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Optimal Design of Compliant Trailing Edge for
Shape Changing
Abstract: Adaptive wings have long used smooth morphing technique of compliant leading an d trailing edge to improve their aerodynamic characteristics．This paper introduces a systematic approach to design compliant structures to carry out required shape changes under distributed pressure loads．In order to minimize the deviation of the deformed shape from the target shape，this method uses M ATLAB and ANSYS to optimize the distributed compliant mechanisms by way of the ground approach and genetic algorithm (GA)to remove the elements possessive of very low stresses．In the optimization process，man y factors should be considered such as air loads，input displacements，and geometric nonlinearities。Direct search method is used to locally optimize the dimension an d input displacement after the GA optimization。The resultant structure could make its shape change from 0 to 9．3degreesTheexperimental data of the model confirm s the feasibility of this approach．
Keywords： adaptive wing；compliant mechanism；genetic algorithm ；topology optimization；distributed pressure load；geometric nonlinearity
1 Introduction：
As conventional airfoil contours are usually designed with specific lift coefficients and M ach numbers，they could not change in accordance with the environment changing．Siclari and Austin indicated that the variable camber trailing edge would produce the drag about sixty percent less than the conventional fixed camber airfoil
There are three methods used to design able camber wings．Of them．one is conventional hinged mechanism，which，however, will create discontinuities over the wings surface leading to earlier airflow separation an d drag increase． The others are smart material and the compliant mechanism，of which both could realize smooth shape changing．Nevertheless，compared to the compliant mechanism，the smart—material—made actuators have many disadvantages，such as deficient in energy ,slow in response，strong in hysteresis，limited by temperature，and difficult to control too many actuators．Musolff from Industry University of Berlin used Ni—Ti shape—memory—alloy wire to make an adaptive variable camber wing，which could quickly change its shape，but could not perform highly frequent alteration because of its resilience depend en ton the heat exchange with the outside environment。
Compliant mechanism is a kind of one-piece flexible structure，which can transfer motion and power through its own elastic deformation．It is not only flexible enough to deform，but also has enough stiffness to withstand external loads．Thanks to its joint—free nature，it does not have the trouble some problems confronted by conventional mechanism such as friction，lubrication，noise and recoiling，thereby achieving smooth shape changing.
In 1 994，Kota，a professor from University of Michigan，firstly pointed out that compliant mechanism could be used to control static shape changing under the sponsorship of the Air Force Of ice of Scientific Research in USA．Saggere and Kota
suggested a new method to design compliant adaptive structures，which made the least square errors between the shape—changed curve and the target curve as the objective function for optimization．Based on their work，Lu put forward a load path
representation method．However, her work was limited to only linear analysis under consideration of nodal loads．Good[ from Virginia Polytechnic Institute of State University used the compliant mechanism and the Moving Asymptotes method to design the fuselage tail within the allowable range of its tip maximal deflection．Kota and He trick in2004 designed a compliant trailing edge on the base
of the F16s data，which can change from 0。to 15。and obtained a patent．Campanile from German Aerospace Center presented a modal procedure to design synthetic flexible mechanisms for airfoil shape control，and pointed out that the future re—search should take into account the air load and the geometric nonlinearity．Buhl from Riso National Laboratory of the Wind Energy Department in Denmark used the SIM P method and geometrically nonlinear finite element method to design compliant trailing edge flaps．FlxSys Inc in 2006 produced an adaptive compliant wing，which stood the test on the White Knight airplane．The results indicated that the compliant trailing edge could change+10 ．In China，the research of adaptive wing has been concentrated on smart material and conventional mechanism．Few people，it seems，have worked on designing adaptive wings with the compliant mechanism．Yang is an exception．He analyzed the active aero—elastic wings based on the aero—servo—
elasticity technology．Chen and Huang separately investigated the morphing of the compliant leading edge from the viewpoints of discreteness and continuity.
This paper presents a method to design the shape changeable structure by MATLAB and AN—SYS associated with distributed compliant mechanism on the base of the ground structure approach and genetic algorithm (GA)taking into account the external distributed loads and geometric nonlinearity.
