基于多相材料的穩(wěn)態(tài)熱傳導(dǎo)結(jié)構(gòu)輕量化設(shè)計(jì)1)

龍 凱 王 選 韓 丹

?(華北電力大學(xué)新能源電力系統(tǒng)國(guó)家重點(diǎn)實(shí)驗(yàn)室，北京102206)

?(大連理工大學(xué)工業(yè)裝備結(jié)構(gòu)分析國(guó)家重點(diǎn)實(shí)驗(yàn)室，遼寧大連116024)

基于多相材料的穩(wěn)態(tài)熱傳導(dǎo)結(jié)構(gòu)輕量化設(shè)計(jì)1)

龍 凱?,2)王 選?韓 丹?

?(華北電力大學(xué)新能源電力系統(tǒng)國(guó)家重點(diǎn)實(shí)驗(yàn)室，北京102206)

?(大連理工大學(xué)工業(yè)裝備結(jié)構(gòu)分析國(guó)家重點(diǎn)實(shí)驗(yàn)室，遼寧大連116024)

在多相材料的結(jié)構(gòu)拓?fù)鋬?yōu)化問題中，通常給定各相材料體積約束或材料總重量約束作為材料的控制用量.在結(jié)構(gòu)輕量化設(shè)計(jì)的實(shí)際工程背景下，以結(jié)構(gòu)總重量最小化為目標(biāo)的優(yōu)化模型具有明確的工程意義.針對(duì)含多相材料的穩(wěn)態(tài)傳熱結(jié)構(gòu)拓?fù)鋬?yōu)化問題，提出了以結(jié)構(gòu)總重量最小化為目標(biāo)和給定熱柔順度為約束的多工況連續(xù)體結(jié)構(gòu)拓?fù)鋬?yōu)化建模方法.遵循獨(dú)立連續(xù)映射建模方式，采用兩類獨(dú)立拓?fù)渥兞糠謩e表征單元熱傳導(dǎo)矩陣和單元重量狀態(tài).推導(dǎo)了熱柔順度和總重量對(duì)設(shè)計(jì)變量的敏度，基于一階和二階泰勒展開得到各自的近似表達(dá)式.通過求解偏微分方程，實(shí)現(xiàn)了約束函數(shù)一次項(xiàng)過濾，消除了棋盤格現(xiàn)象和網(wǎng)格依賴性問題，并保證了約束方程在過濾后嚴(yán)格成立.建立的近似優(yōu)化模型具有二次函數(shù)形式的目標(biāo)函數(shù)和一次函數(shù)形式的約束函數(shù).基于對(duì)偶序列二次規(guī)劃方法對(duì)優(yōu)化模型進(jìn)行求解直至收斂.通過四個(gè)三維結(jié)構(gòu)數(shù)值算例分析對(duì)比了熱柔順度約束限值、不同材料混合及多工況、多約束條件對(duì)優(yōu)化結(jié)果的影響.數(shù)值算例結(jié)果表明，本文提出的優(yōu)化方法在基于多相材料的多工況穩(wěn)態(tài)熱傳導(dǎo)結(jié)構(gòu)輕量化設(shè)計(jì)中具有可行性和有效性.

拓?fù)鋬?yōu)化,多相材料,穩(wěn)態(tài)熱傳導(dǎo),獨(dú)立連續(xù)映射法,序列二次規(guī)劃

引言

自Bendsoe和Kikuchi[1]提出連續(xù)體結(jié)構(gòu)拓?fù)鋬?yōu)化方法以來，各類方法如均勻化法、基于固體各向同性懲罰(solid isotropic material with penalization,SIMP)模型的變密度法、雙向進(jìn)化法 (bi-directional evolutionary structural optimization,BESO)、水平集法、相場(chǎng)法、可動(dòng)變形構(gòu)件法等被相繼提出，各類方法在不同時(shí)間階段的發(fā)展可參考綜述類文獻(xiàn)[2-6].作為結(jié)構(gòu)輕量化設(shè)計(jì)的工具，拓?fù)鋬?yōu)化方法不僅在機(jī)械或力學(xué)領(lǐng)域得到應(yīng)用，傳熱結(jié)構(gòu)輕量化設(shè)計(jì)方法研究也得以開展，尤其在航空、電氣、電子產(chǎn)品研發(fā)中的需求日益緊迫.Li等[7-8]較早應(yīng)用BESO方法實(shí)現(xiàn)了穩(wěn)態(tài)熱傳導(dǎo)結(jié)構(gòu)的拓?fù)鋬?yōu)化設(shè)計(jì).在SIMP方法和水平集方法中，類似的研究有所體現(xiàn)[9-11].在優(yōu)化模型方面，劉書田等[12-14]提出溫度方差、幾何平均溫度最小化模型，給出了熱載荷相關(guān)和克服BESO法敏度誤差的解決方案，實(shí)現(xiàn)了自發(fā)熱體冷卻的內(nèi)置導(dǎo)熱路徑最優(yōu)設(shè)計(jì).高彤等[15]提出了設(shè)計(jì)變量相關(guān)熱載荷作用下的BESO方法.莊春剛等[16-18]提出了瞬態(tài)熱傳導(dǎo)拓?fù)鋬?yōu)化的全局衡量指標(biāo)，采用SIMP法和水平集法實(shí)現(xiàn)了基于三角形網(wǎng)格、單點(diǎn)溫度約束和多相材料的瞬態(tài)傳熱結(jié)構(gòu)拓?fù)鋬?yōu)化設(shè)計(jì).

