A fast boundary element method for the solution of periodic many-inclusion problems via hierarchical matrix techniques

Abstract

Abstract. Our work is motivated by numerical homogenization of materials such as concrete, modeled as composites structured as randomly distributed inclusions imbedded in a matrix. In this paper, we propose a method for the approximation of the periodic corrector problem based on boundary integral equations. The fully populated matrices obtained by the discretization of the integral operators are successfully dealt with using the H-matrix format.Résumé. Nous nous intéressonsà l'homogénéisation de matériaux de type béton, c'està dire composés d'agrégats distribués aléatoirement dans une matrice. Nous proposons une méthode pour la résolution du problème de correcteur périodique en s'appuyant sur une formulation intégrale surfacique. Les matrices pleines obtenues par la discrétisation des opérateurs intégraux sont traitées avec succèsà l'aide de matrices hiérarchiques.

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

“…We believe that an elastostatic version [12,36] would be very similar to what we present for Stokes. We believe that an elastostatic version [12,36] would be very similar to what we present for Stokes.…”

“…The macroscopic response of a given microscopic periodic composite medium can often be summarized by an effective material property (e.g., a conductivity or permeability tensor), a fact placed on a rigorous footing by the field of homogenization (for a review see [14]). Application areas span all of the major elliptic PDEs, including the Laplace equation (thermal/electrical conductivity, electrostatics and magnetostatics of composites [12,27,35,38]); the Stokes equations (porous flow in periodic solids [18,26,50,75], sedimentation [1], mobility [69], transport by cilia carpets [16], vesicle dynamics in microfluidic flows [58]); elastostatics (microstructured periodic or random composites [29,36,61,64]); and the Helmholtz and Maxwell equations (phononic and photonic crystals, bandgap materials [42,65]). Application areas span all of the major elliptic PDEs, including the Laplace equation (thermal/electrical conductivity, electrostatics and magnetostatics of composites [12,27,35,38]); the Stokes equations (porous flow in periodic solids [18,26,50,75], sedimentation [1], mobility [69], transport by cilia carpets [16], vesicle dynamics in microfluidic flows [58]); elastostatics (microstructured periodic or random composites [29,36,61,64]); and the Helmholtz and Maxwell equations (phononic and photonic crystals, bandgap materials [42,65]).…”

“…Furthermore, to obtain accurate average material properties for random media or suspensions via the Monte Carlo method, thousands of simulation runs may be needed [38]. Integral equation methods [45,49,55,67] are natural, since the PDE has piecewise constant coefficients, and are very popular [1,12,26,27,36,50,64,69]. 2.8]), including particular solutions (starting in 1892 with Rayleigh's method for cylinders and spheres [70], and, more recently, in Stokes flow [71]), eigenfunction expansions [75], lattice Boltzmann and finite differencing [48], and finite element methods [29,61].…”

Lippmann‐Schwinger solvers for the explicit jump discretization for thermal computational homogenization problems

“…The jump equation (8) serves as the starting point for boundary integral techniques. 24,25 A major benefit lies in the reduction of the degrees of freedom to the interface. To enable processing of image data, Wiegmann and Zemitis 26 introduced the explicit jump immersed interface method (EJIIM), a finite difference discretization on a regular grid building upon earlier work of LeVeque and Li.…”

“…The jump equation serves as the starting point for boundary integral techniques . A major benefit lies in the reduction of the degrees of freedom to the interface.…”