28-08-2024 | Posted by Joaquín Martí
Prior to prototyping, simulation helps engineers create and test designs that should remain viable under many different conditions and can be optimized for various factors. An essential tool for simulation is the finite element method (FEM), a numerical technique for finding approximate solutions to boundary value problems for partial differential equations.
The roots of FEM can be traced back to the 19th century, with the development of mathematical methodologies for solving differential equations. This period saw advances in calculus and numerical analysis that laid the groundwork for FEM.
In the early 20th century Boris Galerkin proposed methods to convert a continuous operator problem, such as a differential equation, to a discrete problem by applying linear constraints determined by finite sets of basis functions. But the modern development of FEM began in the 1940s and Richard Courant is often credited with its early conceptual framework. In 1943, he used piecewise linear approximations over triangular subregions to solve torsion problems, setting the stage for FEM.
The 1950s marked significant progress with contributions from engineers working on structural analysis problems. Ray Clough is often called the “father of FEM.” His 1956 paper, “The Finite Element Method in Plane Stress Analysis,” formalized many aspects of the method and introduced the term “finite element.”
The 1960s saw rapid development and application of FEM, particularly in civil and aerospace engineering. John Argyris, Olgierd (Olek) Zienkiewicz, and others made substantial contributions during this period. Zienkiewicz’s 1967 book, “The Finite Element Method in Structural and Continuum Mechanics,” became a seminal text.
Incidentally, I met Olek in the early 1980s at the Athenaeum, a gentlemen’s club in London’s Pall Mall with 51 Nobel Laureates among its members. I was modelling extrusion of aluminium alloys using adaptive meshing and Alcan, my client, wanted Olek to approve my methodology. I passed the exam, but also kept vivid memories of the setting, of Olek’s rich voice, his pipe, and his view that, apart from himself, the club basically gathered archbishops and generals.
Proceeding with history, during the 1970s FEM expanded beyond structural mechanics to other fields such as heat transfer, fluid dynamics, and electromagnetics. The theoretical foundations were further solidified, with mathematicians like Gilbert Strang and George Fix contributing to its mathematical rigor.
Over the last few decades of the 20th century, software packages like NASTRAN, ANSYS, ABAQUS, COMSOL, etc. made FEM accessible to a broader range of engineers and scientists. The method started to be applied to a wider variety of problems, including biomechanical analysis, climate modelling, and the design of complex systems in automotive and aerospace industries.
During the 21st century, FEM has continued to advance, driven by both theoretical innovations and practical applications. On the one hand, high-performance, parallel, and distributed computing techniques have significantly reduced computation times. Also, the use of graphics processing units (GPUs) for FEM computations has further accelerated speeds, allowing for real-time analysis and more detailed models.
From the viewpoint of methodology, adaptive mesh refinement techniques have become more sophisticated, dynamically adjusting the mesh based on the solution, and improving accuracy and efficiency. Isogeometric analysis integrates CAD and FEM, using the same functions to represent geometry and approximate solutions. Extended FEM (XFEM) allows modelling discontinuities without requiring mesh modifications, which is particularly useful in fracture mechanics.
FEM software now often includes capabilities for multiphysics simulations, where multiple interacting physical phenomena (e.g., thermal, structural, fluid flow) are modelled simultaneously. And techniques have been developed to bridge different scales (e.g., atomic, molecular, macroscopic), providing more comprehensive insights into material behaviour and system performance. Modern FEM software has also become more user-friendly, with better graphical interfaces and pre- and post-processing tools. Collaborative platforms further increase the usefulness of simulations. And a great variety of application areas, some previously hard to imagine, benefit from all those developments.
The future is likely to see further advances in computational power, as well in numerical techniques, all contributing to improved accuracy and speed, and, also, to broader areas of application. FEM is likely to remain a critical tool in scientific research and engineering, driving innovation and enabling more accurate and efficient simulations.
