Integral Equation Method Research Articles

Legendre spectral projection methods for Fredholm integral equations of first kind

Abstract In this paper, we discuss the Legendre spectral projection method for solving Fredholm integral equations of the first kind using Tikhonov regularization. First, we discuss the convergence analysis under an a priori parameter strategy for the Tikhonov regularization using Legendre polynomial basis functions, and we obtain the optimal convergence rates in the uniform norm. Next, we discuss Arcangeli’s discrepancy principle to find a suitable regularization parameter and obtain the optimal order of convergence in uniform norm. We present numerical examples to illustrate the theoretical results.

Recovery of an infinite rough surface by a nonlinear integral equation method from phaseless near-field data

Abstract This paper is concerned with the two-dimensional inverse acoustic scattering by an unbounded, sound-soft rough surface. We propose a nonlinear integral equation method using multi-frequency phaseless near-field data associated with point sources to reconstruct the shape and location of the rough surface, yielding a fast imaging algorithm. Numerical examples are presented to show the effectiveness of the inverse algorithm.

Effects of a vertical wall on wave power absorption with wave energy converters arrays

The application of in-house transient wave-body interaction ITU-WAVE computational tool is extended to predict the wave power absorption with Wave Energy Converters (WECs) arrays in front of a vertical wall using time dependent boundary integral equation method. The vertical wall effect is taken into account with method of images which considers the perfect reflection of incident waves from a vertical wall. The effects of separation distance between WECs as well as a vertical wall and WECs, and heading angles are studied to predict wave power absorption, mean and individual interaction factors which determine the performances of WECs arrays. The numerical results of WECs arrays in front of vertical wall show that both radiation and exciting force parameters are quite different from those of without a vertical wall. The numerical investigations also demonstrate that wave power absorption with an array system in front of a vertical wall are significantly greater than those of without a vertical wall. This is due to nearly trapped and standing waves between a vertical wall and WECs. The prediction of hydrodynamic parameters in front of a vertical wall with present ITU-WAVE are validated against other published numerical, analytical, and experimental results which show satisfactory agreements.

Wave Scattering in Layered Orthotropic Media I: A Stable PML and a High-Accuracy Boundary Integral Equation Method

In anisotropic media, the standard perfectly matched layer (PML) technique suffers irrevocable instability in terminating the unbounded problem domains. It remains an open question whether a stable PML-like absorbing boundary condition exists. For wave scattering in a layered orthotropic medium, this question is affirmatively answered for the first time in this paper. In each orthotropic medium, the permittivity tensor uniquely determines a change of coordinates, that transforms the governing anisotropic Helmholtz equation into an isotropic Helmholtz equation in the new coordinate system. This leads us to propose a novel Sommerfeld radiation condition (SRC) to rigorously characterize outgoing waves in the layered orthotropic medium. Naturally, the SRC motivates a regionalized PML (RPML) to truncate the scattering problem, in the sense that a standard PML is set up in the new coordinate system in each orthotropic region. It is revealed that the RPML is unconditionally stable compared with the unstable uniaxial PML. A high-accuracy boundary integral equation (BIE) method is developed to solve the resulting boundary value problem. Numerical experiments are carried out to validate the stability of the RPML and the accuracy of the BIE method, showing exponentially decaying truncation errors as the RPML parameters increase.

Dynamic Response of Twin Circular Unlined Tunnels Subjected to Blasting P Waves

The dynamic response of twin circular unlined tunnels is studied by indirect boundary integral equation method (IBIEM). The twin tunnels are assumed to lie in an unbounded elastic space subjected to blasting P waves. The influence of incident angle on the distribution of DSCF around the twin tunnels is analyzed. Besides, the influences of normalized wave number αa and normalized distance d* (ratio of spacing distance to tunnel radius) on the reference DSCF, that is, the peak DSCF for different incident angles with an assurance rate of 95%, are discussed. Results show that 1) the IBIEM is a high-precision method analyzing the DSCF around twin circular unlined tunnels induced by blasting P waves; 2) the incident angle θ0, αa, and d* all have significant influences on the distribution of DSCF; 3) for a constant d*, the peak DSCF around the right tunnel is slightly greater than the left one; 4) the reference DSCF decreases exponentially with the increasing of αa or d*, and the corresponding fitting functions are proposed.

