1、第一章 引言1.1 本課題的意義鍛造操作機是鍛造車間實現鍛造自動化的關鍵設備，用于夾持鍛件配合壓機完成鍛造工藝動作。在大鍛件生產中，鍛造操作機更是必不可少的設備。鍛造操作機在20世紀60年代初就已問世，近二、三十年更是得到了迅速的發展。最早是在美國、前蘇聯，而后在德國、英國、日本等國發展起來，并成為系列化產品進入工業生產。最初的操作機多為全機械傳動，隨著科學技術的發展，到60、70年代出現了混合傳動和全液壓傳動、結構緊湊、操作靈活的鍛造操作機。它與壓機配合使用，提高了生產效率及最大鍛件質量。80年代以后，隨著大型裝備制造的快速發展，對大鍛件生產又提出了更高的要求，促進了鍛造操作機技術的發展，主

2、要表現在對鍛造操作機的需求量不斷增加，對鍛造操作機的最大鍛件質量要求大大提高，引起了各國對鍛造操作機在鍛造生產作用的重視。我國鍛造操作機起步于70年代，開始只能由一些鍛造廠自己制造有軌鍛造操作機，這些操作機結構簡單，鉗子的張合夾緊靠與吊鉗分離開的電動方頭扳手來完成，因而夾緊鍛件不方便，只能用于鋼錠開坯、撥料。隨著國民經濟的發展，80年代開始研制出全機械傳動和少數液壓傳動有軌操作機。隨后，小型液壓傳動有軌操作機得到發展，并出現了液壓傳動無軌操作機。90年代初期我國自行設計制造的100kN鍛造操作機主要技術性能已達到世界80年代水平，該臺鍛造操作機于1992年5月在太原試制成功。近年來，核電、造船

3、、化工、國防等領域的大型鍛件精確高效制造迫切需要重載鍛造操作機。重載鍛造操作機發展水平的落后制約了我國的大裝備制造能力，部分大型裝備的關鍵構件完全依賴進口。重載鍛造操作機直接影響國家重大工程的實施和國民經濟的發展，開展重載鍛造操作機的研究具有重要戰略意義。1.2 鍛造操作機的國內外發展現狀大型鍛造操作機屬于當前世界最大的多自由度重載機器人，屬于機、電、液高度一體化的復雜裝備，它是萬噸鍛造壓機的重要配套設備，也是國家經濟建設急需的重大機械裝備之一。并且，大型鍛件制造業是裝備制造業的基礎行業，是關系到國家安全和國家經濟命脈的戰略性行業，其發展水平是衡量國家綜合國力的重要標志。通過深入開展大型鍛造操

4、作機的研究工作，將逐步實現大型鍛造操作機的國產化，對提升我國大型裝備及關鍵零部件的自主設計和制造能力、滿足國家經濟建設的需求結束我國不能設計大型鍛造操作機的歷史都具有重要的社會意義和經濟效益。一、大型鍛造操作機的發展歷史鍛造操作機最早出現在美國和原蘇聯，而后在日本、英國、奧地利等國發展起來，并成為系列化產品進入工業性生產。最初的操作機多為全機械傳動，60、70年代出現了混合傳動和全液壓傳動、結構緊湊、操作靈活的鍛造操作機。到了80年代，各國對鍛造操作機的設計、制造、技術改造方面又有了更高的要求，不斷改進結構及生產工藝，促進了鍛壓技術的發展。特別是鍛造操作機的需求量不斷增加，引起了國內外大、中型

5、企業對鍛造操作機在生產中作用的重視。90年代中期，國外大型鍛造操作機技術已經成熟，大型操作機與30000kN自由鍛造水壓機聯動操作，不斷提高了水壓機生產能力。我國鍛造操作機起步于60年代，開始只能由某些工廠自己制造有軌操作機。90年代初期，我國自行設計制造的100kN鍛造操作機于1992年5月在太原試制成功，其主要技術性能已達到世界80年代水平，能替代同類進口產品。至今，我國自主研發投產的全液壓鍛造操作機最大夾持能力也只有500kN。世界上裝備的萬噸級自由鍛造壓機近30臺，最大的模鍛水壓機載荷能力高達7.5萬噸，最大的六自由度鍛造操作機操作力矩達7500kN·m，最大承載能力高達25

6、00kN。目前，我國已具備了萬噸級鍛壓裝備的設計與制造能力，如中國一重自主設計、自主制造的世界上最先進的150MN自由鍛造水壓機，2006年末已經投產使用，但與之配套的大型鍛造操作機仍在研發當中。二、大型鍛造操作機的研究現狀國內外大型鍛造操作機的研究現狀鍛造操作機作為進行鍛造工藝的重要設備，眾多國外公司對其進行了系統化研究，目前，德國DDS公司、韓國HBE PRESS公司以及捷克ZDAS公司的鍛造操作機的制造水平處在世界前列。其中，德國DDS公司和WEPUKO公司是世界著名的鍛造操作機專業研發、制造企業，在重型鍛造操作機研制領域具有70多年的歷史。此外，日本三菱長琦生產的操作機因擁有高速、高精

