Radial Point Interpolation Research Articles

Meshfree radial point interpolation method for the vibration and buckling analysis of FG-GPLRC perforated plates under an in-plane loading

A meshfree radial point interpolation method (RPIM) based on a higher-order shear deformation theory (HSDT) is here developed to investigate the free vibration and buckling behavior of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) perforated plates under an in-plane edge loading. The non-uniform initial stresses in the multilayer FG-GPLRC perforated plates are evaluated accurately by solving the equilibrium equations derived from the same approach. The mixed collocation-Lagrange multiplayer technique is employed to enforce the various boundary conditions. The multilayer FG-GPLRC plates are built up as perfectly-bonded composite layers reinforced with a uniform distribution of graphene platelets (GPLs), randomly oriented. Perforated plates with three different cut-out shapes are analyzed. Several numerical examples are provided to demonstrate the high performance, the fast convergence rate and the high accuracy of the proposed method. Then, we study the influence of different GPLs distribution patterns, GPLs weight fraction, shape and size of cut-outs on the free vibration and buckling behavior of the multilayer FG-GPLRC perforated plates under an in-plane edge loading.

An efficient and accurate hybrid weak-form meshless method for transient nonlinear heterogeneous heat conduction problems

Our purpose is to establish a numerical method meeting the requirements of efficiency, stability, accuracy and easy-using. Faced with these demands, in this paper, a hybrid method combining the advantages of 2 weak-form meshless methods is proposed for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In the process of the calculation for an arbitrary model discretized by the hybrid method, two different weak-form methods are used for the nodes categorized as: (1) boundary nodes, (2) near boundary interior nodes and (3) interior nodes. A Galerkin free element method (GFREM) is proposed for the interior nodes of the Cartesian grid; and the local radial point interpolation method (LRPIM) is used for the other two types of nodes. The GFREM using the Cartesian grid can greatly improve the accuracy and efficiency of the hybrid scheme. Special decomposition technology to determine the irregular integral domain for LRPIM can increase the flexibility for complex models. In addition, the full implicit finite difference scheme and Newton–Raphson method are used for solving transient nonlinear problems. And non-crossing interface integral domain and support domain are employed for heterogeneous media. Compared with other methods, numerical results show better accuracy, stability and efficiency from the numerical experiments.

A coupled meshless-SBFEM-FEM approach in simulating soil-structure interaction with cross-scale model

This paper presents an integrated, customized routine that uses object-oriented programming and a “super element class” to apply an integrated approach to simulate soil-structure interaction (SSI) problems. The scaled boundary finite element method (SBFEM) and finite element method (FEM) are applied together with the quad tree technique to implement cross-scale modeling for solid elements. Meanwhile, the meshless interface seamlessly connects mismatched and differing sizes of meshes between the soil and the structure, thus allowing cross-scale modeling at the interface level. In addition, through modify the radial point interpolation function (RPIM) and the method of searching supporting nodes, the precision near the boundaries of meshless interface are improved, and circular (cambered) type interfaces can be simulated. Thus, through a coupled Meshless-SBFEM-FEM approach, a fine analysis of soil-structure interaction problem with multilevel cross-scale model can be conducted.

Transient responses of functionally graded magneto-electro-elastic structures with holes in thermal environment using stabilized node-based smoothed radial point interpolation method

In the present work, a stabilized node-based smoothed radial point interpolation method (SNS-RPIM) is proposed for analysing the behaviour of functionally graded magneto-electro-elastic (FGMEE) structures with holes in thermal environment. By introducing the stable item related to the gradient variance of field variables, a close-to-exact SNS-RPIM model is constructed, and the temporal instability of traditional node-based smoothed radial point interpolation method (NS-RPIM) is cured. Several numerical examples validate that SNS-RPIM performs well in efficiency and convergence than finite element method (FEM) and NS-RPIM and could solve magneto-electro-thermo-elastic (METE) multi-physics coupling problems more accurately. It is also found that the position of the holes has a greater influence on the displacement and magnetic potential of the structure and has less influence on the electric potential. This provides a reference for the design and application of FGMEE structures with holes working in thermal environment in practical engineering.

