Optimizacija forme zakrivljenog štapa

master's thesis

• Title (english): Optimizing form of an elastic curved rod
• Author: Mislav Jelašić
• Mentor: Josip Tambača (mentor)
• Committee member: Josip Tambača (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-11-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu prezentiramo optimizaciju forme elastičnog zakrivljenog štapa. Najprije uvodimo jednodimenzionalni Antman-Cosseratov model, definiran za neproduljive i nesmičljive štapove, koji je zadan sustavom diferencijalnih jednadžbi. Zakrivljeni štap je zatim aproksimiran konačnom unijom ravnih štapića. Uz određene rubne uvjete, izvodimo slabu (varijacijsku) formulaciju našeg problema (prvo za pojedini ravni štapić, a onda i za cijeli zakrivljeni štap). Glavni zadatak je pronaći optimalnu debljinu štapa, za koju je štap najkrući, kada na njega djeluje vanjska sila. Iz mješovite formulacije, definiramo minimizacijsku zadaću koristeći podatljivost kao funkciju cilja. Također pokazujemo da postoji rješenje zadanog problema. Za računanje optimalne debljine prezentiramo iterativni algoritam. Na kraju, definiramo metodu konačnih elemenata za zakrivljeni štap, koju koristimo za numeričko određivanje rješenja.

Abstract (english)

In this paper we present the optimization of the form of an elastic curved rod. First, we introduce a one-dimensional Antman-Cosserat model, defined for inextensible and unshearable rods, which is given by a system of differential equations. The curved rod is then approximated by a finite union of straight rods. Using certain boundary conditions, we derive a weak (variational) formulation of our problem (firstly for a single straight rod, and then for the entire curved rod). The main goal is to find the thickness of rod for which the rod is the most stiff, when an external force acts on it. From the mixed formulation, we prescribe minimization problem using compliance as a cost function. We also show that there is a solution to the given problem. For calculating the optimal thickness, we present an iterative algorithm. Finally, we define the finite element method for a curved rod, which we use to numerically calculate the solution.

• Keywords: jednodimenzionalni Antman-Cosseratov model; iterativni algoritam; metoda konačnih elemenata
• Keywords (english): one-dimensional Antman-Cosserat model; iterative algorithm; finite element method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:611704
• 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: 2022-12-15 15:25:11