A cell-by-cell thermally driven mushy cell tracking algorithm for phase-change problems

The finite volume based numerical approach is used to simulate phase-change processes including natural convection. This approach is based on a cell-by-cell, thermally driven mushy cell tracking equation, developed in Part I [20], to trace the front at which phase-change occurs. A mushy cell is a specialized cell where the interface between liquid and solid phases is located. In this paper, the mushy cell tracking equation and the associated boundary condition around the mushy cells are derived in a general manner and shown to have the same form as that used in Part I. The SIMPLE algorithm is adopted to solve the flow, including pressure field, as well, in the liquid phase and a conjugate gradient method is used when solving the system of discretized equations. To reduce computational time, an acceleration technique, based on a justified quasi-steady state assumption, is adopted. The proposed numerical method is applied to simulate the solidification and melting of Tin with natural convection. The numerical predictions are compared well with the available experimental data and previously published numerical results. Specifically, these comparisons demonstrate that the proposed methodology is capable of predicating the location of moving fronts and the temperature distributions for phase-change processes with natural convection.     

An approximate method for evaporation problem with mushy zone

In the present paper, a simple mushy zone model is used to track the moving boundaries in an evaporation problem in which the vapor is removed upon formation. Two main parameters for the mushy zone model are analyzed as well as their effect on the movement of the moving boundaries and the thickness of the mushy zone. A new approximate method is developed for analysis and tracking the moving boundaries appears throughout the process. The proposed method mainly based on applying the boundary integral equation corresponding to each phase in such a way that the associated boundary and initial conditions as well as energy equations at the moving boundaries achieved with minimum error and low number of iterations. The results of the present paper seem to be good because there are neither analytical or numerical solutions available.     

An explicit unconditionally stable variable time-step method for one-dimensional stefan problems

Modelling of heat conduction processes with phase changes benefits from the application of variable time-step methods when the behaviour of the moving boundary is not known a prioiri. Due to convergence and stability constraints only implicit difference equations have been used with these methods. Implicit methods show a significant loss of accuracy and exhibit convergence difficulties when used for relatively slow or rapid moving-boundary problems. To overcome these problems an improved explicit variable time-step method which combines the explicit exponential difference equation and a variable time-step grid network with virtual subspace increments around the moving boundary is presented and tested for both a solidification and a melting problem. A virtual subinterval time-step elimination technique is incorporated to ensure that stability is automatically maintained for any mesh size. Unlike the implicit variable time-step methods, the accuracy of the resulting method is not affected by the velocity of the moving boundary. For both test problems numerical results are in better agreement with known analytical solutions than results predicted by other numerical methods.     

Numerical modelling of chemical effects of magma solidification problems in porous rocks

The solidification of intruded magma in porous rocks can result in the following two consequences: (1) the heat release due to the solidification of the interface between the rock and intruded magma and (2) the mass release of the volatile fluids in the region where the intruded magma is solidified into the rock. Traditionally, the intruded magma solidification problem is treated as a moving interface (i.e. the solidification interface between the rock and intruded magma) problem to consider these consequences in conventional numerical methods. This paper presents an alternative new approach to simulate thermal and chemical consequences/effects of magma intrusion in geological systems, which are composed of porous rocks. In the proposed new approach and algorithm, the original magma solidification problem with a moving boundary between the rock and intruded magma is transformed into a new problem without the moving boundary but with the proposed mass source and physically equivalent heat source. The major advantage in using the proposed equivalent algorithm is that a fixed mesh of finite elements with a variable integration time‐step can be employed to simulate the consequences and effects of the intruded magma solidification using the conventional finite element method. The correctness and usefulness of the proposed equivalent algorithm have been demonstrated by a benchmark magma solidification problem. Copyright © 2005 John Wiley & Sons, Ltd.     

An adjoint method for the inverse design of solidification processes with natural convection

This paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes. In particular, the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved. This is the first time that an infinite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse/design problems with moving boundaries and Boussinesq incompressible flow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspecified boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidification design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as finding the appropriate spatial-temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed. © 1998 John Wiley & Sons, Ltd.     

