Meshless Local Boundary Integral Equation Research Articles

On optimal radius of sub-domains in meshless LBIE method

Local weak based meshless methods construct weak form of governing equations on local sub-domains. In two dimensional domains, for the simplicity of computations, these sub-domains are taken as circles. In these methods, the optimal radius of sub-domains has been an open problem yet. This paper aims at solving this problem for meshless local boundary integral equation (LBIE) method to enhance its performance. It is proved that a sub-domain for which the Lebesgue constant takes its minimum over its boundary is the optimal sub-domain. In other words, the optimal sub-domain is one for which the solution of PDE is approximated on its boundary as accurate as possible. A comprehensive numerical study confirms the theory.

Meshless velocity – vorticity local boundary integral equation (LBIE) method for two dimensional incompressible Navier-Stokes equations

PurposeThis paper aims to describe the 2D meshless local boundary integral equation (LBIE) method for solving the Navier–Stokes equations.Design/methodology/approachThe velocity–vorticity formulation is selected to eliminate the pressure gradient of the equations. The local integral representations of flow kinematics and transport kinetics are derived. The integral equations are discretized using the local RBF interpolation of velocities and vorticities, while the unknown fluxes are kept as independent variables. The resulting volume integrals are computed using the general radial transformation algorithm.FindingsThe efficiency and accuracy of the method are illustrated with several examples chosen from reference problems in computational fluid dynamics.Originality/valueThe meshless LBIE method is applied to the 2D Navier–Stokes equations. No derivatives of interpolation functions are used in the formulation, rendering the present method a robust numerical scheme for the solution of fluid flow problems.

An improved local boundary integral equation method implemented by the transformed MLS approximation with the delta property

The meshless local boundary integral equation (LBIE) method is a promising method for solving problems of elasticity with nonhomogeneous material properties. However, the shape functions of the LBIE method obtained by the moving least-squares (MLS) approximation, generally, do not satisfy the Kronecker delta property. To impose boundary conditions, numerical integrations of LBIE method are carried out under consideration of boundary conditions and the fictitious nodal values would be solved out. In this paper, the transformed MLS shape functions can be obtained, satisfying the Kronecker delta property and requiring no singular weight function. And then the potential and flux in 2D potential problems would be regarded as independent variables each other. The numerical integrations of LBIEs can be processed without regard to boundary conditions and the coefficient matrix of LBIE method would be irrelated to boundary conditions. Finally boundary conditions would be directly imposed and the unknown nodal potentials or the unknown fluxes on the boundary nodes would be solved out. Three numerical examples are computed to verify this feasibility. The coincidence of the numerical results obtained by the proposed method with the traditional LBIE method shows the feasibility of the proposed method.

A LBIE-RBF solution to the convected wave equation for flow acoustics

In this paper, the meshless local boundary integral equation (LBIE) method is implemented for the solution of convected wave equation in frequency domain. Such a problem plays an important role for the noise prediction in flow acoustics. The case of a uniform flow in x-direction is analyzed. For the resultant convected Helmholtz equation, the free-space Green's function has a more complicated form than the Green's function for the classical Helmholtz equation, though it reduces to the latter when a coordinate (Prandtl–Glauert) transformation is applied. Furthermore, the LBIE method requires the determination of the solution of the corresponding Dirichlet's problem in local subdomains (companion solution). Therefore, the relevant companion solution is derived. Radial basis functions are used for the interpolations of the field variable. Numerical examples are shown for cases up to frequency of 5000 Hz and Mach number of 0.5, and the results are compared with FEM software (COMSOL).

Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation

In this paper, the local boundary integral equation (LBIE) method based on generalized moving least squares (GMLS) is proposed for solving extended Fisher---Kolmogorov (EFK) equation. Since the local weak forms are directly approximated from nodal data using a GMLS approximation, this method is called direct local boundary integral equation (DLBIE) method. In addition to DLBIE method, three other meshless local boundary integral equation methods such as LBIE method based on moving least squares, LBIE method based on radial point interpolation, and LBIE method based on moving Kriging are applied to find the numerical solution of EFK equation. A comparison between these methods is done from the perspective of accuracy and computational efficiency. The computational efficiency is the most significant advantage of the DLBIE method in comparison with the other local boundary integral equation methods. DLBIE shifts the numerical integrations over low-degree polynomials instead of over complicated shape functions and this reduces the computational costs, significantly. The main purpose of this paper is to show that the local boundary integral equation methods can be used for solving the fourth-order non-linear partial differential equations especially EFK equation. The numerical results confirm the good efficiency of the proposed methods for solving our model.

