机械毕业设计英文外文翻译177估计导致工程几何分析错误的一个正式理论

附录I 外文文献翻译

估计导致工程几何分析错误的一个正式理论

SankaraHariGopalakrishnan，KrishnanSuresh

机械工程系，威斯康辛大学，麦迪逊分校，2006年9月30日

摘要：几何分析是著名的计算机辅助设计/计算机辅助工艺简化“小或无关特征”在CAD 模型中的程序，如有限元分析。然而，几何分析不可避免地会产生分析错误，在目前的理论框架实在不容易量化。

本文中，我们对快速计算处理这些几何分析错误提供了严谨的理论。尤其，我们集中力量解决地方的特点，被简化的任意形状和大小的区域。提出的理论采用伴随矩阵制定边值问题抵达严格界限几何分析性分析错误。该理论通过数值例子说明。

关键词:几何分析;工程分析;误差估计;计算机辅助设计/计算机辅助教学

1.介绍

机械零件通常包含了许多几何特征。不过，在工程分析中并不是所有的特征都是至关重要的。以前的分析中无关特征往往被忽略，从而提高自动化及运算速度。

举例来说，考虑一个刹车转子，如图1(a)。转子包含50多个不同的特征，但所有这些特征并不是都是相关的。就拿一个几何化的刹车转子的热量分析来说，如图1(b)。有限元分析的全功能的模型如图1(a)，需要超过150,000度的自由度，几何模型图1(b)项要求小于25，000个自由度，从而导致非常缓慢的运算速度。

图1(a)刹车转子图1(b)其几何分析版本除了提高速度，通常还能增加自动化水平，这比较容易实现自动化的有限元网格几何分析组成。内存要求也跟着降低，而且条件数离散系统将得以改善;后者起着重要作用迭代线性系统。

但是，几何分析还不是很普及。不稳定性到底是“小而局部化”还是“大而扩展化”，这取决于各种因素。例如，对于一个热问题，想删除其中的一个特征，不稳定性是一个局部问题:(1)净热通量边界的特点是零。(2)特征简化时没有新的热源产生; [4]对上述规则则例外。展示这些物理特征被称为自我平衡。结果，同样存在结构上的问题。

从几何分析角度看，如果特征远离该区域，则这种自我平衡的特征可以忽略。但是，如果功能接近该区域我们必须谨慎，。

从另一个角度看，非自我平衡的特征应值得重视。这些特征的简化理论上可以在系统

任意位置被施用,但是会在系统分析上构成重大的挑战。

目前，尚无任何系统性的程序去估算几何分析对上述两个案例的潜在影响。这就必须依靠工程判断和经验。

在这篇文章中，我们制定了理论估计几何分析影响工程分析自动化的方式。任意形状和大小的形体如何被简化是本文重点要解决的地方。伴随矩阵和单调分析这两个数学概念被合并成一个统一的理论来解决双方的自我平衡和非自我平衡的特点。数值例子涉及二阶scalar偏微分方程，以证实他的理论。

本文还包含以下内容。第二节中，我们就几何分析总结以往的工作。在第三节中，我们解决几何分析引起的错误分析，并讨论了拟议的方法。第四部分从数值试验提供结果。第五部分讨论如何加快设计开发进度。

2.前期工作

几何分析过程可分为三个阶段:

识别:哪些特征应该被简化；

简化：如何在一个自动化和几何一致的方式中简化特征；

分析:简化的结果。

第一个阶段的相关文献已经很多。例如，企业的规模和相对位置这个特点，经常被用来作为度量鉴定。此外，也有人提议以有意义的力学判据确定这种特征。

自动化几何分析过程，事实上，已成熟到一个商业化几何分析的地步。但我们注意到，这些商业软件包仅提供一个纯粹的几何解决。因为没有保证随后进行的分析错误，所以必须十分小心使用。另外，固有的几何问题依然存在，并且还在研究当中。

本文的重点是放在第三阶段，即快速几何分析。建立一个有系统的方法，通过几何分析引起的误差是可以计算出来的。再分析的目的是迅速估计改良系统的反应。其中最著名的再分析理论是著名的谢尔曼-Morrison和woodbury公式。对于两种有着相似的网状结构和刚度矩阵设计，再分析这种技术特别有效。然而，过程几何分析在网状结构的刚度矩阵会导致一个戏剧性的变化，这与再分析技术不太相关。

3.拟议的方法

3.1问题阐述

我们把注意力放在这个文件中的工程问题，标量二阶偏微分方程式(pde): +

?

