form finding for complex structures using evolutionary structural optimization method

Form finding for complex structuresusing evolutionary structuraloptimization method

Y.M. Xie, P. Felicetti and J.W. Tang, School of Civil and Chemical

Engineering, RMIT University, GPO Box 2476V, Melbourne 3001,

Australia

M.C. Burry, Spatial Information Architecture Laboratory, School of

Architecture and Design, RMIT University, GPO Box 2476V,

Melbourne 3001, Australia

This paper demonstrates the effectiveness of the Evolutionary Structural

Optimization (ESO) method in developing conceptual forms of complex

structures. A three-dimensional ESO computer code has been developed

which is capable of analysing and optimizing structures of any geometries

and loading conditions. The technique has been tested on a wide range of

examples. In this paper the ESO method is compared to the hanging

model approach used by the architect Antonio Gaudı́.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: artificial evolution, design technique, engineering design,

computer aided design

The renowned architect Antonio Gaudı́ (1852e1926) developed

a number of his architectural designs through the use of

funicular structural systemsdthe so called hanging models.

There has been extensive research on the hanging model approach. One

example is shown in Figure 1 below (Tomlow, 1989).

To describe the concept of funicular structural systems, it is worth

quoting a paragraph from Schodek (1992):

‘A cable subjected to external loads will obviously deform in a way

dependent on the magnitude and location of the external forces. The form

acquired is often called the funicular shape of the cable (the term

funicular is derived from the Latin word for ‘rope’). Only tension forces

will be developed in the cable. Inverting the structural form obtained will

yield a new structure that is exactly analogous to the cable structure

except that compression rather than tension forces are developed.

Theoretically, the shape found could be constructed of simply stacked

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elements that are non rigidly connected (a ‘compression chain’) and the

resultant structure would be stable.’

The funicular shape of the ‘cable’ is load dependent. Variations of the

shape will occur as the cable is loaded with point loads, uniformly

distributed loads etc. For example, a cable supporting a uniformly

distributed loading will deform to a catenary shape, whilst that

supporting point loads only will deform to a series of straight line

segments.

There has been considerable debate as to whether Gaudı́’s complex

forms were based on rational structural methods or based on aesthetics.

However, there is evidence that he utilized geometrical methods of

resolving forces and proportioning the elements of his building designs.

Also Gaudı́ did not take horizontal action into account, but designed

and built the towers of the Nativity for example of sections that made

equilibrium possible without the need for buttresses, regardless of any

wind load.

It is the objective of the authors of this paper to demonstrate that the

method of Evolutionary Structural Optimization (ESO) (Xie and

Steven, 1997), developed approximately 100 years after Gaudı́’s first

experiments with hanging models, is a contemporary digital procedure

that can produce novel forms of funicular structures. Other digital

Figure 1 Reconstruction of

Gaudı́’s hanging model (Tom-

low, 1989) (Photo courtesy

of Institute for Lightweight

Structures, Stuttgart.)

56 Design Studies Vol 26 No. 1 January 2005

Form finding for comp

methods of structural topological optimization, such as homogenization

method (Bendsøe and Kikuchi, 1988; Suzuki and Kikuchi, 1991), can

also be used for form finding of structures. Comparatively, ESOmethod

is simple in concept and effective in application. The authors of this

paper modified the original ESO method pioneered by Xie and Steven

during the early 1990s. The original ESO method consisted of removing

redundant material at each iteration using von Mises stress criterion.

The modifications are enhancements to the method. Instead of von

Mises criterion, there are now options of optimizing structures such that

all remaining elements are in compression, or optimizing such that all

remaining elements are in tension.

1 ESO methodThe automated procedure of ESO is such that each iteration consists of

a finite element analysis (to determine stresses) and then removing

inefficient/inappropriate elements. The process is summarized as

follows:

(1) Execute finite element analysis and output element stresses;

(2) Define element stress se using different design criteria

se ¼

sVM; von Mises criterion

s11Cs22Cs33; tension criterion

�s11 �s22 �s33; compression criterion

8>><>>:

ð1Þ

where sVM is element von Mises stress, s11, s22 and s33 are element

principle stresses, and s11Rs22Rs33.

(3) Sort all elements in design domain by se in descending order. Then

the maximum stress is

smax ¼ seð1Þ ð2Þ

(4) Find the first threshold stress s1th according to volume removal rate

VR. If n � NEð1� VRÞ, then element n is the threshold element

from the total number of elements NE according to VR, and

s1th ¼ seðnÞ ð3Þ

(5) Find the second threshold stress s2th according to rejection ratio

RR.

lex structures using evolutionary structural optimization method 57

s2th ¼ smax �RR ð4Þ

(6) The actual threshold stress that satisfies both RR and VR is

sth ¼minðs1th;s

2thÞ ð5Þ

(7) Delete elements if se!sth;

(8) Repeat steps (1)e(7) until the maximum iteration number is

reached, or a given percentage of volume has been removed from

the ground structure.

