Surface Growth Driven by an Optimality Criterion

TL;DR

This paper introduces a variational framework for modeling surface growth driven by an optimality principle, using constrained minimization and gradient flow concepts, with applications to elastic beams.

Contribution

It develops a novel variational approach for accretive surface growth that replaces kinetic laws with an optimality-based minimization framework.

Findings

Growth modeled as a constrained minimization problem.

Residual stresses can cause nonuniqueness and localization.

A formal limit yields a constrained gradient flow.

Abstract

We propose a variational framework for accretive surface growth driven by an optimality principle. Rather than prescribing a kinetic law, the configuration at each time step is obtained, within a time-discrete setting, as the solution of a constrained minimization problem. Growth is modeled as an irreversible surface deposition process subject to a global mass constraint, while the driving mechanism is encoded in an objective functional, here taken to be the structural mean compliance. The approach is illustrated on a linearly elastic cantilever beam whose cross-sectional height evolves through layered accretion, possibly involving prestrain and precurvature. Growth-induced residual stresses can alter the convexity of the compliance functional, leading to nonuniqueness and localization phenomena. We explore the possibility of adding a regularization term penalizing deviations from the previous-step configuration. Finally, through a formal limiting procedure, we derive from the time-discrete formulation a time-continuous limit in the form of a constrained gradient flow.