Asymptotic expansions for currents caused by small interface changes of an electromagnetic inclusion

We consider solutions to the Helmholtz equation in two dimensions. The aim of this article is to advance the development of high-order asymptotic expansions for boundary perturbations of currents caused by small perturbations of the shape of an inhomogeneity with 𝒞²-boundary. The work represents a natural completion of Ammari et al. [H. Ammari, H. Kang, M. Lim, and H. Zribi, Conductivity interface problems. Part I: Small perturbations of an interface, Trans. Am. Math. Soc. 363 (2010), pp. 2901–2922], where the solution for the Helmholtz equation is represented by a system and the proof of our asymptotic expansion is radically different from Ammari et al. (2010). Our derivation is rigorous and is based on the field expansion method. Its proof relies on layer potential techniques. It plays a key role in developing effective algorithms to determine certain properties of the shape of an inhomogeneity based on boundary measurements.     

Reconstructing small perturbations of an obstacle for acoustic waves from boundary measurements on the perturbed shape itself

We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigorous by using a systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann (DNO) and Neumann-to-Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower order Fourier coefficients of the shape perturbation of the obstacle.     

An asymptotic expansion for perturbations in the displacement field due to the presence of thin interfaces

We rigorously derive an asymptotic expansion for two-dimensional displacement field associated with thin elastic inclusion having no uniform thickness. Our approach is based on layer potential techniques through integral representation formulas of the fields. We extend these techniques to determine a relationship between traction–displacement measurements and the shape of the thin inclusion.     

Reconstructing small perturbations of scatterers from electric or acoustic far‐field measurements

In this paper, we consider the problem of determining the boundary perturbations of an object from far‐field electric or acoustic measurements. Assuming that the unknown scatterer boundary is a small perturbation of a circle, we develop a linearized relationship between the far‐field data and the shape of the object. This relationship is used to find the Fourier coefficients of the perturbation of the shape. Copyright © 2008 John Wiley & Sons, Ltd.     

Asymptotic expansions for the voltage potentials with two‐dimensional and three‐dimensional thin interfaces

We derive asymptotic formulae for two‐dimensional and three‐dimensional steady state voltage potentials associated with thin conductivity imperfections having no uniform thickness. These formulae recover highly conducting inclusions and those with interfacial resistance. Our calculations are rigorous and based on layer potential techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

An effective asymptotic method in the axisymmetric frictionless contact problem for an elastic layer of finite thickness

A brief review of asymptotic methods to deal with frictionless unilateral contact problems for an elastic layer of finite thickness is presented. Under the assumption that the contact radius is small with respect to the layer thickness, an effective asymptotic method is suggested for solving the unilateral contact problem with a priori unknown contact radius. A specific feature of the method is that the construction of an asymptotic approximation is reduced to a linear algebraic system with respect to integral characteristics (polymoments) of the contact pressure. As an example, the sixth‐order asymptotic model has been written out. Copyright © 2015 John Wiley & Sons, Ltd.     

Non‐classical boundary value problems of elastostatics and hydrostatics on Lipschitz domains

The main goal of this paper is to undertake a systematic study of the Stokes and Lamé systems with non‐classical boundary conditions in arbitrary Lipschitz subdomains of R ³. These include prescribing 〈 n , u 〉 in concert with n ×curl u on the boundary of the domain in question. Copyright © 2001 John Wiley & Sons, Ltd.     

A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method (IPFEM) is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.     

Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials

In this article, we present a strategy of using rectangular and triangular Bézier surface patches for nonelement representation of 3D boundary geometries for problems of linear elasticity. The boundary generated in this way is directly incorporated in the parametric integral equation system (PIES), which has been developed by the authors. The boundary values on each surface patch are approximated by Lagrange polynomials. Three illustrative examples are presented to confirm the effectiveness of the proposed boundary representation in connection with PIES and to show good accuracy of numerical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 51–79, 2018     

Simplified immersed interface methods for elliptic interface problems with straight interfaces

In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one‐dimensional problems or two‐dimensional problems with circular interfaces, we propose a conservative second‐order finite difference scheme whose coefficient matrix is symmetric and definite. For two‐dimensional problems with straight interfaces, we first propose a conservative first‐order finite difference scheme, then use the Richardson extrapolation technique to get a second‐order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 188–203, 2012     

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

A second‐order immersed interface technique for an elliptic Neumann problem

A second‐order finite difference scheme for mixed boundary value problems is presented. This scheme does not require the tangential derivative of the Neumann datum. It is designed for applications in which the Neumann condition is available only in discretized form. The second‐order convergence of the scheme is proven and the theory is validated by numerical examples. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 400–420, 2007     

Wave front sets of the Riemann function of elastic interface problems

We obtain an inner and an outer estimates of wave front sets and analytic wave front sets of the Riemann function of elastic interface problems by using the localization method due to Wakabayashi. In our problem the outer estimate of wave front sets and analytic wave front sets of the Riemann function coincides with the inner estimate of those. The strong point of our results is to catch the lateral wave as well as the incident, the reflected, and the refracted waves. Copyright © 2004 John Wiley & Sons, Ltd.     