除了傳統(tǒng)的材料布局優(yōu)化問題外，Thomsen[19]較早研究了多相材料結(jié)構(gòu)的拓?fù)鋬?yōu)化問題.Sigmund和Torquato[20]運(yùn)用多相材料實(shí)現(xiàn)了負(fù)熱膨脹系數(shù)的材料微結(jié)構(gòu)優(yōu)化設(shè)計(jì).早期的多相材料研究多采用SIMP法，其后在水平集法[21-25]、相場(chǎng)法[26-27]和BESO法[28-29]中也有所體現(xiàn).Yin等[30]提出了峰值函數(shù)法，用單變量描述多相材料屬性以減少設(shè)計(jì)變量數(shù)目.高彤等[31-33]提出了線性對(duì)等混合材料插值模型，實(shí)現(xiàn)了重量約束下的多相材料布局優(yōu)化，方法具有變量可分離特征.賈嬌等[34]將該重量約束模型推廣到多相材料/結(jié)構(gòu)的一體化設(shè)計(jì)中.Tavakoli和Mohseni[35]將多相布局優(yōu)化問題轉(zhuǎn)化為序列兩相材料優(yōu)化問題進(jìn)行求解.Ramani[36-37]提出了離散變量的多相材料拓?fù)鋬?yōu)化方法.左文杰等[38]考慮了重量和成本約束，提出了序列SIMP方法.

隋允康[39]于1996年提出獨(dú)立連續(xù)映射(independent continuous mapping,ICM)法，該方法以獨(dú)立于單元具體物理參數(shù)的變量表征單元的有無.區(qū)別于其他方法，拓?fù)渥兞康莫?dú)立性主要體現(xiàn)在拓?fù)渥兞颗c物理量間的關(guān)系上，并涌現(xiàn)出磨光函數(shù)、過濾函數(shù)等新概念[40].其次，ICM方法以重量最小化為目標(biāo)，將應(yīng)力、節(jié)點(diǎn)位移、頻率為約束并納入到序列二次規(guī)劃求解體系下，處理多約束問題上具有穩(wěn)健高效的優(yōu)點(diǎn).基于對(duì)約束條件的理性處理，提出全局應(yīng)力概念并解決了強(qiáng)度拓?fù)鋬?yōu)化問題[41-42].近年來ICM方法研究成果集中體現(xiàn)在專著[43]中.

目前，多相材料的布局優(yōu)化通常采用各相材料多體積比約束或所有相材料的總重量約束建立優(yōu)化模型，而以重量最小化為目標(biāo)的拓?fù)鋬?yōu)化模型具有明確的工程意義.本文將針對(duì)穩(wěn)態(tài)熱傳導(dǎo)拓?fù)鋬?yōu)化問題，基于ICM方法建立多相材料布局的拓?fù)鋬?yōu)化模型,定義兩類獨(dú)立拓?fù)渥兞恳员碚鲉卧挠袩o和所對(duì)應(yīng)材料的強(qiáng)弱，推導(dǎo)了熱柔順度值和結(jié)構(gòu)總重量的敏度表達(dá)式.通過泰勒展開得到近似表達(dá)式，將原優(yōu)化模型轉(zhuǎn)換為二次規(guī)劃模型并求解.基于求解偏微分方程的方法實(shí)現(xiàn)了約束方程一次項(xiàng)過濾，消除了棋盤格現(xiàn)象和網(wǎng)格依賴性問題.最后通過三維數(shù)值算例驗(yàn)證了方法的可行性和有效性.

