1. Meshless modeling of thermomechanics of lowfrequency electromagnetic direct chill castingBoštjan Mavrič, 2017, doctoral dissertation Abstract: The aim of this dissertation is to devise a meshless model describing the thermomechanical phenomena, which occur during DC casting of aluminium alloys under the influence of electromagnetic stirring. The thermoemchanical phenomena are important, because they can cause several type of defects, which can significantly deteriorate the quality of the resulting billet. The two most important of them are the hot tearing, which causes cracks to appear in the mushy zone, and the porosity, which demonstrates itself as micrometer sized voids in the microstructure of the billet.
To calculate the stresses and strains, a computational model of the stationary state of the process, stated in axial symmetry, is formulated. It uses Eulerian formulation by fixing the computational domain to the mold of the casting device allowing the material to move through the computational domain. The stresses are calculated from the stress equilibrium equations. The small strain approximation is used to consider the three contributions to strain. The strain consists of the thermal strain, which is caused by the inhomogeneous thermal profile in the billet, the viscoplastic strain, which is caused by the irreversible deformation because of the large stresses occurring in the billet, and the elastic strain.
The spatial discretization of the governing equations is performed by local radial basis function collocation method (LRBFCM) and the temporal discretization is achieved by the method of lines with implicit Euler formula. The method used for spatial discretization uses radial basis functions augmented by monomials to approximate the solution values on localized stencils. This approximation is used to construct the discretization coefficients of the differential operators present in the model. A flexible framework for formulation of multiphysics problems is developed to use the obtained discretization coefficients to construct the temporal discretization of the governing equations. The node arrangement, on which the spatial discretization is performed, was generated by a pointrepel algorithm.
The performance of the method is tested on several benchmark test cases. The accuracy of the discretization is estimated by comparing the analytic and the numerical solution to several stationary problems in thermomechancis. Of special interest is the performance of the method with respect to the choice of the shape parameter, which determines the spatial scale of the radial basis functions. Besides this, the dependence of the condition number of the interpolation matrix on the shape parameter is studied. The condition number is found fit to replace the condition number as the shapedetermining free parameter of the method.
The implementation of the solver of time dependent problems is tested on problem of thermoelasticity, which couples the thermal transport with the elastic waves. The results of the problem are compared with the finite element method, showing good agreement of the two methods. The results are also compared with the results obtained by meshless local PetrovGalerkin method and the proposed local collocation method demonstrated significantly better solution quality in the studied case.
The performance of the solver used to solve the system of nonlinear equations given by the viscoplastic constitutive equations is estimated on a quasi zerodimensional problem. The results are found to match perfectly. Solution of a more complicated problem is obtained with the proposed method and the finiteelement method, both methods giving practically the same solution, although some serious limitations of the chosen finite element solver are exposed during the selection of the problem parameters.
Finally, the devised method is applied to the problem of DC casting of aluminium alloys. The thermomechanical model relies on a model of heat and mass transfer to obtain the input fields needed in the solver. The required fields are: temperature, pressure, liquid Found in: ključnih besedah Summary of found: ...method, showing good agreement of the two methods. The results are also compared with the... Keywords: thermomechanics, viscoplasticity, aluminium alloys, directchill casting, electromagnetic stirring, hot tearing, porosity, meshless methods, local collocation method, radial basis functions, shape parameter Published: 28.06.2017; Views: 3161; Downloads: 114 Fulltext (21,30 MB) 
2. Modelling of Macrosegregation of a LowFrequency Electromagnetic Direct Chill Casting by a Meshless MethodVanja Hatić, 2019, doctoral dissertation Abstract: The main aim of the dissertation is to develop a meshless model that describes the solidification and macrosegregation phenomena during the direct chill casting (DCC) of aluminium alloys under the influence of a lowfrequency electromagnetic field. Macrosegregation is an undesired consequence of alloy solidification. It represents one of the major casting defects and substantially reduces the quality of the finished product. On the other hand, lowfrequency electromagnetic casting (LFEC) is a process that promises to increase greatly the product quality, including the reduction of macrosegregation. The modelling of both processes is of tremendous importance to the metallurgical industry, due to the high costs of experiments during production.
