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

Abstract

SUMMARYA computational method for the design of directional alloy solidiÿcation processes is addressed such that a desired growth velocity v f under stable growth conditions is achieved. An externally imposed magnetic ÿeld is introduced to facilitate the design process and to reduce macrosegregation by the damping of melt ow. 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 L 2 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 ÿelds such that the gradient of the L 2 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 uxes for the directional growth of a germanium melt with dopant impurities in the presence of an externally applied magnetic ÿeld. The design is shown to achieve a stable interface growth at a prescribed desired growth rate.

Numerical Study of Convection in the Directional Solidification of a Binary Alloy Driven by the Combined Action of Buoyancy, Surface Tension, and Electromagnetic Forces

“…As one can note from , solidification under reduced gravity and sufficiently strong magnetic field ensures that I is almost vertical. Achieving a flat interface growth in the presence of melt convection has been a very important objective in the processing of advanced materials [17,32]. In order to quantify the deviation from "flat-interface growth," the standard deviation σ s (t) (see Eq.…”

“…Even though the applicability of this model is limited, it results in a mathematically tractable framework that can be used to provide a detailed insight into the complex interaction of heat, mass, and momentum transport in the solidification system [27][28][29][30]. For materials for which this assumption of a thin mushy zone is not applicable, sharp interface models have been used to pose optimal control problems in which the interface stability is enforced explicitly by a proper selection of the process conditions (e.g., by a boundary heat flux design [31,32]). …”

Control of solidification of non-conducting materials using tailored magnetic fields

“…Most of these design and control problems are posed as inverse problems in which incomplete conditions are available on one part of the boundary, whereas over-specified boundary conditions are available on another part of the boundary or inside the domain [2]. The main emphasis of our earlier work has been the design of mold or furnace heating/cooling conditions in order to achieve a solid-liquid front growth with a desired interfacial flux G and freezing front velocity v f [4].…”

“…In the context of thermal design, inverse heat conduction problems [2], inverse convection and fluid flow control problems [3] have been well studied. These developed methodologies have simultaneously been extended to solidification process design [4]. Most of these design and control problems are posed as inverse problems in which incomplete conditions are available on one part of the boundary, whereas over-specified boundary conditions are available on another part of the boundary or inside the domain [2].…”

Error and stability analysis of an anisotropic phase-field model for binary-fluid mixtures in the presence of magnetic-field

“…Moreover, some experimental observations show that the control on dendrite growth can be achieved in the during the solidification process by applying electric and magnetic fields externally, see for example [14,24] and the references therein. For similar other applications wherin authors have discussed the effect of magnetic field on the metals and alloys, refer to the studies, e.g., for the semi-conductor flow in the melt crystal evolution, [7], for the MHD flows [12], [11], [32], [9] and for the processes of dendritic solidification, [23], [24], [25] and the references therein.…”