Singular Kernel Function Research Articles

Free vibration analysis of thin-walled folded structures employing Galerkin-based RKPM and stabilized nodal integration methods

A Galerkin-based meshfree flat shell formulation is chosen to study natural frequency and eigenmode of thin-walled folded structures. Reproducing kernel is used as the interpolation function. Stabilized conforming nodal integration is employed for numerical integration of the weak form. Additionally, sub-domain stabilized conforming integration is adopted for the folded region to integrate the stiffness matrix accurately. The first order shear deformation theory is utilized considering in-plane deformation, out-of-plane deformation and drilling components. The singular kernel function is introduced to effectively handle the folded geometry. An advanced free vibration simulation can be achieved. Accuracy and effectiveness of the meshfree modeling are demonstrated through the numerical examples.

Implicitly linear Jacobi spectral-collocation methods for two-dimensional weakly singular Volterra-Hammerstein integral equations

Weakly singular Volterra integral equations of the second kind typically have nonsmooth solutions near the initial point of the interval of integration, which seriously affects the accuracy of spectral methods. We present Jacobi spectral-collocation method to solve two-dimensional weakly singular Volterra-Hammerstein integral equations based on smoothing transformation and implicitly linear method. The solution of the smoothed equation is much smoother than the original one after smoothing transformation and the spectral method can be used. For the nonlinear Hammerstein term, the implicitly linear method is applied to simplify the calculation and improve the accuracy. The weakly singular integral term is discretized by Jacobi Gauss quadrature formula which can absorb the weakly singular kernel function into the quadrature weight function and eliminate the influence of the weakly singular kernel on the method. Convergence analysis in the L∞-norm is carried out and the exponential convergence rate is obtained. Finally, we demonstrate the efficiency of the proposed method by numerical examples.

Kernel representation of long-wave dynamics on a uniform slope.

Long-wave propagation on a uniformly sloping beach is formulated as a transient-response problem, with initially stationary water subjected to an incident wave. Both the water surface elevation and the horizontal flow velocity on the slope can be represented as convolutions of the rate of displacement of the water surface at the toe of the slope with a singular kernel function of time and space. The kernel, which is typically expressed in the form of an infinite series, accommodates the dynamic processes of long waves, such as shoaling, reflection, and multiple reflections over the slope and yields exact solutions of the linear shallow water equations for any smooth incident wave. The kernel convolution can be implemented numerically by using double exponential formulas to avoid the kernel singularity. The kernel formulation can be extended readily to nonlinear dynamics via the hodograph transform, which in turn enables the instantaneous prediction of nonlinear wave properties and of the occurrence of wave breaking in the near-shore area. This general description of long-wave dynamics provides new insights into the long-studied problem.

Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces

We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.

On the numerical solution of a class of linear fractional integro-differential algebraic equations with weakly singular kernels

This paper focuses on providing a novel high-order algorithm for the numerical solution of a class of linear semi-explicit fractional order integro-differential algebraic equations with weakly singular kernels using the discrete Legendre collocation method. For this purpose, we investigate the existence and uniqueness as well as the regularity properties of this problem and show that some derivatives of its solutions suffer from discontinuity at the left endpoint of the integration domain dependence on both fractional derivative order and the weakly singular kernel function. To remove the singularity, using a new smoothing transformation the main equation is converted into an equivalent equation with better regularity properties and the Legendre collocation discretization method is designed for the transformed equation. The convergence properties of the proposed method are also studied, and we show that the new strategy exhibits a high-order of convergence even when the exact solutions are non-smooth. Some numerical examples are given to illustrate the performance of the approach.

On existence solution for Schrödinger–Kirchhoff-type equations involving the fractional p-Laplacian in ℝN

ABSTRACTThe aim of this paper is to study the existence solution for Schrödinger–Kirchhoff-type equations involving nonlocal p-fractional Laplacianwhere is a real positive parameter, is a continuous function, is a singular kernel function, is a nonlocal fractional operator, with f is a Carathéodory function on satisfying the Ambrosetti–Rabinowitz-type condition. Using Mountain Pass Theorem, we obtain the existence of above equations. Our result is a extension the problem studied by Pucci–Xiang–Zhang [1].