1 基于多相材料的拓?fù)鋬?yōu)化建模與求解

1.1 兩類獨(dú)立拓?fù)渥兞亢屯負(fù)鋬?yōu)化建模

在ICM方法中定義兩類獨(dú)立拓?fù)渥兞縮i(0≤si≤1)和ti(0≤ti≤1)用于表征單元的有無和所對(duì)應(yīng)材料的強(qiáng)弱，變量與單元熱傳導(dǎo)矩陣和單元重量的關(guān)系為

式中kI和kII分別表示材料I和II對(duì)應(yīng)的單元熱傳導(dǎo)矩陣；wI和wII分別表示材料I和II對(duì)應(yīng)的單元重量.si=1或0表示單元保留或空洞；ti=1或0表示單元選用材料I或材料II.α和β分別為熱傳導(dǎo)矩陣和單元重量的懲罰因子，本文數(shù)值算例α=4，β=1；N為設(shè)計(jì)域內(nèi)的單元數(shù)目.

遵循ICM建模方式，建立以結(jié)構(gòu)總重量W最小化為目標(biāo)，結(jié)構(gòu)熱柔順度為約束的多工況拓?fù)鋬?yōu)化模型

式中，cj和j分別表示j工況的熱柔順度和對(duì)應(yīng)的限值.對(duì)于多相材料布局優(yōu)化問題，當(dāng)設(shè)計(jì)區(qū)域全部由導(dǎo)熱性較好的材料組成，通過有限元分析得到的熱柔順度值為則熱柔順度值限值不小于時(shí)，傳熱結(jié)構(gòu)具有可優(yōu)化的余地.拓?fù)渥兞肯孪抻糜诒苊庥邢拊治龌蛎舳确治鲆鸬臄?shù)值不穩(wěn)定性.

基于有限元法的穩(wěn)態(tài)熱傳導(dǎo)方程為

式中,K為整體熱傳導(dǎo)矩陣，?為節(jié)點(diǎn)溫度列陣，P為溫度載荷列陣.由此可得

1.2 敏度分析與熱柔順度的顯式表達(dá)

由伴隨法易得熱柔順度對(duì)拓?fù)渥兞康拿舳葹?/p>

與SIMP方法不同，ICM采用拓?fù)渥兞?類似SIMP方法的密度變量)的倒變量函數(shù)作為設(shè)計(jì)變量，其好處在于：其一，當(dāng)設(shè)計(jì)變量增大時(shí)，常見的結(jié)構(gòu)響應(yīng)量如節(jié)點(diǎn)位移、熱柔順度值增大，這樣結(jié)構(gòu)響應(yīng)量與設(shè)計(jì)變量之間可以采用線性函數(shù)描述，并具有較好的精度；其二，倒變量表示的重量函數(shù)，其一階泰勒近似精度較差，但作為設(shè)計(jì)變量的顯式函數(shù)，可以采用二階泰勒近似來彌補(bǔ)其精度不足的缺點(diǎn).且重量函數(shù)的一階、二階導(dǎo)數(shù)可以解析出來，計(jì)算量?。黄淙?，當(dāng)分別采用一階、二階泰勒近似得到約束函數(shù)和目標(biāo)函數(shù)的顯式表達(dá)式后，采用序列二次規(guī)劃法求解，優(yōu)化迭代穩(wěn)健高效.這里定義兩類設(shè)計(jì)變量xi和yi

由式(6)可得

熱柔順度采用一階泰勒近似得到顯式表達(dá)式

式中上標(biāo)為k的各變量代表第k輪優(yōu)化迭代中的變量取值.

式中，向量B和矩陣H由W對(duì)設(shè)計(jì)變量的一階、二階導(dǎo)數(shù)計(jì)算得到.

令γ=-β/α，由式(1b)和式(6)可得

由式(11)可得重量函數(shù)的一階、二階導(dǎo)數(shù)

忽略重量函數(shù)常數(shù)項(xiàng)，由此可得優(yōu)化模型(2)的近似模型為

模型 (14)是標(biāo)準(zhǔn)的二次規(guī)劃模型，為了克服拓?fù)鋬?yōu)化中設(shè)計(jì)變量龐大的困難，通常轉(zhuǎn)化為對(duì)偶模型并采用序列二次規(guī)劃求解，具體過程可參考文獻(xiàn)[44-45].

1.3 消除拓?fù)鋬?yōu)化中的數(shù)值不穩(wěn)定性

棋盤格現(xiàn)象和網(wǎng)格依賴性問題是拓?fù)鋬?yōu)化中常見的數(shù)值不穩(wěn)定性現(xiàn)象，本文將通過求解偏微分方程的過濾方法來消除上述這些問題.

以設(shè)計(jì)變量xi為例，定義式(8)中含偏導(dǎo)數(shù)的一次項(xiàng)

以dij,x為過濾對(duì)象，過濾后的熱柔順度敏度為

這里通過求解偏微分方程(partial di ff erential equation, PDE)實(shí)現(xiàn)過濾[46].設(shè)dij,x構(gòu)成的場(chǎng)函數(shù)在過濾前后記為滿足Neumann邊界條件

式中r為PDE參數(shù)，與傳統(tǒng)過濾方法中的過濾半徑具有對(duì)應(yīng)關(guān)系[46].

