On the geometry and the deformation of shape represented by a piecewise continuous B\'ezier curve with application to shape optimization

TL;DR

This paper introduces a framework using piecewise Bézier curves for shape deformation and optimization, reducing complex shape problems to finite-dimensional integrals, and demonstrating practical algorithmic applications.

Contribution

It presents a novel approach to shape optimization via piecewise Bézier curves, simplifying the problem to finite-dimensional integrals and providing efficient algorithms.

Findings

Framework effectively models shape deformation.

Reduction to finite-dimensional integral problems.

Practical algorithms demonstrate efficiency.

Abstract

In this work, we develop a framework based on piecewize B\'ezier curves to plane shapes deformation and we apply it to shape optimization problems. We describe a general setting and some general result to reduce the study of a shape optimization problem to a finite dimensional problem of integration of a special type of vector field. We show a practical problem where this approach leads to efficient algorithms.