Analysis and control of shapes and systems

该计划的主题是研究物理和技术系统建模、设计和控制中形状和几何的理论和数值方面。该领域的研究非常广泛、丰富和引人入胜，在许多不同领域具有极其重要的应用潜力：汽车工业机械部件的优化设计，传感器和执行器的定位，材料科学中自由边界位置的控制，噪声的主动控制，图像处理，自由和移动边界问题，生物医学设备设计，薄结构的设计和控制，通过飞机机翼形状的微小变化控制阻力等。确实需要理论和方向来理解所遇到的建模问题，新的现象学研究带来的挑战，并解释大量的计算结果。该程序集中于以下五个相互交织的项目：1)形状和几何的数学分析和优化，2)为欧几里德空间子流形上的微积分和泛函分析开发适当的内在方法，3)薄的和渐近的机械壳、梁和曲杆、压电壳和薄流体壳(例如海洋)的内在建模和分析，4)图像处理、裂解集、脑成像，5)内在化的建模、设计和控制，N-D截面的(N+1)-D图像重建，其中一些工作对加拿大空间计划的热扩散器和散热器的设计产生了影响，以调节卫星的热环境，并对第三代卫星的传感器和执行器的定位产生了影响。此外，研究的基本成果也用于目前的医疗器械设计项目(介入心脏病学中的支架)。

The theme of this program is the study of the theoretical and numerical aspects of shapes and geometries in the modeling, design, and control of physical and technological systems.  This area of research is very broad, rich and fascinating with an extremely important potential for applications in many different areas: optimal design of mechanical parts for the automotive industry, positioning of sensors and actuators, control of the position of the free boundary in material sciences, active control of noise, image processing, free and moving boundary problems, design of biomedical devices, design and control of thin structures, control of the drag by small changes in the shape of the wing of an aircraft, etc... There is a definite need for theory and directions to understand the modelling issues encountered, the challenges arising from new phenomenological studies, and to interpret the overwhelming number of computational results.   The program concentrates on the following five intertwined projects: 1) the mathematical analysis and optimization of shapes and geometries, 2) the development of appropriate intrinsic methods for the differential calculus and functional analysis on submanifolds of the Euclidean space, 3) the intrinsic modelling and analysis of thin and asymptotic mechanical shells, beams, and curved rods, piezoelectric shells, and thin fluid shells (e.g. oceans), 4) image processing, cracked sets, brain imaging, 5) modelling, design, and control of endoprotheses, (N+1)-D image reconstruction from N-D sections,   Some of this work has had an impact on the Canadian Space Program for the design of thermal diffusers and radiators to condition the thermal environment of satellites and on the positioning of sensors and actuators for satellites of the third generation.  The fundamental results of the research are also used on current projects on the design of medical devices (stents in interventional cardiology).