An inherently consistent reproducing kernel gradient smoothing framework toward efficient Galerkin meshfree formulation with explicit quadrature

https://doi.org/10.1016/j.cma.2019.02.029Get rights and content

Highlights

  • A reproducing kernel gradient smoothing framework is proposed for meshfree methods.

  • Integration consistency is an inherent property of the proposed methodology regardless of integration schemes.

  • Explicit quadrature rules referring to 2D triangular and 3D tetrahedral integration cells are developed.

  • Total number of sample points for the present quadrature rules is minimized from a global point of view.

  • The proposed framework enables efficient Galerkin meshfree analysis with optimal convergence, particularly for higher order basis functions.

Abstract

A reproducing kernel gradient smoothing framework with explicit quadrature rules is proposed for efficient Galerkin meshfree formulation. In this framework, the meshfree smoothed gradients are formulated via a reproducing kernel representation of the standard gradients of field variables. It is interesting to find that this reproducing kernel construction of smoothed gradients of meshfree shape functions enables an identical satisfaction of the integration constraint derived from Galerkin meshfree formulation. In other words, the integration consistency is an inherent property of the proposed reproducing kernel gradient smoothing methodology regardless of integration schemes. Consequently, without violation of the integration constraint, the conventional normal low order Gauss quadrature rules in finite element analysis now can be used to properly integrate the meshfree stiffness matrix simply through replacing the standard meshfree gradients with the reproducing kernel smoothed gradients at Gauss points. In order to efficiently compute the reproducing kernel smoothed gradients, a set of explicit quadrature rules referring to 2D triangular and 3D tetrahedral integration cells are systematically developed. The total number of sample points associated with these quadrature rules is minimized from a global point of view through introducing an equivalent number of sample points, which implies as many as sample points are shared by neighboring integration cells. The proposed methodology recovers the stabilized conforming nodal integration when linear basis function is employed, while the focus of the present work is multidimensional higher order basis functions such as quadratic and cubic ones. Superior convergence, accuracy as well as efficiency performances of the proposed reproducing kernel gradient smoothing framework are thoroughly demonstrated by a series of numerical examples.

Introduction

Galerkin meshfree methods have attracted significant research attention and been widely used due to their excellent accuracy and sound stability characteristics [1], [2]. An earlier Galerkin meshfree formulation is the diffuse element method developed by Nayroles et al. [3], which employs the moving least squares (MLS) [4] approximation and diffuse derivatives of shape functions. Later on, Belytschko et al. [5], [6] proposed the element-free Galerkin method via introducing full derivatives of MLS shape functions, background cell-based higher order Gauss integration and Lagrange multiplier enforcement of essential boundary conditions. On the other hand, through incorporating the correction function and reproducing conditions into the kernel estimation, Liu et al. [7] presented a reproducing kernel (RK) meshfree approximation and the related reproducing kernel particle method, whose discrete consistency was illustrated by Chen et al. [8]. It turns out that MLS and RK approximants are equivalent when monomial basis functions are used. Since then, Galerkin meshfree formulations have enjoyed great popularity and many versatile methods have been developed over the years, for instance, h-p cloud method [9], partition of unity method [10], natural element method [11], meshless local Petrov–Galerkin method [12], finite sphere method [13], radial point interpolation method [14], generalized finite-element method (GFEM) [15], reproducing kernel interpolation method [16], moving kriging interpolation element-free Galerkin method [17], reproducing kernel element method [18], max-entropy meshfree method [19], [20], smoothed finite element method [21], optimal transportation meshfree method [22], semi-Lagrangian reproducing kernel particle method [23], generalized meshfree approximation method [24], reproducing kernel meshfree peridynamics [25], quasi-convex reproducing kernel meshfree method [26], direct displacement smoothing method [27], hierarchical partition of unity element-free Galerkin method [28], quasi-linear reproducing kernel particle method [29] and smoothed particle Galerkin method [30], among others. More detailed summaries on the advances of Galerkin meshfree methods can be found in [1], [2], [31], [32], [33], [34], [35], [36].

