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附錄I 外文文獻(xiàn)翻譯
估計(jì)導(dǎo)致工程幾何分析錯(cuò)誤的一個(gè)正式理論
SankaraHariGopalakrishnan，KrishnanSuresh
機(jī)械工程系，威斯康辛大學(xué)，麥迪遜分校，2006年9月30日
摘要：幾何分析是著名的計(jì)算機(jī)輔助設(shè)計(jì)/計(jì)算機(jī)輔助工藝簡(jiǎn)化 “小或無(wú)關(guān)特征”在CAD模型中的程序，如有限元分析。然而，幾何分析不可避免地會(huì)產(chǎn)生分析錯(cuò)誤，在目前的理論框架實(shí)在不容易量化。
本文中，我們對(duì)快速計(jì)算處理這些幾何分析錯(cuò)誤提供了嚴(yán)謹(jǐn)?shù)睦碚?。尤其，我們集中力量解決地方的特點(diǎn)，被簡(jiǎn)化的任意形狀和大小的區(qū)域。提出的理論采用伴隨矩陣制定邊值問(wèn)題抵達(dá)嚴(yán)格界限幾何分析性分析錯(cuò)誤。該理論通過(guò)數(shù)值例子說(shuō)明。
關(guān)鍵詞:幾何分析;工程分析;誤差估計(jì);計(jì)算機(jī)輔助設(shè)計(jì)/計(jì)算機(jī)輔助教學(xué)
1. 介紹
機(jī)械零件通常包含了許多幾何特征。不過(guò)，在工程分析中并不是所有的特征都是至關(guān)重要的。以前的分析中無(wú)關(guān)特征往往被忽略，從而提高自動(dòng)化及運(yùn)算速度。
舉例來(lái)說(shuō)，考慮一個(gè)剎車轉(zhuǎn)子，如圖1(a)。轉(zhuǎn)子包含50多個(gè)不同的特征，但所有這些特征并不是都是相關(guān)的。就拿一個(gè)幾何化的剎車轉(zhuǎn)子的熱量分析來(lái)說(shuō)，如圖1(b)。有限元分析的全功能的模型如圖1(a)，需要超過(guò)150,000度的自由度，幾何模型圖1(b)項(xiàng)要求小于25，000個(gè)自由度，從而導(dǎo)致非常緩慢的運(yùn)算速度。
圖1(a)剎車轉(zhuǎn)子 圖1(b)其幾何分析版本
除了提高速度，通常還能增加自動(dòng)化水平，這比較容易實(shí)現(xiàn)自動(dòng)化的有限元網(wǎng)格幾何分析組成。內(nèi)存要求也跟著降低，而且條件數(shù)離散系統(tǒng)將得以改善;后者起著重要作用迭代線性系統(tǒng)。
但是，幾何分析還不是很普及。不穩(wěn)定性到底是“小而局部化”還是“大而擴(kuò)展化”，這取決于各種因素。例如，對(duì)于一個(gè)熱問(wèn)題，想刪除其中的一個(gè)特征，不穩(wěn)定性是一個(gè)局部問(wèn)題:(1)凈熱通量邊界的特點(diǎn)是零。(2)特征簡(jiǎn)化時(shí)沒(méi)有新的熱源產(chǎn)生; [4]對(duì)上述規(guī)則則例外。展示這些物理特征被稱為自我平衡。結(jié)果，同樣存在結(jié)構(gòu)上的問(wèn)題。
從幾何分析角度看，如果特征遠(yuǎn)離該區(qū)域，則這種自我平衡的特征可以忽略。但是，如果功能接近該區(qū)域我們必須謹(jǐn)慎，。
從另一個(gè)角度看，非自我平衡的特征應(yīng)值得重視。這些特征的簡(jiǎn)化理論上可以在系統(tǒng)任意位置被施用,但是會(huì)在系統(tǒng)分析上構(gòu)成重大的挑戰(zhàn)。
目前，尚無(wú)任何系統(tǒng)性的程序去估算幾何分析對(duì)上述兩個(gè)案例的潛在影響。這就必須依靠工程判斷和經(jīng)驗(yàn)。
在這篇文章中，我們制定了理論估計(jì)幾何分析影響工程分析自動(dòng)化的方式。任意形狀和大小的形體如何被簡(jiǎn)化是本文重點(diǎn)要解決的地方。伴隨矩陣和單調(diào)分析這兩個(gè)數(shù)學(xué)概念被合并成一個(gè)統(tǒng)一的理論來(lái)解決雙方的自我平衡和非自我平衡的特點(diǎn)。數(shù)值例子涉及二階scalar偏微分方程，以證實(shí)他的理論。