Modified Integral Equation Method With Time-Varying Factors for Low-Frequency Ground-Wave Propagation Delay Prediction

A large amount of long-term measurements illustrate that the propagation delay of low-frequency (LF) ground-wave exhibits complex temporal variation properties. In this letter, we present a modified integral equation (IE) method to be capable of predicting the temporal variation of propagation delay. The modified method introduces suitable time-varying conductivity and refractivity models to characterize the temporal variation of propagation channel in terms of two aspects of ground and atmosphere. Comparisons between the modified method and experiment data along two actual propagation paths show good agreement. The effect of the time-varying factors controlling the conductivity and refractivity on propagation delay is further analyzed. It is found that compared to the refractivity, the time-varying factors in the conductivity have greater influence on the propagation delay.

A problem with shift for a mixed-type model equation of the second kind in an unbounded domain

This article studies a problem with shift in the characteristics of different families in an unbounded domain for a mixed-type model equation of the second kind. The elliptic part of this problem is the vertical halfstrip; the hyperbolic part is the characteristic triangle bounded by the characteristics of the equation. Using the extremum principle we prove the uniqueness of the solution. With the integral equations method we prove the existence of the solution.

Applications of the integral equation method on nonlinear multi degree of freedom vibration systems

This paper compares the accuracy of the integral equation method and the multiple scales method in solving the amplitude equation of multi degree of freedom nonlinear vibration systems. We consider three examples: a two-degree-of-freedom stainless-steel beam system controlled by a saturation controller, a three-degree-of-freedom rotating compressor blade model and a four-degree-of-freedom horizontally supported Jeffcott-rotor system controlled by a PPF controller. The amplitude equations are obtained by applying the integral equation method and the method of multiple scales. The stability analysis is achieved based on the Floquet theory together with Routh–Hurwitz criterion. Furthermore, we modified the iterative procedure of the integral equation method to make the analytic approximate solution more accurate. Finally, the analyses show that in most cases, the analytical solutions obtained by the integral equation method are more excellent agreement with the numerical solutions than the most commonly used method of multiple scales. Therefore, the integral equation method is worth popularizing to obtain the approximate analytical solutions of the multi-degree-of-freedom nonlinear vibration system.

Design and Fabrication of Silicon-Blazed Gratings for Near-Infrared Scanning Grating Micromirror

Blazed gratings are the critical dispersion elements in spectral analysis instruments, whose performance depends on structural parameters and topography of the grating groove. In this paper, high diffraction efficiency silicon-blazed grating working at 800–2500 nm has been designed and fabricated. By diffraction theory analysis and simulation optimization based on the accurate boundary integral equation method, the blaze angle and grating constant are determined to be 8.8° and 4 μm, respectively. The diffraction efficiency is greater than 33.23% in the spectral range of 800–2500 nm and reach the maximum value of 85.62% at the blaze wavelength of 1180 nm. The effect of platform and fillet on diffraction efficiency is analyzed, and the formation rule and elimination method of the platform are studied. The blazed gratings are fabricated by anisotropic wet etching process using tilted (111) silicon substrate. The platform is minished by controlling etching time and oxidation sharpening process. The fillet radius of the fabricated grating is 50 nm, the blaze angle is 7.4°, and the surface roughness is 0.477 nm. Finally, the blazed grating is integrated in scanning micromirror to form scanning grating micromirror by MEMS fabrication technology, which can realize both optical splitting and scanning. The testing results show that the scanning grating micromirror has high diffraction efficiency in the spectral range of 810–2500 nm for the potential near-infrared spectrometer application.

