arXiv subjects

Jun Lai

Publications and source records attributed to Jun Lai.

At least 37 records · Page 2Linked to original sources

arXiv2022

Fast inverse elastic scattering of multiple particles in three dimensions

Many applications require recovering the geometry information of multiple elastic particles based on the scattering information. In this paper, we consider the inverse time-harmonic elastic scattering of multiple rigid particles in three dimensions. We measure the far field information and apply the time reversal method to recover the unknown elastic particles. Two regimes are considered depending on the size and distance among particles. First, an asymptotic analysis for the imaging of small and distant particles is given based on the scattering property of a single particle, which can be used for selective focusing. Second, when particles are not small but well-separated, a fast algorithm, based on the combination of multiple scattering theory and fast multipole method, is proposed to efficiently simulate the forward multiple scattering problem and applied in the inverse elastic scattering. Numerical experiments demonstrate the proposed method can determine the locations and shapes of multiple particles instantly.

math-ph↗

arXiv2022

A spectral boundary integral method for the elastic obstacle scattering problem in three dimensions

In this paper, we consider the scattering of a plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in three dimensions. Based on the Helmholtz decomposition, the elastic scattering problem is reduced to a coupled boundary value problem for the Helmholtz and Maxwell equations. A novel system of boundary integral equations is formulated and a spectral boundary integral method is developed for the coupled boundary value problem. Numerical experiments are presented to demonstrate the superior performance of the proposed method.

math.NA↗

arXiv2021

A fast solver for elastic scattering from axisymmetric objects by boundary integral equations

Fast and high-order accurate algorithms for three dimensional elastic scattering are of great importance when modeling physical phenomena in mechanics, seismic imaging, and many other fields of applied science. In this paper, we develop a novel boundary integral formulation for the three dimensional elastic scattering based on the Helmholtz decomposition of elastic fields, which converts the Navier equation to a coupled system consisted of Helmholtz and Maxwell equations. An FFT-accelerated separation of variables solver is proposed to efficiently invert boundary integral formulations of the coupled system for elastic scattering from axisymmetric rigid bodies. In particular, by combining the regularization properties of the singular boundary integral operators and the FFT-based fast evaluation of modal Green's functions, our numerical solver can rapidly solve the resulting integral equations with a high-order accuracy. Several numerical examples are provided to demonstrate the efficiency and accuracy of the proposed algorithm, including geometries with corners at different wave number.

math.NA↗

arXiv2020

A fast algorithm for the electromagnetic scattering from a large rectangular cavity in three dimensions

The paper is concerned with the three-dimensional electromagnetic scattering from a large open rectangular cavity that is embedded in a perfectly electrically conducting infinite ground plane. By introducing a transparent boundary condition, the scattering problem is formulated into a boundary value problem in the bounded cavity. Based on the Fourier expansions of the electric field, the Maxwell equation is reduced to one-dimensional ordinary differential equations for the Fourier coefficients. A fast algorithm, employing the fast Fourier transform and the Gaussian elimination, is developed to solve the resulting linear system for the cavity which is filled with either a homogeneous or a layered medium. In addition, a novel scheme is designed to evaluate rapidly and accurately the Fourier transform of singular integrals. Numerical experiments are presented for large cavities to demonstrate the superior performance of the proposed method.

math.NA↗

arXiv2020

A highly accurate boundary integral method for the elastic obstacle scattering problem

Consider the scattering of a time-harmonic plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in two dimensions. In this paper, a novel boundary integral formulation is proposed and its highly accurate numerical method is developed for the elastic obstacle scattering problem. More specifically, based on the Helmholtz decomposition, the model problem is reduced to a coupled boundary integral equation with singular kernels. A regularized system is constructed in order to handle the degenerated integral operators. The semi-discrete and full-discrete schemes are studied for the boundary integral system by using the trigonometric collocation method. Convergence is established for the numerical schemes in some appropriate Sobolev spaces. Numerical experiments are presented for both smooth and nonsmooth obstacles to demonstrate the superior performance of the proposed method.

