On the Design of Continuum Thermal Transport Systems with Applications to Solidification Processes

A general class of design problems is examined in thermal transport systems. We mainly address inverse problems with overspecified coupled boundary conditions in one part of the boundary and unknown thermal conditions in another part of the boundary. The methods of choice for the solution of the above inverse problems are functional optimization methods using appropriately defined continuum sensitivity and adjoint problems. Conjugate gradient techniques, preconditioning and regularization are considered within an innovative object-oriented finite element framework. Our particular interest for examining such inverse transport systems arises from the desire to address the design of directional solidification processes that lead to desired microstructures. As the main application of this paper, we will address the calculation of the mold/furnace heat flux conditions such that a desired solidification state (growth rate and temperature gradient) is achieved at the freezing interface. Changes in growth rate and thermal gradient at the freezing interface are known to alter the relative importance of thermal/mass transport and interfacial energy effects, and the magnitude of this partitioning of available driving forces dictates the formation of specific microstructures. INTRODUCTION We are here emphasizing the solution of inverse problems for continuum transport systems with simultaneous heat, momentum and mass transfer. Additional coupling mechanisms induced by the presence of electromagnetic fields are also of interest. In a typical inverse problem with such coupled transport mechanisms, one for example can calculate the thermal boundary conditions in part of the boundary of a cavity containing a convecting fluid such that a desired temperature field is achieved inside the domain or in parts of the boundary with known heat flux conditions. Solidification processes provide a number of examples where inverse design of coupled transport systems arises. For example, the following technologically important design solidification problems can be posed as inverse problems: • Design of solidification processes to achieve desired growth velocity and freezing interface heat fluxes. This is an important topic in furnaces for single-crystalgrowing of electronic materials and in directional solidification processes for cast turbine blades (Flemings, 1974). • Find the means of delaying or eliminating the morphological instability of the planar front so that one could grow crystals without radial segregation (Davis et al., 1992). One can achieve the above objectives by proper adjustment of the cooling/heating mold/furnace conditions and/or with appropriate design of sources of forced convection. Directional solidification processes with thin-mushy zones are considered here in order to simplify the mathematical design framework. Even though the physical applicability of such solidification systems is limited, thin-mushy zone 1 Copyright c © 2001 by ASME solidification models using front tracking techniques provide us with direct access to the growth velocity and interface temperature gradients and thus a convenient framework to test the ability of controlling the interface conditions. This control will here be achieved with a proper design of the furnace/mold cooling/heating conditions but other means of control e.g. with the use of magnetic fields (Bourgeois et al., 1992; Series and Hurle, 1991; Meir and Schmidt, 1999) can be examined as well. Strong magnetic fields are applied extensively in the processing of advanced materials. In the remaining of this paper, we will present infinite dimensional optimization schemes for the design of transport systems that involve simultaneous momentum, heat and mass transfer as well as electromagnetic fields with emphasis given to design problems that arise in directional solidification processes. DESIGN OF CONTINUUM THERMAL SYSTEMS Inverse problem theory provides the mathematical structure for obtaining a solution to a physical problem that has been mathematically casted as an ill-posed problem (Kirsch, 1996). Most of the advances in inverse theory are in the area of parameter identification in parabolic and elliptic problems and in the estimation of boundary conditions from incomplete measurements within a body (Alifanov, 1994). Inverse heat conduction has been a major area of research due to the relative simplicity of the mathematical models and the potential engineering applications (Alifanov, 1994). In a typical problem, overspecified boundary conditions are supplied in part of the boundary ΓI (temperature and heat flux) whereas in another part of the boundary Γh0 an unknown heat flux qo) must be estimated. Extension of inverse conduction problems to inverse Stefan problems has been considered by Barbu (1990) and Zabaras et al. (1992) where emphasis was given to the design of Stefan processes that lead to a desired growth and/or freezing interface heat fluxes. Very few references have addressed inverse convection problems (Zabaras et al., 1995; Gunzburger, 1995; Zabaras and Yang, 1997; Yang and Zabaras, 1998; Sampath and Zabaras, 2001). The development of practical computational methods for the solution of these inverse problems relies on cascading simulation based software into optimization based algorithms. The inverse problem is usually stated as an optimization problem in an appropriate space and a quasisolution is calculated. The gradient of the objective function is calculated analytically in an appropriate infinite dimensional space or in an approximate finite dimensional subspace (Luenberger, 1968). In the infinite dimensional formulation, the calculation of the gradient of the cost functional is performed analytically after continuum or discrete sensitivity and adjoint problems have been appropriately defined. In the finite-dimensional approach, the gradient of the objective function can be calculated using either the sensitivity or the adjoint fields. Infinite dimensional methods require no quantitative apriori knowledge on the solution and can result in designs that cannot be accurately approximated in arbitrary finite dimensional spaces (Marchuck, 1995). To account for the ill-posedness of the above stated inverse problems, the cost functional is stated over the whole time domain. In this way, all available data is used to evaluate qo at all times. In many cases, it is also important that Tikhonov regularization and other a priori information are introduced to further restrict the set of allowable solutions (Engl et al., 1996). FUNCTIONAL OPTIMIZATION METHODS FOR TRANSPORT SYSTEMS To set forward the ideas for the implementation of complex transport systems, we next define a novelle inverse magneto-convection problem and highlight an adjoint method based solution scheme in L2. This problem is sufficient to clarify the mathematical structure of the problems of interest. An Inverse Magneto-Convection Problem Let Ω be a closed bounded region in nsd with a piecewise smooth boundary Γ (Fig. 1). The region is occupied by an incompressible electrically conducting fluid that is subject to an external magnetic field. The fluid motion is driven by temperature induced density gradients and by the presence of the magnetic field via a Lorentz force. The induced magnetic field by the electric currents in the melt is neglected with respect to the imposed constant field Bo. The current density J can be eliminated from the governing equations (using the conservation of electric current) and the effect of the magnetic field can be solely expressed in terms of the electric field potential φ. In the part Γh of Γ, a heat flux boundary condition is applied, while in the remaining part of the boundary, Γg, a temperature boundary condition is considered (see Fig. 1). However, the distribution of the boundary heat flux on Γh0 ⊂ Γh is not known (Γh0 ∪ Γh1 = Γh,Γh0 ∩ Γh1 = ∅). The objective here is to calculate the flux qo(x, t), (x, t) ∈ Γh0 × [0, tmax] assuming that the temperature field is apriori known in a subset ΓI of Γh1 where the flux q1(x, t) is known, i.e. θ(x, t) = θm(x, t), (x, t) ∈ ΓI × [0, tmax] (1) The over-specified thermal boundary condition on ΓI to2 Copyright c © 2001 by ASME