Showing papers on "Meshfree methods published in 2018"

Smoothed Finite Element Methods (S-FEM): An Overview and Recent Developments

Abstract: The smoothed finite element methods (S-FEM) are a family of methods formulated through carefully designed combinations of the standard FEM and some of the techniques from the meshfree methods. Studies have proven that S-FEM models behave softer than the FEM counterparts using the same mesh structure, often produce more accurate solutions, higher convergence rates, and much less sensitivity to mesh distortion. They work well with triangular or tetrahedral mesh that can be automatically generated, and hence are ideal for automated computations and adaptive analyses. Some S-FEM models can also produce upper bound solution for force driving problems, which is an excellent unique complementary feature to FEM. Because of these attractive properties, S-FEM has been applied to numerous problems in the disciplines of material mechanics, biomechanics, fracture mechanics, plates and shells, dynamics, acoustics, heat transfer and fluid–structure interactions. This paper reviews the developments and applications of the S-FEM in the past ten years. We hope this review can shed light on further theoretical development of S-FEM and more complex practical applications in future.

Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation

Abstract: In the present paper, a spectral meshless radial point interpolation (SMRPI) technique is applied to solve the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in two dimensional case. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on 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 act as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in $$H^1$$ and convergent with the order of convergence $$\mathcal {O}(\delta t^{1+\beta })$$ , $$0<\beta <1$$ . In the current work, the thin plate splines (TPS) are used as the basis functions. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme. Two numerical examples show that the SMRPI has reliable accuracy in general shape domains.

An isogeometric-meshfree coupling approach for analysis of cracks

Abstract: Summary 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 two sub-domains which are formulated with the IGA and meshfree method, respectively. In the meshfree sub-domain, the moving least squares (MLS) shape function is adopted for the discretization of the area around crack tips, and the IGA sub-domain is adopted in the remaining area. Meanwhile, the interface region between the two sub-domains 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 utility and robustness of the proposed method. This article is protected by copyright. All rights reserved.

Reproducing kernel particle method for two-dimensional time-space fractional diffusion equations in irregular domains

Abstract: In recent years, the fractional differential equations have attracted a lot of attention due to their interested characteristics. Meshfree methods are highly accurate and have been extensively explored in engineering and mechanics fields. However, there is few research to develop the reproducing kernel particle method (RKPM), one of the widely used meshfree approach, for fractional partial differential equations. In this work, we solve time-space fractional diffusion equations in 2D regular and irregular domains. The temporal Caputo fractional derivatives are discretized by the L1 finite difference scheme and the spatial Laplacian fractional derivatives are discretized by RKPM based on the matrix transfer method. Especially, the corrected weighted shifted Grunwald–Letnikov scheme is utilized for temporally non-smooth solutions. Numerical examples in rectangular, circular, sector and human brain-like irregular domains are given to assess the efficiency and accuracy of the proposed numerical scheme. The spatial Laplacian fractional derivatives discretized by conventional finite difference method in the rectangular domain are also presented for comparison. The results indicate that RKPM is very effective for analyzing the considered fractional equations in various domains, which lays a concrete foundation for our further research of real application of human brain modeling.

A unified numerical framework for rigid and compliant granular materials

Abstract: A numerical framework for the simulation of granular materials composed of mixed rigid and compliant grains is presented in this paper. This approach is based on a multibody meshfree technique, coupled in a very natural way with classic concepts from the discrete element method. The equations of motion (for the rigid grains) and of continuum mechanics (for the compliant ones) are solved using an adaptive explicit scheme, in fully dynamic conditions. The parallelization strategy is described and tested on an illustrative simulation involving both kinds of grains.

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

Abstract: Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.

Homogenization technique for heterogeneous composite materials using meshless methods

Abstract: Due to its random fibre distribution across the cross-section and their anisotropic and heterogeneous characteristic, the prediction of the mechanical behaviour of fibre composite materials is complex. Multi-scale approaches have been proposed in the literature to more accurately predict their mechanical properties using computational homogenization procedures. This work is based on existing multi-scale numerical transition techniques suitable for simulating heterogeneous materials and makes use of two meshless methods—the Radial Point Interpolation Method (RPIM) and the Natural Neighbour Radial Point Interpolation Method (NNRPIM)—and the Finite Element Method (FEM). Representative volume elements (RVEs) are modelled and discretized using the three numerical methods. Prescribed microscopic displacements are imposed on different RVEs whose boundaries are periodic and, from the obtained stress field, the average stresses are determined. Consequentially, the effective elastic properties of a heterogeneous material are obtained for different fibre volume fractions. In the end, the numerical solutions are compared with the solutions proposed in the literature and it is proved that the NNRPIM achieve more accurate solutions than the RPIM and the FEM.