math.NA↗

arXiv2020

Inverse scattering reconstruction of a three dimensional sound-soft axis-symmetric impenetrable object

In this work, we consider the problem of reconstructing the shape of a three dimensional impenetrable sound-soft axis-symmetric obstacle from measurements of the scattered field at multiple frequencies. This problem has important applications in locating and identifying obstacles with axial symmetry in general, such as, land mines. We present a two-part framework for recovering the shape of the obstacle. In part 1, we introduce an algorithm to find the axis of symmetry of the obstacle by making use of the far field pattern. In part 2, we recover the shape of the obstacle by applying the recursive linearization algorithm (RLA) with multifrequency measurements of the scattered field. In the RLA, a sequence of inverse scattering problems using increasing single frequency measurements are solved. Each of those problems is ill-posed and nonlinear. The ill-posedness is treated by using a band-limited representation for the shape of the obstacle, while the nonlinearity is dealt with by applying the damped Gauss-Newton method. When using the RLA, a large number of forward scattering problems must be solved. Hence, it is paramount to have an efficient and accurate forward problem solver. For the forward problem, we apply separation of variables in the azimuthal coordinate and Fourier decompose the resulting problem, leaving us with a sequence of decoupled simpler forward scattering problems to solve. Numerical examples for the inverse problem are presented to show the feasibility of our two-part framework in different scenarios, particularly for objects with non-smooth boundaries.

math.NA↗

arXiv2019

On Wellposedness and Convergence of UPML method for analyzing wave scattering in layered media

This paper proposes a novel method to establish the wellposedness and convergence theory of the uniaxial-perfectly-matched-layer (UPML) method in solving a two-dimensional acoustic scattering problem due to a compactly supported source, where the medium consists of two layers separated by the horizontal axis. When perfectly matched layer (PML) is used to truncate the vertical variable only, the medium structure becomes a closed waveguide. The Green function due to a primary source point in this waveguide can be constructed explicitly based on variable separations and Fourier transformations. In the horizontal direction, by properly placing periodical PMLs and locating periodic source points imaged by the primary source point, the exciting waveguide Green functions by those imaging points can be assembled to construct the Green function due to the primary source point for the two-layer medium truncated by a UPML. Incorporated with Green's identities, this UPML Green function directly leads to the wellposedness of the acoustic scattering problem in a UPML truncation with no constraints about wavenumbers or UPML absorbing strength. Consequently, we firstly prove that such a UPML truncating problem is unconditionally resonance free. Moreover, we show, under quite general conditions, that this UPML Green function converges to the exact layered Green function exponentially fast as absorbing strength of the UPML increases, which in turn gives rise to the exponential convergence of the solution of the UPML problem towards the original solution.

math.NA↗

arXiv2019

An inverse acoustic-elastic interaction problem with phased or phaseless far-field data

Consider the scattering of a time-harmonic acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous acoustic medium. This paper concerns an inverse acoustic-elastic interaction problem, which is to determine the location and shape of the elastic obstacle by using either the phased or phaseless far-field data. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The jump relations are studied for the second derivatives of the single-layer potential in order to establish the corresponding boundary integral equations. The well-posedness is discussed for the solution of the coupled boundary integral equations. An efficient and high order Nyström-type discretization method is proposed for the integral system. A numerical method of nonlinear integral equations is developed for the inverse problem. For the case of phaseless data, we show that the modulus of the far-field pattern is invariant under a translation of the obstacle. To break the translation invariance, an elastic reference ball technique is introduced. We prove that the inverse problem with phaseless far-field pattern has a unique solution under certain conditions. In addition, a numerical method of the reference ball technique based nonlinear integral equations is also proposed for the phaseless inverse problem. Numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed methods.

