Control of the freezing interface morphology in solidification processes in the presence of natural convection

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q₀ in the fixed mold wall, δΩ₀, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history q_ms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous. The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, Ω_L(t), before solving an ill-posed inverse design conduction problem in the solid phase in an a priori known growing region, Ω_S(t). The direct convection problem is used to evaluate the flux q_ml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux q_ml can be used together with the Stefan condition to provide the freezing interface flux q_ms in the solid side of the front. As such, two boundary conditions (flux q_ms and freezing temperature θ_m) are especified along the (known) freezing interface δΩ_I(t). The developed design technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ∥θ_m – θ( x , t; q₀)∥, that expresses the square error between the calculated temperature θ( x , t; q₀) in the solid phase along δΩ_I(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.     

On the solution of an inverse natural convection problem using various conjugate gradient methods

The inverse problem of determining the time‐varying strength of a heat source, which causes natural convection in a two‐dimensional cavity, is considered. The Boussinesq equation is used to model the natural convection induced by the heat source. The inverse natural convection problem is solved through the minimization of a performance function utilizing the conjugate gradient method. The gradient of the performance function needed in the minimization procedure of the conjugate gradient method is obtained by employing either the adjoint variable method or the direct differentiation method. The accuracy and efficiency of these two methods are compared, and a new method is suggested that exploits the advantageous aspects of both methods while avoiding the shortcomings of them. Copyright © 2000 John Wiley & Sons, Ltd.     

First-order and second-order adjoint methods for parameter identification problems with an application to the elasticity imaging inverse problem

We devise a fast and reliable computational framework for the elasticity imaging inverse problem of detecting cancerous tumors in the human body using an output least-squares (OLS) approach. From a mathematical standpoint, this inverse problem requires identifying a parameter in a mixed variational problem. We develop, in a continuous setting, a first-order adjoint method and two second-order adjoint methods. The continuous formulae are then used to devise a scheme for an efficient computation of the gradient and the Hessian of the OLS objective. We give detailed numerical examples.     

Inverse-heat-conduction problems with boundary elements: analysis of a corner effect

A transient two-dimensional inverse-heat-conduction problem is investigated. It consists in the determination of both temperature and heat-flux density in the vicinity of an angle (≤ 180°) when some internal temperatures are known. Such a problem is solved by using a boundary-element approach with a time- and space-dependent fundamental solution. It uses a time-marching scheme that involves future time steps and a regularization procedure. An exhaustive study of sensitivity to the leading parameters of the problem is produced, and it is especially pointed out that the accuracy of the resolution is strongly affected as the corner angle decreases. In order to overcome this difficulty, a localized regularization procedure is suggested.     

An adjoint method in inverse problems of chromatography

How to determine adsorption isotherms is an issue of significant importance in chromatography. A modern technique of obtaining adsorption isotherms is to solve an inverse problem so that the simulated batch separation coincides with actual experimental results. In this work, as well as the natural least-square approach, we consider a Kohn–Vogelius type formulation for the reconstruction of adsorption isotherms in chromatography, which converts the original boundary fitting problem into a domain fitting problem. Moreover, using the first momentum regularizing strategy, a new regularization algorithm for both the Equilibrium-Dispersive model and the Transport-Dispersive model is developed. The mass transfer resistance coefficients in the Transport-Dispersive model are also estimated by the proposed inverse method. The computation of the gradients of objective functions for both of the two models is derived by the adjoint method. Finally, numerical simulations for both a synthetic problem and a real-world problem are given to show the robustness of the proposed algorithm.     

Pb-Sn合金侧向凝固过程A偏析的数值模拟

基于二元系凝固过程热溶质的传输行为,建立了描述A偏析形成及演化的数学模型,给出了固相分数与温度场及浓度场的耦合关系.先用已有的实验结果验证了模型的正确性,然后模拟计算了Pb-Sn合金侧向凝固过程A偏析的形成及演化过程,并研究了浮力数对A偏析形成位置及偏析程度的影响.结果表明,在糊状区中双扩散对流引起的密度变化,导致局部流动,形成偏析通道;为了维持偏析通道中局部液体的流动,枝晶间的液体通过糊状区从液相区得到补充.在相同的凝固条件下,浮力数越小,A偏析的形成时间愈迟,偏析的程度也越小.     

基于情感共生理念的宠物清洁产品创新设计

目的 通过对宠物行为、心理状态及饲养人员情感的综合分析,进行了猫砂盆的设计,使主人清理猫咪厕所更便捷、高效,并在清理时不会嗅到异味。方法 从人与宠物和谐共处的角度出发对宠物猫的心理、行为及饲主情感需求进行分析,解决了宠物清洁类产品设计的现有问题。结果 通过分析研究现有猫砂盆设计,总结归纳出猫砂盆设计的创新点,设计出一款宠物与人更为相处融洽的宠物清洁类产品设计。结论 运用情感共生理念解决问题,设计出一款能够解决猫咪如厕异味重、饲养者清理难及外观问题的猫砂盆,为情感共生理念在解决实际问题中的应用提供了思路和参考。     

SIMULTANEOUS OPTIMIZATION OF SHAPE AND FLOW PARAMETERS FOR COMBINED FREE AND FORCED CONVECTION IN VERTICAL CHANNELS

Simultaneous optimization of shape and flow parameters is performed for a combined free and forced convection flow through vertical rectangular channels with moving walls. The laminar flow is assumed to be fully developed in the axial direction. The wall velocity, the axial pressure gradient and the channel height in the transverse plane are taken as the optimization parameters. The sensitivity expressions of both the objective function and the flow rate constraint of optimization are obtained in terms of the relevant physical variables, as well as adjoint variables which satisfy additional p.d.e.'s. All equations are discretized using the finite element method. Numerical results are provided for the present constrained optimization problem for various values of the problem parameters which include the moving wall segment size and the Rayleigh number. The results indicate that with increased Rayleigh number the optimal values of the wall velocity and the axial pressure gradient are increased, while the optimal value of the channel height is decreased. General sensitivity expressions are also presented in the appendix which might be utilized for arbitrary boundary variations along with arbitrary optimization objectives in other investigations. © 1997 by John Wiley & Sons, Ltd.