Material and Shape Derivative Method for Quasi-Linear Elliptic Systems with Applications in Inverse Electromagnetic Interface Problems
Ivan Cimrák · SIAM Journal on Numerical Analysis · 2012
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 method. In this concept a vector field is introduced that deforms the unknown shape toward the optimum. We characterize the elliptic interface problems whose solutions give the material and the shape derivatives. In particular, we show the existence of weak as well as strong material derivatives. Further, we employ the adjoint variable method to obtain an explicit expression for the gradient of the shape functional. This gradient is then used for the actual implementation of the minimization algorithm. In simulations we use the level set method for the representation of the interface. We present the simulation results showing the reconstructed voids in the nonlinear ferromagnetic material from the near-boundary measurements of magnetic induction.