Meshless Numerical Method Research Articles

Maximization of fundamental frequency of layered composites using differential evolution optimization

Differential evolution optimization is used to find staking sequences for maximization of the natural frequency of symmetric and asymmetric laminates. Clamped and simply supported boundary condition are considered. A meshless numerical method is used to compute free vibrations of square and rectangular plates. Obtained staking sequences for symmetric laminates are in accordance with the literature. Optimized staking sequences for 8 ply asymmetric laminates are also presented. Asymmetric staking sequences produced higher first natural frequencies. Posterior analysis shows that optimized asymmetric staking sequences are hygrothermally stable.

Validation of GPD to Model Rock Fragmentation by TBM Cutters

In this paper, a novel meshless numerical method known as general particle dynamics (GPD) is developed to reveal the mechanism of the rock fragmentation using tunnel-boring machine (TBM) cutters. Rock fragmentation by one cutter and by two cutters is investigated using GPD. The numerical results obtained from GPD well agree with the previous experimental and numerical results. Moreover, the effects of arrays of flaws on rock fragmentation are also studied using GPD as well as stress fields and rock fragmentation efficiency. It is found from the numerical results that arrays of flaws significantly influence the drivage efficiency of rock fragmentation as well as the depth of chipping and the coalescence modes of flaws.

Analysis of Shallow Water Problems Using Element-Free Galerkin Method

A numerical meshless method is proposed to investigate the solution to shallow water equations. The numerical solution to pure convection equations, such as shallow water equations, is an important component of many problems. An element-free Galerkin (EFG) method has been developed and implemented to solve these equations. In this method, there is no need for nodal connectivity; it simply uses nodal data that are similar to those used in the finite element method. The essential boundary condition is enforced by the penalty method and the moving least squares approximation is used for the interpolation scheme. The numerical efficiency of the proposed method is demonstrated by solving several benchmark examples. Sensitivity analysis on the parameters of the EFG method was carried out and the results confirm its ability to solve shallow water equations and reveal that increasing the number of field nodes and Gauss quadrature points can improve the accuracy of the proposed method. It can be concluded that the average distance of the Gauss points should be less than one-third of the average distance of the nodal points. To produce the best results, the radius of influence domain should be three times the distance of the nodal points.

Busca pelo mínimo resíduo para escolha do parâmetro de forma utilizado em RBF’s aplicadas no método assimétrico livre de malhas (MESHLESS) para simulação de fluxo subterrâneo

Este trabalho visa à aplicação do método assimétrico numérico livre de malha, um tipo de Meshless, em dois casos de fluxo hídrico subterrâneo para simular a variação da carga hidráulica da região. O modelo desenvolvido utiliza a função de base radial multiquádrica como função de aproximação e uma nova metodologia, a qual utiliza os resíduos do contorno e do domínio para determinar o parâmetro de forma ‘c’, usado nesse Meshless. A metodologia foi aplicada em dois casos: Para o Caso 1, com solução analítica conhecida, foi possível comparar a solução exata à solução numérica obtida usando a metodologia dos resíduos proposta. O Caso 2 é um cenário real no qual a metodologia foi aplicada para simular o fluxo hídrico subterrâneo de uma pequena região dentro da Bacia Sedimentar do Araripe, situada no Estado do Ceará-Brasil. Nessa região específica a maior parte do abastecimento público de água provém de fontes subterrâneas. No caso 1, os resultados indicaram que o parâmetro ‘c’ obtido se aproximou do valor ideal. No caso 2, a equipotencial do poço utilizado como referência obteve um erro percentual de aproximadamente 5%.

Numerical Simulation of Failure Process of Rock-Like Materials Subjected to Impact Loads

AbstractIn this research, a novel meshless numerical method, termed general particle dynamics (GPD), to simulate crack initiation, propagation, and coalescence in rock-like materials subjected to impact loads was developed. In the GPD code, interaction among discrete particles was formulated with use of the virtual-bond method. Fractures of virtual bonds among particles were determined through a damage evolution law of rock-like materials. An elastobrittle damage model was applied to reflect the initiation and growth of cracks and the macrofailure of the rock-like materials. A damage coefficient, determined by the number of fractures of virtual bonds around one particle, was introduced into the constitutive equations to modify the damaged particles. The damaged virtual bonds were considered the initiation of cracks. The growth path of cracks was captured through the sequence of the damaged virtual bonds. Three numerical cases are presented here. The numerical results were in good agreement with the experi...

