Shape optimization problems are ubiquitous in science, engineering and industrial applications. Indeed, starting with the foundation of PDE-based optimization (see Lions, 1971), shape design has became one of the most frequent application in technologies and it is nowadays one main focus of aerodynamic and fluidynamic simulation.
One of the most remarkable advances in shape design has been to replace the approach of parametric optimization with the concept of continuous shape design. In fact, in the former approach the control variable (i.e., the shape) is restricted to belong to a finite dimensional space spanned by suitable basis functions, while in the latter case it is an element of an infinite-dimensional space. This second approach opens enormous perspective in the formulation of more accurate and sophisticated shape optimization problems.
The possibility of formulating the shape optimization problems at the infinite-dimensional level poses new challenges to the design and implementation of numerical optimization schemes that properly accommodate the infinite dimensionality of the control function. In particular, a successful and effective algorithm must allow the control function to be adaptively approximated and optimized to any desired degree of accuracy.
View a list of journal publications and MOX Reports related to shape optimization problems.
View a list of theses related to shape optimization problems.