Three-dimensional Integral Equation Research Articles

Time-domain simulation of nonlinear motions of moored floating body in irregular waves using rankine source method

This study introduces a three-dimensional time-domain Indirect Boundary Integral Equation Method (IBIEM) for simulating the motion of the floating body in both regular and irregular waves. The method systematically addresses second-order wave loads, focusing on low-frequency motions induced by low-frequency wave loads, crucial for the stability of offshore structures. Unlike traditional methods, the time-domain IBIEM simplifies the analysis by simultaneously considering all wave components of irregular waves, making the calculation of nonlinear wave loads more straightforward and intuitive. An approximate second-order equation of motion is proposed, simplifying the computational process by eliminating the need for solving boundary value problems for second-order radiation forces. The IBIEM is coupled with a quasi-static mooring model based on the elastic catenary equation, accurately capturing the influence of mooring systems on the response of the floating body. Validation against model tests and established numerical methods demonstrates the accuracy of the developed code in predicting the dynamic behavior of offshore structures. Applied to various scenarios, including a truncated cylinder, the barge, H2FPSO, and K-Semi, the results show strong agreement with experimental data, confirming the reliability and effectiveness of the method in simulating complex wave-structure interactions, making it a valuable tool for offshore structure design and analysis.

Efficient Collocation Algorithm for Solving Three-Dimensional Linear and Nonlinear Volterra Integral Equations

Efficient Collocation Algorithm for Solving Three-Dimensional Linear and Nonlinear Volterra Integral Equations

To the Theory of Three-Dimensional Model Integral Volterra-Type Equations with Boundary Singular, Weakly Singular, and Strongly Singular Kernels

To the Theory of Three-Dimensional Model Integral Volterra-Type Equations with Boundary Singular, Weakly Singular, and Strongly Singular Kernels

Three-dimensional Wiener-Hopf integral equation in mixed problems of anisotropic media

For the first time, an exact solution of the three-dimensional Wiener-Hopf integral equation is given, one-dimensional and two-dimensional versions of which are widely used in mixed, including contact problems. The equation in question can be abstractly represented as the action of a three-dimensional stamp on a four-dimensional half-space or layer. In the work, in relation to contact problems of mechanics, as well as seismology, an application of this method was found. It consists in the fact that along with two geometric coordinates, the time coordinate extending along the entire axis is taken as the third. It allows you to introduce a change of condition at zero for stresses, moving in the contact zone from their values to their velocities. Attempts at an analytical or numerical solution of this problem are not known to the authors. The results may be useful in the fields of using the Wiener-Hopf equation in probability theory and statistics, where one-dimensional variants are used, as well as in seismology. Just as in the two-dimensional case, a universal modeling method based on block elements is used in the study.

Solving Three-Dimensional Volterra Integral Equations by Using Operational Matrix with Block-Pulse Functions

A simple direct method for solving three-dimensional linear Volterra integral equation of the second kind was introduced in this paper. Our method was demonstrated by applying three-dimensional block-pulse functions (3D-BPFs) and their operational matrix of integration. Indeed, we converted an integral equation to a linear system that can be easily solved. The convergence analysis of the method was discussed by convergence of 3D-BPFs and we found a bound for the error. Finally, some numerical examples illustrated that our method is feasible and efficient.

