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

A thermally driven mushy cell tracking algorithm for phase-change problems with a moving boundary is presented. The equation used to track the moving boundary is based on energy balance over the mushy cell and is applied to advance a moving front in a cell-by-cell manner. The efficacy of the tracking algorithm is demonstrated on specific problems solved using the finite volume method. An implicit scheme is adopted to ensure that the numerical solution is unconditionally stable in time. A preconditioned conjugated gradient (P-CG) solver is implemented to ensure that solutions converge in a finite number of iterations. Four benchmark cases are used to validate the algorithm including solidification in one dimensional space (two-region problem), melting of pure aluminum in two-dimensional (2D) space, solidification with periodic boundary conditions, and solidification of one-region problem. The results obtained show that the current algorithm is capable of converging to accurate solutions for moving fronts and the numerical predications are in excellent agreement with corresponding analytical solutions.     

NUMERICAL SOLUTION OF TRANSIENT,FREE SURFACE PROBLEMS IN POROUS MEDIA

The focus of this paper is the development of numerical schemes for tracking the moving fluid surface during the filling of a porous medium (e.g., polymer injection into a porous mold cavity). Performing a mass balance calculation on an arbitrarily deforming control volume, leads to a general governing filling equation. From this equation, a general, fully time implicit, numerical scheme based on a finite volume space discretization is derived. Two numerical schemes are developed: (1) a fully deforming grid scheme, which explicitly tracks the location of the filling front, and (2) a fixed grid scheme, that employs an auxiliary variable to locate the front. The validity of the two schemes is demonstrated by solving a variety of one- and two-dimensional problems; both approaches provide predictions with similar accuracy and agree well with available analytical solutions.     

The boundary element‐free method for elastoplastic implicit analysis

A meshless procedure, based on boundary integral equations, is proposed to analyze elastoplastic problems. To cope with non‐linear problems, the usual boundary element method introduces domain discretization cells, often considered a ‘drawback’ of the method. Here, to get rid of the standard element and cell, i.e. boundary and domain discretization, the orthogonal moving least squares (also known as improved moving least squares) method is used. The algorithm adopted to solve these particular inelastic non‐linear problems is a well‐established, criterion‐independent implicit procedure, previously developed by the authors. Comparative results are presented at the end to illustrate the effectiveness of the proposed techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

An implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

In this work, a new comprehensive method has been developed which enables the solution of large, non‐linear motions of rigid bodies in a fluid with a free surface. The application of the modern Eulerian–Lagrangian approach has been translated into an implicit time‐integration formulation, a development which enables the use of larger time steps (where accuracy requirements allow it). Novel features of this project include: (1) an implicit formulation of the rigid‐body motion in a fluid with a free surface valid for both two or three dimensions and several moving bodies; (2) a complete formulation and solution of the initial conditions; (3) a fully consistent (exact) linearization for free surface flows valid for any boundary elements such that optimal convergence properties are obtained when using a Newton–Raphson solver. The proposed framework has been completed with details on implementation issues referring mainly to the computation of the complete initial conditions and the consistent linearization of the formulation for free surface flows. The second part of the paper demonstrates the mathematical and numerical formulation through numerical results simulating large free surface flows and non‐linear fluid structure interaction. The implicit formulation using a fully consistent linearization based on the boundary element method and the generalized trapezoidal rule has been applied to the solution of free surface flows for the evolution of a triangular wave, the generation of tsunamis and the propagation of a wave up to overturning. Fluid–structure interaction examples include the free and forced motion of a circular cylinder and the sway, heave and roll motion of a U‐shaped body in a tank with a flap wave generator. The presented examples demonstrate the applicability and performance of the implicit scheme with consistent linearization. Copyright © 2001 John Wiley & Sons. Ltd.     