GeoMFree[formula omitted]: A package of meshfree local Radial Point Interpolation Method (RPIM) for geomechanics

Mesh-based numerical methods such as the Finite Element Method (FEM) and Finite Difference Method (FDM) are widely used in various fields of science and engineering. However, the shortcomings of mesh-based methods are becoming increasingly obvious mainly due to the distorted or broken elements when analyzing the problems of large deformation or crack propagation. To overcome the above problems, meshfree methods such as the Meshless Local Petrov–Galerkin (MLPG) and Radial Point Interpolation Method (RPIM) were proposed, which can eliminate or partially eliminate the difficulties triggered by mesh-dependence. In geotechnics, the numerical modeling of deformations and failures of soil and rock masses is one of the most important approaches to understand and prevent geohazards, such as landslides and ground subsidence. In this paper, a local RPIM-based meshfree software package GeoMFree3D is specifically developed for analyzing the deformations and failures of soil and rock masses. To enhance the computing capability of GeoMFree3D, (1) effective memory storage strategies are adopted to solve the memory requirement bottleneck; and (2) parallel computing strategies are utilized and heterogeneously implemented on multi-core CPU and many-core GPU to improve the computational efficiency. To demonstrate the performance of GeoMFree3D, three simple verification examples and one application case are presented in this paper. The verification examples indicate that the computational accuracy and efficiency of the package GeoMFree3D is acceptable. Furthermore, the application case indicates that GeoMFree3D is capable of analyzing the displacements of complex rock slopes. GeoMFree3D is currently under intensive development, plans for future work on the development of GeoMFree3D are also introduced.

Meshfree thermomechanical crack growth simulations with new numerical integration scheme

This paper presents for the first time an improved meshfree particle method without the need for background cells in terms of numerical integration for thermal-mechanical crack growth analysis. In this work, both adiabatic and isothermal crack surfaces are considered. Asymptotic solutions-based enriched functions are incorporated into the approximation scheme to mathematically capture jump across crack surfaces of both the temperature and displacements, as well as the singularities of heat fluxes and stresses in the vicinity of crack tip. Once the stress intensity factors (SIFs) are evaluated, the direction of crack growth can be determined. The meshfree analysis is based on radial point interpolation method (RPIM), in which Cartesian transformation method (CTM) is adopted for numerical integration, instead of the conventional Gaussian quadrature. The utilization of Gaussian scheme requires background cells in shape of quadrilaterals or triangles. In contrast, the CTM is advantageous in the manner that the background cells are no longer required, leading to truly meshfree formulation. The novel contribution of the current work is the extension of CTM scheme in which the integration domain is updated according to crack growth. The crack surfaces are viewed as part of the boundaries of the problem domain. Therefore, it is guaranteed that no discontinuities exist within the domain. The accuracy of the proposed approach is demonstrated by comparison of computed results with reference ones from analytical solution, and other existing numerical methods.

A meshless computational approach for solving two-dimensional inverse time-fractional diffusion problem with non-local boundary condition

This paper is devoted to investigating a two-dimensional inverse anomalous diffusion problem. The missing solely time-dependent Dirichlet boundary condition is recovered by imposing an additional integral measurement over the domain. An efficient computational technique based on a combination of a time integration scheme and local meshless Petrov–Galerkin method is implemented to solve the governing inverse problem. Firstly, an implicit time integration scheme is used to discretize the model in the temporal direction. To fully discretize the model, the primary spatial domain is represented by a set of distributed nodes and data-dependent basis functions are constructed by using the radial point interpolation method. Then, the local meshless Petrov–Galerkin method is used to discretize the problem in the spatial direction. Numerical examples are presented to verify the accuracy and efficiency of the proposed technique. The stability of the method is examined when the input data are contaminated with noise.

Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn–Hilliard conditions

In this study, we develop an approximate formulation for a generalization form of biharmonic problem based on pseudospectral meshless radial point interpolation (PSMRPI). The boundary conditions are considered as Cahn–Hilliard type boundary conditions with application to spinodal decomposition. Since the rigorous steps to analyze such a problem is of high-order derivatives, implementing multiple boundary conditions and especially when the geometry of domain of the problem is complex. In PSMRPI method, the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high-order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. Furthermore, it is observed that the multiple boundary conditions can be imposed by an erudite application of PSMRPI on nodal points near the boundaries of the domain. The main results of generalized biharmonic problem are demonstrated by some examples to show validity and trustworthy of PSMRPI technique.

Inverse Multiquadric Radial Basis Functions in Eigenvalue Analysis of a Circular Waveguide Using Radial Point Interpolation Method

Meshless time domain methods allow flexible distribution of nodes in the computational domain. The radial point interpolation method (RPIM) in electromagnetics with Gaussian radial basis functions (RBFs) requires global interpolation matrices with large condition numbers even for simple problems such as eigenvalue analysis of a circular waveguide. In addition, nonorthogonal grids in such problems demand divergence-free conditions to be enforced. In this letter, a divergence-free inverse multiquadric RBF has been proposed to efficiently solve such electromagnetic (EM) problems. Real parts in eigenvalues are nearly eliminated by this approach, thus removing spurious solutions. In addition, the proposed approach results in longtime stability of numerical analysis.

A Time-Domain Meshless Method Without Discretization in Time

In this letter, we propose a new time-domain meshless method without discretization in time based on eigenexpansions of numerical solutions. As a result, numerical time-stepping is not required, and its associated errors are eliminated. Analytical solutions of the radial point interpolation meshless (RPIM) method are derived, and the solutions become always stable. In other words, electric and magnetic field values are available at any time, making late-time electromagnetic field analysis convenient and efficient. Numerical experiments are conducted to verify the effectiveness and accuracy of the proposed method.

A novel stabilized node-based smoothed radial point interpolation method (SNS-RPIM) for coupling analysis of magneto-electro-elastic structures in hygrothermal environment

In this article, a stabilized node-based smoothed radial point interpolation method (SNS-RPIM) was proposed to simulate the magneto-electro-elastic (MEE) structures in hygrothermal environment. The node-based smoothing domains were constructed based on triangular background elements, and the approximate strain, electric field intensity and magnetic flux density were derived using node-based strain smoothing operation. In the present formulation, the stabilization term related to the gradient variance of the field variables was used to obtain the system stiffness matrices. Then, by using the smoothed Galerkin weak form, discrete equations of the system were established to solve the generalized displacement of the structure. The numerical experiments validated the feasibility of SNS-RPIM for solving hygrothermo-magneto-electro-elastic (HMEE) coupling problems with different hygrothermal loadings and boundary conditions. Results proved that the introduction of the stabilization term made the SNS-RPIM had a ‘close-to-exact’ stiffness, thereby solving the ‘overly-soft’ in the traditional node-based smoothed radial point interpolation method (NS-RPIM). SNS-RPIM was in good agreement with the reference solution when using the same fewer nodes as finite element method (FEM), and the errors were much smaller than those of FEM. Moreover, SNS-RPIM performed better than NS-RPIM and FEM when processing distorted meshes. Therefore, the developed method showed great potential in simulating MEE structure in hygrothermal environment.

New insight into meshless radial point Hermite interpolation through direct and inverse 2-D reaction–diffusion equation

In this paper, we propose an effective method to solve partial differential equations dependent on time with Neumann boundary condition, by examining its effectivity on direct and inverse reaction–diffusion equation. This method merges the radial point interpolation and the Hermite-type interpolation techniques to provide us suitable tools to impose the boundary condition. This technique is called meshless radial point Hermite interpolation “MRPHI” which utilizes the radial basis function and its derivative to prepare suitable shape functions that are the key for expanding the high-order derivative. This procedure is tested on some types of two-dimensional diffusion equations to show stability through the time in different arbitrary domains.

