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Title:Numerical modelling of dendritic solidification based on phase field formulation and adaptive meshless solution procedure : dissertation
Authors:ID Dobravec, Tadej (Author)
ID Šarler, Božidar (Mentor) More about this mentor... New window
Files:URL http://repozitorij.ung.si/IzpisGradiva.php?id=6364
 
.pdf Tadej_Dobravec.pdf (6,11 MB)
MD5: 2833A551416C652B00FB37B47D186741
 
Language:English
Work type:Doctoral dissertation
Typology:2.08 - Doctoral Dissertation
Organization:FPŠ - Graduate School
Abstract:The main aim of the dissertation is to develop a novel numerical approach for an accurate and computationally efficient modelling of dendritic solidification, which is one of the most commonly observed phenomena in the industrial casting of the metallic alloys. The size and the morphology of dendritic structures as well as the distribution of the solute within them critically effect the mechanical and the electro-chemical properties of the solidified material. The numerical modelling of dendritic solidification can be applied for an in-depth understanding and optimisation of the casting process under various solidification conditions and chemical compositions of the alloy under consideration. The dendritic solidification of pure materials and dilute multi-component alloys is considered in the dissertation. The phase field formulation is applied for the modelling of dendritic solidification. The formulation is based on the introduction of the continuous phase field variable that is constant in the bulk of the solid and liquid phases. The phase field variable has a smooth transition from the value denoting the solid phase to the value denoting the liquid phase at the solid-liquid interface over the characteristic interface thickness. A phase field model yields a system of coupled non-linear parabolic partial differential equations that govern the evolution of the phase field and other thermodynamic variables. The meshless radial basis function-generated finite-differences (RBF-FD) method is used for the spatial discretisation of the system of partial differential equations. The forward Euler scheme is applied for the temporal discretisation. Fifth-degree polyharmonic splines are used as the shape functions in the RBF-FD method. A second-order accurate RBF-FD method is achieved by augmenting the shape functions with monomials up to the second degree. The adaptive solution procedure is developed in order to speed-up the calculations. The procedure is based on the quadtree domain decomposition of a rectangular computational domain into rectangular computational sub-domains of different sizes. Each quadtree sub-domain has its own regular or scattered distribution of computational nodes in which the RBF-FD method and the forward Euler scheme apply for the discretisation of the system of partial differential equations. The adaptive solution procedure dynamically ensures the prescribed highest density of the computational nodes at the solid-liquid interface and the lowest-possible density in the bulk of the solid and liquid phases. The adaptive time-stepping is employed to further speed-up the calculations. The stable time step in the forward Euler scheme depends on the density of the computational nodes; hence, different time steps can be used in quadtree sub-domains with different node densities. The main originality of the present work is the use of the RBF-FD method for the thorough analysis of the impact of the type of the node distribution and the size of a local sub-domain to the accuracy when the phase field modelling of dendritic solidification for arbitrary preferential growth directions is considered. It is shown how the use of the scattered node distribution reduces the undesirable mesh-induced anisotropy effects, present when the partial differential equations are discretisied on a regular node distribution. The main advantage of the RBF-FD method for the phase field modelling of dendritic growth is the simple discretisation of the partial differential equations on the scattered node distributions. The RBF-FD method is, for the first time, used in combination with the spatial-temporal adaptive solution procedure based on the quadtree domain decomposition. The adaptive solution procedure successfully speeds-up the calculations; however, the advantages of the use of the scattered node distribution are partly compromised due to the impact of regularity in the quadtree domain decomposition.
Keywords:dendritic solidification, phase field method, meshless methods, RBF-FD, adaptive solution procedure
Place of publishing:Nova Gorica
Place of performance:Nova Gorica
Publisher:T. Dobravec
Year of publishing:2021
Year of performance:2021
Number of pages:XXXVI, 172 str.
PID:20.500.12556/RUNG-6364 New window
COBISS.SI-ID:58503939 New window
UDC:531/533
NUK URN:URN:SI:UNG:REP:VOOW1FLQ
Publication date in RUNG:07.04.2021
Views:5414
Downloads:154
Metadata:XML DC-XML DC-RDF
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DOBRAVEC, Tadej, 2021, Numerical modelling of dendritic solidification based on phase field formulation and adaptive meshless solution procedure : dissertation [online]. Doctoral dissertation. Nova Gorica : T. Dobravec. [Accessed 23 April 2025]. Retrieved from: https://repozitorij.ung.si/IzpisGradiva.php?lang=eng&id=6364
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Secondary language

