The answer above by Simon does address Steiner's much newer and immensely more computable structure, the Augmented Directed Complexes, but he doesn't really cover its benefits over all of the previous methods. ADCs have been used to extend the lax/oplax tensor product to strict $\omega$-categories as well as the lax/oplax join and its various slice adjoints by Dimitri Ara and Georges Maltsiniotis because working with chain complexes of free modules is, all told, pretty easy.
It reduces a lot of the combinatorics of pasting to doing linear algebra in a chain complex of free abelian groups with a particular basis. The morphisms are just morphisms that preserve the free commutative monoids of 'positive elements' generated by the basis. Steiner does dip back into some of his earlier directed complex work in order to prove that Augmented Directed Complexes with a unital loop-free basis embed fully and faithfully into strict $\omega$-categories and are dense in it.
This answer is mostly supplementary to Simon's but if you're looking into working with pasting diagrams, I would highly highly recommend you look into the work of Steiner.
Also important is the fact that in Steiner's case, because the embedding is full and faithful, you can find 'relevant' elements as the images of maps from the ADC associated with an $n$-globe.