Structural Morphology Optimization by Evolutionary Procedures

A. Baseggio1, F. Biondini2, F. Bontempi3, M. Gambini1, P.G. Malerba4

ABSTRACT

The paper deals with the identification of optimal structural morphologies through evolutionary procedures.Two main approaches are considered. The first one simulates the Biological Growth (BG) of naturalstructures like the bones and the trees. The second one, called Evolutionary Structural Optimization (ESO),removes material at low stress level. Optimal configurations are addressed by proper optimality indexesand by a monitoring of the structural response. Design graphs suitable to this purpose are introduced andemployed in the optimization of a pylon carrying a suspended roof and of a bridge under multiple loads.

1. INTRODUCTION

One of the most promising research field which has been recently applied to the identification of optimalstructural morphology deals with evolutionary procedures which operate on the basis of some analogieswith the growing and the evolutionary processes of natural systems. Such methods are based on thesimple concept that by slowly removing and/or reshaping regions of inefficient material, belonging to agiven over-designed structure, its shape and topology evolve toward an optimum configuration.

Two main approaches among those proposed in literature are here considered. In the first one, the structuralmorphology is modified by simulating the Biological Growth (BG) of natural structures like the bones andthe trees (Mattheck & Burkhardt 1990, Mattheck & Moldenhauer 1990). In the second one, called EvolutionaryStructural Optimization (ESO), material at low stress level is removed by degrading its constitutiveproperties (Xie and Steven 1993, 1994). The basic steps of these procedures should be repeated untiloptimal configurations appear. However, to this regards, no well established convergence criteria exist.

In this work, the better structural solutions emerging from the evolutionary process are identified on thebasis of proper optimality indexes and by monitoring the actual structural response. In particular, designgraphs suitable to this purpose are firstly introduced. Subsequently, both BG and ESO methods are brieflyrecalled and these graphs are usefully employed in the selection of the optimal morphology of a pyloncarrying a suspended roof and of a bridge type structure under a multiple load condition. These structuresare considered to be in plane stress and made of linear elastic material having symmetric or non symmetricbehavior in tension and in compression. The structural analyses needed during the evolutionary process arecarried out by a LST-based (Linear Strain Triangle) finite element technique (Baseggio 1999, Gambini 2000).

2. OPTIMALITY INDEXES AND DESIGN GRAPHS

As mentioned, at each step of the evolutionary process the present structure is modified in such a waythat a better configuration with respect to given evolutionary criteria is hopefully achieved. However, theoptimality of such solutions needs often to be judged with respect to design criteria which are not necessarilycoincident with those which regulate the evolution. In this work, design criteria are synthesized by one ormore optimality indexes able to measure the quality of the present solution with respect to the initial one.

It is generally recognized that Nature tends to build structures in such a way that the internal strain energy,or the external work done by the applied loads, is minimum. Based on such consideration, a properoptimality index may be represented by the following Performance Structural Index (Zhao et al. 1997):

WV

WVPSI

⋅⋅

= 00

1 Structural Engineers, Milan, Italy (marco.gambini@tiscalinet.it, ambrogio.baseggio@infostrada.it).2 PhD, Department of Structural Engineering, Technical University of Milan, Italy (biondini@stru.polimi.it).3 Professor, Dept. of Structural and Geotechnical Engineering, University of Rome "La Sapienza", Italy

(bontempi@scilla.ing.uniroma1.it).4 Professor, Department of Civil Engineering, University of Udine, Italy (piergiorgio.malerba@dic.uniud.it).