A critical Kirchhoff type problem involving the fractional Laplacian in

In this paper, we are concerned with the existence of solutions for a critical p-Kirchhoff type problem driven by a nonlocal integro-differential operator:where is a continuous function, is a singular kernel function, is a nonlocal fractional operator, with , , f is a Carathéodory function on satisfying the Ambrosetti–Rabinowitz type condition. Under some suitable assumptions, we obtain the existence of nontrivial solutions for above problem by applying the mountain pass theorem. A distinguished feature of this paper is that M(0) may be zero, which means that the problem is degenerate. Consequently, the main theorem extends in several directions the recent results of Autuori, Fiscella and Pucci [Nonlinear Anal. 2015;125:699–714].

Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations

The main concern of this paper is to develop and analyze an operational Tau method for obtaining the numerical solution of fractional weakly singular integro-differential equations when the Jacobi polynomials are used as natural basis functions. This strategy is an application of the matrix–vector–product approach in Tau formulation of the problem. We first study the regularity of the exact solution and show that some derivatives of the exact solution have a singularity at origin dependence on both order of fractional derivative and weakly singular kernel function which makes poor convergence results for the Tau discretization of the problem. In order to recover high-order of convergence, we propose a new variable transformation to regularize the given functions and then to approximate the solution via a satisfactory order of convergence using an operational Tau method. Convergence analysis of this novel method is presented and the expected spectral rate of convergence for the proposed method is established. Numerical results are given which confirm both the theoretical predictions obtained and efficiency of the proposed method.

A method of treating the nearly singular integral in calculation of sound radiation with BEM

For the nearly singular integral of three-dimensional acoustic boundary element method (BEM), based on the 6-noded triangular isoparametric element, a new semi-analytical algorithm of three-dimensional high order element is proposed in this paper. Using Taylor expansion of trigonometric functions in the three dimensional acoustic fundamental solutions, the singular part of the fundamental solutions is separated. Based on the geometric characteristics of the 6-noded triangular element, an approximate singular kernel function is constructed which has the same singularity as singular integral kernel function. Subtracting the approximate kernel function from the kernel function of the singular integral, the latter is decomposed into a regular kernel function and an approximate singular kernel function. The integral of the regular kernel function can be calculated accurately by using the conventional Gauss numerical quadrature. The integral of the new singular part is calculated by the semi-analytic formula derived in this paper. In the surface of the integral element, the local coordinate system ρθ is established and the approximate singular integral is transformed into the integrals of variables ρ and θ which are already separated in ρθ system. The integral with respect to polar variable ρ is expressed by the analytic formulations first. Then the new singular integral which is a surface integral is transformed into the line integral with respect to variable θ , which can be evaluated by the Gaussian quadrature. Consequently, the new semi-analytic algorithm is established to calculate the nearly singular surface integrals in 3D acoustic BEM. Some examples are given in the last part of this paper to show the accuracy and the effectiveness of the present algorithm. The computed results demonstrate that the semi-analytic algorithm with high order element presented in this paper is more effective than linear regularization BEM to solve nearly singular integrals for 3D acoustic BEM.

Komparativna analiza uticaja tečenja betona spregnute grede čelik - beton modelima zasnovanim na integralnoj jednačini Volterra

The paper presents analysis of the stress-strain behaviour and deflection changes due to creep in statically determinate composite steel-concrete beam according to EUROCODE 2, ACI209R-92 and Gardner&Lockman models. The mathematical model involves the equation of equilibrium, compatibility and constitutive relationship, i.e. an elastic law for the steel part and an integral-type creep law of Boltzmann - Volterra for the concrete part considering the above mentioned models. On the basis of the theory of viscoelastic body of Maslov-Arutyunian-Trost-Zerna-Bažant for determining the redistribution of stresses in beam section between concrete plate and steel beam with respect to time 't', two independent Volterra integral equations of the second kind have been derived. Numerical method based on linear approximation of the singular kernel function in the integral equation is presented. Example with the model proposed is investigated.

