Meshless Formulation Research Articles

A 3-D Dispersive Time-Domain Meshless Formulation for Frequency-Dependent Materials

This communication presents a novel transient meshless formulation for analyzing wave propagation through linear dispersive materials. Many real materials are linear dispersive; thus, their constitutive parameters are not constant and change with frequency. However, the conventional numerical techniques are not proper tools for numerical analysis of dispersive materials, considering that they are formulated based on the assuming constant constitutive parameters for the materials. Therefore, until now, a number of numerical techniques have been proposed for modeling frequency behavior of dispersive materials. On the other hand, meshless methods are new and powerful numerical techniques which their capability in simulating the problem domain without using connection information among nodes makes them efficient techniques for modeling problems with complex geometries. In this communication, we have demonstrated that by incorporating some approaches into the meshless methods, we can turn them into suitable tools for modeling dispersive media. In order to obtain a meshless analysis of a dispersive medium, we have considered the scalar radial basis function meshless method, and for taking the frequency behavior of the medium into account, the auxiliary differential equations method has been used. Hence, the proposed dispersive meshless method not only models the frequency behavior of the dispersive materials but also provides more flexibility in simulating problems with complex geometries. In addition, the efficiency and accuracy of our proposed method are investigated by two numerical examples.

Strong-form approach to elasticity: Hybrid finite difference-meshless collocation method (FDMCM)

• FDMCM combines the finite difference and the strong form meshless method. • FDMCM scheme uses a uniform Cartesian grid embedded in complex geometry. • The method is suited for problems with geometries known imperfectly from images. • FDMCM method combines the advantages of both FD and meshless methods. We propose a numerical method that combines the finite difference (FD) and strong form (collocation) meshless method (MM) for solving linear elasticity equations. We call this new method FDMCM. The FDMCM scheme uses a uniform Cartesian grid embedded in complex geometries and applies both methods to calculate spatial derivatives. The spatial domain is represented by a set of nodes categorized as (i) boundary and near boundary nodes, and (ii) interior nodes. For boundary and near boundary nodes, where the finite difference stencil cannot be defined, the Discretization Corrected Particle Strength Exchange (DC PSE) scheme is used for derivative evaluation, while for interior nodes standard second order finite differences are used. FDMCM method combines the advantages of both FD and DC PSE methods. It supports a fast and simple generation of grids and provides convergence rates comparable to weak formulations. We demonstrate the appropriateness and robustness of the proposed scheme through various benchmark problems in 2D and 3D. Numerical results show good accuracy and h -convergence properties. The ease of computational grid generation makes the method particularly suited for problems where geometries are very complicated and known only imperfectly from images, frequently occurring in e.g. geomechanics and patient-specific biomechanics, where the proposed FDMCM method, after its extension to non-linear regime, appears to be a promising alternative to the traditional weak form-based numerical schemes used in the field.

The analysis of composite laminated beams using a 2D interpolating meshless technique

Laminated composite materials are widely implemented in several engineering constructions. For its relative light weight, these materials are suitable for aerospace, military, marine, and automotive structural applications. To obtain safe and economical structures, the modelling analysis accuracy is highly relevant. Since meshless methods in the recent years achieved a remarkable progress in computational mechanics, the present work uses one of the most flexible and stable interpolation meshless technique available in the literature—the Radial Point Interpolation Method (RPIM). Here, a 2D approach is considered to numerically analyse composite laminated beams. Both the meshless formulation and the equilibrium equations ruling the studied physical phenomenon are presented with detail. Several benchmark beam examples are studied and the results are compared with exact solutions available in the literature and the results obtained from a commercial finite element software. The results show the efficiency and accuracy of the proposed numeric technique.

Simulation of a crack in stiffened plates via a meshless formulation and FSDT

A meshless formulation for fracture analysis of stiffened plates is introduced in this paper. Based on an improved meshless model of stiffened plates proposed by the authors, in which flat plate and ribs are combined by implementing the displacement compatibility conditions between them, a crack is introduced by the diffraction method. The expanded basis function and the weight function based on t−1 distribution are employed. Inheriting the meshless advantages from the model, the ribs in our formulation can be set at any location on the flat plate, and the remeshing of the flat plate is naturally avoided when rib location changes. Some numerical examples are investigated by the proposed formulation and the commercial FEM software ANSYS, and the accuracy of the proposed formulation is verified. The effects of ribs on the displacement, stress and SIFs of the stiffened plates that have cracks are discussed.