-

?

c=

.

)

.(f

au

u

许多工程技术问题，如热，流体静磁等问题，可能简化为上述公式。

作为一个说明性例子，考虑散热问题的二维模块Ω如图2所示。

图2二维热座装配

热量q从一个线圈置于下方位置列为Ωcoil。半导体装置位于Ωdevice。这两个地方都属于Ω，有相同的材料属性，其余Ω将在后面讨论。特别令人感兴趣的是数量，加权温度Tdevice内Ωdevice(见图2)。一个时段，认定为Ωslot缩进如图2，会受到抑制，

其对Tdevice 将予以研究。边界的时段称为Γslot 其余的界线将称为Γ。边界温度Γ假定为零。两种可能的边界条件Γslot 被认为是:(a)固定热源，即(-k ?t)?n=q，(b)有一定温度，即T=Tslot 。两种情况会导致两种不同几何分析引起的误差的结果。

设T(x ，y)是未知的温度场和K 导热。然后，散热问题可以通过泊松方程式表示:

)

1()().)((00).(?????????Γ=Γ=?-Γ=???Ω-ΩΩ=?-?+slct slct slct

coil coil T T b or on q h k a on T in in in Q T k BC PDE )2(),(),(?????Ω=??Ωdevice d y c T y x H T Compute device

其中H(x ，y)是一些加权内核。现在考虑的问题是几何分析简化的插槽是简化之前分析，如图3所示。

图3defeatured 二维热传导装配模块

现在有一个不同的边值问题，不同领域t(x ，y):

)3(ΩΓon 0t ΩΩ0in ΩQ ).(-k BC PDE coil slot coil ?????=???-+=??+in t

)4(),(),(???Ω

=??Ωdevide device d y x t y x H t Compute

观察到的插槽的边界条件为t(x ，y)已经消失了，因为槽已经不存在了（关键性变化）! 解决的问题是:

设定tdevice 和t(x ，y)的值，估计Tdevice 。

这是一个较难的问题，是我们尚未解决的。在这篇文章中，我们将从上限和下限分析Tdevice 。这些方向是明确被俘引理3、4和3、6。至于其余的这一节，我们将发展基本概念和理论，建立这两个引理。值得注意的是，只要它不重叠，定位槽与相关的装置或热源没有任何限制。上下界的Tdevice 将取决于它们的相对位置。

3.2伴随矩阵方法

我们需要的第一个概念是，伴随矩阵公式表达法。应用伴随矩阵论点的微分积分方程，包括其应用的控制理论，形状优化，拓扑优化等。我们对这一概念归纳如下。

相关的问题都可以定义为一个伴随矩阵的问题，控制伴随矩阵t_(x ，y)，

必须符合下

列公式计算〔23〕:

Γ

=???Ω-Ω+ΩΩ=?-?on t in in H t k device slot device

0)5(0).(**

伴随场t_(x ，y)基本上是一个预定量，即加权装置温度控制的应用热源。可以观察到，伴随问题的解决是复杂的原始问题;控制方程是相同的;这些问题就是所谓的自身伴随矩阵。大部分工程技术问题的实际利益，是自身伴随矩阵，就很容易计算伴随矩阵。

另一方面，在几何分析问题中，伴随矩阵发挥着关键作用。表现为以下引理综述: 引理3.1已知和未知装置温度的区别，即(Tdevice-tdevice)可以归纳为以下的边界积分比几何分析插槽:

?????Γ?--+Γ-?--=-??ΓΓslot

slot d n t k t T d n t T k t t T device device ).)((]).([^*^* 在上述引理中有两点值得注意:

1、积分只牵涉到边界гslot;这是令人鼓舞的。或许，处理刚刚过去的被简化信息特点可以计算误差。

2、右侧牵涉到的未知区域T(x ，y)的全功能的问题。特别是第一周期涉及的差异，在正常的梯度，即涉及[-k(T-t)] ?n;这是一个已知数量边界条件[-k ?t]?n 所指定的时段，未知狄里克莱条件作出规定[-k ?t]?n 可以评估。在另一方面，在第二个周期内涉及的差异，在这两个领域，即T 管; 因为t 可以评价，这是一个已知数量边界条件T 指定的时段。因此。

引理3.2、差额(tdevice-tdevice)不等式

Γ-?-Γ≤Γ-?---Γ-Γ?-≤Γ-?-+-???

???ΓΓΓΓΓΓd n t T k d t d t T n t k t T and d t T d n t k d n t T k t t T slot

slot slot device device slot slot slot device device 2^2*^*22^*^

*]).([)()().()()().()).(()(

然而，伴随矩阵技术不能完全消除未知区域T(x ，y)。为了消除T(x ，y)我们把重点转向单调分析。

3.3单调性分析

单调性分析是由数学家在19世纪和20世纪前建立的各种边值问题。例如，一个单调定理:

"添加几何约束到一个结构性问题，是指在位移(某些)边界不减少"。

观察发现，上述理论提供了一个定性的措施以解决边值问题。

后来，工程师利用之前的“计算机时代”上限或下限同样的定理，解决了具有挑战性的问题。当然，随着计算机时代的到来，这些相当复杂的直接求解方法已经不为人所用。但

是，在当前的几何分析，我们证明这些定理采取更为有力的作用，尤其应当配合使用伴随理论。

我们现在利用一些单调定理，以消除上述引理T(x ，y)。遵守先前规定，右边是区别已知和未知的领域，即T(x ，y)-t(x ，y)。因此，让我们在界定一个领域E(x ，y)在区域为:

e(x ，y)=t(x ，y)-t(x ，y)。

据悉，T(x ，y)和T(x ，y)都是明确的界定，所以是e(x ，y)。事实上，从公式(1)和

(3)，我们可以推断，e(x ，y)的正式满足边值问题:

????

?????Γ-=Γ+?-Γ=Ω=?-?slot slot on t T e b or on q n e k a on e in e k Solve )().)((00).(^ 解决上述问题就能解决所有问题。但是，如果我们能计算区域e(x ，y)与正常的坡度超过插槽，以有效的方式，然后(Tdevice-tdevice)，就评价表示e(X ，Y)的效率，我们现在考虑在上述方程两种可能的情况如(a)及(b)。

例(a)边界条件较第一插槽，审议本案时槽原本指定一个边界条件。为了估算e(x ，y)，考虑以下问题:

)6(,0),(....0).(22^^???????∞→+→Γ+?=?-Ω=?-?∞∞∞

∞y x as y x e on q n t k n e k in e k Solve slot slot

因为只取决于缝隙，不讨论域，以上问题计算较简单。经典边界积分/边界元方法可以引用。关键是计算机领域e1(x ，y)和未知领域的e(x ，y)透过引理3.3。这两个领域e1(x ，y)和e(x ，y)满足以下单调关系:

2

22)(max )()??? ??ΓΓ+Γ≤Γ∞Γ∞Γ??slot slot measure e d e d e slot slot 把它们综合在一起，我们有以下结论引理。

引理3.4未知的装置温度Tdevice ，当插槽具有边界条件，东至以下限额的计算，只要求:(1)原始及伴随场T 和隔热与几何分析域(2)解决e1的一项问题涉及插槽:

Γ?--Γ????--+=≥ΓΓ?d n t k g d n t k q t T T slot slot device lower

device device 2^*^).().(

??

