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• A novel two-level nesting smoothed meshfree method (NSMM) is proposed. • The NSMM is applied for structural dynamic analysis. • The NSMM is directly extended to solving dynamic fracture problems. • The NSMM is efficient, accurate and stable. This paper presents a novel two-level nesting smoothed meshfree method (NSMM), which significantly improves the computational efficiency of the meshfree Galerkin methods without losing their accuracy, thus facilitates the employment of meshfree methods in applications where background integration cells would be prohibitively expensive. In the NSMM, the system stiffness matrix is calculated using the general smoothing strain technique over the two-level nesting smoothing sub-domains where fewer integration points are used and the costly derivative computation of meshfree shape functions is avoided. The accuracy, efficiency and stability of the present method are assessed by virtue of several numerical examples for problems involving free and forced vibration analysis of the linear elastic continua and dynamic crack response of elastic solid. The results reveal that the NSMM stands out and achieves better performance compared to other existing approaches in the literature. [ABSTRACT FROM AUTHOR]

Highlights • Greatly improving the efficiency of the meshfree method by transforming background cell integral into boundary integration. • Directly imposing the boundary condition without any effort. • Completely avoiding the derivation of meshfree shape functions. • Improving the accuracy of the stress field in the vicinity of the crack tips. Abstract A smoothed enriched meshfree Galerkin method with two-level nesting triangular sub-domains (NSEMM) is presented for linear elastic fracture mechanics. In the present method, displacement fields are constructed by using the linear MLS approximation in linear region away from the crack tip and the enriched MLS approximation in enriched region around the crack tip respectively. To overcome the discontinuous problem of the coupling approximations and accelerate meshfree computations, two-level nesting triangular sub-domains based on the triangular background integration cells are designed and used to construct the smoothing strain. Each two-level nesting sub-domains includes the triangular background cell itself and four equal-area triangular sub-cells. The discretized system equations are created by the Generalized smoothed Galerkin (GS-Garlerkin) weak form and Richardson extrapolation method. The values of SIFs are evaluated by using the domain form of interaction integral. Numerical examples have been simulated to demonstrate the effectiveness and robustness of the present method for crack problems with complicated configurations, and very satisfactory results have been obtained. The NSEMM takes advantages of both EFGM and generalized strain smoothing technique such as lower computational cost, high accurate and easy to implementation. [ABSTRACT FROM AUTHOR]

In this paper, a meshless collocation method with barycentric rational interpolation is proposed to find and represent the solution of the two-dimensional hyperbolic telegraph equation. Barycentric rational interpolation functions are used for spatial variable and its partial derivatives, which produce a system of second order ordinary differential equations based on collocation method. The resulting system has been solved by central differential scheme. The accuracy and efficiency of proposed approach are demonstrated by several numerical experiments, where comparison is made with some earlier work. It is clear that the proposed method produces good results in comparison with those available in literature. Moreover, CPU time taken in our computation is much less. It is proved that the proposed method is very simple, fast and accurate. [ABSTRACT FROM AUTHOR]

• A coupled extended meshfree–smoothed meshfree method (CXSMM) is presented for simulating crack growth in 2D solids. • Division the analysis domain into discontinuous and continuous regions is available. • The CXSMM has the advantages of both extended meshfree method and smoothed meshfree method. • Several problems are used to evaluate the accuracy, efficiency and robustness of the CXSMM. A coupled extended meshfree–smoothed meshfree method (CXSMM) is presented for simulating crack growth in 2D solids. In terms of the relationship between the crack surface and the background integration cells, the analysis domain is easy to be divided into discontinuous and continuous regions, in which the meshfree Galerkin method and the smoothed meshfree method are employed to construct the discretized system equations, respectively. To precisely capture the singular stress field near the crack tip, higher order Gauss integration rules are used in the discontinuous region. At the same time, the general smoothed strain technique is utilized in the continuous region for efficient evaluation of the domain integration. In this work, the initiation and propagation of the crack are treated by the aid of the level set algorithm. The CXSMM has the advantages of both extended meshfree method and smoothed meshfree method and can accurately and efficiently model the linear elastic fracture. Several benchmark problems are used to evaluate the accuracy, efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, an efficient and accurate meshfree method, named the two-level nesting smoothed extended meshless method (S-XMM-N), is developed for simulating the static and dynamic fracture behaviors of two-dimensional (2D) functionally graded materials (FGMs). To accurately capture the displacement discontinuity along the crack face and the stress singularities at the crack tips in FGMs, we improve the extended meshless approximation by adding the Heaviside function and four crack tip enrichment functions to the approximation of RPIM. This approximation not only inherits all the advantages of traditional extended meshless approximation but also has interpolation properties. Furthermore, to avoid the complex and expensive integration calculation when constructing the global stiffness matrix, we use the recently developed two-level nesting smoothing integration algorithm, which is based on the first and second level smoothed strains and Richardson extrapolation theory. The static and dynamic stress intensity factors (SIFs) are calculated by the interaction integral. Several static and dynamic problems of FGM thin plates with the elastic modulus varying with respect to the exponential or hyperbolic tangent functions are solved by S-XMM-N. By investigating the SIFs given by various methods, the feasibility and effectiveness of the proposed methods are validated. [ABSTRACT FROM AUTHOR]

• S-XMM-N formulation for 2D dynamic crack analysis is presented. • Numerical integration of two-level nesting smoothing sub-domains is presented. • Static and dynamic stress intensity factors are analyzed. • Effect of nodes, domain integration sizes, loading type, loading peak intensity and time step is investigated. A two-level nesting smoothed extended meshfree method (S-XMM-N) is presented for modeling dynamic fracture behaviors in two-dimensional linear elastic solids. The extended meshfree method based on the partition of unity allows for arbitrary discontinuities, but with low computational efficiency needs improvement. Meanwhile, directly introducing the generalized smoothed strain technique into the extended meshfree method can save more computational times, and with low accuracy also improvement. The proposed method combined the best feature of both the recently developed two-level nesting smoothing meshfree method and the extended radial point interpolation method (XRPIM). On the one hand, the S-XMM-N saves much more computational time than XRPIM by using the proposed two-level nesting triangular sub-domains smoothing integration, where only fewer integration points are employed and the derivatives of shape functions do not need to be calculated for constructing the stiffness matrix. On the other hand, the computational accuracy can be ensured due to the employment of Richardson extrapolation method. The results for the static and dynamic SIFs are compared with the analytical and other numerical solutions by several benchmark fracture problems. [ABSTRACT FROM AUTHOR]