槽形托輥帶式輸送機(jī)設(shè)計(jì)【30米】【說(shuō)明書(shū)+CAD】

槽形托輥帶式輸送機(jī)設(shè)計(jì)【30米】【說(shuō)明書(shū)+CAD】,30米,說(shuō)明書(shū)+CAD,槽形托輥帶式輸送機(jī)設(shè)計(jì)【30米】【說(shuō)明書(shū)+CAD】,槽形托輥帶式,輸送,設(shè)計(jì),30,說(shuō)明書(shū),仿單,cad 南昌航空大學(xué)科技學(xué)院學(xué)士學(xué)位論文 偽形的機(jī)械結(jié)構(gòu)優(yōu)化構(gòu)形理論 學(xué)生姓名：郭亮 班級(jí)：0781052 指導(dǎo)老師：封立耀 Jean Luc Marcelin 2007年1月10日收到 /接受:2007 年5月1日/在線發(fā)表：2007年5月25日。 2007年斯普林格出 版社倫敦有限公司 摘要 這項(xiàng)工作提供了偽構(gòu)形理論的一些應(yīng)用程序，機(jī)械結(jié)構(gòu)的形狀優(yōu)化技術(shù)。在本文構(gòu)形理論的發(fā)展中， 優(yōu)化的主要目標(biāo)是最終總勢(shì)能的最小化 。其他目標(biāo)優(yōu)化使用的機(jī)械結(jié)構(gòu)優(yōu)化通常被用來(lái)限制或優(yōu)化約束。在這里介紹二種應(yīng)用：第一個(gè)是使用遺傳算法與偽構(gòu)形技術(shù)對(duì)一水滴形狀優(yōu)化和第二個(gè)是對(duì)一個(gè)液壓錘后軸承的形狀優(yōu)化處理。 關(guān)鍵詞 形狀優(yōu)化 結(jié)構(gòu) 遺傳算法 1引言 本文介紹一種偽構(gòu)形方法來(lái)達(dá)到物體形狀優(yōu)化基于總勢(shì)能的最小化。我們將介紹減少結(jié)構(gòu)總勢(shì)能尋找最優(yōu)形狀，這可能在某些情況下是個(gè)好主意。該參考的構(gòu)形理論可以以某種方式合理的理由解釋如下。 據(jù)Bejan [1]，在工程設(shè)計(jì)和自然性能中，形狀和結(jié)構(gòu)一直在演變?yōu)楦玫男阅?；在工程設(shè)計(jì)中用到的目標(biāo)和制約因素是從該幾何相同的機(jī)制在自然流動(dòng)系統(tǒng)中出現(xiàn)。Bejan [1]開(kāi)始設(shè)計(jì)和工程系統(tǒng)優(yōu)化，并從自然系統(tǒng)中發(fā)現(xiàn)了幾何形式的確定原則。這種發(fā)現(xiàn)是新的構(gòu)形理論的根據(jù)。優(yōu)化配置注定是不完善的。該系統(tǒng)的不完善到處蔓延時(shí)，效果最差，使越來(lái)越多的內(nèi)部點(diǎn)和硬件工作部件被壓?？此破毡榈膸缀涡问綀F(tuán)結(jié)工程與自然的流動(dòng)系統(tǒng)。Bejan [1]采用了一種新的理論，他毫不掩飾地說(shuō)明，與熱力學(xué)第二定律是相同的理論，因?yàn)橐粋€(gè)簡(jiǎn)單的理論意圖 預(yù)測(cè)地球上任何活著的幾何形式。 許多構(gòu)形理論的應(yīng)用程序在機(jī)械流體中開(kāi)發(fā)，特別是在優(yōu)化流動(dòng)[2-10]中。另一方面，據(jù)我們所知，在固體或結(jié)構(gòu)力學(xué)中有一些應(yīng)用實(shí)例。因此，我們至少有一半的參考文獻(xiàn) 在流體動(dòng)力學(xué)論文引用（同作者），因?yàn)闃?gòu)形方法是先由同一作者發(fā)展，只有阿德里安Bejan在論文中提到流體動(dòng)力學(xué)。構(gòu)形理論基于所有自然的創(chuàng)作，整體最佳的理論相比，該控制的演變與自然系統(tǒng)適應(yīng)。構(gòu)形分配原則由不完善的地方以及盡可能把最小的規(guī)模擴(kuò)到最大組成?？偟暮暧^構(gòu)形理論工程結(jié)構(gòu)從基本結(jié)構(gòu)組裝開(kāi)始，通過(guò)與自然的規(guī)則相一致最佳分布不完善。我們的目標(biāo)是研究降低成本。 但是，從這里長(zhǎng)期偽構(gòu)形來(lái)看，機(jī)械結(jié)構(gòu)優(yōu)化的全局宏觀解決方案已經(jīng)成本降低，目標(biāo)非常接近構(gòu)形理論。那個(gè)構(gòu)形理論是預(yù)測(cè)的理論，只有一 單一的原則，從優(yōu)化所有上升。這篇文章的題目同樣適用于偽構(gòu)形的步驟。偽構(gòu)形理論單一的優(yōu)化原則是最小化總勢(shì)能。此外，在我們以下提出的例子中偽構(gòu)形原則和遺傳算法有相關(guān)性，其結(jié)果是我們的優(yōu)化將非常接近自然理論。 本文的目是展示偽構(gòu)形步驟用來(lái)適用力學(xué)結(jié)構(gòu)，特別是對(duì)形狀優(yōu)化機(jī)械結(jié)構(gòu)。其基本思想是非常簡(jiǎn)單：處于平衡狀態(tài)的機(jī)械結(jié)構(gòu)對(duì)應(yīng)最小總勢(shì)能。以同樣的方式，最佳的機(jī)械結(jié)構(gòu)也必須符合最低限度總勢(shì)能。這目標(biāo)必須首先對(duì)其他的東西進(jìn)行干預(yù)。正是這種構(gòu)想，在這篇文章中發(fā)展。 兩個(gè)例子將會(huì)在后面提到。 最小化總勢(shì)能以優(yōu)化一機(jī)械結(jié)構(gòu)不是全新的的想法。已經(jīng)有很多文件解決了這一問(wèn)題，使這一方法系統(tǒng)化。最優(yōu)化的唯一目標(biāo)是使能量最小化。 高斯林[11]用一個(gè)簡(jiǎn)單的方法提出了硬件案件形式的有線網(wǎng)絡(luò)團(tuán)體和膜發(fā)現(xiàn)結(jié)構(gòu)。該方法是根據(jù)基本的能源概念。剪應(yīng)力變表達(dá)式是用來(lái)定義總勢(shì)能。最后的能源形式是盡量減少使用鮑威爾算法。在菅野和大崎[12]中，最低的互補(bǔ)能源原則是建立網(wǎng)絡(luò)作為變量應(yīng)力組件幾何非線性彈性。為了顯示總勢(shì)能和能量互補(bǔ)之間的強(qiáng)對(duì)偶問(wèn)題，這些問(wèn)題的凸配方被進(jìn)行調(diào)查，可嵌入到原始的二階規(guī)劃問(wèn)題中。 Taroco [13]進(jìn)行分析形成一個(gè)彈性固體的敏感性平衡問(wèn)題。第一階形中，域、邊界積分和總勢(shì)能的第二階形表達(dá)衍生物已經(jīng)建立。在華納[14]中，最優(yōu)設(shè)計(jì)問(wèn)題是根據(jù)其自身的重量解決了一個(gè)彈性懸掛桿。他已經(jīng)發(fā)現(xiàn)截面在一個(gè)均衡狀態(tài)中總勢(shì)能的面積最小化分布。在相類似的設(shè)計(jì)問(wèn)題中，在相同的約束條件潛在最大的能量也已解決了。在文圖拉[15]中，邊界條件控制的問(wèn)題已經(jīng)用網(wǎng)格方法解決。在文圖拉[15]中，在總勢(shì)能功能的彈性固體問(wèn)題中介紹移動(dòng)最小化近似值了，廣義的拉格朗日術(shù)語(yǔ)被添加到滿足本質(zhì)邊界的條件中。 總的潛在能量最小化原理除了在一般有限元基礎(chǔ)上制定，還找到一個(gè)未知的最佳目標(biāo)結(jié)點(diǎn)因素[16]。 2所使用的方法 在本文的偽構(gòu)形理論中，最優(yōu)化的主要目標(biāo)是盡量減少總勢(shì)能。在優(yōu)化機(jī)械結(jié)構(gòu)里其他的物體通用被限制或優(yōu)化約束。例如，一個(gè)東西可能在重量上有限制，或不超過(guò)其應(yīng)力值。 在本文中這種想法是很簡(jiǎn)單的。機(jī)械結(jié)構(gòu)通過(guò)兩種參數(shù)類型來(lái)描述：已知的離散變量（例如，用有限元方法的自由度的位移）以及幾何變量設(shè)計(jì)（例如參數(shù)，使人們有可能描述機(jī)械結(jié)構(gòu)形狀）?？倽摿δ茉谕粫r(shí)間里通過(guò)一個(gè)確定隱含或明確的方式離散設(shè)計(jì)變量。因此，進(jìn)行雙重優(yōu)化機(jī)械結(jié)構(gòu)相比離散化設(shè)計(jì)變量，其目的是減少整體的總勢(shì)能。