The prediction of the crack path is still challenging in the field of fracture mechanics. A simple experiment firstly performed by Yuse and Sano (1993) has attracted many attentions because of the reproducible controlled crack propagation. In this experiment, a glass plate was moved between a heater and a water bath at a constant velocity. Due to the thermal gradients in a glass plate, an initial crack placed on the bottom of a glass plate propagated under the Rayleigh wave speed. Depending on the controlling parameters (e.g., size of a glass plate, quenching speed, temperature gradient), the different crack morphology such as straight cracks, oscillating cracks, and branching cracks were observed; increasing the quenching speed and the temperature gradient changed crack morphology from simple to complex.In order to explain the mechanism for this instability of the crack growth, many experimental and theoretical investigations have been proposed. Although the experiment is very simple, the effective explanation for the multiple transitions of the crack morphology has not been proposed. It is therefore the numerical analysis is essential for fully understanding of the mechanism of the crack transition in a quenched glass plate.In this study, we perform the numerical analysis to reproduce the multiple transitions of the crack morphology concerning of this experiment. We employ PDS-FEM (Particle Discretization Scheme Finite Element Method) which is a fracture analysis method which can treat deformation and fracture of the solid continuum in the framework of linear elastic fracture mechanics to simulate the crack propagation in a quenched glass plate.In the results of the numerical analysis, the transitions of the crack morphology (i.e., straight cracks, oscillating cracks, and branching cracks) were observed by controlling the quenching speed and temperature gradient. In addition, the change in the amplitude of the oscillating cracks depending on the temperature gradient was also reproduced. These features of the crack morphology coincide with the experimetal observations. These results indicate that the proposed numerical analysis method captures the fundamental mechanism for the crack instability in a glass plate with thremal gradients.