An analysis of least-squares oversampled collocation methods for compactly perturbed boundary integral equations in two dimensions

In recent work (Maierhofer and Huybrechs, 2022, Adv. Comput. Math.), the authors showed that least-squares oversampling can improve the convergence properties of collocation methods for boundary integral equations involving operators of certain pseudo-differential form. The underlying principle is that the discrete method approximates a Bubnov–Galerkin method in a suitable sense. In the present work, we extend this analysis to the case when the integral operator is perturbed by a compact operator K which is continuous as a map on Sobolev spaces on the boundary, K:Hp→Hq for all p,q∈R.This study is complicated by the fact that both the test and trial functions in the discrete Bubnov–Galerkin orthogonality conditions are modified over the unperturbed setting. Our analysis guarantees that previous results concerning optimal convergence rates and sufficient rates of oversampling are preserved in the more general case. Indeed, for the first time, this analysis provides a complete explanation of the advantages of least-squares oversampled collocation for boundary integral formulations of the Laplace equation on arbitrary smooth Jordan curves in 2D. Our theoretical results are shown to be in very good agreement with numerical experiments.

Electromagnetic response and optical properties of anisotropic CuSbS2 nanoparticles

We investigate the electromagnetic response of anisotropic (non-spherical) copper antimony disulfide ( C u S b S 2 ) nanoparticles and layers embedded with them using computational methods. To this end, we calculate the scattering and absorption efficiencies of oblate spheroidal C u S b S 2 nanoparticles using the surface integral equation method. We find strong dependence of the response depending on the anisotropy of the spheroids and their orientation with respect to the electric field polarization of incoming radiation. Thin spheroids display a sharp plasmonic resonance in the ultraviolet, which is observed only for the electric field polarization along the short axis. Fano resonances that appear in the near infrared (NIR) blueshift when the short axis length is reduced, and they can be either strongly suppressed or enhanced depending on the relative orientation of the spheroid. We further investigate the optical response of thin layers containing C u S b S 2 spheroids at a low volume fraction using a Monte Carlo method. We find that the response of these layers can be considerably modified by changing the short axis length and the orientation of particles within the layer with respect to polarization. Our results demonstrate the potential of anisotropic dielectric particles for polarization-dependent-response applications such as solar devices and NIR sensors.

Extension of the Tricomi problem for a loaded parabolic–hyperbolic equation with a characteristic line of change of type

In this work, we investigate a generalization of the Tricomi problem for a loaded mixed‐type equation with the Riemann–Liouville fractional differential operator. By using the method of integral equations, a unique solvability of the formulated problem given in piecewise non‐parallel characteristics is proven.

Dynamic earthquake sequence simulation with an SBIEM accounting for interseismic poroelastic rebound

Afterslip inside a coseismic slip patch is rarely observed, though some previous studies suggest that it is driven by poroelastic rebound (PER). These studies assume constant frictional strength, whereas time-dependent strengthening (healing) of a fault is expected from laboratory experiments, which provide a basis for a rate- and state-dependent friction law (RSF). In this study, quasistatic poroelasticity (PE) was implemented in a dynamic earthquake sequence simulation using a spectral boundary integral equation method, and the effect of PER on the behavior of a fault governed by RSF was examined. Spatio-temporal convolution for PE would significantly affect the resolution of the numerical simulation affordable. This problem has been resolved by numerical approximation of the time dependency of Green’s function of PE in the wavenumber domain, definition of memory variables, and reformulation of the temporal convolution into ordinary differential equations of them. In the novel method, the additional numerical costs due to PE are negligible. A planar fault with a rate-weakening patch embedded in the rate-strengthening region was simulated. Because it is the healing of the fault that competes against PER, both the aging law and slip law were examined, which have different characteristics in the evolution of the fault strength. The simulation results indicate that PER causes postseismic loading to the patch, but the healing efficiently suppresses afterslip not only for the aging law, but also for the slip law. When cases with different friction laws are compared, the healing is more significant for the aging law, which has log-t strengthening at a limit of Vto 0. However, the effect of PER on the slip rate is minor for the slip law. The slip law yields additional healing if the fault is accelerated by loading owing to PER. The simulation results are consistent with the absence of afterslip within the coseismic slip patches in the observations.Graphical

The factorization method for a mixed inverse elastic scattering problem

Abstract This paper is concerned with the extension of the factorization method to the inverse elastic scattering problem by a mixed scatterer which is the union of a penetrable obstacle and an impenetrable object. Having established the well-posedness of the direct problem by the boundary integral equation method, we study the factorization method to reconstruct the shape of the mixed obstacle, that is, recovering the shape of the mixed scatterer from a knowledge of the far-field pattern at a fixed frequency. Some numerical examples are also presented to demonstrate the feasibility and effectiveness of the factorization method.

