Showing papers on "Meshfree methods published in 2010"

A theoretical study on the smoothed FEM (S‐FEM) models: Properties, accuracy and convergence rates

Abstract: Incorporating the strain smoothing technique of meshfree methods into the standard finite element method (FEM), Liu et al. have recently proposed a series of smoothed finite element methods (S-FEM) for solid mechanics problems. In these S-FEM models, the compatible strain fields are smoothed based on smoothing domains associated with entities of elements such as elements, nodes, edges or faces, and the smoothed Galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix. We present in this paper a general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S-FEM models. First, an assumed strain field derived from the Hellinger–Reissner variational principle is shown to be identical to the smoothed strain field used in the S-FEM models. We then define a smoothing projection operator to modify the compatible strain field and show a set of properties. We next establish a general error bound of the S-FEM models. Some numerical examples are given to verify the theoretical properties established. Copyright © 2010 John Wiley & Sons, Ltd.

Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation

Abstract: In this paper a numerical approach based on the truly meshless methods is proposed to deal with the second-order two-space-dimensional telegraph equation. In the meshless local weak–strong (MLWS) method, our aim is to remove the background quadrature domains for integration as much as possible, and yet to obtain stable and accurate solution. The MLWS method is designed to combine the advantage of local weak and strong forms to avoid their shortcomings. In this method, the local Petrov–Galerkin weak form is applied only to the nodes on the Neumann boundary of the domain of the problem. The meshless collocation method, based on the strong form equation is applied to the interior nodes and the nodes on the Dirichlet boundary. To solve the telegraph equation using the MLWS method, the conventional moving least squares (MLS) approximation is exploited in order to interpolate the solution of the equation. A time stepping scheme is employed to approximate the time derivative. Another solution is also given by the meshless local Petrov-Galerkin (MLPG) method. The validity and efficiency of the two proposed methods are investigated and verified through several examples.

A meshless Total Lagrangian explicit dynamics algorithm for surgical simulation

Abstract: A method is presented for computing deformation of very soft tissue. The method is motivated by the need for simple, automatic model creation for real-time simulation. The method is meshless in the sense that deformation is calculated at nodes that are not part of an element mesh. Node placement is almost arbitrary. Fully geometrically nonlinear Total Lagrangian formulation is used. Geometric integration is performed over a regular background grid that does not conform to the simulation geometry. Explicit time integration is used via the central difference method. As an example the simple but fully nonlinear Neo-Hookean material model is employed. The results are compared with a finite element simulation to verify the usefulness of the method. Copyright © 2010 John Wiley & Sons, Ltd.

Meshless Methods in Solid Mechanics

Abstract: Finite element method has been the dominant technique in computational mechanics in the past decades, and it has made significant contributions to the developments in engineering and science. Nevertheless, finite element method is not well suited to problems having severe mesh distortion owing to extremely large deformations of materials, encountering moving discontinuities such as crack propagation along arbitrary and complex paths, involving considerable meshing and re-meshing in structural optimization problems, or having multidomain of influence in multi-phenomenon physical problems. It is impossible to completely overcome those mesh-related difficulties by a mesh-based method. The highly structured nature of finite element approximations imposes severe penalties in the solutions of those problems. Distinguishing with finite element, finite difference and finite volume methods, meshless method discretizes the continuum body only with a set of nodal points and the approximation is constructed entirely in terms of nodes. There is no need of mesh or elements in this method. It does not posses the mesh related difficulties, eliminates at least part of the FE structure, and provides an approach with more flexibility in the applications in engineering and science. The meshless method started to capture the interest of a broader community of researchers only several years ago, and now it becomes a growing and evolving field. It is showing that this is a very rich area to be explored, and has great promise for many very challenging computational problems. On the one hand, great developments on meshless methods have been achieved. On the other hand, there are many aspects of meshless methods that could be benefit from improvements. A broader community of researchers can bring divergent skills and backgrounds to bear on the task of improving this method. The main objective of this book is to provide a textbook for graduate courses on the computational analysis of continuum and solid mechanics based on meshless (also known as mesh free) methods. It can also be used as a reference book for engineers and scientists who are exploring the physical world through computer simulations. Emphasis of this book is given to the understanding of the physical and mathematical characteristics of the procedures of computational solid mechanics. It naturally brings the essence, advantages and challenging problems of meshless methods into the picture. The subjects in this book cover the fundamentals of continuum mechanics, the integral formulation methods of continuum problems, the basic concepts of finite element methods, and the methodologies, formulations, procedures, and applications of various meshless methods. It also provides general and detailed procedures of meshless analysis on elastostatics, elastodynamics, non-local continuum mechanics and plasticity with a large number of numerical examples. Some basic and important mathematical methods are included in the Appendixes. For the readers who want to gain knowledge through hands-on experience, the meshless programs for elastostatics and elastodynamics are also introduced in the book.

