An Optimization Method of Stochastic Variable to Solve Uncertain Models in Engineering Design

D0I:10.13374/j.issm1001-053x.1991.05.009 北京科技大学学报 第13卷第5(I)期 Vo1.13No.5() 1991年9月 Journal of University of Science and Technology Beijing Sept.1991 An Optimization Method of Stochastic Variable to Solve Uncertain Models in Engineering Design Chen Lizhou He Xiaofeng' (陈立周) (何晓峰) ABSTRACT:The radical concepts of engineering stochastic optimization are introduced and a method for solving uncertain problems is developed according to the satisfied estimation of probabilistic values to stochastic con- straints and the use of a quasi-gradient searching direction.The method is applied to engineering designs with linear or nonlinear stochastic constraints. KEY WORDS:engineering design,stochastic variable,mechanical design In mechanical design,many factors are of stochastic natures,such as loads, working temperature,property of material,machining sizes,etc.That is to say,in a stochastic treatment,many stochastic design variables and parameters must be considered.Due to some or all of the design specifications may also be sufficiently indeterminate,it is required that they be treated as stochastic variables or stochastic functions (but not as design variables).Therefore,this kind of optimization can be generally formulated as the following form: min。E{f°(X,W)}: X=〔X1,X2,…,Xn〕，W=〔W1,形2，…W]r X,W∈(Q,T,P) s.t.E{f(X,W)}≤0 i=1,m1 (1) P{fi(X,W)≤0}≥a j=m1+1,m In the formulation(1),X and W respectively represents a column vector conta- 1990-05-05收稿 ,机幟工程系(Department of Mcchanica】Enginceri血g) 452

第 卷 第 。 期 北 京 科 技 大 学 学 报 。 。 年 。 月 。 。 ， 红 了 陈立 周 何 晓 峰 一 。 ， 。 ， ， ， ， ， ， ， ， ， ， 。 ， ， 。 “ ， 甲 〔 ， ， … ， 。 〕 ， ， 牙 〔平 ， ， 牙 ， … 牙 。 〕 ， 牙 任 口 ， ， 。 。 ‘ ， 班 毛 ， ’ ， 牙 镇 》 价 ， ， ， 牙 一 一 收稿 机械工程 系 DOI ：10．13374／j ．issn1001－053x．1991．05．009

ining N stochastic design variables and K stochastic parameters,f(X,W)repre- sents the objective specification which gives an indication of the quality of the de- sign,and f'(X,W)(i=1,2,..,m)represent design specifications which are linear or nonlinear stochastic functions,E represents expected value of stochastic specifications and P represents the probabilitiy that satisfies stochastic spe- cifications;a (j=m+1,m+2,.,m)are the specific probabilistic values which the stochastic constraints must satisfy.In the process of optimization, the expected values of stochastic variables are chosen to make iterations.That is mainly due to the expected values of stochastic variables are similar to the nominal values in traditional design methods.Also the stochastic optimization criteria such as min.Var..}and min.P(.}can be adopted according to the re- quirements of various designs.So it becomes more complicated to build the models when dealing with stochastic optimal design problems.As usual,only when all stochastic variables and parameters are normal distribution and the design specifications are simple functions,the stochastic optimization can be slo- ved by the use of the Chance Technique). In general cases,it must determine how the probabilistic distribution of each x varies as its mean varies,and at every iterative step in a numerical search,it is necessary to make a probabilistic analysis to obtain the mean of f (X,W)and the joint probability of failure associated with const:ained function. 1 General Principles of Algorithm The optimum of problem (1)may be described as follows: Dp and Ds(X)represent stochastic feasible region of problem (1)and a neighborhood of X. Supposing theX·∈Ds(X)cD,c(2,T,P),for allX∈Ds(X),if there exists fo(X,W)<f(X,W),then X is a constrained local stochastic optimum;and if for all XEDC(Q,T,P),there exists o(X.,W)f(X,W), then X.is a contrained global stochastic optimum. Several arbitrary assumptions are presented before the following discussions. The expected value of f (X,W)has been used in the optimization,and the distribution form of each stochastic variable dose not vary as the mean varies. In the execution of optimization,some characteristics of stochastic specfications are tried to be approximated step by step in order to save the solving time. A complete algorithm for stochastic optimization has to include two parts, one is the probabilitic analysis for the stochastic variables and stochastic design specifications,another is the search strategy of stochastic variables.In many 453

