Boštjan Mavrič

Local radial basis function collocation method for linear thermoelasticity in two dimensions

Purpose – The purpose of this paper is to upgrade our previous developments of Local Radial Basis Function Collocation Method (LRBFCM) for heat transfer, fluid flow and electromagnetic problems to thermoelastic problems and to study its numerical performance with the aim to build a multiphysics meshless computing environment based on LRBFCM. Design/methodology/approach – Linear thermoelastic problems for homogenous isotropic body in two dimensions are considered. The stationary stress equilibrium equation is written in terms of deformation field. The domain and boundary can be discretized with arbitrary positioned nodes where the solution is sought. Each of the nodes has its influence domain, encompassing at least six neighboring nodes. The unknown displacement field is collocated on local influence domain nodes with shape functions that consist of a linear combination of multiquadric radial basis functions and monomials. The boundary conditions are analytically satisfied on the influence domains which co...

Simulation of a macrosegregation benchmark with a meshless diffuse approximate method

Purpose The purpose of this paper is to simulate a macrosegregation solidification benchmark by a meshless diffuse approximate method. The benchmark represents solidification of Al4.5wt%Cu alloy in a 2D rectangular cavity, cooled at vertical boundaries. Design/methodology/approach 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 thirteen 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. Findings The results show a very good agreement with...

A cellular automaton – finite volume method for the simulation of dendritic and eutectic growth in binary alloys using an adaptive mesh refinement

Abstract A two-dimensional model to simulate the dendritic and eutectic growth in binary alloys is developed. A cellular automaton 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.

Application of the RBF collocation method to transient coupled thermoelasticity

Purpose In this study, the authors aim to upgrade their 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 The authors solve a thermoelastic benchmark by considering a linear thermoelastic plate under thermal and pressure shock. Spatial discretization is performed by a local collocation with multi-quadrics 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 finite-element method (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. Research limitations/implications The performance of the LRBFCM near the tip of the shock front appears to be suboptimal because 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, the authors 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. Originality/value For the first time, the LRBFCM has been applied to problems of coupled thermoelasticity.

Modelling of stress fields during LFEM DC casting of aluminium billets by a meshless method

Direct Chill (DC) casting of aluminium alloys is a widely established technology for efficient production of aluminium billets and slabs. The procedure is being further improved by the application of Low Frequency Electromagnetic Field (LFEM) in the area of the mold. Novel LFEM DC processing technique affects many different phenomena which occur during solidification, one of them being the stresses and deformations present in the billet. These quantities can have a significant effect on the quality of the cast piece, since they impact porosity, hot-tearing and cold cracking. In this contribution a novel local radial basis function collocation method (LRBFCM) is successfully applied to the problem of stress field calculation during the stationary state of DC casting of aluminium alloys. The formulation of the method is presented in detail, followed by the presentation of the tackled physical problem. The model describes the deformations of linearly elastic, inhomogeneous isotropic solid with a given temperature field. The temperature profile is calculated using the in-house developed heat and mass transfer model. The effects of low frequency EM casting process parameters on the vertical, circumferential and radial stress and on the deformation of billet surface are presented. The application of the LFEM appears to decrease the amplitudes of the tensile stress occurring in the billet.

A multiphysics and multiscale model for low frequency electromagnetic direct-chill casting

Simulation and control of macrosegregation, deformation and grain size in low frequency electromagnetic (EM) direct-chill casting (LFEMC) is important for downstream processing. Respectively, a multiphysics and multiscale model is developed for solution of Lorentz force, temperature, velocity, concentration, deformation and grain structure of LFEMC processed aluminum alloys, with focus on axisymmetric billets. The mixture equations with lever rule, linearized phase diagram, and stationary thermoelastic solid phase are assumed, together with EM induction equation for the field imposed by the coil. Explicit diffuse approximate meshless solution procedure [1] is used for solving the EM field, and the explicit local radial basis function collocation method [2] is used for solving the coupled transport phenomena and thermomechanics fields. Pressure-velocity coupling is performed by the fractional step method [3]. The point automata method with modified KGT model is used to estimate the grain structure [4] in a post-processing mode. Thermal, mechanical, EM and grain structure outcomes of the model are demonstrated. A systematic study of the complicated influences of the process parameters can be investigated by the model, including intensity and frequency of the electromagnetic field. The meshless solution framework, with the implemented simplest physical models, will be further extended by including more sophisticated microsegregation and grain structure models, as well as a more realistic solid and solid-liquid phase rheology.

A meshless approach to thermomechanics of DC casting of aluminium billets

The ability to model thermomechanics in DC casting is important due to the technological challenges caused by physical phenomena such as different ingot distortions, cracking, hot tearing and residual stress. Many thermomechanical models already exist and usually take into account three contributions: elastic, thermal expansion, and viscoplastic to model the mushy zone. These models are, in a vast majority, solved by the finite element method. In the present work the elastic model that accounts for linear thermal expansion is considered. The method used for solving the model is of a novel meshless type and extends our previous meshless attempts in solving fluid mechanics problems. The solution to the problem is constructed using collocation on the overlapping subdomains, which are composed of computational nodes. Multiquadric radial basis functions, augmented by monomials, are used for the displacement interpolation. The interpolation is constructed in such a manner that it readily satisfies the boundary conditions. The discretization results in construction of a global square sparse matrix representing the system of linear equations for the displacement field. The developed method has many advantages. The system of equations can be easily constructed and efficiently solved. There is no need to perform expensive meshing of the domain and the formulation of the method is similar in two and three dimensions. Since no meshing is required, the nodes can easily be added or removed, which allows for efficient adaption of the node arrangement density. The order of convergence, estimated through an analytically solvable test, can be adjusted through the number of interpolation nodes in the subdomain, with 6 nodes being enough for the second order convergence. Simulations of axisymmetric mechanical problems, associated with low frequency electromagnetic DC casting are presented.