Showing papers on "Meshfree methods published in 2012"

Second‐order accurate derivatives and integration schemes for meshfree methods

Abstract: SUMMARY The consistency condition for the nodal derivatives in traditional meshfree Galerkin methods is only the differentiation of the approximation consistency (DAC). One missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms. In this paper, a consistency framework for the meshfree nodal derivatives including the DAC and the discrete divergence consistency (DDC) is proposed. The summation of the linear DDC over the whole computational domain leads to the so-called integration constraint in the literature. A three-point integration scheme using background triangle elements is developed, in which the corrected derivatives are computed by the satisfaction of the quadratic DDC. We prove that such smoothed derivatives also meet the quadratic DAC, and therefore, the proposed scheme possesses the quadratic consistency that leads to its name QC3. Numerical results show that QC3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability, convergence, accuracy, and efficiency among all the tested methods. Particularly, it shows a huge improvement for the existing linearly consistent one-point integration method in some examples. Copyright © 2012 John Wiley & Sons, Ltd.

On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods

Abstract: SUMMARY Meshfree methods (MMs) such as the element free Galerkin (EFG)method have gained popularity because of some advantages over other numerical methods such as the finite element method (FEM). A group of problems that have attracted a great deal of attention from the EFG method community includes the treatment of large deformations and dealing with strong discontinuities such as cracks. One efficient solution to model cracks is adding special enrichment functions to the standard shape functions such as extended FEM, within the FEM context, and the cracking particles method, based on EFG method. It is well known that explicit time integration in dynamic applications is conditionally stable. Furthermore, in enriched methods, the critical time step may tend to very small values leading to computationally expensive simulations. In this work, we study the stability of enriched MMs and propose two mass-lumping strategies. Then we show that the critical time step for enriched MMs based on lumped mass matrices is of the same order as the critical time step of MMs without enrichment. Moreover, we show that, in contrast to extended FEM, even with a consistent mass matrix, the critical time step does not vanish even when the crack directly crosses a node. Copyright © 2011 John Wiley & Sons, Ltd.

Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions

Abstract: A comparison of the performance of the global and the local radial basis function collocation meshless methods for three dimensional parabolic partial differential equations is performed in the present paper. The methods are structured with multiquadrics radial basis functions. The time-stepping is performed in a fully explicit, fully implicit and Crank–Nicolson ways. Uniform and non-uniform node arrangements have been used. A three-dimensional diffusion–reaction equation is used for testing with the Dirichlet and mixed Dirichlet–Neumann boundary conditions. The global methods result in discretization matrices with the number of unknowns equal to the number of the nodes. The local methods are in the present paper based on seven-noded influence domains, and reduce to discretization matrices with seven unknowns for each node in case of the explicit methods or a sparse matrix with the dimension of the number of the nodes and seven non-zero row entries in case of the implicit method. The performance of the methods is assessed in terms of accuracy and efficiency. The outcome of the comparison is as follows. The local methods show superior efficiency and accuracy, especially for the problems with Dirichlet boundary conditions. Global methods are efficient and accurate only in cases with small amount of nodes. For large amount of nodes, they become inefficient and run into ill-conditioning problems. Local explicit method is very accurate, however, sensitive to the node position distribution, and becomes sensitive to the shape parameter of the radial basis functions when the mixed boundary conditions are used. Performance of the local implicit method is comparatively better than the others when a larger number of nodes and mixed boundary conditions are used. The paper represents an extension of our recently made similar study in two dimensions.

Radial basis functions methods for solving Fokker–Planck equation

Abstract: In this paper two numerical meshless methods for solving the Fokker–Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker–Planck equation by using collocation method are applied. The first is based on the Kansa's approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and Ne errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa's approach.

Numerical Analysis of Vibrations of Structures under Moving Inertial Load

Abstract: Introduction.- Analytical solutions.- Semi-analytical methods.- Review of numerical methods of solution.- Classical numerical methods of time integration.- Space-time finite element method.- Space-time finite elements and a moving load.- The Newmark method and a moving inertial load.- Meshfree methods in moving load problems.- Examples of applications.