可以證明，過濾前后滿足由式(18)可得，式(8)表示的約束函數(shù)不等式在過濾前后具有一致性.式(17)將過濾過程視為傳熱傳質(zhì)過程，可利用有限元求解器實(shí)現(xiàn)過濾，其詳細(xì)過程可參考文獻(xiàn)[46].

對(duì)于設(shè)計(jì)變量yi，采用類似的流程得到過濾敏度值，將過濾后的敏度值代入到優(yōu)化模型(14)中進(jìn)行優(yōu)化求解，更新設(shè)計(jì)變量、拓?fù)渥兞亢徒Y(jié)構(gòu).

為了保證拓?fù)鋬?yōu)化構(gòu)型的清晰性，采用文獻(xiàn)[44-45]描述的兩階段優(yōu)化策略，即在第一階段采用過濾措施，目的在于消除棋盤格現(xiàn)象和網(wǎng)格依賴性問題.第二階段不采用過濾措施直至優(yōu)化收斂，得到清晰的拓?fù)鋬?yōu)化構(gòu)型.優(yōu)化收斂判斷表達(dá)式為

式中兩階段收斂率ε分別取值1‰和0.1‰.

2 數(shù)值算例與討論

本節(jié)采用三維結(jié)構(gòu)數(shù)值算例來驗(yàn)證所提方法的可行性和有效性.如圖1所示的平板尺寸為64cm×64cm×4cm，采用八節(jié)點(diǎn)六面體單元離散，單元邊長(zhǎng)為1cm.在不加說明的情況下，整板8個(gè)角點(diǎn)保持相對(duì)恒定溫度為0°C.可選用材料的導(dǎo)熱系數(shù)λ和密度ρ如表1所示.假設(shè)平板全部采用純銅材料,重量為W0，優(yōu)化后的重量為W，以重量比W/W0來說明優(yōu)化效果.

圖1 三維結(jié)構(gòu)示意圖Fig.1 Illustration of 3D structure

表1 材料屬性Table 1 Properties of material

算例1 設(shè)在平板中心處加熱，熱源密度為100W/kg.當(dāng)平板為純銅或純鋁材料時(shí)，其熱柔順度值分別為 2586.9J或 4203.5J.以熱柔順度作為約束條件建立優(yōu)化模型，設(shè)置約束值須大于2586.9J；當(dāng)約束值小于4203.5J時(shí)，純鋁平板無法滿足約束條件.設(shè)熱柔順度約束值在3～6kJ變化，對(duì)于單一組分材料和銅鋁混合材料，不同約束值下的優(yōu)化重量比如圖2所示.圖中紅色代表銅材料，藍(lán)色代表鋁材料.

圖2 不同約束值下的優(yōu)化重量比Fig.2 Weight fraction under di ff erent constraints

由圖2可知，當(dāng)采用多相材料時(shí)，優(yōu)化結(jié)構(gòu)中銅、鋁和空洞并存，具有較好導(dǎo)熱性的銅材料布置在熱源和冷卻點(diǎn)附近.相比銅材料，鋁材料具有更高的比導(dǎo)熱性，在滿足約束條件的前提下，重量最小化目標(biāo)將驅(qū)動(dòng)結(jié)構(gòu)優(yōu)先選擇鋁材.隨著約束值的放松，優(yōu)化結(jié)構(gòu)總重量逐漸減少.相比單相組分材料銅或鋁，基于多相材料的優(yōu)化結(jié)構(gòu)重量更輕，這說明所提方法具有處理多相材料布局優(yōu)化的能力.

算例2 假設(shè)加熱點(diǎn)位于板z向中面內(nèi)，其在xoy平面上的投影如圖3所示，p2和p4點(diǎn)加熱，八角點(diǎn)散熱并形成兩個(gè)獨(dú)立工況，熱源密度為100W/kg，設(shè)熱柔順度約束值為6kJ，不同材料組合為:(1)銀和石墨；(2)純銅和石墨；(3)鈹銅和石墨；(4)鎢和石墨.拓?fù)鋬?yōu)化構(gòu)型如圖4所示，優(yōu)化結(jié)果如表2所示.圖4中藍(lán)色代表石墨材料，紅色代表其他材料.