2 Optimization Process:
2．1 Defining the trailing edge model and objective function
As shown in Fig．1，both curves represent two ideal shapes of the trailing edge in the different flying states．One side point)of the structure is supposed to be fixed，and the other side point) to be sliding horizontally. Firstly, the design domain should be defined by the initial curve shape．the input location and the boundary conditions．Then．it is divided with abeam element network simulating the bird’s feather as shown in Fig．2．This is termed the partial ground structure method.
Fig．1 Initial shape and target shape Fig．2 Discretization of the design domain
The simplest and most effective way to manufacture the planar compliant mechanism is to use wire—cutting technology．In the optimization pro—gram，all the elements are of rectangular beams with the same width equal to the thickness of the material，every beams height being a design variable．
In order to make the structure’s deformation come close to the target shape curve，the least square error(LSE)between the deformed curve and the target curve is defined as the objective function．LSE is the sum of squares of position differences of various points along the curves Its expression is
where I (=1，2，?，P)is the number of the points along the curves，P is the total number of points．a(chǎn)ndare the coordinates of it h node on the target and deformed boundary curve respectively．
The constraints are
Where J (=1，2，?， )is the number of elements，miss the tota1 number of elements，，hi the dimension variable，hmin and hmax are the lower and upper bounds of the element beam height for all elements with the value dependent on manufacturing，hb the height of the boundary elements， the maximumnoda1 deformation of the nodes on the curve boundary when the input point is inactive，and should be smaller than[d]to ensure structure stiffness，[d] the allowable maximum displacement when the input point is inactive，O'max the maximum stress of al1 the elements which must be smaller than Tj to prevent yielding，Tj the topology variable equal to 1，or else0 when the element is eliminated．
2．2 GA optimization
GA is an optimization method which simulates the heuristic selection rule in nature，where the fit．test living things have the most chance to survive，but the inferior ones also have the opportunity to exist． Different from the continuous optimization method，it does not require the gradient-based in—formation of the objective function.
Every element could be expressed as a topology variable and a dimension variable． There—fore，each individua1 could be coded as follows
where ，2 is the number of elements except the boundary ones．With the same heights，the boundary elements throughout the optimizing process are
represented by only one variable，hb.
The fitness is the criterion of the GA optimization．It could be transformed from the objective function into
where βis a coefficient deciding the compulsive selection of the betterindividua1．The smaller the value，the more different would be between the two individuals’fitness thus increasing the compulsiveness of choosing the individual of higher fitness．
The selection of control parameters plays an important role in the convergence of the GA．Generally speaking．the cross probability ranges 0．40—0．99；the mutation probability is 0．000 01-0．01．a(chǎn)nd the number of individuals 1 0．200．
The variable would be updated through the crossover and mutation，so the possible design could generate in the GA process．
2．3 Finite element analysis(FEA)
Because of the limited design variables and the target function，the optimization module of FEA software could not be used to design the compliant morphing mechanism．Therefore，this paper programmed the GA in MATLAB and the FEA in ANSYS．In the FEA，taking only account of geo—metric nonlinearities and the material being of linear elasticity, ANSYS could solve the node displacements and the element stresses．Then by deleting the elements with low stress，the fitness could be calculated．Fig．3 shows the detailed process．
Fig．3 Flowchart of the structural optimization program．
2．4 Second optimization
Although the GA could optimize the topology and dimension simultaneously in a large solution space，the dimension usually could not directly converge to the optimization．In order to solve this problem，after the GA，the Direct Search method
should be used to find the best values of the input displacement and the dimensions of the elements which remain in the results after the GA．
For morphing of compliant mechanism，F(xiàn)ig．3describes the whole optimization process．It mainly contains initialization of the design domain，F(xiàn)EA，GA optimization and second optimization.
3 Presentation of Results:
Adopted from Ref，the sizes of the initial and the target trailing edge are reduced by sixty percent．，I1ab1e 1 lists the design parameters.
Because the displacement is used as the input，the nonlinear analysis could hardly converge and the stress of the initia1 solutions is very large．Which should be considered after thirtieth generation.
