【正文】 adjoint formulation and monotonicity analysis, are bined into a unifying theory to address both sel fequilibrating and nonselfequilibrating features. Numerical examples involving 2nd order scalar partial differential equations are provided to substantiate the theory. The remainder of the paper is anized 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 byproduct 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 i n 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 mercial defeaturing /healing packages are now available [11,12]. But note that these mercial 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, ., to develop a systematic methodology through which defeaturing induced errors can be puted. We should mention here the related work on reanalysis. The objective of reanalysis is to swiftly pute 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 putation of the inverse of a perturbed stiffness matrix。 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 and . 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. . 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.。 the governing equations are identical。 this is encouraging … perhaps, errors can be puted by processing information just over the feature being suppressed. 2. The right hand side however involves the unknown field T (x, y) of the fullfeatured problem. In particular, the first term involves the difference in the normal gradients, .,involves [?k (T ? t)]. ?n。 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 . The difference (Tdevice ? tdevice) satisfies the inequalities Unfortunately, that is how far one can go with adjoint techniques。 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 postprocessing 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.