Showing papers on "Meshfree methods published in 2008"
TL;DR: In this paper, the authors present an analysis of the quadrature errors found in the computation of (material) internal force in MPM and use this analysis to direct proposed improvements.
Abstract: SUMMARY The Material Point Method (MPM) has demonstrated itself as a computationally effective particle method for solving solid mechanics problems involving large deformations and/or fragmentation of structures which are sometimes problematic for finite element methods. However, like most methods which employ mixed Lagrangian (particle) and Eulerian strategies, analysis of the method is not straightforward. The lack of an analysis framework for MPM, as is found in finite element methods, makes it challenging to explain anomalies found in its employment and makes it difficult to propose methodology improvements with predictable outcomes. In this paper we present an analysis of the quadrature errors found in the computation of (material) internal force in MPM and use this analysis to direct proposed improvements. In particular, we demonstrate that lack of regularity in the grid functions used for representing the solution of the equations of motion can hamper spatial convergence of the method. We propose the use of a quadratic B-spline basis for representing solutions on the grid, and we demonstrate computationally and explain theoretically why such a small change can have significant impact on the reduction of the internal force quadrature error (and corresponding “grid crossing error”) often experienced when using the material point method. Copyright c 2008 John Wiley & Sons, Ltd.
214 citations
TL;DR: In this article, a simple methodology to model shear bands as strong displacement discontinuities in an adaptive mesh-free method is presented, where the shear band is represented by a displacement jump at discrete particle positions.
157 citations
15 Jun 2008
TL;DR: The mathematical background of the radial point interpolation method and a two-dimensional implementation are presented and it is shown that solutions converge much faster using the ability of conformal modeling compared to a similar analysis in rectangular grids.
Abstract: A meshless numerical technique based on radial point interpolation is introduced for electromagnetic simulations in time domain. The general class of meshless methods presents very attractive properties for addressing future challenges of electromagnetic modeling. Among the interesting aspects, the ability to handle arbitrary node distributions for conformal and multi-scale modeling can be mentioned first. Furthermore, the possibility of modifying the node distribution dynamically opens new perspectives for adaptive computations and optimization. The mathematical background of the radial point interpolation method and a two-dimensional implementation are presented here. The advantages of this meshless method are discussed and applied to a model consisting of a 90 degree H-plane waveguide bend. It is shown that solutions converge much faster using the ability of conformal modeling compared to a similar analysis in rectangular grids.
80 citations
TL;DR: In this paper, the boundary conditions necessary to match the exact solution are not followed, and the effectivity of adaptive procedures is compromised as a test problem for adaptive procedures as the perfect refined mesh is uniform.
69 citations
TL;DR: In this paper, a mesh-less approach based on the smoothed particle hydrodynamics (SPH) method is proposed for the simulation of shell fracture under impact, which relies on an entirely meshless approach.
Abstract: This paper introduces a new modeling method suitable for the simulation of shell fracture under impact. This method relies on an entirely meshless approach based on the smoothed particle hydrodynamics (SPH) method. The paper also presents the SPH shell formulation being used as well as the different test cases used for its validation. A plasticity model of the global type throughout the thickness is also proposed and validated. Finally, in order to illustrate the capabilities of the method, fracture simulations using a simplified fracture criterion are presented.
69 citations
TL;DR: In this article, a simple classical radial basis functions (RBFs) collocation (Kansa) method was proposed for numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers' equations, and quasi-nonlinear hyperbolic equations.
Abstract: This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers’ equations, and quasi-nonlinear hyperbolic equations. Contrary to the mesh oriented methods such as the finite-difference and finite element methods, the new technique does not require mesh to discretize the problem domain, and a set of scattered nodes provided by initial data is required for realization of solution of the problem. Accuracy of the method is assessed in terms of the error norms L 2 , L ∞ , number of nodes in the domain of influence, time step length, parameter free and parameter dependent RBFs. Numerical experiments are performed to demonstrate the accuracy and robustness of the method for the three classes of partial differential equations (PDEs).
67 citations
TL;DR: In this paper, a new concurrent simulation technique was developed to couple the meshfree method with the finite element method (FEM) for the analysis of crack tip fields, which can take advantage of both the mesh-free method and FEM but at the same time can overcome their shortcomings.
65 citations
TL;DR: In this article, the numerical simulation of the 3D incompressible Euler equations is analyzed with respect to different integration methods, including spectral methods with different strategies for dealiasing and two variants of finite difference methods.
62 citations
TL;DR: In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered, and the radial basis functions are used for finding an approximation of the solution of the present problem.
