Sandpiles and superconductors: dual variational formulations for critical-state problems

TL;DR

This paper develops dual variational formulations for critical-state models of sandpiles and superconductors, enabling simultaneous approximation of primary and dual variables, which enhances numerical simulations of these dissipative systems.

Contribution

It introduces dual variational formulations for sandpile and superconductor models, extending existing primary-variable models to include conjugate variables for better analysis.

Findings

Dual formulations facilitate numerical simulations.

Simultaneous approximation of primary and dual variables achieved.

Models applicable to systems with metastability, hysteresis, and long-range interactions.

Abstract

Similar evolutionary variational inequalities appear as convenient formulations for continuous models for sandpile growth, magnetization of type-II superconductors, and evolution of some other dissipative systems characterized by the multiplicity of metastable states, long-range interactions, avalanches, and hysteresis. The origin of this similarity is that these are quasistationary models of equilibrium in which the multiplicity of metastable states is a consequence of a unilateral condition of equilibrium (critical-state constraint). Existing variational formulations for critical-state models of sandpiles and superconductors are convenient for modelling only the "primary" variables (evolving pile shape and magnetic field, respectively). The conjugate variables (the surface sand flux and the electric field) are also of interest in various applications. Here we derive dual variational formulations, similar to mixed variational inequalities in plasticity, for the sandpile and superconductor models. These formulations are used in numerical simulations and allow us to approximate simultaneously both the primary and dual variables.