The simulation of macrosegregation as a consequence of solidification of a binary Al-4.5% 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 sub-domains through collocation by using multiquad-rics 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 Meth-od (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 non-uniform node arrange-ments. The paper probably for the first time shows an application of a meshless method in such a highly non-linear and multi-physics problem.
COBISS.SI-ID: 1905659
In this paper, three kinds of explicit local meshless methods are compared: the local method of approximate particular solutions (LMAPS), the local direct radial basis function collocation method (LDRBFCM) which are both first presented in this paper, and the local indirect radial basis function collocation method (LIRBFCM). In all three methods, the time discretization is performed in explicit way, the multiquadric radial basis functions (RBFs) are used to interpolate either initial temperature field and its derivatives or the Laplacian of the initial temperature field. The five-noded sub-domains are used in localization. Numerical results of simple diffusion equation with Dirichlet jump boundary condition are compared on uniform and random node arrangement, the accuracy and stabilities of these three local meshless methods are asserted. One can observe that the improvement of the accuracy with denser nodes and with smaller time steps for all three methods. All methods provide a similar accuracy in uniform node arrangement case. For random node arrangement, the LMAPS and the LDRBFCM perform better than the LIDRBFCM.
COBISS.SI-ID: 1541371
This paper introduces an effective H-adaptive upgrade to solution of the transport phenomena by the novel Local Radial Basis Function Collocation Method (LRBFCM). The transport variable is represented on overlapping 5-noded influence-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the variable are calculated from the respective derivatives of the RBFs. The transport equation is solved through explicit time stepping. The H-adaptive upgrade includes refinement/derefinement of one to four nodes to/from the vicinity of the reference node. The number of the nodes added or removed depends on the topology of the reference node vicinity. The refinement/derefinement is triggered by an error indicator, which very simply depends on the ratio between the norm of the collocation coefficients and collocation matrix. The refinement/derefinement is proportional with the growth/decay of this indicator. Such adaptivity much increases the accuracy/performance ratio of the method. The performance of the method is numerically tested on two-dimensional Burger’s equation. The results are compared with different numerical solutions, published in literature. Outstanding CPU efficiency and accuracy are clearly demonstrated from the results. The paper probably for the first time shows such a simple and effective H-adaptive meshless method, designed on five noded influence domain. The advantages of the represented meshless approach are its simplicity, accuracy, similar coding in 2D and 3D, straightforward applicability in non-uniform node arrangements, and native parallel implementation.
COBISS.SI-ID: 2177275