Abstract: Nonlocal mathematical models appear in various problems of physics and engineering. In these models the integral term may appear in the boundary conditions. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. These kinds of problems have certainly been one of the fastest growing areas in various application fields. The presence of an integral term in a boundary condition can greatly complicate the application of standard numerical techniques. As a well-known class of meshless methods, the radial basis functions are used for finding an approximation of the solution of the present problem. Numerical examples are given at the end of the paper to compare the efficiency of the radial basis functions with famous finite-difference methods. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008
54 citations
TL;DR: The renormalized meshfree derivatives are studied, which improve the consistency of the original weighted particle methods, and a hybrid particle scheme is built using the Godunov method and is numerically compared to the weak renormalization scheme.
Abstract: This paper is devoted to the study of a new kind of meshfree scheme based on a new class of meshfree derivatives: the renormalized meshfree derivatives, which improve the consistency of the original weighted particle methods. The weak renormalized meshfree scheme, built from the weak formulation of general conservation laws, turns out to be $L^2$ stable under some geometrical conditions on the distribution of particles and some regularity conditions of the transport field. A time discretization is then performed by analogy with finite volume methods, and the $L^1$, $L^\infty$, and $BV$ stabilities of the obtained time discretized scheme are studied. From the same analogy with finite volume methods, a hybrid particle scheme is built using the Godunov method and is numerically compared to the weak renormalized scheme.
54 citations
TL;DR: In this paper, a simplified finite difference interpolation (SFDI) scheme was proposed for the MLPG-R method, which does not need matrix inverse and consumes less CPU time to evaluate.
Abstract: In the MLPG{\_}R (Meshless Local Petrove-Galerkin based on Rankine source solution) method, one needs a meshless interpolation scheme for an unknown function to discretise the governing equation. The MLS (moving least square) method has been used for this purpose so far. The MLS method requires inverse of matrix or solution of a linear algebraic system and so is quite time-consuming. In this paper, a new scheme, called simplified finite difference interpolation (SFDI), is devised. This scheme is generally as accurate as the MLS method but does not need matrix inverse and consume less CPU time to evaluate. Although this scheme is purposely developed for the MLPG{\_}R method, it may also be used for other meshless methods.
TL;DR: This approach has an edge over the traditional methods such as finite-difference and finite-element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method.
Abstract: This paper formulates a meshfree radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the Korteweg-de Vries (KdV) equation. The accuracy of the method is assessed in terms of the errors in L∞, L2 and root mean square (RMS), number of nodes in the domain of influence, parameter-dependent RBFs time and spatial steps length. This approach has an edge over the traditional methods such as finite-difference and finite-element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. Numerical experiments demonstrate the accuracy and robustness of the method when applied to complicated nonlinear partial differential equations. In this work, three test problems are studied.
TL;DR: It is shown that results computed with the MSPH method for the Noh problem agree well with its analytical solution, and the computed solution is found to agree very well with those obtained by analyzing axisymmetric and 3-D transient deformations of the rod with the commercial code LS-DYNA.
TL;DR: In this paper, a finite point method (FPM) is developed and adopted for solving the chloride diffusion equation for prediction of service life of concrete structures and initiation time of corrosion of reinforcements.
TL;DR: In this paper, a posteriori error estimate and adaptive refinement strategy are developed in conjunction with the collocated discrete least-squares (CDLS) meshless method, which is shown to be naturally related to the least squares functional, providing a suitable posterior measure of the error in the solution.
Abstract: Meshless methods are new approaches for solving partial differential equations. The main characteristic of all these methods is that they do not require the traditional mesh to construct a numerical formulation. They require node generation instead of mesh generation. In other words, there is no pre-specified connectivity or relationships among the nodes. This characteristic make these methods powerful. For example, an adaptive process which requires high computational effort in mesh-dependent methods can be very economically solved with meshless methods. In this paper, a posteriori error estimate and adaptive refinement strategy is developed in conjunction with the collocated discrete least-squares (CDLS) meshless method. For this, an error estimate is first developed for a CDLS meshless method. The proposed error estimator is shown to be naturally related to the least-squares functional, providing a suitable posterior measure of the error in the solution. A mesh moving strategy is then used to displace the nodal points such that the errors are evenly distributed in the solution domain. Efficiency and effectiveness of the proposed error estimator and adaptive refinement process are tested against two hyperbolic benchmark problems, one with shocked and the other with low gradient smooth solutions. These experiments show that the proposed adaptive process is capable of producing stable and accurate results for the difficult problems considered. Copyright © 2007 John Wiley & Sons, Ltd.
TL;DR: The FE-LSPIM QUAD4 element as discussed by the authors uses new shape functions that combine the meshfree and finite element shape functions so as to synergize the individual strengths of mesh free and finite elements methods.
