SHANG Gao-feng,ZHANG Ai-feng,WAN Zheng-quan

(China Ship Scientific Research Center,Wuxi 214082,China)

1 Introduction

Underwater structure generally consists of two hulls.The outer hull is for hydrodynamics and the inner hull is for pressure resistance.The pressure hull structure will be subjected to higher hydrostatic pressure as it goes deeper.The requirements of strength and buckling should be matched in order to provide safety guarantee for the crew and devices as the fundamental platform of the equipments normally working in deep sea.Therefore the objective of underwater structural design is how to perform the optimum design under the requirements of strength and buckling so as to increase the effective payload of underwater vehicles.

The stiffened cylindrical shell is one of the most important pressure hull structures and it is widely used for the pressure hull of submarine and underwater vehicles.A series of investigations were carried out on the minimum weight structural design of pressure hull structures(Pappas and Allentuch 1973,1975;Simitses and Aswani 1977;Kendrick 1986;McGrattan 1987;Blachut 1988;Gorman and Louie 1991;Yang et al 1992).Liang et al(1997)conducted a comprehensive review of research works on minimum weight design of submersible pressure hull structures in previous literature and presented a minimum weight design of a submarine pressure hull under hydrostatic pressure with constraints on factors such as general instability,buckling of shell between ring-stiffeners,shell plate and ring-stiffener yielding.

Ross et al(1987,1993)presented an alternative design for submarine pressure hull structure,which has no ring-stiffeners but its local and general instability are resisted by making the shell of a swedged-shaped form.Zhu et al(2005)carried out the theoretical and experimental investigations on the buckling of cylindrical shell ring-stiffened by half-tubes and a rational design method was developed.Wang et al(2008)presented an optimal design of pressure cylindrical shell with middle frames using the modified compound form method with interactive operation.The conception of margin is cited to analyze the constrained functions while getting the minimal results.Gou and Cui(2009)conducted the structural optimization of multiple intersecting spherical pressure hulls based on Kriging model and nonlinear finite element analysis of pressure hulls at the sample points.

Lagaros et al(2004)presented the optimum design of stiffened shell structures subjected to behavior constraints imposed by structural design codes such as Eurocode by using evolutionary strategies one of evolutionary algorithms.Banichuk et al(2006)conducted the application of genetic algorithm to the shape optimization of axis-symmetric shells under stress intensity constraints.Chen et al(2008)carried out optimal design of shell scantlings and stiffener arrangements of ring-stiffened cylindrical shell under the requirements of strength and buckling by using a hybrid model of genetic algorithm and simplex method.

In present paper an optimization program is developed based on the hybrid genetic algorithm and subdivided polyhedron method for the optimal design of ring-stiffened cylindrical shell subjected to hydrostatic pressure.The gradient based method is presented based on ANSYS APDL language for the comparative study of present method and ANSYS optimization and the good agreement is found.ANSYS optimization and present method are employed respectively to perform the optimal design of pressure cylindrical shell under the requirements of the strength and buckling and the basic fabrication technology.The obtained results are useful for practical design of pressure cylindrical shells,especially for shell scantlings and stiffener arrangements of ring-stiffened cylindrical shells.

2 Mathematic model for structural optimization

In general,the optimization problem in structural design under constraints can be represented in the following form.

in which,x is design variable vector,f（x）is objective function,A is feasible domian(it is defined by constraint conditions,such as gi（x）≤0;hi（x）=0;l≤x≤u).

