Shape optimization and free boundary problems

• Grant number: 16/24776-6
• Support Opportunities: Regular Research Grants
• Start date: June 01, 2017
• End date: November 30, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Antoine Laurain
• Grantee: Antoine Laurain
• 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. Free boundary problems are problems where the geometry is the unknown. These problems can be treated using shape optimization techniques.In this research plan we propose to work on modern theoretical and numerical aspects of shape optimization and free boundary problems.The first part of the project is dedicated to investigations on foundations and new applications of shape optimization. In the first section, we discuss the question of topological changes in level set methods.The idea is that there is an important gap between numerical practice and theory in level set methods, in the sense that topological changes of the sets are desired in numerical applications but cannot be analysed with the current definition of level set methods. We propose to change the framework of level set methods so that topological changes can be mathematically analysed. This is an extremely promising topic as this may lead to a convergence analysis of the level set method which is still missing even if the method was introduced 28 years ago.In section 2 of the first part, we also discuss a project for optimal design of a heat sink based on the concept of topological derivative. This is a joint project with the Escola Politecnica of USP. In the second part of the project we will investigate problems of controlling free boundaries. The objective is to analyse optimization problems depending on the solutions of free boundary problems and develop tehory and numerical techniques for these problems. In Section 1, we will describe the problem of controlling a droplet footprint via substrate surface tension. This problem has various applications such as directing the growth of biofilms.In Section 2, 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, thus an analytic approach could improve current designs. (AU)

Scientific publications

(5)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

LAURAIN, ANTOINE. A level set-based structural optimization code using FEniCS. STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION, v. 58, n. 3, p. 1311-1334, SEP 2018. (16/24776-6)

LAURAIN, ANTOINE; WINCKLER, MALTE; YOUSEPT, IRWIN. SHAPE OPTIMIZATION FOR SUPERCONDUCTORS GOVERNED BY H(CURL)-ELLIPTIC VARIATIONAL INEQUALITIES. SIAM JOURNAL ON CONTROL AND OPTIMIZATION, v. 59, n. 3, p. 2247-2272, 2021. (16/24776-6)

LAURAIN, ANTOINE. Distributed and boundary expressions of first and second order shape derivatives in nonsmooth domains. JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, v. 134, p. 328-368, FEB 2020. (16/24776-6)

LAURAIN, ANTOINE. ANALYZING SMOOTH AND SINGULAR DOMAIN PERTURBATIONS IN LEVEL SET METHODS. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, v. 50, n. 4, p. 4327-4370, 2018. (16/24776-6)

ALBUQUERQUE, YURI FLORES; LAURAIN, ANTOINE; STURM, KEVIN. A shape optimization approach for electrical impedance tomography with point measurements. INVERSE PROBLEMS, v. 36, n. 9, p. 27-pg., 2020-09-01. (14/50279-4, 16/24776-6)