7、度的機械手及控制系統而著稱。國內鍛造操作機的研究起步較晚，在一些技術方面與國外相比還有一定的差距。與萬噸壓機配套的大型鍛造操作機全部采用進口設備，自主開發的大型鍛造操作機至今尚未問世，如中國一重與上海交大聯合開發的1600kN鍛造操作機和北方重工自主開發的2000kN鍛造操作機的整機水平還有待于進一步驗證。我國與大型鍛造操作機相關的研究項目為解決我國重大裝備制造中一批關鍵技術和共性技術問題，實現重大裝備及其成套技術的自主研發，科技部在“十一五”國家科技支撐計劃中設立了“大型鑄鍛件制造關鍵技術及裝備研制”項目，在重點完成的工作中明確提出“150MN自由鍛造水壓機及配套設備關鍵技術研究”和“165

8、MN自由鍛造油壓機及配套設備關鍵技術研究”。第一個課題主要開展大型自由鍛造水壓機整機設計、模態分析、預應力框架結構整體振動及疲勞分析，開展快換機構設計和控制系統設計研究，研制配套操作機；第二個課題自主開展大型自由鍛造油壓機整機設計、快換機構設計、控制系統設計技術研究和關鍵部件研制，攻克多功能操作機設計技術、驅動和控制系統設計技術研究和關鍵結構件制造技術等，掌握核心技術，開展壓機與操作機及輔助裝備聯動協調控制技術研究等。上述兩個課題，對掌握大型操作機核心技術、攻破我國重大技術裝備的生產瓶頸、提高特大型自由鍛件的制造技術水平與制造能力起著關鍵性的作用。2006年，上海交通大學、浙江大學、中南大學清

9、華大學、大連理工大學、華中科技大學共同承擔了國家科技部“973”計劃中“巨型重載操作裝備的基礎科學問題”項目，圍繞“多自由度重載操作機構構型與操作性能的映射規律”“重載操作裝備的界面行為與失效機理”“重載操作裝備的多源能量傳遞規律與動態控制”三個基礎科學問題，開展了7個課題研究，包括大型構件制造操作運動軌跡建模、重載裝備多自由度操作性能度量與機構設計原理、低速非連續工況下重載裝備界面行為與力學特征、大尺度重型構件穩定夾持原理與夾持系統驅動策略、大流量電液伺服系統的介質流動規律、重載大慣量裝備的快速協調控制和巨型重載操作裝備的性能仿真與優化等。從基礎研究的角度，揭示了巨型重載操作裝備的操作靈活性

10、、力承載能力、剛度等性能與機構構型的映射規律。此課題為我國巨型重載操作裝備的研究提供了理論基礎，同時，也為配套操作機的研究提供了進一步的可行性。三、大型鍛造操作機的技術特征大型鍛造操作機和萬噸鍛造壓機是配合在一起聯合工作的，工作過程中操作機保持著頻繁的重復動作，對其性能的要求為動作速度高、空行程時間短、精整時定位準確，以達到快速鍛造，并得到尺寸精確的鍛件。與加工裝備相比，大型操作機的特點是載荷大、慣量大、自由度多、操控能力強。大型鍛造操作機的主要技術特征:一是在重載操作條件下，操作機構件的分布式柔性變形直接影響末端執行器的操作精度。因此，在裝備的機構設計中，既要保證操作裝備在整個工作空間中具有

11、理想的剛度特性，又要通過運動學設計對結構變形在裝備運動鏈中的傳遞特性進行控制。此外，鍛造操作機長期在非連續工作條件下進行操作，其動力學性能在空載和負載操作情況下存在顯著差別。二是大型鍛造操作機制造成本高，設計與制造周期長，通常采用單臺制造模式。重載操作機通常很難通過物理樣機實驗對其操作性能進行分析和驗證，因此，計算機數值模擬是鍛造操作機設計、性能評估與優化的重要支撐技術。第二章 鍛造操作機簡介鍛造操作機（manipulator for forging ）用于夾持鋼錠或坯料進行鍛造操作及輔助操作的機械設備。 所謂，“10噸操作機”，是指該操作機可夾持的鋼錠最大重量為十噸。2.1 基本含義用以夾持