本文還包含以下內(nèi)容。第二節(jié)中，我們就幾何分析總結(jié)以往的工作。在第三節(jié)中，我們解決幾何分析引起的錯(cuò)誤分析，并討論了擬議的方法。第四部分從數(shù)值試驗(yàn)提供結(jié)果。第五部分討論如何加快設(shè)計(jì)開(kāi)發(fā)進(jìn)度。
2. 前期工作
幾何分析過(guò)程可分為三個(gè)階段:
識(shí)別:哪些特征應(yīng)該被簡(jiǎn)化；
簡(jiǎn)化：如何在一個(gè)自動(dòng)化和幾何一致的方式中簡(jiǎn)化特征；
分析:簡(jiǎn)化的結(jié)果。
第一個(gè)階段的相關(guān)文獻(xiàn)已經(jīng)很多。例如，企業(yè)的規(guī)模和相對(duì)位置這個(gè)特點(diǎn)，經(jīng)常被用來(lái)作為度量鑒定。此外，也有人提議以有意義的力學(xué)判據(jù)確定這種特征。
自動(dòng)化幾何分析過(guò)程，事實(shí)上，已成熟到一個(gè)商業(yè)化幾何分析的地步。但我們注意到，這些商業(yè)軟件包僅提供一個(gè)純粹的幾何解決。因?yàn)闆](méi)有保證隨后進(jìn)行的分析錯(cuò)誤，所以必須十分小心使用。另外，固有的幾何問(wèn)題依然存在，并且還在研究當(dāng)中。
本文的重點(diǎn)是放在第三階段，即快速幾何分析。建立一個(gè)有系統(tǒng)的方法，通過(guò)幾何分析引起的誤差是可以計(jì)算出來(lái)的。再分析的目的是迅速估計(jì)改良系統(tǒng)的反應(yīng)。其中最著名的再分析理論是著名的謝爾曼-Morrison和woodbury公式。對(duì)于兩種有著相似的網(wǎng)狀結(jié)構(gòu)和剛度矩陣設(shè)計(jì)，再分析這種技術(shù)特別有效。然而，過(guò)程幾何分析在網(wǎng)狀結(jié)構(gòu)的剛度矩陣會(huì)導(dǎo)致一個(gè)戲劇性的變化，這與再分析技術(shù)不太相關(guān)。
3. 擬議的方法
3.1問(wèn)題闡述
我們把注意力放在這個(gè)文件中的工程問(wèn)題，標(biāo)量二階偏微分方程式(pde):
許多工程技術(shù)問(wèn)題，如熱，流體靜磁等問(wèn)題，可能簡(jiǎn)化為上述公式。
作為一個(gè)說(shuō)明性例子，考慮散熱問(wèn)題的二維模塊Ω如圖2所示。
圖2二維熱座裝配
熱量q從一個(gè)線圈置于下方位置列為Ωcoil。半導(dǎo)體裝置位于Ωdevice。這兩個(gè)地方都屬于Ω，有相同的材料屬性，其余Ω將在后面討論。特別令人感興趣的是數(shù)量，加權(quán)溫度Tdevice內(nèi)Ωdevice(見(jiàn)圖2)。一個(gè)時(shí)段，認(rèn)定為Ωslot縮進(jìn)如圖2，會(huì)受到抑制，其對(duì)Tdevice將予以研究。邊界的時(shí)段稱為Γslot其余的界線將稱為Γ。邊界溫度Γ假定為零。兩種可能的邊界條件Γslot被認(rèn)為是:(a)固定熱源，即(-kt)?n=q，(b)有一定溫度，即T=Tslot。兩種情況會(huì)導(dǎo)致兩種不同幾何分析引起的誤差的結(jié)果。
設(shè)T(x，y)是未知的溫度場(chǎng)和K導(dǎo)熱。然后，散熱問(wèn)題可以通過(guò)泊松方程式表示:
其中H(x，y)是一些加權(quán)內(nèi)核?，F(xiàn)在考慮的問(wèn)題是幾何分析簡(jiǎn)化的插槽是簡(jiǎn)化之前分析，如圖3所示。
圖3defeatured二維熱傳導(dǎo)裝配模塊
現(xiàn)在有一個(gè)不同的邊值問(wèn)題，不同領(lǐng)域t(x，y):
觀察到的插槽的邊界條件為t(x，y)已經(jīng)消失了，因?yàn)椴垡呀?jīng)不存在了（關(guān)鍵性變化）!
