Multiscale finite element calculations in Python using SfePy

• Published in: Advances in computational mathematics Vol. 45; no. 4; pp. 1897 - 1921
• Main Authors: Cimrman, Robert, Lukeš, Vladimír, Rohan, Eduard
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2019; Springer Nature B.V
• Subjects: Application programming interface; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Conduction heating; Conductive heat transfer; Finite element method; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Partial differential equations; Source code; Visualization; Finite element method; 74S05; Piezoelasticity; 35Qxx; Multiscale simulations; SfePy; 65M60; 65Y05; 65N30; Python
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09666-0

SfePy (simple finite elements in Python) is a software for solving various kinds of problems described by partial differential equations in one, two, or three spatial dimensions by the finite element method. Its source code is mostly (85%) Python and relies on fast vectorized operations provided by the NumPy package. For a particular problem, two interfaces can be used: a declarative application programming interface (API), where problem description/definition files (Python modules) are used to define a calculation, and an imperative API, that can be used for interactive commands, or in scripts and libraries. After outlining the SfePy package development, the paper introduces its implementation, structure, and general features. The components for defining a partial differential equation are described using an example of a simple heat conduction problem. Specifically, the declarative API of SfePy is presented in the example. To illustrate one of SfePy’s main assets, the framework for implementing complex multiscale models based on the theory of homogenization, an example of a two-scale piezoelastic model is presented, showing both the mathematical description of the problem and the corresponding code.