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PYFETI

PYFETI is a standalone Python Library to solve and implemented parallel FETI-Like solvers using mpi4py.

Dependecies

• NumPy
• Scipy
• MPI4Py
• Dill
• Pandas
• Matplotlib

Installing PyFETI

Before installing PyFETI we stronly recommend the use of ANACONDA and git. PyFETI is suppose to work in both Windows and Linux system, but is not fully supported, so please let us know with you are facing any problem.

Linux installation

mkdir PYFETI
cd PYFETI
git init 
git clone https://username@bitbucket.org/teamsinspace/documentation-tests.git

The command above should copy the remote files to your local system. Now, we must activate the Anaconda virtual environment and install PyFETI.

source activate $you virtual env$
python setup.py install

If you install PyFETI, you should run all unittest to make sure everything is properly working.

cd pyfeti/src/tests
python test_feti_solver.py

Almost every source file also contains unittests, so feel free to run all of them. PyFETI uses mpi4pi and requires the installation of some mpi distriction, see MSMPI, IntelMPI, and OpenMPI. Because multiple MPI implementation are supported, the user must create a environment variable to set MPI path that must be used in PyFETI.

export MPIDIR=/program/mpi

Also, you can have multiple python virtual environments, then you must set a environment variable to specify which python.exe to use:

export 'PYTHON_ENV'=/condaenv/pyfeti

Now, it is time to run python and import pyfeti modules.

python
>>> import pyfeti

Have fun!

Theory behind PyFETI

Solving with Dual Assembly

The PyFETI library is intend to provide easy function in order to solve, the dual assembly problem, namely:

$$ \begin{bmatrix} K & B^{T} \ B & 0
\end{bmatrix} \begin{bmatrix} u \ \lambda \end{bmatrix}

\begin{bmatrix} f \ 0 \end{bmatrix} $$

Generally the block matrix $K$ is singular due to local rigid body modes, then the inner problem is regularized by adding a subset of the inter-subdomain compatibility requeriments:

$$ \begin{bmatrix} K & B^TG^{T} & B^{T} \ GB & 0 & 0 \ B & 0 & 0 \ \end{bmatrix} \begin{bmatrix} u \ \alpha \ \lambda \end{bmatrix}

\begin{bmatrix} f \ 0 \ 0 \end{bmatrix} $$

Where $G$ is defined as $-R^TB^T$.

The Dual Assembly system of equation describe above can be broken in two equations.

\begin{equation} Ku + B^{T}\lambda = f \ Bu = 0 \end{equation}

Then, the solution u can be calculate by:

\begin{equation} u = K^*(B^{T}\lambda + f) + R\alpha \ \end{equation}

Where $K^*$ is the generelize pseudo inverse and $R$ is $Null(K) = {r \in R: Kr=0}$, named the kernel of the K matrix. In order to the solve $u$ the summation of all forces in the subdomain, interface, internal and extenal forces must be in the image of K. This implies the $(B^{T}\lambda + f)$ must be orthonal to the null space of K.

\begin{equation} R(B^{T}\lambda + f) = 0 \ \end{equation}

Phisically, the equation above enforces the self-equilibrium for each sub-domain. Using the compatibility equation and the self-equilibrium equation, we can write the dual interface equilibrium equation as:

$$ \begin{bmatrix} F & G^{T} \ G & 0
\end{bmatrix} \begin{bmatrix} \lambda \ \alpha \end{bmatrix}

\begin{bmatrix} d \ e \end{bmatrix} $$

Where $F = BK^*B^T$, $G = -R^TB^T$, $d = BK^*f$ and $e =- R^Tf $.

ToDo List

• Unittest with more than 10 Domains
• Easy access to the Parallel F operator
• Parallel Coarse Grip Problem