Material and Shape Derivative Method for Quasi-Linear Elliptic Systems with Applications in Inverse Electromagnetic Interface Problems

@article{Cimrk2012MaterialAS,
  title={Material and Shape Derivative Method for Quasi-Linear Elliptic Systems with Applications in Inverse Electromagnetic Interface Problems},
  author={Ivan Cimr{\'a}k},
  journal={SIAM J. Numer. Anal.},
  year={2012},
  volume={50},
  pages={1086-1110}
}
  • I. Cimrák
  • Published 10 May 2012
  • Computer Science, Mathematics
  • SIAM J. Numer. Anal.
We study a shape optimization problem for quasi-linear elliptic systems. The state equations describe an interface problem and the ultimate goal of our research is to determine the interface between two materials with different physical properties. The interface is identified by the minimization of the shape (or the cost) functional representing the misfit between the data and the simulations. For shape sensitivity of the shape functional we elaborate the material and the shape derivative… 
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References

SHOWING 1-10 OF 38 REFERENCES
Introduction to shape optimization : shape sensitivity analysis
This book presents modern functional analysis methods for the sensitivity analysis of some infinite-dimensional systems governed by partial differential equations. The main topics are treated in a
Finite element methods for semilinear elliptic and parabolic interface problems
The purpose of this paper is to study the finite element methods for second-order semilinear elliptic and parabolic interface problems in two dimensional convex polygonal domains. Optimal order error
The finite element method for elliptic equations with discontinuous coefficients
  • I. Babuska
  • Mathematics, Computer Science
    Computing
  • 2005
TLDR
The proposed approach on a model problem — the Dirichlet problem with an interface for Laplace equation with sufficient condition for the smoothnees can be determined, and the boundary of the domain and the interface will be assumed smooth enough.
Adjoint variable method for time‐harmonic Maxwell equations
Purpose – The purpose of this paper is to study the optimization problem of low‐frequency magnetic shielding using the adjoint variable method (AVM). This method is compared with conventional methods
Level set method for the inverse elliptic problem in nonlinear electromagnetism
TLDR
Simulations has been performed showing the robustness of the algorithm and its ability to reconstruct single inhomogeneities, convex and non-convex, as well as multiple inhomogenities.
Level-set function approach to an inverse interface problem
A model problem in electrical impedance tomography for the identification of unknown shapes from data in a narrow strip along the boundary of the domain is investigated. The representation of the
Shape identification for natural convection problems using the adjoint variable method
An inverse geometry problem is investigated to identify the boundary shape of a domain from temperature measurements on the other boundary, where the temperature field is dominated by natural
Finite element methods and their convergence for elliptic and parabolic interface problems
Abstract. In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in two-dimensional convex polygonal domains. Nearly the same optimal
Finite Element Approximation for Optimal Shape, Material and Topology Design
Preliminaries. Abstract Setting of the Optimal Shape Design Problem and Its Approximation. Optimal Shape Design of Systems Governed by a Unilateral Boundary Value State Problem the Scalar Case.
Interface Problems for Quasilinear Elliptic Equations
In this paper we consider interface problems for quasilinear elliptic partial differential equations in two-dimensional spaces. The main result is that the bounded weak solution in the neighborhood
...
1
2
3
4
...