Numerical Modeling of Casting Solidification Using Generalized Finite Difference Method

Abstract

The system casting-mould is considered. The thermal processes proceeding in a casting sub-domain are described using the one domain approach. The model of solidification process is supplemented by the energy equation concerning the mould sub-domain, the continuity conditions given on the contact surface between casting and mould, boundary conditions on the outer surface of the system and the initial ones. To solve the problem the generalized variant of finite difference method (GFDM) is used. Temporary and local values of temperature can be found at the optional set of collocation points from the domain considered. This essential advantage of GFDM allows to locate and thicken nodes at the regions for which the temperature gradients and cooling (heating) rates are considerable. In the final part of the paper, the example of numerical simulation is shown.

Numerical Modeling of Solidification Process

“…In the case of solidification problem the basic BEM algorithm for linear Fourier's equation should be coupled with procedures correcting the temporary solutions for successive levels of time. These procedures are called the temperature field correction method (Mochnacki & Suchy, 1995;Mochnacki & Szopa, 2008), the alternating phase truncation method (Mochnacki & Suchy, 1995;Majchrzak & Mochnacki, 1995;Majchrzak & Mendakiewicz, 1993) and the artificial heat source method (Mochnacki & Majchrzak, 2010;Mochnacki & Szopa, 1998). The basic idea of the BEM can be realized in different ways.…”

Modelling of transient heat transport in metal films using the interval lattice Boltzmann method

“…Generally, this can be achieved with a lattice spacing shorter than the phonon mean free path. The equations (22) and (23) should be supplemented by the boundary conditions [28], for example of the 2 nd type on the left boundary and the 1 st type on the right boundary as follows T is the phonon's initial temperature. Taking into account equations (30,31), the set of four interval partial differential equations is obtained …”

“…In this arithmetic a set of proper intervals is extended by improper intervals, and all arithmetic operations and functions are also extended. The main advantage of the directed interval arithmetic [20,22] upon the usual interval arithmetic [23] is that the obtained temperature intervals are much narrower and their width does not increase in time.In theory as well as in practice it is valuable to develop the interval version of the LBM. …”

Numerical modelling of the transient heat transport in a thin gold film using the fuzzy lattice Boltzmann method with α-cuts

“…It seems more natural to take into account uncertainties related, for example, to the material parameters. Here, the fuzzy lattice Bolzmann method is presented using the α-cuts of fuzzy numbers with the approach of the directed interval arithmetic [14]. The interval values of relaxation times and boundary conditions are taken into account.…”