【正文】 陣 方法 我們需要的第一個概念是，伴隨 矩陣公式表達法 。 可以 觀察到，伴隨問題的 25 解決是復雜的原始問題 。 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 。 右 側(cè) 牽涉到的未知 區(qū) 域 T(x， y)的全功能的問題。 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。 單調(diào)性 分析是由數(shù)學家在 19世紀和 20 世紀前建立的各種邊值問題。當然，隨著計算機時代的到來 ， 這些 相當復雜的直接求解 方法已經(jīng)不為人所用 。因此，讓我們在界定一個領(lǐng)域 E(x， y)在區(qū)域為 : e(x， y)=t(x， y)t(x， y)。 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。這兩個領(lǐng)域 e1(x， y)和 e(x， y)滿足以下單調(diào)關(guān)系 : 222 )(m a x)() ?????? ?????? ?? ?? ?? s l o ts l o t m e a s u r eedede s l o ts l o t 把 它 們綜合 在一起，我們有以下結(jié)論引理。界定一個 區(qū)域 e(x， y)在滿足 : )7(00).(????????????????????s l o ts l o t ontTeoneinekS l o v e 現(xiàn)在建立一個結(jié)果與 e(x， y)及 e(x， y)。 (2) 圍繞插槽解決 失敗 了 的 邊界問題， : ????????????????????????????????????????????????????s l ots l ots l ots l otde v i c eup pe rde v i c ede v i c es l ots l ots l ots l otde v i c el ow e rde v i c ede v i c ednekdtdtTntktTTdnekdtdtTntktTT^22*^*^22*^*].[)()().(].[)()(.( 再次觀察這兩個方向都是獨立的未知領(lǐng)域 T(x， y)。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 )。 Tlowerdevuce ? Tdevice? Tuperdevuce 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。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。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。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。 29 強制 進行有限元分析每個配置。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)學習效果加 孔 在每個位置是我們的目標。 30 附錄 II 外文文獻原文 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 2020。 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 putational speedup. 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 fullfeatured 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 putational speedup. 31 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 ponent [1,2]. Memory requirements also decrease, while condition number of the discretized system improves。 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 selfequilibrating. 3. Proposed methodology . 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): .).( fauuc ????? A large class of engineering problems, such as thermal, fluid and magostatic problems, may be reduced to the above form. As an illustrative example, consider a thermal problem over the 2D heatblock 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, ., (k? rT).?n = q, or (b) fixed temperature, ., T = Tslot. The two cases will lead to two different results for defeaturing induced error estimation. 33 Fig. 2. A 2D 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]: )1()().)((00).(?????????????????????????????s l c ts l c ts l c tc o i lc o i lTTboronqhkaonTinininQTkBCP D E Given the field T (x, y), the quantity of interest is: )2(),(),(????? ?? ??? d e v i c edycTyxHTC om pu t e d e v i c e 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 2D heat block assembly. We now have a different boundary value problem, governing a different scalar field t (x, y): )3(ΩΓo n 0t ΩΩ0 in ΩQ). ( kBCP D E c o i ls l o tc o i l????? ? ??? ?????? int )4(),(),(??? ?? ??? d e v i d ed e v i c e dyxtyxHtC o m p u t e Observe that the slot boundary condition for t (x, y) has disappeared since the slot does not exist