Table 1 Design parameters
Fig．4 and Fig．5 illustrate the results from the GA optimization and the second optimization respectively．
Fig．4 Results after the GA optimization Fig．5 Results after the second
optimization．
Form Table 2，it could be found that through the second optimization of the input displacement and the dimension，the LSE is reduced by 1．352 8mmand improved by 3．13％ ．The altered angle is increased by 1．049 3
Table 2 Results after the two optimization
Fig．6 Stability of final optimal structure
Fig．6 shows the influences of the parameters when the outside distributed pressure load changes from 0 to 1 0 N／mm and the input displacement re—mains 1 1．389 7 mm on the optimal structure．It could be seen that the optimal structure has a good stability if the load is kept in the range Of 0—5 N／mm．As the external load exceeds 5 N／mm，the max stress is likely to exceed the yield stress．
Because this optimization program is based on the M ATLAB and ANSYS．in order to verify the results．a(chǎn)n attempt is made to introduce the analytical results of the optimized structure into ANSYS and PATRAN respectively, and then a comparison is made between them．As shown in Fig．7 and Fig．8，the two altered shapes are in good agreement：for in ANSYS the tip displacement is 54．97mm and in PATRRAN 54．50mm．The minor difference between them is from the software．
Fig．7 Results of FEA in ANSYS Fig．8 Results of FEA in PATRAN
On the other hand，a model is made by wire—cutting technology to verify the analytical results．The material of the mode1．identical with that of the design，is 5 mm thick．In the experiment，the distributed pressure load is assumed to be zero．The input displacement 11．389 7mm with the required input load 146 N．Fig．9 shows the model and the experimental result．The altered angle is measured9．3。．a(chǎn)nd the tip displacement 53mm．The altered shape well accords to the optimized result．If a displacement of 11．3897mm is imposed on the model，the theoretical tip displacement is 54．796 mm． Be．cause of the friction there is between the model and the experiment table a tiny difference will take place between the measured data and the calculated results．