4th International Colloquium on Structural MorphologyAugust 17–19, 2000, Delft, The Netherlands, 264-271

being W the external work per unit of volume V and where 0 denotes the initial configuration. Thisformulation implicitly refers to a single load condition, but it can be easily extended to account for multipleloads, for example by a weighted average of the contributions iPSI of each load condition i=1,…,NC:

∑∑

∑=

=

= ⋅=⋅

=NC

i

iiNC

ii

NC

i

ii

PSIw

PSIwPSI

1

1

1 ω

Of course, depending on the specific problem to be examined, additional indexes may be introduced. Forexample, structures made of material having low tensile strength, like stone or concrete, should bedesigned by limiting the amount of tensioned material. Thus, by denoting cV the portion of the volume Vwhich is compression dominated (see Fig. 4), the following percentage of Compressed Material Volume:

V

VCMV c=

appears to be as well a meaningful optimality index. Moreover, sometimes may be useful to optimize notonly the mechanical behavior, but also some geometrical properties of the structure. A measure of thepresent free Perimeter Γ of the structural boundary with respect to the initial one Γ0:

0

2ΓΓ

=P

gives for instance an idea about the advantages in terms of cost of formworks and structural durability.In addition, being related to the weight of the structure, the percentage of Removed Material Volume:

0

0

V

VVRMV

−=

may be itself an important indicator about the total structural cost.

After some optimality indexes are selected and eventually grouped in a single averaged index, hierarchicalarrangements of the solutions explored during the evolutionary process become possible. However, someadditional design constraints on the structural response, for example in terms of maximum displacementand maximum stress level, are usually needed to assure the feasibility of the solution which seems toappear optimal. Thus, the best morphology requires to be identified by a monitoring of both the optimalityindexes and the structural response. To this aim, design graphs which contemporarily describe theevolution of all such quantities, for example versus the RMV index as shown in Figure 1, are introduced.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60 70 80 90 100

% Removed Material Volume - RMV

OP

TIM

AL

ITY

IND

EX

0

1

2

3

4

5

6

7

8

9

10A

DIM

EN

SIO

NA

L S

TR

UC

TU

RA

L R

ES

PO

NS

E

OPTIMALITY RANGE

ADIMENSIONAL MAX STRESS

σσ//σσ00

ADIMENSIONAL MAX DISPLACEMENT

S/S0

STRUCTURAL RESPONSE LIMIT

OPTIMALITY INDEX

INITIAL DOMAIN

Figure 1. A typical design graph for evolutionary procedures.

PSI

3. BG PROCEDURES (Biological Growth)

The evolutionary procedures considered in the following work by simulating the Biological Growth (BG)of natural structures (Mattheck & Burkhardt 1990, Mattheck & Moldenhauer 1990). Such structures are knownto evolve by adapting themselves to the applied loads according to the axiom of uniform stress, which statesthat in the optimal configuration the stress field distribution tends to be fairly regular over the structure(Mattheck 1998). Thus, structural shape and topology are gradually modified in such a way that material isadded in the zones with high stress concentrations and removed from under-loaded zones (swelling).

The simplest form of a swelling law able to regulate such modifications is assumed as follows:

DFKdtdV

VnREF

nSW =−⋅== )(

1σσε&

being: SWε& the swelling strain rate; V the evolutionary time-dependent volume V=V(t); K an artificial constant;

σ the actual von Mises stress and REFσ its reference, or far-field, value; n a suitable exponent (n=1 for astress-based and n=2 for a energy-based criterion); DF the Driving Force of the evolutionary process. Basedon this law, the numerical simulation of the growth mechanism is obtained through three steps (Figure 2).

F

BASIC STEP

SWELLING STEP

UPDATE STEP

SW

VM(xi )

u SW(xi)

xi,k+1=x i,k+C u SW(xi)

P0

P0

PSW:∆εSW

SV,SW

SV,0

SV,0

L0=DESIGN CONSTRAINT

L0=DESIGN CONSTRAINT

L0=DESIGN CONSTRAINT

Figure 2. Fundamental steps of the BG evolutionary process.

(1) Basic Step. A finite element analysis is performed to obtain the stress distribution σ over the structure.

