Computer model of unidirectional solidification of single crystals of high temperature alloys

Abstract

I t has been common practice to use mould withdrawal unidirectional solidification to produce single crystal castings. To grow single crystals successfully, it is important to control several solidification parameters, such as the morphology of the solidification front (solid/liquid interface), thermal gradient, and growth rate during solidification. It is the aim of this study to develop a solidification model that can predict such solidification parameters for various design and operating conditions. The solidification phenomena in the process modelled are basically controlled by two heat transfer mechanisms: conduction and radiation. A set of heat transfer equations and boundary conditions were employed to describe mathematically the heat transfer phenomena. Then the finite difference method was used numerically to solve these equations for specified boundary conditions to obtain the temperature distribution and temperature variation in the casting. The solidification parameters can subsequently be deduced from these temperature data. Several thin plate castings were tested using the model developed. The following design and operating conditions were evaluated: susceptor temperature (power input), withdrawal speed, changes of cross-sectional area in the casting, and geometrical arrangement of the casting tree.

MST/1422     

Numerical Modeling of Grain Structure in Continuous Casting of Steel

A numerical model is developed for the simulation of solidification grain structure formation (equiaxed to columnar and columnar to equiaxed transitions) during the continuous casting process of steel billets. The cellular automata microstructure model is combined with the macroscopic heat transfer model. The cellular automata method is based on the Nastac's definition of neighborhood, Gaussian nucleation rule, and KGT growth model. The heat transfer model is solved by the meshless technique by using local collocation with radial basis functions. The microscopic model parameters have been adjusted with respect to the experimental data for steel 51CrMoV4. Simulations have been carried out for nominal casting conditions, reduced casting temperature, and reduced casting speed. Proper response of the multiscale model with respect to the observed grain structures has been proved.     

一种参数三次样条曲线光顺优化算法

论文给出了一种基于修改因子和修改角度的平面参数三次样条曲线的优化光顺算法,该算法通过求解一个带有修改因子λ和修改角度θ的目标函数得到光顺后的型值点,插值光顺后的型值点得到光顺曲线。目的是使曲线的曲率变化均匀的同时,使光顺后的曲线与原曲线的偏差尽量小,此算法简单易行,计算量较小。     

Solidification control in continuous casting of steel

An integrated understanding of heat transfer during solidification, friction/lubrication at solid-liquid interface, high temperature properties of the solidifying shell etc. is necessary to control the continuous casting process. The present paper elaborates upon the knowledge developed in the areas of initial shell formation, mode of mould oscillation, and lubrication mechanism. The effect of these issues on the caster productivity and the quality of the product has been discussed. The influence of steel chemistry on solidification dynamics, particularly with respect to mode of solidification and its consequence on strength and ductility of the solidifying shell, has been dealt with in detail. The application of these basic principles for casting of stainless steel slabs and processing to obtain good quality products have been covered.     

Understanding the Heat Transfer and Solidification of Plasma-Sprayed Yttria-Partially Stabilized Zirconia Coatings

A variety of microstructures have been observed in plasma-sprayed yttria-partially stabilized zirconia (YSZ) thermal barrier coatings. Control of the coating microstructures requires a good understanding of the heat transfer and solidification during the process. This article presents a quantitative analysis of heat transfer and solidification of plasma-sprayed YSZ splats. The analysis is based on a simple heat transfer and solidification model that solves a one-dimensional moving boundary problem with consideration of melt undercooling prior to solidification and nonequilibrium crystalline growth kinetics at the moving interface. The solidification morphology is first assumed to be planar, and the stability of the planar interface is examined against the absolute stability velocity calculated from the linear stability theory. Examining the temperature distribution in a solidifying YSZ zirconia splat indicates that a large positive temperature gradient exists in front of the interface, which leads to a stable planar interface and a segregation-free columnar structure, agreeing well with experimental observation. The model also finds that a low interface velocity results from poor heat transfer, which leads to a formation of cells and, therefore, the segregation of yttria. A steady-state dendrite tip growth model is then employed to calculate the radius of the cell tips and thus the cell spacings, which is then compared with experimental observations.     

Mathematical simulation of metal solidification in a wedge-like casting mold with allowance for natural convection

Using the finite difference method, a nonstationary problem of metal solidification in a wedge-like casting mold has been solved in a two-dimensional statement with allowance for natural convection. For isolated instants of time the positions of the solidification front, the profiles of temperature, horizontal and vertical velocity components, and of the stream function have been obtained.     