Buckling analysis of sandwich plates with functionally graded skins using a new quasi-3D hyperbolic sine shear deformation theory and collocation with radial basis functions

Abstract: A hyperbolic sine shear deformation theory is used for the linear buckling analysis of functionally graded plates. The theory accounts for through-the-thickness deformations. The buckling governing equations and boundary conditions are derived using Carrera's Unified Formulation and further interpolated by collocation with radial basis functions. The collocation method is truly meshless, allowing a fast and simple discretization of equations in the domain and on the boundary. A numerical investigation has been conducted considering and neglecting the thickness stretching effects on the buckling of sandwich plates with functionally graded skins. Numerical results demonstrate the high accuracy of the present approach.

Novel adaptive meshfree integration techniques in meshless methods

Abstract: SUMMARY We propose two strategies of novel adaptive numerical integration based on mapping techniques for solving the complicated problems of domain integration encountered in meshfree methods. Several mapping methods are presented in detail that map a complex integration domain to much simpler ones, for example, squares, triangles or circles. The techniques described in the paper can be applied to both global and local weak forms, and the highly nonlinear meshfree integrands are evaluated with controlled accuracy. The necessity of the clumsy procedure of background mesh or cell structures used for integration purpose in existing meshfree methods is avoided, and many meshfree methods that require the domain integration can now become ‘truly meshfree’. Various numerical examples in two dimensions are considered to demonstrate the applicability and the effectiveness of the proposed methods and it shows that the accuracy is improved significantly. Their obtained results are compared with analytical solutions and other approaches and very good agreements are found. Additionally, some three-dimensional cases applied by the present methods are also examined. Copyright © 2012 John Wiley & Sons, Ltd.

Recent developments of the meshless radial point interpolation method for time-domain electromagnetics

Abstract: Meshless methods are a promising new field in computational electromagnetics. Instead of relying on an explicit mesh topology, a numerical solution is computed on an unstructured set of collocation nodes. This allows to model fine geometrical details with high accuracy and facilitates the adaptation of node distributions for optimization or refinement purposes. The radial point interpolation method (RPIM) is a meshless method based on radial basis functions. In this paper, the current state of the RPIM in electromagnetics is reviewed. The localized RPIM scheme is summarized, and the interpolation accuracy is discussed in dependence of important parameters. A time-domain implementation is presented, and important time iteration aspects are reviewed. New formulations for perfectly matched layers and waveguide ports are introduced. An unconditionally stable RPIM scheme is summarized, and its advantages for hybridization with the classical RPIM scheme are discussed in a practical example. The capabilities of an adaptive time-domain refinement strategy based on the experiences on a frequency-domain solver are discussed. Copyright © 2012 John Wiley & Sons, Ltd.

A new weight‐function enrichment in meshless methods for multiple cracks in linear elasticity

Abstract: SUMMARY A new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions. The main novelty consists in modifying the weight function with an intrinsic enrichment which is discontinuous over the finite length of the crack, represented by a segment, but continuous all around the crack tips. An analytical function is used to introduce discontinuities that are incorporated in the kernel in a simple, multiplicative manner. The resulting method allows a more straightforward implementation and simulation of the presence of multiple cracks, crack branching and crack propagation in a meshless framework without using any of the existing algorithms such as visibility, transparency, and diffraction and without using additional unknowns and additional equations for the evolution of the level-sets, as in extrinsic partition of unity-based methods. Stress intensity factors calculated using the J-integral demonstrate excellent agreement with analytical solutions for classical fracture mechanics benchmarks. Copyright © 2011 John Wiley & Sons, Ltd.

On error control in the element-free Galerkin method

Abstract: The paper investigates discretisation error control in the element-free Galerkin method (EFGM) highlighting the differences from the finite element method (FEM). We demonstrate that the (now) conventional procedures for error analysis used in the finite element method require careful application in the EFGM, otherwise competing sources of error work against each other. Examples are provided of previous works in which adopting an FEM-based approach leads to dubious refinements. The discretisation error is here split into contributions arising from an inadequate number of degrees of freedom eh, and from an inadequate basis ep. Numerical studies given in this paper show that for the EFGM the error cannot be easily split into these component parts. Furthermore, we note that arbitrarily setting the size of nodal supports (as is commonly proposed in many papers) causes severe difficulties in terms of error control and solution accuracy. While no solutions to this problem are presented in this paper it is important to highlight these difficulties in applying an approach to errors from the FEM in the EFGM. While numerical tests are performed only for the EFGM, the conclusions are applicable to other meshless methods based on the concept of nodal support.