Methods and effective algorithms for solving multidimensional integral equations

Objectives. Integral equations have long been used in mathematical physics to demonstrate existence and uniqueness theorems for solving boundary value problems for differential equations. However, despite integral equations have a number of advantages in comparison with corresponding boundary value problems where boundary conditions are present in the kernels of equations, they are rarely used for obtaining numerical solutions of problems due to the presence of equations with dense matrices that arise that when discretizing integral equations, as opposed to sparse matrices in the case of differential equations. Recently, due to the development of computer technology and methods of computational mathematics, integral equations have been used for the numerical solution of specific problems. In the present work, two methods for numerical solution of two-dimensional and three-dimensional integral equations are proposed for describing several significant classes of problems in mathematical physics.Methods. The method of collocation on non-uniform and uniform grids is used to discretize integral equations. To obtain a numerical solution of the resulting systems of linear algebraic equations (SLAEs), iterative methods are used. In the case of a uniform grid, an efficient method for multiplying the SLAE matrix by vector is created.Results. Corresponding SLAEs describing the considered classes of problems are set up. Efficient solution algorithms using fast Fourier transforms are proposed for solving systems of equations obtained using a uniform grid.Conclusions. While SLAEs using a non-uniform grid can be used to describe complex domain configurations, there are significant constraints on the dimensionality of described systems. When using a uniform grid, the dimensionality of SLAEs can be several orders of magnitude higher; however, in this case, it may be difficult to describe the complex configuration of the domain. Selection of the particular method depends on the specific problem and available computational resources. Thus, SLAEs on a non-uniform grid may be preferable for many two-dimensional problems, while systems on a uniform grid may be preferable for three-dimensional problems.

Fourier Bases-Expansion for Three-Dimensional Electromagnetic Inverse Scattering Problems

Three-dimensional Fourier bases-expansion contraction integral equation for inversion (FBE-CIE-I) is developed to solve electromagnetic inverse scattering problems (ISPs). Taking the shape of the scatterer into account, the size of the domain of interest (DoI) and thus the number of low-frequency Fourier bases are flexibly chosen in the proposed method. The accuracy and the stability of the proposed method are verified by the numerical tests on irregularly shaped targets. The results indicate that the accuracy and the convergence of the proposed 3-D FBE-CIE-I is insensitive to the parameters and the noises on the scattered fields.

A three-dimensional indirect boundary integral equation method for the scattering of seismic waves in a poroelastic layered half-space

A new indirect boundary integral equation method (IBIEM) is developed to solve the seismic wave scattering problem in a three-dimensional fluid-saturated layered half-space. Based on Biot's theory, the Green's functions of inclined circular loads in porous elastic layered half-space are firstly deduced. According to the single-layer potential theory, when establishing the scattered wave field, the uniform surface loads and fluid sources are located directly on the boundary surfaces of the scatterers, so as to avoid determining the optimal location of fictitious wave sources surface. At the same time, the radiation condition of wave in semi-infinite layered media can be accurately realized by using the dynamic Green's function of concentrated load, and the computational memory can be greatly reduced. The scattering of seismic waves by a canyon topography in saturated layered half-space is examined in detail. The numerical results indicate that with the increase of the porosity of the overlying soil, the amplitude of surface displacement can be amplified by 1∼6 times, and the amplification effect is more significant near the corner of the canyon, which can be attributed to the superposition of wave diffraction effect and resonance amplification effect of saturated soil layer.

A new method for solving three-dimensional nonlinear Fredholm integral equations by Haar wavelet

In this paper, a new iterative method of successive approximations based on Haar wavelets is proposed for solving three-dimensional nonlinear Fredholm integral equations. The convergence of the method is verified. The error estimation and numerical stability of the proposed method are provided in terms of Lipschitz condition. Conducting numerical experiments confirm the theoretical results of the proposed method and endorse the accuracy of the method.

Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model. Mathematical Subject Classification (2010): 45P05, 45G10, 47H99

Approximating the solution of three-dimensional nonlinear Fredholm integral equations

The purpose of this study is to construct a new efficient iterative method of successive approximation based on the three-point Simpson quadrature rule for solving three-dimensional nonlinear Fredholm integral equations. We have also provided the convergence and error analysis of the proposed method. Furthermore, we present the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, some numerical experiments illustrate the accuracy of the method.

A three-dimensional momentum-space calculation of three-body bound state in a relativistic Faddeev scheme

In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body t–matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.

Boundary integral equations for sound radiation from a harmonically vibrating body moving uniformly in a free space.

