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1 INTRODUCTION

1.1 Computational design of shell structures

Shell structures, in their different shape and configurations, from thin concrete shells, to

glass gridshells with steel or timber grid structure, represent an interesting field for the applica-

tion of search and optimization methods. During the 20th century, concrete shells have been an

important field of experimentation for studying the role of form in architectural and structural

design [1]. When only closed form theoretical models and experimentation on small physical

models were available, the interest was mainly oriented to structures with regular geometry, as

spherical domes, revolution surfaces, cylinders, hypars. Thanks to the developments in compu-

tational technique, since last decades of the 19th century architects were more and more inter-

ested to free form structures in order to widen the field of application and the aesthetic potenti-

alities. The relation between form and structure is nowadays focused on the topics of ‘Non-

Standard’, i.e. on new ways of (i) conceiving, (ii) developing and (iii) constructing architectural

projects by means of computer technologies. This involves the issue of free-form shapes, which

are ‘freely’ designed without any inhibition towards traditional structural and construction

principles. Free-form shapes, on the other hand, require computational tools for their represen-

tation, as well as for the analysis of their structural or functional performances because the ef-

fects of geometrical changes on complex shapes are not easily predictable on the basis of a ty-

pological knowledge. The mechanical behavior, as well as other physical performances, related

to acoustics, energy, light control, weight, etc. are all strictly depending on the shape of the

structure, both at the global and at the local level. The geometric configuration influences the

performances as well or much more that the material properties, so that the search of efficient

Multi-Objective Optimization of Concrete Shells

T. Mendez Echenagucia

Department of Architecture and Design, Politecnico di Torino, Italy

A. Pugnale

Faculty of Architecture, Building and Planning University of Melbourne, Australia

M. Sassone

Department of Architecture and Design, Politecnico di Torino, Italy

ABSTRACT: This paper discusses the application of a Multi Objective Search method to the

design of shell structures. Contrasting objectives of light weight and high stiffness are consid-

ered for two case studies with different geometry and constraints. The multiobjective search is

performed in the frame of the Pareto theory of non domination, implemented in the NSGA-II

algorithm. The research focuses on the way a designer can interact with the search computa-

tional procedure, driving it towards the desired solution. Diversely from pure optimization

techniques, search strategies as Multi Objective search produce sets of feasible solutions, that

the designer can know, evaluate and use in is work, accordingly with other architectural re-

quirements and purposes.

configurations, in a multi-objective oriented approach, concerns specifically the work of the ar-

chitect, as responsible of the formal aspects of the design. Search and optimization methods be-

come then a powerful and necessary support for the design process. In previous researches by

the authors [2][3][4], different optimization algorithms have been adopted to design complex

surfaces, such as folded structures, double curvature surfaces or combinations of both. Optimi-

zation algorithms were used considering one performance at time, the structural response or,

separately, the acoustical behavior. In general situations, however, design problems can be in-

terpreted as sets of parallel objectives, frequently in contrast to one another as, to be satisfied at

the same time. In structural design typical contrasting objectives are the reduction of weight

and the increase of stiffness, or strength. When such situation occurs, traditional optimization

algorithms are not suited to find optimal solutions with respect to all objectives. In these cases

Multi-Objective Optimization methods [5] must be used.

2 BENCHMARKS SETUP

2.1 Geometric parametric model

In order to understand how MOGA works, the search process is applied to the curved shell

structure shown in figure The shell is a free form surface, defined by a NURBS, which plan

projection is a 24m long and 4m wide rectangle. The NURBS is build starting from a set of four

spatial curves, two of which are just straight lines, the short sides of the rectangle, while the

other two are NURBS curves laying in vertical planes. The four curves act as four vertical sec-

tions of the surface to be generated. The surface is defined as a NURBS passing through the

section curves (lofted surface), with assigned polynomial degree. Each curve is defined by four

interpolating points, which vertical position, the z coordinate, is variable. By modifying the co-

ordinate of the interpolating point, the section curves change and so the surface. In such a way a

set of eight real number is used to completely define the surface shape. The other NURBS pa-

rameters of the surface, as the degree of interpolating functions or the number and position of

control points, are set constant or directly handled by the algorithm that generates the surface.