12、鍛坯配合水壓機或鍛錘完成送進、轉動、調頭等主要動作的輔助鍛壓機械。鍛造操作機有助于改善勞動條件，提高生產效率。根據需要，操作機也可用于裝爐、出爐，并可實現遙控和與主機聯動。操作機結構分有軌和無軌兩種，其傳動方式有機械式、液壓式和混合式等。此外，還有專門用于某些輔助工序的操作機，如裝取料操作機和工具操作機等。為了配合操作機的工作，有時 圖2-1 鍛造操作機還配置鍛坯回轉臺，以方便鍛坯的調頭。在模鍛和大件沖壓中，機械手的應用已日益普遍，這樣的機械手實際上是一種自動的鍛造操作機。 主要用于750kg空氣錘、1000-2000kg電液錘、模鍛錘或其它相應噸位的鍛錘，是我國鍛造行業最先進的設備之一。 2

13、.2 操作設備采用全液壓傳動，高集成閥塊，超大流量通徑，使系統壓力損失少 密封性能高，油溫控制好。 匠心獨特的油路設計，真正使液壓系統處在最佳狀態，即使在長期大負荷情況下工作，也能輕松勝任。 運動系統采用了擺線齒輪馬達和漸開線減速機組合，完美地實現了大車的無級變速行走、臺架回轉。 三級連動機構使鉗口平行升降，鉗桿傾斜，360度旋轉，三維空間任意靈活轉動。    圖2-2鍛造操作機機械手造型美觀，結構緊湊，轉動極其靈活，能出色地完成龐大的操作機無法完成的動作，讓操作工體驗到人機合一、隨心所欲的感覺，充分體現操作機向機械手轉變的根本意義。 鍛造操作機適用于鍛造和鍛壓行業，與各種

14、自由鍛錘及壓機配合，能完成坯料成型的各種工序；對減輕勞動強度、提高生產效率60%以上；是鍛造鍛壓行業不可缺少的輔助設備。 鍛造操作機分類鍛造操作機分為：直移式、回轉式、平移式等多種運動形式，全機械、全液壓、機械液壓混合等多種驅動形式，可以從各方面滿足不同用戶的需要。 鍛造操作機功能操作機具有以下動作功能：大車在軌道上自由行走；鉗架前后升降、傾斜；鉗頭夾持、松開、旋轉等。大車架采用整體框架式結構，由電機或馬達驅動。鉗架升降有鋼絲繩或油缸帶動，可實現前后同步升降或分別升降，使鉗架到達水平或實現一定角度的傾斜。鉗頭夾緊由大螺距絲桿或油缸帶動夾持拉桿水平移動實現，并且有緩沖保險裝置。鉗頭旋轉由電機減速

15、機帶動，并設有過載保護裝置。鉗架的前后、兩側及鉗架與升降機之間均設有防振動的緩沖裝置（另有大量配件供應）。 2.3 操作機的結構 10噸操作機是由四部分所組成，其結構示意如圖2-3所示。 (1)升降機構：包括前提升油缸12、后提升油缸9、活塞7和13、活塞桿6和14、活動架19、沿塊5以及彈簧24等。(2)夾緊機構：包括旋轉滑閥26、夾緊油缸22活塞23、活塞桿21、鉗殼17、夾緊滑塊18、夾臂16和鉗口15等。圖2-3 10噸操作機結構示意圖 (3)旋轉機構：包括電動機l、制動器2、行星減速器3、減速器4與空心鈾20等。 (4)大車行走機構：包括電動機27、減速器35、車輪28、車體29等。

16、2.3.1 升降機構升降機構主要是為實現柸料的提升、下降、傾斜等動作，以滿足鍛造工藝過程的需要。升降機構由前提升機構、后提升機構、活動架等三部分所組成。2.3.2夾緊機構 夾緊機構主要用來夾持坯料、鍛件或鋼錠。 夾緊機構可以分成鉗頭和夾緊油缸俯兩大部分，它們分別固定在空心軸的兩抵鉗頭在前端，夾緊油缸在后端。 (1)鉗頭 鉗頭的結構如圖2-4所示。兩個鉗口l通過銷軸l0分別與夾臂3的一端鉸接。小軸9穿過夾臂中間的孔，使夾臂小揣固定在鉗殼2上，這樣，夾臂便形成可以繞小軸回轉的杠桿。夾臂的另一端通道銷軸4與連板5鉸接。連板又通過銷軸8與夾緊滑塊6相連。活塞桿7則以螺紋與夾緊滑塊構成一體。圖2-4 鉗