The Meshless Local Petrov–Galerkin Method in Two-Dimensional Electromagnetic Wave Analysis

Abstract: This paper deals with one member of the class of meshless methods, namely the Meshless Local Petrov-Galerkin (MLPG) method, and explores its application to boundary-value problems arising in the analysis of two-dimensional electromagnetic wave propagation and scattering. This method shows some similitude with the widespread finite element method (FEM), like the discretization of weak forms and sparse global matrices. MLPG and FEM differ in what regards the construction of an unstructured mesh. In MLPG, there is no mesh, just a cloud of nodes without connection to each other spread throughout the domain. The suppression of the mesh is counterbalanced by the use of special shape functions, constructed numerically. This paper illustrates how to apply MLPG to wave scattering problems through a number of cases, in which the results are compared either to analytical solutions or to those provided by other numerical methods.

Comparison of local weak and strong form meshless methods for 2-D diffusion equation

Abstract: A comparison between weak form meshless local Petrov–Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.

Boundary particle method for Cauchy inhomogeneous potential problems

Abstract: This article makes the first attempt to apply the boundary particle method (BPM) to the solution of Cauchy inhomogeneous potential problems. Unlike the other boundary discretization meshless methods, the BPM does not require any inner nodes to evaluate the particular solution, since the method uses the recursive composite multiple reciprocity technique to reduce an inhomogeneous problem to a series of higher order homogeneous problems. Thanks to its truly boundary-only meshless merit, the BPM is particularly attractive to solve inverse problems. In this study, the inner source term, i.e., right-hand side of Poisson equation, is of polynomial, exponential and trigonometric functions, or a combination of these functions, which frequently appear in inverse engineering problems. This article investigates numerical convergence and stability of the BPM in conjunction with the truncated singular value decomposition regularization technique, and presents sensitivity analysis with respect to the ratio parameter of...

A fast object-oriented Matlab implementation of the Reproducing Kernel Particle Method

Abstract: Novel numerical methods, known as Meshless Methods or Meshfree Methods and, in a wider perspective, Partition of Unity Methods, promise to overcome most of disadvantages of the traditional finite element techniques. The absence of a mesh makes meshfree methods very attractive for those problems involving large deformations, moving boundaries and crack propagation. However, meshfree methods still have significant limitations that prevent their acceptance among researchers and engineers, namely the computational costs. This paper presents an in-depth analysis of computational techniques to speed-up the computation of the shape functions in the Reproducing Kernel Particle Method and Moving Least Squares, with particular focus on their bottlenecks, like the neighbour search, the inversion of the moment matrix and the assembly of the stiffness matrix. The paper presents numerous computational solutions aimed at a considerable reduction of the computational times: the use of kd-trees for the neighbour search, sparse indexing of the nodes-points connectivity and, most importantly, the explicit and vectorized inversion of the moment matrix without using loops and numerical routines.

Staggered meshless solid-fluid coupling

Abstract: Simulating solid-fluid coupling with the classical meshless methods is an difficult issue due to the lack of the Kronecker delta property of the shape functions when enforcing the essential boundary conditions. In this work, we present a novel staggered meshless method to overcome this problem. We create a set of staggered particles from the original particles in each time step by mapping the mass and momentum onto these staggered particles, aiming to stagger the velocity field from the pressure field. Based on this arrangement, an new approximate projection method is proposed to enforce divergence-free on the fluid velocity with compatible boundary conditions. In the simulations, the method handles the fluid and solid in a unified meshless manner and generalizes the formulations for computing the viscous and pressure forces. To enhance the robustness of the algorithm, we further propose a new framework to handle the degeneration case in the solid-fluid coupling, which guarantees stability of the simulation. The proposed method offers the benefit that various slip boundary conditions can be easily implemented. Besides, explicit collision handling for the fluid and solid is avoided. The method is easy to implement and can be extended from the standard SPH algorithm in a straightforward manner. The paper also illustrates both one-way and two-way couplings of the fluids and rigid bodies using several test cases in two and three dimensions.

