【正文】 is: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):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 nontrivial 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 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 methodsThe 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 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.)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 plex as the primal problem。 the governing equations are identical。 such problems are called selfadjoint. Most engineering problems of practical interest are selfadjoint, making it easy to pute primal and adjoint fields without doubling the putational effort.For the defeatured problem on hand, the adjoint field plays a critical role as the following lemma summarizes:Lemma . The difference between the unknown and known device temperature, ., (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 Гslot。 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 Neumann boundary conditions [?kT ]. ?n are prescribed over the 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,., 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 . The difference (Tdevice ? tdevice) satisfies the inequalitiesUnfortunately, 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.. Monotonicity analysisMonotonicity 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 displacement 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 ‘putational era’, the same theorems were used by engineers to get quick upper or lower bounds to challenging problems by reducing a plex problem to simpler ones [25]. Of course, on the advent of the puter, such methods and theorems took a backseat since a direct numerical solution of fairly plex problems became , 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, ., 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:Solving the above problem is putationally equivalent to solving the fullfeatured problem of Eq. (1). But, if we could pute 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 slotFirst, 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:The a