A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification

A fully-implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2D is presented. In addition we combine a second-order fully-implicit time discretisation scheme with variable steps size control to obtain an adaptive time and space discretisation method. The superiority of this method, compared to widely used fully-explicit methods, with respect to CPU time and accuracy, is shown. Due to the high non-linearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step. We use a nonlinear multigrid solver which shows almost h-independent convergence behaviour

Advanced numerical methods for the simulation of alloy solidification with high Lewis number

A fully-implicit numerical method based upon adaptively refined meshes for the thermal-solutal simulation of alloy solidification in 2D is presented. In addition we combine an unconditional stable second-order fully-implicit time discretisation scheme with variable step size control to obtain an adaptive time and space discretisation method, where a robust and fast multigrid solver for systems of non-linear algebraic equations is used to solve the intermediate approximations per time step. For the isothermal case, the superiority of this method, compared to widely used fully-explicit methods, with respect to CPU time and accuracy, has been demonstrated and published previously. Here, the new proposed method has been applied to the thermalsolutal case with high Lewis number, where stability issues and time step restrictions have been major constraints in previous research

An adaptive, fully implicit multigrid phase-field model for the quantitative simulation of non-isothermal binary alloy solidification

Using state-of-the-art numerical techniques, such as mesh adaptivity, implicit time-stepping and a non-linear multi-grid solver, the phase-field equations for the non-isothermal solidification of a dilute binary alloy have been solved. Using the quantitative, thin-interface formulation of the problem we have found that at high Lewis number a minimum in the dendrite tip radius is predicted with increasing undercooling, as predicted by marginal stability theory. Over the dimensionless undercooling range 0.2–0.8 the radius selection parameter, σ*, was observed to vary by over a factor of 2 and in a non-monotonic fashion, despite the anisotropy strength being constant

Towards a Physically Consistent Phase-Field Model for Alloy Solidification

We give an overview of contributions made to the computational phase-field modelling of alloy solidification from the University of Leeds as part of the LiME project (EPSRC Advanced Manufacturing Hub in Liquid Metal Engineering). The broader look at the more salient features from our research allows the individual contributions to be seen in a wider context than can be seen from each contribution separately. We begin with a general introduction to phase-field and then reference the numerical issues that arise from the solution of the model before outlining contributions to phase-field modelling that we found most interesting or significant. These range from controlling and developing interface-width independent modelling; controlling morphology in both single and multiphase settings; generalising from single to multiphase models; and creating a thermodynamically consistent framework for modelling entropy flow and thereby postulating a temperature field consistent with the concepts of, and applicable in, multiphase and density-dependent settings

Numerično modeliranje dendritskega strjevanja na podlagi formulacije faznega polja in prilagodljivega brezmrežnega rešitvenega postopka

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 nodeshence, 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 calculationshowever, the advantages of the use of the scattered node distribution are partly compromised due to the impact of regularity in the quadtree domain decomposition.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

Anisotropy and natural convection during solidification and melting

The effects of anisotropy of thermal conductivity and natural convection on solidification have been studied numerically. A fixed grid enthalpy-based formulation was developed to model convection and anisotropic conduction during solidification of pure materials and alloys in a rectangular cavity. The time dependent governing equations, describing the conservation of mass, momentum, energy and concentration were solved using a vorticity-stream function formulation. A finite difference-finite volume method was employed, incorporating an improved discretization method and a modified Samarskii-Andreyev ADI scheme with internal iterations. The interface was tracked with the use of an interfacial energy equation. A monotonic second-order upwind scheme (MSOU) was used for convective fluxes with central differences for the diffusion terms of concentration. Comparisons between the present calculations, analytical solutions, existing experimental results and other numerical methods are very good. The improved discretisation method is shown to have an excellent performance as it can solve the discontinuity of temperature, velocity, vorticity and stream function across the solid-liquid interface. Effects of anisotropic conduction on the temperature distribution through a gallium crystal are examined. The results show that anisotropy distorts the isotherms, especially at the adiabatic boundaries, and also decreases the overall heat transfer at the isothermal walls. Effects of aspect ratio, Stefan number, liquid superheat and boundary conditions and anisotropy during solidification are investigated. A study of solidification from either the side wall or the top wall of a cavity containing pure gallium show that natural convection has a significant effect on rate of solidification and the shape of the solid-liquid interface. The results, covering a range of values of Rayleigh number, aspect ratio and anisotropy characteristics, show how anisotropy affects the growth morphology and the flow structure. The effects of liquid aspect ratio on oscillatory convective flow during solidification are studied and compared with those for pure natural convection. Solidification from the side wall of a cavity containing a gallium-0.5% wt indium alloy was considered. The results show that anisotropy distorts the interface shape, and hence the interface shape has an effect on solute redistribution and flow patterns. The code was also used for natural convection driven melting problems of pure gallium where the interface shape is more irregular than in solidification problems. A correlation of the melting rate is given in terms of non-dimensional time, Rayleigh number, Stefan number and aspect ratio

