Quadratically consistent nodal integration for second order meshfree Galerkin methods

Robust and efficient integration of the Galerkin weak form only at the approximation nodes for second order meshfree Galerkin methods is proposed. The starting point of the method is the Hu-Washizu variational principle. The orthogonality condition between stress and strain difference is satisfied by correcting nodal derivatives. The corrected nodal derivatives are essentially linear functions which can exactly reproduce linear strain fields. With the known area moments, the stiffness matrix resulting from these corrected nodal derivatives can be exactly evaluated using only the nodes as quadrature points. The proposed method can exactly pass the quadratic patch test and therefore is named as quadratically consistent nodal integration. In contrast, the stabilized conforming nodal integration (SCNI) which prevails in the nodal integrations for meshfree Galerkin methods fails to pass the quadratic patch test. Better accuracy, convergence, efficiency and stability than SCNI are demonstrated by several elastostatic and elastodynamic examples.     

A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition

In this article, we present a novel nodal integration scheme for meshfree Galerkin methods, which draws on the mathematical framework of the virtual element method. We adopt linear maximum-entropy basis functions for the discretization of field variables, although the proposed scheme is applicable to any linear meshfree approximant. In our approach, the weak form integrals are nodally integrated using nodal representative cells that carry the nodal displacements and state variables such as strains and stresses. The nodal integration is performed using the virtual element decomposition, wherein the bilinear form is decomposed into a consistency part and a stability part that ensure consistency and stability of the method. The performance of the proposed nodal integration scheme is assessed through benchmark problems in linear and nonlinear analyses of solids for small displacements and small-strain kinematics. Numerical results are presented for linear elastostatics and linear elastodynamics and viscoelasticity. We demonstrate that the proposed nodally integrated meshfree method is accurate, converges optimally, and is more reliable and robust than a standard cell-based Gauss integrated meshfree method.     

Novel adaptive meshfree integration techniques in meshless methods

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.     

Second‐order accurate derivatives and integration schemes for meshfree methods

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.     

Consistent and stable meshfree Galerkin methods using the virtual element decomposition

Over the past two decades, meshfree methods have undergone significant development as a numerical tool to solve partial differential equations (PDEs). In contrast to finite elements, the basis functions in meshfree methods are smooth (nonpolynomial functions), and they do not rely on an underlying mesh structure for their construction. These features render meshfree methods to be particularly appealing for higher‐order PDEs and for large deformation simulations of solid continua. However, a deficiency that still persists in meshfree Galerkin methods is the inaccuracies in numerical integration, which affects the consistency and stability of the method. Several previous contributions have tackled the issue of integration errors with an eye on consistency, but without explicitly ensuring stability. In this paper, we draw on the recently proposed virtual element method, to present a formulation that guarantees both the consistency and stability of the approximate bilinear form. We adopt maximum‐entropy meshfree basis functions, but other meshfree basis functions can also be used within this framework. Numerical results for several two‐dimensional and three‐dimensional elliptic (Poisson and linear elastostatic) boundary‐value problems that demonstrate the effectiveness of the proposed formulation are presented. Copyright © 2017 John Wiley & Sons, Ltd.     

kFEM: Adaptive meshfree finite‐element methods using local kernels on arbitrary subdomains

We propose a novel finite‐element method for polygonal meshes. The resulting scheme is hp‐adaptive, where h and p are a measure of, respectively, the size and the number of degrees of freedom of each polygon. Moreover, it is locally meshfree, since it is possible to arbitrarily choose the locations of the degrees of freedom inside each polygon. Our construction is based on nodal kernel functions, whose support consists of all polygons that contain a given node. This ensures a significantly higher sparsity compared to standard meshfree approximations. In this work, we choose axis‐aligned quadrilaterals as polygonal primitives and maximum entropy approximants as kernels. However, any other convex approximation scheme and convex polygons can be employed. We study the optimal placement of nodes for regular elements, ie, those that are not intersected by the boundary, and propose a method to generate a suitable mesh. Finally, we show via numerical experiments that the proposed approach provides good accuracy without undermining the sparsity of the resulting matrices.     

Cracking particles: a simplified meshfree method for arbitrary evolving cracks

A new approach for modelling discrete cracks in meshfree methods is described. In this method, the crack can be arbitrarily oriented, but its growth is represented discretely by activation of crack surfaces at individual particles, so no representation of the crack's topology is needed. The crack is modelled by a local enrichment of the test and trial functions with a sign function (a variant of the Heaviside step function), so that the discontinuities are along the direction of the crack. The discontinuity consists of cylindrical planes centred at the particles in three dimensions, lines centred at the particles in two dimensions. The model is applied to several 2D problems and compared to experimental data. Copyright © 2004 John Wiley & Sons, Ltd.     

