外文翻譯--估計導(dǎo)致工程幾何分析錯誤的一個正式理論

1、<p>　　附錄I 外文文獻(xiàn)翻譯</p><p>　　估計導(dǎo)致工程幾何分析錯誤的一個正式理論</p><p>　　SankaraHariGopalakrishnan，KrishnanSuresh</p><p>　　機(jī)械工程系，威斯康辛大學(xué)，麥迪遜分校，2006年9月30日</p><p>　　摘要：幾何分析是著名的計算機(jī)輔助設(shè)計/

2、計算機(jī)輔助工藝簡化 “小或無關(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產(chǎn)生分析錯誤，在目前的理論框架實在不容易量化。</p><p>　　本文中，我們對快速計算處理這些幾何分析錯誤提供了嚴(yán)謹(jǐn)?shù)睦碚?。尤其，我們集中力量解決地方的特點，被簡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題抵達(dá)嚴(yán)格界限幾何分析性分析錯誤。該理論通過數(shù)值例子說明。</p><p&

3、gt;　　關(guān)鍵詞:幾何分析;工程分析;誤差估計;計算機(jī)輔助設(shè)計/計算機(jī)輔助教學(xué)</p><p><b>　　介紹</b></p><p>　　機(jī)械零件通常包含了許多幾何特征。不過，在工程分析中并不是所有的特征都是至關(guān)重要的。以前的分析中無關(guān)特征往往被忽略，從而提高自動化及運(yùn)算速度。</p><p>　　舉例來說，考慮一個剎車轉(zhuǎn)子，如圖1(a)。

4、轉(zhuǎn)子包含50多個不同的特征，但所有這些特征并不是都是相關(guān)的。就拿一個幾何化的剎車轉(zhuǎn)子的熱量分析來說，如圖1(b)。有限元分析的全功能的模型如圖1(a)，需要超過150,000度的自由度，幾何模型圖1(b)項要求小于25，000個自由度，從而導(dǎo)致非常緩慢的運(yùn)算速度。</p><p>　　圖1(a)剎車轉(zhuǎn)子 圖1(b)其幾何分析版本</p><p>　

5、　除了提高速度，通常還能增加自動化水平，這比較容易實現(xiàn)自動化的有限元網(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)。</p><p>　　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，對于一個熱問題，想刪除其中的一個特征，不穩(wěn)定性是一個局部問題:(1)凈熱通量邊界的特點是零。(2)特征簡化時沒有新的熱源產(chǎn)生;

6、[4]對上述規(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)上的問題。</p><p>　　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，。</p><p>　　從另一個角度看，非自我平衡的特征應(yīng)值得重視。這些特征的簡化理論上可以在系統(tǒng)任意位置被施用,但是會在系統(tǒng)分析上構(gòu)成重大的挑戰(zhàn)。</p><

7、p>　　目前，尚無任何系統(tǒng)性的程序去估算幾何分析對上述兩個案例的潛在影響。這就必須依靠工程判斷和經(jīng)驗。</p><p>　　在這篇文章中，我們制定了理論估計幾何分析影響工程分析自動化的方式。任意形狀和大小的形體如何被簡化是本文重點要解決的地方。伴隨矩陣和單調(diào)分析這兩個數(shù)學(xué)概念被合并成一個統(tǒng)一的理論來解決雙方的自我平衡和非自我平衡的特點。數(shù)值例子涉及二階scalar偏微分方程，以證實他的理論。</p&g

8、t;<p>　　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誤分析，并討論了擬議的方法。第四部分從數(shù)值試驗提供結(jié)果。第五部分討論如何加快設(shè)計開發(fā)進(jìn)度。</p><p><b>　　前期工作</b></p><p>　　幾何分析過程可分為三個階段:</p><p>　　識別:哪些特

9、征應(yīng)該被簡化；</p><p>　　簡化：如何在一個自動化和幾何一致的方式中簡化特征；</p><p><b>　　分析:簡化的結(jié)果。</b></p><p>　　第一個階段的相關(guān)文獻(xiàn)已經(jīng)很多。例如，企業(yè)的規(guī)模和相對位置這個特點，經(jīng)常被用來作為度量鑒定。此外，也有人提議以有意義的力學(xué)判據(jù)確定這種特征。</p><p>　