A local boundary integral method for two-dimensional particle-driven gravity currents simulation

A meshless local boundary integral equation method (LBIEM) is used to analyse gravity currents flow in two-dimensional domains. The method solves the incompressible Navier–Stokes equations with a transport equation for the particle concentration. The characteristic-based split scheme is used to solve the governing equations. The LBIEM basic equations are derived via interpolation using radial basis functions. Two numerical test cases are presented here; both are focused on the typical lock-exchange channel flow. The LBIEM procedure produces stable solutions with results comparable to those of other conventional methods.

Dispersion analysis of the meshless local boundary integral equation and radial basis integral equation methods for the Helmholtz equation

Numerical solutions of the Helmholtz equation suffer from pollution effect especially for higher wavenumbers. The major cause for this is the dispersion error which is defined as the relative phase difference between the numerical solution of the wave and the exact wave. The dispersion error for the meshless methods can be a priori determined at an interior source node assuming that the potential field obeys a harmonic evolution with the numerical wavenumber.In this paper, the dispersion errors, in the solution of 2D Helmholtz equation, for two different meshless methods are investigated, the local boundary integral equation method and the radial basis integral equation method. Radial basis functions, with second order polynomials and frequency-dependent polynomial basis vectors are used for the interpolation of the potential field in both methods. The results have been found to be of comparable accuracy with other meshless approaches reported in the literature.

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

AbstractThis paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.

Analytical formulation of meshless local integral equation method

In this paper, the exact forms of integrals in the meshless local boundary integral equation method are derived and implemented for elastostatic problems. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the radial basis function (RBF) is employed for the implementation of displacements. A completed set of closed forms of the local boundary integrals are obtained. As there are no numerical integrations to be carried out the computational time is significantly reduced. Three examples are presented to demonstrate the application of this approach in solid mechanics.

New implementation of MLBIE method for heat conduction analysis in functionally graded materials

The meshless local boundary integral equation (MLBIE) method with an efficient technique to deal with the time variable are presented in this article to analyze the transient heat conduction in continuously nonhomogeneous functionally graded materials (FGMs). In space, the method is based on the local boundary integral equations and the moving least squares (MLS) approximation of the temperature and heat flux. In time, again the MLS approximates the equivalent Volterra integral equation derived from the heat conduction problem. It means that, the MLS is used for approximation in both time and space domains, and we avoid using the finite difference discretization or Laplace transform methods to overcome the time variable. Finally the method leads to a single generalized Sylvester equation rather than some (many) linear systems of equations. The method is computationally attractive, which is shown in couple of numerical examples for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.

Meshless Local Integral Equation Method with Analytical Formulation and its Application to Computational Fracture Mechanics

In this paper, the exact forms of integrals in the meshless local boundary integral equation method are presented and applications were demonstrated for fracture mechanics. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the Radial Basis Function (RBF) is employed for the implementation of displacements. For cracked plate, opening displacement on the crack surface can be obtained with satisfactory accuracy without special treatment for stresses singularities at crack tip.

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.

Implementation of meshless LBIE method to the 2D non‐linear SG problem

AbstractIn this paper the meshless local boundary integral equation (LBIE) method for numerically solving the non‐linear two‐dimensional sine‐Gordon (SG) equation is developed. The method is based on the LBIE with moving least‐squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. The approximation functions are constructed entirely using a set of scattered nodes, and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes. A time‐stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non‐linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of method to deal with the unsteady non‐linear problems in large domains. Copyright © 2009 John Wiley & Sons, Ltd.

Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes

The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.

