arXiv subjects

Gang Bao

Publications and source records attributed to Gang Bao.

48 records · Page 3Linked to original sources

arXiv2017

An accurate boundary element method for the exterior elastic scattering problem in two dimensions

This paper is concerned with a Galerkin boundary element method solving the two dimensional exterior elastic wave scattering problem. The original problem is first reduced to the so-called Burton-Miller (\cite{BM71}) boundary integral formulation, and essential mathematical features of its variational form are discussed. In numerical implementations, a newly-derived and analytically accurate regularization formula (\cite{YHX}) is employed for the numerical evaluation of hyper-singular boundary integral operator. A new computational approach is employed based on the series expansions of Hankel functions for the computation of weakly-singular boundary integral operators during the reduction of corresponding Galerkin equations into a discrete linear system. The effectiveness of proposed numerical methods is demonstrated using several numerical examples.

math.NA↗

arXiv2017

Inverse source problems in elastodynamics

We are concerned with time-dependent inverse source problems in elastodynamics. The source term is supposed to be the product of a spatial function and a temporal function with compact support. We present frequency-domain and time-domain approaches to show uniqueness in determining the spatial function from wave fields on a large sphere over a finite interval. Stability estimate of the temporal function from the data of one receiver and uniqueness result using partial boundary data are proved. Our arguments rely heavily on the use of the Fourier transform, which motivated inversion schemes that can be easily implemented. A Landweber iterative algorithm for recovering the spatial function and a non-iterative inversion scheme based on the uniqueness proof for recovering the temporal function are proposed. Numerical examples are demonstrated in both two and three dimensions.

math.AP↗

arXiv2017

Stability for the lens rigidity problem

Let $g$ be a Riemannian metric for $\mathbf{R}^d$ ($d\geq 3$) which differs from the Euclidean metric only in a smooth and strictly convex bounded domain $M$. The lens rigidity problem is concerned with recovering the metric $g$ inside $M$ from the corresponding lens relation on the boundary $\partial M$. In this paper, the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non-trapping metric satisfying "strong fold-regular" condition. A metric $g$ is called strong fold-regular if for each point $x\in M$, there exists a set of geodesics passing through $x$ whose conormal bundle covers $T^*_{x}M$. Moreover, these geodesics contain either no conjugate points or only fold conjugate points with a non-degeneracy condition. Examples of strong fold-regular metrics are constructed and are expected to be generic. Our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points. The approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow, which is a weighted geodesic X-ray transform problem for symmetric 2-tensor fields. A key ingredient is to show that the kernel of the X-ray transform on symmetric solenoidal 2-tensor fields is of finite dimension. It remains open whether the kernel space is trivial or not.

math.DG↗

arXiv2016

Direct and Inverse Elastic Scattering From Anisotropic Media

Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show the uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the boundary of the scatterer is derived. As an application, we design a descent algorithm for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are presented to show the validity and accuracy of our methods.

math.AP↗

arXiv2016

Inverse Random Source Scattering for Elastic Waves

This paper is concerned with the direct and inverse random source scattering problems for elastic waves where the source is assumed to be driven by an additive white noise. Given the source, the direct problem is to determine the displacement of the random wave field. The inverse problem is to reconstruct the mean and variance of the random source from the boundary measurement of the wave field at multiple frequencies. The direct problem is shown to have a unique mild solution by using a constructive proof. Based on the explicit mild solution, Fredholm integral equations of the first kind are deduced for the inverse problem. The regularized Kaczmarz method is presented to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

math.AP↗

arXiv2014

Sensitivity Analysis of an Inverse Problem for the Wave Equation with Caustics

The paper investigates the sensitivity of the inverse problem of recovering the velocity field in a bounded domain from the boundary dynamic Dirichlet-to-Neumann map (DDtN) for the wave equation. Three main results are obtained: (1) assuming that two velocity fields are non-trapping and are equal to a constant near the boundary, it is shown that the two induced scattering relations must be identical if their corresponding DDtN maps are sufficiently close; (2) a geodesic X-ray transform operator with matrix-valued weight is introduced by linearizing the operator which associates each velocity field with its induced Hamiltonian flow. A selected set of geodesics whose conormal bundle can cover the cotangent space at an interior point is used to recover the singularity of the X-ray transformed function at the point; a local stability estimate is established for this case. Although fold caustics are allowed along these geodesics, it is required that these caustics contribute to a smoother term in the transform than the point itself. The existence of such a set of geodesics is guaranteed under some natural assumptions in dimension greater than or equal to three by the classification result on caustics and regularity theory of Fourier Integral Operators. The interior point with the above required set of geodesics is called "fold-regular"; (3) assuming that a background velocity field with every interior point fold-regular is fixed and another velocity field is sufficiently close to it and satisfies a certain orthogonality condition, it is shown that if the two corresponding DDtN maps are sufficiently close then they must be equal.

math.AP↗

arXiv2013

Nearly Cloaking the Full Maxwell Equations

The approximate cloaking is investigated for time-harmonic Maxwell's equations via the approach of transformation optics. The problem is reduced to certain boundary effect estimates due to an inhomogeneous electromagnetic inclusion with an asymptotically small support but an arbitrary content enclosed by a thin high-conducting layer. Sharp estimates are established in terms of the asymptotic parameter, which are independent of the material tensors of the small electromagnetic inclusion. The result implies that the `blow-up-a-small-region' construction via the transformation optics approach yields a near-cloak for the electromagnetic waves. A novelty lies in the fact that the geometry of the cloaking construction of this work can be very general. Moreover, by incorporating the conducting layer developed in the present paper right between the cloaked region and the cloaking region, arbitrary electromagnetic contents can be nearly cloaked. Our mathematical technique extends the general one developed in [30] for nearly cloaking scalar optics. In order to investigate the approximate electromagnetic cloaking for general geometries with arbitrary cloaked contents, new techniques and analysis tools must be developed for this more challenging vector optics case.