? ??ΓΓ+Γ=Γ?-+Γ?--+=≤∞Γ∞ΓΓ???)(max )(,

).().(22^*^*slot slot slot slot device upper device device m easure e d e g where d n t k g d n t k q t t T

T slot 观察到两个方向的右侧，双方都是独立的未知区域T(x ，y)。

例(b) 插槽Dirichlet 边界条件

我们假定插槽都维持在定温Tslot 。考虑任何领域，即包含域和插槽。界定一个区域e(x ，y)在满足:

)7(00).(??

???Γ-=Γ=Ω=?-?--

--

-slot slot on t T e on e in e k Slove

现在建立一个结果与e-(x ，y)及e(x ，y)。

引理3.5 ??Γ-ΓΓ?≤Γ?slot slot d n e k d n e k ^2

^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，:

???????Γ?-Γ

+

Γ-?-+=≤???????Γ?-Γ

-

Γ-?-+=≥??????Γ-ΓΓΓ-ΓΓslot slot slot slot device upper device

device slot slot slot slot device lower device device d n e k d t d t T n t k t T T d n e k d t d t T n t k t T T ^22*^*^22*^*].[)()().(].[)()(.(

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

k = 5W/m ?C, Q = 105 W/m3 and H = device)Area(1

Ω。

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

子中，我们可以看到最后一栏则是介于第二和第三列。T lower

devuce≤ Tdevice≤ T

upper

devuce

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 durin g, 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 几何分析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):

+

?

?

c=

-

)

.

.(f

au

u

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 a s 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., (-k ?rT).?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]:

)

1()().)((00).(?????????Γ=Γ=?-Γ=???Ω-ΩΩ=?-?+slct slct slct

coil coil T T b or on q h k a on T in in in Q T k BC PDE Given the field T (x, y), the quantity of interest is:

)2(),(),(?????Ω=??Ωdevice d y c T y x H T Compute device

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):

)3(ΩΓon 0t ΩΩ0in ΩQ ).(-k BC PDE coil slot coil ?????=???-+=??+in t

)4(),(),(???Ω

=??Ωdevide device d y x t y x H t Compute

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.)

Γ

=???Ω-Ω+ΩΩ=?-?on t in in H t k device slot device

0)5(0).(**

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:

?????Γ?--+Γ-?--=-??ΓΓslot

slot d n t k t T d n t T k t t T device device ).)((]).([^*^*

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 *?k ?T +. ?n are prescribed over the slot since *?k ?t+. ?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

Γ-?-Γ≤Γ-?---Γ-Γ?-≤Γ-?-+-???

???ΓΓΓΓΓΓd n t T k d t d t T n t k t T and d t T d n t k d n t T k t t T slot

slot slot device device slot slot slot

device device 2^2*^*22^*^*]).([)()().()()().()).(()(

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 displa cement 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:

????

?????Γ-=Γ+?-Γ=Ω=?-?slot slot on t T e b or on q n e k a on e in e k Solve )().)((00).(^ 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:

)6(,0),(....0).(22^^???????∞→+→Γ+?=?-Ω=?-?∞∞∞

∞y x as y x e on q n t k n e k in e k Solve slot slot

The above exterior Neumann 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 computed 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:

222)(max )()??

? ??ΓΓ+Γ≤Γ∞Γ∞Γ??slot slot measure e d e d e slot slot Proof. Proof exploits triangle inequality.

Piecing it all together, we have the following conclusive lemma.

Lemma 3.4. The unknown device temperature Tdevice, when the slot has Neumann boundary conditions 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 involving the slot:

Γ

?--Γ????--+=≥ΓΓ?d n t k g d n t k q t T T slot slot device lower

device device 2^*^).().(??

? ??ΓΓ+Γ=Γ?-+Γ?--+=≤∞Γ∞ΓΓ???)(max )(,

).().(22^*^*slot slot slot slot device upper device device m easure e d e g where d n t k g d n t k q t t T

T slot Proof. Follows from the above lemmas. _

Observe that the two bounds on the right hand sides are independent of the unknown field T (x, y).

Case (b) Dirichlet boundary condition over slot

Let us now consider the case when the slot is maintained at a fixed temperature Tslot. Consi der any domain ? that is contained by the domain that contains the slot. Define a field e?(x, y) in ? that satisfies:

)7(00).(?????Γ-=Γ=Ω=?-?--

--

-slot slot on t T e on e in e k Slove

We now establish a result relating e?(x, y) and e(x, y). Lemma 3.5.

