A meshless method using the Takagi–Sugeno fuzzy model

This article proposes a new strong-form meshless method using the Takagi–Sugeno fuzzy model (MTSF) for solving differential equations (DEs). Considering the conventional fuzzy model, the fuzzy inference system (FIS) can be categorized into two architectures, a simple rule base using the Euclidean distance in a multidimensional space (Simple-FIS), and an adaptive neuro-fuzzy inference system (ANFIS). Accordingly, MTSF also can be implemented using Simple-FIS and ANFIS. Based on the two architectures, an approximation scheme for continuous functions is drawn out first. In turn, the derivation is further proposed in which the differential functions are approximated using two independent sets of points, one for the collocation point and the other for the rule point. Solving higher-order DEs becomes possible by following the derivations, and eventually numerical solutions can be obtained. Several examples of one-dimensional ordinary and two-dimensional partial DEs (ODEs and PDEs) are presented to demonstrate the performance of the MTSF method. By MTSF, solutions solved using Simple-FIS and using ANFIS are compared. Variations in boundary conditions and membership function parameters are also studied to examine the agreement among numerical solutions.     

A meshfree method: meshfree weak–strong (MWS) form method,for 2-D solids

A novel meshfree weak–strong (MWS) form method is proposed based on a combined formulation of both the strong-form and the local weak-form. In the MWS method, the problem domain and its boundary is represented by a set of distributed points or nodes. The strong form or the collocation method is used for all nodes whose local quadrature domains do not intersect with natural (Neumann) boundaries. Therefore, no numerical integration is required for these nodes. The local weak-form, which needs the local numerical integration, is only used for nodes on or near the natural boundaries. The locally supported radial point interpolation method and the moving least squares approximation are used to construct the meshfree shape functions. The final system matrix will be sparse and banded for computational efficiency. Numerical examples of two-dimensional solids are presented to demonstrate the efficiency, stability, accuracy and convergence of the proposed meshfree method.     

A meshless method for two-dimensional diffusion equation with an integral condition

This paper presents a new approach based on the meshless local Petrov–Galerkin (MLPG) and collocation methods to treat the parabolic partial differential equations with non-classical boundary conditions. In the presented method, the MLPG method is applied to the interior nodes while the meshless collocation method is applied to the nodes on the boundaries, and so the Dirichlet boundary condition is imposed directly. To treat the complicated integral boundary condition appearing in the problem, Simpson's composite numerical integration rule is applied. A time stepping scheme is employed to approximate the time derivative. Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method.     

Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation

In this paper a numerical approach based on the truly meshless methods is proposed to deal with the second-order two-space-dimensional telegraph equation. In the meshless local weak–strong (MLWS) method, our aim is to remove the background quadrature domains for integration as much as possible, and yet to obtain stable and accurate solution. The MLWS method is designed to combine the advantage of local weak and strong forms to avoid their shortcomings. In this method, the local Petrov–Galerkin weak form is applied only to the nodes on the Neumann boundary of the domain of the problem. The meshless collocation method, based on the strong form equation is applied to the interior nodes and the nodes on the Dirichlet boundary. To solve the telegraph equation using the MLWS method, the conventional moving least squares (MLS) approximation is exploited in order to interpolate the solution of the equation. A time stepping scheme is employed to approximate the time derivative. Another solution is also given by the meshless local Petrov-Galerkin (MLPG) method. The validity and efficiency of the two proposed methods are investigated and verified through several examples.     

Radial basis functions methods for solving Fokker-Planck equation

In this paper two numerical meshless methods for solving the Fokker-Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker-Planck equation by using collocation method are applied. The first is based on the Kansa's approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and N_e errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa's approach.     

