有限元法：诞生、演变和未来

注：机翻，未校。

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

Published: 13 June 2022

Volume 29, pages 4431–4453, (2022)

Wing Kam Liu, Shaofan Li & Harold S. Park

A Correction to this article was published on 28 June 2022

Abstract 抽象

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.
本文档全面介绍了自 1941 年以来有限元方法（FEM）的发展，特别强调了与固体力学相关的发展。我们从有限元力学的理论公式和起源开始，对历史进行了概述，同时讨论了使有限元数学成为固体力学中许多问题的首选数值方法的重要发展。

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.
2021 年是有限元方法（FEM）发明 80 周年，它已成为工程设计分析和各种物理过程科学建模的计算主力，包括材料和结构力学、流体流动和热传导、用于医疗诊断和手术规划的各种生物过程、电磁学和半导体电路以及芯片设计和分析，增材制造，以及通常可以用偏微分方程（PDE）描述的所有可以想象的问题。FEM 从根本上彻底改变了我们进行科学建模和工程设计的方式，范围从汽车、飞机、海洋结构、桥梁、高速公路到高层建筑。与 FEM 的发展相关的是一门称为计算力学或计算科学与工程的工程科学学科的同步发展。

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.
在本文中，我们提出了有限元方法发展的历史视角，主要关注其在固体和结构力学中的应用和相关发展，而仅限于其产生重大影响的其他领域，如流体力学、传热和流固耦合。为了获得一个完整的故事情节，我们将有限元方法的发展分为四个时间段：I. （1941–1965） FEM 的早期；II. （1966–1991） FEM 的黄金时代；III. （1992–2017） FEM 的大规模工业应用和材料建模的发展，以及 IV （2018–） FEM 研究当前和未来时代最先进的 FEM 技术。请注意，本文可能并不严格遵循 FEM 发展的时间顺序，因为这些发展往往交织在不同的时期。

1 (1941–1965) The Birth of the Finite Element Method 1 （1941–1965）有限元法的诞生

The origin of the FEM as a numerical modeling paradigm may be traced back to the early 1940s. In 1941, A. Hrennikof, a Russian-Canadian structural engineer at the University of British Columbia, published a paper in the ASME Journal of Applied Mechanics on his membrane and plate model as a lattice framework [1]. This paper is now generally regarded as a turning point that led to the birth of the FEM. In the paper, he discretized the solution domain into a mesh of lattice structure, which was the earliest form of a mesh discretization.
FEM 作为一种数值建模范式的起源可以追溯到 1940 年代初。1941 年，不列颠哥伦比亚大学的俄裔加拿大结构工程师 A. Hrennikof 在 ASME Journal of Applied Mechanics 上发表了一篇论文，将他的膜和板模型作为晶格框架 [1]。这篇论文现在被普遍认为是导致 FEM 诞生的转折点。在论文中，他将解域离散为晶格结构的网格，这是网格离散化的最早形式。

On May 3rd, 1941, the same year that Hrennikoff published his paper, R. Courant of New York University delivered an invited lecture at a meeting of the American Mathematical Society held in Washington D.C. on his numerical treatment using a variational method to solve a second order PDE, which arises from Saint–Venant’s torsion problem of a cylinder. In this work, Courant systematically used the Rayleigh–Ritz method with a trial function defined on finite triangle subdomains, which is a primitive form of of the finite element method [2].
1941 年 5 月 3 日，也就是 Hrennikoff 发表论文的同一年，纽约大学的 R. Courant 在华盛顿特区举行的美国数学学会会议上发表了一篇特邀演讲，主题是他使用变分方法进行数值处理来解决二阶偏微分方程，该偏微分方程由圣维南的圆柱体扭转问题引起。在这项工作中，Courant 系统地使用了 Rayleigh-Ritz 方法，并在有限三角形子域上定义了试验函数，这是有限元方法的原始形式 [2]。

Courant’s presentation was later published as a paper in 1943 [3]. Similar works of discretization and variational formulations were also reported in the literature, including McHenry [4], Prager and Synge [5], and Synge and Rheinboldt [6]. As Ray Clough commented in his 1980 paper, “One aspect of the FEM, mathematical modeling of continua by discrete elements, can be related to work done independently in the 1940s by McHenry and Hrennikoff-formulating bar element assemblages to simulate plane stress systems. Indeed, I spent the summer of 1952 at the Boeing Airplane Company trying to adapt this procedure to the analysis of a delta airplane wing, the problem which eventually led to the FEM”.
Courant 的演讲后来于 1943 年作为论文发表 [3]。文献中也报道了类似的离散化和变分公式工作，包括 McHenry [4]、Prager 和 Synge [5] 以及 Synge 和 Rheinboldt [6]。正如 Ray Clough 在他 1980 年的论文中所评论的那样，“有限元的一个方面，即通过离散元件对连续体进行数学建模，可以与 McHenry 和 Hrennikoff 在 1940 年代独立完成的工作有关，这些工作制定了杆单元组合以模拟平面应力系统。事实上，我在 1952 年夏天在波音飞机公司度过了一段时间，试图将该程序应用于三角洲飞机机翼的分析，这个问题最终导致了 FEM”。

By the early 1950s, several engineers and academics had further developed and extended these early approaches to solve real engineering problems in aeronautical and civil engineering. In parallel but with different emphases, Argyris at the Imperial College London, and M. J. Turner (1950–1956) at Boeing Company, who was later joined by R.W. Clough of UC Berkeley and H.C. Martin of Washington University, developed what we know today as the earliest form of the finite element method (1954), which was called the Matrix Stiffness Method at the time. In a paper published in 1960 [14], R.W. Clough coined the phrase Finite Element Method, and this unassuming and right-to-the-point phrase was an instant hit, bringing out the essence of the method.
到 1950 年代初期，一些工程师和学者进一步发展和扩展了这些早期方法，以解决航空和土木工程中的实际工程问题。同时但侧重点不同的是伦敦帝国理工学院的 Argyris 和波音公司的 MJ Turner（1950-1956），后来加州大学伯克利分校的 R.W. Clough 和华盛顿大学的 H.C. Martin 加入了他们的行列，开发了我们今天所知的有限元方法（1954 年）的最早形式，当时称为矩阵刚度法。在 1960 年发表的一篇论文中 [14]，R.W. Clough 创造了有限元法这个词，这个不起眼、直截了当的短语一炮而红，带出了该方法的精髓。

With his deep insights and profound vision into Courant’s variational approach, J.H. Argyris developed the energy method for engineering structures [6,7,9], a foundational development enabling FEM for solids. While importantly, Turner, Clough, Martin, and Topp successfully developed FEM interpolants for triangular elements [10], which is suitable for structural parts with arbitrary shape. In some senses, the invention of the triangle element was a “quantum leap”, and hence for a large spectrum of the engineering community, the inception of the FEM is the publication of the landmark paper by Turner et al. [10]. The following is an excerpt from a 2014 document that celebrates the 50th anniversary of the formation of National Academy of Engineering, which is an official account of that part of FEM history: To ensure safety and avoid costly modifications after planes entered production, engineers needed a reliable method for determine in advance whether
凭借对 Courant 变分方法的深刻见解和深刻见解，J.H. Argyris 开发了工程结构的能量方法 [6,7,9]，这是一项使固体有限元法成为可能的基础发展。同样重要的是，Turner、Clough、Martin 和 Topp 成功开发了用于三角形单元的 FEM 插值器 [10]，适用于任意形状的结构部件。从某种意义上说，三角形元件的发明是一次 “量子飞跃”，因此，对于工程界的大部分人来说，FEM 的诞生是 Turner 等人 [10] 发表的具有里程碑意义的论文。以下是 2014 年庆祝美国国家工程院成立 50 周年的文件的摘录，该文件是对 FEM 历史的官方描述：为了确保安全并避免在飞机投入生产后进行代价高昂的修改，工程师需要一种可靠的方法来提前确定

