Optimisation of the fast craft hull structure by the genetic algorithm

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 Optimisation of the fast craft hull structure by the genetic algorithm Z. Sekulski* & T. Jastrzebski** Faculty of Maritime Technology, Technical University of Szczecin, Al Piastow, 7-06 Szczecin, Poland EMail: * zbych@shiptech.tuniv.szczecin.pl * * tadjast@shiptech. tuniv. szczecin.pl Abstract The genetic algorithm (GA) was applied to study the minimum weight problem of the fast craft hull structure with several design variables. A computer code was built for optimization of the fast craft hull structure. The crossover strategy with random number of cutting points was proposed. The fitness function was based on loads and strength criteria suggested by the classification rules. Some results of calculations are presented in the paper. In conclusion the GA is recommended for practical application in design of ship hull structures. Introduction Ship structural design generally involve a large number of design parameters. Those parameters can be in a form either of continuous function, discrete values or both and they often include constraints of allowable values. The aim of the optimised ship structural design is to find a solution that represents a global maximum or minimum in the design space with unknown number of the relative extreme. In addition, often the solution area of ship structural problem contains non-differentiable and/or discontinuous regions. More constraints are in non-linear form in terms of design variables. All these features sorely test the capabilities of many of the traditional sequential or enumerative optimisation

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 2 Marine Technology techniques, and often require patches or hybridisation of traditional optimisation methods if these methods are to be applied at all. Many well known disadvantages of the traditional methods of optimisation may be avoided by application of methods which have been developed for some years, such as: methods of simulated annealing, methods of neural nets and genetic methods. Some interesting applications of methods of the last group, which offer so called Genetic Algorithms (GAs), have proven that they are particularly well suited for problems of optimisation in several domains. Trials for GAs application for ship design and ship structural design have been carried out. Okada & Neki^ developed the GA for ship structural design. The optimisation of double hull tanker structure have been presented. Nobukawa & Zhou") developed a discrete optimisation method using GAs for the design of selected models of ship structures. Sommersel^ described application of the GA for ship design. The example of supply ship design have been described. Zhou at al'^ present a GA application for structural optimisation of cargo ship with large hatch openings. Sekulski & Jastrzebski^ developed the GA for fast craft deck structural optimisation. 2 Computer realisation of genetic algorithm for optimisation of structures Genetic Algorithms are computerised search procedures based on principles of the natural evolution and heredity. The GAs were first developed by Holland^ to allow computers to evolve solutions in the function optimisation and in the artificial intelligence. There are many reference textbooks and papers about GAs, such as those by Davis/) Goldberg,^ Davis,*' Forrest,^ Buckles & Petry,^ Michalewicz,^) Back/) The advantages and disadvantages of using GAs for ship structural optimisation were briefly summarised by Sekulski & Jastrzebski.^) For numerical realisation of structural optimisation using GA the computer code has been built. The description, flowchart and the most important features of the code have been presented in the same paper. ^ Structural model The structural model for the optimisation study was selected after the analysis of typical layouts of the SES (Surface Effect Ship) type craft. Finally a model similar to the one proposed by Jang & Seo^ was selected. The vessel and its corresponding cross and longitudinal sections are shown in Fig.l. The main geometrical characteristics of the structure are in Fig.2. The structural material is the marine aluminium alloy of properties shown in Table. The plate thickness and the bulb and tee bar extruded stiffener sections are assumed according to the commercial shipbuilding standards. The formulae for scantling calculation for plate thickness and section moduli of stiffeners and web frames

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 Marine Technology are taken from the UNITAS*^ rules. A minimal structure weight (volume of structure) was assumed as the criterion in the study and it was introduced in the objective function and constrains defined on the base of classification rules. * Side profile ; ; ; VVfeb frame ; ', ', X ', ; ~ BJtt*s&i BJkheeri/*, : Upper deck Inner deck Superstructure _. Midship section ij ;. Wfet-deck Side outboard- Side inboard QOQ_ ^ Figure : Surface Effect Ship (SES) - assumed craft model and its structural idealisation. Figure 2: An example of structural model used in the study - the upper deck region.

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 Marine Technology Table. Assumed properties of structural material - aluminium alloy No. 2 Property 2 Yield stress Young's modulus Poisson's ratio Density Symbol &0.2 E V p Value 2 (for 08 alloy) 20 (for 6082 alloy) 70000 0. 2.66 Unit N/mnf N/mnf t/nf Formulation of the optimisation model For hull section structural model the set of the assumed design variables is presented in Table 2 and may be given as: xy = (x,,%2,...,*a #=29. () The objective function,/*,), for the optimisation of the hull structure was written in the following form: (2) where: */ - fth design variable, R - number of structural regions, SWj - structural weight of they'th structural region, Wj - relative weight (relative importance) of structural weight of regions. The behaviour constraints were formulated for each region according to the UNITAS rules'**, for example: - required plate thickness based on the permissible bending stress, tp^ie'- o, () where: /, is the actual calculated value of plate thickness mjth region, required section moduli of stiffeners, ^, //«,: where: Z,j is the actual calculated value of section modulus of stiffeners myth region. Examples of side constraints for design variables are also given in Table 2. Some of them correspond to the number of elements in the commercial standard. The others have been taken according to the authors' experience. The additional geometrical constraints were introduced due to some fabrication and standardisation reasons, such as: relation between the plate ()