Application of generalized finite difference method to propagation of nonlinear water waves in numerical wave flume

In this paper, a numerical wave flume is formed by combining the generalized finite difference method (GFDM), the Runge–Kutta method, the semi-Lagrangian technique, the ramping function and the sponge layer to efficiently and accurately analyze the propagation of nonlinear water waves. On the basis of potential flow, the mathematical description of wave propagation is a time-dependent boundary value problem, governed by a Laplace equation for velocity potential and two nonlinear free-surface boundary conditions. The incident waves are introduced by imposing horizontal velocity along upstream boundary, as a sponge layer is placed at the end of flume to absorb wave energy and avoid any reflection of waves. The GFDM, a newly-developed meshless numerical method, and the second-order Runge–Kutta method were, respectively, adopted for spatial and temporal discretizations of the moving-boundary problems. The GFDM, which is truly free from mesh generation and numerical quadrature, is easy-to-program, straightforward and efficient, especially for moving-boundary problems. Four numerical examples are adopted in this paper to validate the stability, the efficiency and the accuracy of the proposed meshless numerical wave flume. The GFDM results were compared with other numerical solutions and experimental data to verify the merits and robustness of the proposed meshless numerical model.

A meshless radial basis function method for 2D steady-state heat conduction problems in anisotropic and inhomogeneous media

The paper presents a new meshless numerical method for solving 2D steady-state heat conduction problems in anisotropic and inhomogeneous media. The coefficients of the governing PDEs are spatially dependent functions including the main operator part. The boundary conditions of a most general form for the temperature and the heat flux are considered. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of a RBF and a special correcting function which is chosen to satisfy the homogeneous BC of the problem. The conical radial basis functions, the Duchon splines and the multiquadric RBFs are used in approximation of the PDE. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem with any choice of the free parameters. As a result we separate the approximation of the boundary conditions and the approximation of the PDE inside the solution domain. The numerical experiments are carried out for accuracy and convergence investigations. The comparison of the numerical results obtained in the paper with the exact solutions and with the data obtained with the use of other numerical techniques is performed. The numerical examples demonstrate that the present method is accurate, convergent, stable, and computationally efficient in solving this kind of problems.

3D Numerical Study on the Growth and Coalescence of Pre-existing Flaws in Rocklike Materials Subjected to Uniaxial Compression

AbstractA novel, meshless numerical method, called general particle dynamics (GPD), is proposed to simulate the initiation, propagation, and coalescence of three-dimensional (3D), pre-existing penetrating and embedded flaws as well as size effects and large deformations of rock materials. On the basis of the nonlinear unified strength criterion, an elastic–brittle–plastic damage model was developed to reflect the initiation, growth, and coalescence of the 3D flaws and the macrofailure of rocklike materials by tracing the propagation of the cracks. Then, growth paths of cracks were captured through the sequence of such damaged particles. In this paper, the GPD code is applied to simulate the macrofailure, large deformation, and size effects of the heterogeneous rocklike materials. The present numerical simulations focus on the effects of sample sizes, the nonoverlapping length and types of flaws on the failure, and the complete stress–strain curves of the rocklike materials. The initiation, propagation, an...

Localized radial basis function scheme for multidimensional transient generalized newtonian fluid dynamics and heat transfer

The local radial basis function (RBF) scheme is developed to simulate 2D and 3D heat transfer and flow dynamics of generalized Newtonian fluids (GNF). The local RBF scheme is a meshless numerical method based on radial basis functions with localized technique. The procedure of localization reduces the computational cost more efficiently than the traditional global RBF method. This meshless method does not require mesh generation, numerical integration and only needs point collocation. Besides, it is very easy to interpolate physical values and its derivatives everywhere in the domain. We consider one isothermal and three non-isothermal multidimensional transient GNF fluid and heat problems in this paper. The dynamic viscosity of the GNF is specified as two different models: the power law model (temperature independent) or Cross model (temperature dependent). The viscous heating is also considered in this work. Numerical results show that the local RBF scheme is stable and accurate as far as the four tested cases are concerned.

Numerical simulation of the nonlinear ultrasonic pressure wave propagation in a cavitating bubbly liquid inside a sonochemical reactor

We investigate the acoustic wave propagation in bubbly liquid inside a pilot sonochemical reactor which aims to produce antibacterial medical textile fabrics by coating the textile with ZnO or CuO nanoparticles. Computational models on acoustic propagation are developed in order to aid the design procedures. The acoustic pressure wave propagation in the sonoreactor is simulated by solving the Helmholtz equation using a meshless numerical method. The paper implements both the state-of-the-art linear model and a nonlinear wave propagation model recently introduced by Louisnard (2012), and presents a novel iterative solution procedure for the nonlinear propagation model which can be implemented using any numerical method and/or programming tool. Comparative results regarding both the linear and the nonlinear wave propagation are shown. Effects of bubble size distribution and bubble volume fraction on the acoustic wave propagation are discussed in detail. The simulations demonstrate that the nonlinear model successfully captures the realistic spatial distribution of the cavitation zones and the associated acoustic pressure amplitudes.

