Liping Liu: A tutorial on homogenization method and rigorous bounds

报告时间：2024年7月19日（星期五）9:00-10:00 报告地点：纬地楼414 报 告 人：Liping Liu, Professor 工作单位：Rutgers University 举办单位：土木与水利工程学院 报告简介： In this talk I will give a brief introduction on homogenization theory. In this area, an optimal design problem stands out for practical applications pertaining to the maximum performance within design constraints. For such a nonconvex problems, the direct method can hardly yield any definite answer. We rely on the indirect method. The indirect method for optimal design of composites consists of two steps:  a rigorous bound is first derived for the objective functional properties, and then microstructures are constructed to achieve the rigorous bounds based on some exactly solvable geometries. In this paper, we achieve a few results on both aspects. First, we present a novel method for rigorous Hashin-Shtrikman-type bounds. This method greatly simplifies the classical Hashin-Shtrikman's calculations on variational bounds. In addition, it gives rise to multifunctional cross-property that could potentially characterize the all possible multifunctional properties of two-phase composites. Second, based on the method of variational inequality, we construct manufacturable optimal microstructures which are termed (periodic) E-inclusion. The method of variational inequality turns out to be useful for constructing other exactly solvable geometries, including neutral inclusions for cloaking, structures with minimum stress concentration, and structures that can amplify far applied fields.  Last, based on the concept of null Lagrangian, we obtain a set of necessary conditions for attaining optimal bounds. These conditions are shown to be sufficient for three-phase composites by explicitly constructing microstructures. We anticipate these rigorous results would give useful insight for topological optimization of multifunctional composites. 报告人简介： Dr. Liping Liu (刘利平) received a B.E. degree in Mechanics and Engineering Science from Peking University, Beijing, China, in 2000, and a Ph.D. degree in Aerospace Engineering and Mechanics from the University of Minnesota at Twin Cities, USA, in 2006. Since 2012, he has been a Professor with the Department of Mathematics and the Department of Mechanical Aerospace Engineering at Rutgers, The State University of New Jersey in New Jersey, USA. His research interests include the mechanics, mathematics of materials including: optimal bounds and optimal microstructures for multiphase composites, Eshelby inclusion problems, the predictive modeling of multifunctional composites, continuum theory for multiferroic materials, and multiscale analysis.