2020 MCM Problem B The Longest Lasting Sandcastle(s) Notes

2020 MCM Problem B The Longest Lasting Sandcastle(s)(bk1)

Wherever there are recreational sandy ocean beaches in the world, thereseem to be children (and adults) creating sandcastles on the seashore.(bk1’) Using tools,toys, and imagination, beach goers create sandcastles that range from simplemounds of sand to complicated replicas of actual castles with walls, towers,moats, and other features that mimic real castles.(bk2’) In all these, onetypically forms an initial foundation consisting of a single, nondescriptmound of wetted sand, and then proceeds to cut and shape this base into arecognizable 3-dimensional geometric shape upon which to build the morecastle-defining features.(imp1)

Inevitably, the inflow of ocean waves coupled with rising tides erodessandcastles.(pr1) It appears, however, that not all sandcastles react the same way to wavesand tides, even if built roughly the same size and at roughly the same distancefrom the water on the same beach.(pr1’) Consequently, onewonders if there exists a best 3-dimensional geometric shape to use for asandcastle foundation.(imp2)

Requirements

1.      Construct amathematical model to identify the best 3-dimensional geometric shape to use asa sandcastle foundation that will last the longest period of time on a seashorethat experiences waves and tides under the following conditions:(spm1)

·        built at roughly the same distance from the water onthe same beach, and

·        built using the same type of sand, roughly the sameamount of sand, and the same water-to-sand proportion.

2.      Using your model,determine an optimal sand-to-water mixture proportion for the castlefoundation, assuming you use no other additives or materials (e.g. plastic orwooden supports, stones, etc.).(spm1)

3.      Adjust your modelas needed to determine how the best 3-dimensional sandcastle foundation youidentified in requirement 1 is affected by rain, and whether itremains the best 3-dimensional geometric shape to be used as a castlefoundation when it is raining.(spm2)

4.      What otherstrategies, if any, might you use to make your sandcastle last longer?(spm1,2)

5.      Finally, write aninformative, one- to two-page article describing your model and its results forpublication in the vacation magazine: Fun in the Sun, whose readersare mainly non-technical.(mss1)

Your submission should consist of:

·        One-page Summary Sheet

·        Table of Contents

·        One- to Two-page Article

·        Your solution of no more than 20 pages,for a maximum of 24 pages with your summary, table of contents, and article.

Note: Reference List and any appendices do not count toward the page limitand should appear after your completed solution. You should not make use ofunauthorized images and materials whose use is restricted by copyright laws.Ensure you cite the sources for your ideas and the materials used in yourreport.

 

Analysis:

没有火灾，没有疫情，来了个沙堡。

但是数学建模人看待他们的方式是一模一样的。

本题问题清晰，能够很好地体现一个队伍的真实实力。

关键在于建模清楚一个沙堡自身的稳固以及收到波浪和雨水两种冲击下的稳固程度。两种方法，简单的有离散后的元胞自动机，难一点的直接用微分方程，去寻找最优的沙堡曲线。

前面两段标准的从背景到问题的介绍，其中有两个细节要注意，暗示了沙子的基本性质，以及最终的决策对象是一个三维沙堡，来抵抗潮汐。

 

模型1：

X：沙堡的三维形状，可以是连续曲面方程，也可以是离散的点集合，取好样本空间方便优化；

Y：最长时间的抵抗潮汐侵袭，即模拟出侵袭过程以后算得的X垮掉的时间。

参数：距离d影响侵袭强度，沙子种类c，沙量q，沙水比r。

 

问题1是以X为决策变量的优化，问题2是对沙水比r的灵敏度分析。

 

模型2：

保持X和参数不变，抵抗的变成了雨水，有了新的Y时间，同理建模。

 

问题3就是此模型，问题4其实是鲁棒性分析，前面提到不许用的其他条件可以用了，其他参数也可以再做灵敏度分析，看看是否有所改进。

 

问题5常规完成任务。

 

MCM的题还是一贯的清晰明了，大家加油！

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