A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.

[1]  Christian Kahle,et al.  A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system , 2016, Optimization and Engineering.

[2]  H. Garcke,et al.  Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow , 2014, 1407.5470.

[3]  A. Chambolle,et al.  Design-dependent loads in topology optimization , 2003 .

[4]  Shinji Nishiwaki,et al.  Shape and topology optimization based on the phase field method and sensitivity analysis , 2010, J. Comput. Phys..

[5]  Harald Garcke,et al.  Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method , 2011 .

[6]  Endre Süli,et al.  Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow , 1997 .

[7]  Sébastien Boisgérault.  Shape derivative of sharp functionals governed by Navier-Stokes flow , 2017 .

[8]  M. Hinze,et al.  Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach , 2015, 1504.06402.

[9]  Christian Kahle,et al.  An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system , 2013, J. Comput. Phys..

[10]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[11]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[12]  Luc Tartar,et al.  Problemes de Controle des Coefficients Dans des Equations aux Derivees Partielles , 1975 .

[13]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

[14]  H. Goldberg,et al.  On NEMYTSKIJ Operators in Lp‐Spaces of Abstract Functions , 1992 .

[15]  Congsi Wang,et al.  On distorted surface analysis and multidisciplinary structural optimization of large reflector antennas , 2007 .

[16]  F. Murat.  Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients , 1977 .

[17]  Jan Sokolowski,et al.  Shape Derivative of Drag Functional , 2010, SIAM J. Control. Optim..

[18]  Olivier Pironneau,et al.  Optimal Shape Design , 2000 .

[19]  Enrique Fernández-Cara,et al.  The Differentiability of the Drag with Respect to the Variations of a Lipschitz Domain in a Navier--Stokes Flow , 1997 .

[20]  Ronald F. Gariepy.  FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs) , 2001 .

[21]  J. Zowe,et al.  Regularity and stability for the mathematical programming problem in Banach spaces , 1979 .

[22]  H. Sohr,et al.  The Navier-Stokes Equations: An Elementary Functional Analytic Approach , 2012 .

[23]  Harald Garcke,et al.  Sharp Interface Limit for a Phase Field Model in Structural Optimization , 2016, SIAM J. Control. Optim..

[24]  J. Petersson,et al.  Topology optimization of fluids in Stokes flow , 2003 .

[25]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[26]  S. Schmidt,et al.  Shape derivatives for general objective functions and the incompressible Navier-Stokes equations , 2010 .

[27]  Johan Hoffman,et al.  Adaptive finite element methods for incompressible fluid flow , 2001 .

[28]  M. Rumpf,et al.  A phase-field model for compliance shape optimization in nonlinear elasticity , 2012 .

[29]  Atsushi Kawamoto,et al.  Drag minimization and lift maximization in laminar flows via topology optimization employing simple objective function expressions based on body force integration , 2012 .

[30]  Jacques Simon,et al.  Domain variation for drag in stokes flow , 1991 .

[31]  Jürgen Appell,et al.  Nonlinear Superposition Operators , 1990 .

[32]  Luise Blank,et al.  An Extension of the Projected Gradient Method to a Banach Space Setting with Application in Structural Topology Optimization , 2015, SIAM J. Control. Optim..

[33]  O. Pironneau.  On optimum design in fluid mechanics , 1974 .

[34]  Shiwei Zhou,et al.  Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition , 2006 .

[35]  H. Garcke,et al.  Shape and Topology Optimization in Stokes Flow with a Phase Field Approach , 2016 .

[36]  Harald Garcke,et al.  Numerical Approximation of Phase Field Based Shape and Topology Optimization for Fluids , 2014, SIAM J. Sci. Comput..

[37]  L. Modica.  The gradient theory of phase transitions and the minimal interface criterion , 1987 .

[38]  Fredi Tröltzsch,et al.  Optimal Control of Coupled Systems of Partial Differential Equations , 2009 .

[39]  S. M. Robinson.  Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems , 1976 .

[40]  C. Hecht.  Shape and topology optimization in fluids using a phase field approach and an application in structural optimization , 2014 .

[41]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[42]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[43]  Stefan Ulbrich,et al.  A Continuous Adjoint Approach to Shape Optimization for Navier Stokes Flow , 2009 .

[44]  谷内 靖,et al.  書評 Hermann Sohr : The Navier-Stokes Equations : An Elementary Functional Analytic Approach , 2014 .

[45]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[46]  Michael Hintermüller,et al.  Distortion compensation as a shape optimisation problem for a sharp interface model , 2016, Computational Optimization and Applications.