Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation

Bernauer, Martin K.; Herzog, Roland

doi:10.1137/100783327

Cited by 47 publications

(47 citation statements)

References 24 publications

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“…This basis has the advantage over the standard Lagrangian basis that setting all off-diagonal entries to zero is a conservative mass lumping strategy. Moreover, the Taylor basis allows for the efficient solution of the adjoint systems in an optimal control approach to motion planning for the two-phase Stefan problem in level set formulation, see [10].…”

Section: Taylor Basismentioning

confidence: 99%

Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

Bernauer

Herzog

2011

J Sci Comput

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The classical two-phase Stefan problem in level set formulation is considered. The implementation of a solver on triangular grids is described. Extended finite elements (X-FEM) in space and an implicit Euler method in time are used to approximate the temperature. For the level set equation, a discontinuous Galerkin (DG) and a strong stability preserving (SSP) Runge-Kutta scheme are employed. Polynomial spaces of quadratic order are used. A numerical example with a change of topology is provided, and the order of convergence is studied on the Frank sphere example.

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Section: Taylor Basismentioning

confidence: 99%

Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

Bernauer

Herzog

2011

J Sci Comput

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“…We consider a two-dimensional two-phase Stefan problem as in [2,3] where a domain Ω ∈ R 2 is split into the solid phase Ω s (t) and the liquid phase Ω l (t) as in figure 1. The phases are separated by the moving interface Γ I (t) which we model by the Stefan condition and which we represent as a graph, h : [0, T ] × Γ cool → R, over the lower boundary Γ cool .…”

Section: Two-phase Stefan Problemmentioning

confidence: 99%

Adjoint‐Based Optimal Boundary Control of a Stefan Problem System Fully Coupled with Navier‐Stokes Equations

Baran

Heiland

2016

Proc Appl Math and Mech

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Free boundary and moving boundary problems, that can be used to model crystal growth or the solidification and melting of pure materials, receive growing attention in science and technology. The optimal control of these problems appear even more interesting since certain desired shapes of the boundaries improve, e.g., the material quality in the case of crystal growth. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface.In the work presented, we take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state at a final time as well as the control costs. Following the "optimize-then-discretize" paradigm, we formulate a first order optimality system using the formal Lagrange approach and derive the adjoint PDE system that provides the needed gradient of the cost functional.By means of an example setup of a container with an in-and outflow of water and a cooling unit at the bottom, we illustrate how the derived formulations can be used to achieve a desired interface between the solid and the fluid phase by controlling the flow at the inlet. Two-Phase Stefan ProblemWe consider a two-dimensional two-phase Stefan problem as in [2,3] where a domain Ω ∈ R 2 is split into the solid phase Ω s (t) and the liquid phase Ω l (t) as in figure 1. The phases are separated by the moving interface Γ I (t) which we model by the Stefan condition and which we represent as a graph, h : [0, T ] × Γ cool → R, over the lower boundary Γ cool . Further, Γ cool is a cooling boundary, the inlet Γ in is a heat source and Γ out is the outlet. The temperature distribution is characterized by the heat equation, and the system is fully coupled with Navier-Stokes equations which describe the fluid flow in Ω l (t). A detailed description of the system can be found in [1]. We solve the system numerically with finite elements and mesh movement methods with the software FEniCS. The code is available at https://gitlab.mpi-magdeburg.mpg.de/baran/Stefan_Problem_in_FEniCS.git.

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“…Optimization and optimal control of multi-phase problems governed by Navier-Stokes equations with free boundaries or phase transitions have been applied by, e.g., [1,6,15,21]. Since already a single numerical simulation of such a free boundary problem is a difficult issue, optimal control is a very challenging task.…”

Section: Introductionmentioning

confidence: 99%

Analysis and optimal control for free melt flow boundaries in laser cutting with distributed radiation

Vossen

Hermanns

Schüttler³

2013

Z Angew Math Mech

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This work introduces a mathematical model for laser cutting taking account of spatially distributed laser radiation. The model involves two coupled nonlinear partial differential equations describing the interacting dynamical behaviors of the free boundaries of the melt during the process. The model will be investigated by linear stability analysis to study the occurence of ripple formations at the cutting surface. We define a measurement for the roughness of the cutting surface and introduce an optimal control problem for minimizing the roughness with respect to the laser beam intensity along the free melt surface. Necessary optimality conditions will be deduced. Finally, a numerical solution will be presented and discussed by means of the necessary conditions. physical considerations

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Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation

Cited by 47 publications

References 24 publications

Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

Adjoint‐Based Optimal Boundary Control of a Stefan Problem System Fully Coupled with Navier‐Stokes Equations

Analysis and optimal control for free melt flow boundaries in laser cutting with distributed radiation

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