We describe a new method for solving an inverse Dirichlet problem for harmonic functions that arises in the mathematical modelling of electrostatic and thermal imaging methods. This method may be interpreted as a hybrid of a decomposition method, in the spirit of a method developed by Kirsch and Kress, and a regularized Newton method for solving a nonlinear ill-posed operator equation, in terms of the solution operator that maps the unknown boundary onto the solution of the direct problem. As opposed to the Newton iterations the new method does not require a forward
solver. Its feasibility is demonstrated through numerical examples.
We consider the problem of determining the shape and location of cracks from Cauchy data on the boundary of semi-infinite domains modeling the reconstruction of cracks within a heat conducting medium from temperature and heat flux measurements. Our reconstructions are based on a pair of nonlinear integral equations for the unknown crack and the unknown flux jump on the crack that are linear with respect to the flux and nonlinear with respect to the crack. We propose two different iteration methods employing the following idea: Given an approximate reconstruction for the crack we first solve one of the equations for the flux and subsequently linearize the other equation for updating the crack. The foundations for this approach for solving the inverse problem in semi-infinite domains are provided and numerical experiments exhibit the feasibility of both methods and their stability with respect to noisy data.
Akduman, Haddar and Kress [1, 12, 15] proposed a conformal mapping approach for solving inverse boundary value problems for the two-dimensional Laplace equation in a doubly connected domain D with interior boundary curve Γ0 and exterior boundary curve Γ1. The inverse problem consists of reconstructing the interior boundary Γ0 from the Cauchy data on Γ1 of a harmonic function satisfying a homogeneous Dirichlet or Neumann boundary condition on the unknown interior boundary curve Γ0. The reconstruction method consists of two parts: In the first step, by successive approximations a nonlocal and nonlinear ordinary differential equation is solved to determine the boundary values of a holomorphic function Ψ on the outer boundary circle C1 of an annulus B. Then in the second step via regularizing a Laurent expansion in the sense of Tikhonov an ill-posed Cauchy problem is solved to determine Ψ in the annulus and the unknown Γ0 as the image Ψ(C0) of the interior boundary circle C0 of B. The present paper extends this approach to the case of a homogeneous impedance boundary condition. The analysis and the numerical implementation of the method differ from the limiting cases of the Dirichlet and Neumann conditions since the impedance problem in the annulus B that is associated with the impedance problem in the original domain D depends on the conformal map Ψ.
A new second order Newton method for reconstructing the shape of a sound soft scatterer from the measured far-field pattern for scattering of time harmonic plane waves is presented. This method extends a hybrid between regularized Newton iterations and decomposition methods that has been suggested and analyzed in a number of papers by Kress and Serranho [Inverse Problems 19: 91–104, 2003, Inverse Problems 21: 773–784, 2005, J. Comput. Appl. Math. 204: 418–427, 2007, Inverse Problems 22: 663–680, 2006, Inverse Problems and Imaging 1: 691–712, 2007] and has some features in common with the second degree method for ill-posed nonlinear problems as considered by Hettlich and Rundell [SIAM J.Numer. Anal. 37: 587–620, 2000]. The main idea of our iterative method is to use Huygen's principle, i.e., represent the scattered field as a single-layer potential. Given an approximation for the boundary of the scatterer, this leads to an ill-posed integral equation of the first kind that is solved via Tikhonov regularization. Then, in a second order Taylor expansion, the sound soft boundary condition is employed to update the boundary approximation. In an iterative procedure, these two steps are alternated until some stopping criterium is satisfied. We describe the method in detail and illustrate its feasibility through examples with exact and noisy data.
This paper contributes an analytical nonlinear morphing model for high-amplitude corrugated thinwalled laminates of arbitrary stack-up with a corrugation shape composed of circular sections. The model describes large deformations, the nonlinear relation between line force and global stretch, and the distribution of local line loads. The quarter-unit-cell approach together with assuming small material strains and a plane strain situation contribute to the model’s simplicity. It is explained how the solution procedure minimizes the force and moment residual of the equilibrium of cutting and reaction line loads by using Newton’s optimization method. Deformation results are verified by comparison with FEM simulation. The effects of laminate design and corrugation amplitude on deformations, line-force-stretch diagrams, and bending-curvature-stretch diagrams are presented and discussed.
The problem to determine the location and shape of a perfect conductor within a conducting homogeneous host medium from measured current and voltages on the accessible exterior boundary of the host medium can be modelled by an inverse Dirichlet boundary value problem for the Laplace equation. For this, recently Kress and Rundell suggested a novel inverse algorithm based on nonlinear integral equations arising from the reciprocity gap principle. The present paper extends this approach to the problem to recover the location and shape of a rigid body immersed in a fluid from the measured velocity and traction at the exterior boundary of the fluid, that is, to an inverse boundary value problem for the Stokes equation. The mathematical foundation of this extension is provided and numerical examples illustrate the feasibility of the method.