Shape and Topological Optimization: Analysis and Differential Calculus

本课程的主题是在物理、技术和生物医学系统的建模、设计、控制和识别中研究形状和拓扑的数学方面。基本思想来自系统理论、优化和控制理论，但建模、优化或控制变量不再是一组参数或函数，而是几何对象的形状或结构，或者只是欧几里得空间的一个子集。这一研究领域在许多不同领域的应用和应对挑战性问题方面具有重要的潜力：机械零件的优化设计（汽车工业）、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、医疗器械的设计、药物释放、薄结构的设计和控制、通过飞机机翼形状的微小变化来控制阻力、优化游泳。******考虑优化/设计问题及其数值模拟/解决方案，一个好的分析框架是必不可少的。它必须足够宽，以容纳不能局部识别为欧几里得空间的光滑结构的集合。在这种精神下，已经构造了两种类型的度量空间：对应于固定集合映像的微分同态空间（例如Courant度量）和集参数化函数的空间，例如特征、距离（Hausdorff度量）或允许拓扑变化和裂纹的定向距离函数。第一种类型的空间是无限维的芬斯勒流形，而第二种类型的空间具有更复杂的性质。大多数已知的紧性定理都与第二类空间有关（如一致锥或尖性质，Caccioppoli集合）。******表征最优集的下一个要素是包含链式法则的优秀微分学。欧拉导数（形状导数）是满足第一类空间期望的微分。第二类空间的切空间不是线性的，只能得到半微分，如Hadamard半微分。例如，勒贝格可测集的特征函数的阿贝尔群包含半切线的测度和分布。这些度量与所谓的从点的扩张得到的拓扑导数有关。这一概念通过d维闵可夫斯基内容扩展到费德勒的可整流集和正延伸集的扩张。它们提供了额外必要的最优性条件。大量的数值计算表明了它在力学中的适用性。最后，将拉格朗日极小极大函数的参数导数定理应用于状态约束的目标函数。

The theme of this program is the study of the mathematical aspects of shapes and topologies in the modeling, design, control, and identification of physical, technological, and biomedical systems. Fundamental ideas come from systems theory, optimization, and control theory, but the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object or simply a subset of the Euclidean space. This area of research has an important potential in applications and in responding to challenging issues in many different areas: optimal design of mechanical parts (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 medical devices, drug release, design and control of thin structures, control of the drag by small changes in the shape of the wing of an aircraft, optimal swimming.******To consider optimization/design problems and their numerical simulation/solution, a good analytical framework is essential. It must be sufficiently broad to accommodate sets that are not locally identifiable to the Euclidean space with a smooth structure. In this spirit two types of metric spaces have been constructed: spaces of diffeomorphisms corresponding to images of a fixed set (e.g. Courant metrics) and spaces of set-parametrized functions such as the characteristic, distance (Hausdorff metric), or oriented distance functions that allow topological changes and cracks. The spaces of the first type are infinite dimensional Finsler manifold while those of the second type are of a more complex nature. Most of the known compactness theorems are related to spaces of the second type (e.g. uniform cone or cusp property, Caccioppoli sets). ******The next ingredient to characterize optimal sets is a good differential calculus that includes the chain rule. The Eulerian derivative (shape derivative) is a differential that meets the expectations for the spaces of the first type. The tangent space to spaces of the second type is not linear and only semi-differentials such as the Hadamard semi-differential can be obtained. For instance, the Abelian group of characteristic functions of Lebesgue measurable sets contains measures and distributions that are semi-tangents. The measures are associated with the so-called topological derivative obtained from the dilatation of a point. This notion extends to the dilatation of rectifiable sets and sets of positive reach of Federer via the d-dimensional Minkowski content. They provide additional necessary optimality conditions. Extensive numerical computations show its pertinence in Mechanics. Finally, theorems on the parametric derivative of the minimax of a Lagrangian are used for state constrained objective functions.