math.AP↗

arXiv2013

Nearly Cloaking the Full Maxwell Equations: Cloaking Active Contents with General Conducting Layers

The regularized near-cloak via the transformation optics approach in the time-harmonic electromagnetic scattering is considered. This work extends the existing studies mainly in two aspects. First, it presents a near-cloak construction by incorporating a much more general conducting layer between the cloaked and cloaking regions. This might be of significant practical importance when production fluctuations occur. Second, it allows the cloaked contents to be both passive and active with an applied current inside. In assessing the near-cloaking performance, comprehensive and sharp estimates are derived for the scattering amplitude in terms of the asymptotic regularization parameter and the material tensors of the conducting layer. The scattering estimates are independent of the passive/active contents being cloaked, which implies that one could nearly cloak arbitrary contents by using the proposed near-cloak construction.

math.AP↗

arXiv2007

Improved Simulation of the Mass Charging for ASTROD I

The electrostatic charging of the test mass in ASTROD I (Astrodynamical Space Test of Relativity using Optical Devices I) mission can affect the quality of the science data as a result of spurious Coulomb and Lorentz forces. To estimate the size of the resultant disturbances, credible predictions of charging rates and the charging noise are required. Using the GEANT4 software toolkit, we present a detailed Monte Carlo simulation of the ASTROD I test mass charging due to exposure of the spacecraft to galactic cosmic-ray (GCR) protons and alpha particles (3He, 4He) in the space environment. A positive charging rate of 33.3 e+/s at solar minimum is obtained. This figure reduces by 50% at solar maximum. Based on this charging rate and factoring in the contribution of minor cosmic-ray components, we calculate the acceleration noise and stiffness associated with charging. We conclude that the acceleration noise arising from Coulomb and Lorentz effects are well below the ASTROD I acceleration noise limit at 0.1 mHz both at solar minimum and maximum. The coherent Fourier components due to charging are investigated, it needs to be studied carefully in order to ensure that these do not compromise the quality of science data in the ASTROD I mission.

astro-ph↗

arXiv2007

Simulation of ASTROD I test mass charging due to solar energetic particles

As ASTROD I travels through space, its test mass will accrue charge due to galactic cosmic-rays and solar energetic particles incident on the spacecraft. This test mass charge will result in Coulomb forces between the test mass and the surrounding electrodes. In earlier work using the GEANT4 toolkit, we predicted a net charging rate of nearly 9.0 +e/s from cosmic-ray protons between 0.1 and 1000 GeV at solar maximum, and rising to 26.5 +e/s at solar minimum. Here we use GEANT4 to simulate the charging process due to solar energetic particle events and to estimate the magnitude of acceleration noise due to this charging. The predicted charging rates range from 2840 to 64300 +e/s, at peak intensity, for the 4 largest SEP events in September and October 1989. For the 2 larger events, the acceleration disturbances due to charging exceeds the ASTROD I acceleration noise budget requirement. Continuous discharge should be considered for suppressing this charging noise. The acceleration noise during the 2 small events is well below the design target, although during these events, the net charging rate will be dominated by these solar fluxes.

astro-ph↗

arXiv2007

ASTROD I Charging Simulation and Disturbances

ASTROD I is planned as a single spacecraft mission. It will use interferometric and pulse ranging techniques between the spacecraft and ground stations, to make high precision measurements of the parameters that describe the solar system, and to test relativistic gravity with improved accuracy. At the heart of the spacecraft is a test mass, which the spacecraft will follow using a drag-free control system. The mission critically depends on maintaining the geodesic motion of the test mass. Charging of the test mass due to cosmic rays and solar particles will disturb its geodesic motion. We have modelled the charging process using the GEANT4 toolkit and a simplified, geometrical model and estimate that the ASTROD I test mass will charge positively, at a rate of 24 +/-7 e+/s, due to cosmic ray protons and alpha particles (3He and 4He) at solar minimum. We have used the results of this simulation to estimate the magnitude of disturbances associated with test mass charging, for the worst-case scenario, taking into account uncertainties in the model and potential charging contributions from minor cosmic-ray components.

astro-ph↗

arXiv2000

Models for Accretion Disk Fluctuations through Self-Organized Criticality Including Relativistic Effects

The possibility that some of the observed X-ray and optical variability in active galactic nuclei and galactic black hole candidates is produced in accretion disks through the development of a self-organized critical state is reconsidered. New simulations, including more complete calculations of relativistic effects, do show that this model can produce light-curves and power-spectra for the variability which agree with the range observed in optical and X-ray studies of AGN and X-ray binaries. However, the universality of complete self-organized criticality is not quite achieved. This is mainly because the character of the variations depend quite substantially on the extent of the unstable disk region. If it extends close to the innermost stable orbit then a physical scale is introduced and the scale-free character of self-organized criticality is vitiated. Significant dependence of the power spectrum density slope on the type of diffusion within the disk and a weaker dependence on the amount of differential rotation are noted. When general relativistic effects are incorporated in the models, additional substantial differences are produced if the disk is viewed from directions far from the accretion disk axis.

astro-ph↗