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

Pascal Stiefenhofer

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.

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

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.
Paper Structure (72 sections, 6 theorems, 145 equations, 2 tables)

This paper contains 72 sections, 6 theorems, 145 equations, 2 tables.

Table of Contents

  1. Keywords.
  2. Mathematics Subject Classification (2020).
  3. Introduction
  4. A Dynamical Formulation of Constructal Law
  5. Architectural dynamics
  6. Persistence and viability
  7. Constructal dissipation
  8. Interpretation
  9. Architectural State Space and Conserved Flow
  10. Irreversibility and Architectural Persistence
  11. Constructal Access Functional
  12. Regularity of the resistance functional.
  13. Constructal Dissipation Inequality
  14. Monotone resistance decay.
  15. Asymptotic balance.
  16. ...and 57 more sections

Key Result

Proposition 2.11

Let $K\subset\mathbb{R}^n$ be nonempty, closed, and convex, and let $\mathcal{R}:K\to\mathbb{R}$ be locally Lipschitz. Define the set-valued map where $T_K(x)$ denotes the Bouligand tangent cone of $K$ and $\Pi_{T_K(x)}$ is the Euclidean projection onto it. Define the residual function where $N_K(x)$ denotes the Clarke normal cone to $K$. Then the dissipation inequality holds. Consequently Assu

Theorems & Definitions (14)

  • Proposition 2.11: Projected subgradient sufficiency
  • proof
  • Proposition 2.14: Contraction induced by resistance reduction: constant mobility
  • proof
  • Remark 2.15: Extension across switching via Filippov convexification
  • Theorem 2.17: Global incremental exponential stability
  • proof
  • Theorem 2.18: Constructal architecture selection principle
  • proof
  • Proposition 2.20: Gradient-type structural sufficiency
  • ...and 4 more