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1. A Framework for Simulation of Multiple Elastic Scattering in Two Dimensions

   https://doi.org/10.1137/18M1232814  

   Lai, J. ; Li, P. ( January 2020 , SIAM journal on scientific computing)

   Consider the elastic scattering of a time-harmonic wave by multiple well-separated rigid particles with smooth boundaries in two dimensions. Instead of 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 the Helmholtz equation. 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 is 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 elastic scattering problems with multiple particles. 

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2. Learning domain-independent Green’s function for elliptic partial differential equations

   https://doi.org/10.1016/j.cma.2024.116779  

   Negi, Pawan ; Cheng, Maggie ; Krishnamurthy, Mahesh ; Ying, Wenjun ; Li, Shuwang ( March 2024 , Computer Methods in Applied Mechanics and Engineering)

   Lorenzis, Laura (Ed.)

   Green’s function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green’s function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain independent Green’s function, referred to as BIN-G. We evaluate the Green’s function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green’s function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green’s function for PDEs with variable coefficients. The learned Green’s function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients. 

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3. New scaling relations to compute atom-in-material polarizabilities and dispersion coefficients: part 2. Linear-scaling computational algorithms and parallelization

   https://doi.org/10.1039/c9ra01983a  

   Manz, Thomas A. ; Chen, Taoyi ( October 2019 , RSC Advances)

   We present two algorithms to compute system-specific polarizabilities and dispersion coefficients such that required memory and computational time scale linearly with increasing number of atoms in the unit cell for large systems. The first algorithm computes the atom-in-material (AIM) static polarizability tensors, force-field polarizabilities, and C 6 , C 8 , C 9 , C 10 dispersion coefficients using the MCLF method. The second algorithm computes the AIM polarizability tensors and C 6 coefficients using the TS-SCS method. Linear-scaling computational cost is achieved using a dipole interaction cutoff length function combined with iterative methods that avoid large dense matrix multiplies and large matrix inversions. For MCLF, Richardson extrapolation of the screening increments is used. For TS-SCS, a failproof conjugate residual (FCR) algorithm is introduced that solves any linear equation system having Hermitian coefficients matrix. These algorithms have mathematically provable stable convergence that resists round-off errors. We parallelized these methods to provide rapid computation on multi-core computers. Excellent parallelization efficiencies were obtained, and adding parallel processors does not significantly increase memory requirements. This enables system-specific polarizabilities and dispersion coefficients to be readily computed for materials containing millions of atoms in the unit cell. The largest example studied herein is an ice crystal containing >2 million atoms in the unit cell. For this material, the FCR algorithm solved a linear equation system containing >6 million rows, 7.57 billion interacting atom pairs, 45.4 billion stored non-negligible matrix components used in each large matrix-vector multiplication, and ∼19 million unknowns per frequency point (>300 million total unknowns). 

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

   https://doi.org/10.1088/1361-6420/ab693e  

   Dong, H. ; Lai, J. ; Li, P. ( January 2020 , Inverse problems)

   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 is concerned with 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 deduce 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 proposed for the phaseless inverse problem. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed methods. 

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5. Boundary Integral Equation Methods for Optical Cloaking Models

   Camille Carvalho, Elsie A. ( January 2022 , WAVES 2022)

   Optical cloaking refers to making an object invisible by preventing the light scattering in some directions as it hits the object. There is interest in cloaking devices in radar and other applications. Developing a model to accurately capture cloaking comes with numerical challenges, however. We must determine how light propagates through a medium composed by multiple, thin layers of materials with different electromagnetic properties. In this paper we consider a multi-layered scalar transmission problem in 2D and use boundary integral equation methods to compute the field. The Kress product quadrature rule is used to approximate singular integrals evaluated on boundaries, the Boundary Regularized Integral Equation Formulation (BRIEF) method [1] with Periodic Trapezoid Rule (PTR) is employed to treat nearly singular ones (off boundaries) appearing in the representation formula. Numerical results illustrate the efficiency of this approach, which may be applied to N arbitrary smooth layers. 

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