解決的問(wèn)題是:
設(shè)定tdevice和t(x，y)的值，估計(jì)Tdevice。
這是一個(gè)較難的問(wèn)題，是我們尚未解決的。在這篇文章中，我們將從上限和下限分析Tdevice。這些方向是明確被俘引理3、4和3、6。至于其余的這一節(jié)，我們將發(fā)展基本概念和理論，建立這兩個(gè)引理。值得注意的是，只要它不重疊，定位槽與相關(guān)的裝置或熱源沒(méi)有任何限制。上下界的Tdevice將取決于它們的相對(duì)位置。
3.2伴隨矩陣方法
我們需要的第一個(gè)概念是，伴隨矩陣公式表達(dá)法。應(yīng)用伴隨矩陣論點(diǎn)的微分積分方程，包括其應(yīng)用的控制理論，形狀優(yōu)化，拓?fù)鋬?yōu)化等。我們對(duì)這一概念歸納如下。
相關(guān)的問(wèn)題都可以定義為一個(gè)伴隨矩陣的問(wèn)題，控制伴隨矩陣t_(x，y)，必須符合下列公式計(jì)算〔23〕:
伴隨場(chǎng)t_(x，y)基本上是一個(gè)預(yù)定量，即加權(quán)裝置溫度控制的應(yīng)用熱源?？梢杂^察到，伴隨問(wèn)題的解決是復(fù)雜的原始問(wèn)題;控制方程是相同的;這些問(wèn)題就是所謂的自身伴隨矩陣。大部分工程技術(shù)問(wèn)題的實(shí)際利益，是自身伴隨矩陣，就很容易計(jì)算伴隨矩陣。
另一方面，在幾何分析問(wèn)題中，伴隨矩陣發(fā)揮著關(guān)鍵作用。表現(xiàn)為以下引理綜述:
引理3.1已知和未知裝置溫度的區(qū)別，即(Tdevice-tdevice)可以歸納為以下的邊界積分比幾何分析插槽:
在上述引理中有兩點(diǎn)值得注意:
1、積分只牽涉到邊界гslot;這是令人鼓舞的?；蛟S，處理剛剛過(guò)去的被簡(jiǎn)化信息特點(diǎn)可以計(jì)算誤差。
2、右側(cè)牽涉到的未知區(qū)域T(x，y)的全功能的問(wèn)題。特別是第一周期涉及的差異，在正常的梯度，即涉及[-k(T-t)] ?n;這是一個(gè)已知數(shù)量邊界條件[-kt]?n所指定的時(shí)段，未知狄里克萊條件作出規(guī)定[-kt]?n可以評(píng)估。在另一方面，在第二個(gè)周期內(nèi)涉及的差異，在這兩個(gè)領(lǐng)域，即T管; 因?yàn)閠可以評(píng)價(jià)，這是一個(gè)已知數(shù)量邊界條件T指定的時(shí)段。因此。
引理3.2、差額(tdevice-tdevice)不等式
然而，伴隨矩陣技術(shù)不能完全消除未知區(qū)域T(x，y)。為了消除T(x，y)我們把重點(diǎn)轉(zhuǎn)向單調(diào)分析。
3.3單調(diào)性分析
單調(diào)性分析是由數(shù)學(xué)家在19世紀(jì)和20世紀(jì)前建立的各種邊值問(wèn)題。例如，一個(gè)單調(diào)定理:
"添加幾何約束到一個(gè)結(jié)構(gòu)性問(wèn)題，是指在位移(某些)邊界不減少"。
觀察發(fā)現(xiàn)，上述理論提供了一個(gè)定性的措施以解決邊值問(wèn)題。
后來(lái)，工程師利用之前的“計(jì)算機(jī)時(shí)代”上限或下限同樣的定理，解決了具有挑戰(zhàn)性的問(wèn)題。當(dāng)然，隨著計(jì)算機(jī)時(shí)代的到來(lái)，這些相當(dāng)復(fù)雜的直接求解方法已經(jīng)不為人所用。但是，在當(dāng)前的幾何分析，我們證明這些定理采取更為有力的作用，尤其應(yīng)當(dāng)配合使用伴隨理論。