Fig．9 The model and experimental result
4 Conclusions：
Proved by the simulation and experiments，the proposed method to design morphing compliant mechanism is effectual in turning out a trailing edge with required morphing effects and ability of with—standing external loads．The combination of MAT—LAB and ANSYS in the optimization renders the program simple and universa1．There is no need for frequent changes of the rigid matrix．It also avoids the complexity of programming the nonlinear FEA and the transforming distributed loads into nodal loads．Using the mixed code，the topology and the dimension could simultaneously be optimized by the GA．Removing the free elements after the FEA could speed up the optimization．The second optimization could improve the GA results．
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符合尾隨邊緣形態(tài)變化的優(yōu)化設(shè)計(jì)
摘要：自適應(yīng)機(jī)翼一直使用柔和的技術(shù)指導(dǎo)變形的后緣，以改善他們的氣動性能，本文介紹了一種在分布壓力下，符合形狀變化的結(jié)構(gòu)設(shè)計(jì)的系統(tǒng)化方法。為了使需要的形狀與目標(biāo)形狀偏差盡量最小，這種方法使用MATLAB和ANSYS的方式來優(yōu)化標(biāo)準(zhǔn)分布機(jī)制。這種方式通過局部優(yōu)化和遺傳算法來獲得。在優(yōu)化過程中，許多因素應(yīng)該考慮在內(nèi)，例如：空氣載荷、輸出位移量和幾何非線性。直接搜索法適用于局部優(yōu)化和GA優(yōu)化后的輸入位移量。由此產(chǎn)生的結(jié)構(gòu)可以做出他們在0到90。.之間變化，模型試驗(yàn)已經(jīng)確認(rèn)了這種方法的可行性。
關(guān)鍵詞：自適應(yīng)機(jī)翼，伺服順從機(jī)構(gòu)，遺傳算法，拓?fù)渥顑?yōu)化，分布壓力載荷，幾何非線性
1．說明
由于傳統(tǒng)的機(jī)翼輪廓通常是按照特定的上升系數(shù)和馬赫數(shù)設(shè)計(jì)的。他們不能隨著環(huán)境的變化而變化。Siclar和Austin指出可變的后緣曲面將會產(chǎn)生比傳統(tǒng)的固定傾角機(jī)翼少60%左右的阻力。
有三種去設(shè)計(jì)可變的曲面機(jī)翼的方法。他們中的一種是傳統(tǒng)的鉸鏈機(jī)構(gòu)，然而，他會導(dǎo)致機(jī)翼表面的不連續(xù)性和早期氣流分流與阻力的增加。其它的則是智能材料和順從機(jī)構(gòu)，他們能實(shí)現(xiàn)平穩(wěn)的形狀變化。盡管如此，與順從機(jī)構(gòu)相比較，由智能材料制成的傳動裝置有許多不足之處。例如：能量不足；反應(yīng)緩慢；強(qiáng)烈的滯后性；受溫度的限制；控制太多裝置的難度大。由來自柏林工業(yè)大學(xué)的用鎳鈦記憶合金作出的自適應(yīng)可變拱形的機(jī)翼可以快速改變他的形狀，但他不能執(zhí)行高頻繁的變化，因?yàn)樗膹椥砸蕾囉谂c外部環(huán)境進(jìn)行的熱量交換。
順從機(jī)構(gòu)是一種單件靈活的機(jī)構(gòu)。他可以通過彈性變形傳送運(yùn)動和能量。他不僅具有足夠的變形性，而且具有足夠的剛度來抵御外部的載荷。由于他的連接自由性，他沒有傳統(tǒng)所面臨的棘手問題，例如：摩擦、潤滑、噪聲、反沖。因此可以獲得平穩(wěn)的形狀變化。
