Fourier-Accelerated Nodal Solvers (FANS)

Fourier-Accelerated Nodal Solver (FANS) is an FFT-based homogenization solver for microscale multiphysics problems. FANS is written in C++, built using CMake, and it has MPI parallelization.

Table of contents

• Dependencies
• Building
• Installing
• Input File Format

Dependencies

FANS has the following dependencies:

• A C++ compiler (e.g., GCC, Clang, etc.)
• CMake (version 3.21 or higher)
• Git (for cloning this repo)
• MPI (mpicc and mpic++)
• HDF5 with MPI support
• Eigen3
• FFTW3 with MPI support

Installing dependencies

• On Debian-based systems, we recommend installing the dependencies using apt,

bash apt-get install \ libhdf5-dev \ libopenmpi-dev \ libeigen3-dev \ libfftw3-dev \ libfftw3-mpi-dev

• On macOS, you can obtain the dependencies using brew and set the environment variables:

```zsh brew install gnu-time cmake gcc@14 brew install open-mpi --build-from-source --cc=gcc-14 brew install hdf5-mpi --build-from-source --cc=gcc-14 brew install fftw eigen

export CC=gcc-14 CXX=g++-14 MPICC=mpicc MPICXX=mpicxx ```

Setting up a Python environment

Also, we recommend setting up a Python virtual environment for the FANS_Dashboard.ipynb via pixi with all required Python dependencies in an isolated environment:

```bash

Install Pixi if not done already,

curl -fsSL https://pixi.sh/install.sh | sh

Create and activate the environment

pixi shell ```

We also provide a set of Docker images. For further information, please refer to the Docker README.

Installing dependencies using Spack

Spack is a package manager designed for high-performance computing environments. It simplifies the installation of complex software stacks, making it ideal for setting up FANS on HPC systems.

1. Install Spack by following these installation instructions.

2. Install Dependencies: Once Spack is set up, install the required dependencies:

   bash spack install cmake spack install mpi spack install hdf5 +cxx +mpi spack install eigen spack install fftw +mpi

   Additionally, optimized FFTW implementations can be used depending on your system's architecture, for example amdfftw (For AMD systems) or cray-fftw (For Cray systems), or fujitsu-fftw (For Fujitsu systems).

3. Load Dependencies Once dependencies are installed, load them before building:

   bash spack load cmake mpi hdf5 eigen fftw

Building

1. Clone the repository:

   bash git clone https://github.com/DataAnalyticsEngineering/FANS.git cd FANS

2. Configure the build using CMake:

   bash mkdir build cd build cmake ..

3. Compile:

   bash cmake --build . -j

The compilation symlinks the generated FANS binary into the test/ directory for convenience.

Configuring a build

The following CMake configuration options exist:

• CMAKE_BUILD_TYPE: Sets the build type. Common values are Debug, Release, RelWithDebInfo, and MinSizeRel.

  • Default: NONE

• FANS_BUILD_STATIC: Build static library instead of shared library.

  • Default: OFF

• CMAKE_INTERPROCEDURAL_OPTIMIZATION: Enable inter-procedural optimization (IPO) for all targets.

  • Default: ON (if supported)
  • Note: When you run the configure step for the first time, IPO support is automatically checked and enabled if available. A status message will indicate whether IPO is activated or not supported.

Installing

Install FANS (system-wide) using the following options:

1. Using CMake (sudo required if --prefix is omitted):

   bash cmake --install . [--prefix <install-dir>]

Install using Conda

FANS is also available as a conda package on conda-forge/fans. No dependencies have to be manually installed for it to work. It can be installed via

bash conda install conda-forge::fans

exposing the executable FANS.

Input File Format

FANS requires a JSON input file specifying the problem parameters. Example input files can be found in the test/input_files directory. It is recommended to use these files as a reference to create your input file.

Microstructure Definition

json "microstructure": { "filepath": "microstructures/sphere32.h5", "datasetname": "/sphere/32x32x32/ms", "L": [1.0, 1.0, 1.0] }

• filepath: This specifies the path to the HDF5 file that contains the microstructure data.
• datasetname: This is the path within the HDF5 file to the specific dataset that represents the microstructure.
• L: Microstructure length defines the physical dimensions of the microstructure in the $x$, $y$, and $z$ directions.

Problem Type and Material Model

json "matmodel": "LinearElasticIsotropic", "material_properties": { "bulk_modulus": [62.5000, 222.222], "shear_modulus": [28.8462, 166.6667] }

• problem_type: This defines the type of physical problem you are solving. Common options include thermal problems and mechanical problems.

• matmodel: This specifies the material model to be used in the simulation. Examples include

  • LinearThermalIsotropic for linear isotropic conductive material model
  • LinearThermalTriclinic for linear triclinic conductive material model
  • GBDiffusion for diffusion model with transversely isotropic grain boundary and isotropic bulk for polycrystalline materials
  • LinearElasticIsotropic for linear isotropic elastic material model
  • LinearElasticTriclinic for linear triclinic elastic material model
  • PseudoPlasticLinearHardening / PseudoPlasticNonLinearHardening for plasticity mimicking model with linear/nonlinear hardening
  • J2ViscoPlastic_LinearIsotropicHardening / J2ViscoPlastic_NonLinearIsotropicHardening for rate-independent / dependent J2 plasticity model with kinematic and linear/nonlinear isotropic hardening.

• material_properties: This provides the necessary material parameters for the chosen material model. For thermal problems, you might specify conductivity, while mechanical problems might require bulk_modulus, shear_modulus, and more properties for advanced material models. These properties can be defined as arrays to represent multiple phases within the microstructure.

Solver Settings

json "method": "cg", "error_parameters":{ "measure": "Linfinity", "type": "absolute", "tolerance": 1e-10 }, "n_it": 100,

• method: This indicates the numerical method to be used for solving the system of equations. cg stands for the Conjugate Gradient method, and fp stands for the Fixed Point method.
• error_parameters: This section defines the error parameters for the solver. Error control is applied to the finite element nodal residual of the problem.
  • measure: Specifies the norm used to measure the error. Options include Linfinity, L1, or L2.
  • type: Defines the type of error measurement. Options are absolute or relative.
  • tolerance: Sets the tolerance level for the solver, defining the convergence criterion based on the chosen error measure. The solver iterates until the solution meets this tolerance.
• n_it: Specifies the maximum number of iterations allowed for the FANS solver.

Macroscale Loading Conditions

json "macroscale_loading": [ [ [0.004, -0.002, -0.002, 0, 0, 0], [0.008, -0.004, -0.004, 0, 0, 0], [0.012, -0.006, -0.006, 0, 0, 0], [0.016, -0.008, -0.008, 0, 0, 0], ], [ [0, 0, 0, 0.002, 0, 0], [0, 0, 0, 0.004, 0, 0], [0, 0, 0, 0.006, 0, 0], [0, 0, 0, 0.008, 0, 0], ] ],

• macroscale_loading: This defines the external loading applied to the microstructure. It is an array of arrays, where each sub-array represents a loading condition applied to the system. The format of the loading array depends on the problem type:
• For thermal problems, the array typically has 3 components, representing the temperature gradients in the $x$, $y$, and $z$ directions.
• For mechanical problems, the array must have 6 components, corresponding to the components of the strain tensor in Mandel notation (e.g., $[\varepsilon{11}, \varepsilon{22}, \varepsilon{33}, \sqrt{2}\varepsilon{12}, \sqrt{2}\varepsilon{13}, \sqrt{2}\varepsilon{23}]$).

In the case of path/time-dependent loading, as shown, for example, in plasticity problems, the macroscale_loading array can include multiple steps with corresponding loading conditions.

FANS also supports mixed boundary conditions, where some components can be strain-controlled while others are stress-controlled:

json "macroscale_loading": [{ "strain_indices" : [2,3,4,5], "stress_indices" : [0,1], "strain" : [[0.005 , 0.0, 0.0, 0.0], [0.010 , 0.0, 0.0, 0.0]], "stress" : [[0.0, 0.0], [0.0, 0.0]] }]

Results Specification

json "results": ["stress_average", "strain_average", "absolute_error", "phase_stress_average", "phase_strain_average", "microstructure", "displacement", "displacement_fluctuation", "stress", "strain"]

• results: This array lists the quantities that should be stored in the results HDF5 file during the simulation. Each string in the array corresponds to a specific result:

  • stress_average and strain_average: Volume averaged- homogenized stress and strain over the entire microstructure.
  • absolute_error: The L-infinity error of the finite element nodal residual at each iteration.
  • phase_stress_average and phase_strain_average: Volume averaged- homogenized stress and strain for each phase within the microstructure.
  • microstructure: The original microstructure data.
  • displacement: The displacement field (for mechanical problems) and temperature field (for thermal problems) at each voxel in the microstructure.
  • displacement_fluctuation: The periodic displacement fluctuation field (for mechanical problems) and periodic temperature fluctuation field (for thermal problems at each voxel in the microstructure).
  • stress and strain: The stress and strain fields at each voxel in the microstructure.

• Additional material model-specific results can be included depending on the problem type and material model.

Acknowledgements

Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2075 – 390740016. Contributions by Felix Fritzen are funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Heisenberg program - DFG-FR2702/8 - 406068690; DFG-FR2702/10 - 517847245 and through NFDI-MatWerk - NFDI 38/1 - 460247524. We acknowledge the support of the Stuttgart Center for Simulation Science (SimTech).

Contributors

• Sanath Keshav
• Florian Rieg
• Ishaan Desai
• Moritz Sigg
• Claudius Haag
• Felix Fritzen