According to the general multiplier method(Hestenes 1969;Powell 1969 and Rockafellar 1973),the constrained optimization problem can be transferred to the unconstrained optimization problem as follows,

In which,

2.1 Design variable and constraint conditions

Pressure hull structure generally consists of cylindrical shell,conic shell and sphere shell.The main scantlings are pre-determined by overall system design and general arrangement.The objective of optimization structural design is to determine the shell thickness,ring-stiffener arrangements and sectional parameters of ring-stiffener and inertia modulus under the requirements of strength and buckling.For ring-stiffened cylindrical shell,the radius of cylinder,R,length of compartment,L are assumed as design constants while the shell thickness,t,spacing of ring-stiffeners,l,section area,F and inertia modulus of ring-stiffener,I are selected as the design variables.The non-dimensional parameters relating to the design variables and constants are defined according to theory and experiments on modern submarine structure strength(Xu et al,2007)as follows:

The strength assessment of ring-stiffened cylindrical shell is performed by the structural design formulae based on axisymmetric solution of circular cylindrical shell.The local and general buckling of ring-stiffened cylindrical shell is analyzed by the classical elastic shell buckling theory considering the effects of initial imperfections and material nonlinearities on the collapse pressure of hull structure.

(1)Axial stress of inner surface of shell at the intersection of shell and ring-stiffener,σ1

(2)Loop stress of mid-section of shell between the two adjacent ring-stiffeners,σ20=K20

(3)Nominal stress of ring-stiffener,

(4)Critical pressure of local buckling,

(5)Critical pressure of general buckling,pcr′=0.9CsCg′pE′.

in which,K1,K20and Kfare stress coefficients;Cs,Cgand Cg′are knock down factors accounting for the effect of material nonlinearities and initial imperfections on local and general buckling respectively.The design diagrams have been provided in theory and experiments on modern submarine structure strength(Xu et al,2007).

It is inevitable that the differences between designed geometrical dimension and produced structural components are caused by the welding and assembling in the manufacturing process.All structures and their elements are produced with the allowed manufacturing tolerance.The extreme value of initial geometrical imperfection,0.2t for shell and 0.002 5R for ring-stiffener,is permissible in manufacturing pressure cylindrical structures.

For the simplicity and efficiency of the calculation non-dimensional parametersare introduced in this paper.It is well known that the ratio of structural weight and its displacement volume is a considered key factor for deepsea pressure structures.The rational strength grade of pressure structural steel is selected in pressure shell design because of the limitation of the structural weight and displacement volume ratio.Therefore the non-dimensional parameters {ζ,u,β,U,f }are selected as design variables x={x1,x2,x3,x4,x5}while the non-dimensional parameters{α,η }are assumed as design constants in present optimization procedure.

The constraint equations gk（x）≤0 are obtained by assessing the strength and stability of ring-stiffened cylindrical shell for structural safety and considering the basic requirement of fabrication technologies.

2.2 Objective function and fitness function

The objective function of ring-stiffened cylindrical shell can be expressed as following form.

in which,the first term of above expressionis a rational approach to the ratio of structural weight and displacement volume of ring-stiffened cylindrical shell and the second termis panel function based on the general multiplier method.

The sum of the values that fails the constraint equation,is defined to represent the extent that the optimization trial departures from the structural design constraints.The fitness function is constructed according to the following equation.

in which,a,b are algorithm parameter respectively.

The objective function of the structural optimization problem is mapped into the fitness function,which is used to evaluate the performance of the individual strings.This fitness function is developed by Chen et al(2008)to perform the optimization of the Sphere function and the Ackley function as well as the ring-stiffened cylindrical shell and reasonable results are obtained.

2.3 Optimization strategies

In recent years,genetic algorithms,which are optimization algorithms based on the theory of biological evolution and adaptation,have been applied to solve the optimal solution to a variety of problems.In genetic algorithms,the fitness is used to represent the objective value of the optimization problems,and artificial chromosomes represent the design variables.Genetic algorithms find an optimal solution by generating population of solution strings randomly and improving the solutions in succeeding generations based on the mechanics of natural selection and natural evolution such as reproduction,crossover and mutation,etc.It is different from optimization algorithm based on the gradient information,which requires only the function information.The employed genetic algorithm in this paper is different from one of the original genetic algorithms.It is modified and implemented by using hybrid genetic algorithm and subdivided polyhedron method in order to enhance the capability of optimization strategy.The basic procedure of hybrid genetic algorithm and subdivided polyhedron method in this paper for structural optimization problem can be stated as follows:

(1)Modeling the structural response and design variables based on structural mechanics.The constraint equation is represented by the requirements of strength and buckling.

