Application of Particle Swarm Optimization Theory in the Hydrofoil Design

XU Wei-bao,WANG Chao,HE Bao,HUANG Sheng ,ZHOU Bin

(1 College of Shipbuilding Engineering,Harbin Engineering University,Harbin 150001,China;2 China Ship Scientific Research Center,Wuxi 214082,China)

1 Introduction

The problem of hydrofoil section optimization has a long history(such as the design of the rudder section and propeller blade section which is highly required on the hydrofoil liftdrag ratio)and the solution also has continuous development[1].Traditional optimization methods include steepest descent method,conjugate gradient method and Newton method[2].However,in practical engineer applications,all these traditional methods need to calculate the gradient of the objective function,and they require a lot of computation.In recent years,because the process of processing information such as genetic life forms,neural network and so on has developed a lot,these methods are applied to the field of hydrofoil section optimization successfully[3].Although genetic algorithm and some numerical optimization algorithms can find the global optimal solutions,they need a wide range of choices,cross and mutation.These algorithms not only require a huge computation,but are easily influenced by group size and evolution generation,so they often converge to some local solution[4].Particle swarm optimization arithmetic is a newly developed optimization algorithm in recent years,which compared with genetic algorithm can get more better results in solving some complex nonlinear optimization problems[5-6]。

The Particle Swarm Optimization(PSO)arithmetic was put into the design process of hydrofoil optimization in this paper.The lift and drag of hydrofoil were calculated by panel method combined with PSO.At the some time,CFD method was adopted to do numerical simulation for the original hydrofoil and design hydrofoil,whose results were used to validate that the optimization method using PSO is feasible.

2 Mathematical model of PSO algorithm

Particle Swarm Optimization(PSO),also known as fine grain swarm optimization,had been developed into an evolutionary computation technique by Kennedy and Ebehtart,and so on in 1995,which comes from simulation of a simplified social model[7].The mathematical description is:

Assume the search space is D-dimensional,and the total number of particle is n,position of i-particle is expressed as vector xi=（xi1,xi2,…,xi D）.So far the optimal location of i-particle is pbest=（pi1,pi2,…,pi D）and the optimal location of the whole particle swarm is gbest=（g1,g2,…,g D）and the position variation of i-particle is vector vi=（vi1,vi2,…,viD）.Particle velocity of each dimension and position in the evolutionary process changes as following formula:

where w is inertia weight which reflects the choices of algorithm between global search and local search;while c1 and c2 called as cognitive and social parameters are non-negative constant;r1 and r2 are the random numbers between 0,［］1;and k is compressibility factor which limits the speed of particles.Usually a range for each dimension of particle position and velocity needs to set,if exceeding this range,a boundary value must be set.Initial location and velocity of particle swarm generate randomly.

3 PSO algorithms for hydrofoil section optimization

3.1 Function of hydrofoil section

In order to keep the smooth shape of hydrofoil section,the linear superposition of Hicks-Henne functions family can be used for re-expression of hydrofoil surface[8]during the optimization process.Seven Hicks-Henne functions are selected here.Finally,hydrofoil section can be expressed as:

where yup（x）,ylow（x）means the vertical axis of upper and lower surfaces respectively,while youp（x）,yolow（x）are the original hydrofoil.Ckis a variable quantity between-0.000 5 and 0.000 5 which controls the change of hydrofoil and fk（x）is a Hicks-Henne function,whose expression is:

where k=2,3,4,5,6,7;xk=0.15,0.3,0.45,0.60,0.75,0.9,respectively.

The number of particles n is the candidate for Hydrofoil shape,which called as population size.After the specific concepts,we can optimize the hydrofoil section according to particle Swarm Optimization theory.

3.2 The process of particle swarm optimization algorithm

3.2.1 Selections and calculations of Fitness functions

The lift-drag ratio of three-dimensional hydrofoil in a certain angle of attack and uniform stream is taken when the fitness function is selected.This paper is based on potential flow theory panel method to calculate the three-dimensional hydrodynamic hydrofoil performance.Due to its fast development,panel method is accurate and relatively mature,so it is widely used in the field of hydrodynamic calculations.Compared with using CFD software to solve N-S function,panel method can not only greatly reduce the computational time,but also get more accurate results with viscous correction,which can compare the advantages and disadvantages of various design as the optimal design process.Fitness(y)is selected as the liftdrag ratio of hydrofoil section optimization,where y means the control variable quantity Ck.The hydrofoil optimization is also taken hydrofoil cavitation number into account.So in the optimization process it can not only pursue a large lift-drag ratio,but should be stopped when the minimum negative pressure coefficient is close to critical cavitation number.

3.2.2 Initialization of particle swarm

According to population size n,random variables Ckfor each group are set and n groups of new hydrofoil on the basis of the original ones are generated.So at the moment of t,position function is an array yt（1:k,1:n）,while velocity function is an array vt（1:k,1:n）.The best position array of initial particle is p（1:k,1:n）=yt（1:k,1:n）.The largest value of fitness function p （1:k,nt_best）is selected to assign to the array of best global position pg （1:k）.

