Nonlocal damage modelling using the element-free Galerkin method in the frame of finite strains

Prediction of size effects has been a challenging problem since some experiments found the size effect in material damage. Both material model and numerical algorithm have to be improved to consider the complex damage process. In the present paper we implement element-free Galerkin (EFG) method for a strain-gradient based nonlocal damage model and use it to analyze ductile material damage process. The EFG algorithm overcomes some drawbacks of the FEM in convergence of numerical iteration due to large deformations as well as evaluation of the higher-order gradients of the plastic strain. The numerical benchmarks show that the EFG method for the nonlocal damage model provides more stable numerical results. The size effect in notched specimens can be predicted in the computations. Both ductile fracture in tensile specimens as well as their size effects are investigated and the computational results agree very well with experiments.     

Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ? J/σ₀) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.     

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.     

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. The strong and weak formulations of this 3D static problem are derived on the basis of the Reissner mixed variational theorem (RMVT) where the strong formulation consists of the Euler–Lagrange equations of the problem and its associated boundary conditions, and the weak formulation represents a weighted-residual integral in which the differentiation is equally distributed among the primary field variables and their variations. The early proposed DRK interpolation is used to construct the primary field variables where the Kronecker delta properties are satisfied, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation. In the illustrative examples, it is shown that the solutions obtained from these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.     

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) free vibration analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. Based on the Reissner Mixed Variational Theorem (RMVT), the strong and weak formulations of this problem are derived, in which the material properties of each individual FGM layer, constituting the plate, are assumed to obey the power-law distributions of the volume fractions of the constituents. The system motion equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation, in which the shape functions of the unknown functions satisfy the Kronecker delta properties, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. In the illustrative examples, the natural frequencies and their corresponding modal field variables varying along the thickness coordinate of the plate are studied. It is shown that the solutions obtained using these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.     

无网格Galerkin法的理论进展及其应用研究

无网格Galerkin(Element-free Galerkin,EFG)法是无网格方法中应用比较广泛的一种,在介绍其基本特点和原理的基础上,对其移动最小二乘近似过程中涉及到的基函数、权函数的选择、影响域半径的确定等方面取得的新进展进行了介绍.并针对本征边界条件的满足,离散和积分方案的实施,自适应分析及误差分析的应用等一系列相关问题的研究现状及取得的成果进行了详细阐述.同时以受均布载荷的悬臂梁为例,编制了EFG平面弹性程序,验证了EFG法的可行性.最后针对EFG法存在的不足,提出了几个研究方向.     

The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems

In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.     

Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method

 An efficient meshfree formulation based on the first-order shear deformation theory (FSDT) is presented for the static analysis of laminated composite beams and plates with integrated piezoelectric layers. This meshfree model is constructed based on the element-free Galerkin (EFG) method. The formulation is derived from the variational principle and the piezoelectric stiffness is taken into account in the model. In numerical test problems, bending control of piezoelectric bimorph beams was shown to have the efficiency and accuracy of the present EFG formulation for this class of problems. It is demonstrated that the different boundary conditions and applied actuate voltages affects the shape control of piezolaminated composite beams. The meshfree model is further extended to study the shape control of piezo-laminated composite plates. From the investigation, it is found that actuator patches bonded on high strain regions are significant in deflection control of laminated composite plates. Received: 23 October 2001 / Accepted: 29 July 2002     

Smoothed Galerkin methods using cell-wise strain smoothing technique

A smoothed Galerkin method (SGM) using cell-wise strain smoothing operation is formulated in this paper. In present method, the field nodes can be divided into two types: boundary field nodes and interior field nodes. The background cells are divided into SC smoothing cells, and the strains in each smoothing cell are obtained using a gradient smoothing technique which can avoid evaluating derivatives of shape functions at integration point. The field variables of boundary points are approximated using linear interpolation of neighbour boundary field nodes, and the shape functions possess the Kronecker Delta property and facilitate the impositions of essential boundary conditions. The field variables of interior points are approximated using moving least-squares approximation using the support field nodes around them. A number of numerical examples are studied and confirm the significant features of the present methods: (1) can pass the standard patch test; (2) can easily impose essential boundary conditions as those in finite element method; (3) can avoid evaluating derivatives of shape functions; (4) no numerical parameter is required.     

