毕业论文（设计）--2013年美赛a题h奖论文.docVIP

Summary We build two basic models for the two problems respectively: one is to show the distribution of heat across the outer edge of the pan for different shapes, rectangular, circular and the transition shape; another is to select the best shape for the pan under the condition of the optimization of combinations of maximal number of pans in the oven and the maximal even heat distribution of the heat for the pan. We first use finite-difference method to analyze the heat conduct and radiation problem and derive the heat distribution of the rectangular and the circular. In terms of our isothermal curve of the rectangular pan, we analyze the heat distribution of rounded rectangle thoroughly, using finite-element method. We then use nonlinear integer programming method to solve the maximal number of pans in the oven. In the even heat distribution, we define a function to show the degree of the even heat distribution. We use polynomial fitting with multiple variables to solve the objective function For the last problem, combining the results above, we analyze how results vary with the different values of width to length ratio W/L and the weight factor p. At last, we validate that our method is correct and robust by comparing and analyzing its sensitivity and strengths /weaknesses. Based on the work above, we ultimately put forward that the rounded rectangular shape is perfect considering optimal number of the pans and even heat distribution. And an advertisement is presented for the Brownie Gourmet Magazine. Contents 1 Introduction 3 1.1 Brownie pan 3 1.2 Background 3 1.3 Problem Description 3 2. Model for heat distribution 3 2.1 Problem analysis 3 2.2 Assumptions 4 2.3 Definitions 4 2.4 The model 4 3 Results of heat distribution 7 3.1 Basic results 7 3.2 Analysis 9 3.3 Analysis of the transition shape—rounded rectangular 9 4 Model to select the best shape 11 4.1 Assumptions 11 4.2 Definitions 11 4.3 The model 12 5 Comparision and Degree of fitting 19 6 Sensit