(2)Transferring the constraint optimization to unconstraint optimization by using generalized multiplier method.

(3)Mapping the design variables with the individual strings.The design variables of the structural optimization problems correspond to the individual strings of artificial evolution process.

(4)Constructing the objective function and the fitness function to evaluate the performance of the design variables and the individual strings respectively.

(5)Initializing the design variables and corresponding the population of individual strings.The values of constraint equation,objective function and fitness function are calculated.

(6)Employing the subdivided polyhedron method for each individual to search the local minimum value of objective function by reflecting,expanding,compressing and shrinking operation.

(7)Continuing the optimization process by the subdivided polyhedron method until the convergence is adopted.The local minimum value of objective function is obtained and the value of fitness function is calculated and evaluated for each individual string.

(8)Generating the next generation of population by means of reproduction,crossover and mutation according to the predefined probability and the fitness value of each individual string.Continuing the artificial evolution until the convergence is satisfied.

(9)Continuing the procedure of the hybrid strategy until the optimal solution of structural optimization problem is obtained.

3 Numerical examples

Two examples are presented below.The first one is for comparative study of present method and ANSYS optimization.The second one deals with the optimization problem of ringstiffened cylindrical shell under hydrostatic pressure.In both examples the ring-stiffened cylindrical shell is made of high tensile strength steel and material properties such as elastic modulus and yielding stress,etc are predefined.The maximum calculated pressure,the radius and overall length of cylindrical shell are assumed to be design constants.

3.1 Comparative study

The ring-stiffened cylindrical shell is a good structure to resist the effects of external hydrostatic pressure and widely used in submersible vehicle.The typical structural form is shown in Fig.1.For A1 and A2 ring-stiffened cylindrical shells the non-dimensional design constant η is predefined as η =93 according to the yielding stress and maximum calculated pressure and the non-dimensional design constant α is predefined as α=0.73 and α=0.71 respectively according to the different radius and overall length.

The optimization of ring-stiffened cylindrical shells is carried out by present method and ANSYS optimization.It is shown in Tab.1 that the ratio of structural weight and its displacement volume WVobtained by present method is in good agreement with that by ANSYS opti-mization while the non-dimensional parameters have some difference between the present method and ANSYS optimization.The discrepancy of the ratio of structural weight and its displacement volume is less than 3%and the obtained result by present method is a little more than that by ANSYS optimization because the optimization variables are much less in present method than in ANSYS optimization.Compared with ANSYS calculation,the present method provides a simple and efficient optimization tool for the initial design of ring-stiffened cylindrical shell subjected to hydrostatic pressure.The buckling wave of A1 ring-stiffened cylindrical shell is presented in Fig.2 and it is found that the first failure mode is local buckling between adjacent ring stiffeners.

Fig.1 Ring-stiffened cylindrical shell

Fig.2 Buckling wave of A1 Cylindrical shell

Tab.1 Comparative results of present method and ANSYS software

Fig.3 Relationship between WVand η

3.2 Optimum calculation

A series of optimum calculation is performed by present method for the optimization of the ring-stiffened cylindrical shell subjected to hydrostatic pressure.The optimization results are obtained and shown in Fig.3 to Fig.4.It is found that the ratio of structural weight and its displacement volume is very sensitive to the ratio of yielding stress and maximum calculated pressure while the non-dimensional parameter α,which represents the ratio of the radius and overall length of cylindrical shell has little influence on the ratio of structural weight and its displacement volume.So it is very important to select the strength grade of high tensile strength steel according to the maximum calculated pressure because of the limitation of the ratio of structural weight and its displacement volume.

Fig.4 Relationship between WVand α

4 Discussion and conclusions

An optimization program is developed based on the hybrid genetic algorithm and subdivided polyhedron method for the optimal design of pressure cylindrical shell under the requirements of the strength and buckling and the basic fabrication technology.The gradient based method based on ANSYS APDL language is employed to carry out the comparative study of present method and ANSYS optimization and the good agreement is found.