基于遗传算法和自适应的平面线轮廓度误差评定方法

 为了解决在测量平面线轮廓度中由于存在被测轮廓与其测量基准间存在位置误差而影响评定精度的问题，提出了一种基于遗传算法和自适应的计算平面线轮廓度误差的新方法。该方法满足最小条件原理，它利用样条插值函数拟合理论轮廓，并在评定过程中能自动地实现被测轮廓与理论轮廓之间的适应性调整，从而能够分离并消除被测轮廓与其测量基准之间的位置误差对轮廓误差评定结果的影响，在遗传优化中获得全局最优解。实例计算验证了这一结果。这种算法简单明确，具有精度高、收敛速度快、易于计算机程序实现、易于推广应用等特点。     

Microwave imaging of parallel perfectly conducting cylinders

This paper considers microwave imaging of parallel perfectly conducting cylinders using a solution of the scattering problem by the point‐matching method. A cubic B‐spline, real‐coded genetic algorithm and an adaptive hybrid algorithm are proposed to solve the inverse problem. Previous shape functions in trigonometric series with arbitrary coefficients are nondefinite, which intensify the ill‐posedness and slow the early time convergence of the algorithm. A novel shape function based on cubic B‐splines is developed and the real‐coded genetic algorithm is modified accordingly. Numerical simulation examples show that the early time convergence of the real‐coded genetic algorithm is improved significantly. Next, the adaptive hybrid algorithm is developed to improve the late time convergence of the cubic B‐spline real‐coded genetic algorithm. © 2001 John Wiley & Sons, Inc. Int J Imaging Syst Technol 11, 365–371, 2000     

Index profiling of anisotropic graded-index planar waveguides from effective indices

We present a method for recovering the refractive-index profile of an anisotropic graded-index waveguide from the effective indices by using a cubic spline interpolation function. The first and second derivatives of cubic splines are continuous to ensure a smooth index profile, which is consistent with practical graded-index distributions. A straightforward iteration with an exact dispersion equation to verify the interpolated profile makes this method easy and reliable for application. This approach is proved by numerical analysis of several typical index distributions and experimental examples showing that the refractive-index profiles in anisotropy can be reconstructed close to the exact profile. Waveguides with both more modes (more than four guiding modes) and fewer modes (two to four) can be universally profiled with good accuracy.     

On the computation of numerical derivatives by cubic splines

A method for improving the computational accuracy of numerical derivatives based on cubic splines and computed from a given set of function values with possible errors is described. The method incorporates smoothing, a derivative profile and/or numerical derivatives in the computational process by solving a pentadiagonal system of linear equations. Results comparing the derivatives computed by the present method, cubic spline and finite differences are given.     

基于三次样条插值的精细积分法

结合指数矩阵的精细算法,提出了一类基于三次样条插值的精细积分方法。针对结构动力学方程一般解中的积分项,考虑在一个时间步长内激励为线性和正余弦两种变化形式,通过对积分项中的指数矩阵进行三次样条插值函数模拟,得到一组新的被积函数,最后通过多次分部积分,构造了一类新的高精度计算格式。在三次样条插值函数构造过程中引入了指数矩阵的精细算法,有效避免了中间过程中有效数字的丢失,同时还有效解决了HPD-F算法中涉及的矩阵求逆问题,大大增加了算法的数值稳定性。数值算例显示了该方法的有效性。     

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.     

Phase field simulations of coupled microstructure solidification problems via the strong form particle difference method

This paper presents the development of a strong form-based collocation method called the particle difference method (PDM), capable of predicting the spatiotemporal evolution of polycrystalline material solidification through coupling of multi-phase and temperature fields. Cross coupled phase field evolution and heat transfer equations are discretized via the PDM to obtain the interface kinematics of polycrystalline boundary during solidification. A distinct feature of the PDM is its ability to represent derivative operators via a moving least-square approximation of the Taylor expansion through point-wise computations at collocation points. The method discretizes directly the strong forms using the pre-computed derivative operators at each collocation point and elegantly overcomes the topological difficulty in modeling intricate moving interfaces. To verify the efficacy of the PDM, numerical results are compared with those obtained from the conventional finite difference method for uniform and irregular distributions of the collocation points. The scalability of the parallelized PDM is tested by measuring its efficiency with increasing the number of processors. We also provide a solidification simulation with two ellipsoidal inclusions to demonstrate the capability of the PDM in complex moving interface problems with high curvature.     

