2020美賽建模B題思路和理解

思路和理解

• 問題中心：沙堡生存期仿真建模

題目要求：

1. 基礎沙堡模型。
   第一問不同三維形狀的模型需要滿足相同的基本條件①距離水的距離②沙子類型③水沙比例。
   問題細節：需要定義水域與沙堡的距離（下稱水堡距）、沙堡毀滅的標準、沙堡的形狀類型、水浪衝刷的量化影響（同時考慮每次沖刷對上次沖刷的相互影響）、不同水沙組合可以獲得的最大應力(需要有土建專業基礎)等，構建沙堡狀態與上述變量的非線性模型，每個變量是關於時間的函數。

2. 優化模型
   第二問相對較簡單，簡單問題要做的漂亮。控制水堡距、日照、風蝕等相關因素，分別對每種形狀的沙堡尋優，求取最佳水沙比例的具體範圍，注意上述變量有可能是水沙比例的函數。

3. 考慮雨水影響的沙堡模型。構建雨水衝擊力的影響模型，同樣建立爲時間的函數，加入到問題一的模型中即可，評價不同三維形狀沙堡在考慮該因素前後的生命長度。（圓錐形或許在該情況下是最佳結構，而在問題一種並非最佳）

4. 那就再考慮下其他想法呀！不同形狀組合、給沙堡加個圍牆、排水溝…建議沒有沙灘經歷的同學先開頭腦風暴聊一聊。具體模型改進無非完善模型，使得沙堡生命週期更持久。

5. 寫一份1-2頁的文章，深入淺出通俗易懂的向雜誌讀者表達你們的結論，什麼樣的沙堡給孩子建立更有成就感，不受到人爲干預可以存在多少多少年…注意嚴謹又不失內涵的表達方式。

• 該問題的亮點在於模型的可視表達，如每個三維沙堡不同生命週期的演化，單獨某因素（降雨）的影響演化。

原題目翻譯

原文

2020 MCM Weekend 2
Problem B: The Longest Lasting Sandcastle(s)
Wherever there are recreational sandy ocean beaches in the world, there seem to be children (and
adults) creating sandcastles on the seashore. Using tools, toys, and imagination, beach goers
create sandcastles that range from simple mounds of sand to complicated replicas of actual
castles with walls, towers, moats, and other features that mimic real castles. In all these, one
typically forms an initial foundation consisting of a single, nondescript mound of wetted sand,
and then proceeds to cut and shape this base into a recognizable 3-dimensional geometric shape
upon which to build the more castle-defining features.
Inevitably, the inflow of ocean waves coupled with rising tides erodes sandcastles. It appears,
however, that not all sandcastles react the same way to waves and tides, even if built roughly the
same size and at roughly the same distance from the water on the same beach. Consequently, one
wonders if there exists a best 3-dimensional geometric shape to use for a sandcastle foundation.
Requirements

1. Construct a mathematical model to identify the best 3-dimensional geometric shape to use as a
   sandcastle foundation that will last the longest period of time on a seashore that experiences
   waves and tides under the following conditions:
    built at roughly the same distance from the water on the same beach, and
    built using the same type of sand, roughly the same amount of sand, and the same waterto-sand proportion.
2. Using your model, determine an optimal sand-to-water mixture proportion for the castle
   foundation, assuming you use no other additives or materials (e.g. plastic or wooden supports,
   stones, etc.).
3. Adjust your model as needed to determine how the best 3-dimensional sandcastle foundation
   you identified in requirement 1 is affected by rain, and whether it remains the best 3-dimensional
   geometric shape to be used as a castle foundation when it is raining.
4. What other strategies, if any, might you use to make your sandcastle last longer?
5. Finally, write an informative, one- to two-page article describing your model and its results for
   publication in the vacation magazine: Fun in the Sun, whose readers are mainly non-technical.
   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 limit and should appear
   after your completed solution. You should not make use of unauthorized images and materials
   whose use is restricted by copyright laws. Ensure you cite the sources for your ideas and the
   materials used in your report.