The numerical example shows that the ratio of yielding stress and maximum calculated pressure has an effect on the ratio of structural weight and its displacement volume while the ratio of the radius and overall length of cylindrical shell has little influence.So it is very important to select the strength grade of high tensile strength steel according to the maximum calculated pressure.The obtained results are useful for practical design of pressure cylindrical shells,especially for shell scantlings and stiffener arrangements of ring-stiffened cylindrical shells.

[1]Banichuk N V,Serra M,Sinitsyn A.Shape optimization of quasi-brittle axisymmetric shells by genetic algorithm[J].Computers and Structures,2006,84:1925-1933.

[2]Blachut J.Optimum shaped torisphere with respect to buckling and their sensitivity to axisymmetric imperfections[J].Computers and Structures,1988,29:975-981.

[3]Blachut J.Search for optimal torispherical end closures under buckling constraint[J].International Journal of Mechanical Sciences,1988,31:623-633.

[4]Chen A Z,Wan Z Q,Zhu B J.An application of hibrid GA and SM algorithm in optimal design of pressure structure[J].Journal of Ship Mechanics,2008,12(2):283-289.(in Chinese)

[5]Gou P,Cui W C.Structural optimization of multiple intersecting spherical pressure hulls based on Kriging model[J].Journal of Ship Mechanics,2009,13(1):100-106.(in Chinese)

[6]Gorman J J,Louie L L.Submersible pressure hull design parametrics[J].SNAME Translations,1991,99:146-149.

[7]Hestenes M R.Multiplier and gradient methods[J].Journal of Optimization Theory and Application,1969,4:303-320.

[8]Kendrick S B.Design of submarine structures[C].Advanced in Marine Structure,1986.

[9]Lagaros N D,Fragiadakis M,Papadrakakis M.Optimum design of shell structures with stiffening beams[J].AIAA Journal,2004,42(1):175-184.

[10]Liang C C,Hsu C Y,Tsai H R.Minimum weight design of submersible pressure hull under hydrostatic pressure[J].Computers and Structures,1997,63(2):187-201.

[11]McGrattan R J.Weight optimization of stiffened cylindrical panels[J].Journal of Pressure Vessel Technology,1987,109:1-9.

[12]Pappas M,Allentuch A.Automated optimal design of frame reinforced submersible,circular,cylindrical shells[J].Journal of Ship Research,1973,17:208-216.

[13]Pappas M,Allentuch A.Pressure hull optimization using general instability equation admitting more than one longitudinal buckling half-wave[J].Journal of Ship Research,1975,19(1):18-22.

[14]Powell M J D.A method for nonlinear constraints in minimization problems,optimization[M].Edited by Fletcher R.Academic Press,New York,1969:283-298.

[15]Rockafellar R T.The multiplier method of Hestense and Powell applied to convex programming[J].Journal of Optimization Theory and Application,1973,12:555-562.

[16]Ross C T F.A noval submarine pressure hull design[J].Journal of Ship Research,1987,31(3):186-188.

[17]Ross C T F,Palmer A.General instability of swedge-stiffened circular cylinders under uniform external pressure[J].Journal of Ship Research,1993,37(1):77-85.

[18]Simitses G J,Aswani M.Minimum-weight design of stiffened cylinders under hydrostatic pressure[J].Journal of Ship Research,1977,21:217-224.

[19]Wang L,Li F,Yu M H.An optimal design of pressure cylindrical shell with middle frames[J].Journal of Ship Mechanics,2008,12(6):986-993.

[20]Xu B H,Zhu B J,Ouyang L W,Pei J H.Theory and experiments on modern submarine structure strength[M].China National Defense Industry Press,2007.(in Chinese)

[21]Zhu B J,Wan Z Q,Xu B H,Shao D Y.On the buckling of cylindrical shell ring-stiffened by half-tubes under hydrostatic pressure[J].Journal of Ship Mechanics,2005,9(1):79-83.(in Chinese)

[22]Yang M F,Liang C C,Chen C H.A rational shape design of externally pressurized torispherical dome ends under buckling constraints[J].Computers and Structures,2002,43:839-851.