圖3 加熱點(diǎn)示意圖Fig.3 Illustration of heated point

圖4 不同材料組合優(yōu)化拓?fù)錁?gòu)型Fig.4 Optimal topological configuratio under di ff erent combination of base material

表2 拓?fù)鋬?yōu)化結(jié)果Table 2 Results of topology optimization

相同的熱柔順度約束值對(duì)應(yīng)為相同的傳熱性能要求，由圖4可知，不同多相材料組合下的拓?fù)鋬?yōu)化構(gòu)型和結(jié)果不同.與算例1結(jié)果一致，在4種情況下，導(dǎo)熱性較強(qiáng)的材料均布置在加熱點(diǎn)和冷卻點(diǎn)附近，以獲得最佳的導(dǎo)熱效果，其余地方則布置較輕的石墨材料.純銅、銀、鈹銅和鎢的比導(dǎo)熱性依次下降，優(yōu)化結(jié)構(gòu)重量依次增加.在實(shí)際工程應(yīng)用中，應(yīng)根據(jù)實(shí)際情況選擇材料，實(shí)現(xiàn)結(jié)構(gòu)輕量化的目的.

算例3 圖3中的p2點(diǎn)和p4點(diǎn)為加熱點(diǎn)，與算例2不同的是，當(dāng)p4點(diǎn)加熱時(shí)，圖1中左側(cè)平面A四角點(diǎn)冷卻并構(gòu)成工況1；當(dāng)p2點(diǎn)加熱時(shí)，圖1中右側(cè)平面B四角點(diǎn)冷卻并構(gòu)成工況2，加熱點(diǎn)熱源密度和熱柔順度約束值同算例2.設(shè)熱柔順度值在5～7kJ之間變化.當(dāng)采用單一組分材料或銅鋁混合材料時(shí)，不同約束值下的優(yōu)化重量比如圖5所示.圖中紅色代表銅材料，藍(lán)色代表鋁材料.

圖5 不同約束值下的優(yōu)化重量比Fig.5 Weight fraction under di ff erent constraints

該算例中的加熱點(diǎn)和邊界條件的不同組合構(gòu)成了兩個(gè)不同工況，故而建立了多工況下的多約束拓?fù)鋬?yōu)化模型.由圖5可知，多相材料的分布具有同算例1類似的規(guī)律，相比單相材料銅或鋁，基于多相材料的優(yōu)化結(jié)構(gòu)重量更小，表明了所提方法在處理多工況多相材料布局優(yōu)化上的可行性.

算例4 加熱點(diǎn)位于板中面.如圖3所示，p1～p4點(diǎn)單獨(dú)加熱并形成4個(gè)獨(dú)立工況，選用純銅和鋁兩種材料實(shí)現(xiàn)傳熱結(jié)構(gòu)輕量化設(shè)計(jì)，設(shè)熱柔順度約束值為4kJ.(1)p1～p4點(diǎn)的熱源密度均為100W/kg；(2)p1～p4點(diǎn)熱源密度分別為100W/kg,102W/kg, 104W/kg和106W/kg.兩種情況下的優(yōu)化拓?fù)錁?gòu)型如圖6所示，材料顏色表示與算例1相同，拓?fù)鋬?yōu)化結(jié)果如表3所示.

圖6 優(yōu)化拓?fù)錁?gòu)型Fig.6 Optimal topological configuratio

表3 拓?fù)鋬?yōu)化結(jié)果Table 3 Results of topology optimization

由圖6可知，在熱源密度不同和相同的熱柔順度約束值下的拓?fù)鋬?yōu)化構(gòu)型有所不同.在情況2下，材料呈現(xiàn)非對(duì)稱構(gòu)型分布.由表3可知，2種情況下，4種工況的熱柔順度均略小于設(shè)定的約束值，說明優(yōu)化結(jié)構(gòu)充分發(fā)掘了優(yōu)化余地.提出的方法在單一重量最小目標(biāo)下，可以協(xié)調(diào)不同工況的熱源載荷得到優(yōu)化結(jié)構(gòu)，具有處理多材料、多工況、多約束的能力.

3 結(jié)論

以重量最小化為目標(biāo)、各類結(jié)構(gòu)響應(yīng)量為約束的優(yōu)化模型具有實(shí)際工程意義.本文提出基于多相材料的多工況穩(wěn)態(tài)傳熱結(jié)構(gòu)拓?fù)鋬?yōu)化的ICM方法.方法采用兩類不同拓?fù)渥兞勘碚鲉卧獰醾鲗?dǎo)矩陣與單元重量.遵循ICM方法，建立了重量最小化和以熱柔順度限值為約束的拓?fù)鋬?yōu)化模型，推導(dǎo)了熱柔順度和重量函數(shù)的敏度，顯式化表達(dá)了目標(biāo)與約束函數(shù)，采用序列二次規(guī)劃法優(yōu)化求解，主要得到以下結(jié)論：

(1)在相同的熱柔順度約束條件下，基于多相材料下的拓?fù)鋬?yōu)化結(jié)構(gòu)比單一材料的拓?fù)鋬?yōu)化結(jié)構(gòu)重量更小，能夠獲得更優(yōu)的材料布局設(shè)計(jì).