The improved element-free Galerkin method for two-dimensional elastodynamics problems

In this paper, we derive an improved element-free Galerkin (IEFG) method for two-dimensional linear elastodynamics by employing the improved moving least-squares (IMLS) approximation. In comparison with the conventional moving least-squares (MLS) approximation function, the algebraic equation system in IMLS approximation is well-conditioned. It can be solved without having to derive the inverse matrix. Thus the IEFG method may result in a higher computing speed. In the IEFG method for two-dimensional linear elastodynamics, we employed the Galerkin weak form to derive the discretized system equations, and the Newmark time integration method for the time history analyses. In the modeling process, the penalty method is used to impose the essential boundary conditions to obtain the corresponding formulae of the IEFG method for two-dimensional elastodynamics. The numerical studies illustrated that the IEFG method is efficient by comparing it with the analytical method and the finite element method.     

Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method

The nonlocal elasticity theory of Eringen is used to study bending, buckling and free vibration of Timoshenko nanobeams. A meshless method is used to obtain numerical solutions. Results are compared with available analytical solutions. Two different collocation techniques, global (RBF) and local (RBF-FD), are used with multi-quadrics radial basis functions.     

Probabilistic fracture mechanics by Galerkin meshless methods – part II: reliability analysis

 This is the second in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper, a stochastic meshless method is presented for probabilistic fracture-mechanics analysis of linear-elastic cracked structures. The method involves an element-free Galerkin method for calculating fracture response characteristics; statistical models of uncertainties in load, material properties, and crack geometry; and the first-order reliability method for predicting probabilistic fracture response and reliability of cracked structures. The sensitivity of fracture parameters with respect to crack size, required for probabilistic analysis, is calculated using a virtual crack extension technique described in the companion paper [1]. Numerical examples based on mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that the predicted probability of fracture initiation based on the proposed formulation of the sensitivity of fracture parameter is accurate in comparison with the Monte Carlo simulation results. Since all gradients are calculated analytically, reliability analysis of cracks can be performed efficiently using meshless methods. Received 20 February 2001 / Accepted 19 December 2001     

Element-free Galerkin method using directed graph and its application to creep problems

The element-free Galerkin method (EFGM) is one of the meshless methods proposed by Belytschko et al. Since node-element connectivities used in the finite element method (FEM) are not needed in the EFGM, the EFGM is expected to be applied to many problems of the continuum mechanics and to be utilized for a tool in a CAE system instead of the FEM. However the EFGM requires more CPU time to search nodes of the MLSM than the FEM. In this paper, the method of the directed graph and the Delaunay triangulation are respectively used for searching nodes and the division of the integral domain respectively. These techniques are useful for saving the CPU time and the simplification of the analysis for the EFGM. Furthermore, the EFGM has not been applied to nonlinear problems such as creep problems under elevated temperature. In this paper, the EFGM using the method of the directed graph and the Delaunay triangulation is applied to several creep problems. The CPU times for the analyses are reduced by the proposed EFGM. The results obtained from the EFGM analyses agree well with those of the FEM.     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

Probabilistic fracture mechanics by Galerkin meshless methods – part I: rates of stress intensity factors