Advanced operator splitting based semi-implicit spectral method to solve the binary and single component phase-field crystal model

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We present extensive testing in order to find the optimum balance among errors associated with time integration, spatial discretization, and splitting for a fully spectral semi implicit scheme of the phase field crystal model. The scheme solves numerically the equations of dissipative dynamics of the binary phase field crystal model proposed by Elder et al. [Elder et al, 2007]. The fully spectral semi implicit scheme uses the operator splitting method in order to decompose the complex equations in the phase field crystal model into sub-problems that can be solved more efficiently. Using the combination of non-trivial splitting with the spectral approach, the scheme leads to a set of algebraic equations of diagonal matrix form and thus easier to solve. Using this method developed by the BCAST research team we are able to show that it speeds up the computations by orders of magnitude relative to the conventional explicit finite difference scheme, while the costs of the pointwise implicit solution per timestep remains low. Comparing both the finite difference scheme used by Elder et al [Elder et al, 2007] to the spectral semi implicit scheme, we are also able to show that the finite differencing cannot compete with the spectral differencing in regards to accuracy. This is mainly due to numerical dissipation in finite differencing. In addition the results show that this method can efficiently be parallelized for distributed memory systems, where an excellent scalability with the number of CPUs. We have applied the semi-implicit spectral scheme for binary alloys to explore polycrystalline dendritic solidification. The kinetics of transformation has been analysed in terms of Johnson-Mehl-Avrami-Kolmogorov formalism. We show that Avrami plots are not linear, and the respective Avrami-Kolmogorov exponents (PAK) vary with the transformed fraction (or time). Using the semi-implicit spectral scheme we have been able to provide extensive numerical testing of methods in solving the single component case. This has been demonstrated by using unconditional time stepping with comparable simulations using conditional time stepping. We show the accuracy of the solution for unconditional time stepping is not compromised and furthermore computational efficiency can be significantly increased with the introduction of this scheme. Finally we have investigated how the composition of the initial liquid phase influences the eutectic morphology evolving during solidification. This is the first study that addresses this question using the dynamical density functional theory.EPSR

Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification

The single-domain, Darcy-Brinkman model is applied to some analytically tractable flows through adjacent porous and pure-fluid domains and is compared systematically with the multiple-domain, Stokes-Darcy model. In particular, we focus on the interaction between flow and solidification within the mushy layer during binary alloy solidification in a corner flow and on the effects of the chosen mathematical description on the resulting macrosegregation patterns. Large-scale results provided by the multiple-domain formulation depend strongly on the microscopic interfacial conditions. No satisfactory agreement between the single- and multiple-domain approaches is obtained when using previously suggested conditions written directly at the interface between the liquid and the porous medium. Rather, we define a viscous transition zone inside the porous domain, where Stokes equation still applies, and we impose continuity of pressure and velocities across it. This new condition provides good agreement between the two formulations of solidification problems when there is a continuous variation of porosity across the interface between a partially solidified region (mushy zone) and the melt

