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The purpose of this paper is to investigate the ways to diminish or eliminate numerical diffusion and dispersion. Numerical dispersion and diffusion are present in the predicted macrosegregation profiles reported in the literature and they hinder the interpretation of the simulation results. With the motivation to eliminate these numerical problems by employing appropriate meshes, simulations of macrosegregation in a billet direct‐chill cast from a multi‐component aluminium alloy has been performed.

First the idea that numerical dispersion could be alleviated by refining the structured mesh size is tested and the extent of this mesh refining to overcome these numerical problems is discussed. Second the link of numerical dispersion and diffusion to the type of mesh used is investigated.

Unstructured mesh eliminates the numerical dispersion present in the structured mesh while it introduces the numerical diffusion. It is concluded by performing calculations with the same settings but different meshes that, although refining the structured mesh could alleviate the numerical oscillation, it increases the computation time dramatically. Therefore the best solution to overcome these numerical problems is the employment of a hybrid mesh consisting of both structured and unstructured mesh.

This work reveals the reasons behind the numerical dispersion and diffusion in macrosegregation modelling and gives a practical solution.

The axisymmetric steady‐state convective‐diffusive thermal field problem associated with direct‐chill, semi‐continuously cast billets has been solved using the dual reciprocity boundary element method. The solution is based on a formulation which incorporates the one‐phase physical model, Laplace equation fundamental solution weighting, and scaled augmented thin plate splines for transforming the domain integrals into a finite series of boundary integrals. Realistic non‐linear boundary conditions and temperature variation of all material properties are included. The solution is verified by comparison with the results of the classical finite volume method. Results for a 0.500[m] diameter Al 4.5 per cent Cu alloy billet at typical casting conditions are given.

A rigorous Finite Element (FE) formulation based on an enthalpy technique is developed for solving coupled nonlinear heat conduction/mass diffusion problems with phase change. The FE formulation consists of a fully coupled heat conduction and solute diffusion formulation, with solid‐liquid phase change, where the effects of pressure and convection are neglected. A full enthalpy method is employed eliminating singularities which result from abrupt changes in heat capacity at the phase interfaces. The FE formulation is based on the fixed grid technique where the elements are two dimensional, four noded quadrilaterals with the primary variables being enthalpy and average solute concentration. Temperature and solid mass fraction are calculated on a local level at each integration point of an element. A fully consistent Newton‐Raphson method is used to solve the global coupled equations and an Euler backward difference scheme is used for the temporal discretization. The solution of the enthalpy‐temperature relationship is carried out at the integration points using a Newton‐Raphson method. A secant method employing the regula falsi technique takes into account sudden jumps or sharp changes in the enthalpy‐temperature behaviour which occur at the phase zone interfaces. The Euler backward difference integration rule is used to calculate the solid mass fraction and its derivatives. A practical example is analysed and results are presented.

The purpose of this paper is to reveal the solute segregation behavior in the molten and solidified regions during direct chill (DC) casting of a large-size magnesium alloy slab under no magnetic field (NMF), harmonic magnetic field (HMF), pulsed magnetic field (PMF) and two types of out-of-phase pulsed magnetic field (OPMF).

A 3-D multiphysical coupling mathematical model is used to evaluate the corresponding physical fields. The coupling issue is addressed using the method of separating step and result inheritance.

The results suggest that the solute deficiency tends to occur in the central part, while the solute-enriched area appears near the fillet in the molten and solidified regions. Applying magnetic field could greatly homogenize the solute field in the two-phase region. The variance of relative segregation level in the solidified cross-section under NMF is 38.1%, while it is 21.9%, 18.6%, 16.4% and 12.4% under OPMF2 (the current phase in the upper coil is ahead of the lower coil), HMF, PMF and OPMF1 (the current phase in the upper coil lags behind the lower coil), respectively, indicating that OPMF1 is more effective to reduce the macrosegregation level.

There are few reports on the solute segregation degree in rectangle slab under magnetic field, especially for magnesium alloy slab. This paper can act a reference to make clear the solute transport behavior and help reduce the macrosegregation level during DC casting.

This paper aims to point out the critical problems in numerical verification of solidification simulation codes and the complexity of the verification and to propose and apply a procedure of generalized verification for macrosegregation simulation.

A partial verification of a finite‐volume computational model of macrosegregation in direct chill (DC) casting of binary aluminum alloys, including the coupled transport phenomena of heat transfer, fluid flow and species transport, is performed. The verification procedure is conducted on numerical test problems, defined as subproblems with respect to the complexity of the physical model, geometry, and boundary conditions. The studied cases are thermal convection with solidification in DC casting, thermal natural convection of a low‐Prandtl‐number liquid metal in a rectangular cavity and 1D directional solidification of a binary Al‐Cu alloy. Grid‐convergence studies, code comparison with an alternative Chebyshev‐collocation method, and comparison with a reference similarity solution are used for verification.

