Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

We consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold $\overline{M}$. We first assume the pull back Weitzenböck operator of $\overline{M}$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of $\overline{M}$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when $\overline{M}$ is the space form.     

An explicit series solution of the squeezing flow between two infinite plates by means of the homotopy analysis method

In this paper, we investigated an axisymmetric Newtonian fluid squeezed between two parallel plates. The steady nonlinear governing equations are reduced to a single differential equation using integrability condition. Homotopy analysis method (HAM) is used to solve the nonlinear differential equation analytically. Numerical solutions indicate this method is satisfactory.     

基于机器学习的增材制造合金材料力学性能预测研究进展与挑战

增材制造是现代高端装备制造领域的革命性突破技术之一. 其中, 增材构件的大批量生产和高可靠应用, 关键在于制造可重复性、质量可靠性与性能可预测性. 而在各向异性组织、广域分布缺陷、深部残余应力和复杂表面粗糙度等诸多因素的共同影响下, 基于传统经验模型和有限数据的增材制造金属力学性能预测效率与准确性面临着严峻挑战. 近年来, 作为大数据与人工智能发展到一定阶段的必然产物, 机器学习(machine learning, ML)方法为有效处理高维物理量之间的复杂非线性关系提供了契机, 在增材制造合金材料力学性能预测领域得到持续关注. 文章综述了机器学习在增材制造材料及构件力学性能预测中的国内外研究进展. 首先简述了常见的机器学习算法和通用的机器学习流程, 重点分析了融合物理信息的机器学习(physics-informed machine learning, PIML)方法的特点与构造方式; 然后概述了增材制造合金材料力学性能4大影响因素的形成原因及机器学习在这些影响因素预测中的应用现状; 重点介绍了ML和PIML在拉伸性能和疲劳断裂性能预测中的代表性研究成果; 最后指出当前机器学习在力学性能预测中的局限性, 并探讨了发展趋势和技术前景.