顯然，機(jī)械結(jié)構(gòu)的優(yōu)化問(wèn)題用下列方法處理： - 目標(biāo)：減少總勢(shì)能 - 變量的優(yōu)化：同時(shí)確定離散變量（在結(jié)構(gòu)力學(xué)的有限元法的傳統(tǒng)用例），描述設(shè)計(jì)變量的形狀結(jié)構(gòu) - 優(yōu)化限制： - 重量或體積 - 位移或限制 - 壓力 - 頻率 機(jī)械結(jié)構(gòu)的優(yōu)化問(wèn)題將用以下方法解決，如果需要的話需要在這些階段中重申（按照問(wèn)題的本質(zhì)）： 第1階段 最小化機(jī)械結(jié)構(gòu)總勢(shì)能和唯一的離散結(jié)構(gòu)變量相比較（度的有 限元）。它的作用在這里作為一個(gè)優(yōu)化不優(yōu)化的限制。在此階段唯一的限制是 純粹的機(jī)械原點(diǎn)，并涉及邊界條件，適用于結(jié)構(gòu)外部的作用。 在第一階段，設(shè)計(jì)變量保持不變，根據(jù)設(shè)計(jì)變量1獲得的隱含或明確表述的自由度（可以是變量，使人們有可能描述形狀，以外形的優(yōu)化為例子）。大家可以看到在下面部分的例子中，這些表現(xiàn)形式可以是有形或無(wú)形的，且這是適當(dāng)?shù)闹委熀蟮那闆r。在案件1中用有限元計(jì)算方法，這一階段1是在有限元計(jì)算的基礎(chǔ)上，以獲取機(jī)械結(jié)構(gòu)的自由度。事實(shí)上，有限元，位移與節(jié)點(diǎn)、機(jī)械結(jié)構(gòu)網(wǎng)格，獲得了最小化總勢(shì)能[16]。 第2階段中 機(jī)械結(jié)構(gòu)自由度的表達(dá)根據(jù)設(shè)計(jì)前先獲得的變量，然后注入機(jī)械結(jié)構(gòu)總勢(shì)能（你會(huì)看到下面的部分中第二個(gè)例子它是如何影響自由度在隱含的職能設(shè)計(jì)變量的情況下）。然后得到一個(gè)表達(dá)式總的潛在能量，它依賴于（以明示或默示的形式）設(shè)計(jì)變量。 第3階段 隨后進(jìn)行的第二次和新的最小化總勢(shì)能通過(guò)前面的形式取得，但這次比設(shè)計(jì)變量同時(shí)遵循技術(shù)限制或優(yōu)化約束的問(wèn)題。這種方法按問(wèn)題的本質(zhì)可以使用或多或少的設(shè)備。這點(diǎn)很明顯，例如，如果離散變量按照設(shè)計(jì)變量可以表示為一明確的方式，在2到3個(gè)階段是可以立即設(shè)置的，并無(wú)迭代。 如果離散變量不能按照設(shè)計(jì)變量明確的方式表達(dá)，或者如果結(jié)構(gòu)拓?fù)洳皇枪潭ǖ模蛘呷绻袨椴皇蔷€性的，這將有必要通過(guò)分階段進(jìn)行1至3連續(xù)迭代。這將在下面部分的例子中提到，一會(huì)我們會(huì)看到戰(zhàn)略的場(chǎng)合，可以采用一種類型為這些迭代?？偨Y(jié)，偽構(gòu)形的步驟，主要目的只是盡量減少潛在總能源，其他可能的目標(biāo)是限制或優(yōu)化約束。 在我們的例子中使用的最優(yōu)化方法是遺傳算法（遺傳算法），如[17]。，我們也可以找到很多書(shū)籍有典型類似的教學(xué)價(jià)值，例如在[18]中。這種方法是非常先進(jìn)方便于我們的偽構(gòu)形方法。撰文已在天然氣方面廣泛地開(kāi)展了工作，關(guān)于這一主題出版的期刊被譽(yù)為期刊[19-31]。由于天然氣問(wèn)題在社會(huì)結(jié)構(gòu)力學(xué)上還比較新，在這里我們提供的一些細(xì)節(jié)正是使用這里的算法——多點(diǎn)交叉使用，而不是一單點(diǎn)交叉。在甄選計(jì)劃上，每年的使用完全是隨機(jī)生成。在我們的例子中，幾代人是等同的銜接使用。我們提供例子的結(jié)果是不斷地通過(guò)使用不同的遺傳算法。一個(gè)比較標(biāo)準(zhǔn)的遺傳算法已經(jīng)被證明是我們足夠的榜樣。 3范例 盡管潛在的能源可能是一個(gè)好的舉措對(duì)于一些優(yōu)化問(wèn)題，勢(shì)能不是賦予形成水滴的能量，也沒(méi)有定義錘子的最佳形狀，這就是為什么勢(shì)能不是唯一的、客觀的，但最優(yōu)化問(wèn)題是多目標(biāo)的和用公式明確表示的兩個(gè)例子的目標(biāo)函數(shù)。 3.1例1：對(duì)一滴水形狀優(yōu)化 第一個(gè)測(cè)試?yán)邮菍?duì)下降的水滴形狀優(yōu)化（圖1）。這個(gè)問(wèn)題是等同于抵抗坦克的膜理論計(jì)算。其目的是看看偽構(gòu)形理論給出了大自然的優(yōu)化設(shè)計(jì)。 3.1.1使用的方法 該一水滴幾何的定義是：產(chǎn)生的軸對(duì)稱殼薄線。此行描述于連續(xù)直線或圓形段描述在特定意義和輸入數(shù)據(jù)定義點(diǎn)上的坐標(biāo)和半徑值。初始數(shù)據(jù)是一個(gè)由直線段連接結(jié)點(diǎn)的集合。 每一個(gè)結(jié)點(diǎn)是確定它兩個(gè)圓柱坐標(biāo)上（R，z），和真正的R代表的半徑圓相切的兩個(gè)交叉直線段的這一點(diǎn)。另一臺(tái)計(jì)算機(jī)的計(jì)算給出任何邊界的坐標(biāo)點(diǎn)，特別是切點(diǎn)必須界定圓弧長(zhǎng)度。 水式設(shè)計(jì)描述了三個(gè)弧圓如圖1所示。 通過(guò)有限元方法采用三節(jié)點(diǎn)拋物原理運(yùn)用基爾霍夫殼體理論分析。自動(dòng)網(wǎng)格生成器建立每個(gè)直線或圓形段的有限元網(wǎng)格 ，它們被視為宏觀有限元。 我們的目標(biāo)是獲得一個(gè)水滴形狀形成最低總勢(shì)能（這是主要目標(biāo)）和平等的抵抗坦克（這是唯一約束或限制的問(wèn)題）。 事實(shí)上，為了水滴的問(wèn)題，目標(biāo)是多對(duì)象的，兩個(gè)目標(biāo)（F1=最低總額的F1 勢(shì)能和f2 =等于電阻）的合并多目標(biāo)：F1=F1 + F2。 馬塞蘭指出，在總勢(shì)能的減少中約束或限制的問(wèn)題被考慮進(jìn)去，在[19]中。 3.1.2結(jié)果 在水滴外形設(shè)計(jì)中描述了三個(gè)弧圓（圖1），他們的中心和半徑是設(shè)計(jì)變量。因此，有9個(gè)設(shè)計(jì)變量，其中：r1，Z1，R1為 圓1;r2, Z2,R2為圓2; r3，Z3,R3為圓3 。在遺傳算法中，其中每個(gè)設(shè)計(jì)變量通過(guò)3個(gè)二進(jìn)制數(shù)字編碼. 所有這些二進(jìn)制數(shù)字編碼是端到端地形成27個(gè)二進(jìn)制數(shù)字的染色體長(zhǎng)度。 GA是運(yùn)行了30個(gè)，一個(gè)數(shù)字對(duì)50代，一個(gè)穿越的概率為0.8，而突變概率為0.1。 對(duì)應(yīng)的染色體最優(yōu)解是 100 100 011 011 010 011 100 011 101 這給出了圖1的解決方案。其中： - r1 = 18，z1 = 17，R1 =- 0.065 - R2= 13.75，z2= 12.2，R2 =- 7.7 - r3 = 4.1，Z3= 21.4，R3 =- 21 這是關(guān)于一水滴的外形非常接近自然的最佳解決方案。通過(guò)三個(gè)圓的弧模式的水滴模型并非十全十美。但是，構(gòu)形理論用于優(yōu)化不完善的地方，并發(fā)現(xiàn)最接近自然的解決辦法。因此，構(gòu)形原則包括盡可能的分配不完善的地方。 3.2例2：軸對(duì)稱結(jié)構(gòu)的形狀優(yōu)化 在這一部分，呈現(xiàn)了液壓錘后軸承傳統(tǒng)的最優(yōu)化影響。相對(duì)于較少的周期操作軸承問(wèn)題（圖2）漸漸體現(xiàn)出來(lái)。 對(duì)于軸對(duì)稱結(jié)構(gòu)，分析是通過(guò)有限元方法進(jìn)行的，遺傳算法優(yōu)化的過(guò)程中的特殊字符一直用來(lái)緩解計(jì)算和節(jié)省計(jì)算機(jī)的時(shí)間。