FElupe documentation
====================
FElupe is a Python 3.9+ ๐ finite element analysis package ๐ฆ focusing on the formulation and numerical solution of nonlinear problems in continuum mechanics of solid bodies ๐ง. This package is intended for scientific research ๐ป, but is also suitable for running nonlinear simulations ๐ in general ๐๏ธ. In addition to the transformation of general weak forms into sparse vectors and matrices, FElupe provides an efficient high-level abstraction layer for the simulation of the deformation of solid bodies.
.. grid::
.. grid-item-card:: ๐ Beginner's Guide
:link: tutorials
:link-type: ref
New to FElupe? The Beginner's Guide contains an introduction to the concept of FElupe.
.. grid-item-card:: ๐ API reference
:link: felupe-api
:link-type: ref
The reference guide contains a detailed description of the FElupe API. It describes how the methods work and which parameters can be used. Requires an understanding of the key concepts.
.. grid::
.. grid-item-card:: โ How-To
:link: how-to
:link-type: ref
Step-by-step guides for specific tasks or problems with a focus on practical usability instead of completeness. Requires an understanding of the key concepts.
.. grid-item-card:: ๐ Examples
:link: examples
:link-type: ref
A gallery of examples.
.. admonition:: Highlights
:class: admonition
+ 100% Python package built with `NumPy `_ and `SciPy `_
+ easy to learn and productive high-level :ref:`API `
+ nonlinear deformation of :class:`solid bodies `
+ interactive views on meshes, fields and solid bodies (using `PyVista `_)
+ typical :ref:`finite elements `
+ cartesian, axisymmetric, plane strain and mixed fields
+ :ref:`hyperelastic material models ` with :class:`automatic differentiation `
Installation
------------
Install Python, fire up ๐ฅ a terminal and run ๐
.. image:: https://img.shields.io/pypi/v/felupe.svg
:target: https://pypi.python.org/pypi/felupe/
.. code-block:: shell
pip install felupe[all]
where ``[all]`` is a combination of ``[autodiff,io,parallel,plot,progress,view]`` and installs all optional dependencies. FElupe has minimal requirements, all available at PyPI supporting all platforms.
* `numpy `_ for array operations
* `scipy `_ for sparse matrices
In order to make use of all features of FElupe ๐๐ฐ๐๐๐, it is suggested to install all optional dependencies.
* `einsumt `_ for parallel (threaded) assembly
* `jax `_ for automatic differentiation in material formulations (JAX-based)
* `h5py `_ for writing XDMF result files
* `matplotlib `_ for plotting graphs
* `meshio `_ for mesh-related I/O
* `pyvista `_ for interactive visualizations
* `tensortrax `_ for automatic differentiation in material formulations (NumPy-based)
* `tqdm `_ for showing progress bars during job evaluations
The development version may contain not yet released bug fixes and features. Consider using the ``--user`` option to install the package into your home directory (see `pip documentation `_ for more details). To install FElupe from the latest development branch, use
.. code-block:: shell
pip install git+https://github.com/adtzlr/felupe.git@main
or clone the repository and install the package in editable mode.
.. code-block:: shell
git clone https://github.com/adtzlr/felupe.git
cd felupe
pip install --editable .
Extension Packages
------------------
The capabilities of FElupe may be enhanced with extension packages created by the community.
+-----------------------------------------------------------+---------------------------------------------------------+
| Package | Description |
+===========================================================+=========================================================+
| `hyperelastic `_ | Constitutive hyperelastic material formulations |
+-----------------------------------------------------------+---------------------------------------------------------+
| `matadi `_ | Material Definition with Automatic Differentiation (AD) |
+-----------------------------------------------------------+---------------------------------------------------------+
| `tensortrax `_ | Differentiable Tensors based on NumPy Arrays |
| | (bundled with FElupe) |
+-----------------------------------------------------------+---------------------------------------------------------+
| `feplot `_ | A visualization tool for FElupe |
+-----------------------------------------------------------+---------------------------------------------------------+
Performance
-----------
This is a simple benchmark to compare assembly times for linear elasticity and hyperelasticity on tetrahedrons.
