Constructal design under discontinuities: Emergence and stability of flow architectures under irreversible constraints

Constructal Law states that a finite-size flow system that persists in time must evolve its configuration so as to provide progressively easier access to the currents that flow through it. Classical Constructal theory typically derives hierarchical transport architectures from static resistance minimization under finite-size constraints. However, irreversible transport limits may generate switching regimes and nonsmooth transport responses for which smooth variational formulations are insufficient. This paper develops a dynamical formulation of Constructal Law within the framework of Filippov differential inclusions. The evolving flow architecture is represented as a nonsmooth dissipative dynamics on a compact forward-invariant admissible configuration set. Irreversible constraints generate switching manifolds across which the governing architectural vector field becomes discontinuous. A resistance dissipation inequality formalizes the principle of progressively easier access as a nonsmooth Lyapunov condition, while regime-wise contraction conditions provide verifiable spectral bounds for the induced transport dynamics. Under these assumptions, we establish global exponential convergence of all admissible trajectories toward a unique invariant architecture. Finite size, irreversibility, and resistance dissipation therefore imply existence, uniqueness, and asymptotic stability of persistent flow configurations through admissible nonsmooth evolution. As an application, the classical area–to–point transport hierarchy of Bejan–Bădescu–De Vos is embedded within the proposed framework. The classical Constructal scaling relations emerge as sliding invariant states of the Filippov inclusion, and their intersection defines a unique globally attracting transport architecture. Constructal hierarchy is therefore characterized not only as a variational optimum, but as the stable asymptotic outcome of dissipative regime-dependent transport dynamics.