to generate geodesi s on impli itly represented surfa es.

The validity of the proposed, novel, geodesi DR method is demonstrated with a ben hmark ase

study of a pre-stressed, ridge-valley system supported by six masts.

Although the equality onstraint onditions in the ase study were limited to length onstraints of

the masts, stay ables and membrane radial ables, the geodesi DR method ould be enhan ed with

other types of equality onstraint onditions. For example, when total volume of air (in the ase of

a pneumati system) or the angle of hinge type joint are onstrained, the Lagrange multipliers an

generate the air pressure a ting on the membrane or moment a ting on the hinge, respe tively.

Finally, be ause energy onservation was observed in the geodesi DR method, we would like to

suggest that the geodesi DR method ould be used to solve dynami problems, in whi h ase the

mass matrix should not be ignored in its formulation.

ACKNOWLEDGMENTS

This resear h was partially supported by the Ministry of Edu ation, Culture, Sports, S ien e and

Te hnology (MEXT) in Japan, Grant-in-Aid for JSPS Fellows, 10J09407, from 2010 to 2011, and

by the Japan Agen y of S ien e and Te hnology / Erato, Igarashi design interfa e proje t from 2012

to 2013. We would like to thank Kendall S hmidt for editing the text.

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A. A BENCHMARK CASEWITHOUT CONSTRAINT CONDITIONS

Although this paper mainly des ribed the extension of the DR method to problems of equilibrium

with onstraint onditions, we provided a exible framework for damping, whi h is also bene ial

for problems without onstraint onditions. However, other than for kineti damping, the DRmethod

hanges its performan e a ording to the adjustable parameters in the vis ous and drift damping

formulations (equations (13) and (15) respe tively). Those adjustable parameters are (i) the onstant

value for γ in the vis ous damping and (ii) the γ − θ relation in the drift damping.

In this se tion, by using a simple ase study, we further larify the performan e of the DR

method with different parameter hoi es in ea h damping approa h. Figure 21 (a) shows the initial

onguration of the ben hmark model, whi h lies in the x-y plane. The model has 25×25=625 nodes

(in luding the 4 orner nodes thatare pinned in all dire tions) and 24×25×2=1200 linear elements.

The energy fun tion to be minimized is,

f (x) =∑

j

L2j (x) → min . (56)

This ase study has a unique solution whi h is shown by Figure 21 (b). As f (x) is a quadrati

fun tion of x, the same problem an be solved by a single omputation of a linear system of

equations. This linear method is known as the For e Density Method (FDM) [45. Using this

method the same geometry shown in Figure 21 (b) is obtained. The result by FDM is thought to

be suf iently pre ise and the absolute value of f (x) at the solution by FDM is 6130.0579.

For this ase study, Figure 22 (a) shows the history urves of the total and kineti energy and the

norm of gradient of f in a DR pro ess with kineti damping. We used β = 0.1 for the time step and

the same number was used for the other damping approa hes. The absolute value of f (x) at the step2000 is 6130.058. Be ause the kineti damping has no parameters, there is only a unique trial. Sin e

there is no need to adjust parameters, this damping approa h, with solely kineti damping, might

be preferred to the other approa hes. However, the other damping approa hes, espe ially the drift

damping, have the potential to be superior to the kineti damping in terms of onvergen e ef ien y

due to its adjustable parameter.

Figure 22 (b) and ( ) show plots of the time step versus the total and kineti energy and the

norm of gradient for the un onstrained ase study for 4 variations for both the vis ous and the drift

damping approa hes. The absolute value of f (x) at the step 2000 is 6130.072, for Figure 22 (b-1)and (b-2), and 6130.058 for ( -3), (a-1), (a-2) and (a-3). Figure 22 (b-4) and ( -4) are not judged

as onverged at time step 2000. In addition, FDM solved the problem in 280ms while 8 se onds

of omputation was needed for 2000 step iterations of DR. Hen e, FDM is mu h faster than DR



28

(a) Initial configuration

Linear element

A node that is pinned

in all directions.

80

(b) Solution

80

Figure 21. The ase study model: (a) Initial onguration. (b) Minimization result.

method. However, FDM an only solve types of quadrati problems like Equation (56) while DR

an solve more general nonlinear problems. Furthermore, while FDM reates 621×621 matrix, DR

only reates 1242×1 ve tors. Hen e, DR is not as memory onsuming as FDM. Comparing and

ontrasting the results of these 9 analyses (1 kineti , 4 vis ous and 4 drift damping), yield the

following observations:

Although the analysis using the kineti damping approa h demonstrates superior performan e

in terms of redu ing the norm of gradient, the peak of kineti energy is 4 to 6 times larger

ompared to the peaks in drift damping ( -1) and ( -2), whi h exhibit lose onvergen e

ef ien y to the kineti damping.

As shown in Figure 22 (b-2), an approa h that adopts vis ous damping, with a spe i

parameter giving the best performan e in the rst stage of the minimization, performs poorly

in the later steps. This observation suggest that the dynami adjustment in luded in the kineti

and drift damping approa h is desirable in the DR pro ess.

Although the drift damping requires an adjustment of γ − θ relation urve, it is possible to

a hieve a performan e lose to the one obtained by kineti damping. However, employing

this lose drift damping approa h yields kineti energy peaks that are onsiderably smaller

than the peaks obtained using kineti damping.



29

Total Energy E=f+K

Kinetic energy K

(a) Kinetic Damping

(b) Viscous Damping

(c) Drift Damping

(c-2) γ=θ/20+0.95(c-1) γ=θ/10+0.90 (c-3) γ=θ/50+0.98 (c-4) γ=θ/500+0.998

(b-1) γ=0.90 (b-2) γ=0.95 (b-3) γ=0.98 (b-4) γ=0.998

Energy

Norm of

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Norm of

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Figure 22. Results of a ben hmark test: (a) History urves with kineti damping. (b) History urves with

vis ous damping with different parameters. ( ) History urves with drift damping with different γ − θrelations.



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