An analysis of solidification problems by the boundary element method

The problem of interest in this paper is the calculation of the motion of the solid–liquid interface and the time-dependent temperature field during solidification of a pure metal. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time-dependent Green's functions and convolution integrals. The BEM approach requires discretization of only the surface of the solidifying body. Thus, the numerical method closely follows the physics of the problems and is intuitively very appealing. The formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems. Emphasis is given to two-dimensional problems. Comparison with existing semi-analytical solutions and other numerical solutions from the literature reveals that the method is fast, accurate and without major time step limitations.     

Modelling convection in solidification processes using stabilized finite element techniques

Solidification of dendritic alloys is modelled using stabilized finite element techniques to study convection and macrosegregation driven by buoyancy and shrinkage. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. A single domain model is considered with a fixed numerical grid and without boundary conditions applied explicitly on the freezing front. The mushy zone is modelled here as a porous medium with either an isotropic or an anisotropic permeability. The stabilized finite‐element scheme, previously developed by authors for modelling flows with phase change, is extended here to include effects of shrinkage, density changes and anisotropic permeability during solidification. The fluid flow scheme developed includes streamline‐upwind/Petrov–Galerkin (SUPG), pressure stabilizing/Petrov–Galerkin, Darcy stabilizing/Petrov–Galerkin and other stabilizing terms arising from changes in density in the mushy zone. For the energy and species equations a classical SUPG‐based finite element method is employed with minor modifications. The developed algorithms are first tested for a reference problem involving solidification of lead–tin alloy where the mushy zone is characterized by an isotropic permeability. Convergence studies are performed to validate the simulation results. Solidification of the same alloy in the absence of shrinkage is studied to observe differences in macrosegregation. Vertical solidification of a lead–tin alloy, where the mushy zone is characterized by an anisotropic permeability, is then simulated. The main aim here is to study convection and demonstrate formation of freckles and channels due to macrosegregation. The ability of stabilized finite element methods to model a wide variety of solidification problems with varying underlying phenomena in two and three dimensions is demonstrated through these examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Continuously deforming finite elements for the solution of parabolic problems,with and without phase change

A number of transport problems are complicated by the presence of physically important transition zones where quantities exhibit steep gradients and special numerical care is required. When the location of such a transition zone changes as the solution evolves through time, use of a deforming numerical mesh is appropriate in order to preserve the proper numerical features both within the transition zone and at its boundaries. A general finite element solution method is described wherein the elements are allowed to deform continuously, and the effects of this deformation are accounted for exactly. The method is based on the Galerkin approximation in space, and uses finite difference approximations for the time derivatives. In the absence of element deformation, the method reduces to the conventional Galerkin formulation. The method is applied to the two-phase Stefan problem associated with the melting and solidification of A substance. The interface between the solid and liquid phase form an internal moving boundary, and latent heat effects are accounted for in the associated boundary condition. By allowing continuous mesh deformation, as dictated by this boundary condition, the moving boundary always lies on element boundaries. This circumvents the difficulties inherent in interpolation of parameters and dependent variables across regions where those quantities change abruptly. Basis functions based on Hermite polynomials are used, to allow exact specification of the flux-latent heat balance condition at the phase boundary. Analytic solutions for special cases provide tests of the method.     

Control of Unsteady Solidification Via Optimized Magnetic Fields

This article presents a numerical procedure for automatically controlling desired features of a melt undergoing solidification by applying an external magnetic field whose time-varying intensity and spatial distribution are obtained by the use of a hybrid optimization algorithm. The intensities of the magnets along the boundaries of the container were discretized by using B-splines. The inverse problem is then formulated to find the magnetic boundary conditions (the coefficients of the B-splines) in such a way that the gradients of temperature along the gravity direction are minimized at each instant as the solidification front advances through a moving melt. For this task, a hybrid optimization code was used that automatically switches among the following six optimization modules; the Davidon-Fletcher-Powell (DFP) gradient method, a genetic algorithm (GA), the Nelder-Mead (NM) simplex method, quasi-Newton algorithm of Pshenichny-Danilin (LM), differential evolution (DE), and sequential quadratic programming (SQP). Transient Navier-Stokes and Maxwell's equations were discretized by using a finite volume method in a generalized curvilinear nonorthogonal coordinate system. For the phase change problems, an enthalpy formulation was used. The computer code was validated against analytical and numerical benchmark results with very good agreements in both cases.     