119 Singular kernelを用いた組合せ構造の非線形解析に関する研究

119 Singular kernelを用いた組合せ構造の非線形解析に関する研究

Some benchmark problems and basic ideas on the accuracy of boundary element analysis

Abstract Taken the linear elasticity problems as examples, some benchmark problems are presented to investigate the influence of calculation error and discretization error on the accuracy of boundary element analysis. For the conventional boundary element analysis based on singular kernel function of fundamental solution and using Gaussian elimination method, the main calculation error comes from the integration of kernel and shape function product on each element. Based on some benchmark problems of “simple problem” without discretization error, it can be observed that sometimes a large number of integration points in Gaussian quadrature should be adopted. To guarantee the integration accuracy reliably, an improved adaptive Gaussian quadrature approach is presented and verified. The main error of boundary element analysis is the discretization error, provided the calculation error has been reduced effectively. Based on some benchmark problems, it can be observed that for the bending problems both the constant and linear element are not efficient, the quadratic element with a reasonable refined mesh is required to guarantee the accuracy of boundary element analysis. An error indicator to guide the mesh refinement in the convergence test towards the converged accurate results based on the distribution of boundary effective stress is presented and verified.

저매개변수 요소를 사용한 2차원 선형탄성 직접 경계요소법의 Kernel 적분법

본 논문은 2차원 선형탄성 직접 경계요소법에서 저매개변수 요소를 사용할 때 Kernel의 적분방법에 대하여 논의하였다. 일반적으로 등매개변수 요소의 경우 형상함수로 통칭되는 해의 기저함수와 요소의 적분을 위해 사용되는 사상함수를 동일하게 사용한다. 그러나 본 논문에서는 사상함수의 차수를 낮게 취하여 순수기저절점을 도입하고 그때 직접 경계요소의 Kernel을 적분하기 위한 방법이 모색되었다. 일반적으로 경계요소법의 적분 Kernel의 경우 Log수치적분과 코쉬주치(Cauchy principal value) 등을 통해 해결하는데, 본 논문에서는 대수적 조작을 통해 적분값의 정확도를 높일 수 있도록 새로운 수식을 유도하였다. 본 연구에서 저매개변수 기반의 직접 경계요소에 대한 강건성과 정확도를 검증하기 위해 2차원 타원형 편미분방정식으로 표현되는 평면응력과 평면변형문제에 대해 적용하였다. 적용 예제로는 단순연결영역(simple connected region)의 대표적 문제인 캔틸레버보와 다중연결영역(multiple connected region)의 대표적인 문제인 개구부가 있는 사각평면에 대해 각각 수치해석을 수행한 결과 대폭적인 자유도의 감소에 비해 정확도 측면에는 기존의 방법과 차이가 없음을 볼 수 있었다. 본 논문에서 제시된 방법은 기저함수 고차화 저매개변수 직접 경계요소법(subparametric high order boundary element)과 이에 기초를 둔 저매개변수 고차 이중경계요소법(subparametric high order dual boundary element)의 초석이 될 수 있을 것이다. In this study, the Kernel integration scheme for 2D linear elastic direct boundary element method has been discussed on the basis of subparametric element. Usually, the isoparametric based boundary element uses same polynomial order in the both basis function and mapping function. On the other hand, the order of mapping function is lower than the order of basis function to define displacement field when the subparametric concept is used. While the logarithmic numerical integration is generally used to calculate Kernel integration as well as Cauchy principal value approach, new formulation has been derived to improve the accuracy of numerical solution by algebraic modification. The subparametric based direct boundary element has been applied to 2D elliptical partial differential equation, especially for plane stress/strain problems, to demonstrate whether the proposed algebraic expression for integration of singular Kernel function is robust and accurate. The problems including cantilever beam and square plate with a cutout have been tested since those are typical examples of simple connected and multi connected region cases. It is noted that the number of DOFs has been drastically reduced to keep same degree of accuracy in comparison with the conventional isoparametric based BEM. It is expected that the subparametric based BEM associated with singular Kernel function integration scheme may be extended to not only subparametric high order boundary element but also subparametric high order dual boundary element.