Figure 1. Waved shell used in the optimization process

2.2 Parameters domain

The ranges of variability of each parameter define the set of potential solution. As it has been

already said [5], the width of the solutions domain influences the search process: when the do-

main is just a narrow layer, round an initial surface, the process is a form-improving, while

when it is a large box, the process is rather a form-finding. In the proposed application the

search is performed considering a range spanning from -10m to 10m for each variable, corre-

sponding to a true form-finding.

Figure 2: Effect of domain size: form-improving and form-finding

2.3 Finite element model

The construction of a finite element model finalized to structural optimization presents some

differences with respect to models used in normal analyses. The repetition of the analysis for

many times is time consuming and, when the problem is relatively complex, it represents a

“bottle neck” in the flux of operations. Hence, the first requirement of the model is to be the

simplest as possible, with a number of elements strictly necessary to catch the pertinent aspect

of structural behavior and with a mesh correctly defined. Even with powerful hardware devices,

the repetition for hundreds or thousands analyses can make the optimization problem a hard

task, if the model is not efficient. There are basically two possibilities: the use of customized

finite elements solvers, developed in the same environment and the use of external applications,

as commercial software. Both the alternatives have advantages and disadvantages, but in this

part we will consider the first one as the more suitable in order to check the effects of different

models on the efficiency of the optimization process. In shell analysis an important issue is the

choice of the elements to use: in fact, even for a simple non-layered elastic shell, different

formulations and approaches can be adopted.

In the proposed application, the shell is approximated by a net of one dimensional beam

elements, which geometric properties are defined in order to reproduce the characteristics of a

continuous shell. This allowed to use a custom Python FEM code developed to interact with the

parametric modeler.

2.4 Multiobjective optimization

Multiobjective optimization is based on the concept of Pareto Dominance between solutions.

Shape and characteristics of Pareto fronts can be explained considering a set of benchmarks,

in which the fitness landscape is known for every objective and can be expressed by an analyti-

cal function.

Figure 3: Two variables - two fitness problem: first fitness landscape (top left); second fitness landscape

(top right); Pareto front in fitness space (bottom right), Pareto front in variables space (bottom left).

2.5 NSGA-II algorithm

Deb, Agrawal, Pratap and Meyarivan, in their article “A fast elitist multi-objective genetic

algorithm: NSGA-II”, proposed the Non-dominated Sorting Multi-Objective Algorithm known

as NSGA-II. The goal of this algorithm is to generate approximated Pareto fronts of a given

multiobjective problem through a genetic optimization process. In order to find an evenly

distributed range of solutions in the Pareto front, the algorithm combines the methods

traditionally present in a Genetic Algorithm, with the Non-dominated Sorting (NS) algorithm

and the Crowding Distance procedure (CD). Considering a finite number of solutions, produced

during a given generation step, the Non-dominated Sorting (NS) groups them in a series of sub-

groups. The elements of a single group are all non dominated between them, they dominate part

of the other groups and are in turn dominated by the rest of the groups. This allows us to

organize the sub-groups in a domination relationship, starting with the group who’s elements

dominate all of the elements of all other groups, to the group which elements are dominated by

the elements of all other groups. Since each sub-group contains elements that do not dominate

each other, it can be seen as a Pareto front inside the complete population. The sub-group that

occupies the highest position in the non dominated order is the Pareto front of the entire

population, and it can be regarded as the current (for the given generation) approximation of the

Pareto front of the given problem. Figure xx shows how the NS organizes the solutions.

The CD subroutine guarantees that current solutions are evenly spaced on the Pareto front,

avoiding unwanted concentrations. In technical words, can be regarded as a ‘niching’ operator,

that preserves diversity of equivalent solutions along all front. After solutions are organized and

sorted in sub-groups by NS, CD assigns a further fitness value to each configuration taking into

account the distance from the neighbors. The higher is such a distance, the higher is the fitness.

Figure 4. NSGA-II algorithm flowchart

The flow chart in Figure 3 shows the distinguishing characteristics of the NSGA-II. It works

with multiple fitness functions, it uses a non dominated sorting algorithm, an explicit diversity

preservation operator (Crowding Distance).