时域径向积分边界元法在平面单相凝固问题中的应用

该文将时域精细积分边界元方法与界面追踪法相结合,给出平面单相凝固热传导问题的一个有效数值分析方法。首先,利用稳态热传导问题的基本解和径向积分法给出瞬态传热问题的边界积分方程,并采用精细积分方法求解离散的微分方程组,获得相变界面的热流密度。然后应用相变界面上的能量守恒方程,采用界面追踪法来预测相变边界的移动位置,从而给出相关问题数值模拟的结果。最后,为验证该文方法的有效性,给出两个数值算例并与解析解进行了对比。结果表明,该文方法具有较高的求解精度,是求解相变热传导问题的一种有效数值方法。     

Boundary elements for half‐space problems via fundamental solutions: A three‐dimensional analysis

An efficient solution technique is proposed for the three‐dimensional boundary element modelling of half‐space problems. The proposed technique uses alternative fundamental solutions of the half‐space (Mindlin's solutions for isotropic case) and full‐space (Kelvin's solutions) problems. Three‐dimensional infinite boundary elements are frequently employed when the stresses at the internal points are required to be evaluated. In contrast to the published works, the strongly singular line integrals are avoided in the proposed solution technique, while the discretization of infinite elements is independent of the finite boundary elements. This algorithm also leads to a better numerical accuracy while the computational time is reduced. Illustrative numerical examples for typical isotropic and transversely isotropichalf‐space problems demonstrate the potential applications of the proposed formulations. Incidentally, the results of the illustrative examples also provide a parametric study for the imperfect contact problem. Copyright © 2001 John Wiley & Sons, Ltd.     

Modified enthalpy method for the simulation of melting and solidification

Enthalpy method is commonly used in the simulation of melting and solidification owing to its ease of implementation. It however has a few shortcomings. When it is used to simulate melting/solidification on a coarse grid, the temperature time history of a point close to the interface shows waviness. While simulating melting with natural convection, in order to impose no-slip and impermeability boundary conditions, momentum sink terms are used with some arbitrary constants called mushy zone constants. The values of these are very large and have no physical basis. Further, the chosen values affect the predictions and hence have to be tuned for satisfactory comparison with experimental data. To overcome these deficiencies, a new cell splitting method under the framework of the enthalpy method has been proposed. This method does not produce waviness nor requires mushy zone constants for simulating melting with natural convection. The method is then demonstrated for a simple one-dimensional melting problem and the results are compared with analytical solutions. The method is then demonstrated to work in two-dimensions and comparisons are shown with analytical solutions for problems with planar and curvilinear interfaces. To further benchmark the present method, simulations are performed for melting in a rectangular cavity with natural convection in the liquid melt. The solid–liquid interface obtained is compared satisfactorily with the experimental results available in literature.     

A meshless method for solving 1D time-dependent heat source problem

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) and the heat polynomials is proposed for recovering a time-dependent heat source and the boundary data simultaneously in an inverse heat conduction problem (IHCP). We will transform the problem into a homogeneous IHCP and initial value problems for the first-order ordinary differential equation. An improved method of MFS is used to solve the IHCP and a finite difference method is applied for solving the initial value problems. The advantage of applying the proposed meshless numerical scheme is producing the shape functions which provide the important delta function property to ensure that the essential conditions are fulfilled. Numerical experiments for some examples are provided to show the effectiveness of the proposed algorithm.     

Hybrid fundamental-solution-based FEM for piezoelectric materials

In this paper, a new type of hybrid finite element method (FEM), hybrid fundamental-solution-based FEM (HFS-FEM), is developed for analyzing plane piezoelectric problems by employing fundamental solutions (Green’s functions) as internal interpolation functions. A modified variational functional used in the proposed model is first constructed, and then the assumed intra-element displacement fields satisfying a priori the governing equations of the problem are constructed by using a linear combination of fundamental solutions at a number of source points located outside the element domain. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The proposed methodology is assessed by several examples with different boundary conditions and is also used to investigate the phenomenon of stress concentration in infinite piezoelectric medium containing a hole under remote loading. The numerical results show that the proposed algorithm has good performance in numerical accuracy and mesh distortion insensitivity compared with analytical solutions and those from ABAQUS. In addition, some new insights on the stress concentration have been clarified and presented in the paper.     