Solution of quasi-static large-strain problems by the material point method

Abstract: SUMMARY A procedure for solving quasi-static large-strain problems by the material point method is presented. Owing to the Lagrangian–Eulerian features of the method, problems associated with excessive mesh distortions that develop in the Lagrangian formulations of the finite element method are avoided. Threedimensional problems are solved utilizing 15-noded prismatic and 10-noded tetrahedral elements with quadratic interpolation functions as well as an implicit integration scheme. An algorithm for exploiting the numerical integration procedure on the computational mesh is proposed. Several numerical examples are shown. Copyright 2010 John Wiley & Sons, Ltd.

Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems

Abstract: A stabilization procedure for curing temporal instability of node-based smoothed finite element method (NS-FEM) is proposed for dynamic problems using linear triangular element. A stabilization term is added into the smoothed potential energy functional of the original NS-FEM, consisting of squared-residual of equilibrium equation. A gradient smoothing operation on second order derivatives is applied to relax the requirement of shape function, so that the squared-residual can be evaluated using linear elements. Numerical examples demonstrate that stabilization parameter can “tune” NS-FEM from being “overly soft” to “overly stiff”, so that eigenvalue solutions can be stabilized. Numerical tests provide an empirical value range of stabilization parameter, within which the stabilized NS-FEM can still produce upper bound solutions in strain energy to the exact solution of force-driven elastostatics problems, as well as lower bound natural frequencies for free vibration problems.

A new boundary meshfree method with distributed sources

Abstract: A new boundary meshfree method, to be called the boundary distributed source (BDS) method, is presented in this paper that is truly meshfree and easy to implement. The method is based on the same concept in the well-known method of fundamental solutions (MFS). However, in the BDS method the source points and collocation points coincide and both are placed on the boundary of the problem domain directly, unlike the traditional MFS that requires a fictitious boundary for placing the source points. To remove the singularities of the fundamental solutions, the concentrated point sources can be replaced by distributed sources over areas (for 2D problems) or volumes (for 3D problems) covering the source points. For Dirichlet boundary conditions, all the coefficients (either diagonal or off-diagonal) in the systems of equations can be determined analytically, leading to very simple implementation for this method. Methods to determine the diagonal coefficients for Neumann boundary conditions are discussed. Examples for 2D potential problems are presented to demonstrate the feasibility and accuracy of this new meshfree boundary-node method.

Modelling brain deformations for computer-integrated neurosurgery

Abstract: In this review paper we discuss Intelligent Systems for Medicine Laboratory's contributions to mathematical and numerical modelling of brain deformation behaviour for neurosurgical simulation and brain image registration. These processes can be reasonably described in purely mechanical terms, such as displacements, strains and stresses and therefore can be analysed using established methods of continuum mechanics. We advocate the use of fully non-linear theory of continuum mechanics. We discuss in some detail modelling geometry, boundary conditions, loading and material properties. We consider numerical problems such as the use of hexahedral and mixed hexahedral–tetrahedral meshes as well as meshless spatial discretization schemes. We advocate the use of total Lagrangian formulation of both finite element and meshless methods together with explicit time-stepping procedures. We support our recommendations and conclusions with two examples: computation of the reaction force acting on a biopsy needle, and computation of the brain shift for image registration. Copyright © 2009 John Wiley & Sons, Ltd.

A new method for meshless integration in 2D and 3D Galerkin meshfree methods

Abstract: A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.

Eigenvalue Analysis and Longtime Stability of Resonant Structures for the Meshless Radial Point Interpolation Method in Time Domain

Abstract: A meshless collocation method based on radial basis function (RBF) interpolation is presented for the numerical solution of Maxwell's equations. RBFs have attractive properties such as theoretical exponential convergence for increasingly dense node distributions. Although the primary interest resides in the time domain, an eigenvalue solver is used in this paper to investigate convergence properties of the RBF interpolation method. The eigenvalue distribution is calculated and its implications for longtime stability in time-domain simulations are established. It is found that eigenvalues with small, but nonzero, real parts are related to the instabilities observed in time-domain simulations after a large number of time steps. Investigations show that by using global basis functions, this problem can be avoided. More generally, the connection between the high matrix condition number, accuracy, and the magnitude of nonzero real parts is established.

On the optimum support size in meshfree methods: a variational adaptivity approach with maximum-entropy approximants

Abstract: We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle. The method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. We consider local maximum-entropy approximation schemes, which exhibit smooth basis functions with respect to both space and the discretization parameters (the node location and the locality parameters). We illustrate by the Poisson, linear and non-linear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions. Copyright © 2009 John Wiley & Sons, Ltd.

Dynamic fracture with meshfree enriched XFEM

Abstract: The enrichment of the extended finite element method (XFEM) by meshfree approximations is studied. The XFEM allows for modeling arbitrary discontinuities, but with low order elements the accuracy often needs improvement. Here, the meshfree approximation is used as an enrichment in a cluster of nodes about the crack tip to improve accuracy. Several numerical examples show that this leads to more accuracy for stress intensity factors computations, and to the capability to capture the branching point of a propagating crack from the stresses.