TL;DR: In this paper, the authors address the problem of numerically simulating the Friction Stir Welding (FSW) process using the Natural Element Method (NEM) and present some interesting characteristics such as the ease of imposition of essential boundary conditions and coupling with FEM codes.
Abstract: In this work we address the problem of numerically simulating the Friction Stir Welding process. Due to the special characteristics of this welding method (i.e., high speed of the rotating pin, very large deformations, etc.) finite element methods (FEM) encounter several difficulties. While Lagrangian simulations suffer from mesh distortion, Eulerian or Arbitrary Lagrangian Eulerian (ALE) ones still have difficulties due to the treatment of convective terms, the treatment of the advancing pin, and many others. Meshless methods somewhat alleviate these problems, allowing for an updated Lagrangian framework in the simulation. Accuracy is not affected by mesh distortion (and hence the name meshless), but the price to pay is the computational cost, higher than in the FEM. The method used here, the Natural Element Method (NEM), presents some interesting characteristics, such as the ease of imposition of essential boundary conditions and coupling with FEM codes. Even more, since the method is formulated in a Lagrangian setting, it is possible to track the evolution of any material point during the process and also to simulate the Friction Stir Welding (FSW) of two slabs of different materials. The examples shown in this paper cover some of the difficulties related with the simulation of the FSW process: very large deformations, complex nonlinear and strongly coupled thermomechanical behaviour of the material and mixing of different materials.
TL;DR: Two Galerkin-based meshless methods to determine the light exitance on the surface of the diffusive tissue using moving least squares approximation, which can simplify the processing of boundary conditions in comparison with the finite element method.
Abstract: As an important small animal imaging technique, optical imaging has attracted increasing attention in recent years. However, the photon propagation process is extremely complicated for highly scattering property of the biological tissue. Furthermore, the light transport simulation in tissue has a significant influence on inverse source reconstruction. In this contribution, we present two Galerkin-based meshless methods (GBMM) to determine the light exitance on the surface of the diffusive tissue. The two methods are both based on moving least squares (MLS) approximation which requires only a series of nodes in the region of interest, so complicated meshing task can be avoided compared with the finite element method (FEM). Moreover, MLS shape functions are further modified to satisfy the delta function property in one method, which can simplify the processing of boundary conditions in comparison with the other. Finally, the performance of the proposed methods is demonstrated with numerical and physical phantom experiments.
TL;DR: The scaling and wavelet functions of the DB wavelet are used as basis functions to approximate the unknown field functions, so there is no need to construct costly shape functions as in the finite element method (FEM) and other meshless methods.
TL;DR: In this paper, a residual based error estimator using radial basis functions (RBFs) is proposed to evaluate the residual in the strong-form governing equation in the local domain through direct integration.
TL;DR: The accuracy already obtained with this meshless implementation of the Cell Method makes it a good candidate for some clinical applications, especially considering the full automation of the method, which does not require any data pre-processing.
TL;DR: In this article, the application of the element-free Galerkin method to the study of 2D electromagnetic-wave scattering problems was discussed and validated by comparison with a standard finite-element approach, focusing attention on three kinds of model problems.
Abstract: This paper presents and discusses the application of the element-free Galerkin method to the study of 2-D electromagnetic-wave scattering problems. The proposed formulation is validated by comparison with a standard finite-element approach, focusing attention on three kinds of model problems, which give rise to localized steep gradients.
TL;DR: In this paper, a numerical scheme based on the mesh-free plane wave method applied to inverse boundary value problems associated with Helmholtz-type equations is investigated, and the resulting ill-conditioned system of linear algebraic equations is solved in a stable manner by employing the truncated singular value decomposition, while the optimal truncation number, i.e., the regularization parameter, is determined using the L-curve criterion.
Abstract: In this paper, a numerical scheme based on the meshfree plane wave method applied to inverse boundary value problems associated with Helmholtz-type equations is investigated. The resulting ill-conditioned system of linear algebraic equations is solved in a stable manner by employing the truncated singular value decomposition, while the optimal truncation number, i.e. the regularization parameter, is determined using the L-curve criterion. Numerical results are presented for two- and three-dimensional problems in smooth and piecewise smooth geometries, with both exact and noisy data. The accuracy, convergence and stability of the numerical method are analysed and, furthermore, a comparison with other meshless methods is also performed.
TL;DR: A Kruskov technique is used to obtain an norm comparison between the approximate solution and the solution of a regularized conservation law, the pseudoviscous problem, and establishes a convergence rate of the renormalized meshfree numerical scheme applied to scalar conservation laws.
