Advertisement

Adaptive Path Following Primal Dual Interior Point Methods for Shape Optimization of Linear and Nonlinear Stokes Flow Problems

  • Ronald H. W. Hoppe
  • Christopher Linsenmann
  • Harbir Antil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)

Abstract

We are concerned with structural optimization problems in CFD where the state variables are supposed to satisfy a linear or nonlinear Stokes system and the design variables are subject to bilateral pointwise constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. The discretization is taken care of by Taylor-Hood elements with respect to a simplicial triangulation of the computational domain. The efficient numerical solution of the discretized problem relies on adaptive path-following techniques featuring a predictor-corrector scheme with inexact Newton solves of the KKT system by means of an iterative null-space approach. The performance of the suggested method is documented by several illustrative numerical examples.

Keywords

Design Variable Outlet Boundary Curve Representation Shape Optimization Problem Reference Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allaire, G.: Shape Optimization by the Homogenization Method. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Antil, H., Hoppe, R.H.W., Linsenmann, C.: Path-following primal-dual interior-point methods for shape optimization of stationary flow problems. Journal of Numerical Mathematics (to appear 2007)Google Scholar
  3. 3.
    Bendsøe, M.P.: Optimization of Structural Topology, Shape, and Material. Springer, Berlin (1995)CrossRefzbMATHGoogle Scholar
  4. 4.
    Biros, G., Ghattas, O.: Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part i: The Krylov-Schur solver. SIAM J. Sci. Comp (to appear 2004)Google Scholar
  5. 5.
    Biros, G., Ghattas, O.: Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part ii: The Lagrange-Newton solver and its application to optimal control of staedy viscous flows. SIAM J. Sci. Comp (to appear 2004)Google Scholar
  6. 6.
    Delfour, M.C., Zolesio, J.P.: Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM, Philadelphia (2001)zbMATHGoogle Scholar
  7. 7.
    Deuflhard, P.: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer, Berlin (2004)zbMATHGoogle Scholar
  8. 8.
    Griewank, A.: Evaluating Derivatives, Principles and Techniques of Automatic Differentiation. SIAM, Phildelphia (2000)zbMATHGoogle Scholar
  9. 9.
    Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization: Theory, Approximation, and Computation. SIAM, Philadelphia (2004)zbMATHGoogle Scholar
  10. 10.
    Hoppe, R.H.W., Linsenmann, C., Petrova, S.I.: Primal-dual Newton methods in structural optimization. Comp. Visual. Sci. 9, 71–87 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hoppe, R.H.W., Litvinov, W.G.: Problems on electrorheological fluid flows. Communications in Pure and Applied Analysis 3, 809–848 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hoppe, R.H.W., Petrova, S.I.: Primal-dual Newton interior point methods in shape and topology optimization. Numerical Linear Algebra with Applications 11, 413–429 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids. Oxford University Press, Oxford (2001)zbMATHGoogle Scholar
  14. 14.
    Rozvany, G.: Structural Design via Optimality Criteria. Kluwer, Dordrecht (1989)CrossRefzbMATHGoogle Scholar
  15. 15.
    Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ronald H. W. Hoppe
    • 1
    • 2
  • Christopher Linsenmann
    • 1
    • 2
  • Harbir Antil
    • 1
  1. 1.Department of MathematicsUniversity of HoustonUSA
  2. 2.Institute for MathematicsUniversity of AugsburgGermany

Personalised recommendations