Multiscale modelling of twin roll casting

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel UniversityTwin roll casting (TRC) is an emerging material processing technique to manufacture thin metal strips directly from liquid metal. Formation of unfavourable microstructural features during TRC, such as centreline segregation and columnar grains, prompted experimental and modelling studies on the effects of various casting conditions on the as-cast microstructures of different alloys. Previous modelling work focuses on low speed TRC with large rolling effect. By increasing nuclei density in the melt, via addition of grain refiners or melt conditioning, the effect of solidification can be significantly improved. This thesis concerns the development and application of multiscale multiphysics modelling techniques to provide an insight into the evolution of microstructure during solidification processing of metals, with a focus on twin-roll casting of thin strips of light alloys. The effects of various casting conditions on the as-cast microstructures are investigated using the multiscale model. The first part of the multiscale model is a macroscale thermal-mechanical model, which predicts the temperature and stress distributions developed in the metal strips during TRC, under the influence of various casting conditions, including casting speed, roll temperature and air convection. The theoretical maximum casting speed is calculated from a quasi-1D solidification model, which can be used to give a guideline for casting conditions used in models and experiments. The second part is a microscale model which predicts the as-cast microstructure, via a phase field-Potts model which simulates the evolution of phase field, alloy concentration and grain orientation during solidification, coupled with a Bingham lattice Boltzmann model which simulates the velocity field in fluid flow. The temperature profile obtained from the macroscale model is used as the thermal boundary condition of the microscale model. To validate the model, temperature data obtained from experiments of Al and Mg TRC is compared with the results of the macroscale model. More experimental data of the as-cast microstructure and texture is required to validate the microscale model

Multiscale modelling of the influence of convection on dendrite formation and freckle initiation during vacuum arc remelting

Vacuum Arc Remelting (VAR) is employed to produce homogeneous ingots with a controlled, fine, microstructure. It is applied to reactive and segregation prone alloys where convection can influence microstructure and defect formation. In this study, a microscopic solidification model was extended to incorporate both forced and natural convection. The Navier-Stokes equations were solved for liquid and mushy zones using a modified projection method. The energy conservation and solute diffusion equations were solved via a combined stochastic nucleation approach along with a finite difference solution to simulate dendritic growth. This microscopic model was coupled to a 3D transient VAR model which was developed by using a multi-physics modelling software package, PHYSICA. The multiscale model enables simulations covering the range from dendrites (in microns) to the complete process (in meters). These numerical models were used to investigate: (i) the formation of dendritic microstructures under natural and forced convections; (ii) initiation of solute channels (freckles) in directional solidification in terms of interdendritic thermosolutal convection; and (iii) the macroscopic physical dynamics in VAR and their influence on freckle formation. 2D and 3D dendritic microstructure were simulated by taking into account both solutal and thermal diffusion for both constrained and unconstrained growth using the solidification model. For unconstrained equiaxed dendritic growth, forced convection was found to enhance dendritic growth in the upstream region while retarding downstream growth. In terms of dimensionality, dendritic growth in 3D is faster than 2D and convection promotes the coarsening of perpendicular arms and side branching in 3D. For constrained columnar dendritic growth, downward interdendritic convection is stopped by primary dendritic arms in 2D; this was not the case in 3D. Consequently, 3D simulations must be used when studying thermosolutal convection during solidification, since 2D simulations lead to inappropriate results. The microscopic model was also used to study the initiation of freckles for Pb-Sn alloys, predicting solute channel formation during directional solidification at a microstructural level for the first time. These simulations show that the local remelting due to high solute concentrations and continuous upward convection of segregated liquid result in the formation of sustained open solute channels. High initial Sn compositions, low casting speeds and low temperature gradients, all promote the initiation of these solute channels and hence freckles. to study the initiation of freckles for Pb-Sn alloys, predicting solute channel formation during directional solidification at a microstructural level for the first time. These simulations show that the local remelting due to high solute concentrations and continuous upward convection of segregated liquid result in the formation of sustained open solute channels. High initial Sn compositions, low casting speeds and low temperature gradients, all promote the initiation of these solute channels and hence freckles