Phase Field Approach to Optimal Packing Problems and Related Cheeger Clusters

Abstract

In a fixed domain of R N we study the asymptotic behaviour of optimal clusters associated to α-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter α converges to the "critical" value N −1 N + , optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute α-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions.

Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid

“…On the numerical aspects, there are several methods available in the Fig. 1 Relation between stress and strain in a Bingham fluid literature for the computation of limit loads [8] and Cheeger sets [6,7,9], and both of these problems are closely related to ours, as we shall see below.…”

“…with p z the pressure gradient along the vertical direction. The second equation in (6) expresses that for a steady fall motion, the gravity and buoyancy forces should be in equilibrium with the shear forces exerted by the fluid on each particle [32].…”

“…Notice that since we work inĤ , the no-slip boundary condition (7) and solid constitutive law (5) are automatically satisfied, and adequate testing directions are constant on connected components ofˆ s , which leads to the force balance condition in the second part of (6). Condition (8) is implied (in an appropriate weak form) by the fact thatω ∈ H 1 (ˆ ).…”

“…In this paper we consider anti-plane shear flow in an infinite cylinder, where an ensemble of inclusions move under their own weight inside a Bingham fluid of lower density, and in which the gravity and viscous forces are in equilibrium [cf (6)], therefore inducing a flow which is steady or stationary, that is, in which the velocity does not depend on time. For such a configuration, we are interested in determining the ratio between applied forces and the yield stress such that the Bingham fluid stops flowing completely.…”

Fast and Efficient Numerical Finite Difference Method for Multiphase Image Segmentation

“…Phase-field models have been actively studied until recently in various fields [3,4]. Bogosel et al [5] proposed an efficient phase-field method based on a multiphase Γ-convergence. Oudet and Santambrogio [6] presented the Modica-Mortola approximation for branched transport.…”