their designs could withstand the stresses of fight. M. Jon Turner, head of Boeing’s Structural Dynamics Unit, addressed that problem in the early 1950s by bringing civil engineering professor Ray Clough of the University of California, Berkeley, and Harold Martin of the University of Washington to Boeing for summer ``faculty internships,’’ Collectively, they created a method of structural analysis that Turner applied at Boeing using computers to perform the myriad calculations need to predict real-world performance. That fruitful collaboration led to Clough’s development a few years later of what he named the finite element method (FEM). Clough formed a research group at UC Berkeley that used FEM in a host of analytical and experimental activities, from designing buildings and structures to withstand nuclear blasts or earthquakes to analyzing structural requirements for spacecraft and deep-water offshore drilling. By revolutionizing the application of computer technologies in engineering, FEM continues to help engineers design to this day all sorts of durable, cost-effective structures. Meanwhile, Turner’s efforts at Boeing contributed to the success of its renowned line of commercial jets, beginning in 1958 with the 707 and continuing in 1964 with the 727, which could land on shorter runways and serve more airports. Equipped with three fuel-efficient turbofan engines, the 727 became the workhorse of commercial aviation and helped achieve a threefold increase in U.S. passenger air traffic in the’60s.
他们的设计可以承受战斗的压力。波音公司结构动力学部门负责人 M. Jon Turner 在 1950 年代初期解决了这个问题，他邀请加州大学伯克利分校的土木工程教授 Ray Clough 和华盛顿大学的 Harold Martin 到波音公司进行暑期 “教师实习”，他们共同创造了一种结构分析方法，Turner 在波音公司应用了这种方法，使用计算机执行预测实际性能所需的无数计算。这种富有成效的合作导致 Clough 在几年后开发了他称之为有限元的方法（FEM）。Clough 在加州大学伯克利分校成立了一个研究小组，将 FEM 用于一系列分析和实验活动，从设计能够承受核爆炸或地震的建筑物和结构，到分析航天器和深水海上钻探的结构要求。通过彻底改变计算机技术在工程中的应用，FEM 至今仍在帮助工程师设计各种耐用、经济高效的结构。与此同时，特纳在波音的努力为其著名的商用喷气式飞机系列的成功做出了贡献，从 1958 年的 707 开始，到 1964 年的 727 继续，这些飞机可以在更短的跑道上降落，并为更多的机场提供服务。727 配备了三个省油的涡扇发动机，成为商业航空的主力军，并帮助美国客运航空在 60 年代实现了三倍的增长。

Independently and separately, in the early 1960s, Kang Feng of the Chinese Academy of Science also proposed a discretization-based numerical method for variational principles for solving elliptic partial differential equations [11]. As Peter Lax [11, 12] commented, “Independently of parallel developments in the West, he (Feng) created a theory of the finite element method. He was instrumental in both the implementation of the method and the creation of its theoretical foundation using estimates in Sobolev spaces….”, which was one of the first convergence studies of FEMs.
在 1960 年代初期，中国科学院的康峰还独立地提出了一种基于离散化的变分原理数值方法，用于求解椭圆偏微分方程 [11]。正如 Peter Lax [11， 12] 所评论的那样，“独立于西方的平行发展，他（Feng）创造了有限元方法的理论。他在该方法的实施和使用索博列夫空间中的估计创建其理论基础方面都发挥了重要作用…“，这是有限元的最早收敛研究之一。

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.
在此期间，几位伟大的工程人才都专注于开发 FEM。J.H. Argyris 和他的同事在斯图加特大学；R. Clough 及其同事，如加州大学伯克利分校的 E. L. Wilson 和 R.L. Taylor;O.C. Zienkiewicz 和他的同事，如斯旺西大学的 E. Hinton 和 B. Irons; 巴黎大学 XI 的 P. G. Ciarlet; 康奈尔大学的 R. Gallager 和他的团队、Philco Corporation 的 R. Melosh、列日大学的 B. Fraeijs de Veubeke 和空军技术学院的 J. S. Przemieniecki 为有限元方法的早期发展做出了一些重要和重大的贡献。

To understand what happened sixty years ago, we quote an excerpt from a FEM history paper by Clough and Wilson [13], in which they recalled:
为了理解 60 年前发生的事情，我们引用了 Clough 和 Wilson [13] 的一篇 FEM 历史论文的摘录，他们在其中回忆道：

When Clough presented the first paper using the finite element terminology in 1960 it attracted the attention of his friend, Professor O. C. Zienkiewicz, who was then on the faculty at Northwestern University. A few weeks after the presentation of the paper Zienkiewicz invited Clough to present a seminar on the finite element method to his students. Zienkiewicz was considered one of the world’s experts on the application of the finite difference method to the solution of continuum mechanics problems in Civil Engineering; therefore, Clough was prepared to debate the relative merits of the two methods. However, after a few penetrating questions about the finite element method, Zienkiewicz was almost an instant convert to the method.
当 Clough 在 1960 年提出第一篇使用有限元术语的论文时，它引起了他的朋友 O. C. Zienkiewicz 教授的注意，当时他在西北大学任教。论文发表几周后，Zienkiewicz 邀请 Clough 为他的学生举办一个关于有限元方法的研讨会。Zienkiewicz 被认为是应用有限差分法解决土木工程中连续介质力学问题的世界专家之一；因此，Clough 准备辩论这两种方法的相对优点。然而，在对有限元方法提出了一些尖锐的问题后，Zienkiewicz 几乎立即就皈依了该方法。

Z. Bazant, then a visiting associate research engineer at UC Berkeley, recalled: …… The founding of FEM was paper in the ASCE Conf. on Electronic Computation Clough [14], which was the first to derive, by virtual work, the finite element stiffness matrix of an element of a continuum (a triangular constant strain element). ……. I recall Ray Clough showing to me his 1962 report to the U.S. Engineer District, Little Rock, Corps of Engineers on his analysis of a crack observed in Norfolk Dam [15], during my stay in Berkeley in 1969. I was mesmerized by seeing that 1962 report. It presented a 2D stress analysis of large crack observed in Norfork dam. The dam was subdivided into about 200 triangular elements and provided stress contours for a number of loading cases. …… Clough was at that time way ahead of anybody else.
当时在加州大学伯克利分校担任访问副研究工程师的 Z. Bazant 回忆说：… 有限元元的成立是 ASCE 电子计算会议 [14] 上的论文，该论文首次通过虚拟工作推导出连续体单元（三角形恒定应变单元）的有限元刚度矩阵。……. 我记得 Ray Clough 向我展示了他 1962 年向美国小石城工程兵团工程师区提交的报告，该报告是他对 1969 年在伯克利逗留期间在诺福克大坝观察到的裂缝的分析 [15]。看到 1962 年的那篇报告，我被迷住了。它提出了在 Norfork 大坝中观察到的大裂缝的 2D 应力分析。大坝被细分为大约 200 个三角形单元，并为许多加载情况提供了应力等值线。…… 那时 Clough 远远领先于其他人。

To collaborate Z. Bazant’s recollection, we cite E.L. Wilson’s recounting of that part of FEM history:
为了配合 Z. Bazant 的回忆，我们引用了 E.L. Wilson 对 FEM 那段历史的叙述：

In 1956, Ray, Shirley, and three small children spent a year in Norway at the Ship Research Institute in Trondheim. The engineers at the institute were calculating stresses due to ship vibrations to predict fatigue failures at the stress concentrations. This is when Ray realized his element research should be called the Finite Element Method which could solve many different types of problems in continuum mechanics. Ray realized the FEM was a direct competitor to the Finite Difference Method. At that time FDM was being used to solve many problems in continuum mechanics. His previous work at Boeing, the Direct Stiffness Method, was only used to calculate displacements not stresses.
1956 年，Ray、Shirley 和三个年幼的孩子在挪威的特隆赫姆船舶研究所度过了一年。该研究所的工程师正在计算船舶振动引起的应力，以预测应力集中时的疲劳失效。这时 Ray 意识到他的元素研究应该被称为有限元方法，它可以解决连续介质力学中许多不同类型的问题。Ray 意识到 FEM 是有限差分法的直接竞争对手。当时，FDM 被用于解决连续介质力学中的许多问题。他之前在波音公司的工作是直接刚度法，仅用于计算位移，而不是应力。

