This paper explores optimization of structural morphologies using evolutionary procedures, focusing on two main approaches: biological growth simulation and evolutionary structural optimization (ESO). The study illustrates how these methods can effectively refine structural designs by removing inefficient material and adapting shapes based on stress distribution, with applications demonstrated through the optimization of a pylon and a bridge. The results highlight the importance of iterative processes in achieving optimal configurations, emphasizing the role of proper optimality indexes and design graphs.

1. 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 natural
structures 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 indexes
and by a monitoring of the structural response. Design graphs suitable to this purpose are introduced and
employed 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 optimal
structural morphology deals with evolutionary procedures which operate on the basis of some analogies
with the growing and the evolutionary processes of natural systems. Such methods are based on the
simple concept that by slowly removing and/or reshaping regions of inefficient material, belonging to a
given 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 structural
morphology is modified by simulating the Biological Growth (BG) of natural structures like the bones and
the trees (Mattheck&Burkhardt 1990, Mattheck&Moldenhauer 1990). In the second one, called Evolutionary
Structural Optimization (ESO), material at low stress level is removed by degrading its constitutive
properties (Xie and Steven 1993, 1994). The basic steps of these procedures should be repeated until
optimal 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 the
basis of proper optimality indexes and by monitoring the actual structural response. In particular, design
graphs suitable to this purpose are firstly introduced. Subsequently, both BG and ESO methods are briefly
recalled and these graphs are usefully employed in the selection of the optimal morphology of a pylon
carrying a suspended roof and of a bridge type structure under a multiple load condition. These structures
are considered to be in plane stress and made of linear elastic material having symmetric or non symmetric
behavior in tension and in compression. The structural analyses needed during the evolutionary process are
carried 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 way
that a better configuration with respect to given evolutionary criteria is hopefully achieved. However, the
optimality of such solutions needs often to be judged with respect to design criteria which are not necessarily
coincident with those which regulate the evolution. In this work, design criteria are synthesized by one or
more 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 proper
optimality index may be represented by the following Performance Structural Index (Zhao et al. 1997):
WV
WV
PSI
⋅
⋅
= 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 Morphology
August 17–19, 2000, Delft, The Netherlands, 264-271

2. being W the external work per unit of volume V and where 0 denotes the initial configuration. This
formulation implicitly refers to a single load condition, but it can be easily extended to account for multiple
loads, for example by a weighted average of the contributions i
PSI of each load condition i=1,…,NC:
∑
∑
∑
=
=
=
⋅=
⋅
=
NC
i
i
iNC
i
i
NC
i
i
i
PSI
w
PSIw
PSI
1
1
1
ω
Of course, depending on the specific problem to be examined, additional indexes may be introduced. For
example, structures made of material having low tensile strength, like stone or concrete, should be
designed by limiting the amount of tensioned material. Thus, by denoting cV the portion of the volume V
which is compression dominated (see Fig. 4), the following percentage of Compressed Material Volume:
V
V
CMV c
=
appears to be as well a meaningful optimality index. Moreover, sometimes may be useful to optimize not
only the mechanical behavior, but also some geometrical properties of the structure. A measure of the
present free Perimeter Γ of the structural boundary with respect to the initial one Γ0:
0
2
Γ
Γ
=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
VV
RMV
−
=
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, hierarchical
arrangements of the solutions explored during the evolutionary process become possible. However, some
additional design constraints on the structural response, for example in terms of maximum displacement
and maximum stress level, are usually needed to assure the feasibility of the solution which seems to
appear optimal. Thus, the best morphology requires to be identified by a monitoring of both the optimality
indexes and the structural response. To this aim, design graphs which contemporarily describe the
evolution 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
OPTIMALITYINDEX
0
1
2
3
4
5
6
7
8
9
10 ADIMENSIONALSTRUCTURALRESPONSE
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. 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 known
to evolve by adapting themselves to the applied loads according to the axiom of uniform stress, which states
that 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 is
added 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:
DFK
dt
dV
V
n
REF
n
SW =−⋅== )(
1
σσε&
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 a
stress-based and n=2 for a energy-based criterion); DF the Driving Force of the evolutionary process. Based
on 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=xi,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 the
following isotropic swelling strain increment vector T
2
1
]011[SWSW ε∆=∆e is considered for the time
increment ∆t. Based on such strain distribution, the load vector SW
f∆ equivalent to swelling is derived
and the corresponding incremental displacement vector SWu∆ is evaluated as follows:
∫ ∆=
V
SW
T
SW
dVeDBfÄ ⇒ SWsw
fuK ∆=∆ ⇒ SWsw fKu ∆=∆ −1
being B the compatibility matrix of the finite element, D the constitutive matrix of the material and K
the stiffness matrix of the structure. It is worth noting that, in this work, additional geometrical design
constraints are accounted directly by replacing the actual boundary conditions of the swelling model
in such a way that swelling displacements which violate the constraints are not allowed. This concept is
shown 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:

