Lippmann-Schwinger Type Research Articles

Two-step scattering theory method for designing metamaterials

Abstract A two-step method for designing an acoustic metamaterial with desired properties is proposed. At the first step, the scattering matrices of its elements are calculated using the Lippmann-Schwinger type equation. For scatterers of small wave size, the problem is reduced to determining the phases of several scattering coefficients. At the second step, the particular design of metamaterial elements is determined. This problem can be considered an inverse scattering problem or a problem of optimizing the geometric parameters of a metamaterial element. The second approach turns out to be more practically applicable. As an example, a metamaterial in the form of a reflective layer with a narrow waveguide channel is considered, and the design of its elements is determined with the proposed method.

A fast numerical method for the conductivity of heterogeneous media with Dirichlet boundary conditions based on discrete sine–cosine transforms

The aim of this work is to develop a fast numerical method for solving conductivity problems in heterogeneous media subjected to Dirichlet boundary conditions. The method is based on a fixed-point iterative solution to an integral Lippmann–Schwinger type equation that is obtained by a Galerkin discretization of the cell problem using an approximation space spanned by sine series; the solution field is split between a known term verifying the Dirichlet boundary conditions and an unknown term described by sine series, which is null on the boundary by construction. With a suitable numerical integration scheme of the elementary integrals involved in the Galerkin formulation, based on discrete sine–cosine transforms, the method relies on the numerical complexity of fast Fourier transforms. The method is assessed in several problems including composite materials and fibrous networks.

Single Ionization of Helium by Protons of Various Energies in the Parabolic Quasi-Sturmians Approach

The parabolic quasi-Sturmian approach, recently introduced for the calculation of ion–atom ionizing collisions, is adapted and applied here to the single ionization of helium induced by an intermediate-energy proton impact. Within the method, the ionization amplitude is represented as the sum of the products of the basis amplitudes associated with the asymptotic behavior of the continuum states of the two noninteracting hydrogenic subsystems (e−,He+) and (p+,He+). The p−e interaction is treated as a perturbation in the Lippmann–Schwinger-type (LS) equation for the three-body system (e−,He+,p+). This LS equation is solved numerically using separable expansions for the proton–electron potential. We examine the convergence behavior of the transition amplitude expansion as the number of terms in the representation of the p−e interaction is increased and find that, for some kinematic regimes, the convergence is poor. This difficulty, which is absent for a higher proton energy impact, is solved by varying the momentum of the auxiliary proton plane wave introduced into the basis function. Fully differential cross-sections are calculated and compared with experimental data for 75 keV protons and the results obtained with the 3C model.

Seismic and medical ultrasound imaging of velocity and density variations by nonlinear vectorial inverse scattering.

We present an iterative nonlinear inverse scattering algorithm for high-resolution acoustic imaging of density and velocity variations. To solve the multi-parameter nonlinear direct scattering problem, the acoustic wave equation for inhomogeneous media in the frequency domain is transformed into a vectorial integral equation of the Lippmann-Schwinger type for the combined pressure and pressure-gradient field. To solve the multi-parameter nonlinear inverse scattering problem, we use the Newton-Kantorovich method in conjunction with matrix-free representations of the Fréchet derivative operators and their adjoints. The approximate Hessian information that is accounted for in our iterative solution of the (nonlinear) multi-parameter inverse scattering problem is essential for the mitigation of multi-parameter cross talk effects. Numerical examples related to seismic and medical ultrasound breast imaging illustrate the performance of the new algorithm for multi-parameter acoustic imaging.

Stability analysis of the Lippmann–Schwinger equation

The Lippmann–Schwinger type equation for Bruckner's reaction operator is analysed with respect to its convergence properties for the case of small deviations from the fixed point. It is found that the eigenvalues of the Jacobian of the iterative equation predict the convergence to a good approximation.

Homotopy scattering series for seismic forward modelling with variable density and velocity

ABSTRACTWe have derived a convergent scattering series solution for the frequency‐domain wave equation in acoustic media with variable density and velocity. The convergent scattering series solution is based on the homotopy analysis of a vectorial integral equation of the Lippmann–Schwinger type. By using the Green's function and partial integration, we have derived the vectorial integral equation of the Lippmann–Schwinger type that involves the pressure gradient field as well as the pressure field from the wave equation. The vectorial Lippmann–Schwinger equation can in principle be solved via matrix inversion, but the computational cost of matrix inversion scales like , where is the number of grid blocks. The computational cost can be significantly reduced if one solves the vectorial Lippmann–Schwinger equation iteratively. A simple iterative solution is the Born series, but it is only convergent when the scattering potential is sufficiently small. In this study, we have used the so‐called homotopy analysis method to derive an iterative solution for the vectorial Lippmann–Schwinger equation which can be made convergent even in strongly scattering media. The computational cost of our convergent scattering series scales as . Our algorithm, which is based on the homotopy analysis method, involves a convergence control operator that we select using hierarchical matrices. We use a three‐layer model and a resampled version of the SEG/EAGE salt model to show the performance of the developed convergent scattering series.

