# Eighty Years of the Finite Element Method: Birth, Evolution, and Future

arXiv: https://arxiv.org/abs/2107.04960 · HTML: https://mlsanity.com/paper/2107.04960/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2021.

## Built on by (most cited first, 21 of 21)

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- [VFEAgent: A Multimodal Agent Framework for End-to-End Automated Finite Element Analysis](https://mlsanity.com/paper/2605.28978/) (2026)
- [Mesh Graph Neural Network Framework for Accelerating Finite Element Simulation for Arbitrary Geometries](https://mlsanity.com/paper/2606.08287/) (2026)
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## Abstract (arXiv)

This year marks the eightieth anniversary of the invention of the finite element method (FEM). FEM has become the computational workhorse for engineering design analysis and scientific modeling of a wide range of physical processes, including material and structural mechanics, fluid flow and heat conduction, various biological processes for medical diagnosis and surgery planning, electromagnetics and semi-conductor circuit and chip design and analysis, additive manufacturing, i.e. virtually every conceivable problem that can be described by partial differential equations (PDEs). FEM has fundamentally revolutionized the way we do scientific modeling and engineering design, ranging from automobiles, aircraft, marine structures, bridges, highways, and high-rise buildings. Associated with the development of finite element methods has been the concurrent development of an engineering science discipline called computational mechanics, or computational science and engineering. In this paper, we present a historical perspective on the developments of finite element methods mainly focusing on its applications and related developments in solid and structural mechanics, with limited discussions to other fields in which it has made significant impact, such as fluid mechanics, heat transfer, and fluid-structure interaction. To have a complete storyline, we divide the development of the finite element method into four time periods: I. (1941-1965) Early years of FEM; II. (1966-1991) Golden age of FEM; III. (1992-2017) Large scale, industrial applications of FEM and development of material modeling, and IV (2018-) the state-of-the-art FEM technology for the current and future eras of FEM research. Note that this paper may not strictly follow the chronological order of FEM developments, because often time these developments were interwoven across different time periods.