In the fall semester of 1957, Ray returned from his sabbatical leave in Norway and immediately posted a page on the student bulletin board asking students to contact him if they were interested in conducting finite element research for the analysis of membrane, plate, shell, and solid structures. Although Ray did not have funding for finite element research, a few graduate students who had other sources of funds responded. At that time, the only digital computer in the College of Engineering was an IBM 701 that was produced in 1951 and was based on vacuum tube technology. The maximum number of linear equations that it could solve was 40. Consequently, when Ray presented his first FEM paper in September 1960, “The Finite Element Method in Plane Stress Analysis,” at the ASCE 2**nd Conference on Electronic Computation in Pittsburgh, Pennsylvania, the coarse-mesh stress-distribution obtained was not very accurate. Therefore, most of the attendees at the conference were not impressed. After the improvement of the speed and capacity of the computers on the Berkeley campus, Professor Clough’s paper was a very fine mesh analysis of an existing concrete dam. The paper was first presented in September 1962 at a NATO conference in Lisbon, Portugal. Within a few months, the paper was republished in an international Bulletin, which had a very large circulation, as “Stress Analysis of a Gravity Dam by the Finite Element Method”, (with E. Wilson), International Bulletin RILEM, No. 10, June 1963.
1957 年秋季学期，Ray 从挪威休假回来，并立即在学生公告板上发布了一个页面，询问学生是否有兴趣进行有限元研究以分析膜、板、壳和固体结构，请与他联系。尽管 Ray 没有用于有限元研究的资金，但一些有其他资金来源的研究生做出了回应。当时，工程学院唯一的数字计算机是 IBM 701，它于 1951 年生产，基于真空管技术。它可以求解的最大线性方程数为 40。因此，当 Ray 于 1960 年 9 月在宾夕法尼亚州匹兹堡举行的 ASCE 第 2 届电子计算会议上提交他的第一篇 FEM 论文 “平面应力分析中的有限元方法” 时，获得的粗网格应力分布不是很准确。因此，大多数与会者都没有留下深刻的印象。在伯克利校区计算机的速度和容量得到提高后，Clough 教授的论文是对现有混凝土坝进行了非常精细的网格分析。该论文于 1962 年 9 月在葡萄牙里斯本举行的北约会议上首次发表。几个月内，该论文被重新发表在发行量非常大的国际公报上，标题为 “用有限元方法对重力坝进行应力分析”，（与 E. Wilson 合著），国际公报 RILEM，第 10 期，1963 年 6 月。

The Lisbon paper reported on the finite element analysis of the 250-foot-high Norfork Dam in Arkansas, which had developed a vertical crack during construction in 1942. The FEM analysis correctly predicted the location and size of the crack due to the temperature changes and produced realistic displacements and stresses within the dam and foundation for both gravity and several hydrostatic load conditions. Because of this publication, many international students and visiting scholars came to Berkeley to work with Professor Clough. Also, he freely gave the FORTRAN listing of their finite element analysis computer program to be used to evaluated displacement and stresses in other two-dimensional plane structures with different geometry, materials, and loading. Therefore, professional engineers could easily use the powerful new FEM to solve for the stress distributions in their structural engineering problems in continuum mechanics. However, he did not capitalize on his success in the development of the FEM. He returned to the task of building the earthquake engineering program at Berkeley – the task he given when he was first hired in 1949.
里斯本的论文报道了阿肯色州 250 英尺高的诺福克大坝的有限元分析，该大坝在 1942 年的施工过程中产生了垂直裂缝。有限元分析正确预测了温度变化引起的裂缝的位置和大小，并在重力和几种静水载荷条件下在大坝和地基内生成了真实的位移和应力。因为这本书的出版，许多国际学生和访问学者来到伯克利与 Clough 教授一起工作。此外，他还免费提供了 FORTRAN 的有限元分析计算机程序列表，该程序用于评估具有不同几何形状、材料和载荷的其他二维平面结构中的位移和应力。因此，专业工程师可以轻松使用强大的新 FEM 来解决连续介质力学中结构工程问题中的应力分布。然而，他并没有利用他在 FEM 发展中的成功。他回到了在伯克利建立地震工程项目的任务 —— 这是他在 1949 年第一次被聘用时交给的任务。

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**.**
由于他们对 FEM 的创建和发展做出的决定性贡献，RW Clough 于 1994 年被当时的美国副总统 Al Gore 授予国家科学奖章，而 O. C. Zienkiewicz 则被授予大英帝国司令勋章（CBE）。今天，人们的共识是，J.H. Argyris、R.W. Clough 和 O. C. Zienkiewicz 对有限元方法的诞生和早期发展做出了最关键、最关键和最重要的贡献，此前他为 R. Courant 的数学基础做出了早期贡献。

It is worth noting that E.L. Wilson of UC Berkeley was the first person to develop finite element open-source software. An excerpt from Clough and Wilson’s paper in 1999 stated 13: In 1958 Wilson, under the direction of Clough, initiated the development of an automated finite element program based on the rectangular plane stress finite element developed at Boeing. After several months of learning to program the IBM 701, Wilson produced a limited capacity, semiautomated program which was based on the force method. An MS research report was produced, which has long since been misplaced, with the approximate title of Computer Analysis of Plane Stress Structures. …… In 1959 the IBM 704 computer was installed on the Berkeley Campus. It had 32K of 32-bit memory and a floating-point arithmetic unit which was approximately 100 times faster than the IBM 701. This made it possible to solve practical structures using fine meshes.
值得注意的是，加州大学伯克利分校的 E.L. Wilson 是开发有限元开源软件的第一人。1999 年 Clough 和 Wilson 论文的摘录指出 13：1958 年，Wilson 在 Clough 的指导下，开始开发基于波音公司开发的矩形平面应力有限元的自动化有限元程序。在学习了几个月的 IBM 701 编程后，Wilson 制作了一个基于力法的有限容量的半自动程序。一份 MS 研究报告被制作出来，该报告早已被放错地方，大致标题为平面应力结构的计算机分析。……1959 年，IBM 704 计算机安装在伯克利校区。它具有 32K 的 32 位内存和一个浮点运算单元，比 IBM 701 快大约 100 倍。这使得使用细网格求解实际结构成为可能。

It is also worth mentioning that, Oden [16], a senior structural engineer in the research and development division of General Dynamics Corporation at Fort Worth at the time, wrote a 163-page comprehensive technical report with G.C. Best, in which they developed a long list of solid and structural finite elements, including tetrahedral, hexahedral, thin plate, thick plate, plate element with stringers or stiffeners, composite sandwich plate elements, and shallow shell elements [17]. In fact, Oden and Best wrote one of the first general purpose FEM computer codes at the time. The Fortran FEM code developed by Oden and Best had an element library that includes elements for 3D elasticity, 2D plane elasticity, 3D beam and rod elements, composite layered plate and shell elements, and elements for general composite materials. Their work also included hybrid methods and stress based FEMs, which may be even earlier than those of Pian [18]. Moreover, Oden and Best’s FEM code was also able to handle FEM eigenvalue modal analysis, and numerical integration over triangle and tetrahedra elements, and it had linear system solvers for general FEM static analysis that were among the most effective at that time (see [19]). This FEM computer code was used for many years in aircraft analysis and design in the aerospace and defense industry.
还值得一提的是，当时位于沃斯堡的通用动力公司研发部门的高级结构工程师 Oden [16] 与 G.C. Best 一起撰写了一份 163 页的综合技术报告，其中他们开发了一长串实体和结构有限元，包括四面体、六面体、薄板、厚板、带纵梁或加劲肋的板单元、复合夹芯板单元和浅壳单元 [17]。事实上，Oden 和 Best 编写了当时最早的通用 FEM 计算机代码之一。由 Oden 和 Best 开发的 Fortran FEM 代码有一个单元库，其中包括 3D 弹性单元、2D 平面弹性单元、3D 梁和杆单元、复合多层板和壳单元以及通用复合材料单元。他们的工作还包括混合方法和基于应力的有限元法，这可能比 Pian [18] 的工作还要早。此外，Oden 和 Best 的 FEM 代码还能够处理 FEM 特征值模态分析，以及三角形和四面体单元的数值积分，并且它具有用于一般 FEM 静态分析的线性系统求解器，这是当时最有效的求解器之一（参见 [19]）。该 FEM 计算机代码在航空航天和国防工业的飞机分析和设计中使用多年。

It appears that, I.C. Taig [20] of English Electric Aviation first introduced the concept of the isoparametric element when he used the matrix-displacement method to conduct stress analysis of aerospace structures, which was later formally dubbed ``isoparametric element’’ by Ergatoudis et al. [21].
英国电气航空的 I.C. Taig [20] 在使用矩阵 - 位移法对航空航天结构进行应力分析时，首次引入了等参数单元的概念，后来 Ergatoudis 等 [21] 将其正式称为 “等参数单元”。

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]).
值得一提的是，还有其他一些先驱者在有限元技术的早期发展中做出了一些重大贡献，如 Levy [22]、Comer [23]、Langefors [24]、Denke [25]、Wehle 和 Lansing [26]、Hoff 等 [27] 和 Archer [28] 等。这些人聚集在一起，为有限元方法的创建做出了非凡的历史性贡献。其中，JS Przemieniecki （Janusz Stanisław Przemieniecki）做出了一些显着贡献，他是一名波兰工程师，1961 年至 1995 年在美国俄亥俄州空军技术学院担任教授和院长。Przemieniecki 开展了一系列开创性的研究工作，涉及使用有限元法对航空航天结构（如板、壳和柱）进行应力和屈曲分析（参见 Przemieniecki [29]、Przemieniecki 和 Denke [30]、Przemieniecki [28,29,30,31,32,33]）。

