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Eighty Years of the Finite Element Method: Birth, Evolution, and Future
Wing Kam Liu, Shaofan Li, Harold Park
TL;DR
The paper addresses how FEM developed over eighty years into a broadly applicable framework for scientific modeling and engineering design. It presents a historical synthesis centered on solid and structural mechanics, organized into four periods and connected to advances in materials modeling, adaptive methods, multiscale methods, and emerging data-driven approaches. The history culminates in FEM research shifting toward machine-learning-based methods and reduced-order models for industrially relevant simulation demands.
Problem
The paper examines the historical development of FEM and its expanding role across engineering and scientific problems described by PDEs.
Method
The authors provide a historical perspective organized into four periods, emphasizing applications and related developments in solid and structural mechanics.
Results
FEM evolved from conventional discretization into industrial, multiscale, adaptive, machine-learning-based, and reduced-order approaches.
Takeaways & Limitations
Current FEM research focuses on data-driven methods and reduced-order models to support design, optimization, and faster engineering simulations.
Abstract
from arXiv · showhide
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.
III. (1992-2017) Broad Industrial Applications and Materials Modeling
From 1992 to 2017, FEM expanded through industrial applications, materials modeling, adaptive accuracy control, and formulations able to handle complex geometries and multiscale behavior.
- Accuracy and verification: The Zienkiewicz-Zhu error estimator introduced a major advance in finite-element approximation and supported adaptive mesh refinement for solution quality control.Posteriori error estimation enables computational resources to be allocated through adaptive refinement.
- Industrial integration: Isogeometric analysis integrated FEM with CAD by using NURBS functions as shape functions in a Galerkin formulation on the control mesh.The approach was developed to blend finite-element analysis directly into computer-aided design workflows.
- Multiscale materials modeling: Multiscale FEM coupled atomistic methods, including molecular dynamics and DFT, with continuum-scale finite-element methods for nanotechnology applications.Related homogenization methods obtained continuum-scale properties from smaller-scale microstructures for periodic or random composites.
- Materials modeling: Crystal plasticity FEM incorporated crystal slip, dislocation, orientation, and texture information to model anisotropic deformation, roughness, and fracture.CPFEM was introduced in 1982 and became an important advance in finite-element materials modeling.
- Advanced discretizations: Virtual element, generalized, cohesive-zone, meshfree, and extended finite-element methods broadened FEM to irregular geometries, fracture, localization, and crack growth without remeshing.These approaches address difficult meshes, localized refinement, mesh bias, and discontinuities through generalized or enriched approximations.
IV. (2018- present) Coming of A New Era
The new era of finite element research combines machine learning, mechanistic data science, and reduced-order modeling to address data-rich problems and the need for fast simulations. Recent work also constructs neural-network-based finite element approximations and develops data-driven methods for nonlinear, noisy, and inverse problems.
- Machine learning and mechanistic data science: Finite element research is shifting toward machine-learning methods that process data, extract mechanistic features, reduce dimensions, and learn hidden relationships for design and optimization.These methods include active deep learning and hierarchical neural networks, producing reduced-order forms for new scientific and engineering systems.
- Neural-network-based FEM: Deep neural networks have become a state-of-the-art approach for solving FEM, following earlier neural-network representations developed for boundary-value problems such as Poisson equations.HiDeNN constructs finite element shape functions from hierarchical deep networks, while related work includes RKPM, NURBS, and IGA interpolation functions.
- Reduced-order modeling: Two-stage data-driven methods reduce FE computational cost by generating a finite-element database offline and computing final solutions online.Related approaches include hyper-reduction and dimensional reduction of nonlinear finite element dynamic models.
- Data-driven applications: Data-driven finite element research now addresses dynamics, noisy databases, inverse collision problems, and physics-constrained reconstruction.Examples include data-driven finite elements for dynamics, physics-constrained RKPM, and machine-learning-based prediction of pre-crash car-collision data.
- Reduced-order modeling: Fast, nearly real-time simulations are increasingly needed for online control, structural and vehicle health monitoring, manufacturing feedback, and automated driving.Such applications require intensive interaction among sensors, control algorithms, and simulation tools.