， ” ， 平 ， ‘ ， 牙 ， ，… ， 切 五 ， 。 。 ’ ， ， ， 一 ， 五 。 ， 。 呈 一 。 。 ， ， ， ’ ， 五 ， ， ， 班 ， 。 尸 。 任 。 · 二 ， 压 口 ， ， ， 任 。 ’ ， “ ‘ ，平 “ ，牙 ， ‘ 任 二 口 ， ， ， 。 ， ，详 二 。 ，详 ， ， 。 ， 。 ， 亏 ， ， ，

cases,their distributive forms can be determined by experiment,by comparision, by judgement of designer,Also their stochastic natures may be determined by the method of Maximum Entropys)and Moment Principle method.It is very convenient to use the last two methods to determine distributive nature of a stochastic function,For a function of 3M s=50t3 where the independent random variables are M and f.Their density functions are defined numerically as follows,and plotted in Fig.1. 1D0 7000.090 Bending moment/N.cm 0.314 Thickness/cm Fig.I Density functions of stochastic variables By the Moment Principle method,the expression of density function f(s)was got,and plotted in Fig.2.The expression is: 1030 50 70 90 110 Stress/N.cm-2 Fig.2 Density function from moment principle methocl f(s)=exp(-17.238+3.995e4s-6.957e-9.s2+3.143e-14.s8) In a general way,the design specifications are linear or nonlinear functions of stochastic variables and parameters,which may be represented as Y=f(X,W)=f(X1,X2…,Xw,W1,W2,…,W.), (2) f(X,形)，X,W∈(2，T,P) Suppose that s is a specific value and is not necessarily constant,then the probability that satisfies Y>S can be calculated as 454

， ， ， 〔 〕 位 〔 充〕 。 五 。 几 几 刀伟︸工名‘ 口‘伟 几口一心 。 一﹄ 五 八 。 ， 。 令 ‘ 孑 了 口 功 七 ，爪 。 卜主。 已 ， 〔 〕 ， “ ， 。 升 ， 咖 一 垃 亡 一 。 ， 一 一 。 一 。 “ 。 一 “ 一 “ 叭 ， 。 ， ， 牙 ， 一 ， 二 ， 平 ， 平 ， ， 平 ， ， 牙 ， ， 牙 〔 口 ， ， ， ， 乒 拜 林 魂 拜

PY-S≥0}=∫/.(s)Jif)ays (3) In Fig.3,suppose the shady area is A,then f() (s) 0 30 50 70 90 110 Stress/N.cm-2 Fig.3 Probability that satisfics design requirement P{f{Y-S≥0}=1-A (4) As stated above,stochastic design specification is really different from deter- ministic one. Now it should consider how to deal with stochastic constraint in formulation (1),namely P{f(X,)≤0}≥a (5) According to the Chance Techniqce3,,formulation (5)may be transferred into f(X,W)+Φ-1(a)a≤0 (6) But formulation (6)is no longer equivalent to (5)in general cases.Beca- use the transformation from formulation (5)to (6)is under the assumptions of normal distributive natures and the Taylor's series expansion of sim ple stochastic fanctions.If formulation (6)is used to make the transformation of formulation (5),the transferred errors will increase when design specifications are nonlinear functions and stochastic factors are of abnormal natures. In this paper the Chebyshev inequality is used to estimate probabili- stic constraint and make it satisfy a specific probabilistic value, The Chebyshev inequality has the following form 1 1 PyX≥x+bo:≤1+62orPX≤+b0.}≥1-1+6： (7) where X is a stochastic variable of any distributicn,X and ox respectively represents the expected value and variance of X. In the same way,following formulation may be obtained from formulation (7) 455

‘ 一 “ 。 ， 二 ， · · ， 夕 夕 。 ， ， 几 笼 几‘劝 训入。 多 ‘ 代找 刃 厂 任 仃 一 李 一 ， ， 尸 了 ， 牙 毛。 五 ‘ “ ’ ， 扭 耳 ， 牙 必 一 ’ 『， 往 ， 。 ， 川 ， 鱼 。 了 ， 二 十 吞， 二 一 ， ， 叮 山产、 了 丹户舀‘ ﹄ 任、、 万 兮 二 ， 。 舆

