In the laser-assisted fiber cleaving process, an ultrashort, high-energy laser is used to introduce a flaw on the optical fiber surface, and then the fiber is cleaved under a tension or bending load. The quality of the cleaved end depends on the crack propagation from the artificial flaw. To understand the cleaving process and the effects of flaw parameters on cleaving quality, crack propagation is modeled by considering a pre-existing surface flaw using a graph-based finite element method (GraFEA). GraFEA is based on the nonlocal multiple cracking simulation framework for brittle and quasi-brittle materials. First, a three-point bending test is conducted to calibrate material parameters in GraFEA for commercial fused silica glass. Subsequently, the model is validated by four-point bending and ring-on-ring tests. After validation, the fiber cleaving process is investigated by parametric simulations in which different loading types (bending or tension) are considered. Finally, a beneficial process window is obtained and recommended for improved cleaving quality.