Three-dimensional shape optimization in stokes flow problems

Michael Zabarankin, Anton Molyboha

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

An integral equation constrained optimization approach to finding minimum-drag shapes for solid bodies of revolution in Stokes flows subject to constraints on body's volume and shape has been developed. An axially symmetric Stokes flow problem has been formulated in the form of an integral equation (state equation), and finding an optimal shape has been reduced to an integral equation constrained optimization problem. The total variation of the Lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation-based method, and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically. It has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula, which eliminates the need for solving the adjoint equation. The approach has been illustrated in finding the minimum-drag shape for a solid unit volume particle and in finding the optimal shape for the fixed-volume nose of a solid torpedo-shaped body encountering minimum drag along its axis of revolution. In the later problem, the dependence of the nose's optimal shape on the torpedo's length has been investigated.

Original languageEnglish
Pages (from-to)1788-1809
Number of pages22
JournalSIAM Journal on Applied Mathematics
Volume70
Issue number6
DOIs
StatePublished - 2010

Keywords

  • Adjoint equation-based method
  • Drag minimization
  • Generalized analytic functions
  • Integral equation
  • Optimal shape
  • Steepest descent method
  • Stokes flows

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