 This is the first in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper (Part I), a Galerkin-based meshless method is presented for predicting first-order derivatives of stress-intensity factors with respect to the crack size in a linear-elastic structure containing a single crack. The method involves meshless discretization of cracked structure, domain integral representation of the fracture integral parameter, and sensitivity analysis in conjunction with a virtual crack extension technique. Unlike existing finite-element methods, the proposed method does not require any second-order variation of the stiffness matrix to predict first-order sensitivities, and is, consequently, simpler than existing methods. The method developed herein can also be extended to obtain higher-order derivatives if desired. Several numerical examples related to mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that first-order derivatives of stress-intensity factors using the proposed method agree very well with reference solutions obtained from either analytical (mode I) or finite-difference (mixed mode) methods for the structural and crack geometries considered in this study. For mixed-mode problems, the maximum difference between the results of proposed method and finite-difference method is less than 7. Since the rates of stress-intensity factors are calculated analytically, the subsequent fracture reliability analysis can be performed efficiently and accurately. Received 20 February 2001 / Accepted 19 December 2001     

Meshless analysis of two-dimensional Stokes flows with the Galerkin boundary node method

In this paper, the Galerkin boundary node method (GBNM) is developed for the solution of stationary Stokes problems in two dimensions. The GBNM is a boundary only meshless method that combines a variational form of boundary integral formulations for governing equations with the moving least-squares (MLS) approximations for construction of the trial and test functions. Boundary conditions in this approach are included into the variational form, thus they can be applied directly and easily despite the MLS shape functions lack the property of a delta function. Besides, the GBNM keeps the symmetry and positive definiteness of the variational problems. Convergence analysis results of both the velocity and the pressure are given. Some selected numerical tests are also presented to demonstrate the efficiency of the method.     

A priori and a posteriori analysis of the meshless Galerkin boundary node method for three-dimensional elasticity

The meshless Galerkin boundary node method is presented in this paper for boundary-only analysis of three-dimensional elasticity problems. In this method, boundary conditions can be implemented directly and easily despite the employed moving least-squares shape functions lack the delta function property, and the resulting system matrices are symmetric and positive definite. A priori error estimates and the consequent rate of convergence are presented. A posteriori error estimates are also provided. Reliable and efficient error estimators and an efficient and convergent adaptive meshless algorithm are then derived. Numerical examples showing the efficiency of the method, confirming the theoretical properties of the error estimates, and illustrating the capability of the adaptive algorithm, are reported.     

The concept of control points in hybrid discontinuous Galerkin methods—Application to geometrically nonlinear crystal plasticity

A new concept for hybrid discontinuous Galerkin (DG) methods is presented: control points. These are defined on the interelement boundaries. The concept makes it possible to formulate element shape functions without nodes. Moreover, the theory is not restricted to certain element shapes. Furthermore, one can formulate the discrete model such that the displacement is either continuous or discontinuous at the control points. Classical continuous isoparametric elements are included as special case. As an additional new feature, a regularization technique for very high strain rate sensitivity exponents up to 1000 in finite single crystal viscoplasticity is presented and implemented into the new hybrid DG framework. In addition, the numerical linearization used in an earlier work is carried out analytically in this work. To the knowledge of the authors, this work presents the first hybrid DG implementation of geometrically nonlinear plasticity, here in the context of single crystal plasticity. The regularization method in combination with the DG formulations facilitates a very simple implementation leading to a numerically efficient, robust, and locking‐free model. Two examples are investigated: the deformation of a planar double slip single crystal exhibiting localization in the form of shear bands and an oligocrystal under uniaxial load.     

Thermal Analysis of CNT-Based Nano-Composites by Element Free Galerkin Method

This paper deals with the thermal analysis of carbon nanotube (CNT) based composites by meshless element free Galerkin method. Cylindrical representative volume element (cylindrical RVE) has been chosen to evaluate the thermal properties of nano-composites using multi-domain and simplified approaches. The values of temperature have been calculated at different points and plotted against RVE length and RVE radius. A sensitivity analysis of RVE as well as CNT dimensions has been carried out in detail. The present computations show that the equivalent thermal conductivity is a function of CNT length, CNT radius, RVE length and RVE radius. Based on present numerical simulations, an approximate formula is proposed to calculate the equivalent thermal conductivity of nano-composites. The results obtained by simplified approach have been found in good agreement with those obtained by multi-domain approach.