In terms of worldwide research interest, by 1965, FEM research had become a highly active field, with the total number of papers published in the literature exceeding 1000. During this period, there were two seemingly unrelated events for FEM development, which significantly affected future FEM developments. These events were the discovery of mixed variational principles in elasticity. In 1950, E. Reissner [34] rediscovered E. Hellinger’s mixed variational principle from 1914 [35], in which both the displacement field and the stress field are the primary unknowns. This variational principle is called the Hellinger–Reissner variational principle. Shortly after, Hu [36] and Washizu [37] proposed a three-field mixed variational principle in elasticity, which was called the Hu-Washizu variational principle. As early as 1964, Pian [18] recognized the potential of using these variational principles to formulate Galerkin weak form-based FE formulations and proposed the assumed stress FEM. This began the use of mixed variational principles to formulate Galerkin FEMs, which was followed by the assumed strain FEM developed by J. Simo and his co-workers in the later period of FEM developments.
就全球研究兴趣而言，到 1965 年，有限元研究已成为一个非常活跃的领域，文献中发表的论文总数超过 1000 篇。在此期间，FEM 发展发生了两个看似无关的事件，这对未来的 FEM 发展产生了重大影响。这些事件是弹性中混合变分原理的发现。1950 年，E. Reissner [34] 重新发现了 1914 年的 E. Hellinger 混合变分原理 [35]，其中位移场和应力场都是主要的未知数。这种变分原理称为 Hellinger-Reissner 变分原理。不久之后，胡 [36] 和鹫津 [37] 提出了一种弹性的三场混合变分原理，称为胡 - 鹫津变分原理。早在 1964 年，Pian [18] 就认识到使用这些变分原理来制定基于伽辽金弱形式的有限元公式的潜力，并提出了假设应力有限元。这开始使用混合变分原理来构建 Galerkin FEM，随后 J. Simo 和他的同事在 FEM 发展的后期开发了假设应变 FEM。

2 (1966–1991) The Golden Age of the Finite Element Method 2 （1966–1991）有限元法的黄金时代

The mid 1960s saw rapid developments in finite element method research and applications. As T.J.R. Hughes recalled, “I first heard the words ‘the Finite Element Method” in 1967—which changed my life. I started to read everything that was available and convinced my boss to start the Finite Element Method Development Group, which he did. Dr. Henno Allik was Group Leader, and I was the Group, then we added programmers. In one year, we had a 57,000-line code, GENSAM (General Structural Analysis and Matrix Program, or something like that). That was 1969, and the code was continually developed thereafter and may still be in development and use at GD/Electric Boat and General Atomics, originally a division of General Dynamics.”
1960 年代中期见证了有限元方法研究和应用的快速发展。正如 T.J.R. Hughes 回忆的那样，“我第一次听到 ’ 有限元方法 ’ 这个词是在 1967 年，它改变了我的生活。我开始阅读所有可用的资料，并说服我的老板成立有限元方法开发小组，他做到了。Henno Allik 博士是组长，我是组长，然后我们增加了程序员。在一年内，我们有一个 57,000 行的代码，GENSAM（General Structural Analysis and Matrix Program，或类似的东西）。那是 1969 年，此后代码不断开发，可能仍在 GD/Electric Boat 和 General Atomics（最初是 General Dynamics 的一个部门）开发和使用。

Starting from the end of 1960s, the rigorous approximation theory that underpins the FEM started to be developed (e.g. Aubin [38], Zlamal [39], Birkhoff [40], Nitsche [41], Aziz [42], Bubuska [43], Bubuska and Aziz [44], Dupont [45], Douglas and Dupont [46]. Nitsche and Schatz [47], and Bubuska [48]). This movement was first highlighted by the proof of optimal and superconvergence of FEMs. This attracted the interest of some distinguished mathematicians all over the world, including G. Birkhoff, M.H. Schultz, R.S. Varaga, J. Bramble, M. Zlamal, J. Cea, J.P. Aubin, J. Douglas, T. Dupont, L.C. Goldstein, LR. Scott, J. Nitsche, A.H. Schatz, P.G. Cialet, G. Strang, G. Fix, JL. Lions, M. Crouzeix, P.A. Raviart, and I. Babuska, A.K. Aziz, and J.T. Oden (see Bramble, Notsche and Schatz [49], Bubuska, Oden, and Lee [50]) . Some notable results developed for the proof of FEM convergence are the Cea lemma and the Bramble-Hubert lemma. The fundamental work of Nitsche [41] on L∞ estimates for general classes of linear elliptic problems also stands out as one of the most important contributions for mathematical foundation of FEM in 1970s. It may be noted that unlike other mathematics movements, the convergence study of FEMs was an engineering-oriented movement. The mathematicians soon found that, in practice, engineers were using either non-conforming FEM interpolants or numerical quadrature that violates variational principles or the standard bilinear form in Hilbert space. G. Strang referred to these numerical techniques as “variational crimes”. To circumvent complicated convergence proofs, the early FEM patch tests were invented by B. Iron and R. Melosh, which were proven to be instrumental for ensuring convergence to the correct solution.
从 1960 年代末开始，支撑有限元的严格近似理论开始发展（例如 Aubin [38]、Zlamal [39]、Birkhoff [40]、Nitsche [41]、Aziz [42]、Bubuska [43]、Bubuska and Aziz [44]、Dupont [45]、Douglas and Dupont [46]。Nitsche 和 Schatz [47] 以及 Bubuska [48]）。这一运动首先通过证明 FEM 的最优和超收敛来突出。这引起了世界各地一些杰出数学家的兴趣，包括 G. Birkhoff、MH Schultz、R.S. Varaga、J. Bramble、M. Zlamal、J. Cea、JP Aubin、J. Douglas、T. Dupont、L.C. Goldstein、LR。斯科特，J. Nitsche，AH Schatz，PG Cialet，G. Strang，G. Fix，JL。Lions， M. Crouzeix， P.A. Raviart， and I. Babuska， A.K. Aziz，和 J.T. Oden （参见 Bramble， Notsche and Schatz [49]， Bubuska， Oden， and Lee [50]）。为证明有限元收敛性而开发的一些值得注意的结果是 Cea 引理和 Bramble-Hubert 引理。Nitsche [41] 对线性椭圆问题一般类别的 L∞ 估计的基础工作也是 1970 年代对有限元数学基础的最重要贡献之一。可以指出的是，与其他数学运动不同，FEM 的收敛研究是一个以工程为导向的运动。数学家很快发现，在实践中，工程师要么使用不符合标准的 FEM 插值器，要么使用违反变分原理或希尔伯特空间中标准双线性形式的数值正交。G. Strang 将这些数字技术称为 “变分犯罪”。为了规避复杂的收敛证明，早期的 FEM 斑贴测试由 B. Iron 和 R. Melosh 发明，事实证明，这些测试有助于确保收敛到正确的解决方案。

Following T.H.H. Pian’s invention of the assumed stress element, attention shifted to the mixed variational principle based FEM [51, 52]). In 1965, L.R. Hermann [53] proposed a mixed variational principle for incompressible solids. However, most mixed variational principles are not extreme variational principles, and thus suffer from numerical instability. In early 1970s, I. Babuska and F. Brezzi discovered their groundbreaking results, known today as the Babuska-Brezzi condition, or the LBB condition, giving tribute to O. Ladyzhenskya—a Russian mathematician, who provided the early insight of this problem. The so-called LBB condition, or the inf–sup condition, provides a sufficient condition for a saddle point problem to have a unique solution that depends continuously on the input data; thus, it provides a guideline to construct shape functions for the mixed variational principle-based FEM (see: [54]).
在 T.H.H. Pian 发明了假设应力单元之后，人们的注意力转移到了基于混合变分原理的有限元 [51， 52]）。1965 年，L.R. Hermann [53] 提出了不可压缩固体的混合变分原理。然而，大多数混合变分原理不是极端变分原理，因此存在数值不稳定性。在 1970 年代初期，I. Babuska 和 F. Brezzi 发现了他们的开创性结果，今天被称为 Babuska-Brezzi 条件或 LBB 条件，以向俄罗斯数学家 O. Ladyzhenskya 致敬，他提供了这个问题的早期见解。所谓的 LBB 条件或 inf-sup 条件为鞍点问题提供了一个充分的条件，使其具有持续依赖于输入数据的唯一解；因此，它为基于混合变分原理的有限元法构建形状函数提供了指南（参见：[54]）。

