FEniCS-Shells: a modern open-source extensible finite element

FEniCS-Shells: a modern open-source extensible finite element
implementation of linear and nonlinear plate and shell models
Corrado Maurini / Matteo Brunetti
Sorbonne Universits, UPMC Univ. Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert
CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005, Paris
F-75005, Paris, France
Jack S. Hale
Faculte´ des Sciences, de la Technologie et de la Communication, Universit du Luxembourg
Campus Kirchberg, 6, rue Coudenhove-Kalergi, L-1359
Luxembourg
Stéphane P.A. Bordas
Faculte´des Sciences, de la Technologie et de la Communication, Universite´du Luxembourg,
Campus Kirchberg, 6, rue Coudenhove-Kalergi, L-1359, Luxembourg.
School of Engineering, Cardiff University, Cardiff, United Kingdom.
Intelligent Systems for Medicine Laboratory, The University of Western Australia, Australia.
Simulating thin deformable structures implies solving complex non-linear partial differential equations.
Many commercial and open-source softwares successfully allow the user to solve a few standard thinstructural models. However, if the user wants to model the behaviour of a novel multi-physics structural
problem that arises at the frontier between mechanics, physics and applied mathematics, they must usually
implement a custom routine in a low-level language. This is a notably difficult and error-prone task, due
to the complexity and diversity of mathematical models and the advanced discretisation techniques that are
required. We present FEniCS-Shells [1], an open-source Python library based on the FEniCS Project [2]
for simulating a variety of thin structural models using the finite element method. Leveraging the expressiveness of the Unified Form Language (UFL) [3] for declaring finite element discretisations of variational
forms, FEniCS-Shells allows for the seamless formulation of thin structural models with a clear and direct link between the underlying mathematical formulation and the practical implementation in code. We
demonstrate the efficacy and reliability of our approach on various thin structural models, including linear
plates, composite plates, von-Karman plates, and Madare-Naghdi shell models using modern finite element
solution techniques. Our discretisation is based on mixed formulation [4] and MITC reduction operators,
for which we present a special implementation within FEniCS using UFL syntax. We release the library as
open-source software (LGPL) in the hope that it will be of use in research and teaching, forming a common
platform for extensible simulations of thin structures. Building on FEniCS and PETSc, fenics-shells runs
transparently and efficiently on modern parallel computing hardware.
Keywords(optional): thin structures, plates, shells, finite element methods, domain specific
language, MITC, mixed finite elements, Reissner-Mindlin, Kirchhoff-Love, von-Karman.
References(optional)
[1]FEniCS-Shells, https://bitbucket.org/unilucompmech/fenics-shells
[2]M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E.
Rognes and G. N. Wells (2015). The FEniCS Project Version 1.5, Archive of Numerical Software,
3(100).
[3]M. S. Alnaes, A. Logg, K. B. lgaard, M. E. Rognes, G. N. Wells (2014). Unified Form Language: A
domain-specific language for weak formulations of partial differential equations, ACM Transactions
on Mathematical Software, 40(2)
[4]R.Duran, E.Liberman (1992), On mixed finite element methods for the reissner-Midlin plate model,
Mathematics of Computations 58 (198) pp.561-573.