Lippmann-Schwinger Integral Equation Research Articles

Microstructural material database for self-consistent clustering analysis of elastoplastic strain softening materials

Multiscale modeling of heterogeneous material undergoing strain softening poses computational challenges for localization of the microstructure, material instability in the macrostructure, and the computational requirement for accurate and efficient concurrent calculation. In the paper, a stable micro-damage homogenization algorithm is presented which removes the material instability issues in the microstructure with representative volume elements (RVE) that are not sensitive to size when computing the homogenized stress–strain response.The proposed concurrent simulation framework allows the computation of the macroscopic response to explicitly consider the behavior of the separate constituents (material phases), as well as the complex microstructural morphology. A non-local material length parameter is introduced in the macroscale model, which will control the width of the damage bands and prevent material instability.The self-consistent clustering analysis (SCA) recently proposed by Liu et al. [1] provides an effective way of developing a microstructural database based on a clustering algorithm and the Lippmann–Schwinger integral equation, which enables an efficient and accurate prediction of nonlinear material response. The self-consistent clustering analysis is further generalized to consider complex loading paths through the projection of the effective stiffness tensor. In the concurrent simulation, the predicted macroscale strain localization is observed to be sensitive to the combination of microscale constituents, showing the unique capability of the SCA microstructural database for complex material simulations.

A Fast Integral Equation-Based Method for Solving Electromagnetic Inverse Scattering Problems With Inhomogeneous Background

A family of difference integral equations (IEs), consisting of difference Lippmann–Schwinger IE and difference new IE (D-NIE), are proposed to solve the electromagnetic inverse scattering problems (ISPs) with inhomogeneous background medium bounded in a finite domain. Without resorting to Green’s function for an inhomogeneous background medium, in the frame of the difference IE methods, Green’s function with a homogeneous medium is utilized such that not only fast algorithms (referring to those used in forward scattering problems, such as conjugate-gradient fast Fourier transform and fast multipole method) can be adopted but also the burdensome calculation of the numerical Green’s function for the inhomogeneous background medium is avoided. Especially, to tackle the ISPs with a strong nonlinearity, those with large contrast and/or large dimensions, a low-pass filter-matching regularization is introduced, which aims to stably match the information from the background medium to the unknown scatterers. Together with the D-NIE model, the proposed inversion method can efficiently tackle the ISPs with strong nonlinearity while a bounded inhomogeneous medium is present. Against both synthetic and experimental data, several representative numerical tests illustrate the efficacy of the proposed inversion method.

Separable representation of multichannel nucleon-nucleus optical potentials

One important ingredient for many applications of nuclear physics to astrophysics, nuclear energy, and stockpile stewardship are cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not feasible, indirect methods, e.g. (d,p) reactions, should be used. Those (d,p) reactions may be viewed as three-body reactions and described with Faddeev techniques. Faddeev equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. Optical potentials representing the effective interactions in the neutron (proton) nucleus subsystem are usually non-Hermitian as well as energy-dependent. Including excitations of the nucleus in the calculation requires a multichannel optical potential. The purpose of this paper is to introduce a separable, energy-dependent multichannel representation of complex, energy-dependent optical potentials that contain excitations of the nucleus and that fulfill reciprocity exactly. Momentum space Lippmann-Schwinger integral equations are solved with standard techniques to obtain the form factors for the separable representation. Starting from energy-dependent multichannel optical potentials for neutron and proton scattering from $^{12}$C, separable representations based on a generalization of the Ernst-Shakin-Thaler (EST) scheme are constructed which fulfill reciprocity exactly. Applications to n$+^{12}$C and p$+^{12}$C scattering are investigated for energies from 0 to 50~MeV.

