Metoda nivo skupa u optimizaciji oblika

master's thesis

• Author: Petar Kunštek
• Mentor: Marko Vrdoljak (mentor)
• Committee member: Marko Vrdoljak (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-07-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U radu se promatra optimizacijski problem u kojemu je cilj pronaći particiju domene na dva skupa koja minimizira zadani funkcional integralnog tipa. Pritom funkcional kao argument uzima rješenje parcijalne diferencijalne jednadžbe definirane na elementu particije domene, označen sa $\Omega$. Neka je $C = C(\Omega)$ opisani funkcional. Prvo poglavlje obrađuje osnovna svojstva Banachovog prostora $W^{k,\infty}(\mathbb{R}^d, \mathbb{R}^d)$. Kao direktna posljedica prostor... More $W^{1,\infty}(\mathbb{R}^d, \mathbb{R}^d)$ možemo poistovjetiti sa ograničenim Lipschitz-neprekidnim funkcijama. Uloga prostora $W^{k,\infty}(\mathbb{R}^d, \mathbb{R}^d)$ leži u opisivanju malih promjena skupa, a da pritom ostanu sačuvana neka dobra svojstva skupa kao što su otvorenost ili regularnosti ruba. Promjena skupa je skup $\Omega' = (Id + \theta)\Omega$, gdje je Id identiteta, a $\theta \in W^{k,\infty}(\mathbb{R}^d, \mathbb{R}^d)$. Fiksiranjem $\Omega$ promatramo preslikavanje $C(\Omega; \theta) := C(\Omega')$. Od interesa je objasniti kako male promjene skupa $\Omega$ utječu na vrijednost funkcionala koristeći pojam derivacije oblika: $C'(\Omega, \theta)= \lim_{t \to 0^+} \frac{C(\Omega ; t\theta)-C(\Omega ; 0)}{t}$. Drugo poglavlje sadrži tehničke rezultate o Fréchet-diferencijabilnosti koji se koriste u narednom poglavlju. Treće poglavlje definira pojam lokalne diferencijabilnosti, pomoću kojeg se može detaljnije objasniti ponašanje rješenja diferencijalne jednadžbe pri malim promjenama skupa $\Omega$. Pritom dolazimo do dovoljnih uvjeta za postojanje derivacije oblika $C'(\Omega, \theta)$. U posljednjem poglavlju demonstrirana je primjena teorije na modelu kondenzatora. Less

Abstract (english)

This thesis studies the optimization problem in which the objective is to find the bipartition of domain that minimizes a given integral functional. The functional explicitly depends on the solution of a partial differential equation defined on a bipartion's element denoted with $\Omega$. Let $C = C(\Omega)$ be mentioned functional. In the first chapter basic properties of Banach space $W^{k,\infty}(\mathbb{R}^d, \mathbb{R}^d)$ are introduced. Essentially,... More $W^{1,\infty}(\mathbb{R}^d, \mathbb{R}^d)$ can be identified as a space of a bounded Lipschitz continuous functions. That space is used to explain small changes of the set $\Omega$, while preserving important properties like openness and regularity of the border. Change of a set $\Omega$ is the set $\Omega' = (Id + \theta)\Omega$, where Id is an identity on $\mathbb{R}^d$ and $\theta \in W^{k,\infty}(\mathbb{R}^d, \mathbb{R}^d)$. Fixing $\Omega$ we can introduce mapping $\theta \longmapsto C(\Omega; \theta) := C(\Omega')$. To explain how small changes on $\Omega$ affect the value of the functional one can introduce shape derivative: $C'(\Omega, \theta)= \lim_{t \to 0^+} \frac{C(\Omega ; t\theta)-C(\Omega ; 0)}{t}$. The second chapter deals with technical results on a differentiability which are used in the next chapter. In the third chapter, the term local differentiability is introduced. It is used to better explain how solution can be differentiated with respect to small changes of a set $\Omega$. Within this chapter, a sufficient conditions for existence of the shape derivative $C'(\Omega, \theta)$ are given. All theory is applied in the last chapter, on a model of electric capacitor to prove existence of the local derivative. Less

• Keywords: particija domene; funkcional integralnog tipa; parcijalna diferencijalna jednadžba; Banachov prostor; ograničene Lipschitz neprekidne funkcije; Fréchet diferencijabilnost; model kondenzatora
• Keywords (english): bipartition of domain; integral functional; partial differential equation; Banach space; bounded Lipschitz continuous functions; Fréchet differentiability; model of an electric capacitor
• Language: croatian
• URN:NBN: urn:nbn:hr:217:513591
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-21 11:56:20