Local radial basis function collocation method for bending analyses of quasicrystal plates

• Meshless formulations are developed for static and dynamic bending of QC plates. • The phonon-phason coupling effects are studied numerically. • The influence of elastic foundation and variable plate thickness is investigated. • The response of QC plates to impact loading is simulated. The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.

A radial point interpolation meshless method extended with an elastic rate-independent continuum damage model for concrete materials

ABSTRACTAn advanced discretization meshless technique, the radial point interpolation method (RPIM), is applied to analyze concrete structures using an elastic continuum damage constitutive model. Here, the theoretical basis of the material model and the computational procedure are fully presented. The plane stress meshless formulation is extended to a rate-independent damage criterion, where both compressive and tensile damage evolutions are established based on a Helmholtz free energy function. Within the return-mapping damage algorithm, the required variable fields, such as the damage variables and the displacement field, are obtained. This study uses the Newton–Raphson nonlinear solution algorithm to achieve the nonlinear damage solution. The verification, where the performance is assessed, of the proposed model is demonstrated by relevant numerical examples available in the literature.

Analysis of meshless weak and strong formulations for boundary value problems

This paper introduces a weak meshless procedure combined with a multi-resolution numerical integration and its comparison with a strong local meshless formulation for approximating displacement and strain modeled in the form of Elliptic Boundary Value Problems (EBVPs) in one- and two-dimensional spaces. Assets and losses of both strong and weak meshless approaches are considered in detail. The meshless weak formulation considered in the current paper is the well-known Element Free Galerkin (EFG) method whereas the Local Radial Basis Functions Collocation Method (LRBFCM) is taken as a strong formulation. First aspect of the current work is implementation of the new numerical integration techniques introduced in Siraj-ul-Islam et al. (2010) and Aziz et al. (2011) [1,2] in the EFG method and its comparison with numerical integration based on standard Gaussian quadrature, adaptive integration and stabilized nodal integration techniques used in the context of EFG and other allied weak meshless formulations. Second aspect of the current work is analysis of comparative performance of the localized versions of strong and weak meshless formulations. Standard numerical tests are conducted to validate performance of both the approaches.

Local radial basis function interpolation method to simulate 2D fractional‐time convection‐diffusion‐reaction equations with error analysis

In this article, a kind of meshless local radial point interpolation (MLRPI) method is proposed to two‐dimensional fractional‐time convection‐diffusion‐reaction equations and satisfactory agreements are archived. This method is based on meshless methods and benefits from collocation ideas but it does not belong to the traditional global meshless collocation methods. In MLRPI method, it does not need any kind of integration locally or globally over small quadrature domains which is essential in the finite element method and those meshless methods based on Galerkin weak form. Also, it is not needed to determine shape parameter which plays important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of this kind of MLRPI method is less expensive. The stability and convergence of this meshless approach are discussed and theoretically proven. It is proved that the present meshless formulation is very effective for modeling and simulation of fractional differential equations. Furthermore, the numerical studies on sensitivity analysis and convergence analysis show the stability and reliable rates of convergence. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 974–994, 2017

Bending Analysis of FGM Plates Under Thermal Load

In this paper, the bending of thin functionally graded plates under thermal load is considered within the classical theory of thermoelasticity. The variable material properties of plate (such as the Young's modulus, thermal expansion coefficient, etc.) are allowed to be continuous functions of the position. The governing equations which are given by the 4th order partial differential equations are decomposed into the 2nd order partial differential equations in order to overcome the inaccuracy of approximation of high order derivatives of field variables. The strong form meshless formulations for solution of thin plate bending problem is developed in combination with Moving Least Squares approximation scheme. The attention is paid to the numerical investigation of the influence of several parameters of gradations of material coefficients on the deflection of the plate.

SOLUTION OF 1D AND 2D POISSON'S EQUATION BY USING WAVELET SCALING FUNCTIONS

The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs). Therefore, the use of wavelet scaling functions as a basis in computational analysis holds some promise due to their compact support, orthogonality and localization properties. Daubechies and Deslauriers-Dubuc functions have been successfully used as basis functions in several schemes like the Wavelet- Galerkin Method (WGM) and the Wavelet Finite Element Method (WFEM). Another possible advantage of their use is the fact that the calculation of integrals of inner products of wavelet scaling functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of using approximations by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness matrices and load vectors. In this work, some mathematical foundations regarding wavelet scaling functions, their derivatives and connection coefficients are reviewed. A scheme based on the Galerkin Method is proposed for the direct solution of Poisson's equation (potential problems) in a meshless formulation using interpolating wavelet scaling functions (Interpolets). The applicability of the proposed method and some convergence issues are illustrated by means of a few examples.