Entering the 1970s, FEM development began to focus on using FEM to simulate the dynamic behavior of structures, including crashworthiness in the automotive industry. Various time integration methods had been developed, including the Newmark-beta method, Wilson-theta method [55], the Hilbert-Hughes-Taylor alpha method [56], the Houbolt integration algorithm, and the explicit time integration algorithm [57].
进入 1970 年代，FEM 开发开始专注于使用 FEM 来模拟结构的动态行为，包括汽车行业的耐撞性。已经开发了各种时间积分方法，包括 Newmark-beta 方法、Wilson-theta 方法 [55]、Hilbert-Hughes-Taylor alpha 方法 [56]、Houbolt 积分算法和显式时间积分算法 [57]。

Since the early 1970s, FEM explicit time integration methods had been used to solve various engineering problems, e.g., dynamic contact problems [58], dynamics problems of solid structure [59], fluid dynamics problems [60], and it was extensively used in large scale FEM Lagrangian hydrocodes developed in Lawrence Livermore National Laboratory [61]. However, an important development came in the late 1970s when T. Belytschko, K. C. Park, and later TJR Hughes proposed using explicit or implicit-explicit, or implicit time integration with damping control to solve nonlinear structural dynamics problems. Other notable contributors to this key development are A. Combescure, A. Gravouil among others (see e.g., [62]). It turned out that the explicit time integration was a game changer for the automotive industry, establishing FEM technology as the main tool of passenger vehicle design and crashworthiness analysis. By the end of 1980s, there were thousands of workstations running explicit time integration-based FEM codes in the three major automakers in the United States.
自 1970 年代初以来，有限元显式时间积分方法已被用于解决各种工程问题，例如动态接触问题 [58]、固体结构动力学问题 [59]、流体动力学问题 [60]，并被广泛用于劳伦斯利弗莫尔国家实验室开发的大规模有限元拉格朗日水文规范 [61]。然而，一个重要的发展出现在 1970 年代后期，当时 T. Belytschko、KC Park 和后来的 TJR Hughes 提议使用显式或隐式 - 显式或隐式时间积分与阻尼控制来解决非线性结构动力学问题。这一关键发展的其他著名贡献者是 A. Combescure、A. Gravouil 等（参见 [62]）。事实证明，显式时间积分改变了汽车行业的游戏规则，使 FEM 技术成为乘用车设计和耐撞性分析的主要工具。到 1980 年代末，美国三大汽车制造商有数千个工作站运行基于显式时间集成的 FEM 代码。

One of the main FEM research topics in the 1980s was using FEM techniques to solve the Navier–Stokes equation as an alternative to finite difference and finite volume methods. Starting the early 1970s, J.T. Oden and his co-workers had begun working on FEM solutions for fluid dynamics [63]. The main challenge in using FEM solving fluid dynamics problems is that the Navier–Stokes equation is not an elliptical partial different equation, and the minimization or variational principle-based Petrov–Galerkin procedure may suffer both stability as well as convergence issues. To resolve these issues, T.J.R. Hughes, and his co-workers such as A.N. Brooks and T.E. Tezduyar developed the streamline upwind/Petrov Galerkin method and later Stabilized Galerkin FEM to solve Navier–Stokes equations under various initial and boundary conditions [64]. Furthermore, Hughes and co-workers later developed space–time FEMs and variational multiscale FEMs (see Hughes and Tezdyuar [65] Brooks and Hughes [66], Mizukami and Hughes [67], and Hughes [68]). For this work, from 1986 to 1991, Hughes and his co-workers such as L.P. Franca and others wrote a ten-part series on FEM formulations for computational fluid dynamics Hughes et al. [69,70,71,71] and Shakib et al. [72]. Several years later, Hughes developed a Green function-based subgrid model acting as a stabilized finite element method [68].
1980 年代 FEM 的主要研究课题之一是使用 FEM 技术求解 Navier-Stokes 方程，作为有限差分和有限体积方法的替代方案。从 1970 年代初开始，J.T. Oden 和他的同事开始研究流体动力学的有限元解决方案 [63]。使用 FEM 求解流体动力学问题的主要挑战是 Navier-Stokes 方程不是椭圆偏差分方程，基于最小化或变分原理的 Petrov-Galerkin 程序可能会同时遇到稳定性和收敛问题。为了解决这些问题，T.J.R. Hughes 和他的同事，如 A.N. Brooks 和 T.E. Tezduyar，开发了流线逆风 / 彼得罗夫加辽金方法，后来又开发了稳定伽辽金有限元法，以求解各种初始和边界条件下的纳维 - 斯托克斯方程 [64]。此外，Hughes 及其同事后来开发了时空有限元和变分多尺度有限元（参见 Hughes 和 Tezdyuar [65]、Brooks 和 Hughes [66]、Mizukami 和 Hughes [67] 以及 Hughes [68]）。为了这项工作，从 1986 年到 1991 年，Hughes 和他的同事，如 L.P. Franca 等人，Hughes 等 [69,70,71,71] 和 Shakib 等 [72] 撰写了关于计算流体动力学有限元公式的十部分系列文章。几年后，Hughes 开发了一种基于格林函数的子网格模型，作为稳定的有限元方法 [68]。

Towards the mid-1980s, advanced FEM mesh generation techniques have been developed, which incorporate various solid modeling techniques by using interactive computer graphics and adaptive mesh generator [73], Bennett and Botkin [74] as well as improved quadtree approach to generate FEM meshes for complex geometric shapes and objects [75]. Today, FEM mesh generation has become an integrated part of solid modeling and engineering design, which has the capabilities of automatic node insertion and refinement Wordenweber [76] and Ho-Le [77].
到 1980 年代中期，已经开发了先进的 FEM 网格生成技术，通过使用交互式计算机图形和自适应网格生成器 [73]、Bennett 和 Botkin [74] 以及改进的四叉树方法来生成复杂几何形状和对象的 FEM 网格 [75]。如今，FEM 网格生成已成为实体建模和工程设计不可或缺的一部分，它具有自动节点插入和优化 Wordenweber [76] 和 Ho-Le [77] 的功能。

Among the many advances in FEM technologies in 1980s, the most notable may belong to J. Simo at the University of California, Berkeley and later at Stanford University. Simo and Taylor [78] developed the consistent tangent operator for computational plasticity, which was a milestone after the original concept of consistent linearization proposed by Hughes and Pister [79]. Moreover, after the Hughes-Liu 3D degenerated continuum shell and beam elements and the Belytchko-Tsay single-point element, Simo and his co-workers, such as L. Vu-Quoc and D.D. Fox, developed geometrically exact beam and shell theories and their FEM formulations (see Simo and Vu-Quoc [80], Simo and Fox [81], and Simo et al. [82]. Moreover, Simo and his co-workers such as MS Rifai and F. Armero also developed various assumed strain or enhanced strain methods for mixed variational formulations (see Simo and Rifai [83] and Simo and Armero [84]). It should be noted that E. Ramm and his colleagues at the University of Stuttgart have also made significant contributions on geometrically nonlinear shell element formulations over a span of more than thirty years e.g., Andelfinger and Ramm [85] and Bischoff and Ramm [86].
在 1980 年代 FEM 技术的众多进步中，最引人注目的可能属于加州大学伯克利分校和后来斯坦福大学的 J. Simo。Simo 和 Taylor [78] 开发了用于计算塑性的一致切算子，这是继 Hughes 和 Pister [79] 提出的一致线性化的原始概念之后的一个里程碑。此外，在 Hughes-Liu 3D 退化连续体壳和梁单元以及 Belytchko-Tsay 单点单元之后，Simo 和他的同事，如 L. Vu-Quoc 和 D.D. Fox，开发了几何精确的梁和壳理论及其有限元公式（参见 Simo 和 Vu-Quoc [80]、Simo 和 Fox [81] 以及 Simo 等人 [82]。此外，Simo 和他的同事，如 MS Rifai 和 F. Armero 还开发了各种用于混合变分公式的假设应变或增强应变方法（参见 Simo 和 Rifai [83] 以及 Simo 和 Armero [84]）。应该指出的是，斯图加特大学的 E. Ramm 和他的同事在 30 多年的时间里也对几何非线性壳单元公式做出了重大贡献，例如 Andelfinger 和 Ramm [85] 以及 Bischoff 和 Ramm [86]。

