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The purpose of this paper is to carry out a finite element simulation of a physically non-linear phase change problem in a two-dimensional space without adaptive remeshing or moving-mesh algorithms. The extended finite element method (XFEM) and the level set method (LSM) were used to capture the transient solution and motion of phase boundaries. It was crucial to consider the effects of unequal densities of the solid and liquid phases and the flow in the liquid region.

The XFEM and the LSM are applied to solve non-linear transient problems with a phase change in a two-dimensional space. The model assumes thermo-dependent properties of the material and unequal densities of the phases; it also allows for convection in the liquid phase. A non-linear system of equations is derived and a numerical solution is proposed. The Newton-Raphson method is used to solve the problem and the LSM is applied to track the interface.

The robustness and utility of the method are demonstrated on several two-dimensional benchmark problems.

The novel procedure based on the XFEM and the LSM was developed to solve physically non-linear phase change problems with unequal densities of phases in a two-dimensional space.

Of particular interest is the ability of the extended finite element method (XFEM) to capture transient solution and motion of phase boundaries without adaptive remeshing or moving-mesh algorithms for a physically nonlinear phase change problem. The paper aims to discuss this issue.

The XFEM is applied to solve nonlinear transient problems with a phase change. Thermal conductivity and volumetric heat capacity are assumed to be dependent on temperature. The nonlinearities in the governing equations make it necessary to employ an effective iterative approach to solve the problem. The Newton-Raphson method is used and the incremental discrete XFEM equations are derived.

The robustness and utility of the method are demonstrated on several one-dimensional benchmark problems.

The novel procedure based on the XFEM is developed to solve physically nonlinear phase change problems.

A method has been developed for the solution of inverse heat diffusion problems to find the initial condition, boundary condition, and the source and sink function in the heat diffusion equation. The method has been used in the development of a source‐and‐sink method to find the boundary conditions in inverse Stefan problems. Green's functions have been used in the solution, and the problems are solved by using two approaches: a series solution approach, and a time incremental approach. Both can be used to find the boundary conditions without reliance on the flux information to be supplied at both sides of the interface. The methods are efficient in that they require less equations to be solved for the conditions. The numerical results have shown to be accurate, convergent, and stable. Most of all, the results do not degrade with time as in other time marching schemes reported in the literature. Algorithms can also be easily developed for the solution of the conditions.

Presents a new method for the numerical simulation of diffusion with phase‐change. The method is able to handle hysteresis and finite‐rate kinetics in the phase‐change reaction. Such phenomena are frequent in solid‐solid phase transitions. The model problem discussed concerns hydrogen migration and hydride precipitation in zirconium and its alloys, a problem of interest to the nuclear industry. With respect to previous ones, our method is the first to incorporate an implicit treatment of diffusion, thus avoiding mesh‐dependent stability limits in the time step. The CPU time can in this way be reduced by a factor of 10–20 in applications. Addresses, through numerical studies, convergence with respect to mesh refinement and reduction of the time step. Also reports on an application of the method to the simulation of laboratory experiments. Shows that the method is a powerful tool to deal with general phase‐change problems, extendable to other physical systems.

This paper gives a review of the finite element techniques (FE) applied in the area of material processing. The latest trends in metal forming, non‐metal forming, powder metallurgy and composite material processing are briefly discussed. The range of applications of finite elements on these subjects is extremely wide and cannot be presented in a single paper; therefore the aim of the paper is to give FE researchers/users only an encyclopaedic view of the different possibilities that exist today in the various fields mentioned above. An appendix included at the end of the paper presents a bibliography on finite element applications in material processing for 1994‐1996, where 1,370 references are listed. This bibliography is an updating of the paper written by Brannberg and Mackerle which has been published in Engineering Computations, Vol. 11 No. 5, 1994, pp. 413‐55.

A two‐dimensional, macroscopic, stationary, finite element model is presented for both laser remelting and laser cladding of material surfaces. It considers, in addition to the heat transfer, the important fluid motion in the melt pool and the deformation of the liquid—gas interface. The velocity field in the melt is driven by thermocapillary forces for laser remelting, but also by forces due to powder injection for laser cladding. For a given velocity field within the liquid region, the stationary enthalpy (or Stefan) equation is solved. An efficient scheme allows the LU decomposition of the finite element matrix to be performed only once at the first iteration. Then, the velocity is updated using the Q₁—P₀ element with penalty methods for treating both the incompressibility condition and the slip boundary conditions. Numerical results for three different processing speeds for both laser remelting and laser cladding demonstrate the efficiency and robustness of the numerical approach. The influence of the thermocapillary and powder injection forces on the fluid motion and subsequently on the melt pool shape is seen to be important. This kind of calculations is thus necessary in order to predict with precision the temperature gradients across the solidification interface, which are essential data for microstructure calculations.

The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method.

The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem.

The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat.

The paper presents a new solution algorithm for the Stefan problem.

Since Bram Stoker’s tale of Count Dracula struck a chord with a sensation-hungry public, vampires have remained a popular part of horror in cinema. Since the turn of the millennium, vampires have now become a mainstay of horror TV. Programmes like True Blood (2008–2014) and The Vampire Diaries (2009–2017) have propelled the vampire into the home.

This chapter will investigate the problematic, but often sympathetic relationship between vampires and humans in The Vampire Diaries.