Contact mechanics with maximum-entropy meshfree approximants blended with isogeometric analysis on the boundary

This paper explores the simulation of small deformation contact problems with a coupled isogeometric-meshless formulation, based on local maximum-entropy (LME) approximants. The framework of isogeometric analysis (IGA) is employed on a thin region close to the boundary of the domain and the meshless formulation in its interior. The method maintains the advantages of IGA in the simulation of contact and introduces a higher flexibility in the discretization of complex shapes. In addition, a new integration strategy is presented for the IGA-LME blending and its advantages respect to the original approach are discussed.

NURBS-enhanced maximum-entropy schemes

In this paper, we combine high order local maximum-entropy schemes (HOLMES) with the integration framework developed in the NURBS-enhanced finite element method (NEFEM). We focus on the two-dimensional case where, given a domain described by some NURBS curves, a meshless formulation based on the HOLMES approximants is employed for the discretization and, at the same time, the geometric fidelity given by the NURBS boundary is preserved thanks to the NEFEM integration. Since HOLMES basis functions are not interpolatory on the boundary, different techniques are considered for the imposition of essential boundary conditions. The efficiency and the accuracy of the proposed methodology are confirmed with supportive numerical examples.

On the Convergence of Iterative Methods and Pseudoinverse Approaches in Global Meshless Collocation

In the context of numerical approximation of partial differential equations, meshless methods are recent developments, which have been reported to be relatively straightforward, yet provide better convergence and accuracy as compared to the conventional mesh-based approaches, for some specific problems like stress–strain analysis and modeling in a deforming media. Among several proposed schemes, strong-form collocation schemes using radial basis functions are easy-to-program, require no mesh in the domain or at the boundary, avoid numerical integration, and have similar formulations for all dimensions. Some of the radial basiss functions like the Gaussian, multiquadric and inverse-multiquadric are infinitely smooth. The standard global meshless formulations based on such radial basis functions are often found to lead towards ill-conditioned systems of linear equations. For such formulations, we investigate the degree of ill-conditioning against degrees of freedom for different radial basis functions. Also, the convergence of four iterative methods and pseudoinverse approaches has been tested for the solution of such ill-conditioned systems. In order to compute the pseudoinverse, two different approaches, i.e., singular value decomposition and full rank Cholesky factorization have been used. A set of numerical experiments, performed here, demonstrate that pseudoinverse approach based on singular value decomposition performs better than the iterative methods. Although, pseudoinverse computation via full rank Cholesky factorization is faster, it is not recommended in global meshless collocation schemes due to poor convergence at relatively large degrees of freedom.

Weak-form collocation – A local meshless method in linear elasticity

This paper is concerned with the formulation of local meshfree methods, for the solution of two-dimensional problems in linear elasticity, in the framework of the theory of structures.Local meshfree methods are derived through a weighted-residual formulation which leads to a local weak form that is the well known work theorem of the theory of structures. In an arbitrary local region, the work theorem establishes an energy relationship between a statically-admissible stress field and an independent kinematically-admissible strain field. Based on the independence of these two fields, this paper presents two new meshless formulations that aim a reduction of the computational effort. While in the first formulation the local form of the work theorem is reduced to regular boundary terms only, in the second formulation the local form of the work theorem is simply an integration-free formula.The moving least squares (MLS) approximation of the elastic field is used in this paper to implement both local meshless formulations.Several problems were analyzed with these techniques, in order to assess the accuracy and efficiency of the formulations. The results obtained in this work are in perfect agreement with those of the available analytical solutions. The accuracy and efficiency of the integration-free formulation make this a reliable and robust local meshfree method, generated in the framework of the theory of structures.

Pricing American options under jump-diffusion models using local weak form meshless techniques

ABSTRACTRecently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert, A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth, A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 (2012), pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.

