muFFTTO

FFT-based micro-scale topology optimization on periodic unit cells using the muGrid / muFFT framework.

Features

• Topology optimization: Phase-field based optimization with adjoint sensitivity analysis
• Multi-dimensional: Supports 1D, 2D, and 3D problems
• FEM discretization: Triangular and hexahedral finite elements with quadrature-based integration
• FFT solvers: Matrix-free conjugate gradient solvers accelerated by FFT (via muGrid)
• MPI parallelism: Distributed computing support through muGrid's MPI infrastructure
• Microstructure library: Built-in parametric geometries (laminates, inclusions, lattices, etc.)

Dependencies

• Python >= 3.10
• numpy
• scipy
• muGrid
• NuMPI
• mpi4py
• JAX
• matplotlib

Installation

To install muFFTTO, you can use pip:

git clone https://github.com/imtek-simulation/muFFTTO.git
cd muFFTTO
pip install .

For development, you can install it in editable mode:

pip install -e .

Installing dependencies

The core dependencies numpy and scipy will be installed automatically. muGrid must be installed separately. You can install them from source:

pip install git+https://github.com/muSpectre/muGrid.git

Package Structure

Module	Description
domain.py	Core classes PeriodicUnitCell and Discretization for setting up the computational domain, fields, and operators
topology_optimization.py	Objective functions, sensitivity analysis (adjoint method), and phase-field potentials for elasticity
topology_optimization_conductivity.py	Topology optimization for thermal conductivity problems
solvers.py	Preconditioned conjugate gradient (PCG) and Adam optimizer for solving linear systems and optimization problems
solvers_nonlinear.py	Nonlinear solver implementations for advanced optimization
discretization_library.py	Shape function gradient matrices for various element types (linear triangles, bilinear rectangles, trilinear hexahedra)
material_models.py	Abstract base classes and implementations for material constitutive models
geometry.py	Periodic microstructure geometries: composable shapes, phase fields and named geometries (MPI-parallel)
microstructure_library.py	Backward-compatible get_geometry wrapper used by the examples
grid_adaptation_methods.py	Methods for adaptive mesh refinement and grid adaptation
grid_adaptation_arbitrary.py	Arbitrary grid adaptation strategies
analytical_grid_adaptation.py	Analytical solutions for grid adaptation
tensor_operations.py	Utility functions for tensor operations and indexing
visualization_utils.py	Visualization and post-processing utilities
io_utils.py	Save/load fields to NetCDF (MPI-parallel, muGrid FileIONetCDF): snapshots, time series, numpy reader

Documentation

The docs/ folder explains the mathematics behind the code and how each example works:

• Theory and numerics: the periodic cell problem, FE discretization on a pixel grid, the FFT-preconditioned CG solver, effective properties, adjoint topology optimization and grid adaptation
• Example walkthroughs in docs/examples/, one page per example folder
• Known issues: suspected bugs found while documenting the code

Examples

The examples/ directory contains working examples organized by topic:

Homogenization (examples/homogenization/)

• Thermal conductivity: 2D and 3D examples with various discretization methods
• Small-strain elasticity: 2D and 3D homogenization with FEM

Topology Optimization (examples/topology_optimization/)

• 2D and 3D topology optimization for elasticity and conductivity
• Grid tiling and adaptive refinement strategies

Analytical Solutions (examples/analytical_solutions/)

• Validation against Hashin analytical bounds for composite materials

Grid Adaptation (examples/grid_adaptation/)

• Examples of mesh refinement and coordinate transformations

Internal Contact (examples/internal_contact/)

• Contact mechanics and third-medium interaction problems

Phase-Field Fracture (examples/phase_field_fracture/)

• AT2 brittle fracture of a periodic porous cell under macroscopic strain (staggered scheme, history field)

Authors

• Martin Ladecky (University of Freiburg)
• Lars Pastewka (University of Freiburg)

License

MIT - see LICENSE.md

Funding

This development has received funding from the European Commission (Marie Sklodowska-Curie Fellowship 101106585 — microFFTTO).