Cell-based maximum entropy approximants for three-dimensional domains: Application in large strain elastodynamics using the meshless total Lagrangian explicit dynamics method

We present the cell-based maximum entropy (CME) approximants in E³ space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the maximum entropy approximants and the compact support of element-based interpolants. The method is evaluated in problems of large strain elastodynamics for three-dimensional (3D) continua using the well-established meshless total Lagrangian explicit dynamics method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary condition (EBC) imposition in noninterpolating meshless methods (eg, moving least squares) is eliminated in CME due to the weak Kronecker-delta property. The EBCs are imposed directly, similar to the finite element method. CME is proven a valuable alternative to other meshless and element-based methods for large-scale elastodynamics in 3D. A naive implementation of the CME approximants in E³ is available to download at https://www.mountris.org/software/mlab/cme .     

A meshfree weak-strong (MWS) form method for the unsteady magnetohydrodynamic (MHD) flow in pipe with arbitrary wall conductivity

In this paper a meshfree weak-strong (MWS) form method is considered to solve the coupled equations in velocity and magnetic field for the unsteady magnetohydrodynamic flow throFor this modified estimaFor this modified estimaFor this modified estimaugh a pipe of rectangular and circular sections having arbitrary conducting walls. Computations have been performed for various Hartman numbers and wall conductivity at different time levels. The MWS method is based on applying a meshfree collocation method in strong form for interior nodes and nodes on the essential boundaries and a meshless local Petrov–Galerkin method in weak form for nodes on the natural boundary of the domain. In this paper, we employ the moving least square reproducing kernel particle approximation to construct the shape functions. The numerical results for sample problems compare very well with steady state solution and other numerical methods.     

A concurrent multiscale method based on the alternating Schwarz scheme for coupling atomic and continuum scales with first-order compatibility

As well known, one of the major challenges in developing a multiscale model is how to ensure a seamless interface between the constituent length/time scales. In order to overcome this challenge, a novel concurrent multiscale numerical method is proposed in this paper, which is based on the alternating Schwarz method, to provide the seamless coupling between the atomic and continuum scales. The novelty in this method is the use of the strong-form meshless Hermite–cloud method in the continuum domain for approximation of both the field variable and corresponding first-order derivative simultaneously. As a result, the coupling between the domains is achieved by ensuring the compatibility of both the field variable and the first-order derivative simultaneously across the overlapping transition region. The proposed scheme is validated numerically through both the static and transient benchmark case studies in 1- and 2-D domains. The numerical results show that the proposed method is simple, efficient, and accurate, and also provides a seamless coupling between the two domains.     

A meshless local boundary integral equation method for solving transient elastodynamic problems

A new local boundary integral equation (LBIE) method for solving two dimensional transient elastodynamic problems is proposed. The method utilizes, for its meshless implementation, nodal points spread over the analyzed domain and employs the moving least squares (MLS) approximation for the interpolation of the interior and boundary variables. On the global boundary, displacements and tractions are treated as independent variables. The local integral representation of displacements at each nodal point contains both surface and volume integrals, since it employs the simple elastostatic fundamental solution and considers the acceleration term as a body force. On the local boundaries, tractions are avoided with the aid of the elastostatic companion solution. The collocation of the local boundary/volume integral equations at all the interior and boundary nodes leads to a final system of ordinary differential equations, which is solved stepwise by the -Wilson finite difference scheme. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

A regularized least-squares radial point collocation method (RLS-RPCM) for adaptive analysis

This paper presents a stabilized meshfree method formulated based on the strong formulation and local approximation using radial basis functions (RBFs). The purpose of this paper is two folds. First, a regularization procedure is developed for stabilizing the solution of the radial point collocation method (RPCM). Second, an adaptive scheme using the stabilized RPCM and residual based error indicator is established. It has been shown in this paper that the features of the meshfree strong-form method can facilitated an easier implementation of adaptive analysis. A new error indicator based on the residual is devised and used in this work. As shown in the numerical examples, the new error indicator can reflect the quality of the local approximation and the global accuracy of the solution. A number of examples have been presented to demonstrate the effectiveness of the present method for adaptive analysis.     

