Dynamics and Optimization of Structures

该方案研究了最佳结构材料。这些问题需要发展出非标准的变分演算方法。第一个项目涉及具有内部不稳定性的材料或结构。在弹道冲击下，这些材料以异常高的速率耗散和辐射能量；效果可用于各种保护装置。第二个项目旨在寻找多组分复合材料的最佳结构。即使在最简单的情况下，这个问题也无法解决；最优结构与最优双材料结构明显不同。第三个项目处理“逆优化”的具体问题。它旨在了解自然结构的最佳性，如树干或人类骨骼。这里我们要从已知的结构和外部条件确定结构优化的目标函数。提案的三个章节处理理论材料科学中需要发展新的数学方法的新问题。一切问题都源于科学和工程。第一个项目涉及具有非常高的抗冲击能力的新结构。这种材料可用于各种保护装置或容器。第二个项目涉及由三个或更多组件组成的复合材料的最佳结构。由于大多数天然或人工复合材料都是由两种以上的材料制成的，因此许多工程应用都需要对这种结构进行研究。最后一个项目旨在开发评估生物结构最优性的数学方法。这里的目标是了解生物学在什么意义上优化其结构，并在工程应用中使用生物学“解决方案”来应对进化挑战。

The proposal studies optimal structured materials. These problems require non-standard methods of calculus of variations that are developed. The first project deals with materials or structures with inner instabilities. Under ballistic impact, these materials dissipate and radiate the energy in an abnormal high rate; the effect can be used in various protective devices.The second project aims to find the optimal structures for multi-component composites. This problem is not solved even for the simplest cases; the optimal structures are significantly different from optimal two-material structures.The third project deals with specific problem of ``inverse optimization". It aims on understanding of optimality of natural structures, as trunks of trees or human bones. Here we want to determine the goal functional of the structural optimization from the known structure and external conditions.The three chapters of the proposal treat novel problem in theoretical material science that require development of new mathematical methods. All problems originate in science and engineering.The first projects is concerned with new structures with a very high ability to resist shocks. Such materials can be used in various protective devices or containers. The second project addresses optimal structures of composites that are made of three or more components. The study of such structures is demanded by many engineering applications because the majority of natural or artificial composites are made of more than two materials. The last project aims to develop mathematical methods for evaluating the optimality of biological structures. The goal here is to understand in what sense biology is optimizing its structures and to use biological "solutions" to evolutionary challenge in engineering applications.