Meshfree and finite element nodal integration methods

Nodal integration can be applied to the Galerkin weak form to yield a particle‐type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions. This particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods. To a lesser degree, nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods. In this work, aspects regarding stability, consistency, efficiency and explicit time integration are explored within the context of nodal integration. Both small and large deformation numerical examples are provided. Copyright © 2007 John Wiley & Sons, Ltd.     

An accelerated,convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics

Convergent and stable domain integration that is also computationally efficient remains a challenge for Galerkin meshfree methods. High order quadrature can achieve stability and optimal convergence, but it is prohibitively expensive for practical use. On the other hand, low order quadrature consumes much less CPU but can yield non‐convergent, unstable solutions. In this work, an accelerated, convergent, and stable nodal integration is developed for the reproducing kernel particle method. A stabilization scheme for nodal integration is proposed based on implicit gradients of the strains at the nodes that offers a computational cost similar to direct nodal integration. The method is also formulated in a variationally consistent manner, so that optimal convergence is achieved. A significant efficiency enhancement over a comparable stable and convergent nodal integration scheme is demonstrated in a complexity analysis and in CPU time studies. A stability analysis is also given, and several examples are provided to demonstrate the effectiveness of the proposed method for both linear and nonlinear problems. Copyright © 2015 John Wiley & Sons, Ltd.     

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

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.     

On the completeness of meshfree particle methods

The completeness of Smooth Particle Hydrodynamics (SPH) and its modifications is investigated. Completeness, or the reproducing conditions, in Galerkin approximations play the same role as consistency in finite-difference approximations. Several techniques which restore various levels of completeness by satisfying reproducing conditions on the approximation or the derivatives of the approximation are examined. A Petrov–Galerkin formulation for a particle method is developed using approximations with corrected derivatives. It is compared to a normalized SPH formulation based on kernel approximations and a Galerkin method based on moving least-square approximations. It is shown that the major difference is that in the SPH discretization, the function which plays the role of the test function is not integrable. Numerical results show that approximations which do not satisfy the completeness and integrability conditions fail to converge for linear elastostatics, so convergence is not expected in non-linear continuum mechanics. © 1998 John Wiley & Sons, Ltd.     

Numerical integration in Natural Neighbour Galerkin methods

In this paper, issues regarding numerical integration of the discrete system of equations arising from natural neighbour (natural element) Galerkin methods are addressed. The sources of error in the traditional Delaunay triangle‐based numerical integration are investigated. Two alternative numerical integration schemes are analysed. First, a ‘local’ approach in which nodal shape function supports are exactly decomposed into triangles and circle segments is shown not to give accurate enough results. Second, a stabilized nodal quadrature scheme is shown to render high levels of accuracy, while resulting specially appropriate in a Natural Neighbour Galerkin approximation method. The paper is completed with several examples showing the performance of the proposed techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

Smooth,second order,non‐negative meshfree approximants selected by maximum entropy

We present a family of approximation schemes, which we refer to as second‐order maximum‐entropy (max‐ent) approximation schemes, that extends the first‐order local max‐ent approximation schemes to second‐order consistency. This method retains the fundamental properties of first‐order max‐ent schemes, namely the shape functions are smooth, non‐negative, and satisfy a weak Kronecker‐delta property at the boundary. This last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial. The evaluation of the shape functions is not explicit, but it is very efficient and robust. To our knowledge, the proposed method is the first higher‐order scheme for function approximation from unstructured data in arbitrary dimensions with non‐negative shape functions. As a consequence, the approximants exhibit variation diminishing properties, as well as an excellent behavior in structural vibrations problems as compared with the Lagrange finite elements, MLS‐based meshfree methods and even B‐Spline approximations, as shown through numerical experiments. When compared with usual MLS‐based second‐order meshfree methods, the shape functions presented here are much easier to integrate in a Galerkin approach, as illustrated by the standard benchmark problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A multibody meshfree strategy for the simulation of highly deformable granular materials

In this paper, a multibody meshfree framework is proposed for the simulation of granular materials undergoing deformations at the grain scale. This framework is based on an implicit solving of the mechanical problem based on a weak form written on the domain defined by all the grains composing the granular sample. Several technical choices, related to the displacement field interpolation, to the contact modelling, and to the integration scheme used to solve the dynamic equations, are explained in details. A first implementation is proposed, under the acronym Multibody ELement‐free Open code for DYnamic simulation (MELODY), and is made available for free download. Two numerical examples are provided to show the convergence and the capability of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

A study on the convergence of least‐squares meshfree method under inaccurate integration

In the authors' previous work, it has been shown through numerical examples that the least‐squares meshfree method (LSMFM) is highly robust to the integration errors while the Galerkin meshfree method is very sensitive to them. A mathematical study on the convergence of the solution of LSMFM under inaccurate integration is presented. New measures are introduced to take into account the integration errors in the error estimates. It is shown that, in LSMFM, solution errors are bounded by approximation errors even when integration is not accurate. Copyright © 2003 John Wiley & Sons, Ltd.     