For example, in order to design compression-only structures, the

tension-dominant elements are improper for the design condition, and

therefore are first removed. The elements under compression but at low

stress levels are considered as inefficient, and should be gradually

deleted as well. The cycle (iteration) of finite element analysis and

element removal is repeated many times until a desired structural form

is obtained. Typically, the number of iterations ranges from 10 to 100.

2 ExamplesSeveral examples of 2D and 3D continua are presented.

2.1 Example 1d2D tension-only exampleA structure with self-weight is hanging from two pins, as shown in

Figure 2. Using the tension criterion the structure evolves gradually

towards a catenary (Figure 3(a)e(f)). Not only does the final design fit

a theoretical catenary well, the sizes of every part of the structure have

also been determined to achieve a fully stressed design. This is an

example of hanging chain or hanging rope. With the help of the

modified ESO method the funicular structure can be obtained easily.

Figure 2 Catenary model


2.2 Example 2dTension and compression designof a 2D modelA square structure is fixed at its four corners. The point load F is applied

as shown in Figure 4. Gravity is not considered in this case. Using

different design criteria, we can reach completely different results. The

square evolves to two rods connected to the upper pins when the

compression criterion is used (Figure 5(a)e(c)). However, two different

rods joining the lower pins are achieved from the initial square when

using the tension criterion (Figure 6(a)e(c)). The material is distributed

properly according to the design rules. Using the conventional ESO

method based on von Mises stress the resulting structure would have

been connected to all of its four corners.

Figure 3 Evolution of a cate-

nary model

a

2a 2a

F

4a

Figure 4 Tension and com-

pression model

Form finding for complex structures using evolutionary structural optimization method 59

2.3 Example 3dBasic block domain with foundationconstraints and point loads on the top of the towerand on each shoulder corner

2.3.1 Base model dimensions and loadingof Example 3To demonstrate the modified ESO technique a model domain is adopted

that represents a cathedral type building. An accurate representation of

Gaudı́’s model is not necessary. The base building consists of a ‘dome’

sitting atop a large cubic block, which in turn is supported by four

massive cubic pillars. Support foundations are limited to two outer

perimeters and a central heel of each pillar. The bell tower is represented

by a 1000 MN point load acting atop the ‘dome’, and four ‘pinnacles’ at

each corner by 200 MN loads. Overall dimensions of the model are 60 m

by 60 m at the base and 130 m in height. In this example (as well as in

Examples 4 and 5), the compression criterion as defined in Eq. (1) is used.

Loading to the model consisted of gravity load in the vertical direction

and the point loads, as shown in Figure 7. It is noted that the values of

these point loads are excessive, the reason being to make the effect of

gravity load negligible in this case. Design for gravity dominated loading

will be considered in Example 5.

2.3.2 Evolutionary behaviour of Example 3A summary of the results of the process is given below (Figures 8e14):

(i) As the base building domain is not constrained by ‘non-design’

areas, all elements are free to evolve.

Figure 5 Evolution to a com-

pression-only structure

Figure 6 Evolution to a ten-

sion-only structure


(ii) The point loads provide pre-compression to the zones directly

under the loads, and therefore due to the high compressive stresses

they are the zones least affected by the evolution process.

(iii) Similarly to (ii) above the foundation zones are subjected to high

compressive stress and therefore relatively less elements are

removed in these zones. However, it is interesting to note that

between the ‘heel’ and perimeter footing the resultant structure

spans between the two zones through an arch.

Figure 7 Base building do-

main and plan of foundation

constraints. Shaded region on

plan represents region of

building foundation support

Figure 8 Finite element model

of base building domain


Figure 9 Revised finite ele-

ment geometry after a number

of iterations

Figure 10 Continuing evolu-

tion of form. Note that ‘pin-

nacles’ are developing at

locations of point loads at

the shoulders

Figure 11 ‘Final’ optimized

form


Figure 12 View from above of

optimized structure

Figure 13 View from below of

optimized structural form.

Note in particular arched

shaping to accommodate the

locations of the support foun-

dations

Figure 14 Elevation view of

optimized structural form


(iv) Extending the iteration process to that of Figure 11 results in

a structure that is dominated by resolving the forces of the initial

point loads to the foundations. The base of the resultant structure

splays towards the base.

(v) The resultant tower form is a tower of (almost) uniform

compressive stress.

2.4 Example 4dBasic block domain with foundationconstraints, point loads on the top of the tower and oneach shoulder corner, and non-design domainThe base model is the same as that for Example 3. However, ‘non-design

area’ constraints are applied to the outer two faces of each pillar and to

the base line of the ‘dome’. The non-design areas whilst contributing to

the overall structural behaviour of the model, cannot have its elements

removed, and therefore could be subjected to both compression and

tension stresses (Figure 15).

