Skip to main content
SpringerLink
Log in

Coupling of FEM and BEM in Shape Optimization

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape, especially L 2-tracking type functionals. The underlying state equation is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that the shape Hessian is not strictly H 1/2-coercive at the optimal domain which implies ill-posedness of the optimization problem under consideration. Since the adjoint state depends directly on the state, we propose a coupling of finite element methods (FEM) and boundary element methods (BEM) to realize an efficient first order shape optimization algorithm. FEM is applied in the compact region while the rest is treated by BEM. The coupling of FEM and BEM essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bramble J., Pasciak J.E. (1988). A preconditioner technique for indefinite systems resulting from mixed approximation of elliptic problems. Math Comput 50, 1–17

    Article  MathSciNet  MATH  Google Scholar 

  2. Bramble J., Pasciak J.E., Xu J. (1990) Parallel multilevel preconditioners. Math Comput 55, 1–22

    Article  MathSciNet  MATH  Google Scholar 

  3. Chenais D. (1975) On the existence of a solution in a domain identification problem. J Math Anal Appl 52, 189–219

    Article  MathSciNet  MATH  Google Scholar 

  4. Costabel M., Stephan E.P. (1988) Coupling of finite element and boundary element methods for an elasto-plastic interface problem. SIAM J Num Anal 27, 1212–1226

    Article  MathSciNet  Google Scholar 

  5. Dahmen W., Kunoth A. (1992) Multilevel preconditioning. Num Math 63, 315–344

    Article  MathSciNet  MATH  Google Scholar 

  6. Dambrine M., Pierre M. (2000) About stability of equilibrium shapes. M2AN 34(4): 811–834

    Article  MathSciNet  MATH  Google Scholar 

  7. Dambrine M. (2002) On variations of the shape Hessian and sufficient conditions for the stability of critical shapes. RACSAM Rev R Acad Cien Serie A Mat 96(1): 95–121

    MathSciNet  MATH  Google Scholar 

  8. Delfour M., Zolesio J.-P. (2001) Shapes and geometries. SIAM, Philadelphia

    MATH  Google Scholar 

  9. Eppler K. (2000) Boundary integral representations of second derivatives in shape optimization. Discussiones Mathematicae (Differential Inclusion Control and Optimization) 20, 63–78

    MathSciNet  MATH  Google Scholar 

  10. Eppler K. (2000) Optimal shape design for elliptic equations via BIE-methods. J Appl Math Comput Sci 10, 487–516

    MathSciNet  MATH  Google Scholar 

  11. Eppler K., Harbrecht H. (2003) Numerical solution of elliptic shape optimization problems using wavelet-based BEM. Optim Methods Softw 18, 105–123

    Article  MathSciNet  MATH  Google Scholar 

  12. Eppler K., Harbrecht H. (2006) 2nd Order shape optimization using wavelet BEM. Optim Methods Softw 21, 135–153

    Article  MathSciNet  MATH  Google Scholar 

  13. Eppler K., Harbrecht H. (2005) Exterior electromagnetic shaping using wavelet BEM. Math Meth Appl Sci 28, 387–405

    Article  MathSciNet  MATH  Google Scholar 

  14. Eppler K., Harbrecht H. (2005) Fast wavelet BEM for 3d electromagnetic shaping. Appl Numer Math 54, 537–554

    Article  MathSciNet  MATH  Google Scholar 

  15. Eppler K., Harbrecht H. (2005) A regularized Newton method in electrical impedance tomography using shape Hessian information. Control Cybern 34, 203–225

    MathSciNet  MATH  Google Scholar 

  16. Eppler, K., Harbrecht, H. Shape optimization for 3D electrical impedance tomography, WIAS-Preprint 963, WIAS Berlin. In: Glowinski, R., Zolesio, J.P. (eds.) Lecture Notes in Pure and Applied Mathematics, Vol. 252. Dekkar/CRC Press (to appear, 2004).