(2)隨著組分材料的不同，拓?fù)鋬?yōu)化構(gòu)型和優(yōu)化結(jié)構(gòu)的重量不同.

(3)提出的重量最小化模型，將不同工況下的熱柔順度設(shè)置為約束條件，能協(xié)調(diào)不同工況下的熱載荷,得到輕量化結(jié)構(gòu).所提方法具有處理多材料、多工況、多約束的能力，優(yōu)化結(jié)構(gòu)充分發(fā)揮了結(jié)構(gòu)潛力，優(yōu)化迭代穩(wěn)健高效.

(4)所提基于多相材料的傳熱結(jié)構(gòu)重量最小化模型可拓展到其他物理背景的拓?fù)鋬?yōu)化問題中.

1 Bendsoe MP,Kikuchi N.Generating optimal topologies in structural design using a homogenization method.Computer Methods in Applied Mechanics and Engineering,1988,71(2):197-224

2 Eschenauer HA,Olho ffN.Topology optimization of continuum structures:a review.Applied Mechanics Reviews,2001,54(4):331-390

3 Rozvany GIN.A critical review of established methods of structural topology optimization.Structural and Multidisciplinary Optimiza-tion,2009,37(3):217-237

4 Sigmund O,Maute K.Topology optimization approaches.Structural and Multidisciplinary Optimization,2013,48(6):1031-1055

5 Deaton JD,Grandhi RV.A survey of structural and multidisciplinary continuum topology optimization:post 2000.Structural and Multidisciplinary Optimization,2014,49(1):1-38

6 Zhang W,Yuan J,Zhang J,et al.A new topology optimization approach based on Moving Morphable Components(MMC)and the ersatz material model.Structural and Multidisciplinary Optimization,2016,53:1243-1260

7 Li Q,Steven GP,Querin OM,et al.Shape and topology design for heat conduction by evolutionary structural optimization.International Journal of Heat&Mass Transfer,1999,42(17):3361-3371

8 Li Q,Steven GP,Xie YM,et al.Evolutionary topology optimization for temperature reduction of heat conducting fieldsInternational Journal of Heat&Mass Transfer,2004,47(23):5071-5083

9 Hansen A,Bendsoe M,Sigmund O.Topology optimization of heat conduction problems using the finit volume method.Structural and Multidisciplinary Optimization,2006,31(4):251-259

10 Bruns TE.Topology optimization of convection-dominated steadystateheattransferproblems.InternationalJournalofHeatandMass Transfer,2007,50(15-16):2859-2873

11 Zhuang CG,Xiong ZH,Ding H.A level set method for topology optimization of heat conduction problem under multiple load cases.Computer Methods in Applied Mechanics and Engineering,2007, 196(4-6):1074-1084

12 He D,Liu S.BESO method for topology optimization of structures with high efficiency of heat dissipation.International Journal for Simulation&Multidisciplinary Design Optimization,2008,2(1): 43-48

13 喬赫廷,張永存,劉書田.散熱結(jié)構(gòu)拓?fù)鋬?yōu)化目標(biāo)函數(shù)的討論.中國(guó)機(jī)械工程,2011,22(9):1112-1122(Qiao Heting,Zhang Yongcun,Liu Shutian.Discussion of objective functions in heat conduction topology optimization.China Mechanical Engineering,2011, 22(9):1112-1122(in Chinese))

14 陳文炯,劉書田,張永存.基于拓?fù)鋬?yōu)化的自發(fā)熱體冷卻用植入式導(dǎo)熱路徑設(shè)計(jì)方法.力學(xué)學(xué)報(bào),2016,48(2):406-412(Chen Wenjiong,Liu Shutian,Zhang Yongcun.Optimization design of conductive pathways for cooling a heat generating body with high conductive inserts.Chinese Journal of Theoretical and Applied Mechanics, 2016,48(2):406-412(in Chinese))

15 Gao T,Zhang WH,Zhu JH,et al.Topology optimization of heat conductionprobleminvolvingdesign-dependentheatloade ff ect.Finite Elements in Analysis and Design,2008,44(4):805-813

16 Zhuang CG,Xiong ZH,Ding H.Topology optimization of the transient heat conduction problem on a triangular mesh.Numerical Heat Transfer,Part B,2013,64(3):239-262

17 Zhuang CG,Xiong ZH.A global heat compliance measure based topology optimization for the transient heat conduction problem.Numerical Heat Transfer,Part B,2014,45(5):445-471

18 Zhuang CG,Xiong ZH.Temperature-constrained topology optimization of transient heat conduction problems.Numerical Heat Transfer,Part B,2015,68(4):366-385