Abstract: We establish a convergence rate of the renormalized meshfree numerical scheme applied to scalar conservation laws. Renormalization is a tool introduced in order to eliminate the smoothed particle hydrodynamics (SPH) lack of consistency. A conservative scheme, the weak renormalized scheme, is derived from the general conservation laws weak formulation and is time discretized by using finite volume techniques. Because of the new form of the derivative approximation, the convergence proof of the classical SPH method cannot be applied. Thus, we use a Kruskov technique to obtain an $L^1$ norm comparison between the approximate solution and the solution of a regularized conservation law, the pseudoviscous problem.
TL;DR: In this article, local integral equations (LIE) are derived for numerical solution of 3D problems in linear elasticity of functionally graded materials (FGMs) viewed as 2-D axisymmetric problems.
Abstract: Local integral equations (LIE) are derived for numerical solution of 3-D problems in linear elasticity of functionally graded materials (FGMs) viewed as 2-D axisymmetric problems. Two types of the LIEs are considered with three different kinds of approximation of displacements based on: (i) standard finite elements; (ii) point interpolation method, and (iii) moving least-square approximation. The use of the last two implementations offers the possibility to develop meshless methods. In numerical experiments, the convergence and accuracy of these methods are investigated using the exact solution for a hollow cylinder with power-law gradation of Young's modulus in the radial direction and subjected to internal pressure as the benchmark solution. The efficiency is assessed by comparison of CPU-times.
TL;DR: In this paper, an adaptive analysis procedure was proposed to obtain certified solutions of desired accuracy with bounds to the exact solution in energy norm for elasticity problems, and the results converged very fast.
TL;DR: In this article, a mesh-free IML method based on singular weights for the solution of partial differential equations is presented and a stable inverse is obtained when the vanishing regularization parameter is considered.
Abstract: This paper presents a meshfree Interpolating Moving Least Squares (IMLS) method based on singular weights for the solution of partial differential equations. Due to the to the specific singular choice of the weight functions, which is needed to guarantee the interpolation, there arises a problem of the finding the inverse of the singular matrix. We extend the perturbation technique originally presented in [5] to allow the correct evaluation of all necessary derivatives in interpolation points at a reasonable cost. The inverse is carried out using the regularized weight function. It turns out that a stable inverse is obtained when the vanishing regularization parameter is considered.
TL;DR: Some enrichment techniques for the modeling of heterogeneous media in the presence of singularities such as cracks which overcome long-standing problems associated with the assumption of local periodicity in traditional asymptotic homogenization methods are presented.
TL;DR: This paper presents an efficient scheme for the automatic construction of a direct splitting of a PPUM function space into the degrees of freedom suitable for the approximation of the Dirichlet data and the degree of freedom that remain for the approximation of the PDE by simple linear algebra.
Abstract: This paper is concerned with the treatment of essential boundary conditions in meshfree methods. In particular, we focus on the particle-partition of unity method (PPUM). However, the proposed technique is applicable to any partition of unity-based approach. We present an efficient scheme for the automatic construction of a direct splitting of a PPUM function space into the degrees of freedom suitable for the approximation of the Dirichlet data and the degrees of freedom that remain for the approximation of the PDE by simple linear algebra. Notably, our approach requires no restrictions on the distribution of the discretization points nor on the employed (local) approximation spaces. We attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces. Hence, the global splitting can be computed with (sub)linear complexity. Due to this direct splitting of the meshfree PPUM function space, we can implement a conforming local treatment of essential boundary data so that the realization of Dirichlet boundary values in the meshfree PPUM is straightforward. The presented approach yields an optimally convergent scheme, which is demonstrated by the presented numerical results.
TL;DR: In this article, the de Boor algorithm is extended to two or higher dimensions, thus obtaining a new form of interpolation that can be used in a Galerkin framework to develop a new class of meshless methods.
Abstract: The problem of generalizing the natural element method (NEM) in terms of higher-order consistency and continuity is addressed here. It is done by means of the de Boor algorithm, the same employed to obtain B-splines by linear combinations of linear interpolants in one dimension. By noting that any form of natural neighbour interpolation can be considered as a suitable generalization of linear interpolation to two and higher dimensions, the de Boor algorithm is extended to two or higher dimensions, thus obtaining a new form of interpolation that can be used in a Galerkin framework to develop a new class of meshless methods.
This new class of meshless methods closely resembles the isogeometric analysis developed by Hughes et al. (Comput. Methods Appl. Mech. Eng. 2005; 194:4135–4195). However, unlike B-splines, the new class of interpolants does not rely on an underlying tensor-product quadrilateral mesh. It is based on the Delaunay triangulation of the cloud of knots and does not require any regularity on the connectivity.
In addition, the new method conserves many of the attractive features of the NEM, such as strict interpolation on the boundary, and thus directs imposition of essential boundary conditions. After a theoretical description of the proposed method, some numerical examples are shown to test its performance in the context of linear elastostatics. Copyright © 2007 John Wiley & Sons, Ltd.