The prediction of crack propagation is an important engineering problem. In this work, combined with triangular plane stress finite elements, a new remeshing algorithm for crack opening problems was developed. The proposed algorithm extends the crack iteratively until a threshold maximum crack length is achieved. The crack propagation direction is calculated using the maximum tangential stress criterion. In this calculation, in order to smoothen the stress field in the vicinity of the crack tip, a weighted average of the stresses of the integration points around the crack tip is considered. The algorithm also ensures that there are always at least eight elements and nine nodes surrounding the crack tip, unless the crack tip is close to a domain boundary, in which case there can be fewer elements and nodes around the crack tip.

Four benchmark tests were performed showing that this algorithm leads to accurate crack paths when compared to findings from previous research works, as long as the initial mesh is not too coarse. This algorithm also leads to regular meshes during the propagation process, with very few distorted elements, which is generally one of the main problems when calculating crack propagation with the finite element method.

A modified error indicator and an extended locally implicit scheme with anisotropic dissipation model on quadrilateral-triangular mesh are developed to study the supersonic flow over a backward-facing step. In the Cartesian coordinate system, the unsteady Euler equations are solved. The modified error indicator, in which the unified magnitude of density gradient and unified magnitude of gradient of vorticity magnitude are incorporated, is utilized to treat the new node spacing of mesh remeshing. To assess the accuracy of the extended locally implicit scheme with anisotropic dissipation model on quadrilateral-triangular mesh, two flow calculations which include the oblique-shock reflection at a wall and transonic flow around an NACA 0012 airfoil are performed. Based on the comparison with the related numerical and experimental data, the accuracy of the present approach is confirmed. According to the high-resolutional result on the adaptive mesh, the structure of backstep corner vortex, expansion wave and oblique shock wave is distinctly captured.