我們現(xiàn)在利用一些單調(diào)定理，以消除上述引理T(x，y)。遵守先前規(guī)定，右邊是區(qū)別已知和未知的領(lǐng)域，即T(x，y)-t(x，y)。因此，讓我們?cè)诮缍ㄒ粋€(gè)領(lǐng)域E(x，y)在區(qū)域?yàn)?
e(x，y)=t(x，y)-t(x，y)。
據(jù)悉，T(x，y)和T(x，y)都是明確的界定，所以是e(x，y)。事實(shí)上，從公式(1)和(3)，我們可以推斷，e(x，y)的正式滿足邊值問(wèn)題:
解決上述問(wèn)題就能解決所有問(wèn)題。但是，如果我們能計(jì)算區(qū)域e(x，y)與正常的坡度超過(guò)插槽，以有效的方式，然后(Tdevice-tdevice)，就評(píng)價(jià)表示e(X，Y)的效率，我們現(xiàn)在考慮在上述方程兩種可能的情況如(a)及(b)。
例(a)邊界條件較第一插槽，審議本案時(shí)槽原本指定一個(gè)邊界條件。為了估算e(x，y)，考慮以下問(wèn)題:
因?yàn)橹蝗Q于縫隙，不討論域，以上問(wèn)題計(jì)算較簡(jiǎn)單。經(jīng)典邊界積分/邊界元方法可以引用。關(guān)鍵是計(jì)算機(jī)領(lǐng)域e1(x，y)和未知領(lǐng)域的e(x，y)透過(guò)引理3.3。這兩個(gè)領(lǐng)域e1(x，y)和e(x，y)滿足以下單調(diào)關(guān)系:
把它們綜合在一起，我們有以下結(jié)論引理。
引理3.4未知的裝置溫度Tdevice，當(dāng)插槽具有邊界條件，東至以下限額的計(jì)算，只要求:(1)原始及伴隨場(chǎng)T和隔熱與幾何分析域(2)解決e1的一項(xiàng)問(wèn)題涉及插槽:
觀察到兩個(gè)方向的右側(cè)，雙方都是獨(dú)立的未知區(qū)域T(x，y)。
例(b) 插槽Dirichlet邊界條件
我們假定插槽都維持在定溫Tslot?？紤]任何領(lǐng)域，即包含域和插槽。界定一個(gè)區(qū)域e(x，y)在滿足:
現(xiàn)在建立一個(gè)結(jié)果與e-(x，y)及e(x，y)。
引理3.5
注意到，公式(7)的計(jì)算較為簡(jiǎn)單。這是我們最終要的結(jié)果。
引理3.6 未知的裝置溫度Tdevice，當(dāng)插槽有Dirichlet邊界條件，東至以下限額的計(jì)算，只要求:(1)原始及伴隨場(chǎng)T和隔熱與幾何分析。(2) 圍繞插槽解決失敗了的邊界問(wèn)題，:
再次觀察這兩個(gè)方向都是獨(dú)立的未知領(lǐng)域T(x，y)。
4. 數(shù)值例子說(shuō)明
我們的理論發(fā)展，在上一節(jié)中，通過(guò)數(shù)值例子。設(shè)
k = 5W/m?C, Q = 10 W/m3 and H = 。
表1:結(jié)果表
表1給出了不同時(shí)段的邊界條件。第一裝置溫度欄的共同溫度為所有幾何分析模式(這不取決于插槽邊界條件及插槽幾何分析)。接下來(lái)兩欄的上下界說(shuō)明引理3.4和3.6。最后一欄是實(shí)際的裝置溫度所得的全功能模式(前幾何分析)，是列在這里比較前列的。在全部例子中，我們可以看到最后一欄則是介于第二和第三列。T Tdevice T
對(duì)于絕緣插槽來(lái)說(shuō)，Dirichlet邊界條件指出，觀察到的各種預(yù)測(cè)為零。不同之處在于這個(gè)事實(shí):在第一個(gè)例子，一個(gè)零Neumann邊界條件的時(shí)段，導(dǎo)致一個(gè)自我平衡的特點(diǎn)，因此，其對(duì)裝置基本沒(méi)什么影響。另一方面，有Dirichlet邊界條件的插槽結(jié)果在一個(gè)非自我平衡的特點(diǎn)，其缺失可能導(dǎo)致器件溫度的大變化在。
