Existing solutions for continuous systems with localized, non-smooth nonlinearities (such as impacts) focus on exact methods for satisfying the nonlinear constitutive equations. Exact methods often require that the non-smooth nonlinearities be expressed as piecewise-linear functions, which results in a series of mapping equations between each linear regime of the nonlinearities. This necessitates exact transition times between each linear regime of the nonlinearities, significantly increasing computational time, and limits the analysis to only considering a small number of nonlinearities. A new method is proposed in which the exact, nonlinear constitutive equations are satisfied by augmenting the system's primary basis functions with a set of non-smooth basis functions. Two consequences are that precise contact times are not needed, enabling greater computational efficiency than exact methods, and localized nonlinearities are not limited to piecewise-linear functions. Since each nonlinearity requires only a few non-smooth basis functions, this method is easily expanded to handle large numbers of nonlinearities throughout the domain. To illustrate the application of this method, a pinned–pinned beam example is presented. Results demonstrate that this method requires significantly fewer basis functions to achieve convergence, compared with linear and exact methods, and that this method is orders of magnitude faster than exact methods.