19 Thomsen J.Topology optimization of structures composed of one or two materials.Journal of Structural Optimization,1992,5(1-2): 108-115

20 Sigmund O,Torquato S.Design of materials with extreme thermal expansion using a three-phase topology optimization method.Journal of Mechanics and Physics of Solids,1997,45(6):1037-1067

21 Wang MY,Wang X.“Color”level sets:a multi-phase method for structural topology optimization with multiple materials.Computer Methods in Applied Mechanics and Engineering,2004,193(6):469-496

22 Zhuang C,Xiong Z,Ding H.Topology optimization of multimaterial for the heat conduction problem based on the level set method.Engineering Optimization,2010,42(9):811-831

23 Cui M,Chen H,Zhou J.A level-set based multi-material topology optimization method using a reaction di ff usion equation.Computer-Aided Design,2016,73:41-52

24 Wang Y,Luo Z,Kang Z,et al.A multi-material level set-based topology and shape optimization method.Computer Methods in Applied Mechanics and Engineering,2015,283:1570-1586

25 Liu P,Luo Y,Kang Z.Multi-material topology optimization considering interface behavior via XFEM and level set method.Computer Methods in Applied Mechanics and Engineering,2016,308:113-133

26 Wang MY,Zhou SW.Synthesis of shape and topology of multimaterial structures with a phase-fiel method.Journal of Computer-Aided Materials Design,2004,11(2-3):117-138

27 Zhou SW,Wang MY.Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition.Structural and Multidisciplinary Optimization,2007,33(2): 89-111

28 Huang X,Xie YM.Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials.Computational Mechanics,2009,43(3):393-401

29 Ghabraie K.An improved soft-kill BESO algorithm for optimal distribution of single or multiple material phases.Structural and Multidisciplinary Optimization,2015,52(4):773-790

30 Yin L,Ananthasuresh GK.Topology of compliant mechanisms with multiple materials using a peak function material interpolation scheme.Structural and Multidisciplinary Optimization,2001, 23(1):49-62

31 高彤,張衛(wèi)紅,Pierre D.多相材料結(jié)構(gòu)拓?fù)鋬?yōu)化：體積約束還是重量約束？力學(xué)學(xué)報(bào),2011,43(2):296-305(Gao Tong,Zhang Weihong,Pierre D.Topology optimization of structures designed with multiphase materials:volume constraint or mass constraint?Chinese Journal of Theoretical and Applied Mechanics,2011,43(2): 296-305(in Chinese))

32 Gao T,Zhang W.A mass constraint formulation for structural topology optimization with multiphase materials.International Journal for Numerical Methods in Engineering,2011,88(8):774-796

33 Gao T,Xu P,Zhang W.Topology optimization of thermo-elastic structures with multiple materials under mass constraint.Computers and Structures,2016,173:150-160

34 Jia J,Cheng W,Long K,et al.Hierarchical design of structures and multiphase materials cells.Computers and Structures,2016,165: 136-144

35 Takakoli R,Mohseni SM.Alternating active-phase algorithm for multimaterial topology optimization problems:a 115-line MATLAB implementation.Structural and Multidisciplinary Optimization,2014,49(4):621-642

36 Ramani A.A pseudo-sensitivity based discrete-variable approach to structural topology optimization with multiple materials.Structural and Multidisciplinary Optimization,2010,41(6):913-934

37 Ramani A.Multi-material topology optimization with strength constraints.StructuralandMultidisciplinaryOptimization,2011,43(5): 597-615

38 Zuo W,Saitou K.Multi-material topology optimization using ordered SIMP interpolation.Structural and Multidisciplinary Optimization,2016:1-15

39 隋允康.建模變換優(yōu)化--結(jié)構(gòu)綜合方法新進(jìn)展.大連：大連理工大學(xué)出版社，1996(Sui Yunkang.Modelling,Transformation and Optimization--New Developments of Structural Synthesis Method. Dalian:Dalian University of Technology Press,1996(in Chinese))

40 隋允康,宣東海,尚珍.連續(xù)體結(jié)構(gòu)拓?fù)鋬?yōu)化的高精度逼近ICM方法.力學(xué)學(xué)報(bào),2011,43(4):716-724(Sui Yunkang,Xuan Donghai,Shang Zhen.ICM method with high accuracy approximation for topology optimization of continuum structures.Chinese Journal of Theoretical and Applied Mechanics,2011,43(4):716-724(in Chinese))

41 隋允康,葉紅玲,彭細(xì)榮.應(yīng)力約束全局化策略下的連續(xù)體結(jié)構(gòu)拓?fù)鋬?yōu)化.力學(xué)學(xué)報(bào),2006,38(3):364-370(Sui Yunkang,Ye Hongling,Peng Xirong.Topological optimization of continuum structure under the strategy of globalization of stress constraints.ChineseJournalofTheoreticalandAppliedMechanics,2006,38(3): 364-370(in Chinese))