不過(guò)，固定非零槽溫度預(yù)測(cè)范圍為20度到0度。這可以歸因于插槽溫度接近于裝置的溫度，因此，將其刪除少了影響。
的確，人們不難計(jì)算上限和下限的不同Dirichlet條件插槽。圖4說(shuō)明了變化的實(shí)際裝置的溫度和計(jì)算式。
預(yù)測(cè)的上限和下限的實(shí)際溫度裝置表明理論是正確的。另外，跟預(yù)期結(jié)果一樣，限制槽溫度大約等于裝置的溫度。
5. 快速分析設(shè)計(jì)的情景
我們認(rèn)為對(duì)所提出的理論分析"什么-如果"的設(shè)計(jì)方案，現(xiàn)在有著廣泛的影響。研究顯示設(shè)計(jì)如圖5，現(xiàn)在由兩個(gè)具有單一熱量能源的器件。如預(yù)期結(jié)果兩設(shè)備將不會(huì)有相同的平均溫度。由于其相對(duì)靠近熱源，該裝置的左邊將處在一個(gè)較高的溫度，。
圖4估計(jì)式versus插槽溫度圖
圖5雙熱器座
圖6正確特征可能性位置
為了消除這種不平衡狀況，加上一個(gè)小孔，固定直徑;五個(gè)可能的位置見(jiàn)圖6。兩者的平均溫度在這兩個(gè)地區(qū)最低。
強(qiáng)制進(jìn)行有限元分析每個(gè)配置。這是一個(gè)耗時(shí)的過(guò)程。另一種方法是把該孔作為一個(gè)特征，并研究其影響，作為后處理步驟。換言之，這是一個(gè)特殊的“幾何分析”例子，而擬議的方法同樣適用于這種情況。我們可以解決原始和伴隨矩陣的問(wèn)題，原來(lái)的配置(無(wú)孔)和使用的理論發(fā)展在前兩節(jié)學(xué)習(xí)效果加孔在每個(gè)位置是我們的目標(biāo)。目的是在平均溫度兩個(gè)裝置最大限度的差異。表2概括了利用這個(gè)理論和實(shí)際的價(jià)值。
從上表可以看到，位置W是最佳地點(diǎn)，因?yàn)樗凶畹途殿A(yù)期目標(biāo)的功能。
附錄II 外文文獻(xiàn)原文
A formal theory for estimating defeaturing -induced engineering analysis errors
Sankara Hari Gopalakrishnan, Krishnan Suresh
Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706, United States
Received 13 January 2006; accepted 30 September 2006
Abstract
Defeaturing is a popular CAD/CAE simplification technique that suppresses ‘small or irrelevant features’ within a CAD model to speed-up downstream processes such as finite element analysis. Unfortunately, defeaturing inevitably leads to analysis errors that are not easily quantifiable within the current theoretical framework.