1994年，一位來自密歇根大學(xué)的名叫kota教授首先提出順從機(jī)構(gòu)能夠使用在一項(xiàng)由美國空軍科學(xué)研究院辦公室提供贊助的控制靜態(tài)形狀的科學(xué)研究之中。Saggere和Kota提出了一種設(shè)計(jì)順從機(jī)構(gòu)的新方法，他們能夠使優(yōu)化目標(biāo)函數(shù)曲線中的形狀變化和目標(biāo)曲線的形狀誤差最小，基于他們的研究成果，Lu提出了一種載荷路徑代表方法。然而，他的研究僅限于節(jié)點(diǎn)情況下的線性分析。來自于福尼亞州立學(xué)院的Good使用順從機(jī)構(gòu)和運(yùn)動漸近法來設(shè)計(jì)機(jī)翼的尾部，保證誤差在尖端最大偏差范圍之內(nèi)。基于F16的數(shù)據(jù)，Kota和Hetrick在2004年時(shí)間設(shè)計(jì)順從軌跡邊緣，他能在0。到15。之間變化并且獲得了專利證書。來自德國航空航天中心的Companaile提出了模擬靜態(tài)程序設(shè)計(jì)機(jī)翼形狀控制合成靈活機(jī)構(gòu)，并指出今后的研究應(yīng)將空氣載荷和幾何非線性考慮在內(nèi)。來自工業(yè)能源部實(shí)驗(yàn)室的Buhl使用SIMP法和幾何非線性有限元法來設(shè)計(jì)順從軌跡邊緣。Flxsys Inc在2006年生產(chǎn)的自適應(yīng)兼容機(jī)翼。經(jīng)過了在懷特騎士飛機(jī)上的實(shí)驗(yàn)。結(jié)果表明，風(fēng)和標(biāo)準(zhǔn)的能在（-10。—10。）變化。在中國，適應(yīng)性機(jī)翼研究一直集中在智能材料和常規(guī)機(jī)構(gòu)上，幾乎沒有人在從事帶有順從機(jī)構(gòu)的機(jī)翼研究上。楊是個(gè)例外，他分析了基于伺服彈性技術(shù)的活躍航空彈性機(jī)翼，陳和黃分別調(diào)查了兼容的離散和連續(xù)性的前沿變化。
本文介紹了一種基于局部優(yōu)化和遺傳算法形狀可變機(jī)構(gòu)的設(shè)計(jì)方法，通過使用MATLAB和ANSYS設(shè)計(jì)，同時(shí)將外部載荷和幾何非線性考慮在內(nèi)。
2．優(yōu)化步驟
2.1確定后緣模型和目標(biāo)函數(shù)
如圖一所示，兩條曲線代表不同飛行狀態(tài)的軌跡邊緣。其中一邊（A點(diǎn)）的結(jié)構(gòu)形狀是固定的，另一邊（B點(diǎn)）將水平滑動。
圖一 圖二
首先設(shè)計(jì)領(lǐng)域應(yīng)該由最初曲線形狀所定義，包括輸出位置和邊界狀態(tài)，然后如圖二所示的被光線分成的微量網(wǎng)格模仿鳥的羽毛部分，這就是被稱為局部表面結(jié)構(gòu)方法。
最簡單也是最有效的方法制造出平面兼容機(jī)是使用線切割技術(shù)。在優(yōu)化過程中，所有的元素使用同樣的寬度等于其厚度的梁。其中每個(gè)梁的高度是一個(gè)設(shè)計(jì)變量。
為了使結(jié)構(gòu)的變形接近于目標(biāo)曲線形狀，在變形曲線和目標(biāo)曲線間的最小平方差是被定義的客觀職能。LSE的定義是沿曲線上各個(gè)點(diǎn)位置數(shù)字的平方和，他的表達(dá)式是
其中i（i=1,2,…，p）是沿曲線上點(diǎn)的數(shù)量，p是點(diǎn)的總數(shù)。和是目標(biāo)和邊界曲線變形坐標(biāo)的第i個(gè)節(jié)點(diǎn)。約束條件是：
其中j（j=1,2,…，m）是元素的數(shù)量的總和，hi是尺寸變量，hmin 和 hmax是所有元素的下界與上界，hb是邊界元素的極點(diǎn)，dmax是黨邊界曲線上輸入無效節(jié)點(diǎn)時(shí)的最大彎曲，必須小于 [d] 以保證結(jié)構(gòu)的剛度，[d]是當(dāng)輸入處于無效時(shí)所允許的最大彎曲變形，拓?fù)淞縏j等于1，否則當(dāng)元素被淘汰時(shí)為0。
2.2 GA優(yōu)化
遺傳算法是一種在自然界上模擬選擇的優(yōu)化方法。合適的生物能最大可能性存活下來，但是劣質(zhì)品種也有機(jī)會存在。不同于連續(xù)的優(yōu)化方法，他不要求梯度的目標(biāo)函數(shù)信息。
每一個(gè)元素可以表示為一個(gè)拓?fù)淞亢鸵粋€(gè)尺寸變量。因此，每個(gè)個(gè)體科編碼如下：
其中n是除邊界元素之外元素的數(shù)量。有著同樣的高度，在整個(gè)優(yōu)化過程中的邊界元素只有一個(gè)變量代表hb。
適應(yīng)性是遺傳算法優(yōu)化的評價(jià)標(biāo)準(zhǔn)。他可以從目標(biāo)函數(shù)轉(zhuǎn)化為：