首先，由于只是一個(gè)結(jié)構(gòu)幾個(gè)部分必須經(jīng)常修改，子結(jié)構(gòu)的概念是用來(lái)單獨(dú)“固定”和“移動(dòng)”的部分。固定部分計(jì)算兩次：第一次是開(kāi)始，第二次是結(jié)尾的優(yōu)化過(guò)程。只有這些縮減剛度矩陣的子結(jié)構(gòu)被添加到移動(dòng)部分的矩陣。 與此相關(guān)的部分，自動(dòng)發(fā)電機(jī)創(chuàng)建作為每個(gè)子結(jié)構(gòu)的網(wǎng)格宏觀有限元。這些宏量元素不是 三角（六節(jié)點(diǎn)）或四邊形（8個(gè)節(jié)點(diǎn)）。根據(jù)那些著名的技術(shù)，同樣的細(xì)分用于父的空間，以獲取網(wǎng)本身，這顯然是出于作出相同類型的元素。在這個(gè)網(wǎng)優(yōu)化控制過(guò)程，一個(gè)的離散如有必要可以重新選擇。 總之，優(yōu)化問(wèn)題如下： 總的目標(biāo)函數(shù) 最小化潛在能源。需要注意的另一重要 目標(biāo)（馮米塞斯沿等高線的移動(dòng)相當(dāng)于最小應(yīng)力的最大值）是這里作為問(wèn)題的約束。這第二個(gè)目標(biāo)是要實(shí)現(xiàn)液壓錘的后軸承最小化。 設(shè)計(jì)變量 設(shè)計(jì)變量是半徑為r，寬X附近的半徑（圖2）。 制約因素 制約因素是建立在這樣的在幾何方式上，只允許有微小的變化是。它們考慮到技術(shù)的限制。他們包括編碼設(shè)計(jì)變量。另外，重要的制約因素是，米塞斯沿等高線的移動(dòng)的最大值不能超過(guò)一定的值。約束被考慮到總勢(shì)能的降落中，在[19]說(shuō)明。 所有這些二進(jìn)制數(shù)字終端到終端地形成八個(gè)二進(jìn)制數(shù)字的染色體長(zhǎng)度。 GA運(yùn)行的12個(gè)，數(shù)的30代，交叉概率 為0.5，以及變異概率為0.06。 最優(yōu)解對(duì)應(yīng)的染色體 1101 1000 圖2給出了解決方案。其中： ?= 1.95，X = 6.0 在這種產(chǎn)品的形狀自動(dòng)優(yōu)化中，只需簡(jiǎn)單地把形狀修改小，這比計(jì)算更難預(yù)測(cè)（半徑增加外，減少寬度），大大提高了機(jī)械軸承的耐久性：過(guò)壓力正在減少50％。 4討論 本文件中的兩個(gè)例子可以證明偽構(gòu)形理論。第一個(gè)是對(duì)軸對(duì)稱膜下降形外殼形狀優(yōu)化（水滴）。這種結(jié)構(gòu)是用純的張力。果不其然，盡量減少這種結(jié)構(gòu)的總勢(shì)能，所有可能的變量導(dǎo)致的形狀是完全和調(diào)和十分相似的。但是，第二個(gè)例子事實(shí)證明，制定最低的能源不僅可以 工作在最簡(jiǎn)單的情況下，純粹的張力結(jié)構(gòu)還能彎曲，剪切或更為復(fù)雜的結(jié)構(gòu)扭轉(zhuǎn)應(yīng)力。這個(gè)條件是為了增加這一問(wèn)題次要目標(biāo)（通常用于形狀優(yōu)化）的限制或優(yōu)化約束。 然而，在偽形構(gòu)形理論聲明中，最大限度地減少所有可能的變量的機(jī)械結(jié)構(gòu)中的總勢(shì)能， 這不完全能達(dá)到的。自然和機(jī)械也不是都如此簡(jiǎn)單，多年研究的大自然設(shè)計(jì)結(jié)構(gòu)表明，即使在最簡(jiǎn)單的實(shí)例中，多重標(biāo)準(zhǔn)，以復(fù)雜的方式工作。因此，有必要添加其他標(biāo)準(zhǔn)或優(yōu)化問(wèn)題的制約是顯而易見(jiàn)的。最小化總勢(shì)能只是一個(gè)總的原則在優(yōu)化的過(guò)程啟動(dòng)。 5結(jié)論 一個(gè)有趣的方法引入了形狀優(yōu)化的機(jī)械結(jié)構(gòu)。在這個(gè)文件闡述的偽構(gòu)行理論中，優(yōu)化的主要目標(biāo)是最小化總勢(shì)能。其他的目標(biāo)通常使用的形狀 優(yōu)化這里使用了限制或優(yōu)化限制。它給我們的例子很好的效果。 參考文獻(xiàn) 1、從工程到自然形狀和結(jié)構(gòu) 劍橋大學(xué)出版社,劍橋大學(xué),Bejan A主編 　　 2、偽構(gòu)形理論的網(wǎng)絡(luò)的路徑冷卻機(jī) [J].熱能質(zhì)量40:799-816[J]. Bejan A主編 3、自然如何形成 52英格119(10):90-92 Bejan A主編 4、樹(shù)狀構(gòu)形網(wǎng)絡(luò)空間分布的電力 能源轉(zhuǎn)化管理 44:867-891 Arion V,Cojocari,Bejan一個(gè)(2003)Constructa Bejan A主編 5、對(duì)流體幾何內(nèi)部的優(yōu)化 熱能轉(zhuǎn)化120:357-364[J]. Nelson RA, Bejan A主編 　　 6、碟狀區(qū)域構(gòu)形設(shè)計(jì)的冷卻傳導(dǎo) [J].熱能質(zhì)量45:1643-1652 Rocha LAO, Lorente S, Bejan A主編 　　 7、天然裂縫模式的構(gòu)形理論形成快速冷卻 [J].熱能質(zhì)量 反式41:1945-1954 Bejan A, Ikegami Y, Ledezma GA主編 　　 ORIGINAL ARTICLE Pseudo-constructal theory for shape optimization of mechanical structures Jean Luc Marcelin Received: 10 January 2007 /Accepted: 1 May 2007 /Published online: 25 May 2007 # Springer-Verlag London Limited 2007 Abstract This work gives some applications of a pseudo- constructal technique for shape optimization of mechanical structures. In the pseudo-constructal theory developed in this paper, the main objective of optimization is only the minimization of total potential energy. The other objectives usually used in mechanical structures optimization are treated like limitations or optimization constraints. Two applications are presented; the first one deals with the optimization of the shape of a drop of water by using a genetic algorithm with the pseudo-constructal technique, and the second one deals with the optimization of the shape of a hydraulic hammers rear bearing. Keywords Shapeoptimization . Constructal . Geneticalgorithms 1 Introduction This paper introduces a pseudo-constructal approach to shape optimization based on the minimization of the total potential energy. We are going to show that minimizing the total potential energy of a structure to find the optimal shape might be a good idea in some cases. The reference to the constructal theory can be justified in some way for