The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: The Crank–Nicolson scheme and the method of lines (MOL)

MHD equations have many applications in physics and engineering. The model is coupled equations in velocity and magnetic field and has a parameter namely Hartmann. The value of Hartmann number plays an important role in the equations. When this parameter increases, using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem. In the present paper a numerical meshless method based on radial basis functions (RBFs) is provided to solve MHD equations. For approximating the spatial variable, a new approach which is introduced by Bozzini et al. (2015) is applied. The method will be used here is based on the interpolation with variably scaled kernels. The methodology of the new technique is defining the scale function c on the domain Ω⊂Rd. Then the interpolation problem from the data locations xj∈Rd transforms to the new interpolation problem in the data locations (xj,c(xj))∈Rd+1 (Bozzini et al., 2015). The radial kernels used in the current work are Multiquadrics (MQ), Inverse Quadric (IQ) and Wendland’s function. Of course the latter one is based on compactly supported functions. To discretize the time variable, two techniques are applied. One of them is the Crank–Nicolson scheme and another one is based on MOL. The numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem. Also in appendix, we provide a computational algorithm for implementing the new technique in MATLAB software.

The 3D Numerical Simulation for the Propagation Process of Multiple Pre-existing Flaws in Rock-Like Materials Subjected to Biaxial Compressive Loads

General particle dynamics (GPD), which is a novel meshless numerical method, is proposed to simulate the initiation, propagation and coalescence of 3D pre-existing penetrating and embedded flaws under biaxial compression. The failure process for rock-like materials subjected to biaxial compressive loads is investigated using the numerical code GPD3D. Moreover, internal crack evolution processes are successfully simulated using GPD3D. With increasing lateral stress, the secondary cracks keep growing in the samples, while the growth of the wing cracks is restrained. The samples are mainly split into fragments in a shear failure mode under biaxial compression, which is different from the splitting failure of the samples subjected to uniaxial compression. For specimens with macroscopic pre-existing flaws, the simulated types of cracks, the simulated coalescence types and the simulated failure modes are in good agreement with the experimental results.

Progressive failure processes of reinforced slopes based on general particle dynamic method

In order to resolve grid distortions in finite element method (FEM), the meshless numerical method which is called general particle dynamics (GPD) was presented to simulate the large deformation and failure of geomaterials. The Mohr-Coulomb strength criterion was implemented into the code to describe the elasto-brittle behaviours of geomaterials while the solid-structure (reinforcing pile) interaction was simulated as an elasto-brittle material. The Weibull statistical approach was applied to describing the heterogeneity of geomaterials. As an application of general particle dynamics to slopes, the interaction between the slopes and the reinforcing pile was modelled. The contact between the geomaterials and the reinforcing pile was modelled by using the coupling condition associated with a Lennard-Jones repulsive force. The safety factor, corresponding to the minimum shear strength reduction factor “R”, was obtained, and the slip surface of the slope was determined. The numerical results are in good agreement with those obtained from limit equilibrium method and finite element method. It indicates that the proposed geomaterial-structure interaction algorithm works well in the GPD framework.

Differential evolution for optimization of functionally graded beams

Differential evolution optimization is used to find the volume fraction that maximizes the first natural frequency for a functionally graded beam. A formulation using three parameters is used to describe volume fraction. Beams with different ratios of material properties are considered. Two methods are used to compute the natural frequencies, analytical and meshless numerical method. Results show that differential evolution is capable to find distributions for volume fraction that increase the natural frequency of beams. It was also found that the RBF numerical method can be used with differential evolution to solve problems related to maximization of natural frequencies in functionally graded beams.

