Multiscale finite element calculations in Python using SfePy

TL;DR

SfePy is a Python-based finite element software that simplifies solving PDEs across dimensions, featuring declarative and imperative APIs, and supports complex multiscale modeling like homogenization.

Contribution

This paper introduces SfePy's implementation, structure, and features, highlighting its capabilities for multiscale modeling and flexible problem definition.

Findings

SfePy effectively solves PDEs in 1D, 2D, and 3D.

It supports complex multiscale models such as homogenization.

The framework combines ease of use with computational efficiency.

Abstract

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.