Reproducing polynomial particle methods for boundary integral equations

Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen the effectiveness of meshless methods, Oh et al. developed meshfree reproducing polynomial particle (RPP) shape functions, patchwise RPP and reproducing singularity particle (RSP) shape functions with use of flat-top partition of unity. All of these approximation functions satisfy the Kronecker delta property. In this paper, we report that meshfree RPP shape functions, patchwise RPP shape functions, and patchwise RSP shape functions effectively handle boundary integral equations with (or without) domain singularities.     

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.     

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.     

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 local boundary integral-based meshless method for Biot's consolidation problem

Traditional numerical techniques such as FEM and BEM have been successfully applied to the solutions of Biot's consolidation problems. However, these techniques confront some difficulties in dealing with moving boundaries. In addition, pre-designing node connectivity or element is not an easy task. Recently, developed meshless methods may overcome these difficulties. In this paper, a meshless model, based on the local Petrov–Galerkin approach with Heaviside step function as well as radial basis functions, is developed and implemented for the numerical solution of plane strain poroelastic problems. Although the proposed method is based on local boundary integral equation, it does not require any fundamental solution, thus avoiding the singularity integral. It also has no domain integral over local domain, thus largely reducing the computational cost in formulation of system stiffness. This is a truly meshless method. The solution accuracy and the code performance are evaluated through one-dimensional and two-dimensional consolidation problems. Numerical examples indicate that this meshless method is suitable for either regular or irregular node distributions with little loss of accuracy, thus being a promising numerical technique for poroelastic problems.     

Meshless global radial point collocation method for three-dimensional partial differential equations

This study deals with the numerical solution of three-dimensional partial differential equations by the meshless global radial point collocation method based on various radial basis functions. First, second, third, and fourth-order three-dimensional partial differential equations are considered. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples.     

Point collocation methods using the fast moving least‐square reproducing kernel approximation

A pseudo‐spectral point collocation meshfree method is proposed. We apply a scheme of approximating derivatives based on the moving least‐square reproducing kernel approximations. Using approximated derivatives, we propose a new point collocation method. Unlike other meshfree methods that require direct calculation of derivatives for shape functions, with the proposed scheme, approximated derivatives are obtained in the process of calculating the shape function itself without further cost. Moreover, the scheme does not require the regularity of the window function, which ensures the regularity of shape functions. In this paper, we show the reproducing property and the convergence of interpolation for approximated derivatives of shape functions. As numerical examples of the proposed scheme, Poisson and Stokes problems are considered in various situations including the case of randomly generated node sets. In short, the proposed scheme is efficient and accurate even for complicated geometry such as the flow past a cylinder. Copyright © 2003 John Wiley & Sons, Ltd.     

A hybrid radial boundary node method based on radial basis point interpolation

The hybrid boundary node method (HBNM) retains the meshless attribute of the moving least squares (MLS) approximation and the reduced dimensionality advantages of the boundary element method. However, the HBNM inherits the deficiency of the MLS approximation, in which shape functions lack the delta function property. Thus in the HBNM, boundary conditions are implemented after they are transformed into their approximations on the boundary nodes with the MLS scheme.This paper combines the hybrid displacement variational formulation and the radial basis point interpolation to develop a direct boundary-type meshless method, the hybrid radial boundary node method (HRBNM) for two-dimensional potential problems. The HRBNM is truly meshless, i.e. absolutely no elements are required either for interpolation or for integration. The radial basis point interpolation is used to construct shape functions with delta function property. So unlike the HBNM, the HRBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables, and boundary conditions can be applied directly and easily, which leads to greater computational precision. Some selected numerical tests illustrate the efficiency of the method proposed.     

Meshless Local Petrov-Galerkin (MLPG) Method for Laminate Plates under Dynamic Loading

A meshless local Petrov-Galerkin (MLPG) method is applied to solve laminate plate problems described by the Reissner-Mindlin theory. Both stationary and transient dynamic loads are analyzed here. The bending moment and the shear force expressions are obtained by integration through the laminated plate for the considered constitutive equations in each lamina. The Reissner-Mindlin theory reduces the original three-dimensional (3-D) thick plate problem to a two-dimensional (2-D) problem. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding this node. The weak-form on small subdomains with a Heaviside step function as the test functions is applied to derive local integral equations. After performing the spatial MLS approximation, a system of ordinary differential equations of the second order for certain nodal unknowns is obtained. The derived ordinary differential equations are solved by the Houbolt finite-difference scheme as a time-stepping method.     

Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations

A meshfree method namely, discrete least squares meshless (DLSM) method, is presented in this paper for the solution of elliptic partial differential equations. In this method, computational domain is discretized by some nodes and the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is defined as the sum of squared residuals of the differential equation and its boundary condition calculated at a set of points called sampling points, generally different from nodal points. A moving least squares (MLS) technique is used to construct the shape functions. The proposed method automatically leads to symmetric and positive-definite system of equations. The proposed method does not need any background mesh and, therefore, it is a truly meshless method. The solutions of several one- and two-dimensional examples of elliptic partial differential equations are presented to illustrate the performance of the proposed method. Sensitivity analysis on the parameters of the method is also carried out and the results are presented.     

Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse time-dependent force function in the wave equation on regular and irregular domains. The SMRPI is developed for identifying the force function which satisfies in the wave equation subject to the integral overspecification over a portion of the spatial domain or to the overspecification at a point in the spatial domain. This method is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system. Three numerical examples are tested to show that numerical results are accurate for exact data and stable with noisy data.     

An isogeometric‐meshfree coupling approach for analysis of cracks

This paper presents a new concurrent simulation approach to couple isogeometric analysis (IGA) with the meshfree method for studying of crack problems. In the present method, the overall physical domain is divided into 2 subdomains that are formulated with the IGA and meshfree method, respectively. In the meshfree subdomain, the moving least squares shape function is adopted for the discretization of the area around crack tips, and the IGA subdomain is adopted in the remaining area. Meanwhile, the interface region between the 2 subdomains is represented by coupled shape functions. The resulting shape function, which comprises both IGA and meshfree shape functions, satisfies the consistency condition, thus ensuring convergence of the method. Moreover, the meshfree shape functions augmented with the enriched basis functions to predict the singular stress fields near a crack tip are presented. The proposed approach is also applied to simulate the crack propagation under a mixed‐mode condition. Several numerical examples are studied to demonstrate the use and robustness of the proposed method.     

Enrichment of the method of finite spheres using geometry‐independent localized scalable bubbles

In this paper, we report the development of two new enrichment techniques for the method of finite spheres, a truly meshfree method developed for the solution of boundary value problems on geometrically complex domains. In the first method, the enrichment functions are multiplied by a weight function with compact support, while in the second one a floating ‘enrichment node’ is introduced. The scalability of the enrichment bubbles offers flexibility in localizing the spatial extent to which the enrichment field is applied. The bubbles are independent of the underlying geometric discretization and therefore provide a means of achieving convergence without excessive refinement. Several numerical examples involving problems with singular stress fields are provided demonstrating the effectiveness of the enrichment schemes and contrasting them to traditional ‘geometry‐dependent’ enrichment strategies in which one or more nodes associated with the geometric discretization of the domain are enriched. An additional contribution of this paper is the use of a meshfree numerical integration technique for computing the J‐integral using the domain integral method. Copyright © 2006 John Wiley & Sons, Ltd.     

Structural shape and topology optimization using a meshless Galerkin level set method

This paper aims to propose a meshless Galerkin level set method for shape and topology optimization of continuum structures. To take advantage of the implicit free boundary representation scheme, the design boundary is represented as the zero level set of a scalar level set function, to flexibly handle complex shape fidelity and topology changes by maintaining concise and smooth interface. Compactly supported radial basis functions (CSRBFs) are used to parameterize the level set function and construct the shape functions for meshfree approximations based on a set of unstructured field nodes. The meshless Galerkin method with global weak form is used to implement the discretization of the state equations. This provides a pathway to unify the two different numerical stages in most conventional level set methods: (1) the propagation of discrete level set function on a set of Eulerian grid and (2) the approximation of discrete equations on a set of Lagrangian mesh. The original more difficult shape and topology optimization based on the level set equation is transformed into a relatively easier size optimization, to which many efficient optimization algorithms can be applied. The proposed level set method can describe the moving boundaries without remeshing for discontinuities. The motion of the free boundary is just a question of advancing the discrete level set function in time by solving the size optimization. Several benchmark examples are used to demonstrate the effectiveness of the proposed method. The numerical results show that the proposed method can simplify numerical process and avoid numerical difficulties involved in most conventional level set methods. It is straightforward to apply the proposed method to more advanced shape and topology optimization problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Virtual boundary meshless least square integral method with moving least squares approximation for 2D elastic problem

Moving least squares approximation (MLSA) has been widely used in the meshless method. The singularity should appear in some special arrangements of nodes, such as the data nodes lie along straight lines and the distances between several nodes and calculation point are almost equal. The local weighted orthogonal basis functions (LWOBF) obtained by the orthogonalization of Gramm–Schmidt are employed to take the place of the general polynomial basis functions in MLSA. In this paper, MLSA with LWOBF is introduced into the virtual boundary meshless least square integral method to construct the shape function of the virtual source functions. The calculation format of virtual boundary meshless least square integral method with MLSA is deduced. The Gauss integration is adopted both on the virtual and real boundary elements. Some numerical examples are calculated by the proposed method. The non-singularity of MLSA with LWOBF is verified. The number of nodes constructing the shape function can be less than the number of LWOBF and the accuracy of numerical result varies little.     