Direct meshless local Petrov–Galerkin method for elastodynamic analysis

This article describes a new and fast meshfree method based on a generalized moving least squares (GMLS) approximation and the local weak forms for vibration analysis in solids. In contrast to the meshless local Petrov–Galerkin method, GMLS directly approximates the local weak forms from meshless nodal values, which shifts the local integrations over the low-degree polynomial basis functions rather than over the complicated MLS shape functions. Besides, if the method is set up properly, all local integrals have the same value if all local subdomains have the same shape. These features reduce the computational costs, remarkably. The new technique is called direct meshless local Petrov–Galerkin (DMLPG) method. In DMLPG, the stiff and mass matrices are constructed by integration against polynomials. This overcomes the main drawback of meshfree methods in comparison with the finite element methods (FEM). The Newmark scheme is adapted as a time integration method, and numerical results are presented for various dynamic problems. The results are compared with the exact solutions, if available, and the FEM solutions.     

A comparison of three explicit local meshless methods using radial basis functions

In this paper, three kinds of explicit local meshless methods are compared: the local method of approximate particular solutions (LMAPS), the local direct radial basis function collocation method (LDRBFCM) which are both first presented in this paper, and the local indirect radial basis function collocation method (LIRBFCM). In all three methods, the time discretization is performed in explicit way, the multiquadric radial basis functions (RBFs) are used to interpolate either initial temperature field and its derivatives or the Laplacian of the initial temperature field. The five-noded sub-domains are used in localization. Numerical results of simple diffusion equation with Dirichlet jump boundary condition are compared on uniform and random node arrangement, the accuracy and stabilities of these three local meshless methods are asserted. One can observe that the improvement of the accuracy with denser nodes and with smaller time steps for all three methods. All methods provide a similar accuracy in uniform node arrangement case. For random node arrangement, the LMAPS and the LDRBFCM perform better than the LIDRBFCM.     

A posteriori error approximation in EFG method

Recently, considerable effort has been devoted to the development of the so‐called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the ‘a posteriori error’ should be approximated. An ‘a posteriori error’ approximation based on the moving least‐squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least‐squares approximations. Different selection strategies of the moving least‐squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error. The performance of the developed approximation of the error is illustrated by analysing different examples for two‐dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors. Copyright © 2003 John Wiley & Sons, Ltd.     

Truly meshless localized type techniques for the steady-state heat conduction problems for isotropic and functionally graded materials

A numerical solution of steady-state heat conduction problems is obtained using the strong form meshless point collocation (MPC) method. The approximation of the field variables is performed using the Moving Least Squares (MLS) and the local form of the multiquadrics Radial Basis Functions (LRBF). The accuracy and the efficiency of the MPC schemes (with MLS and LRBF approximations) are investigated through variation (i) of the nodal distribution type used, i.e. regular or irregular, ensuring the so-called positivity conditions, (ii) of the number of nodes in the total spatial domain (TD), and (iii) of the number of nodes in the support domain (SD). Numerical experiments are performed on representative case studies of increasing complexity, such as, (a) a regular geometry with a constant conductivity and uniformly distributed heat source, (b) a regular geometry with a spatially varying conductivity and non-uniformly distributed heat source, and (c) an irregular geometry in case of insulation of vapor transport tubes, as well. Steady-state boundary conditions of the Dirichlet-, Neumann-, or Robin-type are assumed. The results are compared with those calculated by the Finite Element Method with an in-house code, as well as with analytical solutions and other literature data. Thus, the accuracy and the efficiency of the method are demonstrated in all cases studied.     