17、頭 當活塞桿在夾緊油缸的拉力作用下，帶動滑塊和連板向后(即向左)移動時，上夾臂繞小軸作順時鐘方向轉動，下夾臂臂繞小軸作逆晌針方向轉動，使兩鉗口間的距離越來越小，坯料被夾緊。當活塞桿在夾緊油缸的推力作用下，推動滑塊、連板向前(即向右)移動時，上夾臂繞小軸作逆時針方向轉動，下夾臂繞小軸作順時針方向轉動，兩鉗口的距離越來越大，于是刨門鉗口便張開。鉗口與夾臀鉸接是為了擴大夾持坯料的尺寸范圍。如當夾持斷面尺寸較大的鋼錠或坯料時，兩個鉗口可以繞銷鈾向鉗頭內轉動，而當夾持斷面尺寸較小的鋼錠或坯料時，兩個鉗口就繞銷軸向鉗頭外轉動，使鉗口與被夾持的鋼錠或坯料始終保持有足夠的接觸面積，被夾持的鋼錠或坯料就不易松脫

18、。 (2)夾緊油缸 夾緊油缸是操作機產生夾緊力的機構，在它的拉力或推力作用下，使鉗頭的鉗口完成對鋼錠、坯料或鍛件的夾緊與張開動作。 夾緊油缸又可分成兩大部分，一部分為油缸，另一部分為旋轉滑閥。詞條圖冊更多圖冊 2.3.3大車行走機構 大車行走機構承擔著操作機自身的全部重量和操作機所夾持的鋼錠、坯料或鍛件的重量而在軌道上運行，完成鍛造時需要坯料進退的動作。 大車行走機構由車體和行走機構兩部分組成。 (1)車體 車體承擔著操作機自身的重量和被夾持件的重量，它的結構如圖13所示。車體的底座1支承在四個車輪9的鈾承上。八個定位塊l o用以保證車輪與車體的相關位置。托扳13焊接在底座尾部，托看行走機構的

19、電動機3、減速器40在底座上固定著兩根前立柱7和兩根后立柱6，四根立柱又都與車頂11固定在一起。在兩根前立柱間有前導板8，為活動架的前部升降導向部位。雨棍后立柱間則裝有后導板12，后提升機構的升降滑塊就在其問上、下滑動。車頂是裝置液壓系統的油箱、電動機、油泵、蓄能器、各種閥類等部件的地方，同時又支承著升降機構的油缸。圖2-5 大車行走機構 第三章 旋轉機構設計3.1 旋轉機構的組成 please contact Q 3053703061 give you more perfect drawings附錄II 外文文獻原文A formal theory for estimating defeatu

20、ring -induced engineering analysis errorsSankara Hari Gopalakrishnan, Krishnan SureshDepartment of Mechanical Engineering, University of Wisconsin, Madison, WI 53706, United StatesReceived 13 January 2006; accepted 30 September 2006AbstractDefeaturing is a popular CAD/CAE simplification technique th

21、at 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 rigorou

22、s 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 ar

23、rive at strict bounds on defeaturing induced analysis errors. The theory is illustrated through numerical examples.Keywords: Defeaturing; Engineering analysis; Error estimation; CAD/CAE1. IntroductionMechanical artifacts typically contain numerous geometric features. However, not all features are cr

24、itical 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 a

25、re 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 c

26、omputational 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 co

27、ndition 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 var

28、ious 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

29、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

30、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 estimati

31、ng 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 featur

32、es 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 o

33、rder 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 fro

34、m 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 workThe defeaturing process can be categorized into three phases:Identification: what f

35、eatures 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 fea

36、ture 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

37、 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

38、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 thro

39、ugh 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 ShermanMorrison and

40、 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 1517. Such reanalysis techniques are particularly effective when the objective is to analyze two designs that share s

41、imilar 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 localglobal analysis 13, where t

42、he objective is to solve the local field around the defeatured region after the global defeatured problem has been solved. An implicit assumption in localglobal analysis is that the feature being suppressed is self-equilibrating.3. Proposed methodology3.1. Problem statementWe restrict our attention

43、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 ther

44、mal problem over the 2-D heat-block assembly illustrated in Fig. 2.The assembly receives heat Q from a coil placed beneath the region identified as coil. A semiconductor device is seated at device. The two regions belong to and have the same material properties as the rest of . In the ensuing discus

45、sion, a quantity of particular interest will be the weighted temperature Tdevice within device (see Eq. (2) below). A slot, identified as slot in Fig. 2, will be suppressed, and its effect on Tdevice will be studied. The boundary of the slot will be denoted by slot while the rest of the boundary wil

46、l be denoted by . The boundary temperature on is assumed to be zero. Two possible boundary conditions on slot 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 estimatio

47、n.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 conside

48、r 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

49、condition for t (x, y) has disappeared since the slot does not exist any morea 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

50、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 re

51、strictions 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 methodsThe first concept that we would need is that of adjoint for

52、mulation. 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 t

53、his 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 th

54、e 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

55、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.,

56、(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 slot; this is encouraging perhaps, errors can be computed by processing information just over the feature being

57、 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

58、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, o