2013美国大学生数学建模竞赛论文

Our solution paper mainly deals with the following problems:

·How to measure the distribution of heat across the outer edge of pans in different

shapes and maximize even distribution of heat for the pan

·How to design the shape of pans in order to make the best of space in an oven

·How to optimize a combination of the former two conditions.

When building the mathematic models, we make some assumptions to get them

to be more reasonable. One of the major assumptions is that heat is evenly distributed

within the oven. We also introduce some new variables to help describe the problem.

To solve all of the problems, we design three models. Based on the equation of

heat conduction, we simulate the distribution of heat across the outer edge with the

help of some mathematical softwares. In addition, taking the same area of all the pans

into consideration, we analyze the rate of space utilization ratio instead of thinking

about maximal number of pans contained in the oven. What’s more, we optimize a

combination of conditions (1) and (2) to find out the best shape and build a function to

show the relation between the weightiness of both conditions and the width to length

ratio, and to illustrate how the results vary with different values of W/L and p.

To test our models, we compare the results obtained by stimulation and our models, to

find that our models fit the truth well. Yet, there are still small errors. For instance, in

Model One, the error is within 1.2% .

In our models, we introduce the rate of satisfaction to show how even the

distribution of heat across the outer edge of a pan is clearly. And with the help of

mathematical softwares such as Matlab, we add many pictures into our models,

making them more intuitively clear. But our models are not perfect and there are some

shortcomings such as lacking specific analysis of the distribution of heat across the

outer edge of a pan of irregular shapes. In spite of these, our models can mainly

predict the actual conditions, within reasonable range of error.

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