A moving mesh fictitious domain approach for shape optimization problems

Raino A.E. Mäkinen; Tuomo Rossi; Jari Toivanen

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 1, page 31-45
  • ISSN: 0764-583X

Abstract

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A new numerical method based on fictitious domain methods for shape optimization problems governed by the Poisson equation is proposed. The basic idea is to combine the boundary variation technique, in which the mesh is moving during the optimization, and efficient fictitious domain preconditioning in the solution of the (adjoint) state equations. Neumann boundary value problems are solved using an algebraic fictitious domain method. A mixed formulation based on boundary Lagrange multipliers is used for Dirichlet boundary problems and the resulting saddle-point problems are preconditioned with block diagonal fictitious domain preconditioners. Under given assumptions on the meshes, these preconditioners are shown to be optimal with respect to the condition number. The numerical experiments demonstrate the efficiency of the proposed approaches.

How to cite

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Mäkinen, Raino A.E., Rossi, Tuomo, and Toivanen, Jari. "A moving mesh fictitious domain approach for shape optimization problems." ESAIM: Mathematical Modelling and Numerical Analysis 34.1 (2010): 31-45. <http://eudml.org/doc/197532>.

@article{Mäkinen2010,
abstract = { A new numerical method based on fictitious domain methods for shape optimization problems governed by the Poisson equation is proposed. The basic idea is to combine the boundary variation technique, in which the mesh is moving during the optimization, and efficient fictitious domain preconditioning in the solution of the (adjoint) state equations. Neumann boundary value problems are solved using an algebraic fictitious domain method. A mixed formulation based on boundary Lagrange multipliers is used for Dirichlet boundary problems and the resulting saddle-point problems are preconditioned with block diagonal fictitious domain preconditioners. Under given assumptions on the meshes, these preconditioners are shown to be optimal with respect to the condition number. The numerical experiments demonstrate the efficiency of the proposed approaches. },
author = {Mäkinen, Raino A.E., Rossi, Tuomo, Toivanen, Jari},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Shape optimization; fictitious domain method; preconditioning; boundary variation technique; sensitivity analysis.; fictitious domain methods; shape optimization; Poisson equation; boundary variation technique; Lagrange multipliers; saddle-point problems; numerical experiments},
language = {eng},
month = {3},
number = {1},
pages = {31-45},
publisher = {EDP Sciences},
title = {A moving mesh fictitious domain approach for shape optimization problems},
url = {http://eudml.org/doc/197532},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Mäkinen, Raino A.E.
AU - Rossi, Tuomo
AU - Toivanen, Jari
TI - A moving mesh fictitious domain approach for shape optimization problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 1
SP - 31
EP - 45
AB - A new numerical method based on fictitious domain methods for shape optimization problems governed by the Poisson equation is proposed. The basic idea is to combine the boundary variation technique, in which the mesh is moving during the optimization, and efficient fictitious domain preconditioning in the solution of the (adjoint) state equations. Neumann boundary value problems are solved using an algebraic fictitious domain method. A mixed formulation based on boundary Lagrange multipliers is used for Dirichlet boundary problems and the resulting saddle-point problems are preconditioned with block diagonal fictitious domain preconditioners. Under given assumptions on the meshes, these preconditioners are shown to be optimal with respect to the condition number. The numerical experiments demonstrate the efficiency of the proposed approaches.
LA - eng
KW - Shape optimization; fictitious domain method; preconditioning; boundary variation technique; sensitivity analysis.; fictitious domain methods; shape optimization; Poisson equation; boundary variation technique; Lagrange multipliers; saddle-point problems; numerical experiments
UR - http://eudml.org/doc/197532
ER -

References

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