Integral Equation Method Research Articles

Метод пересчета результатов модельных испытаний гребного винта на натурные условия

Object and purpose of research. This paper discusses numerical method for simulation of propeller parameters in model and full-scale conditions. The purpose is to develop a similar methods for extrapolation of propeller model test data to the full-scale. Materials and methods. Propeller parameters are simulated as per the improved numerical method of E. Papmel. The flow around cylindrical section profiles of blades and hub is simulated as per the methods of boundary integral equations and integral calculation methods for the boundary layer with consideration of laminary-to-turbulent transition. Main results. Development of the method taking into account scale effect for propellers. The method has been applied to extrapolate model test data to the full scale. Conclusion. The method is of practical value for modern propeller design.

Регуляризация и выбор параметра решений нелинейных интегральных уравнений Вольтерра-Стилтьеса третьего рода

The article considers nonlinear Volterra-Stieltjes integral equations of the third kind, and its solution by regularizing operator according to M. M. Lavrentev. A uniqueness theorem was proved, and a regularization parameter was chosen. The research uses the concept of a derivative with respect to an increasing function, the method of regularization according to M. M. Lavrentevs methods in functional analysis, methods of transformation of equations, methods of integral and differential equations. Proposed methods can be used to study the integral, integral-differential equations of the Volterra-Stieltjes type of high orders, as well as in the qualitative study of some applied processes in the field of physics, ecology, medicine, and the theory of control complex systems. They can be used in the further development of the theory of integral equations in classes of incorrect problems, in numerical solution of Volterra-Stieltjes integral equations of the third kind, and when solving specific applied problems that lead to equations of the third kind.

An equation-solving method based on radiation distribution factor for radiative transfer in participating media with diffuse boundaries

An efficient and accurate method to solve the directional radiation intensity in the participating medium has always been a pursuit in radiative transfer fields, especially in radiation detection and inverse radiative transfer problems. As one of the few algorithms with both high accuracy and strong applicability, the reverse Monte Carlo (RMC) method suffers from the disadvantages of unavoidable random statistical errors and time-consuming ray tracing. Recently, a so-called equation-solving integral equation method based on the radiation distribution factor (ES-RDFIEM) can effectively overcome these shortcomings of RMC, which has comparable accuracy as RMC while taking only a few hundredths or thousandths of the calculation time of RMC. In this paper, ES-RDFIEM was extended to a radiation system with diffuse surfaces by constructing the radiative transfer equation (RTE) about the radiation distribution factor (RDF) of the wall and internal medium, respectively. The mathematical principle and formula were introduced in detail, and the computational performance was examined by several samples with different radiative parameters. The results showed that the RDF and radiation intensity solved by ES-RDFIEM are in good agreement with RMC, and ES-RDFIEM is more stable and reliable. More importantly, the computation time of ES-RDFIEM is significantly shorter than that of RMC, and it is not affected by wall emissivity, internal medium optical thickness and scattering albedo.

Collocation boundary value methods for auto-convolution Volterra integral equations

In this paper, we analyze the collocation boundary value solutions of the second-kind auto-convolution Volterra integral equations. In contrast to classical collocation methods, the full discretization of auto-convolution Volterra integral equations via the boundary value technique leads to a nonlinear system. Thus we reconsider the solvability of the proposed scheme and the boundedness of the numerical solution. Moreover, with the help of the Peano kernel theorem, the convergence order is developed, which is demonstrated by several numerical examples.

Inverse obstacle scattering for elastic waves in the time domain

This paper concerns an inverse elastic scattering problem which is to determine a rigid obstacle from time domain scattered field data for a single incident plane wave. By using the Helmholtz decomposition, we reduce the initial-boundary value problem for the time domain Navier equation to a coupled initial-boundary value problem for wave equations, and prove the uniqueness of the solution for the coupled problem by employing the energy method. The retarded single layer potential is introduced to establish a set of coupled boundary integral equations, and uniqueness is discussed for the solution of these boundary integral equations. Based on the convolution quadrature method for time discretization, the coupled boundary integral equations are reformulated into a system of boundary integral equations in the s-domain, and then a convolution quadrature based nonlinear integral equation method is proposed for the inverse problem. Numerical experiments are presented to show the feasibility and effectiveness of the proposed method.

