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In this paper, the Newton-Krylov method is explored for robust and efficient time-domain simulation of integrated circuits with large amount of parasitic elements. Different from LU-factorization-based direct methods used in SPICE-like circuit simulators, the Newton-Krylov method uses a preconditioned Krylov-subspace iterative method for solving linearizedcircuit equations. A key contribution of this paper is to introduce an effective quasi-Newton preconditioning scheme for Krylovsubspace methods to reduce the number and cost of LU factorization during an entire time-domain circuit simulation. The proposed quasi-Newton preconditioning scheme consists of four key techniques: 1) a systematic method for adaptively controlling time step sizes; 2) automatically generated piecewise weakly nonlinear (PWNL) definition of nonlinear devices to construct quasi-Newton preconditioners; 3) low-rank update techniques for incrementally updating preconditioners; and 4) incomplete-LU preconditioning for efficiency. Experimental results on a collection of digital, analog, and RF circuits have shown that the quasi-Newton preconditioned Krylov-subspace method is as robust and accurate as the direct method used in SPICE. The proposed Newton-Krylov method is attractive for simulating circuits with massive parasitic RLC elements for postlayout verification. For a nonlinear circuit with power/ground networks with tens-ofthousand elements, the CPU time speedup over SPICE3 is over 20X, and it is expected to increase further with the circuit size.

Current practice for determining the scour depth at a bridge crossing is based on the equilibrium scour depth of a design flood (e.g., 50-year, 100-year, and 500-year flood events), which is unnecessarily larger than a real maximum scour depth during a bridge life span since the peak flow period of a flood event is often much shorter than the corresponding scour equilibrium time. The objective of this study was to present a design method for time-dependent scour depth under bridge-submerged flow . To this end, a series of flume experiments on scour depth under bridge-submerged flow were conducted to collect scour data at different times. A semi-empirical model for estimating time-dependent scour depth was then presented based on the mass conservation of sediment, which agrees very well with the collected data. The proposed method can appropriately reduce the design depth of bridge scour according to design flow and a peak flow period, which can translate into significant savings in the construction of bridge foundations.

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