A numerical algorithm to reduce ill-conditioning in meshless methods for the Helmholtz equation

Abstract: Some meshless methods have been applied to the numerical solution of boundary value problems involving the Helmholtz equation. In this work, we focus on the method of fundamental solutions and the plane waves method. It is well known that these methods can be highly accurate assuming smoothness of the domains and the boundary data. However, the matrices involved are often ill-conditioned and the effect of this ill-conditioning may drastically reduce the accuracy. In this work, we propose a numerical algorithm to reduce the ill-conditioning in both methods. The idea is to perform a suitable change of basis. This allows to obtain new basis functions that span exactly the same space as the original meshless method, but are much better conditioned. In the case of circular domains, this technique allows to obtain errors close to machine precision, with condition numbers of order O(1), independently of the number of basis functions in the expansion.

Evaluation of effective elastic properties of 3D printable interpenetrating phase composites using the meshfree radial point interpolation method

Abstract: Interpenetrating phase composites (IPCs) have recently been fabricated using three-dimensional (3D) printing methods. In a two-phase IPC, the two phases are topologically interconnected and mutually reinforced in the three dimensions. As a result, such IPCs exhibit higher stiffness, strength, and toughness than particle- or fiber-reinforced composites. In the current study, three unit cell models for the IPCs with the simple cubic (SC), face-centered cubic (FCC), and body-centered cubic (BCC) microstructures are developed using the meshfree radial point interpolation method. Radial basis functions with polynomial reproduction are applied to construct shape functions, and the Galerkin method is employed to formulate discretized equations. These unit cell-based meshfree models are used to evaluate effective elastic properties of 3D printable IPCs. The simulation results are compared with those based on the finite element (FE) method and various analytical bounding techniques in micromechanics, inclu...

Two algorithms for fast 2D node generation: application to RBF meshless discretization of diffusion problems and image halftoning

Abstract: Mesh generation techniques for traditional mesh based numerical approaches such as FEM and FVM have now reached a good degree of maturity. There is no such an acknowledged background when dealing with node generation techniques for meshless numerical approaches, despite their theoretical simplicity and efficiency; furthermore node generation can be put in connection with some well-known image approximation techniques. Two node generation algorithms are here proposed and employed in the numerical solution of 2D steady state diffusion problems by means of a local Radial Basis Function (RBF) meshless method. Finally, such algorithms are also tested for grayscale image approximation through stippling.

Optimal transportation meshfree method in geotechnical engineering problems under large deformation regime

Abstract: Meshfree methods have been demonstrated as suitable and strong alternatives to the more standard numerical schemes such as finite elements or finite differences. Moreover, when formulated in a Lagrangian approach, they are appropriate for capturing soil behavior under high‐strain levels. In this paper, the optimal transportation meshfree method has been applied for the first time to geotechnical problems undergoing large deformations. All the features employed in the current methodology (ie, F‐bar, explicit viscoplastic integration, and master‐slave contact) are described and validated separately. Finally, the model is applied to the particular case of shallow foundations by using von Mises and Drucker‐Prager yield criteria to find the load at failure in the. The presented methodology is demonstrated to be robust and accurate when solving this type of problems.

Numerical simulation of nonlinear coupled Burgers’ equation through meshless radial point interpolation method

Abstract: In present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of nonlinear coupled Burgers’ equation in two dimensions. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a 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 act as basis functions in the frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. The aim of this paper is to show that the SMRPI method is suitable for the treatment of nonlinear coupled Burgers’ equation. With regard to test problems that have not exact solutions, we consider two strategies for checking the stability of time difference scheme and for survey the convergence of the fully discrete scheme. The results of numerical experiments confirm the accuracy and efficiency of the presented scheme.

The numerical solution for the time-fractional inverse problem of diffusion equation

Abstract: In this study, the spectral meshless radial point interpolation (SMRPI) technique is applied to the Cauchy problem of two-dimensional fractional diffusion equation. We obtain the unknown data on the inner boundary when overspecified boundary data is imposed on the outer boundary. The SMRPI is based on a combination of meshfree methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. Here, similar to other meshless methods, localization in SMRPI can reduce the ill-posedness of the Cauchy problem. However, it does not require to use regularization algorithms and therefore reduces computational time. Two numerical examples, are tested to show that the SMRPI can overcome the ill-posedness of the Cauchy problem and has acceptable accuracy. Also, by adding some large perturbations, the proposed method is still stable.

Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

Abstract: Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent; however, for the datasets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large datasets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.

Groundwater management using a new coupled model of meshless local Petrov-Galerkin method and modified artificial bee colony algorithm

Abstract: To develop sustainable groundwater management strategies, generally coupled simulation-optimization (SO) models are used. In this study, a new SO model is developed by coupling moving least squares (MLS)-based meshless local Petrov-Galerkin (MLPG) method and modified artificial bee colony (MABC) algorithm. The MLPG simulation model utilizes the advantages of meshless methods over the grid-based techniques such as finite difference (FDM) and finite element method (FEM). For optimization, the basic artificial bee colony algorithm is modified to balance the exploration and exploitation capacity of the model more effectively. The performance of the developed MLPG-MABC model is investigated by applying it to hypothetical and field problems with three different management scenarios. The model results are compared with other available SO model solutions for its accuracy. Further, sensitivity analyses of various model parameters are carried out to check the robustness of the SO model. The proposed model gave quite promising results, showing the applicability of the present approach.