Cloud-to-ground lightning discharges in 2D and 3D domains were simulated using a stochastic dielectric breakdown model. The dependency between fractal dimension of the discharge patterns and the power of the local electric field η was critically evaluated. An exponential decrease in fractal dimension was observed as η increases. Fractal dimension of simulated 3D discharge patterns and 2D images of lightning discharges were compared by taking projections of simulated patterns. Discharge patterns similar to actual lightning were obtained when η ≈ 5.2. Influence of ground objects on simulated lightning discharges was also studied by introducing additional boundary conditions to the ground plane. It was observed that pointed structures on the ground have a higher probability of attracting simulated lightning discharges. An extension was introduced to dielectric breakdown model to simulate the development of upward connecting positive leader discharges that occur during the decent of a downward moving negative leader. It was found that the height of the stepped leader tip above the ground (at the time when the upward connecting leader initiation occurs) is dependent on the initial breakdown voltage threshold. The height of the point of interception was found to decrease exponentially as the breakdown threshold is increased.