Interface problems for quasi-linear elliptic equations by material and shape derivative methods - Cimrák, Ivan (Proceedings of the Conference Algoritmy, 2015)
A shape determinaton problem is studied for quasilinear elliptic problems. Such problems describe interface problems. The ultimate goal of our research is to determine the interface between two materials with di®erent physical properties. The interface is identi¯ed by the minimization of the shape (or the cost) functional evaluating the mis¯t between the data and the simulations. We elaborate the material and shape derivative method. We characterize the elliptic interface problems whose solutions give the material and shape derivatives. Further we employ the adjoint variable method to obtain an explicit expression for the gradient of the shape functional. For the representation of the interface we use the level set method. Simulation presented show the reconstruction of voids in a nonlinear ferromagnetic material. Available data are measurements of magnetic induction.