In this paper, we provide a rigorous theory for swiftly computing such defeaturing -induced engineering analysis errors. In particular, we focus on problems where the features being suppressed are cutouts of arbitrary shape and size within the body. The proposed theory exploits the adjoint formulation of boundary value problems to arrive at strict bounds on defeaturing induced analysis errors. The theory is illustrated through numerical examples.
Keywords: Defeaturing; Engineering analysis; Error estimation; CAD/CAE
1. Introduction
Mechanical artifacts typically contain numerous geometric features. However, not all features are critical during engineering analysis. Irrelevant features are often suppressed or ‘defeatured’, prior to analysis, leading to increased automation and computational speed-up.
For example, consider a brake rotor illustrated in Fig. 1(a). The rotor contains over 50 distinct ‘features’, but not all of these are relevant during, say, a thermal analysis. A defeatured brake rotor is illustrated in Fig. 1(b). While the finite element analysis of the full-featured model in Fig. 1(a) required over 150,000 degrees of freedom, the defeatured model in Fig. 1(b) required <25,000 DOF, leading to a significant computational speed-up.
Fig. 1. (a) A brake rotor and (b) its defeatured version.
Besides an improvement in speed, there is usually an increased level of automation in that it is easier to automate finite element mesh generation of a defeatured component [1,2]. Memory requirements also decrease, while condition number of the discretized system improves;the latter plays an important role in iterative linear system solvers [3].
Defeaturing, however, invariably results in an unknown ‘perturbation’ of the underlying field. The perturbation may be ‘small and localized’ or ‘large and spread-out’, depending on various factors. For example, in a thermal problem, suppose one deletes a feature; the perturbation is localized provided: (1) the net heat flux on the boundary of the feature is zero, and (2) no new heat sources are created when the feature is suppressed; see [4] for exceptions to these rules. Physical features that exhibit this property are called self-equilibrating [5]. Similarly results exist for structural problems.
From a defeaturing perspective, such self-equilibrating features are not of concern if the features are far from the region of interest. However, one must be cautious if the features are close to the regions of interest.
On the other hand, non-self-equilibrating features are of even higher concern. Their suppression can theoretically be felt everywhere within the system, and can thus pose a major challenge during analysis.
Currently, there are no systematic procedures for estimating the potential impact of defeaturing in either of the above two cases. One must rely on engineering judgment and experience.
In this paper, we develop a theory to estimate the impact of defeaturing on engineering analysis in an automated fashion. In particular, we focus on problems where the features being suppressed are cutouts of arbitrary shape and size within the body. Two mathematical concepts, namely adjoint formulation and monotonicity analysis, are combined into a unifying theory to address both self-equilibrating and non-self-equilibrating features. Numerical examples involving 2nd order scalar partial differential equations are provided to substantiate the theory.
The remainder of the paper is organized as follows. In Section 2, we summarize prior work on defeaturing. In Section 3, we address defeaturing induced analysis errors, and discuss the proposed methodology. Results from numerical experiments are provided in Section 4. A by-product of the proposed work on rapid design exploration is discussed in Section 5. Finally, conclusions and open issues are discussed in Section 6.
2. Prior work
The defeaturing process can be categorized into three phases:
Identification: what features should one suppress?
Suppression: how does one suppress the feature in an automated and geometrically consistent manner?
Analysis: what is the consequence of the suppression?
The first phase has received extensive attention in the literature. For example, the size and relative location of a feature is often used as a metric in identification [2,6]. In addition, physically meaningful ‘mechanical criterion/heuristics’ have also been proposed for identifying such features [1,7].
