New strategies for some issues of numerical manifold method in simulation of crack propagation

Abstract

Aiming to solve, in a unified way, continuous and discontinuous problems in geotechnical engineering, the numerical manifold method introduces two covers, namely, the mathematical cover and the physical cover. In order to reach the goal, some issues in the simulation of crack propagation have to be solved, among which are the four issues to be treated in this study: (1) to reduce the rank deficiency induced by high degree polynomials as local approximation, a new variational principle is formulated, which suppresses the gradient-dependent DOFs; (2) to evaluate the integrals with singularity of 1/r, a new numerical quadrature scheme is developed, which is simpler but more efficient than the existing Duffy transformation; (3) to analyze kinked cracks, a sign convention for argument in the polar system at the crack tip is specified, which leads to more accurate results in a simpler way than the existing mapping technique; and (4) to demonstrate the mesh independency of numerical manifold method in handling strong singularity, a mesh deployment scheme is advised, which can reproduce all singular locations of the crack with regard to the mesh. Corresponding to the four issues, typical examples are given to demonstrate the effectiveness of the proposed schemes. 987 patches of nodes on the displacement boundary to constants. Having reformulated NMM with more clarity, Lin [9] explored the merits and limitations of the method. Terada et al. [10] suggested calling NMM the finite cover method (FCM), who tested a number of elasticity problems by FCM and FEM respectively and concluded that the performance deterioration due to element distortion is less serious than that in FEM. Tian and Yagawa [11] developed 2D and 3D simplex elements and associated the physical patch with a so-called generalized node having physically meaning variables. In the simulation of soil consolidation, Zhang and Zhou [12] interpolated displacement and pore pressure independently, leading to a numerically stable scheme for the coupling problems, particularly in the nearly incompressible case.Since 1995, eleven international conferences on the NMM have been held, titled 'International Conference on Analysis of Discontinuous Deformation' and abbreviated as 'ICADD-n' with n as the conference number. ICADD-11 was just held in Fukuoka, Japan, during 27th-29th August, 2013.All developments and applications of NMM have been limited to the second order problems where the control partial DEs are second order; see Ma et al. [13] for details. Recently, Zheng et al. [14] constructed an NMM space of Hermitian form and applied it to the fourth order problems. Through the application to Kirchhoff's thin plate bending problems, they demonstrated that NMM is able to rescue those elements earliest developed in the finite element history, such as Zienkiewicz's plate element, and make them regain vigor.The major advantage of NMM is to solve in a unified way the problems involving continuous and discontinuous deformation. Wu and Wong [15], Kurumatani and Terada [16], a...

Dual reciprocity boundary element based block model for discontinuous deformation analysis

“…Either the element subdivision technique [34] or the numerical quadrature scheme [35] can be used to effectively deal with the weak singularity. The element subdivision technique [34] is adopted.…”

Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method

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A high-order numerical manifold method with nine-node triangular meshes

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