10、　自動化幾何分析過程，事實上，已成熟到一個商業(yè)化幾何分析的地步。但我們注意到，這些商業(yè)軟件包僅提供一個純粹的幾何解決。因為沒有保證隨后進(jìn)行的分析錯誤，所以必須十分小心使用。另外，固有的幾何問題依然存在，并且還在研究當(dāng)中。</p><p>　　本文的重點是放在第三階段，即快速幾何分析。建立一個有系統(tǒng)的方法，通過幾何分析引起的誤差是可以計算出來的。再分析的目的是迅速估計改良系統(tǒng)的反應(yīng)。其中最著名的再分析理論是著名的謝

11、爾曼-Morrison和woodbury公式。對于兩種有著相似的網(wǎng)狀結(jié)構(gòu)和剛度矩陣設(shè)計，再分析這種技術(shù)特別有效。然而，過程幾何分析在網(wǎng)狀結(jié)構(gòu)的剛度矩陣會導(dǎo)致一個戲劇性的變化，這與再分析技術(shù)不太相關(guān)。</p><p><b>　　擬議的方法</b></p><p><b>　　3.1問題闡述</b></p><p>　　我們

12、把注意力放在這個文件中的工程問題，標(biāo)量二階偏微分方程式(pde): </p><p>　　許多工程技術(shù)問題，如熱，流體靜磁等問題，可能簡化為上述公式。</p><p>　　作為一個說明性例子，考慮散熱問題的二維模塊Ω如圖2所示。</p><p><b>　　圖2二維熱座裝配</b></p><p>　　熱量q從一個線圈置

13、于下方位置列為Ωcoil。半導(dǎo)體裝置位于Ωdevice。這兩個地方都屬于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(見圖2)。一個時段，認(rèn)定為Ωslot縮進(jìn)如圖2，會受到抑制，其對Tdevice將予以研究。邊界的時段稱為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=Tslo

14、t。兩種情況會導(dǎo)致兩種不同幾何分析引起的誤差的結(jié)果。</p><p>　　設(shè)T(x，y)是未知的溫度場和K導(dǎo)熱。然后，散熱問題可以通過泊松方程式表示:</p><p>　　其中H(x，y)是一些加權(quán)內(nèi)核?，F(xiàn)在考慮的問題是幾何分析簡化的插槽是簡化之前分析，如圖3所示。</p><p>　　圖3defeatured二維熱傳導(dǎo)裝配模塊</p><p&g

15、t;　　現(xiàn)在有一個不同的邊值問題，不同領(lǐng)域t(x，y):</p><p>　　觀察到的插槽的邊界條件為t(x，y)已經(jīng)消失了，因為槽已經(jīng)不存在了（關(guān)鍵性變化）!</p><p><b>　　解決的問題是:</b></p><p>　　設(shè)定tdevice和t(x，y)的值，估計Tdevice。</p><p>　　這是一個

16、較難的問題，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)的裝置或熱源沒有任何限制。上下界的Tdevice將取決于它們的相對位置。</p><p><b>　　3.2伴隨矩陣方法</b></p><p>　　

17、我們需要的第一個概念是，伴隨矩陣公式表達(dá)法。應(yīng)用伴隨矩陣論點的微分積分方程，包括其應(yīng)用的控制理論，形狀優(yōu)化，拓?fù)鋬?yōu)化等。我們對這一概念歸納如下。</p><p>　　相關(guān)的問題都可以定義為一個伴隨矩陣的問題，控制伴隨矩陣t_(x，y)，必須符合下列公式計算〔23〕:</p><p>　　伴隨場t_(x，y)基本上是一個預(yù)定量，即加權(quán)裝置溫度控制的應(yīng)用熱源?？梢杂^察到，伴隨問題的解決是復(fù)雜的

18、原始問題;控制方程是相同的;這些問題就是所謂的自身伴隨矩陣。大部分工程技術(shù)問題的實際利益，是自身伴隨矩陣，就很容易計算伴隨矩陣。</p><p>　　另一方面，在幾何分析問題中，伴隨矩陣發(fā)揮著關(guān)鍵作用。表現(xiàn)為以下引理綜述:</p><p>　　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:</p><p&

19、gt;　　在上述引理中有兩點值得注意:</p><p>　　1、積分只牽涉到邊界гslot;這是令人鼓舞的。或許，處理剛剛過去的被簡化信息特點可以計算誤差。</p><p>　　2、右側(cè)牽涉到的未知區(qū)域T(x，y)的全功能的問題。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;這是一個已知數(shù)量邊界條件[-kt]?n所指定的時段，未知狄里克萊條件作出規(guī)定[-kt]?n