42 隋允康,葉紅玲,彭細(xì)榮等.連續(xù)體結(jié)構(gòu)拓?fù)鋬?yōu)化應(yīng)力約束凝聚化的ICM方法.力學(xué)學(xué)報(bào),2007,39(4):554-563(Sui Yunkang,Ye Hongling,Peng Xirong,et al.The ICM method for continuum structural topology optimization with condensation of stress constraints.ChineseJournalofTheoreticalandAppliedMechanics,2007,39(4): 554-563(in Chinese))

43 隋允康,葉紅玲.連續(xù)體結(jié)構(gòu)拓?fù)鋬?yōu)化的ICM方法.北京：科學(xué)出版社，2013(Sui Yunkang,Ye Honglin.Continuum Topology Optimization Methods ICM.Beijing:Science Press,2013(in Chinese))

44 隋允康，彭細(xì)榮.結(jié)構(gòu)拓?fù)鋬?yōu)化 ICM 方法的改善.力學(xué)學(xué)報(bào), 2005,37(2):190-198(Sui Yunkang,Peng Xirong.The improvement for the ICM method of structural topology optimization.Acta Mechanica Sinica,2005,37(2):190-198(in Chinese))

45 Sui Yunkang,Peng Xirong.The ICM method with objective function transformed by variable discrete condition for continuum structure.Chinese Journal of Theoretical and Applied Mechanics,2006, 22(1):68-75

46 Lazarov BS,Sigmund O.Filters in topology optimization based on Helmholtz-type di ff erential equations.International Journal for Numerical Methods in Engineering,2011,86(6):765-781

STRUCTURAL LIGHT DESIGN FOR STEADY HEAT CONDUCTION USING MULTI-MATERIAL1)

Long Kai?,2)Wang Xuan?Han Dan?

?(State Key Laboratory for Alternate Electrical Power System with Renewable Energy Sources，North China Electric Power University，Beijing102206，China)

?(State Key Laboratory of Structural Analysis for Industrial Equipment，Dalian University of Technology，Dalian116024，Liaoning，China)

In topology optimization problems of structures containing multiphase materials,it is common practice to set the volume constraint of each constituent phase or total mass of entire constituent phase constraint to control the fina material usage.On the practical engineering background for lightweight design,it is of significanc that the minimized weight is taken as the objective in optimal model from the engineering point of view.To solve the topology optimization problem of steady heat conductive with the multiple candidate materials,a new modeling method of weight minimization with the given thermal compliance constraint under multiple load cases is proposed.Following the modeling manner of independent continuous mapping method,two sets of independent topological variables are employed to identify elemental thermal conductive matrix and elemental weight,respectively.The sensitivities of thermal compliance and globalweight with respect to the design variable are derived,and their approximate expressions are calculated based on the first-orde and second-order Taylor expansion.To eliminate checkerboard patterns and mesh-dependence,the firs term of the constraint function is filtere as a solution of the partial di ff erential equation,which also ensures the constraint equation is consistent.The approximate optimal model with the objective and constraint in the form of quadratic and linear function is established.The topological optimization model is solved by dual sequential quadratic programming.Various e ff ects such as the constraint value of thermal compliance,the selection of multiple materials,and the multiple constraints in multiple load cases on the optimal result are discussed in four 3D numerical examples.The results demonstrate the feasibility and e ff ectiveness of the proposed optimization approach regarding structural light design using multi-material in steady heat conduction.

topology optimization,multiple material,steady heat conduction,independent continuous mapping method, sequential quadratic programming

O343

A

10.6052/0459-1879-16-262

2016–09–18收稿，2016–12–02錄用,2016–12–02網(wǎng)絡(luò)版發(fā)表.

1)國(guó)家自然科學(xué)基金(11202078),中央高校基本科研業(yè)務(wù)費(fèi)專項(xiàng)基金(2014ZD16)資助項(xiàng)目.

2)龍凱，博士，副教授，主要研究方向：連續(xù)體結(jié)構(gòu)拓?fù)鋬?yōu)化、材料拓?fù)鋬?yōu)化設(shè)計(jì).E-mail：longkai1978@163.com

龍凱，王選，韓丹.基于多相材料的穩(wěn)態(tài)熱傳導(dǎo)結(jié)構(gòu)輕量化設(shè)計(jì).力學(xué)學(xué)報(bào),2017,49(2):359-366

Long Kai,Wang Xuan,Han Dan.Structural light design for steady heat conduction using multi-material.Chinese Journal of Theoretical and Applied Mechanics,2017,49(2):359-366