.. tab:: Assembly Runtimes
.. image:: _static/benchmark.png
:width: 400px
+----------------+-------------------+
| Analysis | DOF/s |
+================+===================+
| Linear-Elastic | 130039 +/-23464 |
+----------------+-------------------+
| Hyperelastic | 116819 +/-21979 |
+----------------+-------------------+
Tested on: Windows 10, Python 3.11, Intelยฎ Coreโข i7-11850H @ 2.50GHz, 32GB RAM.
.. tab:: Source Code
.. code-block:: python
from timeit import timeit
import matplotlib.pyplot as plt
import numpy as np
import felupe as fem
def pre_linear_elastic(n, **kwargs):
mesh = fem.Cube(n=n).triangulate()
region = fem.RegionTetra(mesh)
field = fem.FieldContainer([fem.Field(region, dim=3)])
umat = fem.LinearElastic(E=1, nu=0.3)
solid = fem.SolidBody(umat, field)
return mesh, solid
def pre_hyperelastic(n, **kwargs):
mesh = fem.Cube(n=n).triangulate()
region = fem.RegionTetra(mesh)
field = fem.FieldContainer([fem.Field(region, dim=3)])
umat = fem.NeoHookeCompressible(mu=1.0, lmbda=2.0)
solid = fem.SolidBody(umat, field)
return mesh, solid
print("# Assembly Runtimes")
print("")
print("| DOF | Linear-Elastic in s | Hyperelastic in s |")
print("| ------- | ------------------- | ----------------- |")
points_per_axis = np.round((np.logspace(3, 5, 6) / 3)**(1 / 3)).astype(int)
number = 3
parallel = False
runtimes = np.zeros((len(points_per_axis), 2))
for i, n in enumerate(points_per_axis):
mesh, solid = pre_linear_elastic(n)
matrix = solid.assemble.matrix(parallel=parallel)
time_linear_elastic = (
timeit(lambda: solid.assemble.matrix(parallel=parallel), number=number) / number
)
mesh, solid = pre_hyperelastic(n)
matrix = solid.assemble.matrix(parallel=parallel)
time_hyperelastic = (
timeit(lambda: solid.assemble.matrix(parallel=parallel), number=number) / number
)
runtimes[i] = time_linear_elastic, time_hyperelastic
print(
f"| {mesh.points.size:7d} | {runtimes[i][0]:19.2f} | {runtimes[i][1]:17.2f} |"
)
dofs_le = points_per_axis ** 3 * 3 / runtimes[:, 0]
dofs_he = points_per_axis ** 3 * 3 / runtimes[:, 1]
print("")
print("| Analysis | DOF/s |")
print("| -------------- | ----------------- |")
print(
f"| Linear-Elastic | {np.mean(dofs_le):5.0f} +/-{np.std(dofs_le):5.0f} |"
)
print(f"| Hyperelastic | {np.mean(dofs_he):5.0f} +/-{np.std(dofs_he):5.0f} |")
plt.figure()
plt.loglog(
points_per_axis ** 3 * 3,
runtimes[:, 1],
"C0",
label=r"Stiffness Matrix (Hyperelastic)",
)
plt.loglog(
points_per_axis ** 3 * 3,
runtimes[:, 0],
"C1--",
label=r"Stiffness Matrix (Linear-Elastic)",
)
plt.xlabel(r"Number of degrees of freedom $\longrightarrow$")
plt.ylabel(r"Runtime in s $\longrightarrow$")
plt.legend()
plt.tight_layout()
plt.savefig("benchmark.png")
.. toctree::
:maxdepth: 1
:caption: Contents:
tutorial/index
examples/index
howto
felupe
License
-------
FElupe - Finite Element Analysis (C) 2021-2024 Andreas Dutzler, Graz (Austria).
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with this program. If not, see ``_.
Indices and tables
==================
* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`