其中β是一個(gè)只包括雙方較差的個(gè)體參數(shù)。他的數(shù)值越小越有價(jià)值，兩個(gè)個(gè)體的適應(yīng)性會有更多的不同，因此增加了雙方選擇的高度適應(yīng)性。
選擇控制參數(shù)在遺傳算法的收斂中扮演一個(gè)重要的角色。總的來講，交叉概率的范圍為0.40-0.99；突變的概率為0.00001-0.01，個(gè)體的數(shù)量為10-200。
該變量將會通過交叉和變異發(fā)生更新，因此，這個(gè)設(shè)計(jì)可能產(chǎn)生遺傳過程。
2.3適應(yīng)性元素的分析
由于設(shè)計(jì)變量和目標(biāo)函數(shù)是有限元的，有限元分析法優(yōu)化模型是不能被用于設(shè)計(jì)符合變形的機(jī)構(gòu)中，因此，本文在MATLAB中的遺傳算法和在ANSYS中的有限元分析法。在有限元分析法中，僅只考慮幾何非線性和材料的彈性，ANSYS能解決節(jié)點(diǎn)位移和元素壓力，通過刪去低應(yīng)力的元素，良好的結(jié)果能被推算出來。圖三顯示了詳細(xì)的過程。
圖三 整個(gè)的優(yōu)化過程
2.4二次優(yōu)化
盡管遺傳算法可以優(yōu)化大型解空間和拓?fù)浣Y(jié)構(gòu)尺寸。尺寸通常不能直接集中于優(yōu)化中，為了解決這個(gè)問題，遺傳算法優(yōu)化后，直接搜索法應(yīng)該被用來在遺傳算法結(jié)果中去尋找。
3.優(yōu)化的結(jié)果
通過參考文獻(xiàn)[5]可以得出，最初的小徑邊緣尺寸減少36%，表一列出了設(shè)計(jì)參數(shù)的大小。
表一 設(shè)計(jì)參數(shù)的大小
由于位移作為輸入的使用，非線性分析難以解決和廚師壓力非常大，但他必須在三十代以后考慮。
圖四 遺傳優(yōu)化的結(jié)果 圖五 二次優(yōu)化的結(jié)果
圖四和圖五說明了遺傳算法優(yōu)化結(jié)果和二次優(yōu)化結(jié)果。
表二 兩次優(yōu)化的比較
從表格中可以發(fā)現(xiàn)，通過輸入位移和尺寸優(yōu)化，LSE減少了1.3528mm和改善了3.13%，變更角度增加1.0493。。
圖六 外部載荷的分布
圖六表示的是外部分布壓力從0到10N/mm，改變輸入位移量在最初結(jié)構(gòu)上保持11.3897mm上的參數(shù)影響結(jié)果。如果載荷保持在0-5N/mm范圍內(nèi)，優(yōu)化結(jié)構(gòu)看起來有良好的穩(wěn)定性。當(dāng)外部載荷超過5N/mm時(shí)，最大壓力可能超過屈服壓力，因?yàn)檫@個(gè)優(yōu)化方法是基于MATLAB和ANSYS的，為了證明結(jié)果，嘗試去通過將分析結(jié)果分別輸入到ANSYS和PATRAN中，然后是他們之間的比較。如圖七和圖八所示，二者的變更有很大的共同點(diǎn)；在ANSYS中是54097mm，在PATRAN中是54.50mm，他們的不同之處來自個(gè)體上。
圖七 在ANSYS上的結(jié)果 圖八 在PATRAN上的結(jié)果
另一方面，一個(gè)使用線切割技術(shù)的模型來證實(shí)分析法的結(jié)果。模型的材料同設(shè)計(jì)的一樣，都為5mm后。在試驗(yàn)中，假設(shè)分布壓力載荷為零，輸入146N的情況下，輸入位移量為11.3897mm，圖九表示的是模型和測量的結(jié)果。變更的溫度為9.3。。尖端為一位53mm，變更的形狀符合設(shè)計(jì)的結(jié)果。如果11.3897mm的位移量強(qiáng)加在模型上，理論的尖端位移量為54.796mm。因?yàn)槟Ｐ秃驮囼?yàn)臺之間存在摩擦力，測量材料和適合的結(jié)果之間會有少許的差異。
圖九 模型和實(shí)驗(yàn)的結(jié)果
4.結(jié)論
通過方針和實(shí)驗(yàn)證明，該方法符合設(shè)計(jì)變形機(jī)制，探索出具有所需的變性效應(yīng)和承受外部載荷的結(jié)果和能力的機(jī)構(gòu)。在優(yōu)化過程中，MATLAB和ANSYS的聯(lián)合呈現(xiàn)程序的簡單和普遍性。堅(jiān)硬的字模沒有必要頻繁的改變，同時(shí)避免有限元法編程的復(fù)雜性和使分布載荷變成節(jié)點(diǎn)載荷，拓?fù)涑叽缈梢酝瑫r(shí)由GA進(jìn)行優(yōu)化，出去再FEA之后的自由元素能加快優(yōu)化，二次優(yōu)化可以提高GA優(yōu)化的結(jié)果。
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