Finite element methods for Maxwell's equations.

Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

Proceedings of the American Mathematical Society

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Integral Equation Methods In Scattering Theory

This chapter is concerned with the introduction of the basic integral equations for two-dimensional elastic linear material problems. It starts by briefly reviewing the partial differential equations for linear elastic material and introducing the necessary notations involved in the formulation. These governing equations are also extended to deal with problems in which initial stress and strain type loads are applied. Such kind of loads are not only important to take into account temperature or other similar loads, but also to model nonlinear material behaviour when used in conjunction with a well established successive elastic solution technique.

Boundary Integral Equations

The nearly cloaking via transformation optics approach for the isotropic elastic wave fields is considered. This work extends the study of the nearly cloaking scheme to the elasticity system with residual stress in \mathbb{R}^{N} for N=2,3, which is an anisotropic elasticity system. It is worth to mention that there are no minor symmetric properties for the elastic tensor with residual stress and this system is invariant under a coordinate transformation. Therefore, we think the elasticity system residual stress model is more natural than the isotropic elasticity system in designing the elastic cloaking medium in the physical sense. In addition, the main difficulty of treating this problem lies in the fact that there are no layer potential theory for the residual stress system. Instead, we will derive suitable elliptic estimates for the elasticity system residual stress by comparing with the Lam\'e system to achieve desired results.

Nearly cloaking for the elasticity system with residual stress

We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lam\'e moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.

Boundary determination of electromagnetic and Lam\'e parameters with corrupted data

We consider the scattering of elastic waves by highly oscillating anisotropic periodic media with bounded support. Applying the two-scale homogenization, we first obtain a constant coefficient second-order partial differential elliptic equation that describes the wave propagation of the effective or overall wave field. We further pursue a higher-order homogenization with the help of complimentary boundary correctors and provide a detailed analysis on the rate of higher-order convergence. Finally we provide preliminary numerical examples to demonstrate the higher-order homogenization.

Leading and Second Order Homogenization of an Elastic Scattering Problem for Highly Oscillating Anisotropic Medium

In this paper we propose a domain sampling type reconstruction scheme for an inverse boundary value problem to identify an unknown cavity by single measurement on the accessible boundary of a known electric or heat conductive medium. Here the single measurement is to give single current or heat flux which can have a small support over the boundary, and we measure the corresponding voltage or temperature over the whole boundary. For this inverse boundary value problem, we adapted the single NRT (no response test) introduced by Luke and Potthast for inverse scattering problem and show that it can provide such a domain sampling type reconstruction scheme.

Reconstruction of unknown cavity by single measurement

A direct reconstruction algorithm based on Calderon's linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known \emph{a priori}, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderon-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.

Yi-Hsuan Lin

Papers

Nearly cloaking for the elasticity system with residual stress

Boundary determination of electromagnetic and Lam\'e parameters with corrupted data

Leading and Second Order Homogenization of an Elastic Scattering Problem for Highly Oscillating Anisotropic Medium

Reconstruction of unknown cavity by single measurement

A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calder\'on's method in the plane