A fixed grid method for the numerical solution of phase change problems

A fixed grid method using an updated iterative implicit scheme is developed to solve one-dimensional phase change problems. The temperature field is deduced from the resolution of the governing equations whose discretization takes into account the discontinuous variation of the temperature derivative at the phase change front. At each iteration an updated position of the moving front is found from the resolution of the energy conservation at the solid-liquid interface. The accuracy of the proposed numerical method has been checked on three test problems.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

Scattering of electromagnetic pulse from a moving and vibrating perfect plane using characteristic‐based algorithm

Abstract

One‐dimensional numerical simulation results of plane Gaussian electromagnetic pulses reflected from constantly moving and vibrating perfect conductors are provided in this paper. The computational data were obtained using the characteristic‐based algorithm with the aid of relativistic boundary conditions and characteristic variable boundary conditions. Since the perfect conductor can travel and vibrate simultaneously, the size of the grid cell immediately next to the boundary and the corresponding numerical time‐step are time‐dependent. The present numerical method has been shown to accurately handle such problems. In this paper both the reflected electric field intensities and the corresponding spectra are illustrated. The calculated Doppler shifts are found to have good agreement with the theoretical values.     

Simulation of micro flows with moving boundaries using high-order upwind FV method on unstructured grids

 Numerical methods are presented for the simulation of steady and unsteady micro gas flows with moving boundaries found in micro scale fluidic devices. Both steady and unsteady flows are calculated by using an implicit real-time discretization and a dual-time stepping scheme implemented in a high-order upwind finite-volume unstructured-grid Navier–Stokes solver. For moving boundary problems, a new dynamic mesh method has been developed which is shown to be robust in handling large mesh deformation. Micro-scale flows studied with the methods developed include flow in micro channels, unsteady flow around a micro cylinder in oscillation and transport processes in micro pumps. The simulation is based on the continuum fluid model (the compressible Navier–Stokes equations) with slip boundary conditions implemented in the context of unstructured grids as the micro flows studied are all in the slip flow regime. Results are presented to validate the methods and demonstrate their applications to the analysis and design of micro fluidic devices. The implicit dual-time stepping scheme is found to be robust and efficient in dealing with both steady and unsteady micro flows. The unstructured-grid solver proves to be very flexible in dealing with complex geometries such as micro pumps. This is the first known report on the use of finite-volume unstructured grid solver for studying micro flows based on the slip boundary condition with moving boundaries.     

On the numerical solution of phase change problems in transient non-linear heat conduction

A finite difference solution for the transient non-linear heat conduction with phase change in a finite slab is proposed. A two-time level implicit method is used while Taylor's forward projection method is employed for taking into account the non-linearities. The stability due to the boundary conditions at the moving front is verified. The numerical solution is compared with the analytical solutions and found to be in close agreement.     

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。     

A fixed-grid numerical modelling of transient liquid phase bonding and other diffusion-controlled phase changes

A transient liquid phase (TLP), in which a liquid layer is formed and subsequently solidifies, and other diffusion-controlled phase changes are generally associated with moving phase-change interfaces. Both fixed and variable grid discretization models have been formulated to investigate these diffusion-controlled problems. However, all numerical efforts to date have employed one of the approaches explicitly to track the moving interfaces across which there exist step changes in concentrations. In this article, the fixed-grid source-based method originally developed to simulate the temperature fields for melting-solidification phase change processes has been adopted to simulate diffusion-controlled dissolution and solidification. This method solves a unique diffusion equation for the different phases and the moving interfaces using implicit time integration. Compared with previously developed models, it is not only simpler in numerical formulation and procedure, but also more convenient to extend to many phases and high-dimensional problems. We report here the detailed formulation of the relevant equations, and compare and validate the model using experimental data and previous modelling predictions for several systems available from the existing literature.     

The H-gradient method for magnetostatic field computations

A new method for approximating magnetostatic field problems is given in this paper. The new method approximates the scalar potential for the magnetic intensity and is based on a volume integral formulation. The corresponding algorithm is similar to that obtained from coupled differential and boundary integral approaches. Convergence results in computations are compared with results for the usual volume integral method used in GFUN3D.     