4. SWuxx ∆+=+ Ci,kki 1,
being C a suitable extrapolation factor which implicitly contains the constant K. Such factor may be
either 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 progressively
decreasing driving forces (Mattheck & Moldenhauer 1990).
Such BG procedure is applied to the shape optimization of a pylon carrying a suspended roof. The
geometry of the initial structure and the load condition are shown in Figure 3.a. Since the distance
between the supports is retained, the swelling model in Figure 3.b is adopted. The design graph in
Figure 3.d shows the progressive convergence of the evolutionary process towards higher level of the
optimality index PSI and lower level of the structural response, while the structural volume remains
practically the same. Noteworthy the end of the pylon tends to lie along the line of action of the resultant
of the applied loads. Finally, Figure 3.c allows us to compare the maps of the von Mises stress
corresponding, respectively, to the initial structure and the optimal one, and to appreciate how the latter
present a nearly uniform distribution of stress having lower maximum intensity.
1000010000
1500
15001500 9000
8571.41428.610000
2PP
375 375
1000
30°
45°
0.0 28.1 0.0 3.2
Mpa Mpa
N=0
V/V0=1.00
PSI=1.00
N=125
V/V0=0.97
PSI=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
Adimensionalstructuralresponse
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
PerformanceStructuralIndex-PSI
σσmean
σσmax
Smax
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)

5. 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). The
initial domain is subdivided in finite elements and a structural analysis is carried out. A representative
quantity of the structural response, say the von Mises stress, is then evaluated at the element level and
compared with a portion RR (Rejection Ratio) of a reference value, for instance the maximum stress
over the whole structure. If such lower limit is not reached (Criterion 1 in Table 1), the material inside the
corresponding elements is considered to be inefficient and it is removed by degrading its constitutive
properties, typically the Young modulus. The parameter RR, who determines the portion of material
which 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 ≅0.0
and A1≅0.005. In this work, however, the rate of the process is controlled also by introducing an upper
limit on the percentage of removed material volume VREM at each step (Rate of Removed Material):
maxRRM
V
V
RRM 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 structural
morphology should be defined by limiting the amount of material subjected to critical stress states, the
concept of tension and compression dominated material has been introduced (Guan et al. 1999). As shown
in Figure 4, material is considered tension (compression) dominated if the maximum (minimum) principal
stress is of tension (compression) type. Based on this concept, the actual domain Ω is subdivided at
each step in two parts, ΩT and ΩC, and in each of them the efficiency of the material is verified by using the
absolute 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 the
case 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 is
performed on a minimum elimination unit formed by a group of elements. Several grouping criteria are
clearly possible. By joining for example two adjacent triangular elements, structural solutions characterized
by more regular boundaries are usually obtained. Moreover, the discrete nature of some structural types
like 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 also
allows to take the case of non homogeneous structures into account. Finally, by introducing a no-rejection
criterion, is possible to freeze a sub-region of the initial domain (Non Design sub-domain) which cannot be
never removed. This is particularly useful with bridge type structures, where the deck level is usually fixed.
Figure 4. Tension and compression dominated material.
CRITERION AND
MATERIAL 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 for
symmetric and asymmetric material.

6. The ESO procedure is here applied to the optimization of the structural morphology of a bridge subjected
to six load conditions (Ito 1996). The position of both the deck and the supports is assumed to be fixed and
a free space for navigation is provided under the deck. The first window of Figure 5 shows the geometric
proportions, 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=0.22B
B
H=0.22B
0.3 B 0.4 B 0.3 B
0.35H0.65B
t
DESIGN DOMAIN
NON DESIGN DOMAIN
EMPTY SPACE
FOR SHIPWAY
A
B
H=0.22B
0.3 B 0.4 B 0.3 B
0.35H0.65H
0.04 B 0.04 B
CRITERION 1
B
B
H=0.22B
0.3 B 0.4 B 0.3 B
0.35H0.65H
0.04 B 0.04 B
CRITERION 3
C
B
H=0.22B
0.3 B 0.4 B 0.3 B
0.35H0.65H
0.04 B 0.04 B
CRITERION 3
CRITERION 2
Figure 5. Some optimal structural morphologies of a bridge type structure.

7. Windows A, B and C of Figure 5 show some layouts obtained during the evolutionary process for different
material types. By assuming symmetric material (Criterion 1), a balanced arch scheme emerges instead
of the expected one (Figure 5A). To be winning, the cable-stayed scheme should favor tensioned fields
and 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 rational
scheme can be achieved if the rejection criteria are differentiated over the structure, for instance by
assuming 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 if
material having low tension strength is used. Figure 6 shows some of the configurations resulting from
the evolutionary process for the case of symmetric material without pylons. The design graph of Figure 7
allows either to appreciate the optimality level of such schemes with reference to several optimality
indexes, 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=0.22B
0.3 B 0.4 B 0.3 B
0.35H0.65H
0.08 B0.08 B
CRITERION 1
Figure 6. Optimal balanced arch schemes for a bridge type structure.

8. 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 to
morphology optimization problems. The swelling step of the BG procedures has been extended to take
geometrical constraints into account. A formulation of ESO suitable to deal with asymmetric (tension and
compression dominated) materials, fixed geometrical boundaries and alignments (non design domains)
and multiple load conditions has been presented. In designing the morphology, F.E. grouping techniques
allow us to drive the final configurations towards either smooth profiles or segmented boundaries as in
case of masonry structures. Such processes may lead to many final optimal choices, as has been shown
by 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, with
reference to design criteria not necessarily coincident with those which control the evolutionary process.
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