The purpose of this paper is to simulate a macrosegregation solidification benchmark by a meshless diffuse approximate method. The benchmark represents solidification of Al 4.5 wt per cent Cu alloy in a 2D rectangular cavity, cooled at vertical boundaries. A coupled set of mass, momentum, energy and species equations for columnar solidification is considered. The phase fractions are determined from the lever solidification rule. The meshless diffuse approximate method is structured by weighted least squares method with the second-order monomials for trial functions and Gaussian weight functions. The spatial localization is made by overlapping 13-point subdomains. The time-stepping is performed in an explicit way. The pressure-velocity coupling is performed by the fractional step method. The convection stability is achieved by upstream displacement of the weight function and the evaluation point of the convective operators. The results show a very good agreement with the classical finite volume method and the meshless local radial basis function collocation method. The simulations are performed on uniform and non-uniform node arrangements and it is shown that the effect of non-uniformity of the node distribution on the final segregation pattern is almost negligible. The application of the meshless diffuse approximate method to simulation of macrosegregation is performed for the first time. An adaptive upwind scheme is successfully applied to the diffuse approximate method for the first time.

COBISS.SI-ID: 1386922

A comprehensive, multiphysics, meshless, numerical model is developed for the simulation of direct chill casting under the influence of a low-frequency electromagnetic field. The model uses mixture-continuum-mass, momentum and energy-conservation equations to simulate the solidification of axisymmetric aluminium-alloy billets. The electromagnetic-induction equation is coupled with the fluid flow and used to calculate the Lorentz force. The involved partial-differential equations are solved with the meshless-diffuse-approximate method by employing second-order polynomial shape functions and a 13-noded local support. An explicit time-stepping scheme is used. The boundary conditions for the heat transfer involve the effects of hot-top, mould chill and direct chill. The use of a meshless method and the automatic node-arrangement generation made it possible to investigate the complicated flow structures in geometrically complex inflow conditions, including sharp and curved edges, in a straightforward way. A time-dependent adaptive mesh is used to decrease the calculation time. The model is demonstrated by casting an Al-5.25wt%Cu aluminium alloy billet with a radius of 120 mm. Results on simplified and realistic inflow geometries are considered and compared. The effect of the low-frequency electromagnetic force on the temperature, liquid fraction and fluid flow are investigated under different current densities and frequencies.

COBISS.SI-ID: 15664923

A two-dimensional model to simulate the dendritic and eutectic growth in binary alloys is developed. A cellular automata method is adopted to track the movement of the solid-liquid interface. The diffusion equation is solved in the solid and liquid phases by using an explicit finite volume method. The computational domain is divided into square cells that can be hierarchically refined or coarsened using an adaptive mesh based on the quadtree algorithm. Such a mesh refines the regions of the domain near the solid-liquid interface, where the highest concentration gradients are observed. In the regions where the lowest concentration gradients are observed the cells are coarsened. The originality of the work is in the novel, adaptive approach to the efficient and accurate solution of the posed multiscale problem. The model is verified and assessed by comparison with the analytical results of the Lipton-Glicksman-Kurz model for the steady growth of a dendrite tip and the Jackson-Hunt model for regular eutectic growth. Several examples of typical microstructures are simulated and the features of the method as well as further developments are discussed.

COBISS.SI-ID: 15618075

Purpose: In this study we upgrade our previous developments of the Local Radial Basis Function Collocation Method (LRBFCM) for heat transfer, fluid flow, electromagnetic problems and linear thermoelasticity to dynamic coupled thermoelasticity problems. Design/methodology/approach: We solve a thermoelastic benchmark by considering a linear thermoelastic plate under thermal and pressure shock. Spatial discretization is performed by a local collocation with multiquadrics augmented by monomials. The implicit Euler formula is used to perform the time stepping. The system of equations obtained from the formula is solved using a Newton-Raphson algorithm with GMRES to iteratively obtain the solution. The LRBFCM solution is compared with the reference FEM solution and, in one case, with a solution obtained using the meshless local Petrov-Galerkin method. Findings: The performance of the LRBFCM is found to be comparable to the FEM, with some differences near the tip of the shock front. The LRBFCM appears to converge to the mesh-converged solution more smoothly than the FEM. Also, the LRBFCM seems to perform better than the MLPG in the studied case. Originality: For the first time, the LRBFCM has been applied to problems of coupled thermoelasticity. Research Limitations: The performance of the LRBFCM near the tip of the shock front appears to be suboptimal, since it does not capture the shock front as well as the FEM. With the exception of a solution obtained using the meshless local Petrov-Galerkin method, there is no other high-quality reference solution for the considered problem in the literature yet. In most cases, therefore, we are able to compare only two mesh-converged solutions obtained by the authors using two different discretization methods. The shock-capturing capabilities of the method should be studied in more detail.

COBISS.SI-ID: 1331626

This paper represents a continuation of numerical results regarding the recently proposed industrial benchmark tests, obtained by a meshless method. A part of the benchmark test, involving turbulent fluid flow with solidification in two dimensions, a preliminary macrosegregation upgrade and a first three dimensional test were recently presented by authors of this paper. Previous tests were bound to calculations in mold and sub-mold regions only. In the present paper, reference calculations in two dimensions are presented for the entire strand. The physical model is established on a set of macroscopic equations for mass, energy, momentum, species, turbulent kinetic energy, and dissipation rate. The mixture continuum model is used to treat the solidification system. The mushy zone is modeled as a Darcy porous media with Kozeny-Karman permeability relation, where the morphology of the porous media is modeled by a constant value. The incompressible turbulent flow of the molten steel is described by the Low-Reynolds-Number k-ε turbulence model, closed by the Abe-Kondoh-Nagano closure coefficients and damping functions. Lever microsegregation model is used. The numerical method is established on explicit time-stepping, collocation with scaled multiquadrics radial basis functions with adaptive selection of its shape on non-uniform five-nodded influence domains. The velocity-pressure coupling of the incompressible flow is resolved by the explicit Chorin's fractional step method. The advantages of the method are its simplicity and efficiency, since no polygonisation is involved, easy adaptation of the nodal points in areas with high gradients, almost the same formulation in two and three dimensions, high accuracy and low numerical diffusion.

COBISS.SI-ID: 3925499