Strukturalna optimizacija metodom nivo skupa

master's thesis

• Title (english): Structural optimization by the set level method
• Author: Goran Lalić
• Mentor: Josip Tambača (mentor)
• Committee member: Josip Tambača (predsjednik povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Committee member: Luka Grubišić (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Optimizacija oblika elastičnih struktura je suvremeno područje koje se koristi u raznim industrijama i područjima. Razvijeno je nekoliko metoda koje bave tim problemom. U ovom radu proučava se metoda nivo skupa. Promatramo optimizaciju oblika u 2D i modeliramo problem u lineariziranoj elastičnosti. Postavljamo sustav linearizirane elastičnosti za elastično tijelo na kojeg djeluju vanjske sile te pomoću njega mjerimo pomak elastičnog tijela. Maksimiziramo krutost tijela što se provodi minimizacijom funkcije cilja nad dopustivim domenama. Problem minimizacije rješavamo gradijentnom metodom koja zahtijeva uvođnje definicije derivacije oblika. Navodimo razne rezultate koji nam omogućuju dobivanje izraza za derivaciju oblika nekih funkcija cilja. Iz derivacije oblika lako dobivamo brzinu pomicanja granice oblika, odabiremo smjer spusta u gradijentnoj metodi i time imamo generalni postupak algoritma. Granica oblika se može reprezentirati na razne načine od kojih mi odabiremo nivo skupove pomoću kojih imamo jednostavnije izraze za normalu i srednju zakrivljenost na granicu oblika. Proširujemo sustav linearizirane elastičnosti na cijelu radnu domenu pritupom \(ersatz\) materijala. Funkciju koja odreduje nivo skupove dobivamo rješavanjem transportne Hamilton–Jacobijeve jednadžbe. Zatim prezentiramo iterativni algoritam u kojem detaljnije opisujemo postupke inicijaliziranja i iteriranja kroz algoritam. Uvodimo i opisujemo postupak ponovne inicijalizacije funkcije koja određuje nivo skupove i postupak ograničavanja volumena. U drugom poglavlju predstavljamo rezultate dobivene u programskom alatu FreeFem++. Navodimo korištene parametre i neke korisne izvode potrebne za implementaciju. Promatramo različite primjere i funkcije cilja. Rezultate prezentiramo na primjerima 2D nosača i 2D mosta, pri čemu ih detaljnije analiziramo.

Abstract (english)

Shape optimization of elastic structures is a popular area used in various industries and fields. Several methods have been developed to address this problem. In this paper, the level set method is studied. We observe shape optimization in 2D and model the problem in linearized elasticity. We set up a linearized elasticity system for an elastic body on which external forces act. The solution of such a system gives us the displacement of the elastic body. We maximize body stiffness which is represented by minimizing the target functions over the allowable domains. We solve the minimization problem with a gradient descent method that requires the introduction of a definition of shape derivative. We list various results that allow us to obtain expressions for shape derivatives of some objective functions. From the shape derivative we easily obtain the velocity of the moving part of shape boundary, we choose the direction of the descent in the gradient descent method and thus we have the general procedure of the algorithm. The shape boundary can be represented in a variety of ways from which we select level sets by which we have simpler expressions for normal and mean curvature. We extend the system of linearized elasticity to the entire working domain by the simple \(ersatz\) material approach. The level set function is obtained by solving the transport Hamilton–Jacobi equation. We propose an iterative algorithm in which we describe in more detail the procedures of initialization and iteration through the algorithm. We introduce and describe the procedure of re-initialization of the level set function and the procedure of volume restriction. In the second chapter we present the results obtained in the software tool FreeFem++. We list used parameters and some useful derivations required for implementation. We observe different examples and objective functions. The results are presented on the examples of 2D cantilever and 2D bridge, where we analyze them in more detail.

• Keywords: linearizirana elastičnost; gradijentna metoda; derivacija oblika; transportna Hamilton–Jacobijeve jednadžba; programska alat FreeFem++
• Keywords (english): linearized elasticity; gradient descent method; shape derivative; transport Hamilton–Jacobi equation; software tool FreeFem++
• Language: croatian
• URN:NBN: urn:nbn:hr:217:117501
• 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: 2021-10-28 12:52:47