the following reasons. According to Bejan 1, shape and structure spring from the struggle for better performance in both engineering and nature; the objective and constraints principle used in engineering is the same mechanism from which the geometry in natural flow systems emerges. Bejan 1 starts with the design and optimization of engineering systems and discovers a deterministic principle for the generation of geometric form in natural systems. This observation is the basis of the new constructal theory. Optimal distribution of imperfection is destined to remain imperfect. The system works best when its imperfections are spread around so that more and more internal points are stressed as much as the hardest working parts. Seemingly universal geometric forms unite the flow systems of engineering and nature. Bejan 1 advances a new theory in which he unabashedly hints that his law is in the same league as the second law of thermodynamics, because a simple law is purported to predict the geometric form of anything alive on earth. Many applications of the constructal theory were developed in fluids mechanics, in particular for the optimization of flows 210. On the other hand, there exists, to our knowledge, little examples of applications in solids or structures mechanics. So we have at least half of the references to papers in fluid dynamics (most of the same author), because the constructal method was developed first by the same author, Adrian Bejan, with only references to papers in fluid dynamics. The constructal theory rests on the assumption that all creations of nature are overall optimal compared to the laws which control the evolution and the adaptation of the natural systems. The constructal principle consists of distributing the imperfections as well as possible, starting from the smallest scales to the largest. The constructal theory works with the total macroscopic structure starting from the assembly of elementary struc- tures, by complying with the natural rules of optimal distribution of the imperfections. The objective is the research of lower cost. Int J Adv Manuf Technol (2008) 38:16 DOI 10.1007/s00170-007-1080-2 J. L. Marcelin (*) Laboratorie Sols Solides Structures 3S, UMR CNRS C5521, Domaine Universitaire, BP n53, 38041 Grenoble Cedex 9, France e-mail: Jean-Luc.Marcelinujf-grenoble.fr However, a global and macroscopic solution for the optimization of mechanical structures having least cost as the objective can be very close to the constructal theory, from where the term pseudo-constructal comes. The constructal theory is a predictive theory, with only one single principle of optimization from which all rises. The same applies to the pseudo-constructal step which is the subject of this article. The single principle of optimiza- tion of the pseudo-constructal theory is the minimization of total potential energy. Moreover, in our examples presented hereafter, the pseudo-constructal principle will be associated with a genetic algorithm, with the result that our optimization will be very close to the natural laws. The objective of this paper is thus to show how the pseudo-constructal step can apply to the mechanics of the structures, and in particular to the shape optimization of mechanical structures. The basic idea is very simple: a mechanical structure in a balanced state corresponds to a minimal total potential energy. In the same way, an optimal mechanical structure must also correspond to a minimal total potential energy, and it is this objective which must intervene first over all the others. It is this idea which will be developed in this article. Two examples will be presented thereafter. The idea to minimize total potential energy in order to optimize a