Mathematical programming methods for large-scale topology optimization problems

Abstract

Denne afhandling undersøger nye optimeringsmetoder for strukturelle topologiske optimeringsproblemer. Målet med topologisk optimering er at finde det optimale design af en struktur. Det fysiske problem er modelleret som et ikke-lineært optimeringsproblem. Dette stærke værktøj var oprindeligt udviklet til mekaniske problemer, men har siden udviklet sig hastigt til andre discipliner såsom strømningsmekanik (fluid dynamics) og biomekaniske problemer. Ikke desto mindre har nytænkningen og forbedringerne af optimieringsmetoderne været meget begrænset. Det er i den grad nødvendigt at udvikle nye optimeringsmetoder til at forbedre det endelige design og på samme tid reducere antallet af funktionsevalueringer. Ikke-lineære optimeringsmetoder, såsom sekvensiel kvadratisk programming og indre punkts metoder, har næsten ikke fået opmærksomhed af det topologiske optimeringsfaglige fællesskab. Derfor fokuserer dette arbejde på at introducere disse anden-ordens løsningsmetoder for at drive feltet fremad. Den første del af afhandlingen introducerer, for første gang, et omfattende benchmark studie af forskellige optimeringsmetoder indenfor strukturel topologisk optimering. Denne sammenligning anvender et stort testsæt og tre forskellige strukturelle optimeringsproblemer. Afhandlingen undersøger desuden, baseret på kontinuerte tilgange, en alternativ formulering af problemet for at reducere risikoen for at ende i et lokalt minimum, og samtidig mindske antallet af iterationer. Den sidste del fokuserer på skrædersyede metoder til det klassiske minimum compliance problem. To af de mest velansete optimeringsalgoritmer er undersøgt og implementeret for dette struturalle optimeringsproblem. En sekvensiel kvadratisk programmerings (TopSQP) og en indre punks metode (TopIP) er udviklet til at udnytte problemets specielle matematiske struktur. I begge løserer bruger vi eksakt Hessian information. En robust iterativ metode er implementeret til effektivt at løse lineære systemer i stor skala. Både TopSQP og TopIP opnår successfulde resultater, både hvad angår konvergens, antallet af iterationer og objektivværdien. Takket været den implementerede iterative metode, kan TopIP løse problemer i stor skala med mere end tre millioner frihedsgrader.This thesis investigates new optimization methods for structural topology optimization problems. The aim of topology optimization is finding the optimal design of a structure. The physical problem is modelled as a nonlinear optimization problem. This powerful tool was initially developed for mechanical problems, but has rapidly extended to many other disciplines, such as fluid dynamics and biomechanical problems. However, the novelty and improvements of optimization methods has been very limited. It is, indeed, necessary to develop of new optimization methods to improve the final designs, and at the same time, reduce the number of function evaluations. Nonlinear optimization methods, such as sequential quadratic programming and interior point solvers, have almost not been embraced by the topology optimization community. Thus, this work is focused on the introduction of this kind of second-order solvers to drive the field forward. The first part of the thesis introduces, for the first time, an extensive benchmarking study of different optimization methods in structural topology optimization. This comparison uses a large test set of instance problems and three different structural topology optimization problems. The thesis additionally investigates, based on the continuation approach, an alternative formulation of the problem to reduce the chances of ending in local minima, and at the same time, decrease the number of iterations. The last part is focused on special purpose methods for the classical minimum compliance problem. Two of the state-of-the-art optimization algorithms are investigated and implemented for this structural topology optimization problem. A Sequential Quadratic Programming (TopSQP) and an interior point method (TopIP) are developed exploiting the specific mathematical structure of the problem. In both solvers, information of the exact Hessian is considered. A robust iterative method is implemented to efficiently solve large-scale linear systems. Both TopSQP and TopIP have successful results in terms of convergence, number of iterations, and objective function values. Thanks to the use of the iterative method implemented, TopIP is able to solve large-scale problems with more than three millions degrees of freedom