【導讀】在CAD模型中的程序，如有限元分析。然而，幾何分析不可避免地會產(chǎn)生分析錯。誤，在目前的理論框架實在不容易量化。本文中，我們對快速計算處理這些幾何分析錯誤提供了嚴謹?shù)睦碚?。我們集中力量解決地方的特點，被簡化的任意形狀和大小的區(qū)域。用伴隨矩陣制定邊值問題抵達嚴格界限幾何分析性分析錯誤。該理論通過數(shù)值例。不過，在工程分析中并不是所有的特。征都是至關重要的。以前的分析中無關特征往往被忽略，從而提高自動化及運算。轉子包含50多個不同的特征，有限元分析的全功能的模型如圖1，需要超過150,000度的。內(nèi)存要求也跟著降低，而且條件數(shù)離散系統(tǒng)將得以改善;展示這些物理特征被稱為自我平衡。可以在系統(tǒng)任意位置被施用,但是會在系統(tǒng)分析上構成重大的挑戰(zhàn)。目前，尚無任何系統(tǒng)性的程序去估算幾何分析對上述兩個案例的潛在影響。伴隨矩陣和單調(diào)分。縮進如圖2，會受到抑制，其對Tdevice將予以研究。邊界的時段稱為Γslot. 是所謂的自身伴隨矩陣。

　　

【正文】 ngle 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 slot Let 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 examples We 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 all the cases, we can see that the last column lies between the 2nd and 3rd column, Tdevice T Observe that the range predicted for the zero Dirichlet condition is much wider than that for the insulatedslot scenario. The difference lies in the fact that in the first example,a zero Neumann boundary condition on the slot resulted in a selfequilibrating feature, and hence its effect on the device was minimal. On the other hand, a Dirichlet boundary condition on the slot results in a nonselfequilibrating 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 pute the upper and lower bounds different Dirichlet conditions for the slot. Fig. 4 illustrates the variation of the actual device temperature and the puted 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 slottemperature 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 “whatif” design scenarios. Consider the design shown in Fig. 5 that now consists of two devices with a single heat 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 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.