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Control of the freezing interface motion in two‐dimensional solidification processes using the adjoint method
Author(s) -
Kang Shinill,
Zabaras Nicholas
Publication year - 1995
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620380105
Subject(s) - adjoint equation , stefan problem , conjugate gradient method , sensitivity (control systems) , work (physics) , inverse problem , temperature gradient , inverse , mathematics , boundary value problem , equations of motion , thermal conduction , domain (mathematical analysis) , minification , boundary (topology) , mechanics , mathematical analysis , mathematical optimization , geometry , partial differential equation , physics , classical mechanics , thermodynamics , engineering , quantum mechanics , electronic engineering
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.