Another highlight of FEM technology is the development of FEM solvers for fluid–structure interaction. From the mid-1970s to early 1990s, there was an urgent need to develop techniques to solve large-scale fluid–structure interaction problems in the aerospace and civil engineering industries. A class of FEM fluid–structure interaction solvers were developed, and some early contributors include J. Donea, A. Huerta (see [87,88,89,89]), and the Hughes-Liu-Zimmermann Arbitrary Lagrangian–Eulerian (ALE) fluid–structure FEM formulation (see [90, 91]), which describes the moving boundary problem. ALE-based FEM simulations were used due to their ability to alleviate many of the drawbacks of traditional Lagrangian-based and Eulerian-based FEM formulations.
FEM 技术的另一个亮点是开发了用于流固耦合的 FEM 求解器。从 1970 年代中期到 1990 年代初，迫切需要开发技术来解决航空航天和土木工程行业中的大规模流固耦合问题。开发了一类有限元流固耦合求解器，一些早期的贡献者包括 J. Donea、A. Huerta（参见 [87,88,89,89]）和描述移动边界问题的 Hughes-Liu-Zimmermann 任意拉格朗日 - 欧拉（ALE）流体 - 结构有限元公式（参见 [90,91]）。使用基于 ALE 的 FEM 仿真是因为它们能够缓解传统基于拉格朗日和基于欧拉的 FEM 公式的许多缺点。

When using the ALE technique in engineering modeling and simulations, the computational mesh inside the domains can move arbitrarily to optimize the shapes of elements, while the mesh on the boundaries and interfaces of the domains can move along with materials to precisely track the boundaries and interfaces of a multi-material system. The invention of the ALE FEM may be credited to Hirt, Amsden, and Cook [92]. C. Farhat was the first person to use a large-scale parallel ALE-FEM solver to compute fluid–structure interaction problems [93]. He and his group systematically applied FEM-based computational fluid dynamics (CFD) solvers for aircraft structure design and analysis. They developed the finite element tearing and interconnecting (FETI) method for the scalable solution of large-scale systems of equations on massively parallel processors. Today, some fundamental ALE concepts have also been applied to numerical modeling engineering fields other than FEM, such as meshfree modeling. For example, to alleviate tensile instability and distorted particle distributions in smoothed particle hydrodynamic (SPH) simulations, a so-called shifting technique has been adopted in many of today’s SPH simulations (e.g., Oger et al. [94]).
在工程建模和仿真中使用 ALE 技术时，域内的计算网格可以任意移动以优化单元的形状，而域的边界和界面上的网格可以随材料一起移动，以精确跟踪多材料系统的边界和界面。ALE FEM 的发明可以归功于 Hirt、Amsden 和 Cook [92]。C. Farhat 是第一个使用大规模并行 ALE-FEM 求解器计算流固耦合问题的人 [93]。他和他的团队系统地将基于 FEM 的计算流体动力学（CFD）求解器应用于飞机结构设计和分析。他们开发了有限元撕裂和互连（FETI）方法，用于在大规模并行处理器上实现大规模方程组的可扩展解决方案。今天，一些基本的 ALE 概念也已应用于 FEM 以外的数值建模工程领域，例如无网格建模。例如，为了减轻平滑粒子流体动力学（SPH）模拟中的拉伸不稳定性和扭曲的粒子分布，当今的许多 SPH 模拟都采用了一种所谓的移位技术（例如，Oger 等人 [94]）。

The FEM fluid–structure interaction research had a major impact on many practical applications, such as providing the foundation for the patient specific modeling of vascular disease and the FEM-based predictive medicine later developed by C.A. Taylor and T.J.R. Hughes and their co-workers in the mid-1990s (see Taylor et al. [95]). Holzapfel, Eberlein, Wriggers, and Weizsäcker also developed large strain FEM formulations for soft biological membranes (see Holzapfel et al. [96]).
FEM 流固耦合研究对许多实际应用产生了重大影响，例如为血管疾病的患者特异性建模以及后来由 C.A. Taylor 和 T.J.R. Hughes 及其同事在 1990 年代中期开发的基于 FEM 的预测医学奠定了基础（参见 Taylor 等 [95]）。Holzapfel、Eberlein、Wriggers 和 Weizsäcker 也开发了用于软生物膜的大应变 FEM 制剂（参见 Holzapfel 等 [96]）。

Another major milestone in the development of FEMs was the invention and the development of nonlinear probabilistic or random field FEMs, which was first developed by W. K. Liu and T. Belytschko in the late 1980s. (e.g., Liu et al. [97]). By considering uncertainty in loading conditions, material behavior, geometric configuration, and support or boundary conditions, the probabilistic FEM provided a stochastic approach in computational mechanics to account for all these uncertain aspects, which could then be applied in structure reliability analysis. The random field FEM research has become crucial in civil and aerospace engineering and the field of uncertainty quantification.
有限元法发展的另一个重要里程碑是非线性概率或随机场有限元法的发明和发展，该法由 W. K. Liu 和 T. Belytschko 在 1980 年代后期首次开发（例如，Liu 等 [97]）。通过考虑载荷条件、材料行为、几何构型和支撑或边界条件的不确定性，概率有限元在计算力学中提供了一种随机方法来解释所有这些不确定性，然后可以应用于结构可靠性分析。随机场有限元研究在土木和航空航天工程以及不确定性量化领域变得至关重要。

In the early 1980s, M.E. Botkin at General Motors research Lab [98], and N. Kikuchi and his group at the University of Michigan developed structural shape optimization FEM for the automotive industry (see [99], [100], [101]). Other contributors include M. H. Imam from Uman Al-Qura University [102]). This preceded the seminal 1988 paper of Bendsoe and Kikuchi, who developed a homogenization approach to finding the optimal shape of a structure under prescribed loading. Later developments in topology optimization were driven by Gengdong Cheng, Martin Bendsoe, and his student, Ole Sigmund.
在 1980 年代初期，通用汽车研究实验室的 M.E. Botkin [98] 和密歇根大学的 N. Kikuchi 和他的团队开发了用于汽车行业的结构形状优化有限元（参见 [99]、[100]、[101]）。其他贡献者包括来自 Uman Al-Qura 大学的 M. H. Imam [102]）。这早于 Bendsoe 和 Kikuchi 在 1988 年发表的开创性论文，他们开发了一种均质化方法，用于在规定载荷下找到结构的最佳形状。后来拓扑优化的开发是由 Gengdong Cheng、Martin Bendsoe 和他的学生 Ole Sigmund 推动的。

One of the driving forces in the FEM development during in the early decades was the safety analysis of big dams, at first, then of concrete nuclear reactor vessels for gas-cooled reactors, [103], nuclear containments, hypothetical nuclear reactor accidents [104, 105] and of tunnels and of foundations for reinforced concrete structures. To simulate concrete failure, the vertical stress drops in FEMs, and progressive softening technique were introduced already in 1968. However, the spurious mesh sensitivity and its impact on strain localization was generally overlooked until demonstrated mathematically by Bazant [105], because strain softening states of small enough test specimens in stiff enough testing frames are stable. Numerically this was demonstrated by crack band FEM calculations in Bazant and Cedolin [106], where it was also shown that spurious mesh sensitivity causing the sudden stress drop can be avoided by adjusting the material strength to ensure the correct energy release rate. Hillerborg et al. [107] avoided mesh sensitivity by using an interelement cohesive softening called the fictitious crack model. Despite the success of the early FEM calculations, the concept of progressive strain-softening damage was not generally accepted by mechanicians until its validity and limitations were demonstrated by Bazant and Belytschko [108] and Bazant and Chang [109]. They showed that the existence of elastic unloading stiffness (previously ignored) makes waves propagation in a strain-softening state possible. In 1989, Lubliner, Oliver, Oller, and Onate ([110] developed a plastic-damage theory-based FEM formulation to model concrete materials by introducing internal variables, which has the capacity of modeling concrete material degradation and cracking. Today, the multiscale based homogenization and damage analysis method is the state-of-the-art FEM modeling for concrete materials [111].
在最初的几十年里，有限元理论发展的驱动力之一是对大型水坝的安全分析，首先是气冷堆的混凝土核反应堆容器 [103]、核安全壳、假设的核反应堆事故 [104,105] 以及钢筋混凝土结构的隧道和地基的安全分析。为了模拟混凝土破坏，FEM 中的垂直应力下降和渐进软化技术早在 1968 年就已经引入。然而，在 Bazant [105] 用数学证明之前，伪网格敏感性及其对应变定位的影响通常被忽视，因为在足够坚硬的测试框架中，足够小的试样的应变软化状态是稳定的。Bazant 和 Cedolin [106] 的裂纹带有限元计算在数值上证明了这一点，其中还表明，通过调整材料强度以确保正确的能量释放速率，可以避免导致突然应力下降的伪网格敏感性。Hillerborg 等 [107] 通过使用称为虚构裂纹模型的单元间内聚软化来避免网格敏感性。尽管早期的有限元计算取得了成功，但渐进式应变软化损伤的概念并未被机械师普遍接受，直到 Bazant 和 Belytschko [108] 以及 Bazant 和 Chang [109] 证明了其有效性和局限性。他们表明，弹性卸载刚度（以前被忽略）的存在使得波在应变软化状态下的传播成为可能。1989 年，Lubliner、Oliver、Oller 和 Onate （[110] 开发了一种基于塑性损伤理论的有限元公式，通过引入内部变量来模拟混凝土材料，该公式具有模拟混凝土材料降解和开裂的能力。如今，基于多尺度的均质化和损伤分析方法是最先进的混凝土材料有限元建模 [111]。

