Axisymmetric semi-analytical finite elements in Python

What is it?

This repository contains the axisafe package which is an implementation of the axisymmetric finite elements in Python, following a publication in Computers & Structures. The code can be used to find mechanical waves propagating axisymmetric structures, such as pipes. These can be free, buried or submerged, and may contain a fluid inside. An arbitrary composition of layers is allowed.

A few examples corresponding to the cases considered in the aforementioned paper are also included.

The code was written ant tested for:

• Python 3.5
• numpy 1.12.1
• scipy 0.19.0
• matplotlib 2.0.2

Whilst it should work fine with older versions as well, plotting code including in the example scripts may not execute owing to changes in color definition in matplotlib. This does not compromise the core functionality of the package.

License

All course materials are licensed under the MIT License:

Copyright (c) 2017 Michał Kalkowski

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Publication

Full bibliographic details of the parent publication are:

Kalkowski MK et al. Axisymmetric semi-analytical finite elements for modelling waves in buried/submerged fluid-filled waveguides. Comput Struct (2017), https://doi.org/10.1016/j.compstruc.2017.10.004

Whenever you use this code, please cite the above mentioned paper as an attribution to the authors.

Erratum

An ambiguity in the fluid element equations in the published version of the paper became apparent. Eq. (35) ommits the $\rho_f$ factor that multiplies both terms in the virtual work equation. This ommision may be not recalled unless one considers structural acoustic coupling. The code has always preserved the dimensions appropriately.

I no short, all matrices related to acoustic elements (Eq. (24)) should be multiplied by $\rho_f$, which would be best done at element integration stage. Practically, this means adding the $\rho_f$ factor to all matrices in Eq. (24).

As mentioned this is an ommision in the printed version of the equations and not in the implementation.

Request

Please note that this software is written by a mechanical engineer, not a programmer, and is far from ideal from the software engineering perspective. For this reason, it is very much open to improvement and any contribution is more than welcome.