Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

Elisabeth Diehl, Johannes Haubner, Michael Ulbrich, Stefan Ulbrich

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on Lp-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting from the Lp-maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. For objective functionals involving the velocity field or the discontinuous pressure or phase indciator field we derive differentiability results with respect to controls and state formulas for the derivative. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.

Original languageEnglish
Pages (from-to)711-751
Number of pages41
JournalComputational Optimization and Applications
Volume87
Issue number3
DOIs
StatePublished - Apr 2024

Keywords

  • 35Q35
  • 49M41
  • 65K10
  • 76D45
  • 76D55
  • Differentiability
  • Navier-Stokes equations
  • Optimal control
  • Sharp interface
  • Surface tension
  • Two-phase flow
  • Volume of fluid

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