mechanical structure is not brand new. Many papers already deal with this problem. What is new, is to make this approach systematic. The only objective of optimization becomes the minimization of energy. In Gosling 11, a simple method is proposed for the difficult case of form-finding of cablenet and membrane structures. This method is based upon basic energy concepts. A truncated strain expression is used to define the total potential energy. The final energy form is minimized using the Powell algorithm. In Kanno and Ohsaki 12, the minimum principle of complementary energy is established for cable networks involving only stress components as variables in geometrically nonlinear elasticity. In order to show the strong duality between the minimization problems of total potential energy and complementary energy, the convex formulations of these problems are investigated, which can be embedded into a primal-dual pair of second-order programming problems. In Taroco 13, shape sensitivity analysis of an elastic solid in equilibrium is presented. The domain and boundary integral expressions of the first and second-order shape derivatives of the total potential energy are established. In Warner 14, an optimal design problem is solved for an elastic rod hanging under its own weight. The distribution of the cross- sectional area that minimizes the total potential energy stored in an equilibrium state is found. The companion problem of the design that stores the maximum potential energy under the same constraint conditions is also solved. In Ventura 15, the problem of boundary conditions enforcement in meshless methods is solved. In Ventura 15, the moving least-squares approximation is introduced in the total potential energy functional for the elastic solid problem and an augmented Lagrangian term is added to satisfy essential boundary conditions. The principle of minimization of total potential energy is in addition at the base of the general finite elements formulation, with an aim of finding the unknown optimal nodal factors 16. 2 The methods used In the pseudo-constructal theory developed in this paper, the main objective of optimization is only the minimization of total potential energy. The other objectives usually used in mechanical structures optimization are treated here like limitations or optimization constraints. For example, one may have limitations on the weight, or to not exceed the value of a stress. The idea which will be developed in this paper is thus very simple. A mechanical structure is described by two types of parameters: variables known as discretization variables (for example, degrees of freedom in displacement for finite elements method), and geometrical variables of design (for example parameters which make it possible to describe the mechanical structure shape). Total potential energy depends on an implicit or explicit way of determin- ing discretization and design variables at the same time. One thus will carry out a double optimization of the mechanical structure, compared to the discretization and design variables; the objective being to minimize total potential energy overall. Clearly, the problem of optimiza- tion of a mechanical structure will be addressed by the following approach: Objective: to minimize total potential energy Variables of optimization: concurrently determining discretization variables (in the case of a traditional use of the finite element method in mechanics of struc- tures), and design variables describing the shape of the structure Optimization limitations: Weight or volume Displacements or strains Stresses Frequencies The problem of optimization of a mechanical structure will be solved in the following way, while reiterating on 2 Int