Applications of Monge-Kantorovich Theory and Michell Trusses

提案：DMS-9970520首席研究员：Wilfrid Gangbo摘要：1904年，Michell在他关于框架结构经济极限的出色论文中，对以下难题进行了研究：给定一个力的平衡系统（载荷和反力），并给定弹性材料的单轴应力的允许范围，在平面内用给定材料的杆件设计桁架，使给定力产生的轴向应力在允许范围内，而杆件所用材料的总体积尽可能小。米歇尔问题及其许多变体和扩展在过去几年中引起了工程师和数学家的广泛关注。在共面的情况下，米歇尔正式证明，在最佳布局，酒吧安排沿着线的主应变的位移，出现了拉格朗日乘子。这些直线在平面上形成一个正交坐标系。满足这些条件的任何最优设计称为米歇尔桁架。钢波计划研究最优应力支撑的几何和拓扑性质。研究这个问题的技术与应用于Monge-Kantorovich问题的技术相似。Gangbo还计划利用Monge-Kantorovich理论，该理论最近被认为是建设性研究半地转浅水问题的有力工具，这是一个三维自由边界问题。米歇尔桁架问题最初的动机是希望设计一种由能够支撑指定力阵列的梁组成的结构，并通过使用最少数量的给定弹性材料以最大的经济性来实现这一目标。当然，在尽可能提高效率的同时，我们还必须设计出一种确保结构安全的设计。Gangbo的兴趣是确定设计这种最小体积框架的好模型。确定框架的杆的端点是这个问题中最困难的部分。该提案的第二部分包括研究浅水流动，这意味着与水平长度尺度相比，垂直长度尺度较小的流动（例如，洋流的流动）。如果一个人在给定的时间收集数据，他希望预测数据将如何随着时间的推移而演变。例如，如果测量今天的水的压力，那么人们希望预测明天的压力。在地转模式中，气压是一个需要知道的重要量。例如，人们可以从水的压力推断出水的速度。Gangbo将使用所谓的Monge-Kantorovich理论，该理论最近在数学的各个领域中成为一个强大的工具。

Proposal: DMS-9970520Principal Investigator: Wilfrid GangboAbstract: In 1904, in his remarkable paper on the limits of economy in framed structures, Michell attacked the following difficult problem: given an equilibrium system of forces (loads and reactions) with specified points of application, and given an elastic material with allowable range for its uniaxial stress, to design a truss in the plane with bars of the given material such that the given forces produce axial stresses within the allowable limits, while the total volume of material used for the bars is as small as possible. Michell's problem and its many variants and extensions have attracted a lot of attention in the last few years among engineers as well as mathematicians. In the coplanar case, Michell formally proved that, in the optimal layout, bars are arranged along the lines of principal strains of a displacement that arises as a Lagrange multiplier. These lines form an orthogonal system of coordinates in the plane. Any optimal design satisfying these conditions is called a Michell truss. Gangbo plans to study geometrical and topological properties of supports of optimal stresses. Techniques for studying this problem are similar to those that apply to the Monge-Kantorovich problem. Gangbo also plans to make use of the Monge-Kantorovich theory, which has recently been recognized as a powerful tool for constructively studying the semigeostrophic shallow water problem, which is a three-dimensional free boundary problem. The Michell truss problem was originally motivated by the desire to engineer a structure built of beams capable of supporting a specified array of forces, and to do this with maximum economy by using the least quantity of a given elastic material. Of course, while trying to be as efficient as possible, one must still produce a design that ensures a safe structure. Gangbo's interest is to determine good models for the design of such least-volume frames. The determination of the endpoints of the bars of the frame is the most difficult part of this problem. The second part of the proposal consists in studying shallow-water flow, meaning flow in which the vertical length scale is small in comparison to the horizontal length scale (e.g., the flow of ocean currents). If one collects data at a given time, one hopes to predict how that data will evolve over time. For instance, if one measures the pressure of the water today, one would like to forecast its pressure tomorrow. In the geostrophic model, pressure is an important quantity to know. For instance, one can infer the speed of the water from its pressure. Gangbo will use the so-called Monge-Kantorovich theory, which has recently surfaced as a powerful tool in various fields of mathematics.