Previous studies on base isolated systems have shown susceptibility to large displacements at the base level, leading to pounding between structure and perimeter moat wall during a maximum considered earthquake. Undesirable pounding can lead to very high acceleration in the superstructure. To prevent pounding, a phased passive control device, or gap damper, has been developed. By introducing an appropriate initial gap, the gap damper does not engage for earthquakes up to design level, but triggers additional energy dissipation after a threshold displacement has been reached to limit the ultimate displacement in large earthquakes. This paper will summarize previous research on the gap damper, which showed that using viscous damping alone or in combination with hysteretic damping can provide desired additional energy dissipation, reducing displacements in the superstructure. Since a trade-off occurs between effort to decrease base level displacement and roof acceleration increase, performance objectives for both displacement and acceleration were introduced. Based on the performance criteria, the parameters of the gap damper that produced the smallest performance index were identified through an optimization study. In this paper, the influence of residual displacement on the performance of the gap damper is examined. Numerical simulation of base isolation systems with gap dampers indicates that base isolators can re-center, while in general the gap dampers have a permanent offset. When the gap damper is activated after reaching the threshold displacement, it will move until the loading direction reverses. The gap damper will not change its position until the threshold displacement is reached in the reverse direction. After the last impact, the isolation system tends to re-center, but the gap damper will be permanently offset. Therefore, residual displacements for the base isolators are small, but considerable residual displacements are observed in the gap dampers in some cases. If not re-centered, such phenomenon would affect the initial gap distance of the system if a second large ground motion were to occur. To observe how residual displacement affects the gap damper performance, the base isolation system with gap damper was simulated with and without residual displacement. A range of initial offsets, corresponding to residual displacement, was assigned during modeling and compared with the case of no initial offset. Displacement and acceleration histories suggested there is a noticeable difference between the two cases only until the gap damper is first activated. Once activated, the gap damper with initial offset converges to the same displacement history of the gap damper with no offset, and the system hysteresis becomes similar to the hysteresis of the system with no initial offset. The optimization study also showed that the residual displacement reduces the performance index by 1-8%, which is minor. This numerical study is part of an overall research program to develop and test a gap damper that can control the displacements of base-isolation systems in MCE ground shaking. A companion paper describes the development and experimental testing of a physical gap damper device.