【正文】 e 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 above exterior Neumann problem is putationally 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 puted field e1(x, y) and the unknown field e(x, y) through the following . The two fields e1(x, y) and e(x, y) satisfy the following monotonicity relationship:Proof. Proof exploits triangle inequality. Piecing it all together, we have the following conclusive lemma.Lemma . The unknown device temperature Tdevice, when the slot has Neumann boundary conditions prescribed, is bounded by the following limits whose putation 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: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 slotLet us now consider the case when the slot is maintained at a fixed temperature Tslot. Consider any domain ? that is contained by the domain that contains the slot. Define a field e?(x, y) in ? that satisfies:We now establish a result relating e?(x, y) and e(x, y).Lemma .Note that the problem stated in Eq. (7) is putationally less intensive to solve. This leads us to the final result.Lemma . The unknown device temperature Tdevice, when the slot has Dirichlet boundary conditions prescribed, is bounded by the following limits whose putation 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:Proof. Follows from the above lemmas.Observe again that the two bounds are independent of the unknown field T (x, y).4. Numerical examplesWe illustrate the theory developed in the previous section through numerical examples.Let k = 5W/m?C, Q = 10 W/m3 and H = .Table 1 shows the numerical results for different slot boundary conditions. The first device temperature column is the mon 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 and . The last column is the actual device temperature obtained from the fullfeatured model (prior to defeaturing),and is shown here for parison al