In this high-speed transit era, the prediction of sound radiation from a moving and vibrating object is an important research topic. A typical example is a high-speed train wheel that not only moves fast in the railway direction but, due to wheel/rail interaction, also vibrates at high frequency, generating a complex sound field. By making use of sound spectra generated by moving harmonic compact sources, three-dimensional (3D) boundary integral equations are established in this paper for sound radiation from a harmonically vibrating body moving uniformly in a free space. The 3D boundary integral equations are reduced to 2D ones for a special case in which the body is axisymmetric and moves in its axial direction. The 2D boundary element method is applied to solve the 2D boundary integral equations and to produce results for a pulsating and moving sphere. Results show that the moving speed of the pulsating sphere has a significant effect on the generated sound field. This means that, for sounds radiated from a high-speed train wheel vibro-acoustically, the motion of the wheel has to be taken into account.

A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general domains

In this article, we study a kernel-based method to solve three-dimensional linear Fredholm integral equations of the second kind over general domains. The radial kernels are utilized as a basis in the discrete collocation method to reduce the solution of linear integral equations to that of a linear system of algebraic equations. Integrals appeared in the scheme are approximately computed by the Gauss–Legendre and Monte Carlo quadrature rules. The method does not require any background mesh or cell structures, so it is mesh free and accordingly independent of the domain geometry. Thus, for the three-dimensional linear Fredholm integral equation, an irregular domain can be considered. The convergence analysis is also given for the method. Finally, numerical examples are presented to show the efficiency and accuracy of the technique.

A noniterative mean-field QM/MM-type approach with a linear response approximation toward an efficient free-energy evaluation.

Mean-field treatment of solvent provides an efficient technique to investigate chemical processes in solution in quantum mechanics/molecular mechanics (QM/MM) framework. In the algorithm, an iterative calculation is required to obtain the self-consistency between QM and MM regions, which is a time-consuming step. In the present study, we have proposed a noniterative approach by introducing a linear response approximation (LRA) into the solvation term in the one-electron part of Fock matrix in a hybrid approach between molecular-orbital calculations and a three-dimensional (3D) integral equation theory for molecular liquids (multicenter molecular Ornstein-Zernike self-consistent field [MC-MOZ-SCF]; Kido et al., J. Chem. Phys. 2015, 143, 014103). To save the computational time, we have also developed a fast method to generate electrostatic potential map near solute and the solvation term in Fock matrix, using Fourier transformation (FT) and real spherical harmonics expansion (RSHE). To numerically validate the LRA and FT-RSHE method, we applied the present approach to water, carbonic acid, and their ionic species in aqueous solution. Molecular properties of the solutes were evaluated by the present approach with four different types of initial wave functions and compared with those by the original (MC-MOZ-SCF). We found that an initial wave function considering solvation effects is needed to appropriately reproduce the properties by MC-MOZ-SCF. Furthermore, a benchmark test for 32 solute molecules was performed to evaluate the accuracy of the present approach for solvation free energy (SFE) and measure the speedup ratio for MC-MOZ-SCF. The error of SFE for MC-MOZ-SCF does not correlate with the SFE but increases in proportion to the electronic reorganization energy. Similar to water and carbonic acid, an initial wave function with solvation effects is also important to make the error small. From the averaged speed up ratio, the present approach is 13.5 times faster than MC-MOZ-SCF. © 2019 Wiley Periodicals, Inc.

A three-dimensional indirect boundary integral equation method for modeling elastic wave scattering in a layered half-space

A new indirect boundary integral equation method (IBIEM) is proposed in this study to solve three-dimensional (3-D) elastic wave scattering by heterogeneities in a multi-layered half-space, employing Green’s function of distributed loads on equivalent circular elements, thus avoiding the element discretization on layer interfaces. The proposed method enables the fictitious loads to be directly distributed on the surfaces of scatterer and the weak singularity to be tackled by analytical integration. Also, the radiation condition in the semi-infinite layered medium can be satisfied accurately, and the memory requirements can also be greatly reduced, especially for a large number of layers or gradient medium. The numerical accuracy was verified through comparisons with existing results and the numerical convergence was also confirmed. The results clearly demonstrate the simplicity and effectiveness of the method, and also reveals the complicated scattering characteristics in a layered half-space that are dominated by the resonant properties of the layered medium.

