Multiscale finite element calculations in Python using SfePy

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.

Phát triển chương trình tính toán kết cấu mặt đường nhựa theo phương pháp phân lớp hữu hạn

“…Nhận biết được những ưu điểm nổi bật của phương pháp FLM trong việc mô hình kết cấu MĐN, nhóm nghiên cứu tập trung xây dựng chương trình FastKM tích hợp thư viện nguồn mở hỗ trợ tính toán phần tử hữu hạn SfePy (Simple Finite Elements in Python) viết bằng ngôn ngữ Python. Dự án SfePy sử dụng Git để quản lý mã nguồn và trang web GitHub để lưu trữ và trao đổi mã nguồn giữa các nhà phát triển [18,19]. Phiên bản 2020.4 hiện tại được lưu trữ tổng cộng 867 tệp với tổng số 178396 dòng và 7540 lần cập nhật bởi 20 tác giả.…”

scikit-fem: A Python package for finite element assembly

“…In contrast to NGSolve (Schöberl, 2014), FEniCS (Alnaes et al, 2015), Firedrake (Rathgeber et al, 2016), SfePy (Cimrman, Lukeš, & Rohan, 2019), and GetFEM (Renard & Poulios, 2020), scikit-fem contains no compiled code making the installation quick and straightforward. We specifically target finite element assembly instead of encapsulating the entire finite element analysis from pre-to postprocessing into a single framework.…”

Homogenization of the vibro–acoustic transmission on perforated plates

“…The homogenized models and the reference model presented in this paper have been implemented in SfePy -Simple Finite Elements in Python [2], a software developed for an efficient solving of multiscale problems by means of the finite element method. In the validation tests and the coupled problem simulation, by the "multiscale simulations" we mean solutions of the homogenized (macroscopic) problem supplemented by the reconstruction procedure which allows us to respect the local fluctuations superimposed to the solutions of the macroscopic problem when the scale parameter ε 0 is given.…”

“…In this paper, we consider a z-symmetric cell Y , so that Proposition 1 applies. However, for the sake of generality, we shall keep the formulations (33) and (35). This will induce a general set of homogenized effective tensors involved in the macromodel.…”

“…For illustration of the microscopic characteristic responses, we show the corrector functions π β , ξ,ς, see the local problems (33) and (34), calculated in the fluid domain Y * and in the elastic inclusion Ξ c .…”