ISSN:
0029-5981
Keywords:
solidification
;
control
;
natural convection
;
Engineering
;
Engineering General
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Mathematics
,
Technology
Notes:
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 qms 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, ΩL(t), before solving an ill-posed inverse design conduction problem in the solid phase in an a priori known growing region, ΩS(t). The direct convection problem is used to evaluate the flux qml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux qml can be used together with the Stefan condition to provide the freezing interface flux qms in the solid side of the front. As such, two boundary conditions (flux qms and freezing temperature θm) are especified along the (known) freezing interface δΩI(t). The developed design technique uses the adjoint method to calculate in L2 the derivative of the cost functional, ∥θm - θ(x, t; q0)∥L22, that expresses the square error between the calculated temperature θ(x, t; q0) in the solid phase along δΩI(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.
Additional Material:
12 Ill.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/nme.1620380907
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