math.NA↗

arXiv2019

An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects

Fast, high-order accurate algorithms for electromagnetic scattering from axisymmetric objects are of great importance when modeling physical phenomena in optics, materials science (e.g. meta-materials), and many other fields of applied science. In this paper, we develop an FFT-accelerated separation of variables solver that can be used to efficiently invert integral equation formulations of Maxwell's equations for scattering from axisymmetric penetrable (dielectric) bodies. Using a standard variant of Müller's integral representation of the fields, our numerical solver rapidly and directly inverts the resulting second-kind integral equation. In particular, the algorithm of this work (1) rapidly evaluates the modal Green's functions, and their derivatives, via kernel splitting and the use of novel recursion formulas, (2) discretizes the underlying integral equation using generalized Gaussian quadratures on adaptive meshes, and (3) is applicable to geometries containing edges. Several numerical examples are provided to demonstrate the efficiency and accuracy of the aforementioned algorithm in various geometries.

math.NA↗

arXiv2018

A Fast Solver for the Elastic Scattering of Multiple Particles

Consider the elastic scattering of a time-harmonic wave by multiple well separated rigid particles in two dimensions. To avoid using the complex Green's tensor of the elastic wave equation, we utilize the Helmholtz decomposition to convert the boundary value problem of the elastic wave equation into a coupled boundary value problem of Helmholtz equations. Based on single, double, and combined layer potentials with the simpler Green's function of the Helmholtz equation, we present three different boundary integral equations for the coupled boundary value problem. The well-posedness of the new integral equations are established. Computationally, a scattering matrix based method is proposed to evaluate the elastic wave for arbitrarily shaped particles. The method uses the local expansion for the incident wave and the multipole expansion for the scattered wave. The linear system of algebraic equations is solved by GMRES with fast multipole method (FMM) acceleration. Numerical results show that the method is fast and highly accurate for solving the elastic scattering problem with multiple particles.

math.NA↗

arXiv2018

Inverse obstacle scattering problem for elastic waves with phased or phaseless far-field data

This paper concerns an inverse elastic scattering problem which is to determine the location and the shape of a rigid obstacle from the phased or phaseless far-field data for a single incident plane wave. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The relation is established between the compressional or shear far-field pattern for the elastic wave equation and the corresponding far-field pattern for the coupled Helmholtz equations. An efficient and accurate Nyström type discretization for the boundary integral equation is developed to solve the coupled system. The translation invariance of the phaseless compressional and shear far-field patterns are proved. A system of nonlinear integral equations is proposed and two iterative reconstruction methods are developed for the inverse problem. In particular, for the phaseless data, a reference ball technique is introduced to the scattering system in order to break the translation invariance. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method.

math.AP↗

arXiv2017

A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers

Consider the scattering of a time-harmonic plane wave by heterogeneous media consisting of linear or nonlinear point scatterers and extended obstacles. A generalized Foldy-Lax formulation is developed to take fully into account of the multiple scattering by the complex media. A new imaging function is proposed and an FFT-based direct imaging method is developed for the inverse obstacle scattering problem, which is to reconstruct the shape of the extended obstacles. The novel idea is to utilize the nonlinear point scatterers to excite high harmonic generation so that enhanced imaging resolution can be achieved. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2016

A Quantitative Study of the Effect of Cladding Thickness on Modal Confinement Loss in Photonic Waveguides

There has been increasing interest in making the photonic devices more and more compact in the integrated photonics industry, and one of the important questions for manufacturers and design engineers is how to quantify the effect of the finite cladding thickness on the modal confinement loss of photonic waveguides. This requires at least six to seven digits accuracy for the computation of propagation constant $β$ since the modal confinement loss is proportional to the imaginary part of $β$ that is six to seven orders of magnitude smaller than its real part by the industrial standard. In this paper, we present an accurate and efficient method to compute the propagation constant of electromagnetic modes of photonic waveguides with arbitrary number of (nonsmooth) inclusions in a layered media. The method combines a well-conditioned boundary integral equation formulation for photonic waveguides which requires the discretization of the material interface only, and efficient Sommerfeld integral representations to treat the effect of the layered medium. Our scheme is capable of calculating the propagation loss of the electromagnetic modes with high fidelity, even for waveguides with corners imbedded in a cladding material of finite thickness. The numerical results, with more than $10$-digit accuracy, show quantitatively that the modal confinement loss of the rectangular waveguide increases exponentially fast as the cladding thickness decreases.

