Speaker
Description
Shape optimization focuses on determining the optimal shape of a domain—such as a boundary or region in space—to minimize or maximize an objective function. This process is typically subject to constraints, often modeled by partial differential equations (PDEs), as the underlying physical phenomena, such as fluid flow, heat conduction, or elasticity, are governed by these equations. Common applications include minimizing drag in fluid dynamics, maximizing structural strength, and optimizing material distribution in a design domain.
A central challenge in this field is the effective modeling of shapes. Currently, there is no universally accepted approach, and several techniques are employed, including level set methods, the method of mappings, and boundary parametrization. Each method has its own strengths and limitations.
In this talk, we will focus on the Riemannian perspective of shape optimization. Within this framework, Riemannian geometry provides a rigorous mathematical foundation for understanding and navigating the space of shapes. The core concept is to treat the space of admissible shapes as a Riemannian manifold, where each shape—such as a curve in 2D or a surface in 3D—represents a point in either an infinite- or finite-dimensional manifold. Riemannian metrics are used to define distances and gradients on this manifold, enabling the application of optimization techniques, such as gradient descent, in a mathematically consistent way.
We will explore various metrics for infinite-dimensional and finite-dimensional shape manifolds, examine commonly used numerical methods within this framework, and present several illustrative numerical examples.