J Adv Manuf Technol (2008) 38:16 these stages, if needed (according to the nature of the problem): Stage 1 Minimization of the total potential energy of the mechanical structure compared to the only dis- cretization variables of the structure (degrees of freedom in finite elements). It acts here as an optimization without optimization limitations. The only limitations at this stage are of purely mechanical origin, and relate to the boundary conditions and to the external efforts applied to the structure. In this stage 1, the design variables remain fixed, and one obtains the implicit or explicit expressions of the degrees of freedom according to the design variables (which can be the variables which make it possible to describe the shape, in the case of a shape optimization, for example). One will see in the examples of the following part that these expressions can be explicit or implicit and which is the suitable treatment following the cases. In the case of a finite elements method of calculation, this stage 1 is the basis of finite elements calculation to obtain the degrees of freedom of the mechanical structure. Indeed, in finite elements, displacements with the nodes of the mechanical structure mesh are obtained by minimization of total potential energy 16. Stage 2 The expressions of the degrees of freedom of the mechanical structure according to the design variables obtained previously are then injected into the total potential energy of the mechanical structure (one will see in the second example of thefollowingparthowonetreatsthecasewherethe degrees of freedom are implicit functions of the design variables). One then obtains an expression of the total potential energy which depends only on the design variables (in explicit or implicit form). Stage 3 One then carries out a second and new minimi- zation of the total potential energy obtained in the preceding form, but this time compared to the design variables while respecting the technolog- ical limitations or the optimization constraints of the problem. This method can be applied with more or less facility according to the nature of the problem. It is clear, for example, that if the discretization variables can be expressed in an explicit way according to the design variables, the setting in of stages 2 to 3 is immediate, and without iterations. If the discretization variables cannot be expressed in an explicit way according to the design variables, or if the topology of the structure is not fixed, or if the behavior is not linear, it will be necessary to proceed by successive iterations on stages 1 to 3. It is the case of the examples presented in the following part, and one will see on this occasion which type of strategy one can adopt for these iterations. To summarize, in the pseudo-constructal step, the main objective is only the minimization of total potential energy, the other possible objectives are treated like limitations or optimization constraints. The optimization method used for our examples is GA (genetic algorithm), as described in 17. Examples with similar instructional value can also be found in many books, e.g. in 18. This evolutionary method is very convenient for our pseudo-constructal method. The author has worked extensively in GAs and published in some reputed journals on this topic 1931. As the topic of GAs is still relatively new in the structural mechanics commu- nity, we provide here some details of exactly what is used in this GA. A multiple point crossover is used rather than a single point crossover. The selection scheme used at each generation is entirely stochastic. For our examples, the number of generations is equal to that used for conver- gence. The results provided for our examples were consistently reproduced by using different seeds in the GA. It has been proved that a rather standard genetic algorithm is sufficient for our examples. 