Mathematical Sciences: Computation, Control, Optimization and Vibration Analysis of Shell Equations

本研究将集中于(1)边界积分方程的建立和动力壳方程边界元格式及软件的开发；(2)动力壳方程的振动特征频率和阻尼率分析；(3)有阻尼时壳的定性稳定性研究；(4)结构形状控制与优化，针对几种薄或浅圆圆柱、圆锥和球壳模型。众所周知，壳具有有利的结构特性。在适当的设计下，即使很薄的外壳也能承受很大的载荷。因此，它们被广泛应用于飞机和船舶结构中，这些结构对轻量化是必不可少的。在其他情况下，如压力容器、屋顶和圆顶，则利用壳体的综合强度和封闭性能。在数学上，壳层用高阶偏微分方程的耦合系统来描述。本研究的方向是：(1)计算航空航天、民用和海洋结构中壳体的变形和应力分布（特别是，计算效率大大超过传统的有限元方法）；(2)提供稳定壳体动态运动的控制系统反馈设计；(3)分析和模拟壳体的“智能结构控制”，如变形镜中球壳曲率半径的自动调整。

9404380 Chen/Zhou This research will focus on (1) the formulation of boundary integral equations and development of boundary element schemes and software for dynamic shell equations; (2) the analysis of vibration eigenfrequencies and damping rates of dynamic shell equations; (3) the study of qualitative stability properties of shells when damping is present; (4) structural shape control and optimization, for several thin or shallow circular cylindrical, conical, and spherical shell models. Shells are known to have advantageous structural properties. Under suitable design, even very thin shells can support large loads. They are therefore widely used in aircraft and ship structures where light weight is essential. In other cases, such as pressure-vessels, roofs and domes, the combined strength and enclosing properties of shells are utilized. Mathematically, shells are described by coupled systems of partial differential equations of higher order. This research is directed towards (1) computing the deformation and stress distribution of shells in aerospace, civil and maritime structures (in particular, with efficiency greatly surpassing the traditional finite element method); (2) providing feedback designs of control systems that can stabilize dynamic shell motion; (3) analyzing and simulating ``smart structure control'' of shells such as the automatic tuning of the radius of curvature of a spherical shell in a deformable mirror.