An excellent ability of the model to accurately resolve the thermal convection in the pertinent range of Prandtl and Rayleigh numbers is shown. Concerns regarding the solution of species transport in the mushy zone remain.

The proposed verification procedure is not completed in its entirety. Further verification of the solutal and thermosolutal convection problems is required.

This paper proposes verification techniques for complex coupled solidification problems involving significant convection in the melt.

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.

In the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.

Numerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.

Accurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.

The present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.

The paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

The purpose of this paper is to study the effect of feeding scheme on melt flow and temperature field during the steady-state of level-pour direct-chill (DC) casting of A390 alloy hollow billet and optimize the design of feeding scheme.

Melt flow and temperature field are investigated by numerical simulation, which is based on a three-dimensional mathematical model and well verified by experiments.

The numerical results reveal that both melt flow and temperature field are obviously affected by the feeding scheme. The homogeneity of melt flow and temperature field in hollow billet with the feeding scheme of modified four inlets are better than the other feeding schemes. Experimental results show that crack can be eliminated by increasing the number of feeding inlets. The primary Si size appears unaffected while the distribution of primary Si particles is highly affected by the change of feeding scheme. Only with the feeding scheme of modified four inlets can fine and uniformly distributed primary Si particles be achieved.

The paper includes implications for the design of feeding scheme in level-pour DC casting of hollow billet for practical use.

This paper develops different feeding schemes for level-pour DC casting of hollow billet and optimizes the design of feeding scheme.

The purpose of this paper is to investigate the effect of pulsed magnetic field (PMF) with different duty cycles on the melt flow and heat transfer behaviors during direct-chill (DC) casting of large-size magnesium alloy billet and find the appropriate range of duty cycle.

A transient two-dimensional mathematical model coupled electromagnetic field, flow field and thermal field, is conducted to study the melt flow and temperature field under PMF and compared with that under the harmonic magnetic field.

The results reveal that melt vibration and fluctuation are generated due to the instantaneous impact of repeated thrust and pull effects of Lorentz force under PMF. The peak of Lorentz force decreases greatly with the increasing duty cycle, but the melt fluctuation region is expanded with higher duty cycle, which accelerates the interior melt velocity and reduces the temperature gradient at the liquid-solid interface. However, PMF with overly high duty cycle has adverse effect on the melt convection and limited influence on the interior melt. A duty cycle of 20% to 50% is a reasonable range.

This paper can provide guiding significance for the setting of duty cycle parameters on DC casting under PMF.

There are few reports on the effect of PMF parameters during DC casting with applying PMF, especially for duty cycle, a parameter unique to PMF. The findings will be helpful for applying the external field of PMF on DC casting.

The purpose of this investigation is to understand the structure of trapped intermetallics particles and localized composition changes in the anodized anodic oxide film on AA1050 aluminium substrates.

The morphology and composition of Fe‐containing intermetallic particles incorporated into the anodic oxide films on industrially pure aluminium (AA1050, 99.5 per cent) has been investigated. AA1050 aluminium was anodized in a 100 ml/l sulphuric acid bath with an applied voltage of 14 V at 20°C ±2°C for 10 or 120 min. The anodic film subsequently was analyzed using focused ion beam‐scanning electron microscopy (FIB‐SEM), SEM, and EDX.

The intermetallic particles in the substrate material consisted of Fe or both Fe and Si with two different structures: irregular and round shaped. FIB‐SEM cross‐sectioned images revealed that the irregular‐shaped particles were embedded in the anodic oxide film as a thin strip structure and located near the top surface of the film, whereas the round‐shaped particles were trapped in the film with a spherical structure, but partially dissolved and were located throughout the thickness of the anodic film. The Fe/Si ratio of the intermetallic particles decreased after anodizing.

This paper shows that dual beam FIB‐SEM seems to be an easy, less time consuming and useful method to characterize the cross‐sectioned intermetallic particles incorporated in anodic film on aluminium.

The purpose of this paper is to develop a new local radial basis function collocation method (LRBFCM) for one‐domain solving of the non‐linear convection‐diffusion equation, as it appears in mixture continuum formulation of the energy transport in solid‐liquid phase change systems.

The method is structured on multiquadrics radial basis functions. The collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes in the domain of influence have to be solved for each node. The method does not require polygonisation (meshing). The solution is found only on a set of nodes.

The computational effort grows roughly linearly with the number of the nodes. Results are compared with the existing steady analytical solutions for one‐dimensional convective‐diffusive problem with and without phase change. Regular and randomly displaced node arrangements have been employed. The solution is compared with the results of the classical finite volume method. Excellent agreement with analytical solution and reference numerical method has been found.

A realistic two‐dimensional non‐linear industrial test associated with direct‐chill, continuously cast aluminium alloy slab is presented.

A new meshless method is presented which is simple, efficient, accurate, and applicable in industrial convective‐diffusive solid‐liquid phase‐change problems with non‐linear material properties.