For the weak form-based Galerkin meshfree methods, domain integration is inevitable. However, unlike the conventional polynomial type of finite element shape functions [37], [38], the meshfree shape functions such as MLS/RK meshfree shape functions have a rational nature and their influence or support domains are usually overlapped each other [39]. Consequently, the normal low order Gauss quadrature rules designed for the polynomial shape functions are not well suitable for Galerkin meshfree methods and time-consuming higher order Gauss integration rules are often required to maintain the solution accuracy and convergence behavior [39], [40], [41], [42]. Thus the development of efficient domain integration algorithms has been a very active topic for Galerkin meshfree methods [2], [40], [41], [42], [43], [44]. Bessiel and Belytschko [45] presented a nodal integration method to improve the computational efficiency, where the nodally integrated weak form is stabilized via adding the equilibrium residual term multiplied by an artificial parameter. Employing additional stress points together with meshfree nodes as the integration locations offers another choice to retain the stability of Galerkin meshfree methods [46], [47], [48], [49]. Moreover, the support domain integration method was also investigated in several works [12], [13], [50], [51], [52], [53]. This method conveniently takes the support domains of meshfree shape functions as the integration cells, while in each cell higher order Gauss quadrature rules are still necessary to properly integrate the weak form.

Along a different path, Chen et al. [40], [41] presented the so-called integration constraint corresponding to the linear patch test for Galerkin meshfree methods, where it is shown that besides the linear reproducing or consistency condition of meshfree shape functions, the integration constraint has also to be satisfied for the domain integration of Galerkin meshfree methods in order to exactly pass the linear patch test. As a result, they proposed the stabilized conforming nodal integration (SCNI) by means of strain smoothing [54] for Galerkin meshfree methods. Both efficiency and stability are achieved in SCNI. Subsequently, SCNI has been further developed and generalized to many other problems, for examples, the locking-free plate and shell problems [55], [56], [57], [58], [59] large deformation failure simulations [41], [60], [61], [62] and convection dominated problems [63], etc. A sub-domain stabilized conforming integration in conjunction with Hermite reproducing kernel meshfree approximation was introduced by Wang et al. [64], [65], [66], [67], [68] to deal with fourth order thin plate and shell problems. To accommodate the excessively large deformation analysis, Chen et al. [23], [69] proposed a stabilized non-conforming nodal integration equipped with semi-Lagrangian meshfree formulation, where a regular domain is taken as the nodal representative cell for strain smoothing.

Based upon SCNI, Duan et al. [70], [71] investigated the integration constraints for higher order meshfree approximations and presented a quadratically consistent integration (QCI) method that is second order accurate for Galerkin meshfree formulations. In QCI, the smoothed derivatives of meshfree shape functions are not explicitly constructed like those in SCNI, and a system of equations derived from the quadratic integration constraint need to be solved to attain the smoothed derivatives with respect to a specific coordinate component. For example, in 3D case, the meshfree smoothed gradients at a sampling point require solving three sets of 4 by 4 system of equations, which will introduce undesirable additional computational cost [71]. Meanwhile, an arbitrary order variationally consistent integration (VCI) has been proposed by Chen et al. [72], [73] for Galerkin meshfree methods. The method of VCI enables the construction of various integration schemes up to the order of completeness in the meshfree approximation and it can also be used to correct other integration algorithms such as Gauss integration and stabilized non-conforming nodal integration [74]. Nonetheless, VCI falls into the category of Petrov–Galerkin formulation that involves non-symmetric stiffness matrices. Wang and Wu [75] proposed a quadratically exact nesting sub-domain gradient smoothing integration (NSGSI) algorithm for the stiffness matrix evaluation, which is established through optimally combining the contributions from the two-level triangular nesting sub-domains. It turns out that this approach is quite efficient, but its generalization to three dimensional formulation and higher order shape functions is not straightforward. Moreover, gradient stabilized meshfree nodal integration methods have also been developed, for example, the naturally stabilized nodal integration by Hillman and Chen [76] and the displacement smoothing stabilized nodal integration by Wu et al. [27], etc.

