HOMMX: Heterogeneous Multi-Scale Method with DOLFINx

A Python package implementing the Heterogeneous Multi-Scale Method (HMM) for solving multi-scale PDEs using DOLFINx and PETSc.

Overview

HOMMX provides solvers for multi-scale problems, in particular for the Poisson equation and linear elasticity with rapidly oscillating coefficients. The package implements the HMM to efficiently compute homogenized solutions without fully resolving the microscopic scales.

Poisson

$$ \begin{equation} \mathrm{div}(A_\varepsilon\nabla u) = f \end{equation} $$

where $A_\varepsilon(x) = A(x, x/\varepsilon)$.

Additionally it includes an implementation for a stratified periodic coefficient.

Linear elasticity

$$ \begin{equation} \sum_{jkl}\frac{\partial}{\partial x_j}(A_{\varepsilon, ijkl}\frac{\partial u_k}{\partial x_l} ) = f_i \end{equation} $$

where $A_{\varepsilon, ijkl}(x) = A_{ijkl}(x, x/\varepsilon)$.

Key Features

• PoissonHMM: HMM solver for Poisson problems with periodic micro-structure
• PoissonSemiHMM: HMM solver for Poisson problems with stratified periodic micro-structure
• LinearElasticityHMM: HMM solver for Linear Elasticity with periodic micro-structure
• Support for 2D and 3D domains
• Integration with DOLFINx's modern FEM framework
• Parallelization via MPI

Usage

For now, please refer to the examples or take a look at the tests.

Installation

hommx is available as a conda package on Prefix. To install it simply run

conda install -c https://repo.prefix.dev/flxrcz-forge hommx