A meshless method for numerical simulation of depth‐averaged turbulence flows using a k‐ϵ model

SummaryWe investigate the performance of a meshless method for the numerical simulation of depth‐averaged turbulence flows. The governing equations are shallow water equations obtained by depth averaging of the full Reynolds equations including bed frictions, eddy viscosity, wind shear stresses, and Coriolis forces. As a double‐phase closure turbulence model, we consider the depth‐averaged k‐ϵ model. A truly meshless numerical method based on radial basis functions is employed to obtain an accurate approximation to the solution of the model. We validate the algorithm on a linear shallow water problem where analytical solutions are available. Numerical results are also compared with experimental data for a backward‐facing flow problem. Furthermore, we test the method on a practical problem by simulating tidal flows in the Strait of Gibraltar. The main focus is to examine the performance of the meshless method for complex geometries with irregular bathymetry. The obtained results demonstrate its ability to capture the main flow features. Copyright © 2015 John Wiley & Sons, Ltd.

Using a local radial basis function collocation method to approximate radiation boundary conditions

The local radial-basis-function collocation method (LRBFCM) based on the multiquadric type radial basis function is used to simulate the water wave propagation with oblique incidence and radiation. In this study, the mild-slope equation is applied and the radiation boundary condition is expanded to the third-order approximation. Two numerical cases are considered for verification, which include wave scattering around a vertical cylinder and shallow water wave passed over a submerged shoal. Numerical results based on the third-order radiation boundary condition are found to be in a better agreement than those of the first-order and second-order radiation boundary conditions especially for large incidence or radiation angles. Overall, the LRBFCM is a meshless numerical method and its applications to the high-order radiation boundary condition are much simpler compared with the traditional numerical methods, such as the finite difference method and the finite element method.

Meshless method for the numerical solution of the Fokker–Planck equation

Abstract In this paper numerical meshless method for solving Fokker–Planck equation is considered. This meshless method is based on multiquadric radial basis function and collocation method to approximate the solution. Here we apply θ -weighted finite difference method. The stability analysis of the method is dealt with, using a linearized stability method. Numerical examples illustrate the validity and applicability of the method.

Hydroelastic analysis of fully nonlinear water waves with floating elastic plate via multiple knot B-splines

Hydroelastic analysis of fully nonlinear water waves with the floating elastic plate is a hard mission. Especially, the behavior of the wave would be more complex when water wave encounter the floating elastic plate. In this paper, the meshless numerical method is devoted to solve such a problem. Fundamental solution method is applied to approximate the velocity potential in the fluid domain. When the water wave encounters the plate, the wave function would not be enough smooth in the edge of plate compared to the other points. Hence, to analyze numerically the behavior of wave, the solution space should include the basis functions that are not enough smooth in the edge of plate. Moreover, to decrease computational cost significantly, the basis functions had better to have local compact support. The multiple knot B-spline basis functions are suitable that contain both properties. The number of repeated knots, the degree of B-spline and the spatial points are challengeable that are discussed in this paper. The results are in good agreement with those obtained from other numerical works.

A novel meshless numerical method for modeling progressive failure processes of slopes

A novel meshless numerical method, known as general particle dynamics (GPD3D), is proposed to simulate the failure process of slopes as well as the failure process of 2D and 3D samples. An elasto-brittle damage model based on the Mohr–Coulomb criterion is applied to investigate the initiation, growth and coalescence of cracks as well as the macro-failure of slopes. The Weibull statistical approach is applied to model the heterogeneity of geomaterials. Five macro-fracture modes of samples subjected to uniaxial compressive loads are modeled due to the heterogeneity of geomaterials. The safety factor and slip surface of the slope are determined by using the GPD3D approach. The numerical results from the GPD approach are in good agreement with those from the limit equilibrium method and FEM. The present numerical method is demonstrated to predict the complex fracture processes of slopes.

Efficient visibility criterion for discontinuities discretised by triangular surface meshes

This study proposes a computationally efficient algorithm for determining which pairs of points among many predetermined pairs in three dimensions will maintain straight line visibility between one another in the presence of an arbitrary surface mesh of triangles. This is carried out in the context of meshless numerical methods with the goal of implementing near-real-time discontinuity propagation simulation. A brief overview is given of existing discontinuity modelling techniques for meshless methods. Such techniques necessitate determination of which key pairs of points (nodes and quadrature points) lack straight line visibility due to the discontinuity, which is proposed to be modelled with a surface mesh of triangles. The efficiency of this algorithm is achieved by allocating all quadrature points and surface mesh triangles to the cells of an overlayed three-dimensional grid in order to rapidly identify for each triangle an approximately minimal set of quadrature points whose nodal connectivities may be interrupted due to the presence of the triangle, hence eliminating most redundant visibility checking computations. Triangles are automatically split such that any size of overlayed cubic grid cells can be employed, and the parameters governing triangle splitting and binning have been examined experimentally in order to optimise the visibility algorithm.