A hybrid FE-MLPG method to simulate stationary dynamic and propagating quasi-static cracks

•A coupled FE-MLPG method is introduced to deal with 2D mixed-mode stationary and propagating cracks.•A simple approach is proposed to determine the crack tip and add the computational nodes.•Polygonal test function domains are constructed locally at the crack tip to calculate the contour and domain integrals.•Simple and more complex structures are solved with single and multiple cracks. This article describes a combined way based on FE and MLPG1 methods to numerically solve stationary dynamic fracture and quasi-static linear elastic crack propagation problems. The main goal of this research is to take the advantages of two numerical techniques in fracture mechanics. So, the Local Petrov–Galerkin meshfree method with thin plate spline radial basis function is employed for the discretization of non-classic linear elastic thermoelasticity governing equations. The method is equipped with triangular background cells organized throughout a solid to assist in the construction of the polygonal integration boundary-domain and defining the total geometry. The grid is also used to propose a neighbor point determination rule at Gauss points without using average nodal spacing computations, geometry parameters and the well-known visibility method around the crack tip. Several examples including stationary and propagating cracks are presented to show the effectiveness of the hybrid numerical method. Computational accuracy is investigated by error estimation analyses and qualitatively comparing the results of this method with different analytical and numerical solutions available in the literature. Wave propagation is also considered by providing the temporal distribution of stress and thermal waves in different cracked structures.