Extending a radial point interpolation meshless method to non-local constitutive damage models

This work extends the radial point interpolation method (RPIM) to the analysis of concrete structures using an elastic continuum damage constitutive model. Here, the theoretical basis of the material model and the computational procedure are described with detail. Firstly, the plane stress meshless formulation is adapted to a rate-independent standard local damage criterion where both compressive and tensile damage evolution are settled respecting a Helmholtz free energy function. Afterwards, the non-local damage formalism is established based on the foregoing local model. Subsequently, the internal variable fields (such as local and non-local damage parameters) are obtained within the return-mapping damage algorithm. This study employs the Newton-Raphson algorithm to achieve the non-linear damage solution. The numerical stability and performance of the proposed standard and non-local models are assessed using the results of experimental tests available in the literature.

A comparative analysis of local meshless formulation for multi-asset option models

A local meshless radial basis function collocation differential quadrature (LMRBFCDQ) is proposed for the numerical solution of a single and multi-asset option pricing PDE models arising in computational finance. Spatial discretization is performed by both local and a standard global meshless collocation procedures coupled with a set of different time integrators based on the forward Euler difference formula (FEDF), the fully Implicit method (FIM), the Crank–Nicolson method (CNM), the explicit Runge–Kutta method of order two (ERK2), the Crank–Nicolson Runge–Kutta method of order two (CNRK2), the fully Implicit Runge–Kutta method of order two (IRK2), the Runge–Kutta method of order four (RK4), the Embedded Runge–Kutta method (RK23). Operator splitting techniques like the ordinary operator splitting (OOS), the Lie–Trotter splitting, the additive splitting and the Strang splitting are also tested for time integration. The proposed hybrid schemes are the amalgamation of the meshless differential quadrature procedure and the finite difference approximations. Different types of radial basis functions (RBFs) i.e. the multiquadric (MQ), the inverse quadric (IQ) and the Gaussian (GA) are utilized for the spatial discretization of the PDE models. Numerical analysis of a range of computational finance related models are shown to demonstrate accuracy, efficiency and ease of implementation of the proposed meshless-finite difference procedure.

Finite Element Modelling of High Velocity Impact on Plate Structures

The high velocity impact in heterogeneous materials is a very old and complex problem. Accurate numerical simulation of impact events can provide physical insights that cannot be captured by experiments. The high velocity process is very complex and requires the use of reliable and robust constitutive materials models. Currently, for impact modelling in composite structures at low and high speeds is used mainly Finite Element Method (FEM), Boundary Element Method, Finite Volume Method, meshless formulations and recently connection of FEM and element free based formulations. In this paper the finite element models for normal impact were created in ABAQUS/Explicit software. First, the simulations presented here examine the perforation of steel and aluminium plate specimens and then impact load is applied on composite plate with eight layers reinforced by carbon fibers. The Johnson-Cook model for strain rate effects in the material and its compressibility and Hashin failure criterion for damage of composite plate were used in simulations.

A general discrete element approach for particulate materials

Computational approaches of mechanical systems based on the continuum hypothesis, are sometimes inaccurate and not reliable. As an example, problems involving severe mesh distortion, geometric discontinuities or characterized by an assembly of discrete parts, are not easily solvable within the Lagrangian continuum framework, such as by using the classical finite element method. Computational approaches based on the description of the domain without the need of a mesh connectivity, would be useful to overcome this drawbacks. On the other hand a discrete approach is a particularly suitable tool for modeling materials at the microscale where its particulate nature becomes evident. The Lagrangian-based meshless formulation—known as smoothed particle hydrodynamics (SPH)—has been widely applied to different engineering fields. In the present research a general force potential-based particle method falling within the SPH framework for the mechanical simulation of granular and continuum materials under dynamic condition, is developed. The particle–particle and particle-boundary interaction is modeled through force functionals, tuned according to the nature of the material being analyzed (solid, granular, …). The proposed potential-based formulation allows the description of the forces existing between the discrete elements of generic materials. Thanks to the capability to deal with short and long distance actions (namely mechanical and/or electrostatic), general force–deformation laws, etc. it allows a straightforward mechanical simulation of fine particles assemblies such as powders. The theoretical basis of the computational approach are presented and some examples involving powder motion and a continuum mechanical problem are illustrated and discussed.

Analysis of Beams and Thin Plates Using the Wavelet-Galerkin Method

The use of wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs), especially for problems with local high gradient. In this work, the Galerkin Method has been adapted for the direct solution of differential equations in a meshless formulation using Daubechies wavelets and Deslauriers-Dubuc interpolating functions (Interpolets). This approach takes advantage of wavelet properties like compact support, orthogonality and exact polynomial representation, which allow the use of a multiresolution analysis. Several examples based on typical differential equations for beams and thin plates were studied successfully.