Meshless Galerkin least-squares method

Collocation method and Galerkin method have been dominant in the existing meshless methods. Galerkin-based meshless methods are computational intensive, whereas collocation-based meshless methods suffer from instability. A new efficient meshless method, meshless Galerkin lest-squares method (MGLS), is proposed in this paper to combine the advantages of Galerkin method and collocation method. The problem domain is divided into two subdomains, the interior domain and boundary domain. Galerkin method is applied in the boundary domain, whereas the least-squares method is applied in the interior domain.The proposed scheme elliminates the posibilities of spurious solutions as that in the least-square method if an incorrect boundary conditions are used. To investigate the accuracy and efficiency of the proposed method, a cantilevered beam and an infinite plate with a central circular hole are analyzed in detail and numerical results are compared with those obtained by Galerkin-based meshless method (GBMM), collocation-based meshless method (CBMM) and meshless weighted least squares method (MWLS). Numerical studies show that the accuracy of the proposed MGLS is much higher than that of CBMM and is close to, even better than, that of GBMM, while the computational cost is much less than that of GBMM.Acknowledgements The authors gratefully acknowledge the support of the National Natural Science Foundation of China with grant number 10172052.     

A boundary meshless method with shape functions computed from the PDE

This paper presents a new meshless method using high degree polynomial shape functions. These shape functions are approximated solutions of the partial differential equation (PDE) and the discretization concerns only the boundary. If the domain is split into several subdomains, one has also to discretize the interfaces. To get a true meshless integration-free method, the boundary and interface conditions are accounted by collocation procedures. It is well known that a pure collocation technique induces numerical instabilities. That is why the collocation will be coupled with the least-squares method. The numerical technique will be applied to various second order PDE's in 2D domains. Because there is no integration and the number of shape functions does not increase very much with the degree, high degree polynomials can be considered without a huge computational cost. As for instance the p-version of finite elements or some well established meshless methods, the present method permits to get very accurate solutions.     

Numerical solution of integral equations with a logarithmic kernel by the method of arbitrary collocation points

An integral equation whose kernel presents logarithmic singularity is numerically solved by the method of arbitrary collocation points (ACP). As a first step a Gaussian quadrature of order n (hence of polynomial accuracy 2n? 1) is employed for the numerical approximation of the integral. Until now the collocation, which follows, was performed on special points x?_k, determined as roots of appropriate transcedental functions, in order to retain the 2n ? 1 degree of polynomial accuracy of the Gaussian quadrature. In this paper an appropriate interpolatory technique is proposed, so that x_k may be arbitrary and yet the high (2n ? 1) accuracy of the Gaussian quadrature is retained.     

Meshless local Petrov–Galerkin (MLPG) method for wave propagation in 3D poroelastic solids

A meshless local Petrov–Galerkin (MLPG) method is applied to solve wave propagation problems of three-dimensional poroelastic solids with Biot's theory. The Laplace transform is used to eliminate the time dependence of the field variables for the transient elastodynamic case. A weak formulation with a unit step function transforms the set of governing equations into local integral equations on local subdomains. The meshless approximation based on the radial basis function (RBF) is employed for the implementation. Unknown Laplace-transformed quantities, including displacements of solid frame and pressure in the fluid, are computed from the local boundary integral equations. The time-dependent values are obtained by Durbin's inversion technique. In addition, a one-dimensional poroelasticity analytical solution is derived in this paper and introduced for comparison. Several numerical examples demonstrate the efficiency and accuracy of the proposed method.     

Time‐dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in an FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is finite difference method (FDM), which is usually difficult to handle a complex problem domain, and also difficult to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection–diffusion equations (FADE), which is a typical FPDE The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong‐forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate the accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of local boundary integral equation method into micropolar elasticity

A new meshless method for solving boundary value problems in micropolar elasticity is presented. The method is based on the local boundary integral equation (LBIE) method with the moving least squares approximation of physical quantities. Randomly scattered nodes are utilized for interpolation of field data. Every node is surrounded by a simple surface centered at the collocation point in the LBIE method. On the surface of subdomains the LBIEs are written. Fundamental solutions corresponding to uncoupled governing equations are derived. To eliminate the traction vector in the LBIE, the modified fundamental solution is introduced.