An enhanced flexible time-stepping scheme for the hybrid time-domain finite element method

Recently, a hybrid field-circuit solver that combines the capabilities of the time-domain finite element method (TDFEM) and a SPICE-like circuit analysis was developed for accurate and efficient characterization of complicated microwave circuits [1]. This field-circuit solver employs a system-wide global time-step size and thus samples and couples the signals of all the subsystems in a strict synchronous manner. Such a global time-stepping scheme is not optimal in many realistic applications, where signals in subsystems may have quite distinct temporal variations and require different signal-sampling rates. Numerical simulations may also require the time-step sizes of subsystems to satisfy different stability conditions and/or other practical concerns. Under these circumstances the global time-stepping scheme compromises the overall computational efficiency because it simply forces the coupled transient simulation to march on at a time step limited by the subsystem that has the strictest restriction on the time-step size. Therefore, a more flexible time-stepping scheme that allows local, subsystem-wide time-step sizes is of significant interest in order to alleviate the limitation of the original global time-stepping scheme and further improve the flexibility and efficiency of the existing field-circuit solver. Indeed, the idea of utilizing local time-step sizes for different subsystems and then asynchronously coupling them in time, sometimes referred to as multirate simulation, has been adopted widely in many areas. For a hybrid field-circuit analysis, a related approach for a time-domain integral equation (TDIE)-based hybrid electro-magnetic-circuit simulator was developed for modeling chip-to-package interconnects [2]. This paper presents a flexible time-stepping scheme for the TDFEM-based hybrid field-circuit solver that allows subsystems of different sampling rates to be coupled in time. Because of this flexibility, the proposed time-stepping scheme could significantly improve the computational efficiency of the existing TDFEM-based hybrid field-circuit solver especially when the computational cost associated with the slow subsystems is much larger than that associated with the fast subsystems. Moreover, the efficiency of the hybrid field-circuit simulation with the proposed scheme could be further enhanced when the tree-cotree splitting (TCS) technique [3] is applied to the TDFEM part to reduce the iteration count per time step for a preconditioned iterative solution when the time-step size for the FEM subsystem becomes relatively large.