Optical manipulation of multiple stable states in photorefractive four-wave mixing

Abstract

We propose an approach to manipulate the convergence in multiple solutions of phase conjugate reflectivity in photorefractive four-wave mixing. Although a method forcibly adding a π-phase shift to an incident beam has been already proposed to control the reflectivity, some restrictions have been required in the boundary conditions for the successful operation. Here, we control the reflectivity with the boundary conditions in which the phase shift operation is ineffective by itself. In our method, the phase shift operation is combined with the procedure of turning an incident beam on and off. With a numerical analysis of four-wave mixing, we show that our new approach brings drastic change in the spatial distribution of the index grating and leads the phase conjugate reflectivity which was not manipulated previously.     

A posteriori error estimator and an adaptive technique in meshless finite points method

In this work, an adaptive technique for application of meshless methods in one- and two-dimensional boundary value problems is described. The proposed method is based on the use of implicit functions for the geometry definition, fixed weighted least squares approximation and an error estimation by means of simple formulas and a robust strategy of refinement based on the own nature of the approximation sub-domains utilised. With all these aspects, the proposed method becomes an attractive alternative for the adaptive solutions to partial differential equations in all scopes of engineering. Numerical results obtained from the computational implementation show the efficiency of the present method.     

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.     

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.     

Quantitative metallography of sigma phase precipitates in AISI 347 stainless steel – a comparison between different methods

Abstract

A reliable method to perform volume fraction measurements of sigma (σ) phase in a niobium stabilised steel (AISI 347) has been developed. The most accurate results of the tested methods were obtained using backscattered electrons in a scanning electron microscope (SEM) and samples etched with oxalic acid. Both optical microscopy (OM) and SEM either on polished samples or on etched samples have been evaluated to come to this conclusion. Several etchants were also tested and careful etching with oxalic acid gave a well defined rim. The measurement of σ-phase fraction has been performed using manual point counting and digital image analysis using manual threshold. It was concluded that image analysis is usually to be preferred since it is faster and also results in higher precision The phase boundary caused by etching was evaluated, and it was found that the boundary area should be included in the measurement when using the recommended SEM method.     

Heat transfer during fluidized-bed coating

A fully implicit numerical method for linear parabolic free boundary problems with coupled and integral boundary conditions is described. The partial differential equation and the boundary conditions are time discretized with the method of lines. An auxiliary function is introduced to remove the coupled and integral boundary conditions from the resulting free boundary problem for ordinary differential equations. Once separated boundary conditions are obtained, invariant imbedding is used to solve the free boundary problem numerically. The method is illustrated by solving the heat transfer equations for the fluidized-bed coating of a thin-walled cylinder.     

A wall boundary treatment using analytical volume integrations in a particle method

Numerical treatment of complicated wall geometry has been one of the most important challenges in particle methods for computational fluid dynamics. In this study, a novel wall boundary treatment using analytical volume integrations has been developed for two-dimensional (2D) incompressible flow simulations with the moving particle semi-implicit method. In our approach, wall geometry is represented by a set of line segments in 2D space. Thus, arbitrary-shaped boundaries can easily be handled without auxiliary boundary particles. The wall's contributions to the spatial derivatives as well as the particle number density are formulated based on volume integrations over the solid domain. These volume integrations are analytically solved. Therefore, it does not entail an expensive calculation cost nor compromise accuracy. Numerical simulations have been carried out for several test cases including the plane Poiseuille flow, a hydrostatic pressure problem with complicated shape, a high viscous flow driven by a rotating screw, a free-surface flow driven by a rotating cylinder and a dam break in a tank with a wedge. The results obtained using the proposed method agreed well with analytical solutions, experimental observations or calculation results obtained using finite volume method (FVM), which confirms that the proposed wall boundary treatment is accurate and robust.