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The paper presents a method of numerical modeling of binary alloy solidification process in a permanent mold in a two-dimensional area. The model takes into account the influence of the gas gap of variable width existing between the mold and the casting on the rate of solidification process. The numerical model is based on the finite element method (FEM) with independent spatial discretization of the casting and the mold. The displacements of these regions resulting from a thermally dependent changes of their volumes are also taken into account. The adopted approach involves the use of two separated meshes, where the temporary temperature fields are obtained. The thermal interaction between the casting and the mold is described by the appropriate boundary condition. The solution is obtained sequentially in each time step independently for the each region.

The paper presents an approach of numerical modelling of alloy solidification in permanent mold and transient heat transport between the casting and the mold in two-dimensional space. The gap of time-dependent width called "air gap", filled with heat conducting gaseous medium is included in the model. The coefficient of thermal conductivity of the gas filling the space between the casting and the mold is small enough to introduce significant thermal resistance into the heat transport process. The mathematical model of heat transport is based on the partial differential equation of heat conduction written independently for the solidifying region and the mold. Appropriate solidification model based on the latent heat of solidification is also included in the mathematical description. These equations are supplemented by appropriate initial and boundary conditions. The formation process of air gap depends on the thermal deformations of the mold and the casting. The numerical model is based on the finite element method (FEM) with independent spatial discretization of interacting regions. It results in multi-mesh problem because the considered regions are disconnected.

During solidification process the phenomenon of shrinkage cavity is often observed. It has negative impact on the structure and physical properties of the final product. To minimize the negative effect of shrinkage cavity the riser is commonly introduced to the solidifying system. If the riser is properly designed shrinkage cavity rather appears within its volume than in the other parts of the casting. It makes possible to obtain defect-free main part of the casting. Presented work shows the impact of the different shapes of the riser on the localization of the shrinkage cavity. The results are obtained with the use of an original computer program based on the finite element method (FEM). They are presented for three-dimensional case of solidification of three types of the riser. The comparison of the results is discussed in details.

The numerical approach based on the finite element method (FEM) for modeling of mechanical interaction between three-dimensional objects is presented in the paper. The model of contact is based on the assumption that the nodes of the region which is the source of contact cannot overlap with the nodes of the region being the target. The procedure of the detection of collision between surfaces of the source and the target is discussed in details. The behaviour of surfaces being in contact depends on their rigidity and is numerically modeled in the case of perfectly rigid source and deformable target. Each modeled object has an independent mesh of finite elements. These meshes can be freely moved relative to each other. Example of calculation using original program written in C++ is presented and discussed.

The presented work is focused on the basis of mathematical and numerical descriptions of the binary alloy solidification problem. The mathematical formulation is based on the so-called substitute thermal capacity, which implies a change in the specific heat of solidifying material. In the literature one can find many ways to define this parame- ter. Five models, differing in the description of the substitute thermal capacity as well as the numerical model using the finite element method (FEM) are considered.

Mathematical and numerical models of Fe-C alloy solidification process based on the finite element method (FEM) are presented in the paper. The phenomenon of the defect called "shrinkage cavity" is introduced to the model. It is very important aspect of presented work which makes possible the prediction of the location and shape of mentioned defect depending on the geometry of the casting and cooling condition of the process. An original computer program using Visual C++ environment has been written in order to simulate described process. The results of computer calculations of the three-dimensional solidification problem of the casting with riser are presented and discussed in details.

This paper concerns the mathematical and numerical modeling of solidification process of pure metals. Such process takes place without development of “mushy zone”, which very often occurs during solidification of alloys. The model is based on the finite element method (FEM) and takes into account the existence of a sharp solidification front which moves according to time. The paper discusses how to include the conditions of continuity on the moving interface as well as the technique of front tracking based on the level set method (LSM). The paper also presents the results of computer simulations of the solidification process of pure copper. Two in-home solvers for one- and two-dimensional problems were built. Both of them are based on FEM. The correctness of the front tracking technique was checked by comparing the results obtained from 2D simulation to 1D simulation. Furthermore, the final calculation process was done for the two-dimensional area at different temperatures on the boundaries to demonstrate the effectiveness of the solver for a complex shape of the solidification front.

The method of numerical modeling of heat transfer between three-dimensional objects being in contact is described in the paper. Presented approach is based on the finite element method (FEM) with independent spatial discretization of considered regions. The gap between external surfaces of the interacting objects has variable width and is filled with gas or liquid. The medium in the gap introduces thermal resistance into heat transfer process. The mathematical model of considered problem is based on the equation of heat diffusion supplemented by the appropriate initial and boundary conditions. The deformations of the regions resulting from the mechanical contact and thermally dependent changes of their volumes are also included in the model. The results of numerical simulations are presented and discussed.