NURBS-Enhanced Meshfree Method with an Integration Subtraction Technique for Complex Topology

In this paper, we present an integration subtraction technique to model holes interactively in a predesigned domain for adaptive problems. This technique involves two approaches, the normal subtraction technique and the moving subtraction technique. In the basic normal subtraction technique, the predesigned domain can be meshed using any methods as an initial integration background cell for meshfree analysis. Holes are described using closed non-uniform rational B-spline (NURBS) curves to preserve the exact computer-aided design (CAD) geometry and are meshed alone using the homotopic method, so they can easily be subtracted from the predesigned domain with no refinement. On the other hand, when the hole size is varying, the moving subtraction technique, in which only the changing part between the new and old boundaries needs to be integrated and subtracted, is more efficient. Compared with the standard radial point interpolation method (RPIM) and finite element method (FEM) in three linear elastic examples with different holes, the excellent accuracy and good efficiency of the proposed method are demonstrated, and its feasibility in complex topology problems is verified.

PERFORMANCE OF RADIAL POINT INTERPOLATION METHOD IN SOLVING KINEMATIC WAVE EQUATION FOR HYDROLOGIC MODELLING

This paper presents the solution of the kinematic wave equation using a meshless radial point interpolation method (RPIM). The partial differential equation is discretized using a Galerkin weighted residual method employing RPIM shape functions. A forward difference scheme is used for temporal discretization, while the direct substitution method is employed to solve the nonlinear system at each time step. The formulation is validated against solutions from conventional numerical techniques and physical observation. In all cases, excellent agreements are achieved and hence the validation of the proposed formulation. Optimum values of the multi-quadrics shape parameters were then determined before the assessment of the performance of the method. Based on the convergence rate, it has been shown that the proposed method performs better than the finite difference method and equivalent to the finite element method. This highlights the potential of RPIM as an alternative method for hydrologic modeling.

Meshless hybrid implicit‐explicit radial point interpolation method in scattering of dielectric particles by light beam with orbital angular momentum

AbstractThis letter proposes a meshless hybrid implicit‐explicit radial point interpolation method (HIE‐RPIM) to simulate the scattering of the Laguerre‐Gaussian beam with orbital angular momentum (OAM) by dielectric particles. The results are verified by comparison with the finite‐difference time domain (FDTD) and the radial point interpolation method (RPIM) which shows that the HIE‐RPIM agrees well with the FDTD, and the HIE‐RPIM is about 4.9 times faster than the FDTD method and about 4.2 times faster than the RPIM. Moreover, the scattering angle of forward scattering intensity maximum is about 10° to 35° when the OAM modes are L = 1 and 2 and moves back by 5° to 15° from L = 1 to 2. The HIE‐RPIM has potential to research the scattering properties of OAM beam.

Thermo-elastoplastic analysis of functionally graded sandwich plates using a three-dimensional meshless model

In this paper, for the first time, three-dimensional (3D) thermo-elastoplastic bending analysis of functionally graded (FG) sandwich plates with FG face sheets and FG/homogeneous core subjected to combined thermal and mechanical loads is performed. It is assumed that the functionally graded material (FGM) is composed of two different constituents (metal-ceramic) whose volume fractions vary continuously in the thickness direction according to a power law. The von-Mises yield criterion and isotropic strain hardening rule are employed, while temperature-dependent thermal and mechanical properties of the materials are taken into account. A complete 3D meshless numerical model based on the local radial point interpolation method (LRPIM) is developed and used in all analyses. Several numerical examples for temperature, deflection and stress analysis of FG sandwich plates are presented considering different material gradients, layer thickness ratios, thickness-to-length ratios, and boundary conditions. The numerical results are compared with the existing results of 3D, quasi-3D, and 2D shear deformation theories and an excellent agreement is observed.

