Local Boundary Integral Equation Method Research Articles

A new meshless local integral equation method

This paper proposes a novel meshless local integral equation (LIE) method for numerical solutions of two and three-dimensional Poisson equations. The proposed method can be regarded as a new variant of the meshless local Petrov-Galerkin (MLPG) method which already has six variants, and so the proposed method can be called MLPG7. A variant of MLPG called local boundary integral equation (LBIE) method, uses a localized fundamental solution of Laplace equation as test function. This test function vanishes on local boundaries and therefore, the derived local equations do not involve the gradient of field variables. The motivation for new formulation is using an average of LBIE equation over radius of local sub-domains instead of a single sub-domain. A new kernel is introduced which is a new modification of fundamental solutions of Laplace equation. It is proved that using the new kernel as test function leads to the same local equations as averaging. Infact, the new formulation is a modification of the LBIE method for which the boundary integral is replaced with a domain integral. It is proved that new local weak formulation is equivalent with strong form. The convergence of the method for the case of regularly spaced nodal points is proved. For trial approximation, this paper uses Gaussian radial basis functions (Gaussian RBF), however, derivative free property of the method allows the use of any non-differentiable basis functions. Stability and convergence of the method are numerically studied by considering both 2D and 3D problems with regular and irregular nodal points. The method is compared with LBIE, finite differences (FD) and RBF-FD methods and the numerical results reveal the significance of the proposed method.

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.

A meshless regularized local boundary integral equation method and the selection of weight function and geometrical parameters

The local boundary integral equation (LBIE) method is a pure meshless integral method proposed by Atluri and co-workers. Since the fundamental solutions are used as the test functions, the sub-domain can be very small and the LBIE method is a promising method for solving problems of elasticity with nonhomogeneous material properties. In this paper, the subtraction technique is introduced to remove the strong singularity that results in a regularized local boundary integral equation (RLBIE) method. The formula is rather straightforward and more accurate than other treatment techniques. Compared with other meshless methods, more geometrical parameters need to be assigned in the method. If the parameters change a little bit, the solutions obtained from the method may fluctuate quickly and uncertainly. Therefore the effects of weight function, support domain, sub-domain, and monomial basis are investigated and how to select the parameters is discussed. The appropriate parameters can be determined and the meshless RLBIE method is an accurate and robust method.

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.

A New Test Function for Acoustic Computations with Meshless Methods

The meshless local Petrov–Galerkin (MLPG) and the local boundary integral equation (LBIE) methods has been introduced approximately three decades ago. These methods are based on writing the local weak form of the governing equation and performing subsequent numerical integration and interpolations in the local subdomains. A key step is the choice of the test function in the local weak form, which has historically led to several different formulations regarding the final form of the local integrals. Considering the application of the methods to acoustics, four different test functions have been employed so far in the literature; all of these approaches resulted in formulations which contain domain integrals. In this paper, we present a new test function to be used in meshless methods, which yields a simple form of the local integral equation without domain integrals and provides significant improvement in terms of the computational time and CPU requirements. The efficiency and the accuracy of the new method are presented and compared with the previous methods.

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.

Pricing American options under jump-diffusion models using local weak form meshless techniques

ABSTRACTRecently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert, A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth, A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 (2012), pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.

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.

Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options

For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI) based on Wu’s compactly supported radial basis functions (WCS-RBFs). LBIE and LRPI schemes are the truly meshless methods, because, a traditional non-overlapping, continuous mesh is not required, either for the construction of the shape functions, or for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the θ-weighted scheme is employed for the time derivative. Stability analysis of the methods is analyzed and performed by the matrix method. In fact, based on an analysis carried out in the present paper, the methods are unconditionally stable for implicit Euler (θ=0) and Crank–Nicolson (θ=0.5) schemes. It should be noted that LBIE and LRPI schemes lead to banded and sparse system matrices. Therefore, we use a powerful iterative algorithm named the Bi-conjugate gradient stabilized method (BCGSTAB) to get rid of this system. Numerical experiments are presented showing that the LBIE and LRPI approaches are extremely accurate and fast.

A Parallel Method for Solving Laplace Equations with Dirichlet Data Using Local Boundary Integral Equations and Random Walks

In this paper, a hybrid approach for solving the Laplace equation in general three-dimensional (3-D) domains is presented. The approach is based on a local method for the Dirichlet-to-Neumann (DtN) mapping of a Laplace equation by combining a deterministic (local) boundary integral equation (BIE) method and the probabilistic Feynman--Kac formula for solutions of elliptic partial differential equations. This hybridization produces a parallel algorithm where the bulk of the computation has no need for data communication between processors. Given Dirichlet data of the solution on a domain boundary, a local BIE is established over the boundary of a local region formed by a hemisphere superimposed on the domain boundary. By using a homogeneous Dirichlet Green's function for the whole sphere, the resulting BIE involves only the Dirichlet data (the solution value) over the surface of the hemisphere, while over the patch of the domain boundary intersected by the hemisphere, both Dirichlet and Neumann data are use...

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.

A simplified approach for imposing the boundary conditions in the local boundary integral equation method

A simplified approach for imposing the boundary conditions in the local boundary integral equation (LBIE) method is presented. The proposed approach employs an integral equation derived using the fundamental solution and the Green's second identity when the collocation node is at the boundary of the solution domain (global boundary). The subdomains for the nodes placed at the global boundary preserve their circular shapes; avoiding in this way any integration over the global boundary. Consequently, the difficulties related to evaluation of singular integrals and determination of intersection points between the global and local circular boundaries are avoided. So far, attempts to avoid these issues have focused on using schemes based on meshless approximations. The downside of such schemes is that the weak formulation is abandoned. In this study the interpolation of field variables over the boundaries of the subdomains is carried out using the radial basis function approximation. Numerical examples show that the proposed approach despite its simplicity, achieves comparable accuracy to the classical treatment of the boundary conditions in the LBIE.

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.

Calculating the Band Structure of Photonic Crystals Through the Meshless Local Petrov-Galerkin (MLPG) Method and Periodic Shape Functions

This paper illustrates how to determine the bandgap structure of photonic crystals through MLPG. This method is akin to the Finite Element Method (FEM), as it also deals with the discretization of weak forms and produces sparse global matrices. The major difference is the complete absence of any kind of mesh. We concentrate in a particular version of it, the MLPG4, also known as Local Boundary Integral Equation Method (LBIE). Since the boundary conditions governing the electromagnetic field are periodic in a unit cell, we develop a special scheme to embed this feature on the shape functions used in the discretization process. As a result, boundary conditions do not need to be imposed on the unit cell. Index Terms—Electromagnetic wave propagation, finite element methods, integral equations, photonic crystals.

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 new LBIE method for solving elastodynamic problems

A new local boundary integral equation (LBIE) method for the solution of elastodynamic problems in both frequency and time domain is proposed. Non-uniformly distributed points covering the analyzed domain are used for the interpolation of the involved fields. The key-point of the proposed methodology is that the support domain of each point is divided into parts with the aid of cells formed by connecting the point of interest with the nearby points. Then an efficient radial basis functions (RBF) interpolation scheme is exploited for the representation of displacements in each cell, while on the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary elements. For each point the corresponding LBIE is written in terms of displacements only, since on the boundary of support domains tractions are eliminated with the aid of the elastostatic companion solution. The integration in support domains is performed easily and with high accuracy, while due to cells the extension of the method to three dimensions is straightforward. Transient solutions are obtained after inversion of frequency domain results with the inverse fast Fourier transform (FFT). Two representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.