3 Examples Even though potential energy may be a good measure for some optimizations, potential energy is not what gives the shape to a water droplet, nor defines the optimal shape for a hammer, which is why potential energy is not the only objective; but the optimization problem is a multiobjective one and the objective functions for the two examples are then clearly formulated. 3.1 Example 1: optimization of the shape of a drop of water The first test example is the optimization of the shape of a drop of water (Fig. 1). This problem is equivalent to an equal resistance tank calculated by the membrane theory. The objective is to see if the pseudo-constructal theory gives the natures optimum design. 3.1.1 The methods used The geometry of the drop of water is defined by the generating line of a thin axisymmetric shell. This line is described by successive straight or circular segments described in a given sense and defined by input data of master point coordinates and radius values. The initial data are a set of nodal points connected by straight segments. Each nodal point is identified by its two cylindrical Int J Adv Manuf Technol (2008) 38:16 3 coordinates (r, z), and a real R which represents the radius of the circle tangent to the two straight segments intersect- ing at the point. The other computer calculations give the coordinates of any boundary point and especially the tangent points necessary to define the circular arc lengths. The design of the drop of water is described by three arcs of circles as indicated in Fig. 1. Analysis is performed by the finite element method with three-node parabolic elements using the classical Love- Kirchoff shell theory. An automatic mesh generator creates the finite element mesh of each straight or circular segment considered as a macro finite element. The objective is to obtain a shape for the drop of water giving rise to a minimum total potential energy (which is the main objective) and an equal resistance tank (which is the only constraint or limitation of the problem). In fact, for the drop of water problem, the goal is a multi- objective one, the two objectives ( f 1 =minimum total potential energy and f 2 =equal resistance) are combined in a multi-objective: f=f 1 +f 2 . The constraint or limitation of the problem is taken into account by a penalization of the total potential energy as indicated in Marcelin et al. 19. 3.1.2 The results The design of the drop of water is described by three arcs of a circle (Fig. 1). Their centers and radius are the design variables. So, there are nine design variables: r1, z1, R1 for circle 1; r2, z2, R2 for circle 2; and r3, z3, R3 for circle 3. In the genetic algorithm, each of these design variables is coded by three binary digits. The tables of coding-decoding will be the following: For r1: For z1: For R1: For r2: For z2: For R2: For r3: For z3: For R3: All these binary digits are put end to end to form a chromosome length of 27 binary digits. GA is run for a population of 30 individuals, a number of generations of 50, a probability of crossing of 0.8, and a probability of mutation of 0.1. The optimal solution corresponds to the chromosome 100 100 011 011 010 011 100 011 101 which gives the solution of Fig. 1, for which: r1=18, z1=17,and R1=0.065 r2=13.75, z2=12.2 and R2=7.7 r3=4.1, z3=21.4 and R3=21 It is very close to the natures optimal solution for the shape of a drop of water. The model of the water drop modelled by three arcs of a circle is imperfect. However, the constructal theory optimizes the imperfections, and 105 20 25 r 5 10 20 z 