(2) Swelling Step. The Driving Forces are firstly computed. In particular, for structure in plane state thefollowing isotropic swelling strain increment vector T

21 ] 011 [SWSW ε∆=∆e is considered for the time

increment ∆t. Based on such strain distribution, the load vector SWf∆ equivalent to swelling is derived

and the corresponding incremental displacement vector SWu∆ is evaluated as follows:

∫ ∆=V

SWT

SW dV

eDBfÄ ⇒ SWsw fuK ∆=∆ ⇒ SWsw fKu ∆=∆ −1

being B the compatibility matrix of the finite element, D the constitutive matrix of the material and Kthe stiffness matrix of the structure. It is worth noting that, in this work, additional geometrical designconstraints are accounted directly by replacing the actual boundary conditions of the swelling modelin such a way that swelling displacements which violate the constraints are not allowed. This concept isshown in Figure 2, where the cantilever beam is forced to maintain its initial length during the evolution.

(3) Update Step. The location ki ,x of each node i=1,…,N of the finite element model at the current

generation k is updated according to the swelling displacements SWu∆ just obtained as follows:

SWuxx ∆+=+ Ci,kki 1,

being C a suitable extrapolation factor which implicitly contains the constant K. Such factor may beeither fixed at the first generation and then considered time-independent, or varied during the evolution.In any case, its value should be chosen to assure noticeable shape variations and progressivelydecreasing driving forces (Mattheck & Moldenhauer 1990).

Such BG procedure is applied to the shape optimization of a pylon carrying a suspended roof. Thegeometry of the initial structure and the load condition are shown in Figure 3.a. Since the distancebetween the supports is retained, the swelling model in Figure 3.b is adopted. The design graph inFigure 3.d shows the progressive convergence of the evolutionary process towards higher level of theoptimality index PSI and lower level of the structural response, while the structural volume remainspractically the same. Noteworthy the end of the pylon tends to lie along the line of action of the resultantof the applied loads. Finally, Figure 3.c allows us to compare the maps of the von Mises stresscorresponding, respectively, to the initial structure and the optimal one, and to appreciate how the latterpresent a nearly uniform distribution of stress having lower maximum intensity.

1000

010

000

1500

15001500 9000

8571

.414

28.6

1000

0

2PP375 375

1000

30°45°

0.0 28.1 0.0 3.2Mpa Mpa

N=0V/V0=1.00PSI=1.00

N=125V/V0=0.97PSI=32.9

σVMσVM

Swelling Model

L0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 25 50 75 100 125 150

N

Ad

imen

sio

nal

str

uct

ura

l res

po

nse

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

Per

form

ance

Str

uct

ura

l In

dex

- P

SI

σσmean

σσmaxSmax

PSI

Volume

N=30

N=50

N=90

N=125

Figure 3. Evolutionary shape optimization of a pylon carrying a suspended roof by a BG procedure.

(a) (c)

(b) (d)

4. ESO PROCEDURES (Evolutionary Structural Optimization)

The Evolutionary Structural Optimization (ESO) procedures modify the topology of a given over-designed structure by slowly removing regions of inefficient material (Xie and Steven 1993, 1994). Theinitial domain is subdivided in finite elements and a structural analysis is carried out. A representativequantity of the structural response, say the von Mises stress, is then evaluated at the element level andcompared with a portion RR (Rejection Ratio) of a reference value, for instance the maximum stressover the whole structure. If such lower limit is not reached (Criterion 1 in Table 1), the material inside thecorresponding elements is considered to be inefficient and it is removed by degrading its constitutiveproperties, typically the Young modulus. The parameter RR, who determines the portion of materialwhich is removed at each step of the evolutionary process, is usually assumed as follows:

SSAASSRR ⋅+= 10)(

being SS an integer counter which is added by a unity whenever a Steady State is reached, while A0

and A1 are numerical constants able to assure a gradual evolution. Proper values seems to be A0 ≅0.0and A1≅0.005. In this work, however, the rate of the process is controlled also by introducing an upperlimit on the percentage of removed material volume VREM at each step (Rate of Removed Material):

maxRRMV

VRRM REM ≤=

Criterion 1 is appropriate for materials having good strength in both tension and compression. However,many structures exhibit low strength in tension, like those made of stone or concrete, or in compression,like those subjected to buckling phenomena. To account for such cases, in which the optimal structuralmorphology should be defined by limiting the amount of material subjected to critical stress states, theconcept of tension and compression dominated material has been introduced (Guan et al. 1999). As shownin Figure 4, material is considered tension (compression) dominated if the maximum (minimum) principalstress is of tension (compression) type. Based on this concept, the actual domain Ω is subdivided ateach step in two parts, ΩT and ΩC, and in each of them the efficiency of the material is verified by using theabsolute values of the principal stresses instead of the von Mises stresses (Criteria 2 and 3 in Table 1).

The criteria just introduced implicitly refer to a single load condition, but can they be easily applied to thecase of multiple loads by removing material only if the rejection criterion is verified for every load condition.

In the basic formulation the minimum portion of removable material is identified with a single finite element.However, it is worth noting that a more general formulation can be achieved if the control of efficiency isperformed on a minimum elimination unit formed by a group of elements. Several grouping criteria areclearly possible. By joining for example two adjacent triangular elements, structural solutions characterizedby more regular boundaries are usually obtained. Moreover, the discrete nature of some structural typeslike masonry can be also better modeled, for instance, by building blocks representing one or more bricks.

Of course, since in each group a different rejection criteria can be considered, the previous approach alsoallows to take the case of non homogeneous structures into account. Finally, by introducing a no-rejectioncriterion, is possible to freeze a sub-region of the initial domain (Non Design sub-domain) which cannot benever removed. This is particularly useful with bridge type structures, where the deck level is usually fixed.

Figure 4. Tension and compression dominated material.

CRITERION ANDMATERIAL TYPE FORMULATION

(1)symmetric

VMVM SSRR max)( σσ ⋅≤

(2)asymmetric with low

tensile strength

0.011 ≥σ and

max,2222 )( σσ SSRR≤

(3)asymmetric with low

compressive strength

0.022 ≤σ and

max,1111 )( σσ SSRR≤

Table 1.Efficiency and rejection criteria forsymmetric and asymmetric material.

The ESO procedure is here applied to the optimization of the structural morphology of a bridge subjectedto six load conditions (Ito 1996). The position of both the deck and the supports is assumed to be fixed anda free space for navigation is provided under the deck. The first window of Figure 5 shows the geometricproportions, the design requirements, the load conditions and the initial domain chosen for the procedure.At first, a cable-stayed scheme is searched for by adding two axially rigid pylons to the Non Design domain.

DESIGN REQUIREMENTS

LOAD CONDITIONS

B/4 B/4 B/4 B/4

B/4

LC 1

LC 2

LC 3

LC 4

LC 5

LC 6

B

H=

0.2

2 B

B

H=

0.2

2 B

0.3 B 0.4 B 0.3 B

0.35

H0.

65 B

t

DESIGN DOMAIN

NON DESIGN DOMAIN

EMPTY SPACE FOR SHIPWAY

A

B

H =

0.22

B

0.3 B 0.4 B 0.3 B

0.35

H0.

65 H

0.04 B 0.04 B

CRITERION 1

B

B

H =

0.22

B

0.3 B 0.4 B 0.3 B

0.35

H0.

65 H

0.04 B 0.04 B

CRITERION 3

C

B

H =

0.22

B0.3 B 0.4 B 0.3 B

0.35

H0.

65 H

0 .04 B 0.04 B

CRITERION 3

CRITERION 2

Figure 5. Some optimal structural morphologies of a bridge type structure.