A meshfree-enriched finite element method for compressible and near-incompressible elasticity

Abstract: SUMMARY In this paper, a two-dimensional displacement-based meshfree-enriched FEM (ME-FEM) is presented for the linear analysis of compressible and near-incompressible planar elasticity. The ME-FEM element is established by injecting a first-order convex meshfree approximation into a low-order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker-delta property on the element boundaries. The gradient matrix of ME-FEM element satisfies the integration constraint for nodal integration and the resultant ME-FEM formulation is shown to pass the constant stress test for the compressible media. The ME-FEM interpolation is an element-wise meshfree interpolation and is proven to be discrete divergence-free in the incompressible limit. To prevent possible pressure oscillation in the near-incompressible problems, an area-weighted strain smoothing scheme incorporated with the divergence-free ME-FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near-incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.

A Novel Meshless Scheme for Solving Surface Integral Equations With Flat Integral Domains

Abstract: Numerical solutions for electromagnetic (EM) integral equations rely on the discretization of integral domains and the use of meshes for geometric description. Meshing geometries is very tedious, especially for complicated structures with many details (tiny parts) and geometric discontinuities (corners or edges), and remeshing could be required in many scenarios. To reduce the costs of generating quality meshes, meshless or mesh-free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in EM community. The meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally. In this work, we first address the traditional meshless scheme for solving EM integral equations based on the moving least square (MLS) approximation for unknown currents and the use of background meshes for integrations, and then develop a novel meshless scheme by applying the Green's lemma to the EM surface integral equations with flat domains. The novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain. Since only the domain boundaries are discretized and no background meshes are needed, the scheme is truly meshless. Numerical examples for EM scattering by flat-surface objects are presented to demonstrate the effectiveness of the novel scheme.

Numerical integration of multi-dimensional highly oscillatory, gentle oscillatory and non-oscillatory integrands based on wavelets and radial basis functions

Abstract: In this paper Haar wavelets (HWs), hybrid functions (HFs) and radial basis functions (RBFs) are used for the numerical solution of multi-dimensional mild, highly oscillatory and non-oscillatory integrals. Part of this study is extension of our earlier work [9] , [47] to multi-dimensional oscillatory and non-oscillatory integrals. Second part of the study is focused on coupling Levin's approach [30] with meshless methods. In first part of the paper, application of the numerical algorithms based on HWs and HFs [9] is extended to integrals having a varying oscillatory and non-oscillatory integrands defined on circular and rectangular domains. In second part of the paper, we propose a meshless method based on multiquadric (MQ) RBF for highly oscillatory multi-dimensional integrals. The first approach is directly related to numerical quadrature with wavelets basis. Like classical numerical quadrature, this approach does not need any intermediate numerical technique. The second approach based on meshless method of Levin's type converts numerical integration problem to a partial differential equation (PDE) and subsequently finding numerical solution of the PDE by a meshless method. The computational algorithms thus derived are tested on a number of benchmark kernel functions having varying oscillatory character or integrands with critical points at the origin. The novel methods are compared with the existing methods as well. Accuracy of the methods is measured in terms of absolute and relative errors. The new methods are simple, more efficient and numerically stable. Theoretical and numerical convergence analysis of the HWs and HFs is also given.

Mixed discrete least squares meshless method for planar elasticity problems using regular and irregular nodal distributions

Abstract: A Mixed Discrete Least Square Meshless (MDLSM) method is proposed for the solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet-type which is easily incorporated via a penalty method. Because least squares based algorithm of MDLSM method, the proposed method does not need to be satisfied by the LBB condition. The performance of the proposed method is tested on a benchmark example from theory of elasticity namely the problem of infinite plate with a circular hole and the results are presented and compared with those of the analytical solution and the solutions obtained using the irreducible DLSM formulation. The results indicate that the proposed MDLSM method is more accurate than the DLSM method. The results show that the numerical solutions of the MDLSM method can be obtained with lower computational cost and with higher accuracy. Also its performance is marginally affected by the irregularity of the nodal distribution.