Recurrent localization networks applied to the Lippmann-Schwinger equation

The bulk of computational approaches for modeling physical systems in materials science derive from either analytical (i.e., physics based) or data-driven (i.e., machine-learning based) origins. In order to combine the strengths of these two approaches, we advance a novel machine learning approach for solving equations of the generalized Lippmann-Schwinger (L-S) type. In this paradigm, a given problem is converted into an equivalent L-S equation and solved as an optimization problem, where the optimization procedure is calibrated to the problem at hand. As part of a learning-based loop unrolling, we use a recurrent convolutional neural network to iteratively solve the governing equations for a field of interest. This architecture leverages the generalizability and computational efficiency of machine learning approaches, but also permits a physics-based interpretation. We demonstrate our learning approach on the two-phase elastic localization problem, where it achieves excellent accuracy on the predictions of the local (i.e., voxel-level) elastic strains. Since numerous governing equations can be converted into an equivalent L-S form, the proposed architecture has potential applications across a range of multiscale materials phenomena.

On spurious solutions encountered in Helmholtz scattering resonance computations in [formula omitted] with applications to nano-photonics and acoustics

In this paper, we consider a sorting scheme for potentially spurious scattering resonant pairs in one- and two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can, therefore, be applied to structures with graded materials as well as to configurations including piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to mark potentially spurious solutions in electromagnetic and acoustic scattering resonances computations at a low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

Iterative solution of the Lippmann–Schwinger equation in strongly scattering acoustic media by randomized construction of preconditioners

SUMMARY In this work the Lippmann–Schwinger equation is used to model seismic waves in strongly scattering acoustic media. We consider the Helmholtz equation, which is the scalar wave equation in the frequency domain with constant density and variable velocity, and transform it to an integral equation of the Lippmann–Schwinger type. To directly solve the discretized problem with matrix inversion is time-consuming, therefore we use iterative methods. The Born series is a well-known scattering series which gives the solution with relatively small cost, but it has limited use as it only converges for small scattering potentials. There exist other scattering series with preconditioners that have been shown to converge for any contrast, but the methods might require many iterations for models with high contrast. Here we develop new preconditioners based on randomized matrix approximations and hierarchical matrices which can make the scattering series converge for any contrast with a low number of iterations. We describe two different preconditioners; one is best for lower frequencies and the other for higher frequencies. We use the fast Fourier transform both in the construction of the preconditioners and in the iterative solution, and this makes the methods efficient. The performance of the methods are illustrated by numerical experiments on two 2-D models.

SWANLOP: Scattering waves off nonlocal optical potentials in the presence of Coulomb interactions

We introduce the package SWANLOP to calculate scattering waves and corresponding observables for nucleon elastic collisions off spin-zero nuclei. The code is capable of handling local and nonlocal optical potentials superposed to long-range Coulomb interaction. Solutions to the implied Schrödinger integro-differential equation are obtained by solving an integral equation of Lippmann–Schwinger type for the scattering wavefunctions, ψ=ϕC+GCUSψ, providing and exact treatment to the Coulomb force Arellano and Blanchon (2019). The package has been developed to handle potentials either in momentum or coordinate representations, providing flexible options under each of them. The code is fully self-contained, being dimensioned to handle any A≥4 target for nucleon beam energies of up to 1.1 GeV. Accuracy and benchmark applications are presented and discussed. Program summaryProgram title: SWANLOPCPC Library link to program files:https://doi.org/10.17632/89gw9jdfv4.1Licensing provisions: GNU General Public License, Version 2Programming language: FORTRAN-90Nature of problem: Optical model potentials constitute a valuable tool to investigate the physics involved in nuclear collisions and reactions with nucleonic probes. As such, it becomes essential to obtain accurate results for its associated scattering observables and corresponding scattering waves. An important feature of optical potentials is their nonlocal nature, arising from the fermionic nature of the (A+1)–nucleon problem together with the fact that effective nucleon–nucleon (NN) interactions are nonlocal as well. The superposition of Coulomb interaction to these nonlocal potentials poses non-trivial difficulties to obtain scattering waves and observables in collision processes.Solution method: The code performs the calculation of scattering waves associated to nonlocal potentials in the presence of the long-range Coulomb interactions, solving a Lippmann–Schwinger type integral equation for the scattering wavefunction. The potential can be given either in coordinate or momentum space. Phase-shifts and associated elastic scattering observables are extracted from the asymptotic behavior of the solution.