Windows A, B and C of Figure 5 show some layouts obtained during the evolutionary process for differentmaterial types. By assuming symmetric material (Criterion 1), a balanced arch scheme emerges insteadof the expected one (Figure 5A). To be winning, the cable-stayed scheme should favor tensioned fieldsand then work on asymmetric material having low compression strength (Criterion 3, Figure 5B). However,such a solution tends to anchor some tensioned elements directly on the lateral supports. A more rationalscheme can be achieved if the rejection criteria are differentiated over the structure, for instance byassuming the material under the deck to be asymmetric with low tensile strength (Criteria 2-3, Figure 5C).

Despite of the found solutions, the balanced arch scheme initially obtained should be preferred ifmaterial having low tension strength is used. Figure 6 shows some of the configurations resulting fromthe evolutionary process for the case of symmetric material without pylons. The design graph of Figure 7allows either to appreciate the optimality level of such schemes with reference to several optimalityindexes, or to control the corresponding feasibility of the structural response.

RMV PSI CMV

83% 0.95 1.21

70% 1.42 1.22

75% 1.25 1.22

50% 1.58 1.16

60% 1.57 1.20

23% 1.24 1.08

STRUCTURAL SCHEME

0% 1.00 1.00

H=

0.2

2 B

0.3 B 0.4 B 0.3 B

0.35

H0.

65 H

0.08 B0.08 B

CRITERION 1

Figure 6. Optimal balanced arch schemes for a bridge type structure.

Figure 7. Design graph for the bridge type structure of Figure 6.

5. CONCLUDING REMARKS

The Biological Growth (BG) and the Evolutionary Structural Optimization (ESO) have been applied tomorphology optimization problems. The swelling step of the BG procedures has been extended to takegeometrical constraints into account. A formulation of ESO suitable to deal with asymmetric (tension andcompression dominated) materials, fixed geometrical boundaries and alignments (non design domains)and multiple load conditions has been presented. In designing the morphology, F.E. grouping techniquesallow us to drive the final configurations towards either smooth profiles or segmented boundaries as incase of masonry structures. Such processes may lead to many final optimal choices, as has been shownby an application searching for the optimal structural layout of a bridge having clearance limitations.Among these choices, the final actual optimum may be judged by using suitable design graphs, withreference to design criteria not necessarily coincident with those which control the evolutionary process.

REFERENCES

1. Baseggio A. 1999. A Technique for the Optimization of Structures in Plane Stress. Dissertation.Department of Structural Engineering, Technical University of Milan, Milan, Italy (in Italian).

2. Gambini M. 2000. Identification and Optimization of structural Schemes by Evolutionary Procedures.Dissertation. Dept. of Structural Engineering, Technical University of Milan, Milan, Italy (in Italian).

3. Guan H., Steven G.P., Querin O.M., Xie Y.M. 1999. Optimisation of Bridge Deck Positioning by theEvolutionary Procedure. Structural Engineering and Mechanics 7(6), 551-559.

4. Ito M. 1996. Selection of Bridge Types from a Japanese Experiences. Proc. of IASS InternationalSymposium on Conceptual Design of Structures, University of Stuttgart, Stuttgart, 1, 65-72.

5. Mattheck C., Burkhardt S. 1990. A New Method of Structural Shape Optimization based onBiological Growth. Int. J. of Fatigue. 12(3), 185-190.

6. Mattheck C., Moldenhauer H. 1990. An Intelligent CAD-Method based on Biological Growth. FatigueFract. Engng. Mat. Struct. 13(1), 41-51.

7. Mattheck C. 1998. Design in Nature. Learning from Trees. Springer Verlag.8. Xie Y.M, Steven G.P. 1994. Optimal Design of Multiple Load Case Structures using an Evolutionary

Procedure. Engineering Computation, 11, 295-302.9. Xie Y.M., Steven G.P. 1993. A Simple Evolutionary Procedure for Structural Optimization.

Computers & Structures, 49(5), 885-896.10. Zaho C., Hornby P., Steven G.P., Xie Y.M. 1998. A Generalized Evolutionary Method for Numerical

Topology Optimization of Structures under Static Loading Conditions. Structural Optimization, 15, 251-260.