Pf(x,m)≤f(x,w)+6o,≥1-1+0 (8) In formulation (8)a point C is found (shown in Fig.4)that corresponds to a value 6',and C is called the Chebyshev point, C-J(x,w)+b*.Var(f(x,W)) Fig,4 The Chebyshev point (C) Specially,for a value b',if there are f(X,W)+6o4=0 (9) and a(6)=1-1+0 (10) then formulation (11)from (8)is got as follow P{fX,wW)≤0}≥a(b) (11) It can be seen that formulation (11)has the same form as formulation (6). From above explanation a special value b'is determined that makes formu- lation (9)equivalent to (11)no matter what kind of distribution form f(X,W) will have, So formulation (5)is transferred into f(X,W)+b0,≤0 (12) and the uncertain model in formulation (1)can be transferred into min.E{f°(X,W)}: X=〔X1,X2,…,Xw门T,W=〔W1,W2,…,WR)r, X,W∈(？，T,P) (13) s,t,E{f'(X,W)}≤0 i=1,m1 456

毛 ， 平 毛 ， 平 少 ， 几 ， 五 儿 ， ， ， 平 住 一 ，平 ’ 。 ，平 ， 平 ， 。 ，平 〔 ， ， ， … ， 〕 ， 平 〔犷 ， ， 平 ， … ， 牙 〕 ， 平 〔 口 ， ， ， ‘ ，牙 镇 ‘ ， ， 尽

Eifi(X,W))+6010 j=m+1,m According to paper (the stochastic quasi-gradient may be estimated as follows; 6X)=9-3fX”+4",m)-1X…,W4 ·…s: (14) where vector W and X'is respectively observed value of W.and X in the K-th modelling;vector h'is uniformly distributed over [-1,1)whose components are independent;both s and as represent possitive.value and a.1. When the point is in stochastic feasible region,the stochastic quasi-gradient of the objective function is considered as searching direction,As the point satisfies P{f'(X,w)≤0}≥a -i∈(X,e) a subproblem is needed to determine searching direction.The subproblem may be presented as d4+1=d*+6(0(X1)-0(X*)) (15) max。6 3 s.t。(d",s)+6≤0 (0*(X“),)+6≤0 i∈I(X",e) (16) -1≤1川s11≤1，j=1,N I(X",e:)={i:-ek≤9'(X)≤0，e>0} From the subproblem(16),the searching direction sk1 can be calculated. The step size p.used in each iteration may be chosen between the following two step sizes: (1)The stochastic step size p which assures the convergence of the algo- rithm 2pi=∞，p-0 (17) (2)The maximum step size p which may be determined as follows pi=max.(p:X +ps"ED,} (18) where D,represents stochastic feasible region. The step size p may be chosen from formulation (19) 457

， 牙 卜 ‘ 崔 斌下不丽 〕 一 。 ， ” 』 ， 二 了芬 ‘ 吞 ， ’ ‘ ， 平 告 一 ’少 · ， 平 ‘ 、 ， 。 一 一一 一一 一 一 一 月 刀 兮 一 一 一 矿 为 ‘ “ 犷 · 矿 么 ’ 一 ’ “ 〔 一 ， 〕 。 生 ， ， 已尽 一 几 ‘ ，评 成 。 任 ， · 弓 叮 ‘ 十 ’ 二 ‘ 口 ‘ 千 ’ 一 ‘ 。 。 毛 ， ‘ 吞镇。 几 盖 ， ‘ 毛 一 镇 日 川 毛， 任 盖 ， ￡ ， “ ， ， ￡ ‘ 二 礼 一 。 ‘ 夕 ‘ “ 镇 ， 。 、 ‘ ‘ 套 ， 仁 ， 卜 艺 ， ， 丈 丈 了 户艾 。 食 ‘ 份 “ 任 ， ， 。 ，。 ‘ 犷 ， 扭 。 ， 一 飞 》 立 爹碎

p*=m.ingp!,p' (19) 2 The Construction of the Algorithm The algorithm is as follows: (1)Give initial value such as means and Variances of all ramdom parameters, number of stochastic design variables and constraints,etc, (2)Find initial means of stochastic design variables that satisfies all sto- chastic constraints。 (3)Calculate stochastic quasi-gradient of objective function at the initial point from formulation (12)and take it as searching direction S',then go to step 5. (4)Determinate active constraints set I (x,e),if I (x,e)is empty,then go to step 3.Otherwise,calculate stochastic quasi-gradient of the active stochastic constrains and construct subproblem (16)to seach for iterative direction S., then go to step 5. (5)Calculate step size p,and make iteration X+1=X+pSk (6)If new point X+1 does not satisfy convergent condition,then go to step 4,otherwise go to step 7. (7)Output the final results and stop. 3 Application of the Method Shown in Fig.5 is a dam against ocean waves.It is a kind of risk decision problem,It has been observed that the maximum height of ocean waves may be: H=0.2Vmx (m) where ymx represents the maximum wind speed per year and it is supposed to be of logarithm normal distribu- tion. e88 Fig.5 The cross section of the dam 458