  17. Eppler, K., Harbrecht, H. Efficient treatment of stationary free boundary problems. In: WIAS-Preprint 965, WIAS Berlin. Appl Numer Math (to appear, 2004)

  18. Eppler, K., Harbrecht, H., Schneider, R. On convergence in elliptic shape optimization. In: WIAS-Preprint 1016, Berlin: WIAS 2005. SICON. SIAM J Control Optim (submitted)

  19. Eppler, K., Harbrecht, H. Compact gradient tracking in shape optimization. In: WIAS-Preprint 1054, WIAS Berlin, 2005. Comput Optim Appl (submitted)

  20. Gill P.E., Murray W., Wright M.H. (1981) Practical optimization. Academic, New York

    MATH  Google Scholar 

  21. Grossmann Ch., Terno J. (1993) Numerik der Optimierung. Teubner, Stuttgart

    MATH  Google Scholar 

  22. Harbrecht H., Paiva F., Pérez C., Schneider R. (2002) Biorthogonal wavelet approximation for the coupling of FEM-BEM. Numer Math 92, 325–356

    Article  MathSciNet  MATH  Google Scholar 

  23. Harbrecht H., Paiva F., Pérez C., Schneider R. (2003) Wavelet preconditioning for the coupling of FEM-BEM. Numer Linear Algebra Appl 3, 197–222

    Article  Google Scholar 

  24. Haslinger J., Neitaanmäki P. (1996) Finite element approximation for optimal shape, material and topology design, 2nd edn. Wiley, Chichester

    MATH  Google Scholar 

  25. Han H. (1990) A new class of variational formulation for the coupling of finite and boundary element methos. J Comput Math 8(3): 223–232

    MathSciNet  MATH  Google Scholar 

  26. Heise B., Kuhn M. (1996) Parallel solvers for linear and nonlinear exterior magnetic field problems based upon coupled FE/BE formulations. Computing 56, 237–258

    Article  MathSciNet  MATH  Google Scholar 

  27. Hörmander L. (1983-85) The analysis of linear partial differential operators. Springer, Berlin Heidelberg New York

    Google Scholar 

  28. Jung M., Steinbach O. (2002) A finite element-boundary element algorithm for inhomogeneous boundary value problems. Computing 68, 1–17

    Article  MathSciNet  MATH  Google Scholar 

  29. Khludnev A.M., Sokolowski J. (1997) Modelling and control in solid mechanics. Birkhäuser, Basel

    MATH  Google Scholar 

  30. Maz’ya, V.G., Shaposhnikova, T.O. Theory of multipliers in spaces of differentiable functions. Pitman, Boston, 1985.(Monographs and Studies in Mathematics, 23. Pitman Advanced Publishing Program. Boston: Pitman Publishing Inc. XIII, 344 p. (1985))

  31. Murat, F., Simon, J. Étude de problémes d’optimal design. in Optimization Techniques, Modeling and Optimization in the Service of Man, edited by J Céa, Lect Notes Comput Sci 41, Berlin Heidelberg Newyork: Springer, pp 54–62 1976

  32. Pironneau O. (1983) Optimal shape design for elliptic systems. Springer, Berlin Heidelberg Newyork

    Google Scholar 

  33. Schneider R. (1998) Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur Lösung großer vollbesetzter Gleichungssyteme. B.G. Teubner, Stuttgart

    Google Scholar 

  34. Simon J. (1980) Differentiation with respect to the domain in boundary value problems. Numer Funct Anal Optim 2, 649–687

    Article  MathSciNet  MATH  Google Scholar 

  35. Sokolowski J., Zolesio J.-P. (1992) Introduction to shape optimization. Springer, Berlin Heidelberg Newyork

    MATH  Google Scholar 

  36. Sverák, V. On optimal shape design. J Math Pures Appl (9) 72, 537–551 (1993)

    Google Scholar 

  37. Triebel H. (1983) Theory of function spaces. Birkhäuser, Basel

    Google Scholar 

  38. Zenisek A. (1990) Nonlinear elliptic and evolution problems and their finite element approximation. Academic, London

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117, Berlin, Germany

    Karsten Eppler

  2. Institut für Informatik und Praktische Mathematik, Christian–Albrechts–Universität zu Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Helmut Harbrecht

Authors
  1. Karsten Eppler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helmut Harbrecht
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helmut Harbrecht.

Additional information

This research has been carried out when the second author stayed at the Department of Mathematics, Utrecht University, The Netherlands, supported by the EU-IHP project Nonlinear Approximation and Adaptivity: Breaking Complexity in Numerical Modelling and Data Representation

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eppler, K., Harbrecht, H. Coupling of FEM and BEM in Shape Optimization. Numer. Math. 104, 47–68 (2006). https://doi.org/10.1007/s00211-006-0005-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-006-0005-6

Keywords

Search

Navigation