Diehl, Elisabeth ; Haubner, Johannes ; Ulbrich, Michael ; Ulbrich, Stefan (2022)
Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension.
In: Computational Optimization and Applications
doi: 10.1007/s10589-022-00415-6
Artikel, Bibliographie
Kurzbeschreibung (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.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2022 |
| Autor(en): | Diehl, Elisabeth ; Haubner, Johannes ; Ulbrich, Michael ; Ulbrich, Stefan |
| Art des Eintrags: | Bibliographie |
| Titel: | Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension |
| Sprache: | Englisch |
| Publikationsjahr: | 2022 |
| Ort: | New York |
| Verlag: | Springer |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | Computational Optimization and Applications |
| DOI: | 10.1007/s10589-022-00415-6 |
| URL / URN: | https://link.springer.com/article/10.1007/s10589-022-00415-6 |
| Kurzbeschreibung (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. |
| Freie Schlagworte: | Two-phase flow, Surface tension, Sharp interface, Navier-Stokes equations, Volume of fluid, Differentiability, Optimal control |
| Fachbereich(e)/-gebiet(e): | DFG-Sonderforschungsbereiche (inkl. Transregio) DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen > Projektbereich B: Modellierung und Simulation DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen > Projektbereich B: Modellierung und Simulation > B04: Simulationsbasierte Optimierung und Optimales Design von Experimenten für Benetzungsvorgänge 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Optimierung 04 Fachbereich Mathematik > Optimierung > Nonlinear Optimization |
| Hinterlegungsdatum: | 07 Dez 2023 13:32 |
| Letzte Änderung: | 07 Dez 2023 13:32 |
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