In this study, we aim to develop a general reproducing kernel gradient smoothing framework that can provide explicit quadrature rules for efficient Galerkin meshfree formulation. In this framework, the meshfree smoothed gradients are represented by the standard gradients of field variables with a reproducing kernel formulation. This is different to the gradient reproducing formulation in [77], [78], [79], [80], where the meshfree smoothed gradients are expressed by the field variables themselves. By construction, the smoothed gradients automatically satisfy the integration constraint resulting from Galerkin formalism. Thus an inherent integration consistency is embedded into the proposed meshfree smoothed gradients irrespective of integration methods. Consequently, similar to the finite element stiffness matrix integration, the conventional normal low order Gauss quadrature rules designed for polynomial shape functions [37], [38] now can be employed to evaluate the proposed meshfree stiffness matrix without violation of the integration constraint. The only difference is to replace the standard gradients at the Gauss points with the proposed reproducing kernel smoothed gradients of meshfree shape functions. Subsequently, a set of explicit quadrature rules referring to 2D triangular and 3D tetrahedral integration cells are devised to compute the reproducing kernel smoothed gradients. In these quadrature rules, the number of integration sample points is minimized by introducing an equivalent number of sample points, as significantly improves the computational efficiency since as many as sample points are shared by neighboring integration cells. The proposed integration strategy is termed as reproducing kernel gradient smoothing integration (RKGSI). The formulation of RKGSI is applicable to arbitrary order multidimensional Galerkin meshfree methods. RKGSI will reduce to the method of SCNI when a linear basis function is used, and in this work particular emphasis is the Galerkin meshfree formulations with higher order basis functions such as quadratic and cubic basis functions. The efficacy of RKGSI is validated by numerical results.

The remainder of this paper is organized as follows. Section 2 elaborates the Galerkin meshfree formulation with RK approximation and the integration constraint for numerical integration. Subsequently, a reproducing kernel gradient smoothing framework is presented in Section 3, where the inherent integration consistency is discussed. In Section 4, a set of explicit quadrature rules with particular reference to 2D triangular and 3D tetrahedral integration cells are developed to efficiently compute the reproducing kernel smoothed meshfree gradients, in which the number of sampling points is minimized from a global point of view. A series of numerical examples ranging from 1D to 3D are given in Section 5 to demonstrate the proposed methodology. Concluding remarks are finally drawn in Section 6.

Section snippets

RK meshfree approximation

For convenience of the subsequent development, the continuous form of reproducing kernel (RK) approximation defined in a problem domain Ω is first considered. According to the RK theory [7], [8], the continuous RK approximant of a field variable u(x), denoted by ur(x), can be expressed as: ur(x)=Ω(s)cp(x,s)ϕ(x,s)u(s)dΩin which ϕ(x,s) is the non-negative kernel function located at s, ϕ(x,s) has a nodal influence domain Ω(s) and it will vanish when xΩ(s). In this study, the classical cubic

RKGS formulation

In this section, a reproducing kernel gradient smoothing (RKGS) framework that identically meets the integration constraint of Eq. (27) is developed. In other words, the integration consistency is an inherent property of the present method. The proposed methodology begins with a reproducing kernel gradient smoothing representation of the field variable gradient u,i: u˜,ir(x)=Ω(s)c[p1](x,s)ϕ˜(x,s)u,i(s)dΩwhere c[p1](x,s) is the correction function for the reproducing kernel gradient

Explicit quadrature rules for RKGS

From Eqs. (36), (40), it is clear that GC and giIC involve the domain and boundary integrations. In this section, a set of explicit quadrature rules consisting of sampling points and corresponding weights are developed for efficient computation of GC and giIC used in the proposed RKGS. As mentioned earlier, the 2D triangular and 3D tetrahedral integration cells enable an analytical calculation of the RKGS moment matrix GC, and thus these types of simplex integration cells are adopted for the

Numerical examples

In this section, a series of numerical tests ranging from 1D to 3D are performed to verify the proposed methodology. Both quadratic and cubic basis functions are employed for the reproducing kernel meshfree approximation. For the patch tests and potential problems, kij=δij is used for in Eq. (14) and the following general form of analytical solutions is considered: u=(i,j,k)C(j+1)(k+2)2xiyjzk,C={(i,j,k)|i=n,j=k=0}{(i,j,k)|i+j=n,ji,k=0}{(i,j,k)|i+j+k=n,j,ki}1D2D3Dwhere n=1,2,3 are utilized

Conclusions

A general inherently consistent reproducing kernel gradient smoothing framework with explicit quadrature rules was proposed for efficient Galerkin meshfree analysis. In this framework, the meshfree smoothed gradients were built upon the standard gradients of field variables in the context of reproducing kernel formulation. It was shown that the present reproducing kernel smoothed gradients identically meet the integration constraint required to exactly reproduce any solutions spanned by the

Acknowledgments

The support of this work by the National Natural Science Foundation of China (11772280, 11472233) and the Natural Science Foundation of Fujian Province of China (2014J06001) is gratefully acknowledged.

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