Thermomechanics and the formulation of the Stefan problem for fully faceted interfaces
Quarterly of Applied Mathematics
A crystalline motion: uniqueness and geometric properties
SIAM Journal on Applied Mathematics
Perspectives in Flow Control and Optimization
Perspectives in Flow Control and Optimization
Computational Optimization and Applications
A computational framework for the regularization of adjoint analysis in multiscale PDE systems
Journal of Computational Physics
Efficient treatment of stationary free boundary problems
Applied Numerical Mathematics - Selected papers from the first Chilean workshop on numerical analysis of partial differential equations (WONAPDE 2004)
On Convergence in Elliptic Shape Optimization
SIAM Journal on Control and Optimization
Journal of Computational and Applied Mathematics
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This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to PDE constraints. An advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase. It is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated. We describe an efficient iterative solution method for the Stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional. Performance of the proposed approach is illustrated with sample computations concerning 2D steady solidification phenomena.