[Paper Review] Optimal packing of spheres in $\mathbb R^d$ and extremal effective conductivity

Vladimir Mityushev|arXiv (Cornell University)|Dec 23, 2014

Advanced Mathematical Modeling in Engineering6 references3 citations

TL;DR

This paper establishes a connection between optimal sphere packing in $ℝ^d$ and the minimization of effective conductivity in composites with ideally conducting spherical inclusions. By formulating the problem as a discrete energy minimization over Voronoi-Delaunay structures, it derives upper bounds for sphere packing densities and constructs a constructive algorithm for optimal sphere locations using structural approximations and $p$-Laplacian homogenization.

ABSTRACT

Optimal packing of spheres in $\mathbb R^d$ is studied by optimization of the energy $E$ (effective conductivity) of composites with ideally conducting spherical inclusions. It is demonstrated that the minimum of $E$ over locations of spheres is attained at the optimal packing. The energy is estimated in the framework of structural approximations. This method yields upper bounds and sometimes exact values for the maximal concentrations of spheres in $\mathbb R^d$. A constructive algorithm for the optimal locations of spheres associated to the classes of the Delaunay graphs is constructed.

Motivation & Objective

• To establish a correspondence between optimal sphere packing in $ℝ^d$ and the minimization of effective conductivity in periodic composites.
• To develop a structural approximation framework for $p$-Laplacian homogenization in $ℝ^d$ that enables energy-based estimation of packing efficiency.
• To construct a computational algorithm for determining optimal sphere configurations based on Delaunay graph classes and lattice periodicity.
• To derive upper bounds for maximal sphere packing densities in $ℝ^d$ using discrete energy minimization and validate them for laminated structures.
• To provide a constructive method for computing optimal sphere locations by solving decoupled linear systems derived from the energy functional.

Proposed method

• Formulates the effective conductivity problem as a discrete energy minimization: $\min_{t_j} \sum_{k,j} g_{kj}^{(0)} |t_k - t_j|^2$, where $g_{kj}^{(0)}$ represents the leading-order interparticle flux.
• Applies structural approximation theory to $p$-Laplacian systems in $ℝ^d$, assuming non-linear conductivity with $p > \frac{d+1}{2}$ to ensure flux singularities as interparticle distance $\to 0$.
• Uses a toroidal topology on the periodicity cell $Q_0$ with fundamental vectors $\boldsymbol{\nu}_j$, defining distances via periodic identification to model infinite periodic composites.
• Derives a linear system (4.8) for optimal sphere locations $\mathbf{a}_k^*$ by solving the energy minimization under quasi-periodic potential and periodic flux conditions.
• Decomposes the system into $d$ independent $n$-dimensional linear systems by separating coordinates, enabling efficient numerical solution.
• Constructs optimal configurations via $\mathbf{a}_k = \sum_{\ell=1}^d \mathbf{a}_k^{(\ell)} \boldsymbol{\nu}_\ell$, where $\mathbf{a}_k^{(\ell)}$ are solutions to coordinate-wise systems (5.2), preserving lattice symmetry.

Experimental results

Research questions

• RQ1 Can the optimal sphere packing problem in $\u211d^d$ be solved via minimization of effective conductivity in composites with ideal inclusions?
• RQ2 How does the discrete energy functional based on $p$-Laplacian homogenization relate to geometric packing configurations?
• RQ3 What is the role of Delaunay graph classes in determining optimal sphere arrangements and upper bounds for packing density?
• RQ4 Under what conditions does the solution to the energy minimization problem yield macroscopically isotropic structures?
• RQ5 Can the structural approximation framework be extended to non-periodic or non-spherical inclusions, and what are the computational implications?

Key findings

• The minimum of the discrete energy functional over sphere locations corresponds exactly to the optimal sphere packing configuration in $\u211d^d$.
• For laminated structures, the derived upper bounds on sphere packing density are exact, providing analytical solutions within those classes.
• The energy minimization problem reduces to solving $d$ independent linear systems of size $n$, enabling efficient computation of optimal configurations.
• The optimal sphere locations $\mathbf{a}_k^*$ are obtained as linear combinations of lattice vectors $\boldsymbol{\nu}_\ell$, with coefficients computed via coordinate-wise solutions.
• The method yields upper bounds for maximal sphere packing density in $\u211d^d$, with exact values achievable in structured classes like laminated arrangements.
• A necessary condition for macroscopic isotropy is the existence of $d$ percolation chains connecting opposite faces of the periodic cell $Q_0$, which can be checked numerically.