Wavefield interpolation in 3D large‐step Fourier wavefield extrapolation

ABSTRACTExtrapolating wavefields and imaging at each depth during three‐dimensional recursive wave‐equation migration is a time‐consuming endeavor. For efficiency, most commercial techniques extrapolate wavefields through thick slabs followed by wavefield interpolation within each thick slab. In this article, we develop this strategy by associating more efficient interpolators with a Fourier‐transform‐related wavefield extrapolation method. First, we formulate a three‐dimensional first‐order separation‐of‐variables screen propagator for large‐step wavefield extrapolation, which allows for wide‐angle propagations in highly contrasting media. This propagator significantly improves the performance of the split‐step Fourier method in dealing with significant lateral heterogeneities at the cost of only one more fast Fourier transform in each thick slab. We then extend the two‐dimensional Kirchhoff and Born–Kirchhoff local wavefield interpolators to three‐dimensional cases for each slab. The three‐dimensional Kirchhoff interpolator is based on the traditional Kirchhoff formula and applies to moderate lateral velocity variations, whereas the three‐dimensional Born–Kirchhoff interpolator is derived from the Lippmann–Schwinger integral equation under the Born approximation and is adapted to highly laterally varying media. Numerical examples on the three‐dimensional salt model of the Society of Exploration Geophysicists/European Association of Geoscientists demonstrate that three‐dimensional first‐order separation‐of‐variables screen propagator Born–Kirchhoff depth migration using thick‐slab wavefield extrapolation plus thin‐slab interpolation tolerates a considerable depth‐step size of up to 72 ms, eventually resulting in an efficiency improvement of nearly 80% without obvious loss of imaging accuracy. Although the proposed three‐dimensional interpolators are presented with one‐way Fourier extrapolation methods, they can be extended for applications to general migration methods.

Solution of the Lippmann–Schwinger Equation for a Partial Wave Transition Matrix with Repulsive Coulomb Interaction

A special case where the Lippmann–Schwinger integral equation for the partial wave two-body Coulomb transition matrix for likely charged particles with a negative energy has an analytical solution has been considered. Analytical expressions for the partial s-, p-, and d-wave Coulomb transition matrices for repulsively interacting particles at the ground-state energy have been derived, by using the Fock method of stereographic projection of the momentum space onto a four-dimensional unit sphere.

Imaging with electromagnetic waves in terminating waveguides

We study an inverse scattering problem for Maxwell's equations in terminating waveguides, where localized reflectors are to be imaged using a remote array of sensors. The array probes the waveguide with waves and measures the scattered returns. The mathematical formulation of the inverse scattering problem is based on the electromagnetic Lippmann-Schwinger integral equation and an explicit calculation of the Green tensor. The image formation is carried with reverse time migration and with $\ell_1$ optimization.

Analytical expressions for partial wave two-body Coulomb transition matrices at ground-state energy

Leaning upon the Fock method of the stereographic projection of the three-dimensional momentum space onto the four-dimensional unit sphere the possibility of the analytical solving of the Lippmann–Schwinger integral equation for the partial wave two-body Coulomb transition matrix at the ground bound state energy has been studied. In this case new expressions for the partial p-, d- and f-wave two-body Coulomb transition matrices have been obtained in the simple analytical form. The developed approach can also be extended to determine analytically the partial wave Coulomb transition matrices at the energies of excited bound states.

Solution of the proton-hydrogen scattering problem using a quantum-mechanical two-center convergent close-coupling method

Details of the recently developed quantum-mechanical two-center convergent close-coupling approach (Abdurakhmanov et al 2016 J. Phys. B: At. Mol. Phys. 49 03LT01) to proton-hydrogen scattering are presented. The formulation is based on the exact (fully quantum-mechanical) three-body Schrödinger equation. The total scattering wavefunction is expanded using a two-center pseudostate basis. This allows one to include all underlying processes, namely, direct scattering and ionization, electron capture into bound and continuum states of the projectile. The off-shell integration in the coupled-channel Lippmann–Schwinger integral equations emerging from the three-body Schrödinger equation for the scattering wavefunction is taken analytically which greatly reduces computational effort. While the calculated electron capture cross sections are in a good agreement with experiment, some discrepancy exists for the ionization cross sections.

