CDT

Lead Research Organisation: University of Oxford
Department Name: Partial Diff Equa: Analysis & Apps CDT

Abstract

Topology optimization can be interpreted as an extension of shape optimization which has clear physical uses. From designing the wings on a plane to artificial heart pumps in the body, optimizing power, cost and efficiency is often desired.
Topology optimization problems are inherently nonlinear and can support multiple local minima. In practical applications finding multiple minima is often required. It is well documented that gradient-based optimization methods (as is required in such problems) can often get trapped in the large basins of attraction of certain local minima, with other minima being difficult to discover. Furthermore, a priori, it is difficult to know whether all the minima have been found. Current methods for global optimization are the use of continuation methods, however these are often heuristic and must be used on a case-by-case basis. We propose to apply the modern approach known as deflation that discovers multiple minima from the same initial guess. In doing this, we aim to develop a systematic, high adaptable method for the calculating multiple minima of such problems.

The theoretical and computational framework to implement deflation has already been developed. Furthermore, initially, the first topology optimization problems in which we will find multiple minima will be those already described as `benchmark' problems by the topology optimization community. Once the initial research has been exhausted, there are currently more complicated topology optimization problems being formulated in engineering groups in São Paulo. A collaboration might be wise. I believe developing a rigorous numerical analysis framework is necessary and will perhaps be time-consuming. Little work has been done in developing error bounds for finite element approximations as well as other similar analysis, which would provide insight when methods perform sub optimally.

A rough timetable is given as follows.
Oct 2018 - Dec 2018: Literature review and implement deflaion of benchmark problems to establish its effectiveness in finding multiple minima.
Oct 2018 - April 2019: Develop error bounds and convergence rates for benchmark topology optimization problems.
Jan 2018 - August 2019: Use deflation to find multiple minima in more difficult (but still well understood) topology optimization problems.
September 2019: Transfer of status
September 2019 - September 2020: Apply deflation to find multiple minima on new topology optimization problems which have not been analysed before.
September 2020 - July 2021: Attend conferences and tie up loose ends.
Jan 2020: Confirmation of status
July 2021 - September 2021: Writeup and viva.

This project falls within the EPSRC research areas Mathematical analysis; Numerical analysis.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/R512333/1 01/10/2017 30/09/2021
1945507 Studentship EP/R512333/1 01/10/2017 30/09/2021 Ioannis Prokopios Papadopoulos