Inverse design of directional solidification processes in the presence of a strong external magnetic field

A computational method for the design of directional alloy solidification processes is addressed such that a desired growth velocity ν_f under stable growth conditions is achieved. An externally imposed magnetic field is introduced to facilitate the design process and to reduce macrosegregation by the damping of melt flow. The design problem is posed as a functional optimization problem. The unknowns of the design problem are the thermal boundary conditions. The cost functional is taken as the square of the L₂ norm of an expression representing the deviation of the freezing interface thermal conditions from the conditions corresponding to local thermodynamic equilibrium. The adjoint method for the inverse design of continuum processes is adopted in this work. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, velocity and electric potential fields such that the gradient of the L₂ cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the FE solutions of the continuum direct, sensitivity and adjoint problems. The developed formulation is demonstrated with an example of designing the boundary thermal fluxes for the directional growth of a germanium melt with dopant impurities in the presence of an externally applied magnetic field. The design is shown to achieve a stable interface growth at a prescribed desired growth rate. Copyright © 2001 John Wiley & Sons, Ltd.     

Control of the freezing interface motion in two-dimensional solidification processes using the adjoint method

The aim of this work is to calculate the optimum history of boundary cooling conditions that, in two-dimensional conduction driven solidification processes, results in a desired history of the freezing interface location/motion. The freezing front velocity and heat flux on the solid side of the front, define the obtained solidification microstructure that can be selected such that desired macroscopic mechanical properties and soundness of the final cast product are achieved. The so-called two-dimensional inverse Stefan design problem is formulated as an infinite-dimensional minimization problem. The adjoint method is developed in conjunction with the conjugate gradient method for the solution of this minimization problem. The sensitivity and adjoint equations are derived in a moving domain. The gradient of the cost functional is obtained by solving the adjoint equations backward in time. The sensitivity equations are solved forward in time to compute the optimal step size for the gradient method. Two-dimensional numerical examples are analysed to demonstrate the performance of the present method.     

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 ill-posed design solidification problem using minimization techniques in finite- and infinite-dimensional function spaces

This paper provides a comparative study of two alternative methodologies for the solution of an inverse design solidification problem. It is the one-dimensional solidification problem of calculating the boundary heat flux history that achieves a desired freezing front velocity and desired heat fluxes at the freezing front. The front velocity h(t) and flux history q_mS(t) on the solid side of the front control the obtained cast structure. As such, the potential applications of the proposed methods to the control of casting processes are enormous. The first technique utilizes a finite-dimensional approximation of the unknown boundary heat flux function q₀(t). The second technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ‖T_m – T(h(t), t;q₀)‖, that expresses the square error between the calculated T(h(t), t; q₀) and the given freezing front temperature T_m. Both steepest descent (SDM) and conjugate gradient methods (CGM) are examined. A front tracking FEM technique is used for the discretization of the state space. A detailed numerical analysis of the space and time discretization of the ‘parameter’ and state spaces, of the effect of the end condition of the adjoint problem and of other parameters in the solution are examined.     

Stochastic inverse heat conduction using a spectral approach

An adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L₂ optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L₂ optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.     

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.     

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.     

Parameter estimation in heat conduction using a two-dimensional inverse analysis

This article is concerned with a two-dimensional inverse steady-state heat conduction problem. The aim of this study is to estimate the thermal conductivity, the heat transfer coefficient, and the heat flux in irregular bodies (both separately and simultaneously) using a two-dimensional inverse analysis. The numerical procedure consists of an elliptic grid generation technique to generate a mesh over the irregular body and solve for the heat conduction equation. This article describes a novel sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the thermal conductivity, the heat transfer coefficient, and the heat flux. This sensitivity analysis scheme allows for the solution of inverse problem without requiring solution of adjoint equation even for a large number of unknown variables. The conjugate gradient method (CGM) is used to minimize the difference between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results reveal that the proposed algorithm is very accurate and efficient.     

Application of a boundary integral equation to prediction of freezing fronts in soil

A boundary integral equation formulation based on the complex Cauchy integral theorem is applied to two-dimensional soil-water phase change problems encountered in algid soils. The model assumes that potential theory applies in the estimation of heat flux along a freezing front of differential thickness and that quasi-steady-state temperatures occur along the problem domain boundary. Application of the boundary integral formulation to two-dimensional problems results in predicted locations of the freezing front which are highly accurate. Although the proposed formulation is based on the Cauchy integral theorem, similar models may be developed based on other forms of integration equation methods.     

Shape design sensitivity analysis and optimization of two-dimensional periodic thermal problems using BEM

 A general procedure to perform shape design sensitivity analysis for two-dimensional periodic thermal diffusion problems is developed using boundary integral equation formulation. The material derivative concept to describe shape variation is used. The temperature is decomposed into a steady state component and a perturbation component. The adjoint variable method is used by utilizing integral identities for each component. The primal and adjoint systems are solved by boundary element method. The sensitivity results compared with those by finite difference show good accuracy. The shape optimal design problem of a plunger model for the panel of a television bulb, which operates periodically, is solved as an example. Different objectives and amounts of heat flux allowed are studied. Corresponding optimum shapes of the cooling boundary of the plunger are obtained and discussed. Received 15 August 2001 / Accepted 28 February 2002     

Solution of inverse problem of solidification of a cylindrical ingot

An inverse problem of heat conduction that involves determination of the density of boundary heat flux providing a prescribed velocity of solidification front motion is solved by an integral method. Moscow Institute of Steel and Alloys. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 69, No. 4, pp. July–August, 1996.     