Enforcement of boundary conditions in meshfree methods using interpolating moving least squares

A technique for enforcement of boundary conditions in meshfree methods based on interpolating moving least squares (IMLS) via usage of singular weights for the solution of partial differential equations is presented. Due to the specific singular choice of the weight functions, which is needed to guarantee the interpolation, there is a problem to find the inverse of the singular matrix. We extend the perturbation technique applied to transport equations by Kunle [Entwicklung und Untersuchung von Moving Least Square Verfahren zur numerischen Simulation hydrodynamischer Gleichungen, Dissertation, Fakultät für Physik, Eberhard-Karls-Universität zu Tübingen] to allow the correct evaluation of all necessary derivatives for the meshfree collocation solution of boundary value problems in interpolation points at a reasonable cost. The inverse is carried out using the regularized weight function. It turns out that a stable inverse is obtained when the vanishing regularization parameter is considered. The detailed description of this strategy is presented. In contrast to standard kernel functions used in EFGM, RKPM, etc., the singular kernel functions lead to truly interpolating functions which satisfy the Kronecker-delta property and they can therefore be used for enforcement of Dirichlet boundary conditions when solving boundary value problems. As a confirmation of the mathematical derivations we give a solution to a model BVP with a known analytical solution, as well as the experimental convergence study of the method to analytical solutions.

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

This paper is concerned with a trigonometric Hermite wavelet Galerkin method for the Fredholm integral equations with weakly singular kernel. The kernel function of this integral equation considered here includes two parts, a weakly singular kernel part and a smooth kernel part. The approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric Hermite wavelet constructed by E. Quak [Trigonometric wavelets for Hermite interpolation, Math. Comp. 65 (1996) 683–722] are developed. The use of trigonometric Hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix. It is shown that we only need to compute and store O ( N ) entries for the weakly singular kernel representation matrix with dimensions N 2 which can reduce the whole computational cost and storage expense. The computational schemes of the resulting matrix elements are provided for the weakly singular kernel function. Furthermore, the convergence analysis is developed for the trigonometric wavelet method in this paper.

Meshfree collocation solution of boundary value problems via interpolating moving least squares

AbstractThis paper presents a meshfree interpolating moving least squares (IMLS) method based on singular weights for the solution of partial differential equations. Due to the specific singular choice of weight functions, which is needed to guarantee the interpolation, there arises a problem of finding the inverse of the occurring singular matrix. The inverse is carried out using a regularization of weight functions. It turns out that a stable inverse is obtained by considering the vanishing regularization parameter.Moreover, the use of this perturbation technique allows the correct evaluation of all necessary derivatives in interpolation points at a reasonable cost.Unlike standard kernel functions used in EFGM, RKPM, etc., the singular kernel functions lead to really interpolating functions which satisfy the Kronecker‐delta property. They can be used for enforcement of Dirichlet boundary conditions when solving boundary value problems.Solution to a model BVP as well as an experimental convergence study of the method to analytical solutions, confirming the mathematical derivations, are given. Copyright © 2006 John Wiley & Sons, Ltd.

On the discretisation error of the weighted Simpson rule

A method of estimating the discretisation error for the weighted Simpson rule is given. The results are used to obtain the convergence rates of a numerical method for solving integral equations with various singular kernel functions.

Iterative variants of the Nystr�m method for the numerical solution of integral equations

Some iterative variants of the Nystrom method for the numerical solution of linear and nonlinear integral equations are introduced and discussed. Numerical examples are given; some are for integral equations with singular kernel functions.