Application of localized meshless methods to 2D shallow water equation problems

This study aims to apply the meshless local radial-basis-function differential quadrature (LRBFDQ) method to solve the shallow water equations (SWE). This localized approach is developed from the differential quadrature (DQ) method by employing the radial-basis functions (RBFs) as the trial functions. Comparing with global-type meshless methods, the present method is more appropriate to large-scale problems with complex shapes. Moreover the drawbacks rising from the poor selection of shape parameter and also the full resultant matrix with high condition number are reduced. For real hydraulic-engineering applications located in irregular domains, the LRBFDQ method is very suitable to solve these kinds of shallow-water problems. In this work, the numerical models are applied to simulate three typical 2D SWE problems: (1) a tidal-wave propagation, (2) a dam-break problem and (3) an inverse engineering problem: the numerical analysis of the inflow discharge of the Yuanshantze Flood Diversion (YFD) project in Taiwan. As a result, the adopted meshless method not only shows its algorithm superiority over other mesh-dependent numerical schemes, but also brings more efficiency than several conventional mesh or meshless methods. The application of YFD project also delivers its applicability of this meshless scheme to solve real-world engineering projects.     

Inverse heat conduction problems in three-dimensional anisotropic functionally graded solids

A meshless method based on the local Petrov–Galerkin approach is applied to inverse transient heat conduction problems in three-dimensional solids with continuously inhomogeneous and anisotropic material properties. The Heaviside step function is used as a test function in the local weak form, leading to the derivation of local integral equations. Nodal points are randomly distributed in the domain analyzed, and each node is surrounded by a spherical subdomain in which a local integral equation is applied. A meshless approximation based on the moving least-squares method is employed in the implementation. After performing spatial integrations, we obtain a system of ordinary differential equations for certain nodal unknowns. A backward finite-difference method is used for the approximation of the diffusive term in the heat conduction equation. A truncated singular-value decomposition is used to solve the ill-conditioned linear system of algebraic equations at each time step. The effectiveness of the meshless local Petrov–Galerkin (MLPG) method for this inverse problem is demonstrated by numerical examples.     

Meshless point collocation for the numerical solution of Navier-Stokes flow equations inside an evaporating sessile droplet

The Navier-Stokes flow inside an evaporating sessile droplet is studied in the present paper, using sophisticated meshfree numerical methods for the computation of the flow field. This problem relates to numerous modern technological applications, and has attracted several analytical and numerical investigations that expanded our knowledge on the internal microflow during droplet evaporation. Two meshless point collocation methods are applied here to this problem and used for flow computations and for comparison with analytical and more traditional numerical solutions. Particular emphasis is placed on the implementation of the velocity-correction method within the meshless procedure, ensuring the continuity equation with increased precision. The Moving Least Squares (MLS) and the Radial Basis Function (RBF) approximations are employed for the construction of the shape functions, in conjunction with the general framework of the Point Collocation Method (MPC). An augmented linear system for imposing the coupled boundary conditions that apply at the liquid-gas interface, especially the zero shear-stress boundary condition at the interface, is presented. Computations are obtained for regular, Type-I embedded nodal distributions, stressing the positivity conditions that make the matrix of the system stable and convergent. Low Reynolds number (Stokes regime), and elevated Reynolds number (Navier-Stokes regime) conditions have been studied and the solutions are compared to those of analytical and traditional CFD methods. The meshless implementation has shown a relative ease of application, compared to traditional mesh-based methods, and high convergence rate and accuracy.     

Trefftz interpolation based multi-domain boundary point method

The paper describes meshfree boundary point solution of problems based on Trefftz interpolation functions. The solution is a collocation method, which can use any type of Trefftz functions smooth enough for approximation of domain fields. Polynomial, Kelvin and Boussinesq functions are used in our formulations. A multi-domain boundary formulation with continuity of tractions and displacements over discrete points of the inter-domain surface is shown. The method is applied to linear elastostatics. 2D and 3D examples illustrate the formulation. Discussion about advantages and drawbacks is included. As the conservation equations (force and moment equilibrium) are not satisfied in integral sense, some benchmark tests are proposed for meshless formulations to obtain a view about the accuracy.     

Application of the meshless Galerkin boundary node method to potential problems with mixed boundary conditions

In this paper, the meshless Galerkin boundary node method is developed for boundary-only analysis of two- and three-dimensional potential problems with mixed boundary conditions of Dirichlet and Neumann type. This meshless algorithm leads to a symmetric and positive definite system of linear equations. Additionally, boundary conditions can be implemented directly and easily despite the fact that the employed meshless shape functions lack the delta function property. Theoretical error analysis and numerical results indicate that it is an efficient and accurate numerical method.