Numerical determination of modal dispersion and AE signal characterization in waveguides through LBIE/BEM and time–frequency analysis

In the present paper a meshless local boundary integral equation (LBIE) method is employed for the numerical determination of dispersion curves in plate like waveguides. Next, the LBIE method is properly combined with the boundary element method (BEM) for the characterization of acoustic emission signals derived from nucleating surface-breaking and body cracks in plate-like waveguides being in contact with a fluid. The time history of the propagating guided waves is computed by means of either the LBIE method or the hybrid LBIE/BEM scheme and the representation of mode dispersion is accomplished through the Reassigned Smoothed Pseudo Wigner–Ville time–frequency analysis. The effectiveness of the method is demonstrated through representative numerical examples.

Numerical solution to the unsteady two‐dimensional Schrödinger equation using meshless local boundary integral equation method

AbstractA meshless local boundary integral equation (LBIE) method is proposed to solve the unsteady two‐dimensional Schrödinger equation. The method is based on the LBIE with moving least‐squares (MLS) approximation. For the MLS approximation, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. A time‐stepping method is employed to deal with the time derivative. An efficient method for dealing with singular domain integrations that appear in the discretized equations is presented. Finally, numerical results are considered for some examples to demonstrate the accuracy, efficiency and high rate of convergence of this method. Copyright © 2008 John Wiley & Sons, Ltd.

Improved Meshless Local Boundary Integral Equation Method

Combining the local boundary integral equation with the improved moving least-square method,an improved meshless local boundary integral equation method is presented.In the improved moving least-square method,the weighted orthogonal functions are used as basis ones so that the matrix inverse at each quadrature point is avoided and the algebra equations system is not ill-conditioned.In addition,the improved meshless local boundary integral equation method is applied to linear elasticity problems, the corresponding discrete equations are derived.Some numerical results to demons trate the efficiency of the method are presented. Compared with the conventional local boundary integral equation method,the present method has higher computational efficiency and precision,and will not form ill-conditioned or singular equations.

An advanced meshless LBIE/RBF method for solving two-dimensional incompressible fluid flows

The present work presents a meshless local boundary integral equation (LBIE) method for the solution of two-dimensional incompressible fluid flow problems governed by the Navier–Stokes equations. The method uses, for its meshless implementation, nodal points spread over the analyzed domain and employs in an efficient way the radial basis functions (RBF) for the interpolation of the interior and boundary variables. The inverse matrix of the RBF is computed only once for every nodal point and the interpolation functions are evaluated by the inner product of the inverse matrix with the weight vector associated to the integration point. This technique leads to a fast and efficient meshless approach, the locality of the method is maintained and the system matrices are banded with small bandwidth. The velocity–vorticity approach of the Navier–Stokes equations is adopted and the LBIEs are derived for the velocity and the vorticity field, resulting in a very stable and accurate implementation. The evaluation of the volume integrals is accomplished via a very efficient and accurate technique by triangularizing the local area of the nodal point to the minimum number of well formed triangles. Numerical examples illustrate the proposed methodology and demonstrate its accuracy.

Dual error indicators for the local boundary integral equation method in 2D potential problems

Three relative error measurements in the numerical solution of potential problems are firstly investigated in detail, and then an algorithm based on the proposed dual error indicators is developed for the meshless local boundary integral equation (LBIE) method. Numerical experiments show that a combined use of the two error indicators is necessary to adequately measure the error of the LBIE solutions.

A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials

This paper presents a meshless local boundary integral equation method (LBIEM) for dynamic analysis of an anti-plane crack in functionally graded materials (FGMs). Local boundary integral equations (LBIEs) are formulated in the Laplace-transform domain. The static fundamental solution for homogeneous elastic solids is used to derive the local boundary–domain integral equations, which are applied to small sub-domains covering the analyzed domain. For the sub-domains a circular shape is chosen, and their centers, the nodal points, correspond to the collocation points. The local boundary–domain integral equations are solved numerically in the Laplace-transform domain by a meshless method based on the moving least–squares (MLS) scheme. Time-domain solutions are obtained by using the Stehfest's inversion algorithm. Numerical examples are given to show the accuracy of the proposed meshless LBIEM.