Optimal Multimaterial Composites and Exotic Structures

该奖项支持首席研究员在复合材料优化设计领域的研究计划，复合材料是制造业中的重要结构材料(例如)。3D打印、微米和纳米制造等现代技术能力使人们能够制造出种类繁多的材料结构，人们想知道最好的结构，或者如何优化复合微结构。该项目开发了多材料复合材料的结构优化方法。最优的复合材料导致了利用极端性能的超材料。优化复合材料是结构优化的关键，因为优化设计是由特殊定制的复合材料进行的。许多应用都要求对多材料复合材料进行优化设计，但到目前为止，由于现有理论的局限性，绝大多数相关结果只涉及两种材料的优化设计。这个项目填补了这一空白。多相复合材料的最佳组织与最佳两相复合材料的组织有很大的不同。后者有一种直观的预期拓扑：坚硬的材料总是围绕着弱夹杂物。相比之下，最优的三种材料结构表现出各种各样的模式：它们可能包含一个强大的包络，由各向异性“路径”连接的中间材料的“枢纽”，以及其他揭示最优几何本质的配置。最优结构的拓扑结构不容易被猜测，该研究发展了一种构造最优结构的规则方法。用凸分析方法研究了复合材料的最优结构和有效性能的界。将多材料最优复合材料结构问题转化为构造多井拉格朗日拟凸包络的问题。将开发两种方法：一种新的边界推导方法，它解释了由于最优结构(较弱的场对应于较强的材料)中的梯度场的排序而导致的不平等；以及一种互补的方法，用于构建实现边界的最小化序列以确定最优复合材料的结构。假设材料中的场是根据边界导出的最佳性要求而知道的，并且几何形状的选择使得这些场是兼容的。特别关注多相复合材料的优化三维结构，所开发的结构方法可用于构建超材料。

This award supports the research program of the Principal Investigator in the area of optimal design of composite materials, which are important structural materials in manufacturing (for example). Modern technological capabilities such as 3d-printing, micro- and nano-fabrication, allow a huge variety of material structures to be manufactured, and one wants to know the "best" structure, or how composite microstructures can be optimized. This project develops methods of structural optimization of multimaterial composites. Optimal composites lead to metamaterials that utilize the extreme properties. Optimal composites are crucial to structural optimization because optimal designs are made from specially tailored composites. Numerous applications call for optimal design of multimaterial composites, but so far the vast majority of related results deals only with two-material optimal composites because of limitations of existing theory. This project fills this gap. Optimal microstructures of multiphase composites drastically differ from the structures of optimal two-phase composites. The latter have an intuitively expected topology: a strong material always surrounds weak inclusions. In contrast, optimal three-material structures show a large variety of patterns: they may contain a strong envelope, "hubs" of intermediate material connected by anisotropic "pathways", and other configurations that reveal a geometrical essence of optimality. The topologies of optimal structures cannot be easily guessed, and the proposed research develops a regular way to construct them.Optimal structures and bounds for effective properties of composites are studied by methods of convex analysis. The problem of optimal multimaterial composite structure is formulated as a problem of constructing a quasiconvex envelope of a multi-well Lagrangian. Two methods will be developed: a novel method of bounds derivation that accounts for inequalities due to ordering of gradient fields in optimal structures (weaker fields corresponding to stronger materials) and a complementary method for building minimizing sequences that realize the bounds to determine the structure of an optimal composite. It is assumed that fields in the materials are known from optimality requirements derived from the bounds, and the geometry is chosen so that these fields are compatible. Special attention is paid to optimal 3d structures of multiphase composites; the developed methods for structures can be used for constructing metamaterials.