1.1 Introduction
Aluminum alloys are very interesting material widely used in industry. Modeling and computer simulation are one of the most effective methods of studying difficult problems in foundry and metallurgical manufacture. Numerical simulations are used for optimization of casting production. In many cases they are the only possible techniques for carrying out the experiments whose real statement is complicated. Computer modeling allows to define the major factors of a quality estimation of alloy castings. Simulations help to investigate interaction between solidifying casting and changes of its parameters or initial conditions. That process defines the quality of casting, and the problem of adequate modeling of foundry systems. The process mainly depends on the solution of heat equations [].
Increasing capacity of computer memory makes it possible to consider growing problem sizes. At the same time, increased precision of simulations triggers even greater load. There are several ways to tackle this kind of problems. For instance, one can use parallel computing [].
Solidification may take place at a constant temperature or in the temperature range []. If solidification occurs at a constant temperature, it is then referred to as the so-called Stefan problem or the solidification problem with zero temperatures range. Pure metals or alloys of certain specific chemical compositions (e.g. having an eutectic composition) solidify at a constant temperature. However, most of the metal alloys solidify in certain temperature ranges (so-called temperature intervals of solidification). The temperature at which the alloy starts to solidify is called liquidus temperature (
), and the temperature at which solidification ends is called solidus temperature (
). In the case of alloys with eutectic transformation, in which the solute concentration exceeds its maximum solubility in the solid phase, the temperature of the solidification end is the eutectic temperature. Analytical (rarely) and numerical (commonly) methods are used in the modeling of solidification process. The finite elements method (FEM) is the most commonly used numerical method, but finite difference method (FDM), boundary element method (BEM), the Monte-Carlo and other methods are also used.
The most important heat effect, occurring during solidification, is the emission of (latent) heat of solidification ( L ). It is also the most difficult phenomenon to numerical modeling. The basic division of numerical methods of solidification modeling process relates to modeling of the latent heat emission. These methods can be divided into front-tracking methods and fixed-grid methods. Fixed-grid methods are also divided into temperature formulations (the latent heat of solidification is considered as the temperature-dependent term of heat source) and enthalpy formulations (the latent heat of solidification is considered as the temperature-dependent term of heat capacity) [].
During the comparison, we have used Finite Element Method as numerical method of choice. The FEM was used in our own software, that is used to conduct simulations of solidification. This software is implemented in the C++ programming language with the use of essential libraries, i.e. PETSc (for linear algebra) [].
1.2 Description of the Entalphy Formulation
Solidification is described by a quasi-linear equation of heat conduction, considering a term of heat source
as a latent heat of solidification:
By entering the following designation:
equation () can be written as
where
denotes generalized heat source. By introducing enthalpy, defined as:
where
is the reference temperature, and calculating the derivative with respect to the temperature:
where
is the effective heat capacity. Assuming the heat source is equal to zero, the Eq. () can be converted to the form:
All above equations form the basis of the thermal description of solidification.
1.2.1 The Enthalpy and the Effective Heat Capacity
The enthalpy is the sum of explicit and latent heat []. For the metal solidifying in the temperature range (
) amounts to: