Using Gurtin-Murdoch theory and complex potential method, the problem of secondary multiple rips in one-dimensional hexagonal quasicrystals take nano n-edge polygon orifices is studied. The analytical solutions of phonon field, phasor field and electric field, as well as phonon field stress intensity factors and energy release rate are obtained. Some calculations are given to discuss the effects of secondary crack morphology of nano orifice on field intensity factor and energy release rate. The results indicate that when the defect size asymptotically the nanometer level, the surface effect produced by the coupling of phonon field, phase field and electric field, while the smaller the size of secondary crack at the orifice, the stronger surface effect. The more the number of cracks, the smaller the field intensity factor. With the amplify of defect size, the influence by surface influence will gradually weaken, and eventually tends to the existing outcome.

A local discontinuous Petrov-Galerkin method is developed for nonlinear Schrödingerequations. Several kinds of solitons are simulated and related phenomena are discussed, such as the soliton propagation and collision, birth of solitons including standing soliton and mobile soliton, the bound state of N solitons. The algorithm simulates some narrow structures in soliton related phenomenon. Numerical examples show that the algorithm has high accuracy and can reach the optimal convergence order. Compared with local discontinuous Galerkin method, the local discontinuous Petrov-Galerkin method has high computational efficiency and simple computational formula.

Interreaction between a penetrating lip-shaped crack and a screw dislocation in magnetoelectroelastic media is investigated with Muskhelishvili techniques and perturbation technique. Analytic solutions for stress, magnetic induction and electric displacement caused by dislocation and lip-shaped crack in magnetoelectroelastic solid are derived. By using the generalized Peach-Koehler formula, image forces acting on a dislocation are calculated. Regularities of field intensity factors, generalized stress field and image force at different dislocation positions are obtained with numerical examples.