20、可以評估。在另一方面，在第二個周期內(nèi)涉及的差異，在這兩個領(lǐng)域，即T管; 因為t可以評價，這是一個已知數(shù)量邊界條件T指定的時段。因此。</p><p>　　引理3.2、差額(tdevice-tdevice)不等式</p><p>　　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)我們把重點轉(zhuǎn)向單調(diào)分析。</p><p><b>　　

21、3.3單調(diào)性分析</b></p><p>　　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題。例如，一個單調(diào)定理:</p><p>　　"添加幾何約束到一個結(jié)構(gòu)性問題，是指在位移(某些)邊界不減少"。</p><p>　　觀察發(fā)現(xiàn)，上述理論提供了一個定性的措施以解決邊值問題。</p><p>　　

22、后來，工程師利用之前的“計算機(jī)時代”上限或下限同樣的定理，解決了具有挑戰(zhàn)性的問題。當(dāng)然，隨著計算機(jī)時代的到來，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。</p><p>　　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，讓我們在界定一個領(lǐng)域E(

23、x，y)在區(qū)域為:</p><p>　　e(x，y)=t(x，y)-t(x，y)。</p><p>　　據(jù)悉，T(x，y)和T(x，y)都是明確的界定，所以是e(x，y)。事實上，從公式(1)和(3)，我們可以推斷，e(x，y)的正式滿足邊值問題:</p><p>　　解決上述問題就能解決所有問題。但是，如果我們能計算區(qū)域e(x，y)與正常的坡度超過插槽，以有效的方

24、式，然后(Tdevice-tdevice)，就評價表示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)。</p><p>　　例(a)邊界條件較第一插槽，審議本案時槽原本指定一個邊界條件。為了估算e(x，y)，考慮以下問題:</p><p>　　因為只取決于縫隙，不討論域，以上問題計算較簡單。經(jīng)典邊界積分/邊界元方法可以引用。關(guān)鍵是計算機(jī)領(lǐng)域e1(x，y)和未知領(lǐng)域

25、的e(x，y)透過引理3.3。這兩個領(lǐng)域e1(x，y)和e(x，y)滿足以下單調(diào)關(guān)系:</p><p>　　把它們綜合在一起，我們有以下結(jié)論引理。</p><p>　　引理3.4未知的裝置溫度Tdevice，當(dāng)插槽具有邊界條件，東至以下限額的計算，只要求:(1)原始及伴隨場T和隔熱與幾何分析域(2)解決e1的一項問題涉及插槽:</p><p>　　觀察到兩個方向的右

26、側(cè)，雙方都是獨(dú)立的未知區(qū)域T(x，y)。</p><p>　　例(b) 插槽Dirichlet邊界條件</p><p>　　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。界定一個區(qū)域e(x，y)在滿足:</p><p>　　現(xiàn)在建立一個結(jié)果與e-(x，y)及e(x，y)。</p><p><b>　　引理3.5&

27、lt;/b></p><p>　　注意到，公式(7)的計算較為簡單。這是我們最終要的結(jié)果。</p><p>　　引理3.6 未知的裝置溫度Tdevice，當(dāng)插槽有Dirichlet邊界條件，東至以下限額的計算，只要求:(1)原始及伴隨場T和隔熱與幾何分析。(2) 圍繞插槽解決失敗了的邊界問題，:</p><p>　　再次觀察這兩個方向都是獨(dú)立的未知領(lǐng)域T(x，

28、y)。</p><p><b>　　數(shù)值例子說明</b></p><p>　　我們的理論發(fā)展，在上一節(jié)中，通過數(shù)值例子。設(shè)</p><p>　　k = 5W/m?C, Q = 10 W/m3 and H = 。</p><p><b>　　表1:結(jié)果表</b></p><p>

29、;　　表1給出了不同時段的邊界條件。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)。接下來兩欄的上下界說明引理3.4和3.6。最后一欄是實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</p><p>　　對于絕緣插槽來說，Dirichlet邊界條件指出，觀察到的各種預(yù)測為零。不同

30、之處在于這個事實:在第一個例子，一個零Neumann邊界條件的時段，導(dǎo)致一個自我平衡的特點，因此，其對裝置基本沒什么影響。另一方面，有Dirichlet邊界條件的插槽結(jié)果在一個非自我平衡的特點，其缺失可能導(dǎo)致器件溫度的大變化在。</p><p>　　不過，固定非零槽溫度預(yù)測范圍為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。</p><p>　　的確，人們不難