A New Integral Equation Method to Solve Highly Nonlinear Inverse Scattering Problems

A family of new integral equations (NIE) is proposed in this paper, which are transformed from the original Lippmann–Schwinger integral equation. It can be shown that the NIE can effectively reduce the nonlinearity of inverse scattering problems by reducing the global nonlinear effects, introduced by the multiple scattering behaviors, in estimating the contrast. Equipping the previously proposed twofold subspace-based optimization method with such NIE, the new inversion method is able to solve inverse scattering problems with strong scatterers, like with high contrast and/or large dimensions (in terms of wavelength) ones. Furthermore, such a family of NIE could provide a convenient tool to appraise reconstructed results. Several representative numerical tests are carried out, using both synthetic and experimental data, to verify the efficacy of the new inversion method.

A full-field image conversion method for the inverse conductivity problem with internal measurements

This article investigates a Fourier-based algorithm for computing heterogeneous material parameter distributions from internal measurements of physical fields. Within the framework of the periodic scalar conductivity model, a pair of dual Lippmann–Schwinger integral equations is derived for the sought constitutive parameters based on full intensity or current density field measurements. A numerical method based on the fast Fourier transform and fixed-point iterations is proposed. Convergence, stability and approximation quality of the method are analysed. For materials with small contrast, a first-order Born-like approximation is also obtained. Overall, the proposed reconstruction approach enables a direct conversion of full-field measurement images, possibly noisy, into maps of material conductivity. A set of numerical results is presented to illustrate the performance of the method.

An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

This text proposes a fast, rapidly convergent Nystr\"{o}m method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper.

Calculating Masses of Pentaquarks Composed of Baryons and Mesons

We consider an exotic baryon (pentaquark) as a bound state of two-body systems composed of a baryon (nucleon) and a meson. We used a baryon-meson picture to reduce a complicated five-body problem to simple two-body problems. The homogeneous Lippmann-Schwinger integral equation is solved in configuration space by using one-pion exchange potential. We calculate the masses of pentaquarksθc(uuddc¯)andθb(uuddb¯).

Heavy tetraquarks in the diquark–antidiquark picture

The homogeneous Lippmann–Schwinger integral equation is solved in momentum space to calculate the masses of heavy tetraquarks with hidden charm and bottom. The tetraquark bound states are studied in the diquark–antidiquark picture as a two-body problem. A regularized form of the diquark–antidiquark potential is used to overcome the singularity of the confining potential at large distances or small momenta. Our numerical results indicate that the relativistic effect leads to a small reduction in the mass of heavy tetraquarks, which is less than 2% for charm and less than 0.2% for bottom tetraquarks. The calculated masses of heavy tetraquarks for 1s, 1p, 2s, 1d and 2p states are in good agreement with other theoretical calculations and experimental data. Our numerical analysis predict the masses of heavy tetraquarks for 3s, 2d and 3p states for the first time, and we are not aware of any other theoretical results or experimental data for these states.

Nonlinear electromagnetic inverse scattering in via Frozen or Broyden update of the Fréchet derivative

Microwave imaging methods are useful for non-destructive inspection of dielectric targets. In this work, a numerical technique for solving the 3D Lippmann-Schwinger integral equation of the inverse scattering problem via Gauss-Newton linearization in Banach spaces is analysed. More specifically, two different approximations of the Frechet derivative are proposed in order to speed up the computation. Indeed it is well known that the computation of the Frechet derivative is generally quite expensive in three dimensional restorations. Numerical tests show that the approximations give a faster restoration without loosing accuracy.

Comment on “Tetraquarks as diquark–antidiquark bound systems”

The author argues that the calculated masses of heavy tetraquarks obtained by solution of the spin-independent homogeneous Lippmann-Schwinger integral equation in a diquark-antidiquark picture reported by M. Monemzadeh et al., Phys. Lett. B {\bf741}, 124 (2015), are incorrect. We have reexamined all of the published results and we believe that not only the reported tetraquark masses for states with zero total angular momentum are incorrect, the reported masses for states with non-zero total angular momentum are quite misleading, because these states cannot be predicted by a spin-independent formalism.

