Title Analysis of a nonlinear three-dimensional problem of fluid, mesh and shell interaction Title (croatian) Analiza nelinearnog trodimenzionalnog problema interakcije fluida, stenta i ljuske Author Marija Galić Mentor Boris Muha (mentor)Committee member Josip Tambača (predsjednik povjerenstva)Committee member Boris Muha (član povjerenstva)Committee member Mario Bukal (član povjerenstva)Granter University of Zagreb Defense date and country 2018-12-20, Croatia Scientific / art field, NATURAL SCIENCES Universal decimal classificationUDC ) 51 - Mathematics Abstract In the first part of this thesis, we introduce the fluid-structure interaction problems that we are dealing with, and present a literature review of completed and ongoing research on this matter. In the second part of this thesis we prove the existence of a weak solution to a linear fluid-structure interaction problem modeling the flow of an incompressible, viscous three-dimensional fluid, flowing through a cylinder whose lateral wall is described by the two-dimensional linearly elastic Koiter shell equations coupled with the one-dimensional elastic mesh equations. The fluid and the composite structure are fully coupled via the kinematic and dynamic coupling conditions describing continuity of velocity and balance of contact forces. The methodology of the proof is based on the semi-discretization approach, in which the full, coupled problem is discretized in time, and, at the same time, split into a fluid and a composite structure subproblem using the so-called Lie operator splitting strategy. The third part of this thesis deals with a nonlinear, moving boundary fluid-structure interaction problem between an incompressible, viscous fluid flow and an elastic structure composed of a cylindrical shell supported by a mesh-like elastic structure. The first main difference with regards to the linear case is that the fluid flow, which is driven by the time-dependent dynamic pressure data, is modeled by the Navier-Stokes equations. Furthermore, we had to employ the Arbitrary-Lagrangian Eulerian mapping to deal with the motion of the fluid domain, which introduces an additional nonlinearities in the problem. Finally, we prove the existence of a weak solution to this nonlinear, moving boundary fluid-structure interaction problem by using the same strategy as in the linear case together with the non-trivial compactness results, which enabled us to pass to the limit in the weak formulation. These problems were motivated by studying fluid-structure interaction between blood flow through coronary arteries treated with metallic mesh-like devices called stents.
Abstract (croatian) U prvom dijelu ove radnje uvodimo probleme interakcije fluida i strukture kojima ćemo se baviti i dajemo pregled literature o istraživanjima koja su napravljena u tom području. U drugom dijelu ove radnje dokazujemo egzistenciju slabog rješenja linearnog problema interakcije fluida i strukture koji modelira tok inkompresibilnog, viskoznog, trodimenzionalnog fluida kroz cilindričnu domenu čija lateralna granica je modelirana dvodimenzionalnim jednadžbama linearne elastične Koiterove ljuske spojenima s jednodimenzionalnim jednadžbama elastične mreže. Fluid i složena struktura su potpuno spojeni kinematičkim i dinamičkim rubnim uvjetima koji opisuju neprekidnost brzine i balans kontaktnih sila. Metodologija dokaza se sastoji u tome da naš puni problem podijelimo na dva jednostavnija potproblema, tj. potproblem za fluid i potproblem za strukturu, te ih semi-diskretiziramo, tj. diskretiziramo u vremenu, koristeći Lie operator splitting metodu. Treći dio ove radnje se bavi nelinearnim problemom interakcije fluida i strukture s pomičnom granicom. Tok inkompresibilnog, viskoznog fluida je, za razliku od linearnog slučaja, modeliran Navier-Stokesovim jednadžbama i pokreće ga dinamički tlak na ulazu i izlazu iz cilindrične domene. Lateralna/pomična granica je opisana jednadžbama linearne elastične Koiterove ljuske spojenima s jednadžbama elastične mreže i na njoj su zadani kinematički i dinamički uvjeti spajanja. S obzirom da je lateralna granica pomična, u svakom trenutku je njen položaj jedna od nepoznanica problema, što uvodi dodatne nelinearnosti u problem. Dokazujemo egzistenciju slabog rješenja ovog nelinearnog problema koristeći Lie operator splitting metodu i semi-diskretizaciju, kao i u linearnom slučaju, Arbitrary Lagrangian-Eulerian preslikavanje kojim ”fiksiramo” pomičnu granicu, te netrivijalne rezultate kompaktnosti koji nam omogućuju prelazak na limes u slaboj formulaciji. Motivacija za oba problema dolazi iz primjena u hemodinamici, odnosno iz proučavanja toka krvi kroz koronarne arterije tretirane sa stentovima. Stent je mala, metalna, mrežasta cjevčica koja se postavlja u suženi ili zatvoreni dio koronarne arterije s ciljem otvaranja i uspostavljanja normalnog protoka krvi.
Keywords
fluid-structure interaction
moving boundary problem
incompressible Navier-Stokes equations
linearly elastic Koiter shell
one-dimensional elastic mesh
kinematic coupling conditions
dynamic coupling conditions
operator splitting method
Arbitrary Lagrangian-Eulerian mapping
semi-discretization
weak solutions
compactness
Keywords (croatian)
interakcija fluida i strukture
problem pomične granice
inkompresibilne Navier-Stokesove jednadžbe
linearna elastična Koiterova ljuska
jednodimenzionalna elastična mreža
kinematički uvjeti spajanja
dinamički uvjeti spajanja
operator splitting metoda
Arbitrary Lagrangian-Eulerian preslikavanje
semi-diskretizacija
slaba rješenja
kompaktnost
Language english URN:NBN urn:nbn:hr:217:680042 Study programme Title: Mathematics Type of resource Text Extent viii, 107 str. File origin Born digital Access conditions Open access Terms of use Created on 2019-03-26 11:43:11