Free vibration analysis of thin plates using Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration

Abstract: A Hermite reproducing kernel (HRK) Galerkin meshfree formulation is presented for free vibration analysis of thin plates. In the HRK approximation the plate deflection is approximated by the deflection as well as slope nodal variables. The nth order reproducing conditions are imposed simultaneously on both the deflectional and rotational degrees of freedom. The resulting meshfree shape function turns out to have a much smaller necessary support size than its standard reproducing kernel counterpart. Obviously this reduction of minimum support size will accelerate the computation of meshfree shape function. To meet the bending exactness in the static sense and to remain the spatial stability the domain integration for stiffness as well as mass matrix is consistently carried out by using the sub-domain stabilized conforming integration (SSCI). Subsequently the proposed formulation is applied to study the free vibration of various benchmark thin plate problems. Numerical results uniformly reveal that the present method produces favorable solutions compared to those given by the high order Gauss integration (GI)-based Galerkin meshfree formulation. Moreover the effect of sub-domain refinement for the domain integration is also investigated.

Assessment of smoothed point interpolation methods for elastic mechanics

Abstract: A generalized gradient smoothing technique and a smoothed bilinear form of Galerkin weak form have been recently proposed by Liu et al. to create a wide class of efficient smoothed point interpolation methods (PIMs) using the background mesh of triangular cells. In these methods, displacement fields are constructed by polynomial or radial basis shape functions and strains are smoothed over the smoothed domain associated with the nodes or the edges of the triangular cells. This paper summarizes and assesses bound property, convergence rate and computational efficiency for these methods. It is found that: (1) the incorporation of the PIMs with the node-based strain smoothing operation allows us to obtain an upper bound to the exact solution in the strain energy; (2) the incorporation of the PIMs with the edge-based strain smoothing operation using triangular background mesh can produce a solution of ‘ultra-accuracy’ and ‘super-convergence’; (3) the edge-based strain smoothing operation together with the linear interpolation can provide better computational efficiency compared with other smoothed PIMs and the finite element method when the same triangular mesh is used. These conclusions have been examined and confirmed by intensive examples. Copyright © 2009 John Wiley & Sons, Ltd.

Unsymmetric meshless methods for operator equations

Abstract: A general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well-posed linear operator equations is presented. The results provide optimal convergence rates, if the test and trial spaces satisfy a stability condition. Operators need not be elliptic, and the problems can be posed in weak or strong form without changing the theory. Non-stationary kernel-based trial and test spaces are shown to fit into the framework, disregarding the operator equation. As a special case, unsymmetric meshless kernel-based methods solving weakly posed problems with distributional data are treated in some detail. This provides a foundation of certain variations of the “Meshless Local Petrov-Galerkin” technique of S.N. Atluri and collaborators.

A meshless method based on RBFs method for nonhomogeneous backward heat conduction problem

Abstract: Based on the idea of radial basis functions approximation and the method of particular solutions, we develop in this paper a new meshless computational method to solve nonhomogeneous backward heat conduction problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several benchmark problems in both two- and three-dimensions. Numerical results indicate that this novel approach can achieve an efficient and accurate solution even when the final temperature data is almost undetectable or disturbed with large noises. It has also been shown that the proposed method is stable to recover the unknown initial temperature from scattered final temperature data.

Limit analysis of plates and slabs using a meshless equilibrium formulation

Abstract: A meshless Element-Free Galerkin (EFG) equilibrium formulation is proposed to compute the limit loads which can be sustained by plates and slabs. In the formulation pure moment fields are approximated using a moving least-squares technique, which means that the resulting fields are smooth over the entire problem domain. There is therefore no need to enforce continuity conditions at interfaces within the problem domain, which would be a key part of a comparable finite element formulation. The collocation method is used to enforce the strong form of the equilibrium equations and a stabilized conforming nodal integration scheme is introduced to eliminate numerical instability problems. The combination of the collocation method and the smoothing technique means that equilibrium only needs to be enforced at the nodes, and stable and accurate solutions can be obtained with minimal computational effort. The von Mises and Nielsen yield criteria which are used in the analysis of plates and slabs, respectively, are enforced by introducing second-order cone constraints, ensuring that the resulting optimization problem can be solved using efficient interior-point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plate and slab problems.

The Comparison of Three Meshless Methods Using Radial Basis Functions for Solving Fourth-Order Partial Differential Equations

Abstract: In this paper we apply the newly developed method of particular solutions (MPS) and one-stage method of fundamental solutions (MFS-MPS) for solving fourth-order partial differential equations. We also compare the numerical results of these two methods to the popular Kansa's method. Numerical results in the 2D and the 3D show that the MFS-MPS outperformed the MPS and Kansa's method. However, the MPS and Kansa's method are easier in terms of implementation.