Schrödinger operator with a superposition of short-range and point potentials

We study the class of Schrodinger operators whose potential terms are sums of the short-range V (r) and point potentials. We consider the case where the short-range potential has a singularity on the support r = 0 of the point interaction. The point interaction is constructed using the asymptotic form of the Green’s function of the Schrodinger operator −Δ+V (r) with a short-range potential V as r → 0. We consider potentials with a singularity of the form r−ρ, ρ > 0, at the origin. We use the Lippmann-Schwinger integral equation in our study. We show that if the singularity of the potential is weaker than the Coulomb singularity, then the asymptotic behavior of the Green’s function has a standard singularity. If the singularity of the potential has the form r−ρ, 1 ≤ ρ < 3/2, then an additional singularity arises in the asymptotic behavior of the Green’s function. If ρ = 1, then the additional logarithmic singularity has the same form as in the case of the Coulomb potential. If 1 < ρ < 3/2, then the additional singularity has the form of the polar singularity r−ρ+1.

The time‐domain Lippmann–Schwinger equation and convolution quadrature

We consider time‐domain acoustic scattering from a penetrable medium with a variable sound speed. This problem can be reduced to solve a time‐domain volume Lippmann–Schwinger integral equation. Using convolution quadrature in time and trigonometric collocation in space, we can compute an approximate solution. We prove that the time‐domain Lippmann–Schwinger equation has a unique solution and prove conditional convergence and error estimates for the fully discrete solution for globally smooth sound speeds. Preliminary numerical results show that the method behaves well even for discontinuous sound speeds. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 517–540, 2015

Mass Specta of Heavy Quarkonium with Screened Potential in Momentum Space

In this paper we have solved the homogeneous Lippmann-Schwinger integral equation in the momentum space by using a screened funnel potential which is suggested by unquenched lattice calculations. As an application the mass spectra of heavy quarkonia \(\mathrm {\Upsilon }(b\bar {b})\) and \(\psi (c\bar {c}))\), have been calculated. The results have been compared with non screened potentials and experimental data.

Convergence analyses of different modeling schemes for generalized Lippmann–Schwinger integral equation in piecewise heterogeneous media

For wave propagation simulation in piecewise heterogeneous media, Gaussian-elimination-based full-waveform solutions to the generalized Lippmann–Schwinger integral equation (GLSIE) are highly accurate, but involved with extremely time-consuming computations because of the very large size of the resulting boundary–volume integral equation matrix to be inverted. Several flexible approximations to the GLSIE are scaled in an iterative way to adapt numerical solutions to the smoothness of heterogeneous media in terms of incident wavelengths, with a great saving of computing time and memory. Among various typical iterative schemes to the GLSIE matrix, the generalized minimal residual method (GMRES) is an efficient approach to reduce the computational intensity to some degree. The most efficient approximation can be obtained using a Born series, as an alternative iterative solution, to both the boundary-scattering and volume-scattering waves, leading to the Born-series approximation (BSA) scheme and the improved Born-series approximation (IBSA) scheme. These iteration schemes are validated by dimensionless frequency responses to a heterogeneous semicircular alluvial valley, and then applied to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments, compared with the full-waveform numerical solution, indicate that the convergence rates of these methods decrease gradually with increasing velocity perturbations. The comparison also shows that the BSA scheme has a faster convergence than the GMRES method for velocity perturbations less than 10 percent, but converges slowly and even hardly achieves convergence for velocity perturbations greater than 15 percent. The IBSA scheme gives a superior performance over the other methods, with the least iterations to achieve the necessary convergence.

Foldy–Lax approximation on multiple scattering by many small scatterers

The multiple scattering of time harmonic wave emitted by a localized source through a medium with many scatterers can be approximated by an Foldy–Lax self-consistent system when the relative radius of each scatterer is small and the distribution of scatterers is sparse. The scattering amplitude in the Foldy–Lax self-consistent system will be specified in terms of scatterer volume and scattering strength. By neglecting the self-interaction effect, the difference from the exciting field in the Foldy–Lax formula to the analytic wave field given implicitly by the Lippmann–Schwinger integral equation is compared. An upper bound of the difference is obtained in terms of scaled radius and sparsity of the distribution of the scatterers.