A decomposition technique of generalized degrees of freedom for mixedmode crack problems

Abstract

The numerical manifold method (NMM) builds up a unified framework that is used to describe continuous and discontinuous problems; it is an attractive method for simulating a cracking phenomenon. Taking into account the differences between the generalized degrees of freedom of the physical patch and nodal displacement of the element in the NMM, a decomposition technique of generalized degrees of freedom is deduced for mixed mode crack problems. An analytic expression of the energy release rate, which is caused by a virtual crack extension technique, is proposed. The necessity of using a symmetric mesh is demonstrated in detail by analysing an additional error that had previously been overlooked. Because of this necessity, the local mathematical cover refinement is further applied. Finally, four comparison tests are given to illustrate the validity and practicality of the proposed method. The aforementioned aspects are all implemented in the highorder NMM, so this study can be regarded as the development of the virtual crack extension technique and can also be seen as a prelude to an h-version high-order NMM.to extract the strain-energy release rates was provided by [18]. Under the context of XFEM, a direct analytical method to extract the mixed-mode strain-energy release rates from Irwin's integral was given by [19], and then this method was extended to high-order XFEM [20,21].Alternatively, the stiffness derivative technique (SDT) and virtual crack extension techniques (VCET) were proposed, respectively [22,23]. Whereafter, the VCET was applied to determine the SIFs of mode-I and mode-II by carrying out virtual crack extension along both the parallel and perpendicular directions to crack surface [24]. A combination of the VCET and field decomposition technique (DFDT) was initially implemented to extract mixed-mode SIFs by carrying out virtual crack extension in only the parallel direction to crack surface [25]. A double VCET for crack growth stability assessment was described by [26]. Based on an energy principle and the VCET, an approach that does not require the use of symmetric crack-tip mesh nor crack-tip singular elements was developed [27]. The VCET was used for simulation of the fatigue crack propagation by [28,29]. In order to avoid using finite difference approximation, which can lead to calculation error, an analytical expression for the energy release rate was derived [30]; another explicit expression for energy changes due to VCET was formulated on the basis of a variation of isoparametric element mappings [31]. A new direct-integration technique for the VCET using variational theory was presented in [32]. A blend of the VCET and DFDT was implemented to decompose three-dimensional mixed-mode energy release rates [33].The concept of shape design sensitivity analysis [34] was applied to calculate the strain-energy release rate [35]. And then, the equivalent domain integral [36] and the interaction integral [37] were used for the sensitivity analysis of cracked bodies. From the view of the shape design s...

A new family of high‐order triangular weight functions for mass lumping in vibration analysis

“…As a more general scheme, a manifold-based mass lumping scheme was recently presented by Yang et al 26 in the context of partition of unity (PU) method, 27 specifically the numerical manifold method (NMM), 28 which has been successfully applied to structures, [29][30][31][32][33][34] cracks, [35][36][37][38][39][40][41] fluids, [42][43][44] and large deformation. 45,46 The manifold-based mass lumping scheme is a rigorous and unified approach.…”