The development of a generalized meshfree approximation for solid and fracture analysis Open Access
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This dissertation is concerned with the development of a generalized meshfree (GMF) approximation to improve the accuracy and efficiency of conventional meshfree methods in the solid and fracture analysis. The particular focus of the dissertation is placed on both the mathematical and numerical analysis of the proposed approximation in terms of its convergence and stability properties. The GMF approximation can be considered as a general expression reproducing all the existing meshfree approximations as well as a new approximation utilizing different basis functions. In addition, the GMF approximation possess the so-called weak Kronecker delta property at the boundary that, in turn, allows us to directly impose the essential boundary conditions without complicated treatments as seen in the traditional meshfree approximations and therefore greatly improves the computational efficiency. Furthermore, by choosing certain basis function, the proposed approximation can be extended to a higher-order or specific enriched approximation without adding extra nodes and is naturally conforming. This unique feature leads to high accuracy and enables us to solve some challenging problems such as problem involving large deformation, moving discontinuity and high-gradients. The proposed approximation is further furnished with the element-free Galerkin (EFG) method for the solid and fracture analysis.The proposed method is tested against several benchmark problems with analytical solutions as well as some practical problems with experimental data with regard to its predictive simulation capabilities. For the time-dependent problems, a stability analysis is performed to guarantee that the proposed method is stable and produces a bounded solution whenever the solution of the exact differential equation is bounded. In fracture analysis, the cohesive failure model is adopted and the meshfree visibility approach is utilized to define the crack and its initiation and propagation.