Applications of MLPG method in dynamic fracture problems.

*(English)*Zbl 1232.74086Summary: A new numerical algorithm based on the Meshless Local Petrov-Galerkin approach is presented for analyzing the dynamic fracture problems in elastic media. To simplify the treatment of essential boundary condition, a novel modified Moving Least Square (MLS) procedure is proposed by introducing Lagrange multiplier into MLS procedure, which can perform both MLS approximation and interpolation in one approximation domain. The compact spline function is used as the test function in the local form of elasto-dynamic equations. For the feature of stress wave propagation, the coupled second-order ODEs respect to the time are solved by the explicit central difference method with lumped mass matrix. By adopting the modified MLS techniques, the essential boundary conditions can be simply introduced by direct row cross-out method. In order to improve the accuracy of this algorithm, quadratic basis and Gaussian weight function with a support domain larger than test functions are used in the approximation. Visibility criterion is used for presenting the discontinuous fields caused by cracks. The dynamic stress intensity factors in various modes are evaluated through a path-independent dynamic \({J}'\) integral method, and the complicated diffractions of the stress waves near cracks are investigated in detail. The numerical results show that the present method is easy to implement, highly accurate and efficient for the problems considered.

##### MSC:

74R10 | Brittle fracture |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

74S30 | Other numerical methods in solid mechanics (MSC2010) |