31、計算上限和下限的不同Dirichlet條件插槽。圖4說明了變化的實際裝置的溫度和計算式。</p><p>　　預(yù)測的上限和下限的實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。</p><p><b>　　快速分析設(shè)計的情景</b></p><p>　　我們認(rèn)為對所提出的理論分析"什么-如果"

32、;的設(shè)計方案，現(xiàn)在有著廣泛的影響。研究顯示設(shè)計如圖5，現(xiàn)在由兩個具有單一熱量能源的器件。如預(yù)期結(jié)果兩設(shè)備將不會有相同的平均溫度。由于其相對靠近熱源，該裝置的左邊將處在一個較高的溫度，。</p><p>　　圖4估計式versus插槽溫度圖</p><p><b>　　圖5雙熱器座</b></p><p>　　圖6正確特征可能性位置</p&

33、gt;<p>　　為了消除這種不平衡狀況，加上一個小孔，固定直徑;五個可能的位置見圖6。兩者的平均溫度在這兩個地區(qū)最低。</p><p>　　強(qiáng)制進(jìn)行有限元分析每個配置。這是一個耗時的過程。另一種方法是把該孔作為一個特征，并研究其影響，作為后處理步驟。換言之，這是一個特殊的“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題，原來的配置(無孔)和使用的理論發(fā)展在前兩節(jié)學(xué)

34、習(xí)效果加孔在每個位置是我們的目標(biāo)。目的是在平均溫度兩個裝置最大限度的差異。表2概括了利用這個理論和實際的價值。</p><p>　　從上表可以看到，位置W是最佳地點，因為它有最低均值預(yù)期目標(biāo)的功能。</p><p>　　附錄II 外文文獻(xiàn)原文</p><p>　　A formal theory for estimating defeaturing -induced

35、engineering analysis errors</p><p>　　Sankara Hari Gopalakrishnan, Krishnan Suresh</p><p>　　Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706, United StatesReceive

36、d 13 January 2006; accepted 30 September 2006</p><p><b>　　Abstract</b></p><p>　　Defeaturing is a popular CAD/CAE simplification technique that suppresses ‘small or irrelevant feature

37、s’ 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 frame

38、work.</p><p>　　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

39、 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 i

40、llustrated through numerical examples.</p><p>　　Keywords: Defeaturing; Engineering analysis; Error estimation; CAD/CAE</p><p>　　1. Introduction</p><p>　　Mechanical artifacts typical

41、ly 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

42、 computational speed-up.</p><p>　　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 analys

43、is. 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 <

44、;25,000 DOF, leading to a significant computational speed-up.</p><p>　　Fig. 1. (a) A brake rotor and (b) its defeatured version.</p><p>　　Besides an improvement in speed, there i

45、s 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 sys

46、tem improves;the latter plays an important role in iterative linear system solvers [3].</p><p>　　Defeaturing, however, invariably results in an unknown ‘perturbation’ of the underlying field. The perturbatio

47、n 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 t

48、he 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 </p><p>　　From a defeaturing perspect

49、ive, 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.</p><p>　　On t

50、he 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.</p><p&g

51、t;　　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.</p><p>　　In this p

52、aper, 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

53、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

54、 examples involving 2nd order scalar partial differ</p><p>　　The remainder of the paper is organized as follows. In Section 2, we summarize prior work on defeaturing. In Section 3, we address defeaturing ind

55、uced 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,

56、conclusions and open issues are discussed in Section 6.</p><p>　　2. Prior work</p><p>　　The defeaturing process can be categorized into three phases:</p><p>　　Identification: what f

57、eatures should one suppress?</p><p>　　Suppression: how does one suppress the feature in an automated and geometrically consistent manner?</p><p>　　Analysis: what is the consequence of the suppre

58、ssion?</p><p>　　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, ph

59、ysically meaningful ‘mechanical criterion/heuristics’ have also been proposed for identifying such features [1,7].</p><p>　　To automate the geometric process of defeaturing, the authors in [8] develop a set

60、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 defeaturin

61、g /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 guarantee</p><p>　　

62、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 w

63、ork 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

64、formula [14] that allows the swift computation of the invers</p><p>　　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 fi

65、eld 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.</p><p>　　3. P

66、roposed methodology</p><p>　　3.1. Problem statement</p><p>　　We restrict our attention in this paper to engineering problems involving a scalar field u governed by a generic 2nd order partial di

