document: We shall establish the variational format in the space–time domain S=defΩ×I, for given spatial domain Ω and time domain I=(0,T), for a quite broad class of problems involving a first order time-derivative. In particular, the coupled problem of consolidation of geomaterials falls within this class. Another interesting application is the problem of dynamics, rewritten in first-order form, i.e. through a Hamiltonian description. It is of considerable interest to note from the outset that, due to the forward transport of information in time, it is always possible to consider a set of finite time intervals, whereby the solution at the end of any such interval will act as the initial data for the next one. To this end, we introduce a partition 0=t0<t1<⋯<tN=T of the considered time domain I=(0,T) into time-intervals In=(tn−1,tn) of length Δtn=tn−tn−1.11The abbreviated notation Δt=Δtn will be used henceforth for the current time step associated with In. Hence, we define space–time slabs Sn=defΩ×In such that the space–time domain can be given as S=defΩ×I=S1∪S2⋯∪Sn.
extractive_keyphrases: coupled problem of consolidation of geomaterials hamiltonian description space–time domain space–time slabs spatial domain ω
abstractive_keyphrases: first order time-derivative problem of dynamics, rewritten in first-order form s=defω×i s=defω×i=s1∪s2⋯∪sn sn=defω×in time domain i=(0,t) variational format in the space–time domain