Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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PENALIZED APPROACH AND ANALYSIS OF AN OPTIMAL SHAPE CONTROL PROBLEM FOR THE STATIONARY NAVIER-STOKES EQUATIONS

  • Kim, Hong-Chul (Department of Mathematics, Kangnung National University)
  • Published : 2001.01.01
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Abstract

This paper is concerned with an optimal shape control problem for the stationary Navier-Stokes system. A two-dimensional channel flow of an incompressible, viscous fluid is examined to determine the shape of a bump on a part of the boundary that minimizes the viscous drag. by introducing an artificial compressibility term to relax the incompressibility constraints, we take the penalty method. The existence of optima solutions for the penalized problem will be shown. Next, by employing Lagrange multipliers method and the material derivatives, we derive the shape gradient for the minimization problem of the shape functional which represents the viscous drag.

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References

  1. Sobolev spaces R.Adams
  2. Numer. Math. v.36 Finite dimensional approximation of nonlinear problems. Part Ⅰ: Branches of nonsingular solutions F. Brezzi;J. Rappaz;P.A. Raviart
  3. J. Functional Analysis v.104 Structure of shape derivatives for non-smooth domains M. Delfour;J. Zolesio
  4. SIAM J. Control and Optim. v.24 Necessary conditions for a domain optimization problem in elliptic boundary value problems N. Fujii
  5. Finite element methods for Navier-Stokes equations: Theory and algorithms V. Girault;P.A. Raviart
  6. SIAM J. Numerical Analysis v.29 Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses M. Gunzburger;L. Hou
  7. SIAM J. Cotrol Optim v.36 Existence of an optimal solution of a shpae control problem for the stationary Navier-Stokes equations M. Gunzburger;H. Kim
  8. Kangweon-Kyungki Math. J. v.4 Penalized Navier-Stokes equations with inhomogeneous boundary conditions Hongchul Kim
  9. Optimal shape design for elliptic systems O. Pironneau
  10. Lecture Notes in Control and Information Sciences v.125 Optimisation d'un systeme surfacique In: Control of Boundaries and Stabilization B. Rousselet
  11. Introduction to shape optimizations: Shape sensitivity analysis J. Sokolowski;J. P. Zolesio
  12. Navier-Stokes equations and nonlinear founctional analysis R. Temam
  13. Optimization of Distributed Parameter Structures The material derivative (or speed) method for shape optimization J. Zolesio;E. Haug(ed.);J. Cea(ed.)