A three-point Taylor algorithm for three-point boundary value problems

We consider second-order linear differential equations φ(x)y^″+f(x)y^′+g(x)y=h(x) in the interval (−1,1) with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions given at three points of the interval: the two extreme points x=±1 and an interior point x=s∈(−1,1). We consider φ(x), f(x), g(x) and h(x) analytic in a Cassini disk with foci at x=±1 and x=s containing the interval [−1,1]. The three-point Taylor expansion of the solution y(x) at the extreme points ±1 and at x=s is used to give a criterion for the existence and uniqueness of the solution of the boundary value problem. This method is constructive and provides the three-point Taylor approximation of the solution when it exists. We give several examples to illustrate the application of this technique.     

An asymptotic expansion for 1D steady compressible Navier-Stokes equations under nonuniform enthalpy

The solutions of the one-dimensional (1D) steady compressible Navier-Stokes equations have been thoroughly discussed before, but restrained for uniform total enthalpy, which leads to only a shock wave profile possible in an infinite domain. To date, very little progress has been made for the case with nonuniform total enthalpy. In this paper, we affirm that under nonuniform total enthalpy, there also exists steady solution for the 1D compressible Navier-Stokes equations, but the flow domain must be finite in the positive $x$-axis. The 1D steady compressible Navier-Stokes equations can be reduced to a singular perturbed nonlinear ordinary differential equation (ODE) for velocity with the assumptions of $\text{◂=▸}Pr=3/4$ and a constant viscosity coefficient. By analyzing the mathematical property of the nonlinear ODE for velocity, we propose an asymptotic expansion for the solution of it as an exponential type sequence and also prove the convergence. Unlike the case of uniform total enthalpy, where the solutions for all variables keep monotone, we show that under nonuniform total enthalpy and some specific boundary conditions, there exists extreme inside the thin boundary layer. Numerical results verify the accuracy and convergence of the asymptotic expansion. This asymptotic expansion solution can serve as an important testing to demonstrate the efficiency of numerical methods developed for compressible Navier-Stokes equations at high Reynolds number.     

A method of convolution of Fourier expansions as applied to solving boundary value problems with intersecting interface lines

An efficient method of construction of solutions to a set of boundary value problems with additional interface conditions, more complicated boundary conditions, and so on on the basis of known solutions to classical boundary value problems is proposed. The method is based on the representation of solutions to classical and more complicated problems in the form of expansions into Fourier series with subsequent reduction of one series to the other. As a result, formulas directly expressing solutions to more complicated problems in terms of solutions to classical problems are obtained. On the basis of the well-known solution to the Dirichlet problem on a half plane, solutions to boundary value problems with interface conditions (including generalized conditions of the type of a crack and a screen) on intersecting straight lines for boundary conditions of the first and the third kind are obtained.     

Some new analysis results for a class of interface problems

Interface problems modeled by differential equations have many applications in mathematical biology, fluid mechanics, material sciences, and many other areas. Typically, interface problems are characterized by discontinuities in the coefficients and/or the Dirac delta function singularities in the source term. Because of these irregularities, solutions to the differential equations are not smooth or discontinuous. In this paper, some new results on the jump conditions of the solution across the interface are derived using the distribution theory and the theory of weak solutions. Some theoretical results on the boundary singularity in which the singular delta function is at the boundary are obtained. Finally, the proof of the convergency of the immersed boundary (IB) method is presented. The IB method is shown to be first‐order convergent in L ^∞ norm. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotics of a second-order differential equation with a small parameter in the case when the reduced equation has two solutions

The boundary value problem for a second-order nonlinear ordinary differential equation with a small parameter multiplying the highest derivative is examined. It is assumed that the reduced equation has two solutions with intersecting graphs. Near the intersection point, the asymptotic behavior of the solution to the original problem is fairly complex. A uniform asymptotic approximation to the solution that is accurate up to any prescribed power of the small parameter is constructed and justified.