2D conventional MPM uses a regular grid with uniform square cells resulting in limitation in dealing with inclined cracks and dislocations. A new algorithm with irregular mesh has been developed for Material Point Method to allow the use of arbitrary quadrilateral cells in the background grid mesh. This new algorithm has been applied to solve mixed mode crack problems in computational fracture mechanics. Fracture parameters have also been investigated from both MPM and FEM simulations. A new algorithm is developed to enable MPM to use arbitrary quadrilateral cells in the background grid mesh. Ray crossing algorithm is employed to determine which cell in the background grid mesh a particle belongs after the deformation. Validation was first made with a tension problem and compared with the results obtained from ABAQUS/explicit code. An inclined crack problem is solved as an example to demonstrate the capability of the arbitrary quadrilateral cells in the new algorithm. The same problem is also solved using ABAQUS to validate the new algorithm. For dynamic mixed mode crack fracture, the energy release rates were calculated from MPM. These results are in good agreement with FEM. Mode I and Mode II stress intensity factors were also calculated from MPM simulations.