To automate the geometric process of defeaturing, the authors in [8] develop a set of geometric rules, while the authors in [9] use face clustering strategy and the authors in [10] use plane splitting techniques. Indeed, automated geometric defeaturing has matured to a point where commercial defeaturing /healing packages are now available [11,12]. But note that these commercial packages provide a purely geometric solution to the problem... they must be used with care since there are no guarantees on the ensuing analysis errors. In addition, open geometric issues remain and are being addressed [13].
The focus of this paper is on the third phase, namely, post defeaturing analysis, i.e., to develop a systematic methodology through which defeaturing -induced errors can be computed. We should mention here the related work on reanalysis. The objective of reanalysis is to swiftly compute the response of a modified system by using previous simulations. One of the key developments in reanalysis is the famous Sherman–Morrison and Woodbury formula [14] that allows the swift computation of the inverse of a perturbed stiffness matrix; other variations of this based on Krylov subspace techniques have been proposed [15–17]. Such reanalysis techniques are particularly effective when the objective is to analyze two designs that share similar mesh structure, and stiffness matrices. Unfortunately, the process of 幾何分析 can result in a dramatic change in the mesh structure and stiffness matrices, making reanalysis techniques less relevant.
A related problem that is not addressed in this paper is that of local–global analysis [13], where the objective is to solve the local field around the defeatured region after the global defeatured problem has been solved. An implicit assumption in local–global analysis is that the feature being suppressed is self-equilibrating.
3. Proposed methodology
3.1. Problem statement
We restrict our attention in this paper to engineering problems involving a scalar field u governed by a generic 2nd order partial differential equation (PDE):
A large class of engineering problems, such as thermal, fluid and magneto-static problems, may be reduced to the above form.
As an illustrative example, consider a thermal problem over the 2-D heat-block assembly Ω illustrated in Fig. 2.
The assembly receives heat Q from a coil placed beneath the region identified as Ωcoil. A semiconductor device is seated at Ωdevice. The two regions belong to Ω and have the same material properties as the rest of Ω. In the ensuing discussion, a quantity of particular interest will be the weighted temperature Tdevice within Ωdevice (see Eq. (2) below). A slot, identified as Ωslot in Fig. 2, will be suppressed, and its effect on Tdevice will be studied. The boundary of the slot will be denoted by Γslot while the rest of the boundary will be denoted by Γ. The boundary temperature on Γ is assumed to be zero. Two possible boundary conditions on Γslot are considered: (a) fixed heat source, i.e., (-krT).?n = q, or (b) fixed temperature, i.e., T = Tslot. The two cases will lead to two different results for defeaturing induced error estimation.
Fig. 2. A 2-D heat block assembly.
Formally,let T (x, y) be the unknown temperature field and k the thermal conductivity. Then, the thermal problem may be stated through the Poisson equation [18]:
Given the field T (x, y), the quantity of interest is:
where H(x, y) is some weighting kernel. Now consider the defeatured problem where the slot is suppressed prior to analysis, resulting in the simplified geometry illustrated in Fig. 3.
Fig. 3. A defeatured 2-D heat block assembly.
We now have a different boundary value problem, governing a different scalar field t (x, y):
Observe that the slot boundary condition for t (x, y) has disappeared since the slot does not exist any more…a crucial change!
The problem addressed here is:
Given tdevice and the field t (x, y), estimate Tdevice without explicitly solving Eq. (1).
This is a non-trivial problem; to the best of our knowledge,it has not been addressed in the literature. In this paper, we will derive upper and lower bounds for Tdevice. These bounds are explicitly captured in Lemmas 3.4 and 3.6. For the remainder of this section, we will develop the essential concepts and theory to establish these two lemmas. It is worth noting that there are no restrictions placed on the location of the slot with respect to the device or the heat source, provided it does not overlap with either. The upper and lower bounds on Tdevice will however depend on their relative locations.
3.2. Adjoint methods
The first concept that we would need is that of adjoint formulation. The application of adjoint arguments towards differential and integral equations has a long and distinguished history [19,20], including its applications in control theory [21],shape optimization [22], topology optimization, etc.; see [23] for an overview.We summarize below concepts essential to this paper.