Homotopy analysis of the Lippmann–Schwinger equation for seismic wavefield modelling in strongly scattering media

SUMMARYWe present an application of the homotopy analysis method for solving the integral equations of the Lippmann–Schwinger type, which occurs frequently in acoustic and seismic scattering theory. In this method, a series solution is created which is guaranteed to converge independent of the scattering potential. This series solution differs from the conventional Born series because it contains two auxiliary parameters ϵ and h and an operator H that can be selected freely in order to control the convergence properties of the scattering series. The ϵ-parameter which controls the degree of dissipation in the reference medium (that makes the wavefield updates localized in space) is known from the so-called convergent Born series theory; but its use in conjunction with the homotopy analysis method represents a novel feature of this work. By using H = I (where I is the identity operator) and varying the convergence control parameters h and ϵ, we obtain a family of scattering series which reduces to the conventional Born series when h = −1 and ϵ = 0. By using H = γ where γ is a particular pre-conditioner and varying the convergence control parameters h and ϵ, we obtain another family of scattering series which reduces to the so-called convergent Born series when h = −1 and ϵ ≥ ϵc where ϵc is a critical dissipation parameter depending on the largest value of the scattering potential. This means that we have developed a kind of unified scattering series theory that includes the conventional and convergent Born series as special cases. By performing a series of 12 numerical experiments with a strongly scattering medium, we illustrate the effects of varying the (ϵ, h, H)-parameters on the convergence properties of the new homotopy scattering series. By using (ϵ, h, H) = (0.5, −0.8, I) we obtain a new scattering series that converges significantly faster than the convergent Born series. The use of a non-zero dissipation parameter ϵ seems to improve on the convergence properties of any scattering series, but one can now relax on the requirement ϵ ≥ ϵc from the convergent Born series theory, provided that a suitable value of the convergence control parameter h and operator H is used.

Convergent scattering series solution of the inhomogeneous Helmholtz equation via renormalization group and homotopy continuation approaches

The inhomogeneous Helmholtz equation can be represented by an equivalent integral equation of the Lippmann-Schwinger type. Since it is very costly to solve the Lippmann-Schwinger equation exactly via matrix inversion, researchers often try to use the popular Born series, which represents a physics based iterative solution. However, the popular Born series is only guaranteed to converge if the contrast volume is sufficiently small. We here use renormalization group (RG) and homotopy continuation (HC) approaches to derive novel scattering series solutions of the Lippmann-Schwinger equation which are guaranteed to converge for arbitrary large contrast volumes (perturbations). Our RG approach is based on the use of an auxillary set of scale-dependent scattering potentials and Green's operators which gradually evolve toward the real physical scattering potential and the physical Green's operator, respectively. We show that the auxillary Green's operators satisfies the renormalization group law and we derive the corresponding renormalization group flow equation by using familiar “scaling” arguments. The HC approach comes from pure mathematics and is based on the deformation of a relatively simple reference equation (initial condition) into a more complicated target equation (the Lippmann-Schwinger equation for the physical scattering potential), via a parameter that varies continuously between 0 and 1. Our HC approach is more general than the RG approach since it involves a convergence control operator, but the two approaches are nevertheless closely connected. We show that the RG flow and HC equations can be implemented very efficiently using the Fast Fourier Transform (FFT) algorithm. Also, we present a very efficient iterative scheme based on a combination of the zeroth-order deformation equation with a floating initial. We illustrate our results with a numerical example related with seismic wavefield modeling in strongly scattering media.

