This project uses the coupling library preCICE in order to couple PDE (Partial Differential Equation) solver packages relying on model order reduction techniques as provided by pyMOR. The two coupled PDE packages are in particular FEniCS and deal.II.

The repository contains a ready-to-run example in the example directory. It consists of a simplified and adapted version of the partitioned-heat tutorial, where we split a domain artificially into two parts, solve in each subdomain the same problem and carry out a Dirichlet-Neumann coupling across a common coupling interface in order to recover a global solution. However, instead of the time-dependent heat equation, we solve here a stationary Laplace problem with Dirichlet boundary conditions on the left side of the domain u_D = 3 and homogeneous Dirichlet boundary conditions on u_D = 0 on the right side of the domain. The left side ('Dirichlet participant') is computed using FeniCS and the right side ('Neumann participant') is computed using deal.II.

In a first step, the model order reduction was applied to the deal.II (Neumann) participant. Since deal.II is written in C++ and the model order reduction through pyMOR is carried out through the python programming language, we compile the C++ functions of deal.II into a python compatible library using pybin11, which is already included as a submodule within this project. Therefore, the deal.II source code can be found in the lib directory and the function calls for the deal.II heat problem as well as the pyMOR model order reduction are located in example/neumann-reduced/heat_equation_reduced.py. Although model order reduction with FEniCS is supported by pyMOR, we don't apply any model order reduction on the FEniCS side, i.e., the example code example/dirichlet-fenics/heat.py solves always the full order model.

In order to apply the model order reduction on the Neumann side, we parametrize the diffusion coefficient within this participant: we split the squared domain on the right side once more into a square and an L-shaped remainder. In the offline phase, we perform multiple coupled simulations between the Dirichlet and Neumann participant using a different diffusion coefficient in the sub-square on the Neumann side. The FEniCS side computes during the offline phase always the same computational setup. By default, the sub-square with the varying diffusion coefficient is part of the coupling interface. The motivation for such a setup is the runtime reduction for the reduced order model during the online phase: Building the deal.II code in Release mode results in a speedup factor of around 10 for 8 global refinements, which corresponds to 66,049 degrees of freedom.

The setup requires a variety of software packages. Install preCICE (v2.4.0 or greater), pyMOR (at least version ef3242c as set in the requirements.txt is required), FEniCS (legacy version as set in the requirements.txt) and deal.II (v9.2 or greater). Afterwards, install the pyMOR-deal.II wrapper using

git clone --recurse-submodules https://github.com/pymor/pymor-deal.II.git
python3 -m pip install ./pymor-deal.II

and clone this repository using

git clone --recurse-submodules https://github.com/DavidSCN/mor-coupling.git

The deal.II-based executable can be compiled using

cmake -B ./mor-coupling/build -S ./mor-coupling
cmake --build ./mor-coupling/build

The example setup is located in the example directory. In a first step, the offline phase, multiple coupled simulations need to be performed in order to generate the parametrized reduced basis later on. By default the coupled simulation is performed 5 times using uniform samples of the diffusion coefficient. Afterwards, 5 random diffusion coefficients are used in order to compare the full order model and the reduced order model. In order to execute all simulations, execute

./run.sh -n 15

from the dirichlet-fenics directory and

python3 heat_equation_reduced.py

from the neumann-reduced directory. The Neumann participant prints out statistics regarding the error and speedup of the reduced basis.