Fernando Reitich

Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain

In this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials. This corresponds to a two-dimensional quasi-periodic boundary value problem for the Helmholtz equation. We prove that solutions behave analytically with respect to variations of the interface. The interest of this result is both theoretical – the legitimacy of power series expansions in the parameters of the problem has indeed been questioned – and, perhaps more importantly, practical: we have found that the solution can be computed on the basis of this observation. The simple algorithm that results from such boundary variations is described. To establish the property of analyticity of the solution for the grating with respect to the height δ, we present a holomorphic formulation of the problem using surface potentials. We show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary, or, in other words, that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables ( x , y , δ). This permits us to conclude, in particular, that the partial derivatives of u with respect to δ at δ = 0 satisfy certain boundary value problems for the Helmholtz equation, in regions with plane boundaries, which can be solved in a closed form.

The overall elastic energy of polycrystalline martensitic solids

Abstract We are concerned with the overall elastic energy in martensitic polycrystals. These are polycrystals whose constituent crystallites can undergo shape-deforming phase transitions as a result of changes in their stress or temperature. We approach the problem of calculation of the nonlinear overall energy via a statistical optimization method which involves solution of a sequence of linear elasticity problems. As a case study we consider simulations on a two-dimensional model in which circular randomly-oriented crystallites are arranged in a square pattern within an elastic matrix. The performance of our present code suggests that this approach can be used to compute the overall energies in realistic three-dimensional polycrystals containing grains of arbitrary shape. In addition to numerical results we present upper bounds on the overall energy. Some of these bounds apply to the square array mentioned above. Others apply to polycrystals containing circular, randomly-oriented crystallites with sizes ranging to infinitesimal, and no intergrain matrix. The square-array bounds are consistent with our numerical results. In some regimes they approximate them closely, thus providing an insight on the convergence of the numerical method. On the other hand, in the case of the random array the bounds carry substantial practical significance, since in this case the energy contains no artificial contributions from an elastic matrix. In all the cases we have considered our bounds compare favorably with those obtained under the well-known Taylor hypothesis; they show that, as far as polycrystalline martensite is concerned, calculations of the elastic energy based on the Taylor assumption may lead to substantial overestimates of this quantity.

Maxwell equations in a nonlinear Kerr medium

In this paper we present an exact calculation of the transfer function associated with the nonlinear Fabry–Perot resonator. While our exact result cannot be evaluated in terms of elementary functions, it does permit us to obtain a number of simple approximate expressions of various orders of accuracy. In addition, our derivation yields criteria of validity for the approximate formulae. Our approach is to be compared with others in which approximations are introduced in the model itself, either through the equations or through the boundary conditions. Our lowest order approximate formula turns out to be identical, interestingly, with the result obtained from the slowly varying envelope approximation (SVEA). Thus, our validity criteria apply to the SVEA result, and predict well its domain of validity and its breakdown for short wavelengths and for very high intensities and nonlinearities. The simple higher order formulae we present provide improved estimations in such regimes.

Bounds on the Effective Elastic Properties of Martensitic Polycrystals

We draw attention to the problem of estimation of elastic energies in martensitic polycrystals. In particular we introduce a tensorial parameter η=ηijkl which contains information about the microgeometry and disorder of the polycrystalline structure. Under the assumption of isotropic elasticity and mild hypothesis on the statistics of the polycrystal, this parameter allows for explicit calculation of rigorous and stringent upper bounds on the effective energy. For circular grains in two dimensions η gives the elastic energy resulting from transformation of a single circular inclusion in an elastic matrix and the bounds coincide with those derived recently by Bruno, Reitich and Leo. Consideration of such particular cases shows that our bounds can yield substantial improvements over those obtained under Taylor’s constant strain hypothesis. For arbitrary microgeometries the statistical parameter η can be calculated by means of two-point correlations functions.

New approach to the solution of problems of scattering by bounded obstacles

We introduce a new numerical method, based on rigorous perturbative techniques, for the calculation of the patterns of electromagnetic scattering produced by bounded obstacles. As preliminary examples show, our method can lead to results of good accuracy in a wide variety of challenging problems involving large scatterers.

Accurate calculation of diffraction-grating efficiencies

We present a new numerical method for the solution of problems of diffraction of light by a singly or doubly periodic interface between two materials. Our basic result is that the diffracted fields behave analytically with respect to variations of the interface, so that they can be represented by convergent series in powers of the height of the grating profile. A second element in the theory consists of a simple algebraic recursive formula with allows us to obtain the power series by considering a sequence of diffraction problems with flat interface. Once the Taylor coefficients have been computed, we use Pade approximants to extract the values of the fields from their power series expansions. This results in accurate predictions for the efficiencies in the resonance region; in many cases these values are several orders of magnitude more accurate than those obtained by currently available methods. For three dimensional biperiodic gratings, the performance of our method is of the same quality as far as for the singly periodic case. We demonstrate the wide applicability and accuracy of our algorithm with numerical results for two- and three-dimensional problems, and we compare our predictions with some experimental data.