Gradient method for inverse heat conduction problem in nanoscale

An inverse heat conduction problem for nanoscale structures was studied. The conduction phenomenon is modelled using the Boltzmann transport equation. Phonon‐mediated heat conduction in one dimension is considered. One boundary, where temperature observation takes place, is subject to a known boundary condition and the other boundary is exposed to an unknown temperature. The gradient method is employed to solve the described inverse problem. The sensitivity, adjoint and gradient equations are derived. Sample results are presented and discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Boundary formulations for shape sensitivity of temperature dependent conductivity problems

Used in concert with the Kirchhoff transformation, implicit differentiation of the discretized boundary integral equations governing the conduction of heat in solids with temperature dependent thermal conductivity is shown to generate an accurate and economical approach for computation of shape sensitivities. For problems with specified temperature and heat flux boundary conditions, a linear problem results for both the analysis and sensitivity analysis. In problems with either convection or radiation boundary conditions, a non-linear problem is generated. Several iterative strategies are presented for the solution of the resulting sets of non-linear equations and the computational performances examined in detail. Multi-zone analysis and zone condensation strategies are demonstrated to provide substantive computational economies in this process for models with either localized non-iinear boundary conditions or regions of geometric insensitivity to design variables. A series of non-linear example problems is presented that have closed form solutions. Exact anaytical expressions tor the shape sensitivities associated with these problems are developed and these are compared with the sensitivities computed using the boundary element formulation.     

Boundary integral equation method for shape optimization of elastic structures

A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.     

Direct method of solution for general boundary value problem of the Laplace equation

A general boundary value problem for two-dimensional Laplace equation in the domain enclosed by a piecewise smooth curve is considered. The Dirichlet and the Neumann data are prescribed on respective parts of the boundary, while there is the second part of the boundary on which no boundary data are given. There is the third part of the boundary on which the Robin condition is prescribed. This problem of finding unknown values along the whole boundary is ill posed. In this sense we call our problem an inverse boundary value problem. In order for a solution to be identified the inverse problem is reformulated in terms of a variational problem, which is then recast into primary and adjoint boundary value problems of the Laplace equation in its conventional form. A direct method for numerical solution of the inverse boundary value problem using the boundary element method is presented. This method proposes a non-iterative and unified treatment of conventional boundary value problem, the Cauchy problem, and under- or over-determined problems.     

Optimization of thermal processes using an Eulerian formulation and application in laser surface hardening

A systematic design approach has been developed for thermal processes combining the finite element method, design sensitivity analysis and optimization. Conductive heat transfer is solved in an Eulerian formulation, where the heat flux is fixed in space and the material flows through a control volume. For constant velocity and heat flux distribution, the Eulerian formulation reduces to a steady-state problem, whereas the Lagrangian formulation remains transient. The reduction to a steady-state problem drastically improves the computational efficiency. Streamline Upwinding Petrov–Galerkin stabilization is employed to suppress the spurious oscillations. Design sensitivities of the temperature field are computed using both the direct differentiation and the adjoint methods. The systematic approach is applied in optimizing the laser surfacing process, where a moving laser beam heats the surface of a plate, and hardening is achieved by rapid cooling due to the heat transfer below the surface. The optimization objective is to maximize the rate of surface hardening. Constraints are introduced on the computed temperature and temperature rate fields to ensure that phase transformations are activated and that melting does not occur. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element free boundary formulation for the problem of multiphase heat conduction

The problem of heat conduction in a multiphase medium is a free boundary problem in which the free boundary is the phase transition line. The solution to this problem is highly irregular in that the temperature gradient is discontinuous at the free boundary. Two-dimensional, discontinuous, space–time finite elements are introduced to obtain a finite element, free boundary formulation for this problem. The jump in heat flux and position of the phase transition line become dependent variables in the finite element model. The resulting models give accurate results while allowing the calculation of free boundary problems with a fixed finite element mesh.     

Determination and verification of the gap dependent heat transfer coefficient during permanent mold casting of A356 aluminum alloy

In complex castings, the heat transfer across the casting / mold interface depends on the local gap size and contact pressure. Thus, an experimental setup is constructed to measure and evaluate the air‐gap dependent heat transfer coefficient during solidification of an A356 permanent mold casting. In order to evaluate the heat transfer coefficient, the temperature gradient and air gap development is measured at the casting / mold interface. This allows the interface temperatures and the time‐dependent heat flux across the gap to be calculated as a function of the measured gap size. Furthermore, the heat transfer coefficient and gap size are correlated to the interface temperature of the casting. The experimental setup and the evaluation procedure provide consistent and reproducible results. The heat transfer coefficient thus evaluated is employed to simulate the experimental setup. The temperatures measured are well reproduced. The results of the present work are compared to simulations using two heat transfer coefficient functions found in literature. This comparison shows a substantial improvement over the state of the art. This improvement is due to the exact knowledge of gap formation and the corresponding values of the heat transfer coefficient.