The volumeaveraging formulation is used for the modelling of the solidliquid interaction. The conservation equations for mass, energy, momentum, and species are used to model the solidification of aluminiumalloy billets in axysimmetry. The electromagneticinduction equation is coupled with the melt flow. It is used to calculate the magnetic vector potential and the Lorentz force. The Lorentz force is timeaveraged and included in the momentumconservation equation, which intensifies the melt flow. The effect of Joule heating is neglected in the energy conservation due to its insignificant contribution. The semicontinuous casting process is modelled with the Eulerian approach. This implies that the global computational domain is fixed in space. The inflow of the liquid melt is assumed at the top boundary and the outflow of the solid metal is assumed at the bottom. It is assumed that the whole mushy area is a rigid porous media, which is modelled with the Darcy law. The KozenyCarman relation is used for the permeability definition. The incompressible mass conservation is ensured by the pressure correction, which is performed with the fractional step method. The conservation equations and the induction equation are posed in the cylindrical coordinate system. A linearised eutectic binary phase diagram is used to predict the solute redistribution in the solid and liquid phases. The micro model uses the lever rule to determine the temperature and the liquid fraction field from the transport equations.
The partial differential equations are solved with the meshlessdiffuseapproximate method (DAM). The DAM uses weighted least squares to determine a locally smooth approximation from a discrete set of data. The secondorder polynomials are used as the trial functions, while the Gaussian function is used as the weight function. The method is localised by defining a smooth approximation for each computational node separately. This is performed by associating each node with a unique local neighbourhood, which is used for the minimisation. There are 14 nodes included in the local subdomains for the DCC and LFEC simulations. The stability of the advective term is achieved with a shift of the Gaussian weight in the upwind direction. This approach is called the adaptive upwind weight function and is used in the DAM for the first time. The ExplicitEuler scheme is used for temporal discretisation.
The use of a meshless method and the automatic nodearrangement generation makes it possible to investigate the complicated flow structures, which are formed in geometrically complex inflow conditions in a straightforward way. A realistic inflow geometry and mould can therefore be included in the model. The number of computational nodes is increased in the mushy zone and decreased in the solid phase, due to the optimisation of the computational time and memory. The computational node arrangement is automatically adapted with time, as the position of the mushy zone is changed in shape and position. Found in: ključnih besedah Summary of found: ...casting, macrosegregation, electromagnetic stirring, aluminium alloys, meshless methods, diffuseapproximate method, multiphysics model, solidification... Keywords: lowfrequency electromagnetic casting, direct chill casting, macrosegregation, electromagnetic stirring, aluminium alloys, meshless methods, diffuseapproximate method, multiphysics model, solidification Published: 25.04.2019; Views: 1635; Downloads: 87 Fulltext (28,80 MB) 
3. Numerical modelling of dendritic solidification based on phase field formulation and adaptive meshless solution procedureTadej Dobravec, 2021, doctoral dissertation 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 electrochemical properties of the solidified material. The numerical modelling of dendritic solidification can be applied for an indepth 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 multicomponent 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 solidliquid interface over the characteristic interface thickness. A phase field model yields a system of coupled nonlinear parabolic partial differential equations that govern the evolution of the phase field and other thermodynamic variables.
The meshless radial basis functiongenerated finitedifferences (RBFFD) method is used for the spatial discretisation of the system of partial differential equations. The forward Euler scheme is applied for the temporal discretisation. Fifthdegree polyharmonic splines are used as the shape functions in the RBFFD method. A secondorder accurate RBFFD method is achieved by augmenting the shape functions with monomials up to the second degree.
The adaptive solution procedure is developed in order to speedup the calculations. The procedure is based on the quadtree domain decomposition of a rectangular computational domain into rectangular computational subdomains of different sizes. Each quadtree subdomain has its own regular or scattered distribution of computational nodes in which the RBFFD 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 solidliquid interface and the lowestpossible density in the bulk of the solid and liquid phases. The adaptive timestepping is employed to further speedup 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 subdomains with different node densities.
The main originality of the present work is the use of the RBFFD method for the thorough analysis of the impact of the type of the node distribution and the size of a local subdomain 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 meshinduced anisotropy effects, present when the partial differential equations are discretisied on a regular node distribution. The main advantage of the RBFFD method for the phase field modelling of dendritic growth is the simple discretisation of the partial differential equations on the scattered node distributions. The RBFFD method is, for the first time, used in combination with the spatialtemporal adaptive solution procedure based on the quadtree domain decomposition. The adaptive solution procedure successfully speedsup 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. Found in: ključnih besedah Summary of found: ...dendritic solidification, phase field method, meshless methods, RBFFD, adaptive solution procedure... Keywords: dendritic solidification, phase field method, meshless methods, RBFFD, adaptive solution procedure Published: 07.04.2021; Views: 647; Downloads: 32 Fulltext (0,00 KB) This document has many files! More...