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 Marine Technology thickness and web frame thickness, relation between the plate thickness of plate and stiffener web thickness, minimal distance between the edges of frame flanges. Table 2. Simplified specification of bit representation of design variables No., - \ 2 Symbol ^ f ' * ' *i *2 * % X Description '* -,, '* serial No. of upper deck plate serial No. of upper deck bulb serial No. of upper deck T-bulb number of web frames number of upper deck stiffeners Substring length,;,,,, A, min 0 20 Value max <* 7 28 6 0 Step 26 27 28 29 *26 X27 ^28 X29 serial No. of inner deck plate serial No. of inner deck bulb serial No. of inner deck T-bulb number of inner deck stiffeners 7 20 7 28 0 Description of the genetic model Solving the optimisation problem by GAs calls for formulation of the appropriate optimisation model. Therefore the model described in Section has been reformulated into the optimisation model according to requirements of GAs. In particular, this model has been used to build the suitable procedures in computer code and to define search parameters.. Chromosome structure The space of possible solutions is the space of structural variants of the assumed model. The hull structural model was identified by a set of 29 design variables, %.. Each variable may be represented by a string of bits used as chromosome substring in GAs. A variant of solution is simply represented as a bit string. Chromosome length is equal to the sum of all substring. Number of possible solutions is equal to the product of all variable values. In the work chromosome length is equal to bits and number of possible solutions is equal approximately to 0 individuals..2 Fitness function The design problem defined in this paper is to find the minimum weight of deck structure without violating the constraints. In order to transform the constrained

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 6 Marine Technology problem into unconstrained one and due to the fact that GAs do not depend on continuity and existence of the derivatives, penalty methods have been used. Thus, the augmented objective function of unconstrained minimisation problem was expressed as: where:0(x,) is a augmented objective function of unconstrained minimisation, X*,) is an objective function given by equation (2), f, is a penalty term to violation of they th constraint, \v and w,- are weight coefficients for objective and penalty terms, respectively, n^ is a number of constraints. Weight coefficients are adjusted by trial. Additionally, a transformation of minimisation problem (in which the objective function is formulated for the minimisation) into the maximisation one is needed for the GAs procedures (searching of the best individuals). It can be simply carried out by multiplying the structure weight by (-). In that way, the minimisation of the augmented objective function was transformed into a maximisation search by using: ^=LJb)-#^, (6) where: Fj is the fitness function for jth solution, #,(%,) is the augmented objective function for yth solution, dlnw is the maximum value of the augmented function from all the solutions in the simulation. The role of objective function /(*,), formulated in equation (2), is preserving in the relative assessment of chromosomes. The value of parameter #*,*#,) is arbitrary defined by a user of the software to avoid negative fitness values. Its value should be greater than the expected largest value of #,(%,) in the simulation. In the present work the value #**%(%:) = 00000 was assumed. 6 Optimisation calculations To verify the correctness of the assumed optimisation procedure several test cases have been carried out using the model described in Section. Each experiment was characterised by the 8-tuple (%G, ",, PC, c_strategy, njc_site, p^ Pu, elitism) where no is a number of generations, /?, is a population size, PC is a crossover probability, cjstrategy is crossover strategy flag (equal 0 for fixed, and for random number of cutting points), njc_site is a maximal number of cutting points for each mate individuals, p^ is a mutation probability, PU is a update probability, and elitism is a logical value for elitism strategy switch on or switch off. In Tables and the examples of results of one selected trial are presented. The set of experiment parameters was as follows (00, 0, 0.8,, 7, 0.02, 0., yes). There was 000 checked individuals in whole simulation.

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 Marine Technology 7 The lowest value of the objective function,/*,) = 2.70 t, was found in the 27th generation. The corresponding values of design variables are given in Table. From designer point of view the interesting values are those concerning the whole structural weight and/or the relative structural weight. They are given in Table. The achieved structural weight volume density is 0.06 t/nf. The higher structural area! weight density due to high cargo loads of this region is for inner deck region - 0.00 t/m^. The lower structural areal weight density is achieved for superstructure and upper deck regions - 0.07 t/nf. Table. The optimal values of design variables No. - 2 Symbol Description -, : a -.^: ;/,//,;//, /, $ + ',-,/ // *i *2 * XA * serial No. of upper deck plate serial No. of upper deck bulb serial No. of upper deck T-bulb number of web frames number of upper deck stiffeners Optimal value 2 8 26 27 28 29 *26 *27 %28 *29 serial No. of inner deck plate serial No. of inner deck bulb serial No. of inner deck T-bulb number of inner deck stiffeners 9 2 2 Table. Optimal structural weight values No. r i 2 6 7 8 9 Region description %, S,*» ' ',, <% ' - *, Upper deck Superstructure Side inboard Bottom Side outboard Wet deck Inner deck Total, t Total volume density, t/nf Value, t /, x ',** - #*,*'.09 0.726.7.08.27 2.7 8.798 2.70 0.06 Areal density, t/m^ - 0.07 0.07 0.020 0.0 0.02 0.0 0.00

