We have developed and discussed essentially improved method of fundamental solutions in which there is no need to use the artificial boundary for problems of potential flow. With this the main drawback of the method was eliminated. The method has been structured by the single layer and the double layer fundamental solution with similar accuracy. The developed method currently represents most simple and effective known solution of potential flow problems. (Remark: we have successfully upgraded the method in 2012 to the problems of linear elasticity)

COBISS.SI-ID: 1176059

The aim of this paper is simulation of thermally induced liquid-solid dendritic growth in two dimensions by a coupled deterministic continuum mechanics heat transfer model and a stochastic localized phase change kinetics model that takes into account the undercooling, curvature, kineticand thermodynamic anisotropy. The stochastic model receives temperature information from the deterministic model and the deterministic model receives the solid fraction information from the stochastic model. The heat transfer model is solved on a regular grid by the standard explicit Finite Difference Method (FDM). The phasechange kinetics model is solved by the classical Cellular Automata (CA) approach and a novel Point Automata (PA) approach. The PA method was developed and introduced in this paper to circumvent the mesh anisotropy problem, associated with the classical CA method. Dendritic structures are in the CA approach sensitive on the relative angle between the cell structure and the preferential crystal growth direction which is not physical. The CA approach is established on quadratic cells and Neumann neighborhood. The PA approach is established on randomly distributed points and neighbourhood configuration, similar as appears in meshless methods. Both methods provide same results in case of regular PA node arrangements and neighborhood configuration with five points. A comprehensive comparison between both stochastic approaches has been made with respect to curvature calculations, dendrites with different orientations of crystallographic angles and types of the node arrangements randomness. It has been shown that the new method can be used for calculation of the dendrites in any orientation.

COBISS.SI-ID: 1729275

Development of meshless local collocation method for solution of incompressible turbulent problems of mixed forced and natural convection problems is described in the present work. The advantages of the represented meshfree approach are its simplicity, accuracy, similar coding in 2 and 3 dimensions, and straightforward applicability in nonuniform node arrangements. The results have been verified by achieving reasonable agreement with the direct numerical simulation of Kasagi and Nishimura for Reynolds number 4494, based on the channel width, and Grashof number 9.6x10+5.

COBISS.SI-ID: 1781243

The simulation of macrosegregation as a consequence of solidification of a binary Al4.5% Cu alloy in a 2 dimensional rectangular enclosure is tackled in the present paper. Coupled volume averaged governing equations for mass, energy, momentum and species transfer are considered. The phase properties are resolved from the Lever solidification rule, the mushy zone is modeled by the Darcy law and the liquid phase is assumed to behave like an incompressible Newtonian fluid. Double diffusive effects in the melt are modeled by the thermal and solutal Boussinesq hypothesis. The physical model is solved by the novel Local Radial Basis Function Collocation Method (LRBFCM). The involved physical relevant fields are represented on overlapping 5 noded subdomains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBFs. The fields are solved through explicit time stepping. The pressure-velocity coupling is calculated through a local pressure correction scheme. The evolution of the solidification process is presented through temperature, velocity, liquid fraction and species concentration histories in four sampling points. The fully solidified state is analyzed through final macrosegregation map in three vertical and three horizontal cross-sections. The results are compared with the classical Finite Volume Method (FVM). A surprisingly good agreement of the numerical solution of both methods is shown and therefore the results can be used as a reference for future verification studies. The advantages of the represented meshless approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in nonuniform node arrangements. The paper probably for the first time shows an application of a meshless method in such a highly nonlinear and multiphysics problem.

COBISS.SI-ID: 1905659

Based on Haar wavelets an efficient numerical method is proposed for the numerical solution of system of coupled Ordinary Differential Equations (ODEs) related to the natural convection boundary layer fluid flow problems with high Prandtl number (Pr). The numerical study of these flow models is necessary as the existing literature is more focused on the flow problems with small values of Pr. In this work, the problem of natural convection which consists of coupled nonlinear ODEs is solved simultaneously. The ODEs are obtained from the Navier Stokes equations through the similarity transformations. The effects of variation of Pr on heat transfer are investigated. Performance of the Haar Wavelets Collocation Method (HWCM) is compared with the finite difference method (FDM), Runge–Kutta Method (RKM), homotopy analysis method (HAM) and exact solution for the last problem. More accurate solutions are obtained by wavelets decomposition in the form of a multi-resolution analysis of the function which represents solution of the given problems. Through this analysis the solution is found on the coarse grid points and then refined towards higher accuracy by increasing the level of the Haar wavelets. Neumann boundary conditions, which are problematic for most of the numerical methods, are automatically coped with. A distinctive feature of the proposed method is its simple applicability for a variety of boundary conditions. Efficiency analysis of HWCM versus RKM is performed using Timing command in Mathematica software. A brief convergence analysis of the proposed method is given. Numerical tests are performed to test the applicability, efficiency and accuracy of the method.

COBISS.SI-ID: 1740027