3 RESULTS

Figure 5 shows the results of the search process for the first benchmark. The Pareto front,

represented in the fitness space, contains the best solutions found by the algorithm: the lower

branch of the curve contains solution that privilege lightness to stiffness, while in the left

branch stiffer but heavier configurations can be found. In this benchmark, the best solution

related to each fitness is known: the lightest shape is the flat shape, while the stiffer shape is a

dome with the four central point at the top of the domain and the eight lateral points at the

bottom. During search process, the CD algorithm tries to keep a good spacing between solution

in the front, but the two extremes were not found. The knowledge of such extremes allows to

evaluate the efficiency of CD algorithm in terms of ration between the size of the found front

and the size of the actual front, including extremes.

The solutions spacing in the front is a good indicator of the variety of geometrical shapes and of

the way different shapes answer to multiobjective requirements. A set of such shapes, related to

the position on the Pareto front, is shown in figure xx., together with the extreme cases. A di-

rect representation of the position of shapes in the variable space is not suitable, due to the

number of dimensions, but they can be compared one to another by the designer who is in

charge to handle the produced material.

Figure 5. Pareto Front of Benchmark 1

In figure xx the Pareto front, at different stages of the search process, is depicted for the second

benchmark, the shell bridge. Even in this case the lightest shape is the flat shape, but the stiffest

does not have a theoretical significance. If the longitudinal section of the bridge was an arch

with a shape perfectly corresponding to the pressure curve of the load, then the stiffest solution

would have this shape and straight transverse section. It would be a barrel vault, or “flat arch”.

If the NURBS representation does not allow such a perfect shape, the only way to increase

stiffness is to add some bending stiffness, through a transverse waved section. The shell, in this

case, becomes a kind of ribbed arch, in which ribs increase the arch stiffness.

Besides the limit case of the stiffest shape, this considerations are important for other Pareto

front shapes. Ribs, in fact, increase the stiffness and the weight at the same time.

The set of shapes depicted in figure xx includes some example coming from previous search

steps, instead than from the last only: those shapes do not actually represent local minima, but

simply steps of the search path. However, they can play a role in interactive design, because

they can be chosen as starting points of new search processes, through a redefinition of con-

straints and of the domain, suggested by the designer evaluations.

Figure 6. Pareto Front of Benchmark 2

4 CONCLUSIONS

The analysis of the case study shows that the main result of a multi objective search is a set of

solutions, characterized by the fact that they cannot be ranked on the basis of the adopted fit-

ness functions. A further decision step is then mandatory to define the final design solution. In

the case of performances measures that are in great contrast to each other, the geometrical

shapes corresponding to the solutions of the set can be completely different and not necessarily

belonging to a uniform family, from the geometric point of view. Hence the multi objective

procedure can be regarded as a shrinking of the field of feasible solution, rather than the search

of one best solution. The work of the architect as a decision maker, is aided and influenced, but

not totally determined by the process.

5 REFERENCES

Chiorino, M. A. & Sassone, M. 2010. The morphogenesis of shell structures: a conceptual, computational

and constructional challenge. In: 1st International Conference on Structures and Architecture.

Guimaraes (P), july 22-24, 2010.

Deb, K.. 2001 Multi-Objective Optimization using Evolutionary Algorithms. John Wiley & Sons Ltd,

Chichester, England.

Méndez Echenagucia T. & al. 2008. Architectural, Acoustic and Structural Form, Journal of the Interna-

tional Association for Shell and Spatial Structures, vol. 49, no. 3, pp. 181-186.

Pugnale, A. & Sassone, M. 2007. Morphogenesis and Structural Optimization of Shell Structures with the

aid of a Genetic Algorithm, Journal of the International Association for Shell and Spatial Structures.

vol. 48 n. 155, pp. 161-166.

Sassone M. & al. 2008. On the interaction between architecture and engineering: the acoustic optimization

of a reinforced concrete shell. In: Abel J.F., Cooke R.J. (Eds.), Proceedings of the 6th International

Conference on Computation of Shell & Spatial Structures: IASS- IACM. Ithaca NY.