唐家山堰塞湖分析数学模型.pdf

全国第五届研究生数学建模竞赛 全国第五届研究生数学建模竞赛 题 目 唐家山堰塞湖泄洪问题的建模与解决方案 摘 要： 本文通过建立蓄水模型、溃坝模型、演进模型和人员调度模型来分析和解决 唐家山堰塞湖的泄洪问题。在蓄水模型中，假定堰塞湖横截面面积与时间服从分 段线性关系，利用微积分的方法得到水位高程和蓄水量之间的函数关系表达式， 同时预测出5月25日到6月12 日的水位高程，预测值与实测值十分逼近，说明 该模型是可行的。 在溃坝模型中，利用微元法推导出逐步溃坝模式下的流量公式，并采用迭代 法求出坝址处流量过程线和坝前水位变化趋势。演进模型在溃坝模型的流量过程 线基础上，利用Newton-Raphson迭代法求解圣维南方程组,得到洪水在河道中的 演进过程，即洪峰沿河道到达下游各城镇的时间、流量和水位。根据该演进模型， 确定其可能被淹没的区域以及需要撤离的人员数量，建立人员调度的网络流模 型。人员的撤离遵循“步行为主”和“就近安置”的原则,采用动态模拟的方法， 得到人员撤退的可行调度方案。 在三分之一瞬时溃坝情况下，运用上述演进模型和人员调度模型，估计出堰 塞湖下游共有15个城镇会受到不同程度的淹没，需要撤离的人数约为30万。根 据上述调度原则，设置8个安置点，建立相应的人员调度网络，并给出相应的撤 离方案。 最后对所建立的模型进行了优劣性分析，并给出了模型的改进和扩展方法。 此外，还对地震后各种次生山地灾害的防治以及科研人员的工作提出相应的建 议。 关键词：圣维南方程组；动态模拟；网络流算法；微元法；Newton—Raphson 参赛密码 参赛队号 1069809 （由组委会填写） 目 录 目 录··········································································································2 一、 问题的提出······················································································3 1.1 背景··································································································3 1.2 问题··································································································3 二、 基本假设 ··························································································4 三、 符号说明 ··························································································4 四、 问题分析··························································································5 五、 模型的建立与求解 ··········································································5 5.1 问题一的模型建立与算法设计·······················································5 5.1.1 堰塞湖水位高程和蓄水量的关系模型····························