Convergence analysis of oversampled collocation boundary element methods in 2D

Collocation boundary element methods for integral equations are easier to implement than Galerkin methods because the elements of the discretisation matrix are given by lower-dimensional integrals. For that same reason, the matrix assembly also requires fewer computations. However, collocation methods typically yield slower convergence rates and less robustness, compared to Galerkin methods. We explore the extent to which oversampling can improve both robustness and convergence rates of collocation for singular boundary integral equations. We show that in some cases convergence rates can actually be higher than the corresponding Galerkin method, although this requires oversampling at a faster than linear rate. In most cases of practical interest, oversampling at least lowers the error by a constant factor. This can still be a substantial improvement: we analyse an example where linear oversampling by a constant factor J (leading to a rectangular system of size JN × N) improves the error at a cubic rate in the constant J. Furthermore, the oversampled collocation method is much less affected by a poor choice of collocation points, as we show how oversampling can lead to guaranteed convergence. Numerical experiments are included for the two-dimensional Helmholtz equation.

On the Numerical Solution of a Hyperbolic Inverse Boundary Value Problem in Bounded Domains

We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fréchet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.

Integral equation solutions for the average run length for monitoring shifts in the mean of a generalized seasonal ARFIMAX(P, D, Q, r)s process running on a CUSUM control chart.

The CUSUM control chart is suitable for detecting small to moderate parameter shifts for processes involving autocorrelated data. The average run length (ARL) can be used to assess the ability of a CUSUM control chart to detect changes in a long-memory seasonal autoregressive fractionally integrated moving average with exogenous variable (SARFIMAX) process with underlying exponential white noise. Herein, new ARLs via an analytical integral equation (IE) solution as an analytical IE and a numerical IE method to test a CUSUM control chart's ability to detect a wide range of shifts in the mean of a SARFIMAX(P, D, Q, r)s process with underlying exponential white noise are presented. The analytical IE formulas were derived by using the Fredholm integral equation of the second type while the numerical IE method for the approximate ARL is based on quadrature rules. After applying Banach's fixed-point theorem to guarantee its existence and uniqueness, the precision of the proposed analytical IE ARL was the same as the numerical IE method. The sensitivity and accuracy of the ARLs based on both methods were assessed on a CUSUM control chart running a SARFIMAX(P, D, Q, r)s process with underlying exponential white noise. The results of an extensive numerical study comprising the examination of a wide variety of out-of-control situations and computational schemes reveal that none of the methods outperformed the IE. Specifically, the computational scheme is easier and can be completed in one step. Hence, it is recommended for use in this situation. An illustrative example based on real data is also provided, the results of which were found to be in accordance with the research results.

Accelerated Calderón preconditioning for Maxwell transmission problems

We investigate a range of techniques for the acceleration of Calderón (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation of operator products. Our focus is on the well-known PMCHWT formulation, but the techniques we introduce can be applied generically. By using barycentric meshes only for the preconditioner and not for the original boundary integral operator, we achieve significant reductions in computational cost by (i) using “reduced” Calderón preconditioners obtained by discarding constituent boundary integral operators that are not essential for regularisation, and (ii) adopting a “bi-parametric” approach [1,2] in which we use a lower quality (cheaper) H-matrix assembly routine for the preconditioner than for the original operator, including a novel approach of discarding far-field interactions in the preconditioner. Using the boundary element software Bempp (www.bempp.com), we compare the performance of different combinations of these techniques in the context of scattering by multiple dielectric particles. Applying our accelerated implementation to 3D electromagnetic scattering by an aggregate consisting of 8 monomer ice crystals of overall diameter 1 cm at 664 GHz leads to a 99% reduction in memory cost and at least a 75% reduction in total computation time compared to a non-accelerated implementation.

On the direct and inverse electromagnetic scattering by chiral media

Consider the direct and inverse electromagnetic scattering problems by a penetrable cylindrical chiral medium in a homogeneous achiral background medium. The scattering problem can be modeled by a pair of the two-dimensional Helmholtz equations with coupled transmission boundary conditions. The boundary integral equation method is applied to solve the direct scattering problem in a suitable Sobolev space. Then we investigate the linear sampling method for recovering the unknown scatterer from far-field pattern of the scattered field. The coupled boundary conditions bring new challenges during the procedure, and some novel techniques are needed in the theoretical analysis. Finally, we perform several numerical experiments to demonstrate the efficiency and effectiveness of this method.

