Constructal Evolution as a Nonsmooth Dynamical System: Stability and Selection of Flow Architectures

TL;DR

This paper models Constructal Law as a nonsmooth dynamical system, proving the existence, uniqueness, and stability of optimal flow architectures without static optimization, and embedding classical transport hierarchies within this framework.

Contribution

It introduces a novel dynamical systems approach to Constructal evolution, incorporating irreversibility and regime switching, and proves convergence to a unique, stable architecture.

Findings

Existence and uniqueness of a globally stable flow architecture.

Convergence of all trajectories to the equilibrium architecture.

Classical transport hierarchies are embedded as invariant sets in the dynamical framework.

Abstract

Constructal Law states that a finite-size flow system that persists in time evolves its configuration so as to provide progressively easier access to the currents that flow through it. Classical Constructal theory derives hierarchical flow architectures from static resistance minimization under finite-size constraints, but many transport systems operate under irreversible limits that induce regime switching and discontinuous adjustment laws. We formulate Constructal evolution as an autonomous nonsmooth dynamical system. The architectural configuration is modeled as the state of a Filippov differential inclusion defined on a compact forward-invariant admissible set. Irreversible transport constraints generate switching manifolds across which the adjustment field is discontinuous. A resistance dissipation inequality encodes the Constructal principle of progressively improving access as a nonsmooth Lyapunov condition, while a uniform contraction assumption provides spectral bounds on the generalized Jacobians of the regime-dependent dynamics. Under these conditions we prove that the resulting inclusion admits a unique equilibrium architecture and that every admissible trajectory converges to it exponentially. Finite size, irreversibility, and resistance dissipation therefore imply existence, uniqueness, and global stability of persistent flow configurations without invoking static optimization. As an application, the classical area--to--point transport hierarchy of Bejan et. al. is embedded in the dynamical framework. The optimal assembly ratios appear as switching manifolds, while the classical scaling relations arise as sliding invariant sets of the Filippov inclusion. Their intersection defines the uniquely selected globally attracting architecture.