physics.optics↗

arXiv2016

Robust integral formulations for electromagnetic scattering from three-dimensional cavities

Scattering from large, open cavity structures is of importance in a variety of electromagnetic applications. In this paper, we propose a new well conditioned integral equation for scattering from general open cavities embedded in an infinite, perfectly conducting half-space. The integral representation permits the stable evaluation of both the electric and magnetic field, even in the low-frequency regime, using the continuity equation in a post-processing step. We establish existence and uniqueness results, and demonstrate the performance of the scheme in the cavity-of-revolution case. High-order accuracy is obtained using a Nystrom discretization with generalized Gaussian quadratures.

math.NA↗

arXiv2015

Second Kind Integral Equation Formulation for the Mode Calculation of Optical Waveguides

We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. Our method is highly accurate and thus can be used to calculate the propagation loss of the electromagnetic modes accurately, which provides the photonics industry a reliable tool for the design of more compact and efficient photonic devices. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.

math.NA↗

arXiv2015

A new hybrid integral representation for frequency domain scattering in layered media

A variety of problems in acoustic and electromagnetic scattering require the evaluation of impedance or layered media Green's functions. Given a point source located in an unbounded half-space or an infinitely extended layer, Sommerfeld and others showed that Fourier analysis combined with contour integration provides a systematic and broadly effective approach, leading to what is generally referred to as the Sommerfeld integral representation. When either the source or target is at some distance from an infinite boundary, the number of degrees of freedom needed to resolve the scattering response is very modest. When both are near an interface, however, the Sommerfeld integral involves a very large range of integration and its direct application becomes unwieldy. Historically, three schemes have been employed to overcome this difficulty: the method of images, contour deformation, and asymptotic methods of various kinds. None of these methods make use of classical layer potentials in physical space, despite their advantages in terms of adaptive resolution and high-order accuracy. The reason for this is simple: layer potentials are impractical in layered media or half-space geometries since they require the discretization of an infinite boundary. In this paper, we propose a hybrid method which combines layer potentials (physical-space) on a finite portion of the interface together with a Sommerfeld-type (Fourier) correction. We prove that our method is efficient and rapidly convergent for arbitrarily located sources and targets, and show that the scheme is particularly effective when solving scattering problems for objects which are close to the half-space boundary or even embedded across a layered media interface.

math.NA↗

arXiv2014

A fast and robust solver for the scattering from a layered periodic structure containing multi-particle inclusions

We present a solver for plane wave scattering from a periodic dielectric grating with a large number $M$ of inclusions lying in each period of its middle layer.Such composite material geometries have a growing role in modern photonic devices and solar cells. The high-order scheme is based on boundary integral equations, and achieves many digits of accuracy with ease. The usual way to periodize the integral equation---via the quasi-periodic Green's function---fails at Wood's anomalies. We instead use the free-space Green's kernel for the near field, add auxiliary basis functions for the far field, and enforce periodicity in an expanded linear system; this is robust for all parameters. Inverting the periodic and layer unknowns, we are left with a square linear system involving only the inclusion scattering coefficients. Preconditioning by the single-inclusion scattering matrix, this is solved iteratively in $O(M)$ time using a fast matrix-vector product. Numerical experiments show that a diffraction grating containing $M=1000$ inclusions per period can be solved to 9-digit accuracy in under 5 minutes on a laptop.

math.NA↗

arXiv2014

A fast solver for multi-particle scattering in a layered medium

In this paper, we consider acoustic or electromagnetic scattering in two dimensions from an infinite three-layer medium with thousands of wavelength-size dielectric particles embedded in the middle layer. Such geometries are typical of microstructured composite materials, and the evaluation of the scattered field requires a suitable fast solver for either a single configuration or for a sequence of configurations as part of a design or optimization process. We have developed an algorithm for problems of this type by combining the Sommerfeld integral representation, high order integral equation discretization, the fast multipole method and classical multiple scattering theory. The efficiency of the solver is illustrated with several numerical experiments.

math.NA↗