Evaluation of the 3D Topographic Effect of Homogeneous and Inhomogeneous Media on the Results of 2D Inversion of Electrical Resistivity Tomography Data

This work is devoted to the creation of numerical 3D models for studying 3D topographic effects of homogeneous and inhomogeneous media on the results of 2D inversion. To solve the direct problem, the integral equations method (IEM) was applied to simulate the electric field and calculate the values of apparent resistivity. For the numerical solution of the integral equation, a computational mesh, adapted to the terrain features, was constructed. The calculation of the apparent resistivity of the medium was implemented for a pole-dipole array. Calculations have been performed for a conducting medium with 3D local inhomogeneity and the ground surface topography. The influence of the 3D model with topography, flat ground surface, and immersed inhomogeneity on the results of 2D inversion was estimated. Influencing factors of the 3D model, such as the slope angle of the topography, the resistivity ratio of the inhomogeneity and the host medium, the average size of inhomogeneity and topography, the distance from the inhomogeneity to the measuring line for the medium with and without topography, and the distance between the electrodes of the electrical resistivity tomography (ERT) array, were studied. Based on the research results, recommendations and conclusions that can be useful when conducting geophysical studies by the ERT method are drawn.

Three-dimensional numerical modeling of direct current resistivity methods in mixed space-wavenumber domain

At present, in order to obtain highly accurate solution for complex (geo-electric) conditions, it is necessary to divide the model precisely in the numerical modeling of DC resistivity method, which results in greatly increases the amount of computation and memory. To solve this problem, we propose a 3D numerical simulation method in a mixed space-wavenumber domain. The method transforms the partial differential equation of DC abnormal potential into many independent 1D differential equations with different wavenumbers by performing a 2D Fourier transform along the horizontal direction. It decomposes a large-scale 3D numerical modeling problem into many 1D numerical modeling problems, which greatly reduces the computation and memory requirements of modeling, and each 1D differential equation is independent, so it has high parallelism. The vertical direction is reserved as the spatial domain, which is convenient for the subdivision, taking into account the calculation accuracy and efficiency. A finite-element algorithm combined with a chasing method is used to solve the 1D differential equations with different wavenumbers, and a contraction operator based on electromagnetic integral equation method is used to the iteration of the algorithm for the first time. Taking full advantage of the rapidity of Fourier transform, the high efficiency of solving 1D differential equations and the high parallelism of 1D problems with different wavenumbers, it can greatly reduce the amount of calculation and storage, thus greatly improves the efficiency of the algorithm. The accuracy and convergence of the algorithm are verified by using the model of a sphere in half space. Based on the model with five cubes in half space, the efficiency of the algorithm is verified by comparing with the academic Gimli package based on the finite element method. In addition, a parallel algorithm based on OpenMP is used to verify the parallelism of the proposed algorithm. The examples results show that, the 3D numerical modeling method of DC resistivity in the mixed space-wavenumber domain has the characteristics of high efficiency, high precision and highly parallel. This method provides a new way for 3D numerical simulation of DC resistivity method.

Error analysis of reiterated projection methods for Hammerstein integral equations

In this paper, we consider the Newton-iteration scheme based on Galerkin and multi-Galerkin operators to solve non-linear integral equations of the Fredholm-Hammerstein type for both smooth and weakly singular algebraic and logarithmic type kernels. By choosing as space of piecewise polynomials subspace of degree , we show that for a smooth kernel, Galerkin and multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders and , respectively, where h is the norm of the partition. For weakly singular kernels, we show that the Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders , for algebraic kernels and for logarithmic kernels. Also the multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders for algebraic kernels and for logarithmic kernels. Numerical results are given for the justification of our theoretical results.

Singular transients in the presence of homogeneous reactions at cylindrical wire electrodes: Simulation by the adaptive Huber method for integral equations

Current transients possessing a weak temporal singularity of the t-1/2 type arise in the theory of chronoamperometry and cyclic voltammetry. Simulating them by the method of integral equations (IEs) requires a highly accurate algorithm for computing integrals ∫0tK(t,τ)τ-1/2dτ for any kernel term K(t,τ) present in the IEs. Such an algorithm is presented for the kernel term K(t,τ)=exp[-k(t-τ)]kcylw[ρ(t-τ)1/2] characteristic of IEs arising for diffusion coupled with (pseudo-) first order homogeneous reactions at cylindrical wire electrodes. The formerly described adaptive Huber method, equipped with this algorithm, is applied to three example IEs describing singular transients. These simulations are successful when the parameter k (typically a rate constant of a homogeneous reaction) and the electrode cylindricity parameter ρ are moderately large. For too large k or ρ a divergence of the method was observed, due to inevitable roundoff errors. Substantial amendments to the adaptive Huber method may be necessary in order to overcome this numerical difficulty.