On the Existence of Solutions to Spatial Nonlinear Boundary Value Problems for Arbitrary Elastic Inhomogeneous Anisotropic Body

We study the solvability of a nonlinear boundary value problem for a system of nonlinear partial differential equations of the second order. The goal of this paper is to prove the existence theorem for solutions to the mentioned problem. This problem is reduced to a system of three-dimensional nonlinear singular integral equations, whose solvability can be proved with the use of the symbol of a singular operator and the contraction mapping principle.

Discretization Methods for Three-Dimensional Singular Integral Equations of Electromagnetism

Theorems providing the convergence of approximate solutions of linear operator equations to the solution of the original equation are proved. The obtained theorems are used to rigorously mathematically justify the possibility of numerical solution of the 3D singular integral equations of electromagnetism by the Galerkin method and the collocation method.

Stokes flow of vesicles in a circular tube

The inertialess motion of lipid-bilayer vesicles flowing through a circular tube is investigated via direct numerical simulation and lubrication theory. A fully three-dimensional boundary integral equation method, previously used to study unbounded and wall-bounded Stokes flows around freely suspended vesicles, is extended to study the hindered mobility of vesicles through conduits of arbitrary cross-section. This study focuses on the motion of a periodic train of vesicles positioned concentrically inside a circular tube, with particular attention given to the effects of tube confinement, vesicle deformation and membrane bending elasticity. When the tube diameter is comparable to the transverse dimension of the vesicle, axisymmetric lubrication theory provides an approximate solution to the full Stokes-flow problem. By combining the present numerical results with a previously reported asymptotic theory (Barakat & Shaqfeh, J. Fluid Mech., vol. 835, 2018, pp. 721–761), useful correlations are developed for the vesicle velocity $U$ and extra pressure drop $\unicode[STIX]{x0394}p^{+}$. When bending elasticity is relatively weak, these correlations are solely functions of the geometry of the system (independent of the imposed flow rate). The prediction of Barakat & Shaqfeh (2018) supplies the correct limiting behaviour of $U$ and $\unicode[STIX]{x0394}p^{+}$ near maximal confinement, whereas the present study extends this result to all regimes of confinement. Vesicle–vesicle interactions, shape transitions induced by symmetry breaking, and unsteadiness introduce quantitative changes to $U$ and $\unicode[STIX]{x0394}p^{+}$. By contrast, membrane bending elasticity can qualitatively affect the hydrodynamics at sufficiently low flow rates. The dependence of $U$ and $\unicode[STIX]{x0394}p^{+}$ on the membrane bending stiffness (relative to a characteristic viscous stress scale) is found to be rather complex. In particular, the competition between viscous forces and bending forces can hinder or enhance the vesicle’s mobility, depending on the geometry and flow conditions.

A 3D spatial spectral integral equation method for electromagnetic scattering from finite objects in a layered medium

The generalization of a two-dimensional spatial spectral volume integral equation to a three-dimensional spatial spectral integral equation formulation for electromagnetic scattering from dielectric objects in a stratified dielectric medium is explained. In the spectral domain, the Green function, contrast current density, and scattered electric field are represented on a complex integration manifold that evades the poles and branch cuts that are present in the Green function. In the spatial domain, the field-material interactions are reformulated by a normal-vector field approach, which obeys the Li factorization rules. Numerical evidence is shown that the computation time of this method scales as O(N log N) on the number of unknowns. The accuracy of the method for three numerical examples is compared to a finite element method reference.