In the mid-1980s, the mesh sensitivity issue in calculating strain softening or strain localization problems in computational plasticity became a challenging topic. It was eventually accepted that the partial differential equations that are associated with the classical plasticity become ill-posed after the material passes the yield point and enters the softening stage. This especially became a dire situation when civil engineers applied FEM to solve complex structural and geotechnical engineering problems, which involved complex plastic deformations of concrete, rock, soil, clay, and granular material in general. Bazant and others realized that this is because the classical continuum plasticity theory lacks an internal length scale. To remedy this problem, starting from the middle 1980s, many efforts were devoted to establishing FEM formulations of nonlocal (Bazant et al. [112]), strain-gradient, strain-Laplacian media, micropolar or Cosserat continua, because they provided an internal length scale, allowing FEM simulations to capture, in a mesh-independent manner, the strain softening, strain localization or shear band formation. Pijaudier-Cabot and Bazant [113] developed an effective nonlocal FEM in which nonlocality is applied only to the damage strain. Other influential and representative works in this topic are from De Borst [114], Peerlings, De Borest and others [115]. Steinmann and Willam [116], Dietsche et al. [117], Steinmann [118], and Iordache and Willam [119].
在 1980 年代中期，计算应变软化或计算塑性中的应变局部化问题中的网格敏感性问题成为一个具有挑战性的话题。最终人们认为，在材料通过屈服点并进入软化阶段后，与经典塑性相关的偏微分方程变得病态。当土木工程师使用 FEM 解决复杂的结构和岩土工程问题时，情况尤其严峻，这些问题涉及混凝土、岩石、土壤、粘土和颗粒材料的复杂塑性变形。Bazant 和其他人意识到，这是因为经典的连续体塑性理论缺乏内部长度尺度。为了解决这个问题，从 1980 年代中期开始，人们投入了大量精力来建立非局部（Bazant 等 [112]）、应变梯度、应变拉普拉斯介质、微极或 Cosserat continua 的 FEM 公式，因为它们提供了一个内部长度尺度，允许 FEM 模拟以独立于网格的方式捕获应变软化、应变定位或剪切带形成。Pijaudier-Cabot 和 Bazant [113] 开发了一种有效的非局部有限元，其中非局部性仅应用于损伤应变。该主题中其他有影响力和代表性的著作来自 De Borst [114]、Peerlings、De Borest 等 [115]。Steinmann 和 Willam [116]、Dietsche 等 [117]、Steinmann [118] 以及 Iordache 和 Willam [119]。

In 1976, TJR Hughes, R.L. Taylor, J.L. Sackman, A. Curnier, and W. Kanoknukulchai published a paper entitled “A finite element method for a class of contact-impact problems.” [120]. This is one of the earliest FEM analyses in computational contact mechanics. It is the very first work on FEM modeling of dynamic contact and impact problems, and it plays an important role in the simulation accuracy for engineering problems involving interaction between different continuous objects. Examples include sheet metal forming, target impact and penetration, and interaction between pavement and tires. Developing accurate FEM contact algorithms has been a focal point since the 1980s. Various FEM contact algorithms have been developed, and some main contributors are N. Kikuchi, J.T. Oden, J. Simo, P, Wriggers, R.L. Taylor, P. Papadopoulos, and T.A. Laursen. FEM contact algorithm research remained an active research topic until late 1990s (see Simo et al. [121], Simo and Lausen [122], Kikuchi and Oden [123], and Papadopoulos and Taylor [124]).
1976 年，TJR Hughes、R.L. Taylor、J.L. Sackman、A. Curnier 和 W. Kanoknukulchai 发表了一篇题为 “一类接触 - 冲击问题的有限元方法” 的论文。[120]. 这是计算接触力学中最早的有限元分析之一。这是动态接触和冲击问题 FEM 建模的首项工作，在涉及不同连续对象之间交互的工程问题的仿真精度方面发挥着重要作用。示例包括钣金成型、目标冲击和穿透以及路面和轮胎之间的相互作用。自 1980 年代以来，开发准确的 FEM 接触算法一直是我们的重点。已经开发了各种 FEM 接触算法，一些主要贡献者是 N. Kikuchi、J.T. Oden、J. Simo、P、Wriggers、R.L. Taylor、P. Papadopoulos 和 T.A. Laursen。直到 1990 年代末，FEM 接触算法研究仍然是一个活跃的研究课题（参见 Simo 等 [121]、Simo 和 Lausen [122]、菊池和 Oden [123] 以及 Papadopoulos 和 Taylor [124]）。

In this period, one interesting emerging area was the development of FEM exterior calculus by Arnold et al. [125]. The FEM exterior calculus uses tools from differential geometry, algebraic topology, and homological algebra to develop FEM discretizations that are compatible with the underlying geometric, topological, and algebraic structures of the problems that are under consideration.
在这一时期，一个有趣的新兴领域是 Arnold 等 [125] 对有限元外演算的发展。FEM 外部微积分使用微分几何、代数拓扑和同理代数中的工具来开发与正在考虑的问题的底层几何、拓扑和代数结构兼容的 FEM 离散化。

3 (1992–2017) Broad Industrial Applications and Materials Modeling 3 （1992–2017）广泛的工业应用和材料建模

The first major event in FEM development this period was the formulation of the Zienkiewicz-Zhu error estimator [126], which was a major contribution to the mathematical approximation theory of FEMs in the 1990s. The Zienkiewicz-Zhu posteriori error estimators provide the quality control of a FEM solution with an optimal use of computational resources by refining the mesh adaptively. The idea and spirit of Zienkiewicz-Zhu was further carried out by Ainsworth and Oden (see [127, 128, 129]), and today using posteriori error estimation to improve the quality has been elevated to the height of Bayesian inference and Bayesian update. This research topic is now intimately related with what is now called validation and verification (V&V).
这一时期有限元理论发展的第一个重大事件是 Zienkiewicz-Zhu 误差估计器 [126] 的制定，这是对 1990 年代有限元数学近似理论的重大贡献。Zienkiewicz-Zhu 后验误差估计器通过自适应细化网格，以最佳方式利用计算资源，提供 FEM 解的质量控制。Zienkiewicz-Zhu 的思想和精神由 Ainsworth 和 Oden 进一步实施（参见 [127， 128， 129]），今天使用后验误差估计来提高质量已经提升到贝叶斯推理和贝叶斯更新的高度。这个研究主题现在与现在所谓的验证和验证（V&V）密切相关。

Since the late 1970s, Szabo [130] and Babuska [131, 132] started to develop hp versions of FEMs based on piecewise-polynomial approximations that employ elements of variable size (h) and polynomial degree §. They discovered that the FEM converges exponentially when the mesh is refined using a suitable combination of h-refinements (dividing elements into smaller ones) and p-refinements (increasing their polynomial degree). The exponential convergence makes the method a very attractive choice compared to most other FEMs, which only converge with an algebraic rate. This work continued until the late 1990s, spearheaded by M. Ainsworth, L. Demkowicz, J.T. Oden, C.A. Duarte, O.C. Zienkiewicz, and C.E. Baumann (see Demkowicz et al. [133], Demkowicz et al. [134], Demkowicz et al. [135], Oden et al. [136], Oden et al. [137], Baumann and Oden [138]. J. Fish. [139] also proposed a s-version FEM by superposing additional mesh (es) of higher-order hierarchical elements on top of the original mesh of C0 FEM discretization, so that it increases the resolution of the FEM solution.
自 1970 年代后期以来，Szabo [130] 和 Babuska [131， 132] 开始开发基于分段多项式近似的 hp 版本的 FEM，这些近似采用可变大小（h）和多项式次数（p）的元素。他们发现，当使用 h 细化（将单元分成更小的单元）和 p 细化（增加它们的多项式度）的适当组合细化网格时，FEM 呈指数收敛。与大多数其他有限元法相比，指数收敛性使该方法成为一个非常有吸引力的选择，因为大多数其他有限元法仅以代数率收敛。这项工作一直持续到 1990 年代后期，由 M. Ainsworth、L. Demkowicz、J.T. Oden、C.A. Duarte、O.C. Zienkiewicz 和 C.E. Baumann 牵头（参见 Demkowicz 等 [133]、Demkowicz 等 [134]、Demkowicz 等 [135]、Oden 等 [136]、Oden 等 [137]、Baumann 和 Oden [138]。J. Fish.[139] 还提出了一种 s 版本 FEM，方法是在 C0 FEM 离散化的原始网格上叠加高阶分层元素的额外网格，从而提高 FEM 解决方案的分辨率。