A finite element perspective on nonlinear FFT-based micromechanical simulations

Fourier solvers have become efficient tools to establish structure-property relations in heterogeneous materials. Introduced as an alternative to the Finite Element (FE) method, they are based on fixed-point solutions of the Lippmann-Schwinger type integral equation. Their computational efficiency results from handling the kernel of this equation by the Fast Fourier Transform (FFT). However, the kernel is derived from an auxiliary homogeneous linear problem, which renders the extension of FFT-based schemes to non-linear problems conceptually difficult. This paper aims to establish a link between FE- and FFT-based methods, in order to develop a solver applicable to general history- and time-dependent material models. For this purpose, we follow the standard steps of the FE method, starting from the weak form, proceeding to the Galerkin discretization and the numerical quadrature, up to the solution of non-linear equilibrium equations by an iterative Newton-Krylov solver. No auxiliary linear problem is thus needed. By analyzing a two-phase laminate with non-linear elastic, elasto-plastic, and visco-plastic phases, and by elasto-plastic simulations of a dual-phase steel microstructure, we demonstrate that the solver exhibits robust convergence. These results are achieved by re-using the non-linear FE technology, with the potential of further extensions beyond small-strain inelasticity considered in this paper.

Molecular partners of the X(3872) from heavy-quark spin symmetry: a fresh look

The heavy-quark spin symmetry (HQSS) partners of the X(3872) molecule are investigated in a chiral effective field theory (EFT) approach which incorporates contact and one-pion exchange interactions. The integral equations of the Lippmann-Schwinger type are formulated and solved for the coupled-channel problem for the DD , DD *, and D*D * systems with the quantum numbers J PC = 1 ++ , 1+− , 0 ++ , and 2 ++ . We confirm that, if the X (3872) is a 1 ++ DD * molecular state then, in the strict heavy-quark limit, there exist three partner states, with the quantum numbers 1 +− , 0 ++ , and 2 ++ , which are degenerate in mass. At first glance, this result looks natural only for the purely contact pionless theory since pions contribute differently to different transition potentials and, therefore, may lift the above degeneracy. Nevertheless, it is shown that, by an appropriate unitary transformation, the Lippmann-Schwinger equation in each channel still can be brought to a block-diagonal form, with the same blocks for all quantum numbers, so that the degeneracy of the bound states in different channels is preserved. We stress that neglecting some of the coupled-channel transitions in an inconsistent manner leads to a severe violation of HQSS and yields regulator-dependent results for the partner states. The effect of HQSS violation in combination with nonperturbative pion dynamics on the pole positions of the partner states is discussed.

Numerical prediction of the stiffness and strength of medium density fiberboards

A numerical two scale method for the prediction of tensile and bending stiffness and strength of medium density fiberboards (MDF) is proposed with the aim to study the fiber orientation influence on mechanical properties of MDF. The method requires less experimental data to optimize MDF and to improve industrial manufacturing technology of MDF. A new method for computing orientation tensors of the compressed fiber network is proposed. First, the virtual microstructure is generated by simulations of a fiber lay-down and a subsequent compression to obtain the necessary density. The density profile, fiber length, thickness, and orientation are used for the microstructure generation, which are obtained from μCT images and image analysis tools. Then a new damage model for the wood fiber cell walls and joints is introduced. The microstructural problem is formulated as a Lippmann–Schwinger type equation in elasticity and solved by using Fast Fourier Transformation (FFT). The macroscopic three point bending test is simulated with hexahedral finite elements and analytical methods based on Euler–Bernoulli theory. The difference between bending strength and stiffness numerically obtained and corresponding experimentally measured values is less than 10%. This study lays a foundation for the optimal design of MDF fiber structures and the optimization of industrial manufacturing processes. The first results show an increase of up to 60% for bending stiffness in the case of strongly oriented fibers.

An FFT-based Galerkin method for homogenization of periodic media

In 1994, Moulinec and Suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media. The scheme is based on a fixed-point iterative solution to an integral equation of the Lippmann–Schwinger type, with action of its kernel efficiently evaluated by the Fast Fourier Transform techniques. The aim of this work is to demonstrate that the Moulinec–Suquet setting is actually equivalent to a Galerkin discretization of the cell problem, based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme. For the latter framework and scalar elliptic problems, we prove convergence of the approximate solution to the weak solution, including a-priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration. Moreover, we also show that the variational structure implies that the resulting non-symmetric system of linear equations can be solved by the conjugate gradient method. Apart from providing a theoretical support to Fast Fourier Transform-based methods for numerical homogenization, these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature.