Modelling Steady Convection-Dominated Phenomena by Node-Adaptive Radial Point Interpolation Meshfree Method (RPIM) with Various RBFs

For numerically solving PDEs, node or grid adaptation schemes have been invented, applied, and improved mainly to improve solution accuracy while reducing the burden caused by CPU-limitation and/or users’ interference. This becomes crucial particularly for problems involving strong changes in physical phenomena. This work proposes a means of node-adaptation used in conjunction with a numerical meshfree method called ‘radial point interpolation (RPIM)’, carried out using the concept of interpolation via. radial basis functions and hence no grid or mesh required. The proposed method is developed to tackle one of the most challenging problems with convective force being overwhelming in the system. This phenomena is known not to be easily numerically simulated. The proposed node-adaptive scheme is carried out where locally-estimated Péclet number (Pe) being used to indicate the areas of interest, places where nodes are expected to be added or removed. The whole adaptation algorithm is dynamic requiring on user’s interference. When compared to solutions obtained from fixed-node cases and other benchmark work, the results obtained in this work strongly suggest benefits in terms of both solution accuracy improvement and CPU-storage reduction.

Two-dimensional beams in rectangular coordinates using the radial point interpolation method

Abstract The three-dimensional Theory of Elasticity equations lead to a complex solution for most problems in engineering. Therefore, the solutions are typically developed for reduced systems, usually symmetrical or two-dimensional. In this context, computational resources allow the reduction of these simplifications. The most recognized methods of algebraic approximation of the differential equations are the Finite Differences Method and the Finite Element Method (FEM). However, they have limitations in mesh generation and/or adaptation. As follows, Meshless Methods appear as an alternative to these options. The present work uses the Radial Point Interpolation Method (RPIM) to evaluate the stress in two-dimensional beams in regions close to loading (Saint Venant's Principle). Formulations based on the Fourier Series Theory and the RPIM are presented. Multiquadrics Radial Basis Functions were used to obtain the stiffness matrix. Two numerical examples demonstrate the validity of the RPIM for the proposed theme. The results were obtained from the formulations cited and the Finite Element Method for comparison.

A node-based smoothed radial point interpolation method with linear strain fields for vibration analysis of solids

This paper presents a high-order node-based smoothed radial point interpolation method (NS-RPIM) with linear strain fields in smoothing domains. The linear smoothed strains are constructed by complete order of polynomial functions and normalized with reference to the central point of the smoothing region. The new NS-RPIM is one order higher than those of the existing methods which use piecewise constant strains. This high-order method still uses linear displacements within each triangular background cell, but linear strains are created over smoothing domains using the pick-out theory. Because the smoothed strain and the compatible strain within a local region are equal in an integral sense, the unknown parameters in the reconstructed strain functions can be determined uniquely by using three linearly independent weight functions. The numerical verification and computational efficiency are investigated and compared with the standard node-based smoothing models. It is found that the linear strain NS-RPIM shows improvement on both the convergence and computational efficiency in terms of error norm in displacement and is temporally stable. Since no spurious non-zero energy mode appears, this approach can be employed directly for the vibration analysis of solids.

ENHANCED MESHFREE RPIM WITH NURBS BASIS FUNCTION FOR ANALYSIS OF IRREGULAR BOUNDARY DOMAIN

Radial Point Interpolation Method (RPIM) has become a powerful tool to numerical analysis due to its ability to provide a higher-order approximation function with the Kronecker delta property, by which the field nodes can be fitted exactly. However, one of the major drawbacks of RPIM is the inefficiency in handling irregular domain problems. This paper presents an enhanced RPIM formulation that employs Non-Uniform Rational B-Splines (NURBS) basis functions to represent the exact geometry of the boundary domain. The NURBS is a mathematical model which provides an efficient and numerically stable algorithm to exactly represent all conic sections in engineering modelling. Taking advantage of the flexibility and adaptivity of RPIM approximation and the accuracy of geometric representations by NURBS, this new method is able to improve geometry accuracy and flexibility in numerical analysis, thus providing a better and more rational approach to analyze irregular domain problems. Numerical problem of steady heat transfer considering curved beam is presented to verify the validity and accuracy of the developed method. The essential boundary condition can simply be imposed using direct imposition as in Finite Element Method (FEM). The result shows that the RPIM/NURBS achieved the converged solution much faster than conventional RPIM and FEM, with the number of nodes required only less than 200 for an error of less than 0.01%. This shows the potential of the developed method as a powerful numerical technique for future development.