Thirty years of the finite volume method for solid mechanics

TL;DR

This paper reviews the development and various approaches of the finite volume method in solid mechanics over the past 30 years, comparing it with finite element methods and discussing future research directions.

Contribution

It provides a comprehensive overview, historical perspective, and critical analysis of different finite volume approaches in solid mechanics, highlighting their differences and potential future developments.

Findings

Multiple flavors of finite volume methods exist for solid mechanics.

Comparison with finite element method reveals strengths and limitations.

Future research directions are identified for wider adoption.

Abstract

Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs staggered vs vertex-centred), solution algorithm (implicit vs explicit), and stabilisation strategy (Rhie-Chow vs Jameson-Schmidt-Turkel vs Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finishes with a look towards future research directions and steps required for finite volume solid mechanics to achieve more widespread acceptance.