This document presents comprehensive historical accounts on the developments of finite element methods (FEM) since 1941, with a specific emphasis on developments related to solid mechanics. We present a historical overview beginning with the theoretical formulations and origins of the FEM, while discussing important developments that have enabled the FEM to become the numerical method of choice for so many problems rooted in solid mechanics.
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The year 2021 marks the eightieth anniversary of the invention of the finite element method (FEM), which 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, and in general every conceivable problem that can be described by partial differential equations (PDEs). The 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 FEMs has been the concurrent development of an engineering science discipline called computational mechanics, or computational science and engineering.
During this period, several great engineering minds were focusing on developing FEMs. J.H. Argyris with his co-workers at the University of Stuttgart; R. Clough and colleagues such as E. L. Wilson and R.L. Taylor at the University of California, Berkeley; O.C. Zienkiewicz with his colleagues such as E. Hinton and B. Irons at Swansea University; P. G. Ciarlet at the University of Paris XI; R. Gallager and his group at Cornell University, R. Melosh at Philco Corporation, B. Fraeijs de Veubeke at the Université de Liège, and J. S. Przemieniecki at the Air Force Institute of Technology had made some important and significant contributions to early developments of finite element methods.
For their decisive contributions to the creation and developments of FEM, R. W. Clough was awarded the National Medal of Science in 1994 by the then vice-president of the United States Al Gore, while O. C. Zienkiewicz was honored as a Commander of the Order of the British Empire (CBE). Today, the consensus is that J.H. Argyris, R.W. Clough and O. C. Zienkiewicz made the most pivotal, critical, and significant contributions to the birth and early developments of finite element method following an early contribution to its mathematical foundation from R. Courant.
It should be mentioned that there are some other pioneers who made some significant contributions in the early developments of FEM, such as Levy [22], Comer [23], Langefors [24], Denke [25], Wehle and Lansing [26], Hoff et al. [27], and Archer [28], among others. These individuals came together made remarkable and historic contributions to the creation of finite element method. Among them, some notable contributions were made by J. S. Przemieniecki (Janusz Stanisław Przemieniecki), who was a Polish engineer and a professor and then dean at the Air Force Institute of Technology in Ohio in the United States from 1961 to 1995. Przemieniecki conducted a series pioneering research works on using FEMs to perform stress and buckling analyses of aerospace structures such as plates, shells, and columns (see Przemieniecki [29], Przemieniecki and Denke [30], Przemieniecki [28,29,30,31,32,33]).
A significant breakthrough in computational fracture mechanics and FEM refinement technology came in the late 1990s, when Belytschko and his co-workers, including Black, Moes, and Dolbow, developed the eXtended finite element (X-FEM) (see [160, 161], which uses various enriched discontinuous shape functions to accurately capture the morphology of a cracked body without remeshing. Because the adaptive enrichment process is governed by the crack tip energy release rate, X-FEM provides an accurate solution for linear elastic fracture mechanics (LEFM). In developing X-FEM, T. Belytschko brilliantly utilized the PUFEM concept to solve fracture mechanics problems without remeshing. Entering the new millennium, Bourdin, Francfort, Marigo developed a phase-field approach for modeling material fracture [162]. Almost simultaneously, Karma and his co-workers [163], [164]) also proposed and developed the phase-field method to solve crack growth and crack propagation problems, as the phase field method can accurately predict material damage for brittle fracture without remeshing. The main advantage of the phase field approach is that by using the Galerkin FEM to solve the continuum equations of motion as well as a phase equation, one can find the crack solution in continuum modeling without encountering stress singularity as well as remeshing, and the crack may be viewed as the sharp interface limit of the phase field solution. Some of the leading contributors for this research are Bourdin, Borden, Hughes, Kuhn, Muller, Miehe, Landis, among others (see Bourdin and Chambolle [165], Kuhn and Müller [166]. Miehe et al. [167], and Borden et al. [168], Wilson et al. [169], and Pham et al. [170]).