??Γ-ΓΓ?≤Γ?slot slot d n e k d n e k ^2

^2).().(

Note that the problem stated in Eq. (7) is computationally less intensive to solve. This leads us to the final result.

Lemma 3.6. The unknown device temperature Tdevice, when the slot has Dirichlet boundary conditions 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 e? to a collapsed boundary problem surrounding the slot:

???????Γ?-Γ

+Γ-?-+=≤???????Γ?-Γ

-

Γ-?-+=≥??????Γ-ΓΓΓ-ΓΓslot slot slot slot device upper device

device slot slot slot slot device lower device device d n e k d t d t T n t k t T T d n e k d t d t T n t k t T T ^22*^*^22*^*].[)()().(].[)()(.(

Proof. Follows from the above lemmas.

Observe again that the two bounds are independent of the unknown field T (x, y).

4. Numerical examples We illustrate the theory developed in the previous section through numerical examples.

Let k = 5W/m?C, Q = 105 W/m3 and H = device)Area(1

Ω.

Table 1 shows the numerical results for different slot boundary conditions. The first device temperature column is the common temperature for all defeatured models (it does not depend on the slot boundary conditions since the slot was defeatured).The next two columns are the upper and lower bounds predicted by Lemmas 3.4 and 3.6. The last column is the actual device temperature obtained from the full-featured model (prior to defeaturing),and is shown here for comparison purposes.In all the cases, we can see that the last column lies between the 2nd

and 3rd column, i.e.T lower

devuce ≤ Tdevice ≤ T upper

devuce

Observe that the range predicted for the zero Dirichlet condition is much wider than that for the insulated-slot scenario. The difference lies in the fact that in the first example,a zero Neumann boundary condition on the slot resulted in a self-equilibrating feature, and hence its effect on the device was minimal. On the other hand, a Dirichlet boundary condition on the slot results in a non-self-equilibrating feature whose deletion can result in a large change in the device temperature.

Observe however that the predicted range for a fixed nonzero slot temperature of 20 _C is narrower than that for the zerotemperature scenario. This can be attributed to the fact the slot temperature is closer to the device temperature and therefore its deletion has less of an impact.

Indeed, one can easily compute the upper and lower bounds different Dirichlet conditions for the slot. Fig. 4 illustrates the variation of the actual device temperature and the computed bounds as a function of the slot temperature.

Observe that the theory correctly predicts the upper and lower limits of the actual device temperature. Further, the limits are tightest when the slot-temperature is approximately equal to the device temperature, as expected.

5. Rapid analysis of design scenarios

We consider now a broader impact of the proposed theory in analyzing “what -if” design scenarios. Consider the design shown in Fig. 5 that now consists of two devices with a single

heat source.As expected, the two devices will not be at the same average temperature. The device on the left will be at a higher temperature due to its relative proximity to the heat source.

Fig. 4. Estimated bounds versus slot temperature

Fig. 5. Dual device heat block.

Fig. 6. Possible locations for a correcting feature.

Consider the scenario where one wishes to correct the imbalance by adding a small hole of fixed diameter; five possible locations are illustrated in Fig. 6. An optimal position has to be chosen such that the difference in the average temperatures of the two regions is minimized.

A brute force strategy would be to carry out the finite element analysis for each configuration . . . a time consuming process. An alternate strategy is to treat the hole as a ‘feature’ and study its impact as a post-processing step. In other words,this is a special case of ‘defeaturing’, and the proposed methodology applies equally to the current scenario.

We can solve the primal and adjoint problems for the original configuration (without the hole) and use the theory developed in the previous sections to study the effect of adding the hole at each position on our objective. The objective is to minimize the difference in the average temperature of the two devices.

Table 2 summarizes the bounds predicted using this theory, and the actual values.

From the table, it can be seen that the location W is the optimal location since it gives the lowest mean value for the desired objective function, as expected.