The adjoint method for an inverse design problem in the directional solidification of binary alloys
type: Item
instance of: scholarly article
title: The adjoint method for an inverse design problem in the directional solidification of binary alloys
zbMATH Open document ID: 0926.65097
author: George Yang
author: Nicholas Zabaras
published in: Journal of Computational Physics
publication date: 1999-11-30T00:00:00Z
review text: The authors propose a systematic continuum formulation of the adjoint method to solve an inverse thermo-solutal convection problem. The objective is to control the boundary heating/cooling fluxes such that solidification of dilute binary alloys proceeds with a desired stable interface growth. Such studies identify possible inconsistencies in the previous binary alloy solidification models that a priori assume a macroscopically sharp solid/liquid interface. The stability criterion is here chosen as the absence of constitutional undercooling in the liquid ahead of the interface. Such relation of the interface stability is treated as an overspecified boundary condition in the inverse problem formulation. Based on additional physical arguments of marginal stability and minimum energy input, the constitutional undercooling condition is implemented in the form of an equality. Herein, the cost functional is defined as the thermal deviation of the freezing interface from thermodynamic equilibrium. The inverse design problem is posed as the calculation of the optimum boundary heat flux that leads to the minimization of the cost functional. A close form of the adjoint equation system is derived with coupled thermal, solute, and flow transport mechanism. The definition of the adjoint system is made such that the boundary value of the adjoint thermal variable at the fixed wall is equal to the gradient of the cost functional. The conjugate gradient algorithm is used to solve for the optimum boundary heat flux that minimizes the cost functional. Since such a heat flux is a function varying in space and time, the adjoint method provides an elegant and efficient numerical scheme for the solution of this class of problems. An example case of the solidification of \(\text{NH}_4\text{Cl}\) water solution in a rectangular cavity is performed for moderate strength of thermo-solutal convection. A small amount of `over-stability' is introduced to overcome the problem of loss of stability …
Mathematics Subject Classification ID: 65M32
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 35R30
zbMATH DE Number: 1283488
zbMATH Keywords: inverse design problem
zbMATH Keywords: directional solidification
zbMATH Keywords: binary alloys
zbMATH Keywords: adjoint method
zbMATH Keywords: inverse thermo-solutal convection problem
zbMATH Keywords: interface stability
zbMATH Keywords: conjugate gradient algorithm
MaRDI profile type: Publication
OpenAlex ID: W2037920517
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DOI: 10.1006/JCPH.1998.5893
description: scientific article; zbMATH DE number 1283488