Prediction approach for wave forces on large-scale offshore structures under linear wave action

This paper presents a numerical mothed that is implemented by applying higher-order boundary elements in the hybrid integral equation method for predicting the wave forces on arbitrarily-shaped offshore structures. In this study, the virtual cylindrical boundary (i.e., matching boundary) is set to divide the fluid region into inner and outer subdomains. The velocity potentials in the outer domain are expressed using the eigenfunction expansion method. In the inner domain, the simple Green's function is adopted as the fundamental solution and Green's second theorem is applied to derive the boundary integral equation (BIE). The velocity potentials in the two subdomains are matched on the matching boundary. The higher-order boundary elements are ultimately employed to discretize BIE and the velocity potentials in the inner domain are then solved. The wave forces on the truncated and stepped cylinders are calculated using the presented method, and the results match well with analytical results, which illustrates the validity of the presented method. The effects of element sizes and modal truncation numbers are further explored, and the interaction between waves and square columns is subsequently analyzed. Eventually, the proposed method is extended to the practical engineering application of wave diffractions from quasi-elliptic pile caps for sea-crossing bridges. The results show that the proposed method can deal with boundaries with complex geometries. In addition, satisfactory results can be obtained by setting relatively few elements and control points on boundaries, and this method can also effectively predict wave forces on the multi-column array.

A sigmoid method for some nonlinear Fredholm integral equations of the second kind

In this work, we present a computational method for solving nonlinear Fredholm integral equations of the second kind which are based on the use of smooth sigmoidal functions. The rate of convergence of the numerical solution is obtained by imposing some additional regularity on the solution. Numerical examples are provided to demonstrate the efficacy of the proposed method.

Global bifurcation of capillary-gravity dark solitary waves on the surface of a conducting fluid under normal electric fields

Summary This article is concerned with capillary-gravity waves travelling on the interface of a dielectric gas and a conducting fluid under the effect of a vertical electric field. A boundary integral equation method is employed to compute fully nonlinear steady travelling wave solutions. The global bifurcation diagram of periodic waves, solitary waves, generalised solitary waves and dark solitary waves is presented and discussed in detail.

Tridiagonal and Block Tridiagonal Computed Sparse Preconditioners for Large Electrodynamic Electric Field Integral Equation (EFIE) Solution

In this work, we propose simple and efficient tridiagonal computed sparse preconditioners for improving the condition number for large compressed electric field integral equation (EFIE) method of moment (MoM) matrix. The preconditioner computation is based on the triangle and block triangle interaction and filled tridiagonally. The computed preconditioner is highly sparse and retains the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${O(N)}$ </tex-math></inline-formula> complexity of computation and preconditioner matrix solution time. Numerical results show the efficiency and accuracy of the proposed preconditioner.

Equivalent Models of Critical Current and Magnetization for Calculating the Magnetic Field in Simply Connected Structures Made of HTS Bulk and HTS Tape

This paper proposes a way to represent the anisotropic nonlinear superconducting properties using a discrete model for magnetization with the corresponding critical parameters, equivalent to the Power Law model for currents. This simplified model makes it possible to analyze magnetic systems with superconducting elements in a stationary mode, which reduces the calculation time. Using of proposed model is demonstrated by the calculation of force interactions between high-temperature superconducting (HTS) samples and permanent magnet in zero-field-cooling (ZFC) and field-cooling (FC) modes. HTS bulks in the form of a prism and stack made of HTS tape are considered. The calculation of the electromagnetic field was carried out using the method of integral equations for field sources. The data of calculations are compared with the results of experimental studies.

Iterated collocation methods for nonlinear third-kind Volterra integral equations with proportional delays

Iterated collocation methods for nonlinear third-kind Volterra integral equations with proportional delays