Free Boundary Determination in Nonlinear Diffusion

Free boundary problems with nonlinear diffusion occur in various applications, such as solidification over a mould with dissimilar nonlinear thermal properties and saturated or unsaturated absorption in the soil beneath a pond. In this article, we consider a novel inverse problem where a free boundary is determined from the mass/energy specification in a well-posed one-dimensional nonlinear diffusion problem, and a stability estimate is established. The problem is recast as a nonlinear least-squares minimisation problem, which is solved numerically using the lsqnonlin routine from the MATLAB toolbox. Accurate and stable numerical solutions are achieved. For noisy data, instability is manifest in the derivative of the moving free surface, but not in the free surface itself nor in the concentration or temperature.     

Two-dimensional solidification in a cylindrical mold with imperfect mold contact

A two-dimensional axisymmetric problem of solidification of a superheated liquid in a long cylindrical mold has been studied in this paper by employing a new embedding technique. The mold and the melt has an imperfect contact and the heat transfer coefficient has been taken as a function of space and time. Short-time exact analytical solutions for the moving boundary and temperature distributions in the liquid, solid and mold have been obtained. The numerical results indicate that with the present solution, for some parameter values, substantial solidified thickness can be obtained. The method of solution is simple and straightforward, and consists of assuming fictitious initial temperatures for some suitable fictitious extensions of the actual regions. Sufficient conditions for the commencement of the solidification have been discussed.     

An operator-splitting algorithm for two-dimensional convection–dispersion–reaction problems

An operator-splitting algorithm for the two-dimensional convection–dispersion–reaction equation is developed. The flow domain is discretized into triangular elements which are fixed in time. The governing equation is split into three successive initial value problems: a pure convection problem, a pure dispersion problem and a pure reaction problem. For the pure convection problem, solutions are found by the method of characteristics. The solution algorithm involves tracing the characteristic lines backwards in time from a vertex of an element to an interior point. A cubic polynomial is used to interpolate the concentration and its derivatives on an element. For the pure dispersion problem, an explicit finite element algorithm is employed. Analytical solutions are obtained for the pure reaction problem. The treatment of the boundary conditions is also discussed. Several numerical examples are presented. Numerical results agree well with analytical solutions. Because cubic polynomials are used in the interpolation, very little numerical damping and oscillation are introduced, even for the pure convection problem.     

Investigations on boundary temperature control analysis considering a moving body based on the adjoint variable and the fictitious domain finite element methods

This paper presents an examination of moving‐boundary temperature control problems. With a moving‐boundary problem, a finite‐element mesh is generated at each time step to express the position of the boundary. On the other hand, if an overlapped domain, that is, comprising foreground and background meshes, is prepared, the moving boundary problem can be solved without mesh generation at each time step by using the fictitious domain method. In this study, boundary temperature control problems with a moving boundary are formulated using the finite element, the adjoint variable, and the fictitious domain methods, and several numerical experiments are carried out. Copyright © 2015 John Wiley & Sons, Ltd.     

Efficient numerical approach to unbounded systems subjected to a moving load

The present paper solves numerically the problem of vibrations of infinite structures under a moving load. A velocity formulation of the space–time finite element method was applied. In the case of simplex shaped space–time finite elements, the ‘steady state’ dynamic behaviour of the system was obtained. A properly performed discretization allowed of propagating information in a given direction at a limited velocity. The solutions were obtained under the assumption that the deformation is quasi-stationary, i.e., stationary in the coordinate system that moves with the load. The unbounded Timoshenko beam subjected to a distributed moving load was used as a test example. The dynamical system is placed on an elastic foundation. The matrices describing an infinite dynamical system subjected to a moving load are derived and the stability of the numerical scheme is analysed. The numerical results are compared with the analytical solutions in the literature and the classical numerical method.