Determination of Stress Concentration Near Dies, Holes, and Cracks in the Half Plane Based on the Method of Integral Equations and Green Solutions

Determination of Stress Concentration Near Dies, Holes, and Cracks in the Half Plane Based on the Method of Integral Equations and Green Solutions

Method for Loran-C Additional Secondary Factor Correction Based on Neural Network and Transfer Learning

An additional secondary factor (ASF) correction method is proposed to improve the accuracy of Loran-C navigation and positioning based on a combination of backward propagation neural network (BPNN) and transfer learning. The BPNN is used to train a network model based on the theoretical ASF data source calculated by the integral equation method. Although the trained BPNN model fitted well with the theoretical ASF results, it is not sufficient for ASF prediction in the actual environment involving uncertain topography and geology. To compensate for this deficiency, transfer learning is employed to further finely calibrate the BPNN model against the ASF measurements. By using the advantages of BPNN and transfer learning, the proposed method achieved higher accuracy than the purely theoretical method. Measurement results are used to validate the superiority of the proposed ASF correction method to the theoretical one. Particularly, the mean absolute error of the ASF prediction results decreased from 2.02 to 0.29 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</i> s after introducing the transfer learning in the trained BPNN model.

Numerical investigation of the Fourier–Kochin theory for wave-induced response estimation of floating structures

The Fourier–Kochin theory—an indirect solution approach for the boundary integral representations of free-surface flows that rely on free-surface Green function—is investigated by considering effective computation and reliable prediction. The present study addresses wave–body interaction with zero forward speed by adopting a higher-order approximation. Three distinct hull forms, the DTMB 5415, a scaled model reproduced from it, and a barge are used for practical application. The scaled frigate and barge models are used for comparative analysis to assess the performance of different computation techniques, modeling elements, and solution methods and to find the best course; the purpose of studying DTMB 5415 is to provide a realistic application based on a full-scale hull form by using the findings of the earlier benchmark studies. The analyses that cover both the rigid-body and elastic responses indicate that an adaptive approach for the computation of Fourier components has the potential to eliminate the numerical drawbacks of the presented implementation the Fourier–Kochin theory, but also stringently needs parallelism, preferably using the Single Program Multiple Data model. Using discontinuous elements for free-surface discretization when applying the Extended Boundary Integral Equation Method, the featured technique for irregular frequency suppression, is also promoted.

Numerical investigation on evaluating nonlinear waves due to an air cushion vehicle in steady motion by a higher order desingularized boundary integral equation method

The problem of nonlinear ship waves generated by an air cushion vehicle is studied and modelled in the framework of potential flow theory. A higher order desingularized boundary integral equation method (HODBIEM) using Rankine source is developed. A stationary iterative method is adopted to solve the nonlinear free surface which is discretized into the inner area disturbed by cushion pressure and the outer undisturbed area. Techniques for numerical implementation are described in detail. To save computing time, grid density and domain size are refined.Both straight-ahead and yawed cases are considered for validations. Wave-making resistance versus various speeds in deep and shallow water are calculated for straight-ahead vehicles. The resistance curves obtained by HODBIEM agree well with results of nonlinear CFD simulations. The linear theory predicts higher peak value of resistance than nonlinear methods. The experimental data fit at low Froude number but deviate from numerical predictions when exceeding primary hump speed. Effects of speed and water depth are analyzed by wave patterns and wave profiles. Resistance, side force and yaw moment are calculated at Fn = 0.533 for drift angle from 0° to 90°. Satisfactory results are given by present method. Details about wave patterns and wave angles are discussed.

Stability analysis of the high-order multistep collocation method for the functional integral equations with constant delays

The results on the stability of recurrences play an important role in the theory of dynamical systems and computer science in connection to the notions of shadowing and controlled chaos. In this paper, stability properties of high-order multistep collocation method for functional integral equations of Volterra integral equations with constant delays type with respect to significant test equations are investigated.

Plane Boundary Value Problems of Thermoelasticity and Their Generalized Solutions

Nonstationary boundary value problems of uncoupled thermoelasticity are considered. A method of boundary integral equations in the initial space-time has been developed for solving boundary value problems of thermoelasticity by plane deformation by use general functions method (GFM). The generalized solutions of boundary value problems are constructed in the space of generalized vector functions. Fundamental Green tensor, stress tensor and their antiderivatives over time are used for construction their regular integral representations. By use analogue of Gauss formula for fundamental stress the singular boundary integral equations are constructed to determine the unknown boundary functions. The resulting formulas have important engineering application. They make possible to determine the thermal stressed state of the medium by the boundary values of stresses, displacements, temperature and heat flux, without solving singular boundary integral equations. Because for real engineering problems these physical characteristics can be experimentally measured at the boundary. Moreover, the formulas allow to calculate the influence of each of these characteristics of the process on its stressstrain state. The last one is very important in designing structures made of thermoelastic materials.