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 Ste´phane 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.