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 8 Marine Technology All hull structural weight values for feasible individuals searched in the selected trial are presented in Fig.. The solid line in the figure represents the optimal solutions front. It is composed by minimal (optimal) values of structural weight being received in the following simulations. The optimum solution was achieved in 27th generation. 80 70 i -*- - -. -.. v A * *» *.. _._ k'. \ '>*; r.. *.*. t. + * %2* - *. ^*"J» >\V. j \ ;. % * "/#;; * >>}#$ ^y-,v ^l *<*»,^%.: A i 0 ^\0nfcf rt solutions frartt 0^ c)» DO 90 2DO 290» Generadon c "" ""* g^ JC O) 0 i»- 20 Figure : Evolution of structural weight values over 00 generations. The graphs of thefitnessmaximum, average, minimum and variance values across 00 generations for selected trial are presented in Fig.. The saturation was not achieved in the trial. The achieved fitness was nearly 0.6. The standard deviation value is approximately constant - 0.06 for all generations. It means that heredity of generations is approximately constant over simulations. Fitness function and minimum weight of structure function are shown in Fig.. Correspondence of the diagrams can be seen. Increase of the fitness function values in succesive generations is accompanied by the decrease of structural weight values. 8 Conclusions A practical method for the structural optimisation based on the GAs was presented in the paper. The basic features of the method together with the optimisation model and their application to the model of the ship deck structure of a fast craft of SES type were discussed.

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 Marine Technology 9 The investigated structural model was composed of plates, longitudinal stiffeners and transverse web frames. For that model the feasibility of GAs was demonstrated. Test example calculations were also presented. The study confirmed that GAs can be used as a practical tool for searching global extremum (minimum or maximum, depending on the problem) in ship structural design. Figure : Evolution of fitness maximum, average, minimum and standard deviation values over 00 generations; fitness function values are dimensionless and normalised with extreme value of.0. -rr\,, Maximum fitness F/x00 f*n 60 - _j- 0 T^ 0 0 20 0 s^x Minimal structrural weight, int - ^ ( 0 00 0 200 20 C)0 Generation Figure : Evolution of maximal fitness value and absolutely minimal structural weight over 00 generations; absolutely minimal structural weight for simulation only for feasible solutions.

Transactions on the Built Environment vol 2, 999 WIT Press, www.witpress.com, ISSN 7-09 60 Marine Technology References. Back, T. Evolutionary Algorithms in Theory and Practice, Oxford University Press, 996. 2. Buckles, B.P. & Petry, F.E. (eds.) Genetic Algorithms, IEEE Computer Society Press, Los Alamitos, California, 99.. Davis, L. Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York, 99.. Davis, L. (ed.) Genetic Algorithms and Simulated Annealing, Morgan Kaufmann Publishers, Los Altos, California, 987.. Forrest, S. Genetic Algorithms: Principles of Natural Selection Applied to Computation, Science, 99, 26, 872-878. 6. Goldberg, D.E. Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Publishing Company, Inc, 989. 7. Holland, J.H. Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 97. 8. Hughes, O.F., Mistreee, F. & Zanic, V. A practical method for the rational design of ship structures, Journal ofship Research, 980, 2, 0-. 9. Jang, C.D. & Seo, S.I. A study on the Optimum Structural Design of Surface Effect Ships, Marine Structures, 996, 9, 9-. 0. Michalewicz, Z. Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Berlin Heidelberg, 996.. Nobukawa, H. & Zhou, G. Discrete optimization of ship structures with genetic algorithm, J Soc Naval Arch Japan, 996, 79, 29. 2. Okada, T. & Neki, I. Utilization of genetic algorithm for optimizing the design of ship hull structures, Recent Progress on Science & Technology IffI, 99,, -.. Sekulski, Z. & Jastrzebski, T. Optimisation of the Fast Craft Deck Structure by the Genetic Algorithms, Marine Technology Transactions, 998, 9, 6-88.. Sommersel, T. Application of genetic algorithms in practical ship design, in: IMDC '97, pp. 6 to 626, Proceedings of the 6th International Marine Design Conference, 2-2 June 997, Newcastle upon Tyne, U.K.. UNITAS Rules for the Construction and Classification of High Speed Craft, 99. 6. Zhou, G., Nobukawa, H. & Yang, F. Discrete optimization of cargo ship with large hatch opening by genetic algorithms, in ICC AS'97, pp. 2 to 26, Proceedings of the 9th International Conference on Computer Applications in Shipbuilding, Yokohama, Japan.