‘ 二 。 户 ， 户 ‘ 五乎 三， 一 “ ‘ 么 萝 既 时 牌 己 ， 往 ， ， ， 五 一 山 ， 。 。 ， ， 二 仑 了， 。 ， 一 ， ‘ ， ‘ ‘ ‘ ‘ ‘ ， ， 五 。 五 五 二 二 。 犷二 二 犷 二 二 葬 爪 爪 。 誉冬 五 移 五 稗

To design the dam under the presupposition that the cost is minimum (cost of the dam may be evaluated by the formulation 3h2+B2)and the special design probability is no less than 95%. The stochastic design variables and parameters are presented in as follows: Stochastic design variable (m) B h Distribution form Normal Normal Variation 0.15 0.15 Stochastic parameters Vm.x(km/h) H(m) Distribution form Logarithm Normal Normal Mean 80.0 12.0 Variance 16.0 2.40 So far as the wave height H is concerned,the stochastic constraint is P{H-h≤0}≥95% Suppose the P.represents the pressure suffered by the dam,the Pa.m is the strength of the dam,other stochastic constraint is P{P。en-Pam≤0}≥95% The results obtained by the method are given in the following table, Table The results of dam design Method Cost Optimum R① b② Increment AR③ solution (x) of cost Stochastic 866.67 B=9.74Rp1=95%b1=0.4875△C=4%△R1=87.30% optimum design h=17.47Rp2=95%b2=0.5000 4R2=91.12% Determinate 833.33 B=8.33R41=51.0% optimum design h=16.00R42=49.5% 1 R-actual probability of stochastic constraint 2b-relating to the chebyshev point ③△R-increment of the constraint probability The curves in Fig.6 present the density functions of the above two stochastic constraints f(x,w)and f2(x,w)which show the probabilities of the design. In the Table,it can be seen that probability of the first stochastic constraint increases by 87.30%and probability of the second stochastic constraint increases by 91.12%,and based on deterministic results,the objective value only increas- by 4%.From this point of view,stochastic optimal method is a very mean- ingful method. 459

五 皿 立 垃 了 么 。 一 ’ 。 石 犷 二 住 。 。 。 。 ， 尸 一 蕊。 写 。 。 。 ‘ 。 ， 五 ‘ 沮 ， 。 。 。 。 一 。 簇 。 异 住 。 ‘ ， 劣 ① ② 乙 ⑧ 爪 宜 。 。 。 。 。 。 。 △ 二 △ 二 。 。 。 二 。 二 。 ① 一 ② ‘ 一 ⑧ △ 一 二 。 五 ‘ ，， “ ‘ ，， 。 ， 。 写 。 ， ， 川 环 。 ， ‘ “

Density function Density function Rg,=95.5% p2=95.1g 0,51.0% R0249.5% 0.0 f'OY,W网=POCEAN一6AM f2(x,w=II-h Fig.6 Refference 1 He Xiaofeng.MS's paper of University of Science and Technology Beijing (Chinese).1988;.5 2 Chen Lizhou,et al.Optimal Design for Mechanical Engineering,Shanhai Sci.and Tech,Press (Chinese):1982 3 Rao S.S.,Das G.ASME,1984,18:106 4 Eronoliev Yu M.Stochastic Programming Method,Moscow:1976 5 Siddall J N.Probabilistic Engineering Design,DEKKER,New York: 1983 460

加 向 主。 阶 ‘ ，， 一 ，，叭 二 ， · 山厂、 仓 双飞 二， 尺。 二 七加 瓜丽舀而 一粼 。 厂忱、洲祖未 。 ， 。 五 ， 。 五 ， 。 。 。 。 ， 。 ， ， ， 。 ， ， ‘ 心