This article presents new methods for robustly simulating process flowsheets containing nondifferentiable models using recent advances in exact sensitivity analysis for nonsmooth functions. Among other benefits, this allows flowsheeting problems to be equipped with newly-developed nonsmooth inside-out algorithms for non-ideal vapor-liquid equilibrium calculations that converge reliability even when the phase regime at the results of these calculations is unknown a priori. Furthermore, process models for inherently nonsmooth unit operations may be seamlessly integrated into process flowsheets, so long as computationally-relevant generalized derivative information is computed correctly and communicated to the flowsheet convergence algorithm. These techniques may be used in either sequential-modular simulations or simulations in which the most challenging modules are solved using tailored external procedures while the remaining flowsheet equations are solved simultaneously. This new nonsmooth flowsheeting strategy is capable of solving process simulation problems involving nonsmooth models more reliably and efficiently than the algorithms implemented in existing software, and, in some cases, allows for the solution of problems that are beyond the capabilities of classical approaches. As examples of the latter, it will be shown that the nonsmooth approach is particularly well-suited for highly accurate simulation of natural gas liquefaction processes, in which many nonsmooth modeling elements are present in combination with non-ideal thermodynamic behavior and complex heat transfer considerations.