Representations based on hierarchical partitioning have been widely used in computer sciences (tree-codes) and applied mathematics (FFT). For the past 20 years, such methods have dramatically changed the way N-body simulations in physics and mechanics are conducted. Typical classes of problems involve gravitational attraction, molecular dynamics, and micromechanics of heterogeneous fluids and solids. For all of them, hierarchical partitioning allows one to reduce the operation count from O(N2) to O(N) or O(NlogN).
This lecture will be divided into three parts. First, I will explain basic ideas behind hierarchical partitioning methods, using Coulomb's electrostatic interactions as the model problem. Then, I will demonstrate how those methods can be extended to micromechanical simulations of Stokesian emulsions. Finally, I will present recent result establishing a link between hierarchical partitioning and homogenization. Those results generalize classical Hill's results on macroscopic stress and strain.