3.2.3 The evolutionary computation of particle swarm

According to the speed of the particle swarm and position evolution formula(1),the location and velocity function above are evolved.Here the inertia phase can be set to linear de-gressive adjustment operator.

where wmaxis maximum weight,while wminis minimum weight.nt is the current evolution algebra,while nmaxis the maximum evolution algebra assigned.At the moment of t1,the position and velocity of particle swarm are renovated to the arrays yt1（1:k,1:n）,vt1（1:k,1:n）,then the fitness(yt1（1:k,1:n）)is calculated.For each particle,if fitness(yt1（1:k,1:n）)is bigger than fitness(yt（1:k,1:n）),then update the best position of each particle as p（1:k, i）=yt1（1:k, i）,or else keep the original location.For global,if fitness(p（1:k,1:n）is bigger than fitness(pg （1:k）),then update newly global fitness function and record the largest location of particle as pg （1:k）=p （1:k,nt1_best）.Or else keep the best original position.

4 PSO examples

The aerofoil of Naca66mod is chosen to be optimized.This aerofoil has good cavitation performance and large lift-drag ratio and is adopted in the design of propellers widely[9].There is practical significance to optimize this aerofoil section with PSO.The attack angle is 1.5 degrees during optimizing the section.These symbols of cl,cd and tm represent lifting coefficient,drag coefficient,and maximum thickness ratio,respectively.While calculating,some parameters are set,such as the uniform flow velocity 10m/s,aspect ratio 4,b1=b2=2,wmax=1.0,wmin=0.2 and the velocity constraints k=0.5.The particle swarm scale is set as 20 and the particle dimension is set as 14.It is hoped that the surface of optimized hydrofoil can obtain the uniform pressure distribution and this case（-Cpmin>σ）does not occur.The change of lift-drag ratio in the optimization process is shown in Tab.1.The change of Cpwith eight optimization processes and the Geometry form change of hydrofoil with optimization process are shown in Fig.1 and Fig.2,respectively.

Tab.1 Optimization process of blade section at 1.5° attack angle

The optimization results of Tab.1 show that lift coefficient of hydrofoil is increased by 1.8% at 1.5° attack angle,lift-drag ratio is increased by 2.3% and the variation of thickness is not more than 1.02% compared the optimized hydrofoil with the original hydrofoil.This shows that PSO can improve the lift-drag ratio of the aerofoil section.

In Fig.1,it shows that pressure coefficient changes a lot near the rear of hydrofoil with the improvement of lift-drag ratio(increasing lift force and reducing drag force).In Fig.2,it also shows the geometric shape of the hydrofoil began to change accordingly(the specific change is that the front hydrofoil surface becomes thinner and streamline,the up and low surfaces turn to convex and concave from the middle to the rear of hydrofoil respectively).This method can counteract drag of hydrofoil surface in the flow direction effectively.

5 The checking of CFD method

The above method is a solver of calculating the fitness function applying panel method based on potential flow theory.It assumed that the fluid does not stray from the surface of hydrofoil in the evolution using PSO,so the possible fluid breakdown,cavitation and other phenomenon could not be predictied there.The result shows that the suction surface appears negative pressure peak gradually after 5 iterations.In optimization process,the hydrofoil is adjusted only according to the merit principle of lift-drag ratio but neglecting the appearing of pressure discontinuity or peak.Therefore the selected hydrofoil while evolution stopped is the one with some increase of lift-drag ratio,smooth pressure distribution and accepted inception cavitation number.

To verify the above results of optimization,the hydrofoil section of Naca66mod and the optimized aerofoil section after 8th iterations are simulated numerically using CFD method,which are denoted as Naca66mod,PSO-Naca66mod.In Fig.3,it shows the grid distribution of computing domain through large eddy simulation of viscous flow.The calculation results are shown in Figs.4-5 and Tab.2.

Tab.2 The comparison of calculative result

The pressure distribution of the two adopted hydrofoils is shown in Figs.4-5,which is accord with the calculated result in Fig.1 based on the potential flow theory generally.The lift coefficient and drag coefficient of two hydrofoils in the same condition are shown in Tab.2.Comparing the optimized hydrofoil with the original hydrofoil,drag coefficient decreased by 9.25%,lift coefficient increased by 4.66% and lift-drag ratio increased by 15.33%.The results in Tab.1 and in Tab.2 have a little deviation,but their trend is consistent，which validates that the optimization method using PSO is feasible.

6 Conclusions

PSO is applied extensively in the field of intelligent optimization,which needs small computational capacity and can gain the convergent result easily.This paper offers a method optimizing hydrofoil section,which combines the classical panel method and PSO.The accuracy of this comprehensive optimization is carried out through simulation of FLUENT which is one soft of CFD.The two numerical calculation results are consistent,which both proved that lift coefficient of the optimization hydrofoil increases and drag reduces.At the some time,there is a conformable trend of pressure coefficient distribution of the hydrofoil surface using CFD and panel method.This result can also prove that panel method has a higher accuracy of numerical calculation.

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