??.(?k?e?)?0in???Slove?e??0on???e??T?ton?slotslot?现在建立一个结果与e-(x，y)及e(x，y)。 引理3.5
(7)
?slot?(k?e.n)^2d???slot?(k?e?.n)d?
^2注意到，公式(7)的计算较为简单。这是我们最终要的结果。
引理3.6 未知的装置温度Tdevice，当插槽有Dirichlet边界条件，东至以下限额的计算，只要求:(1)原始及伴随场T和隔热与几何分析。(2) 围绕插槽解决失败了的边界问题，:
^?*??(?k?t.n(Tslot?t)d????slot?tdevice??^*2?2??(t)d??slot?[?k?e.n]d???slot?
^?*(?k?t).n(Tslot?t)d??????slot?tdevice??^*2?2??(t)d??slot?[?k?e.n]d???slot?lowerTdevice?TdeviceupperTdevice?Tdevice再次观察这两个方向都是独立的未知领域T(x，y)。
4. 数值例子说明
我们的理论发展，在上一节中，通过数值例子。设
15k = 5W/m?C, Q = 10 W/m3 and H = Area(?device)。
表1:结果表
表1给出了不同时段的边界条件。第一装置温度栏的共同温度为所有几何分析模式(这不取决于插槽边界条件及插槽几何分析)。接下来两栏的上下界说明引理3.4和3.6。最后一栏是实际的装置温度所得的全功能模式(前几何分析)，是列在这里比较前列的。在全部例
lower子中，我们可以看到最后一栏则是介于第二和第三列。Tdevuce? Tdevice? Tdevuce
upper对于绝缘插槽来说，Dirichlet边界条件指出，观察到的各种预测为零。不同之处在于这个事实:在第一个例子，一个零Neumann边界条件的时段，导致一个自我平衡的特点，因此，其对装置基本没什么影响。另一方面，有Dirichlet边界条件的插槽结果在一个非自我平衡的特点，其缺失可能导致器件温度的大变化在。
不过，固定非零槽温度预测范围为20度到0度。这可以归因于插槽温度接近于装置的温度，因此，将其删除少了影响。
的确，人们不难计算上限和下限的不同Dirichlet条件插槽。图4说明了变化的实际装置的温度和计算式。
预测的上限和下限的实际温度装置表明理论是正确的。另外，跟预期结果一样，限制槽温度大约等于装置的温度。
5. 快速分析设计的情景
我们认为对所提出的理论分析\什么-如果\的设计方案，现在有着广泛的影响。研究显示设计如图5，现在由两个具有单一热量能源的器件。如预期结果两设备将不会有相同的平均温度。由于其相对靠近热源，该装置的左边将处在一个较高的温度，。
图4估计式versus插槽温度图
图5双热器座
图6正确特征可能性位置
为了消除这种不平衡状况，加上一个小孔，固定直径;五个可能的位置见图6。两者的平均温度在这两个地区最低。
强制进行有限元分析每个配置。这是一个耗时的过程。另一种方法是把该孔作为一个特征，并研究其影响，作为后处理步骤。换言之，这是一个特殊的“几何分析”例子，而拟议的方法同样适用于这种情况。我们可以解决原始和伴随矩阵的问题，原来的配置(无孔)和使用的理论发展在前两节学习效果加孔在每个位置是我们的目标。目的是在平均温度两个装置最大限度的差异。表2概括了利用这个理论和实际的价值。
从上表可以看到，位置W是最佳地点，因为它有最低均值预期目标的功能。
附录II 外文文献原文
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.