Language:Slovenian
Title:Numerično modeliranje dendritskega strjevanja na podlagi formulacije faznega polja in prilagodljivega brezmrežnega rešitvenega postopka
Abstract:Glavni cilj disertacije je razvoj novega numeričnega pristopa za natančno in računsko učinkovito modeliranje dendritskega strjevanja. Dendritsko strjevanje je eden najpogosteje opaženih pojavov pri industrijskem ulivanju kovinskih zlitin. Velikost in morfologija dendritskih struktur ter porazdelitev topljencev v njih ključno vplivajo na mehanske in elektro-kemijske lastnosti strjenega materiala. Numerično modeliranje dendritskega strjevanja se lahko uporablja za poglobljeno razumevanje in optimizacijo procesa ulivanja pri različnih pogojih strjevanja in pri različnih kemijskih sestavah obravnavane zlitine. V disertaciji obravnavamo dendritsko strjevanje čistih snovi in razredčenih več-sestavinskih zlitin. Za modeliranje dendritskega strjevanja uporabimo formulacija faznega polja. Formulacija temelji na uvedbi zvezne spremenljivke faznega polja, ki je konstantna v trdni in kapljeviti fazi. Spremenljivka faznega polja ima na medfaznem robu zvezen prehod preko značilne debeline medfaznega roba od vrednosti, ki označuje trdno fazo, do vrednosti, ki označuje kapljevito fazo. Model faznega polja poda sistem sklopljenih nelinearnih paraboličnih parcialnih diferencialnih enačb, ki opisujejo časovni razvoj faznega polja in ostalih termodinamskih spremenljivk. Za krajevno diskretizacijo sistema parcialnih diferencialnih enačb uporabimo brezmrežno metodo z radialnimi baznimi funkcijami generiranih končnih razlik (RBF-KR). Za časovno diskretizacijo uporabimo eksplicitno Eulerjevo shemo. Poliharmonične zlepke petega reda uporabimo kot oblikovne funkcije v metodi RBF-KR. Natančnost drugega reda metode RBF-KR dosežemo z dodajanjem monomov do vključno drugega reda k oblikovnim funkcijam. Za pospešitev izračunov razvijemo prilagodljiv rešitveni postopek. Postopek temelji na razdelitvi pravokotne računske domene na pravokotne računske pod-domene različnih velikosti z uporabo štiriškega drevesa. Vsaka pod-domena na štiriškem drevesu vsebuje svojo lastno regularno ali razmetano porazdelitev računskih točk, v katerih z uporabo metode RBF-KR in eksplicitne Eulerjeve sheme diskretiziramo sistem parcialnih diferencialnih enačb. Prilagodljiv rešitveni postopek dinamično zagotavlja predpisano najvišjo gostoto računskih točk na trdno-kapljevitem medfaznem robu in najmanjšo možno gostoto v notranjosti trdne in kapljevite faze. Za dodatno pohitritev izračunov uporabimo prilagodljivo časovno korakanje. Stabilen časovni korak v eksplicitni Eulerjevi shemi je odvisen od gostote računskih točk, zaradi česar lahko uporabimo različne časovne korake v pod-domenah štiriškega drevesa z različnimi gostotami točk. Glavna novost predstavljenega dela je v uporabi metode RBF-KR za temeljito analizo vpliva tipa porazdelitve računskih točk in velikosti lokalnih pod-domen na natančnost pri modeliranju dendritskega strjevanja pri poljubnih preferenčnih smereh rasti z uporabo metode faznega polja. Pokažemo, kako uporaba razmetanih računskih točk zmanjša neželjen vpliv mrežne anizotropije, ki je prisotna, kadar parcialne diferencialne enačbe diskretiziramo na regularni porazdelitvi računskih točk. Glavna prednost metode RBF-KR za modeliranje dendritskega strjevanja je preprosta diskretizacija parcialnih diferencialnih enačb na razmetanih porazdelitvah računskih točk. Metoda RBF-KR je prvič uporabljena v kombinaciji s krajevno-časovnim prilagodljivim rešitvenim postopkom, ki temelji na razdelitvi računske domene s štiriškim drevesom. Prilagodljiv rešitveni postopek uspešno pohitri izračune, vendar se prednosti uporabe razmetane porazdelitve računskih točk delno zmanjšajo zaradi vpliva regularnosti pri razdelitvi računske domene s štiriškim drevesom.
Keywords:dendritsko strjevanje, metoda faznega polja, brezmrežne metode, RBF-KR, prilagodljiv rešitveni postopek


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