An isogeometric‐meshfree coupling approach for analysis of cracks

This paper presents a new concurrent simulation approach to couple isogeometric analysis (IGA) with the meshfree method for studying of crack problems. In the present method, the overall physical domain is divided into 2 subdomains that are formulated with the IGA and meshfree method, respectively. In the meshfree subdomain, the moving least squares shape function is adopted for the discretization of the area around crack tips, and the IGA subdomain is adopted in the remaining area. Meanwhile, the interface region between the 2 subdomains is represented by coupled shape functions. The resulting shape function, which comprises both IGA and meshfree shape functions, satisfies the consistency condition, thus ensuring convergence of the method. Moreover, the meshfree shape functions augmented with the enriched basis functions to predict the singular stress fields near a crack tip are presented. The proposed approach is also applied to simulate the crack propagation under a mixed‐mode condition. Several numerical examples are studied to demonstrate the use and robustness of the proposed method.     

Gradient and dilatational stabilizations for stress‐point integration in the element‐free Galerkin method

Stabilized stress‐point integration schemes based on gradient stabilization and dilatational stabilization methods are presented for linear elastostaticity problems in the framework of element‐free Galerkin (EFG) method. The instability in stress fields associated with the stress‐point integration is treated by the addition to the Galerkin weak form of stabilization terms which contain product of the gradient of the residual or the trace of the gradient of the residual; the latter is called dilatational stabilization. Numerical results show that the oscillations in the stress fields are successfully removed by the presented stabilization methods, and that the convergence and stability properties of direct stress‐point integration are greatly improved. These stabilization methods are particularly suitable for the solution of non‐linear continua with explicit time integration methods. Copyright © 2008 John Wiley & Sons, Ltd.     

基于能量一致积分的拉索-阻尼器实时混合试验方法EI北大核心CSCD

实时混合试验是一种研究速度相关型试件动力性能的抗震试验方法,可以应用于拉索-阻尼器系统的力学行为研究。由于拉索具有较强的几何非线性,传统的线性无条件稳定积分算法无法保证拉索-阻尼器系统动力计算的稳定性。能量一致积分方法可以实现对非线性系统的无条件稳定,但应用于实时混合试验时,会遇到迭代导致作动器加载速度波动较大的问题。为了将能量一致积分方法应用于实时混合试验中,提出采用固定迭代次数并对迭代位移进行插值来实现平滑加载,然后对测得的试验子结构恢复力进行修正来实现系统能量一致。最后,对一个拉索-阻尼器系统进行了一阶模态振动下的实时混合试验数值仿真,验证了该方法的可行性。     

Floating stress‐point integration meshfree method for large deformation analysis of elastic and elastoplastic materials

A new meshfree formulation of stress‐point integration, called the floating stress‐point integration meshfree method, is proposed for the large deformation analysis of elastic and elastoplastic materials. This method is a Galerkin meshfree method with an updated Lagrangian procedure and a quasi‐implicit time‐advancing scheme without any background cell for domain integration. Its new formulation is based on incremental equilibrium equations derived from the incremental virtual work equation, which is not generally used in meshfree formulations. Hence, this technique allows the temporal continuity of the mechanical equilibrium to be naturally achieved. The details of the new formulation and several examples of the large deformation analysis of elastic and elastoplastic materials are presented to show the validity and accuracy of the proposed method in comparison with those of the finite element method. Copyright © 2012 John Wiley & Sons, Ltd.     

Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation

When numerical methods such as the finite element method (FEM) are used to solve the Helmholtz equation, the solutions suffer from the so‐called pollution effect which leads to inaccurate results, especially for high wave numbers. The main reason for this is that the wave number of the numerical solution disagrees with the wave number of the exact solution, which is known as dispersion. In order to obtain admissible results a very high element resolution is necessary and increased computational time and memory capacity are the consequences. In this paper a meshfree method, namely the radial point interpolation method (RPIM), is investigated with respect to the pollution effect in the 2D‐case. It is shown that this methodology is able to reduce the dispersion significantly. Two modifications of the RPIM, namely one with polynomial reproduction and another one with a problem‐dependent sine/cosine basis, are also described and tested. Numerical experiments are carried out to demonstrate the advantages of the method compared with the FEM. For identical discretizations, the RPIM yields considerably better results than the FEM. Copyright © 2008 John Wiley & Sons, Ltd.