A summary of the results of the process is given below (Figures 16e21):

(i) As the base building domain has zones constrained by ‘non-design’

areas, not all elements are free to evolve. Although some of them

are low stressed they are not allowed to be removed freely. This



constraints. Shaded region on

axonometric represents re-

gion of ‘non-design’. Shaded

region on plan represents re-

gion of building foundation

support


Figure 16 Finite element

model of base building

domain

Figure 17 Revised geometry

after a number of iterations

Figure 18 The ‘non-design’

regions can clearly be seen in

this iteration


Figure 19 The optimized

structural form, with ‘non-

design’ structural elements

Figure 20 View from above

Figure 21 View from below


results in a different form to that of Example 3. In this example the

non-design areas act more as non-structural attachment. But in

some cases the existence of non-design areas could change the

route of structural evolution; therefore the ‘non-design’ could

make the final topology quite different.

(ii) Similar to Example 3, the point loads provided pre-compression to

the zones directly under the loads, and therefore due to the high

compressive stresses they are the zones least affected by the

evolution process.

(iii) Similarly to Example 3, the foundation zones are subjected to high

compressive stress and therefore relatively fewer elements are

removed in these zones. However, it is interesting to note that

between the ‘heel’ and perimeter footing the resultant structure

spans between the two zones through an arch.

(iv) The resultant tower form is a tower of (almost) uniform

compressive stress (except for the non-design areas).

2.5 Example 5dBasic block domain with foundationconstraints, no point loadsThis example is identical to the model in Example 3. However, it is only

subjected to gravity load in the vertical direction. It demonstrates the

ability of ESO in finding different structural forms to suit different

loading conditions (Figures 22e31).



constraints. Note that no point

loads have been applied to

the building domain. Shaded

region on plan represents

region of building foundation

support



main

Figure 24 Revised structure

after a number of iterations

Figure 25 Further evolution of

the form



tion


tion

Figure 28 ‘Final’ optimized

structural form


Figure 29 View from above

Figure 30 View from below

Figure 31 Elevation view of

optimized structure


2.6 Comments on the results of Examples 3e5From these examples, the following observations and discussions are

made:

(i) The ‘non-design’ areas in these examples have little impact on the

resulting form. However, it is a useful feature in modelling

architectural projects. Clearly, a model could be created initially

that distinguished areas that are fixed from the architect’s

perspective and those that can be optimized structurally.

(ii) The resultant shape of the optimized structure is clearly load

dependant. It is therefore important in seeking to analyse/model

behaviour of projects such as Gaudı́’s hanging model that the

initial design domain and loadings are modelled correctly.

(iii) To reduce computational effort, where there are lines of symmetry

in a proposed model, it is possible to reduce the model to the

symmetrical portion and evolve that portion only. Splicing

together of the elements can then be undertaken to create the

full model.

(iv) Rapid prototyping of the models can be performed easily by

sending the ESO results to a 3D wax printer. Figure 32 shows the

wax prototype of the optimized structural form of Example 3 after

6 h of printing.

Figure 32 3D wax printout of

the result shown in Figure 11


72

3 Concluding remarks and further studies

3.1 Comparison to work undertaken by Gaudı́Examples 3e5 (in particular Example 3) show close correlation to the

work of Gaudı́’s buildings. The ESO procedure serves as a quick and

accurate method of developing rational structural formdone may

speculate that it was the same kind of structural rationality that Gaudı́

aspired to achieve with his hanging models (and his graphic calculation

models). The ESO method creates the geometries based on a first

principles structural design process, rather than seeking to input the

actual geometry of Gaudı́’s buildings into a finite element model and

analysing forces/stresses etc.

3.2 The relevance to architecture and opportunitiesfor further studyThe use of the ESO technique provides an exciting new opportunity for

architectural form finding. Complex forms can be created that are based

on rational structural behaviour. Loads are not restricted to those in the

vertical plane such as in Gaudı́’s hanging models, but can be located in

any direction at any point, such that buildings can be optimized for

horizontal forces such as wind and earthquake in addition to gravity.

AcknowledgementThe authors are grateful to Professor Jos Tomlow for his comments on

the paper. They also wish to thank Professor Jos Tomlow and Gabriela

Heim for the photo shown in Figure 1.

ReferencesBendsøe, M P and Kikuchi, N (1988) Generating optimal topology instructural design using a homogenization method Computer Methods in

Applied Mechanics and Engineering Vol 71 No 2 pp 197e224Schodek, D (1992) Structures 2nd edn, Prentice Hall, New YorkSuzuki, K and Kikuchi, N (1991) A homogenization method for shape and

topology optimization Computer Methods in Applied Mechanics andEngineering Vol 93 No 3 pp 291e318Tomlow, J (1989) The model Institute for Lightweight Structures,

University of Stuttgart, GermanyXie, Y M and Steven, G P (1997) Evolutionary structural optimizationSpringer-Verlag, Berlin

Design Studies Vol 26 No. 1 January 2005

form finding for complex structures using evolutionary structural optimization method

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