| Grant number: | 16/24776-6 |
| Support type: | Regular Research Grants |
| Duration: | June 01, 2017 - November 30, 2019 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Applied Mathematics |
| Principal researcher: | Antoine Laurain |
| Grantee: | Antoine Laurain |
| Home 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)
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