67、fferential equation (PDE):</p><p>　　A large class of engineering problems, such as thermal, fluid and magneto-static problems, may be reduced to the above form.</p><p>　　As an illustrative examp

68、le, consider a thermal problem over the 2-D heat-block assembly Ω illustrated in Fig. 2.</p><p>　　The assembly receives heat Q from a coil placed beneath the region identified as Ωcoil. A semiconductor devic

69、e 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 Ωde

70、vice (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</p><p>　　Fig. 2. A 2-D heat block a

71、ssembly.</p><p>　　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]:</p><p>　　Given

72、 the field T (x, y), the quantity of interest is:</p><p>　　where H(x, y) is some weighting kernel. Now consider the defeatured problem where the slot is suppressed prior to analysis, resulting in the simplif

73、ied geometry illustrated in Fig. 3.</p><p>　　Fig. 3. A defeatured 2-D heat block assembly.</p><p>　　We now have a different boundary value problem, governing a different scalar field t (x, y):&l

74、t;/p><p>　　Observe that the slot boundary condition for t (x, y) has disappeared since the slot does not exist any more…a crucial change!</p><p>　　The problem addressed here is:</p><p>

75、;　　Given tdevice and the field t (x, y), estimate Tdevice without explicitly solving Eq. (1).</p><p>　　This is a non-trivial problem; to the best of our knowledge,it has not been addressed in the literature.

76、 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 estab

77、lish 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 </p><p>　　3.2. Adjoint methods&

78、lt;/p><p>　　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], in

79、cluding 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.</p><p>　　Associated with th

80、e 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.)</p>

81、<p>　　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 co

82、mplex 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 fi

83、elds without doubling the computational effort.</p><p>　　For the defeatured problem on hand, the adjoint field plays a critical role as the following lemma summarizes:</p><p>　　Lemma 3.1. The di

84、fference between the unknown and known device temperature, i.e., (Tdevice ? tdevice), can be reduced to the following boundary integral over the defeatured slot:</p><p>　　Two points are worth noting in the a

85、bove lemma:</p><p>　　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.</p>

86、<p>　　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; th

87、is 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 involv

88、es the difference in the two fields,i.e., involves (T ? t); this is a k</p><p>　　Lemma 3.2. The difference (Tdevice ? tdevice) satisfies the inequalities</p><p>　　Unfortunately, that is how far

89、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 analysi

90、s.</p><p>　　3.3. Monotonicity analysis</p><p>　　Monotonicity analysis was established by mathematicians during the 19th and early part of 20th century to establish the existence of solutions to

91、various boundary value problems [24].For example, a monotonicity theorem in [25] states:</p><p>　　“On adding geometrical constraints to a structural problem,the mean displacement over (certain) boundaries do

92、es not decrease”.</p><p>　　Observe that the above theorem provides a qualitative measure on solutions to boundary value problems.</p><p>　　Later on, prior to the ‘computational era’, the same th

93、eorems 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

94、-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 conjunc

95、tion </p><p>　　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 a

96、nd unknown fields, i.e., T (x, y) ? t (x, y). Let us therefore define a field e(x, y) over the region as:</p><p>　　e(x, y) = T (x, y) ? t (x, y) in .</p><p>　　Note that since excludes the slot

97、, 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:</p><p>　　Solving the above probl

98、em 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 e

99、valuated from the previous lemma. To evaluate e(x, y) efficiently, we now consider two possible cases (a) and (b) in the above equation.</p><p>　　Case (a) Neumann boundary condition over slot</p><

100、p>　　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:</p><p>　　The above exterior N

101、eumann problem is computationally inexpensive to solve since it depends only on the slot, and not on the domain . Classic boundary integral/boundary element methods can be used [26]. The key then is to relate the compute

102、d field e1(x, y) and the unknown field e(x, y) through the following lemma.Lemma 3.3. The two fields e1(x, y) and e(x, y) satisfy the following monotonicity relationship:</p><p>　　Proof. Proof exploits trian

103、gle inequality.</p><p>　　Piecing it all together, we have the following conclusive lemma.</p><p>　　Lemma 3.4. The unknown device temperature Tdevice, when the slot has Neumann boundary condition

104、s prescribed, is bounded by the following limits whose computation only requires: (1) the primal and adjoint fields t and t_ associated with the defeatured domain, and (2) the solution e1 to an exterior Neumann problem i