Associated with the problem summarized by Eqs. (3) and (4), one can define an adjoint problem governing an adjoint variable denoted by t_(x, y) that must satisfy the following equation [23]: (See Appendix A for the derivation.)
The adjoint field t_(x, y) is essentially a ‘sensitivity map’ of the desired quantity, namely the weighted device temperature to the applied heat source. Observe that solving the adjoint problem is only as complex as the primal problem; the governing equations are identical; such problems are called self-adjoint. Most engineering problems of practical interest are self-adjoint, making it easy to compute primal and adjoint fields without doubling the computational effort.
For the defeatured problem on hand, the adjoint field plays a critical role as the following lemma summarizes:
Lemma 3.1. The difference between the unknown and known device temperature, i.e., (Tdevice ? tdevice), can be reduced to the following boundary integral over the defeatured slot:
Two points are worth noting in the above lemma:
1. The integral only involves the slot boundary Гslot; this is encouraging … perhaps, errors can be computed by processing information just over the feature being suppressed.
2. The right hand side however involves the unknown field T (x, y) of the full-featured problem. In particular, the first term involves the difference in the normal gradients, i.e.,involves [?k(T ? t)]. ?n; this is a known quantity if Neumann boundary conditions [?kT ]. ?n are prescribed over the slot since [?kt]. ?n can be evaluated, but unknown if Dirichlet conditions are prescribed. On the other hand,the second term involves the difference in the two fields,i.e., involves (T ? t); this is a known quantity if Dirichlet boundary conditions T are prescribed over the slot since t can be evaluated, but unknown if Neumann conditions are prescribed. Thus, in both cases, one of the two terms gets ‘evaluated’. The next lemma exploits this observation.
Lemma 3.2. The difference (Tdevice ? tdevice) satisfies the inequalities
Unfortunately, that is how far one can go with adjoint techniques; one cannot entirely eliminate the unknown field T (x, y) from the right hand side using adjoint techniques. In order to eliminate T (x, y) we turn our attention to monotonicity analysis.
3.3. Monotonicity analysis
Monotonicity analysis was established by mathematicians during the 19th and early part of 20th century to establish the existence of solutions to various boundary value problems [24].For example, a monotonicity theorem in [25] states:
“On adding geometrical constraints to a structural problem,the mean displacement over (certain) boundaries does not decrease”.
Observe that the above theorem provides a qualitative measure on solutions to boundary value problems.
Later on, prior to the ‘computational era’, the same theorems were used by engineers to get quick upper or lower bounds to challenging problems by reducing a complex problem to simpler ones [25]. Of course, on the advent of the computer, such methods and theorems took a back-seat since a direct numerical solution of fairly complex problems became feasible.However, in the present context of defeaturing, we show that these theorems take on a more powerful role, especially when used in conjunction with adjoint theory.
We will now exploit certain monotonicity theorems to eliminate T (x, y) from the above lemma. Observe in the previous lemma that the right hand side involves the difference between the known and unknown fields, i.e., T (x, y) ? t (x, y). Let us therefore define a field e(x, y) over the region as:
e(x, y) = T (x, y) ? t (x, y) in .
Note that since excludes the slot, T (x, y) and t (x, y) are both well defined in , and so is e(x, y). In fact, from Eqs. (1) and (3) we can deduce that e(x, y) formally satisfies the boundary value problem:
Solving the above problem is computationally equivalent to solving the full-featured problem of Eq. (1). But, if we could compute the field e(x, y) and its normal gradient over the slot,in an efficient manner, then (Tdevice ? tdevice) can be evaluated from the previous lemma. To evaluate e(x, y) efficiently, we now consider two possible cases (a) and (b) in the above equation.
Case (a) Neumann boundary condition over slot
First, consider the case when the slot was originally assigned a Neumann boundary condition. In order to estimate e(x, y),consider the following exterior Neumann problem:
The above exterior Neumann problem is computationally inexpensive to solve since it