3 2 1 15 15 Fig. 1 Optimization of the shape of a drop of water 000 001 010 011 100 101 110 111 16 16.5 17 17.5 18 18.5 19 19.5 000 001 010 011 100 101 110 111 15 15.5 16 16.5 17 17.5 18 18.5 000 001 010 011 100 101 110 111 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 000 001 010 011 100 101 110 111 13 13.25 13.5 13.75 14 14.25 14.5 14.75 000 001 010 011 100 101 110 111 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 000 001 010 011 100 101 110 111 7.4 7.5 7.6 7.7 7.8 7.9 8 8.1 000 001 010 011 100 101 110 111 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 000 001 010 011 100 101 110 111 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 000 001 010 011 100 101 110 111 18.5 19 19.5 20 20.5 21 21.5 22 4 Int J Adv Manuf Technol (2008) 38:16 finds the nearest solution to that of nature. So, the constructal principle consists of distributing the imperfec- tions as well as possible. 3.2 Example 2: optimization of the shape of an axisymmetric structure In this part, the very localized optimization of the rear bearing of a hydraulic hammer is presented. The bearing in question (Fig. 2) breaks after relatively few cycles of operation. For axisymmetric structures, analysis is performed by the finite element method in which the special character of a GA optimization process has been considered to ease the calculations and to save computer time. First, because just a few parts of the structure must often be modified, the substructure concept is used to separate the “fixed” and the “mobile” parts. The fixed parts are calculated twice: once at the beginning and also at the end of the optimization process. Only the reduced stiffness matrices of these substructures are added to the matrices of the mobile parts. Related to this division, an automatic generator creates the finite element mesh of each substructure considered as a macro finite element. These macro elements are either triangular (six nodes) or quadrilateral (eight nodes). Following a well-known technique, the same subdivision is used in the parent space to obtain the mesh itself, which is obviously made out of the same types of elements. During the optimization process this mesh is controlled and a new discretization can be chosen if necessary. To summarize, the optimization problem is the following: Objective function Minimization of the total potential energy. It is important to note that another important objective (the minimization of the maximum value of the Von Mises equivalent stress along the mobile contour) is taken here as a constraint of the problem. This second objective is necessary to achieve the minimization of the rear bearing of the hydraulic hammer . Design variables The design variables are radius r and width X near the radius (Fig. 2). Constraints The side constraints are established in such a way that only small changes in geometry are allowed. They take into account the technological constraints. They are included in the coding of the design variables. Another important constraint is that the maximum value of the Von Mises equivalent stress along the mobile contour must not exceed a certain value. The constraints are taken into account by a penalization of the total potential energy as indicated in 19. The tables of coding-decoding are the following: For r: For X: All these binary digits are put end to end to form a chromosome length of eight binary digits. The GA is run for a population of 12 individuals, a number of generations of 30, a probability of crossing of 0.5, and a probability of mutation of 0.06. The optimal solution corresponds to the chromosome 1101 1000 which gives the solution of Fig. 2, for which: r=1.95, X=6.0 The automatic optimization of the shape of this product has,