Numerička metoda za model lineariziranog stenta

master's thesis

• Author: Josip Iveković
• Mentor: Josip Tambača (mentor)
• Committee member: Josip Tambača (predsjednik povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Zvonimir Tutek (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-02-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Stent se obično modelira kao trodimenzionalno elastično tijelo. U ovome radu, mi smo mrežu stenta definirali kao neusmjeren graf, u kojemu tanki metalni štapovi, modelirani lineariziranim Antman-Cosserat modelom za nerastezljive i nesmičljive stapove, predstavljaju bridove toga grafa. Na taj smo način postigli da umjesto problema u tri dimenzije rješavamo problem u jednoj dimenziji, što zahtijeva manji utrošak memorije u procesu računanja. Pokazali smo da uz određene uvjete mješovita formulacija tog modela ima jedinstveno rješenje. Za aproksimaciju pomaka središnje linije bridova te infinitezimalnu rotaciju poprečnog presjeka bridova koristili smo prostore polinoma. Također, radi jednostavnijeg računanja, mi smo pojedine štapove u stentu, koji su savijeni, mi smo aproksimirali ravnim štapovima. Ustanovili smo da numerička metoda koju koristimo konvergira, uz redove konvergencije ovisne o prostoru polinoma kojima aproksimiramo funkcije. Ipak, između P2P1 i P3P2 elemenata nije bilo razlike u konvergenciji metode, a i razlike u dobivenim rezultatima su neznatne. Zanimljivo bi bilo dati teorijsku podlogu navedenim zapažanjima te vidjeti koliko brzu konvergenciju rješenja uopće možemo postići ovim pristupom.

Abstract (english)

Stent is usually modelled as a three-dimensional elastic body. In this paper, we have defined the stent net as a non-directed graph, in which the slender metallic rods, modelled by the linearized Antman-Cosserat model for inextensible, unshearable rods, represent the edges of the graph. In that way, we have transformed a three-dimensional problem into a one-dimensional problem, which is computationally cheaper and easier to handle. We have shown that under certain conditions the mixed formulation of our model has a unique solution. For the approximation of the middle line shift of the stent struts and infinitesimal rotation of the cross-section of the stent struts, we have used polynomial spaces. In addition, to make the calculus simpler, we have taken the stent struts, which are actually curved, to be straight. We have established that the numerical method we use converges, and the scale of convergence depends on the polynomial space we use to approximate the functions. However, there is no difference between the convergence scale of P2P1 elements and that of P3P2 elements. Moreover, the solutions obtained by P2P1 elements are just slightly different from the solutions obtained by P2P1 elements. It would be interesting to give some theoretical background to these observations, and also to see what rate of convergence is actually possible to achieve with this approach.

• Keywords: stent; neusmjeren graf; Antman-Cosserat model za nerastezljive i nesmičljive stapove; prostori polinoma
• Keywords (english): stent; non-directed graph; Antman-Cosserat model for inextensible unshearable rods; polynomial spaces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:401004
• 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: 2019-01-31 10:37:55