Control of free boundary problems

• Grant number: 16/22324-0
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: January 23, 2017
• End date: February 05, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Antoine Laurain
• Grantee: Antoine Laurain
• Visiting researcher: Ravi Prakash
• Visiting researcher institution: Universidad de Concepción (UdeC), Chile
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

Shape optimization and related fields cover a large spectrum of mathematics, from purely theoretical problems to applications in engineering and industrial mathematics, such as structural mechanics, inverse problems, imaging, PDE-constrained optimization or free boundary problems. In this project we will work on theoretical and numerical aspects of the relatively new field of control of free boundaries. Free boundary problems are boundary value problems where the domain of definition of the partial differential equations is also unknown. They usually arise due to overdetermined boundary conditions which can only be achieved on a particular domain of definition. The objective of this research project is to analyze optimization problems depending on the solutions of free boundary problems and develop numerical techniques for these problems. We will study two problems, starting with the control of the Bernoulli free boundary problem. In a recent paper of the principal investigator, a shape optimization method has been devised to drive the free boundary to a target shape. In this project we are interested in the unstable cases for the control problem. We will develop a systematic technique to detect unstable configurations of the free boundaries. This requires the shape sensitivity analysis of a Steklov-type eigenvalue problem with a weight depending on the mean curvature of the free boundary. In the second part of the project we will develop, using tools of shape optimization, a method to control the die swell free boundary, using the design of the extrusion die as the control. An important industrial objective is to achieve a homogeneous die swell. Currently the die design relies heavily on experiments and on the experience of the engineers. (AU)