Effect of casting/mould interfacial heat transfer during solidification of aluminium alloys cast in CO2-sand mould

The ability of heat to flow across the casting and through the interface from the casting to the mold directly affects the evolution of solidification and plays a notable role in determining the freezing conditions within the casting, mainly in foundry systems of high thermal diffusivity such as chill castings. An experimental procedure has been utilized to measure the formation process of an interfacial gap and metal-mould interfacial movement during solidification of hollow cylindrical castings of Al-4.5 % Cu alloy cast in CO₂-sand mould. Heat flow between the casting and the mould during solidification of Al-4.5 % Cu alloy in CO₂-sand mould was assessed using an inverse modeling technique. The analysis yielded the interfacial heat flux (q), heat transfer coefficient (h) and the surface temperatures of the casting and the mould during solidification of the casting. The peak heat flux was incorporated as a dimensionless number and modeled as a function of the thermal diffusivities of the casting and the mould materials. Heat flux transients were normalized with respect to the peak heat flux and modeled as a function of time. The heat flux model proposed was to estimate the heat flux transients during solidification of Al-4.5 % Cu alloy cast in CO₂-sand moulds.     

样条加权残数法分析薄壳

本文用加权残数法分析薄壳,以五次样条函数构造试函数。我们用离散型最小二乘法分析了扁壳和变厚度圆柱壳的轴对称弯曲问题,对等厚度圆柱壳的轴对称弯曲问题分别采用了配点和配线法进行计算此较。各个问题都编制了程序,通过实例计算并与精确解比较,证明上述方法程序编制简单,输入数据少,耗用机时少,计算精度高,较为经济、实用。     

Microstructural characterization of Co-based coating deposited by low power pulse laser cladding

A detailed microstructural study of Stellite 6 coating deposited on a low carbon ferritic steel substrate using preplaced powder method and low power Nd:YAG pulse laser is performed. The grain structure and solidification texture of the coating are investigated by orientation imaging microscopy (OIM) and scanning electron microscopy. In addition, the effect of consecutive pulses on the microstructure of the coating is examined. The orientation relationship (OR) at coating/substrate interface and the solid state phase transformation in heat-affected zone are studied as well as the Vickers microhardness profile measurement in order to support the microstructural observations. An important conclusion is reached that the shape of solidification front during pulsed laser cladding is similar to the shape of solidification front during continuous cladding with a doubled laser beam scanning speed. Further, OIM reveals the Greninger–Troiano OR between the face centered cubic coating and bcc substrate grains. It is concluded that at the moment of solidification epitaxial growth of the grains in the coating occur on the austenitic grains of the substrate and that an austenite–ferrite transformation occurs in the heat-affected zone upon subsequent cooling.     

一种自适应的平面线轮廓度误差评定方法

提出一种基于最小二乘法,结合样条插值函数和优化技术的用于平面线轮廓度误差评定的数据处理方法,该方法的优点在于在轮廓度误差评定过程中能自动地实现被测轮廓与理论轮之间的适应性调整,以此分离并消除被测轮廓与其测量基准之间的位置误差对轮廓误差评定结果的影响。     

Thin plate spline radial basis function for the free vibration analysis of laminated composite shells

In this paper, the free vibration characteristics of laminated composite cylindrical and spherical shells are analyzed by the first-order shear deformation theory and a meshless global collocation method based on thin plate spline radial basis function. The singularity of thin plate spline radial basis function is eliminated by adding infinitesimal to the zero distance. Several numerical examples are used to show convergence of the present method and choose the proper shape parameter. It is found that the natural frequencies computed by thin plate spline radial basis function with shape parameter m = 4 converge most rapidly. In the comparison study, the present results are in good agreement with the results of Reddy and Liu [8] and Ferreira et al. [21].     

基于跖跗围参数的鞋楦曲面重构

针对面向合脚性、舒适性的鞋楦个性化设计中的曲面重构问题,提出并实现了一种基于鞋楦跖跗围截面变换技术的鞋楦曲面参数化重构方法。通过对鞋楦跖跗截面围长和高度的控制,将跖、跗围的截面曲线变换到个性化设计所需求效果;基于三次样条插值函数构造局部、整体两类扩散函数,实现两种将两围截面变换效果扩散到整个鞋楦曲面的方式。此算法不仅可以准确地满足鞋楦围长约束要求,且保持了曲面光顺性。所提出的重构过程在Matlab平台上可以实现可视化交互设计,界面具有操作简单方便的特点。     

复合材料层合板自由振动的薄板样条无网格解

无网格全局配点法分为多项式配点法和径向基函数配点法,国内外很多文献利用径向基函数配点法对复合材料层合板进行了分析。利用一阶剪切变形理论和基于薄板样条径向基函数的无网格配点法计算了复合材料层合板自由振动的固有频率和振型。研究了薄板样条径向基函数中形状参数的选取和本工作方法的收敛性。结果表明:形状参数m=3时收敛性最好,计算精度最高。将本工作计算结果与文献中的实验结果进行了对比,验证了本文方法的精度和效率。