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...

This paper presents the response and the wake modes of a freely vibrating D-section prism with varying angles of attack ( $\alpha = 0^\circ \text {--}180^\circ$ ) and reduced velocity ( $U^* = 2\text {--}20$ ) by a numerical investigation. The Reynolds number, based on the effective diameter, is fixed at 100. The vibration of the prism is allowed only in the transverse direction. We found six types of response with increasing angle of attack: typical vortex-induced vibration (VIV) at $\alpha = 0^\circ \text {--}35^\circ$ ; extended VIV at $\alpha = 40^\circ \text {--}65^\circ$ ; combined VIV and galloping at $\alpha = 70^\circ \text {--}80^\circ$ ; narrowed VIV at $\alpha = 85^\circ \text {--}150^\circ$ ; transition response, from narrowed VIV to pure galloping, at $\alpha = 155^\circ \text {--}160^\circ$ ; and pure galloping at $\alpha = 165^\circ \text {--}180^\circ$ . The typical and narrowed VIVs are characterized by linearly increasing normalized vibration frequency with increasing $U^*$ , which is attributed to the stationary separation points of the boundary layer. On the other hand, in the extended VIV, the vortex shedding frequency matches the natural frequency in a large $U^*$ range with increasing $\alpha$ generally. The galloping is characterized by monotonically increasing amplitude with enlarging $U^*$ , with the largest amplitude being $A^* = 3.2$ . For the combined VIV and galloping, the vibration amplitude is marginal in the VIV branch while it significantly increases with $U^*$ in the galloping branch. In the transition from narrowed VIV to pure galloping, the vibration frequency shows a galloping-like feature, but the amplitude does not monotonically increase with increasing $U^*$ . Moreover, a partition of the wake modes in the $U^*$ – $\alpha$ parametric plane is presented, and the flow physics is elucidated through time variations of the displacement, drag and lift coefficients and vortex dynamics. The angle-of-attack range of galloping is largely predicted by performing a quasi-steady analysis of the galloping instability. Finally, the effects of $m^*$ and ${\textit {Re}}$ , the roles of afterbody and the roles of separation point in determining vibration responses and vortex shedding frequency are further discussed.

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This paper numerically investigates particle saltation in a turbulent channel flow having a rough bed consisting of two to three layers of densely packed spheres. The Shields function is 0.065 which is just above the sediment entrainment threshold to give a bed-load regime. The applied methodology is a combination of three technologies, i.e., the direct numerical simulation of turbulent flow; the combined finite-discrete element modeling of the deformation, movement, and collision of the particles; and the immersed boundary method for the fluid-solid interaction. It is shown that the presence of entrained particles significantly modifies the flow profiles of velocity, turbulent intensities, and shear stresses in the vicinity of a rough bed. The quasi-streamwise-aligned streaky structures are not observed in the near-wall region and the particles scatter on the rough bed owing to their large size. However, in the outer flow region, the turbulent coherent structures recover due to the weakening rough-bed effects and particle interferences. First- and second-order statistical features of particle translational and angular velocities, together with sediment concentration and volumetric flux density profiles, are presented. Several key parameters of the particle saltation trajectory are calculated and agree closely with published experimental data. Time histories of the hydrodynamic forces exerted upon a typical saltating particle, together with those of the particle's coordinates and velocities, are presented. A strong correlation is shown between the abruptly decreasing streamwise velocity and increasing vertical velocity at collision which indicates that the continuous saltation of large-grain-size particles is controlled by collision parameters such as particle incident angle, local bed packing arrangement, and particle density, etc.

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