Variational Bounds in N-Particle Scattering Using the Faddeev–Yakubovskii Equations: Deuteron-Deuteron S=2 Scattering

A variational-bound formulation of the N-particle scattering problem has been developed based on the Yakubovskii–Faddeev chain-of-partition equations. It is shown that the scattering amplitude for the elastic, rearrangement or break-up processes satisfies a Lippmann–Schwinger type integral equation in which the kernel integration is over the open channels and the closed channels enter through the effective potential. In the case where only two (three)-cluster open channels are allowed, the integral equation for the transition amplitude involves integration over only one (two) momentum vector(s). A variational estimate for the effective potential input to the integral equation is obtained when the closed channel partition Green’s functions are estimated variationally. It is shown that the variational estimates for the closed-channel Green’s function also provide upper and lower bounds that can be used as a subsidiary extremum principle to determine optimum parameters in the trial function. Several methods are provided for simplifying the determination of the effective potential. The inclusion of Coulomb potentials in the Yakubovskii–Faddeev (YF) chain-of-partition formalism is also described. In this approach the inter-particle potential is not assumed to be separable as in typical quasi-particle schemes. The many-body dependence of the effective potential is included via expectation values involving an inter-particle potential and spatially decaying trial functions. The N-body scattering problem is therefore reduced to: (a) solving a two-body scattering problem (three-body in the case of break-up) and (b) a bound-state type calculation to determine the effective potential. As an initial application, the method is applied to the case of low-energy elastic deuteron-deuteron scattering (including the Coulomb force) and compared to a recent cluster-reduction calculation.

A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms

Composite materials possess a highly complex material behavior, and thus advanced simulation techniques are necessary to compute their mechanical response. In this regard, especially modeling failure and progressive damage presents a challenging task. Conventional macro mechanical methods and even closed form estimates are in many cases not sufficient to predict the appropriate mechanical material response. Full-field simulations must be resorted to, but these are known to be very expensive from the computational point of view. In this contribution we propose a more efficient multiscale approach similar to FE2. Nonlinear material effects caused by progressive damage behavior are captured directly on the discretized material level using simple isotropic continuum damage laws. In contrast to conventional FE2 methods which use the Finite Element Method (FEM) to solve both scales numerically, the fine scale problem (material level) is rewritten in an integral form of Lippmann–Schwinger type and solved efficiently using the fast Fourier transformation (FFT). The calculation is carried out on a regular voxel grid that can be obtained from 3D images like tomographies. The fine scale problem is integrated in a standard Finite Element framework which is used to solve the macroscopic BVP (component level). In the work at hand, the scale coupling technique and the computation of the macroscopic tangent are described, and in some numerical examples the convergence behavior of the macroscopic Newton algorithm is investigated. Thereby the simulations were considered until localization and softening on the material scale occurred. It is shown that the proposed method presents an effective way to determine the exact physical macroscopic response considering arbitrary microstructures and loading conditions.

Scattering of electromagnetic waves by thin high contrast dielectrics: effects of the object boundary

We study the scattered field from thin high contrast dielectrics modeled by the full time-harmonic Maxwell equations while accounting for material boundaries. We derive a formulation of Lippmann-Schwinger type for a dielectric scatterer; this formulation has an additional surface term to account for the material discontinuities. The layer potential operator resulting from this surface term is shown to converge in a weak sense to an explicitly computable limit as the thickness of the domain approaches zero. By properly accounting for the boundary effects, we show two results about the thin high contrast limit: First, the normal component of the electric field’s interior trace on the lateral boundary approaches zero. Second, the perpendicular component of the electric field goes to zero inside the slab. We propose a new two-dimensional limiting equation as a first-order computational technique.

Zero-range potential for particles interacting via Coulomb potential

The zero-range potential is constructed for a system of two particles interacting via the Coulomb potential. The singular part of the asymptote of the wavefunction at the origin, which is caused by the common effect of the zero-range potential singularity and of the Coulomb potential, is explicitly calculated by using the Lippmann–Schwinger type integral equation. The singular pseudo potential is constructed from the requirement that it enforces the solution to the Coulomb Schrödinger equation to possess the calculated asymptotic behavior at the origin. This pseudo potential is then used for constructing a model of the imaginary absorbing potential for the positron–electron system. This potential allows us to treat the annihilation process in positron–electron collisions on the basis of the non-relativistic Schrödinger equation. The functional form of the pseudo potential constructed in this paper is analogous to the well known Fermi–Breit–Huang pseudo potential. The generalization of the optical theorem in the case of the imaginary absorbing potential in the presence of the Coulomb force is given in terms of the partial wave series.