An important advance of the FEM is the development of the crystal plasticity finite element method (CPFEM), which was first introduced in a landmark paper by Pierce et al. [192]. In the past almost four decades, there are numerous researchers who have made significant contributions to the subject, for example, A. Arsenlis and DM. Parks from MIT and Lawrence Livermore National Laboratory [193], [194], [195], Dawson et al. at Cornell University (Quey et al. [196]; Mathur and Dawson [197]; Raabe et al. at Max-Planck-Institute fur Eisenforschung [196,197,198,201], among others. Based on crystal slip, CPFEM can calculate dislocation, crystal orientation and other texture information to consider crystal anisotropy during computations, and it has been applied to simulate crystal plasticity deformation, surface roughness, fractures and so on. Recently, S. Li and his co-workers developed a FEM-based multiscale dislocation pattern dynamics to model crystal plasticity in single crystal [202, 203]. Yu et al., [204] reformulated the self-consistent clustering analysis (SCA) for general elasto-viscoplastic materials under finite deformation. The accuracy and efficiency for predicting overall mechanical response of polycrystalline materials are demonstrated with a comparison to traditional full-field FEMs.
In 2013, a group of Italian scientists and engineers led by L. Beirão da Veiga and F. Brezzi proposed a so-called virtual element method (VEM) (see Beirao et al. [205, 206]). The virtual element method is an extension of the conventional FEM for arbitrary element geometries. It allows the polytopal discretizations (polygons in 2-D or polyhedra in 3-D), which may be even highly irregular and non-convex element domains. The name virtual derives from the fact that knowledge of the local shape function basis is not required, and it is in fact never explicitly calculated. VEM possesses features that make it superior to the conventional FEM for some special problems such as the problems with complex geometries for which a good quality mesh is difficult to obtain, solutions that require very local refinements, and among others. In these special cases, VEM demonstrates robustness and accuracy in numerical calculations, when the mesh is distorted.
Scientific and engineering problems typically fall under three categories: (1) problems with abundant data but undeveloped or unavailable scientific principles, (2) problems that have limited data and limited scientific knowledge, and (3) problems that have known scientific principles with uncertain parameters, with possible high computational load [228]. In essence, mechanistic data science (MDS) mimics the way human civilization has discovered solutions to difficult and unsolvable problems from the beginning of time. Instead of heuristics, MDS uses machine learning methods like active deep learning and hierarchical neural network(s) to process input data, extract mechanistic features, reduce dimensions, learn hidden relationships through regression and classification, and provide a knowledge database. The resulting reduced order form can be utilized for design and optimization of new scientific and engineering systems (see Liu et. al. [229]). Thus, the new focus of the FEM research has shifted towards the development of machine learning based FEMs and reduced order models.
With the recent development of machine learning and deep learning methods, solving FEM by constructing a deep neural network has become a state-of-the-art technology. Earlier research focused on building up a shallow neural network following the FEM structure to solve boundary value problems. Takeuchi and Kosugi [230] proposed a neural network representation of the FEM to solve Poisson equation problems. Yagawa and Aoki [231] replaced the FEM functional with the network energy of interconnected neural networks (NNs) to solve a heat conduction problem. Due to the limitation of computationliual power and slow convergency rate in shallow neural networks, earlier applications could only solve simple PDE problems. After the 2010s, neural networks for solving computational mechanics problems have become increasingly popular with the rapid growth of deep learning techniques and the development of more sophisticated neural network structures, such as convolutional neural networks (CNN), Generative Adversarial Networks (GAN) and residual neural networks (ResNet). For its high dimensional regression ability, some researchers, for example, Ghavamiana and Simone [232] used deep neural networks as a regression model to learn the material behavior or microstructure response. Other works focus on solving PDEs using deep learning neural networks. G. Karniadakis and his coworkers (see Raissi et al. [233, 234], Karniadakis et al. [235]) proposed a Physics-Informed Neural Networks (PINNs) to solve high dimensional PDEs in the strong form with constraints to accommodate both natural and essential boundary conditions. The idea of constructing deep neural networks following the FEM structure is investigated again with advanced neural network methodologies. Weinan E and his co-workers [236] and B. Yu [237] proposed a Deep Ritz Method for solving variational problems. Sirignano and Spiliopoulos [238] proposed the so-called Deep Galerkin Method (DGM) to solve high-dimensional PDEs. Zabaras and his co-worders proposed a CNN-based physics-constrained deep learning framework for high-dimensional surrogate modeling and uncertainty quantification (see Zhu et al. [239]). Rabczuk [240] systematically explore the potential to use NNs for computational mechanics by solving energetic format of the PDE (see Samaniego et al. [241]). Lee [242] proposed a partition of unity network for deep hp approximation of PDEs and extensively the training and initialization strategy to accelerate the convergence of the solution process (see Lee et al. [242]). The constructing of element shape function by activation functions has been studied by J. Opschoor and his coworkers (See [243, 244]). 2ff7e9595c
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