To solve material and structural failure problems, research work in the 1990s focused on variational principle based discretized methods to solve fracture mechanics problems or strain localization problems. In 1994, Xu and Needleman [140] developed a FEM cohesive zone model (CZM) that can simulate crack growth without remeshing, which was later further improved by M. Ortiz and his co-workers, e.g. Camacho and Ortiz [141] and Ortiz and Pandolfi [142], who later used CZM FEM to solve fragmentation and material fatigue problems [143]. It should be noted that long before the invention of cohesive zone model, Pietruszczak and Mroz [144] developed the first cohesive FEM for shear fracture in soil. Later, Bazant’s group at Northwestern University developed various interface FEMs, such as the microplane model, to study size effects of concrete and other brittle composite materials (Brocca and Bazant [145], Caner and Bazant [146]). These models became a standard tool for simulating missile impact and explosions at, e.g., ERDA Vicksburg. A microplane FEM model for fiber composites has been developed for Chrysler and Ford Co. to compare various designs of automobile crush-cans (Smilauer et al. [147]). An anisotropic poromechanical microplane model has been formulated and used for FEM analysis of hydraulic fracturing (Rahimi et al. [148]).
为了解决材料和结构失效问题，1990 年代的研究工作集中在基于变分原理的离散化方法上，以解决断裂力学问题或应变定位问题。1994 年，Xu 和 Needleman [140] 开发了一种 FEM 内聚区模型（CZM），该模型可以模拟裂纹扩展而无需重新划分网格，后来 M. Ortiz 和他的同事，如 Camacho 和 Ortiz [141] 以及 Ortiz 和 Pandolfi [142] 进一步改进，他们后来使用 CZM FEM 解决了碎裂和材料疲劳问题 [143]。应该指出的是，早在内聚区模型发明之前，Pietruszczak 和 Mroz [144] 就开发了第一个用于土体剪切断裂的内聚有限元。后来，西北大学的 Bazant 小组开发了各种界面 FEM，例如微平面模型，以研究混凝土和其他脆性复合材料的尺寸效应（Brocca 和 Bazant [145]、Caner 和 Bazant [146]）。这些模型成为模拟 ERDA Vicksburg 等导弹撞击和爆炸的标准工具。已经为克莱斯勒和福特公司开发了一种用于纤维复合材料的微平面有限元模型，以比较汽车破碎罐的各种设计（Smilauer 等 [147]）。已经建立了一个各向异性多孔机械微平面模型，并将其用于水力压裂的有限元分析（Rahimi 等 [148]）。

To alleviate mesh bias issues in modeling material fracture and damage problems, T. Belytschko and WK. Liu developed meshfree particle methods, namely the element-free Galerkin (EFG) method [149] and the reproducing kernel particle method (RKPM) [150, 151], which are based on the moving least square method and the wavelet multiresolution analysis, respectively. RKPM provides consistency and thus convergence enhancements as compared to the popular smoothed particle hydrodynamics (SPH) method. Li et al. [152, 153] successfully employed meshfree Galerkin methods to accurately simulate adiabatic shear band formation and propagation with minimum mesh adaptation. At the same time, Duarte and Oden developed the so-called hp-Cloud method, Onate et al. developed a finite point method [154], and Atluri and Zhu [155] proposed a meshless local Petrov–Galerkin (MLPG method, among many other meshfree methods. Fleming and Belytschko also showed that singularity functions could be included in the approximation functions to greatly improve simulations involving fracture mechanics (see Fleming et al. [156]).
为了缓解建模材料断裂和损伤问题中的网格偏置问题，T. Belytschko 和 WK. Liu 开发了无网格粒子方法，即无单元伽辽金（EFG）方法 [149] 和再现核粒子法（RKPM） [150， 151]，它们分别基于移动最小二乘法和小波多分辨率分析。与流行的平滑粒子流体动力学（SPH）方法相比，RKPM 提供了一致性，从而增强了收敛性。Li 等 [152,153] 成功地采用了无网格伽辽金方法，以最小的网格适应精确模拟了绝热剪切带的形成和传播。与此同时，Duarte 和 Oden 开发了所谓的 hp-Cloud 方法，Onate 等人开发了有限点方法 [154]，Atluri 和 Zhu [155] 提出了一种无网格的局部 Petrov-Galerkin（MLPG 方法，以及许多其他无网格方法。Fleming 和 Belytschko 还表明，奇异性函数可以包含在近似函数中，从而大大改善涉及断裂力学的模拟（参见 Fleming 等 [156]）。

One of the most challenging problems in the development of meshfree Galerkin methods is how to integrate the weak form, because the meshfree interpolants are highly irregular and it is difficult to make them variationally consistent. In 2001, J. S. Chen and his co-workers proposed a stabilized conforming nodal integration method for meshfree RKPM method, which is not only simple and stable, but also variationally consistent with the Galerkin weak formulation (see Chen et al. [157]).
在开发无网格伽辽金方法的过程中，最具挑战性的问题之一是如何对弱形式进行积分，因为无网格插值是高度不规则的，很难使它们在变分上保持一致。2001 年，J. S. Chen 和他的同事提出了一种用于无网格 RKPM 方法的稳定一致节点积分方法，该方法不仅简单稳定，而且在变分上与 Galerkin 弱公式一致（参见 Chen 等人 [157]）。

Shortly after the meshfree method developments, I. Babuska and his co-workers developed the partition of unity finite element method (PUFEM), which was later coined generalized finite element method (GFEM) (see Melenk and Babuska [158] and Babuska and Melenk [159]). PUFEM is a powerful method because it can be used to construct FE spaces of any given regularity, which is a generalization of the h, p, and hp versions of the FEM, as well as providing the ability to embed an analytic solution into the FEM discretization instead of relying upon a generic polynomial interpolant.
在无网格方法发展后不久，I. Babuska 和他的同事开发了单位有限元分割法（PUFEM），后来创造了广义有限元法（GFEM）（参见 Melenk 和 Babuska [158] 以及 Babuska 和 Melenk [159]）。PUFEM 是一种强大的方法，因为它可用于构造任何给定规则的有限元空间，这是有限元的 h、p 和 hp 版本的泛化，并且能够将解析解嵌入到 FEM 离散化中，而不是依赖于通用多项式插值。

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]).
1990 年代后期，计算断裂力学和 FEM 细化技术取得了重大突破，当时 Belytschko 和他的同事，包括 Black、Moes 和 Dolbow，开发了扩展有限元（X-FEM）（参见 [160， 161]，它使用各种丰富的不连续形状函数来准确捕获裂纹体的形态，而无需重新划分网格。由于自适应富集过程受裂纹尖端能量释放速率控制，因此 X-FEM 为线弹性断裂力学（LEFM）提供了精确的解决方案。在开发 X-FEM 时，T. Belytschko 出色地利用了 PUFEM 概念来解决断裂力学问题，而无需重新划分网格。进入新千年后，Bourdin、Francfort 和 Marigo 开发了一种相场方法来模拟材料断裂 [162]。几乎同时，Karma 和他的同事 [163]， [164]）也提出并开发了相场法来解决裂纹扩展和裂纹扩展问题，因为相场法可以准确预测脆性断裂的材料损伤，而无需重新划分网格。相场方法的主要优点是，通过使用 Galerkin FEM 求解连续运动方程和相方程，可以在连续建模中找到裂纹解，而不会遇到应力奇异性和重新网格划分，并且裂纹可以被视为相场解的尖锐界面极限。这项研究的一些主要贡献者是 Bourdin、Borden、Hughes、Kuhn、Muller、Miehe、Landis 等（参见 Bourdin 和 Chambolle [165]、Kuhn 和 Müller [166]）。Miehe et al. [167]、Borden et al. [168]、Wilson et al. [169] 和 Pham et al. [170]）。

As mentioned before, the main reason for the huge success of FEMs is their broad applicability to engineering analysis and design across scientific disciplines. On the other hand, most mechanical engineering designs are performed by using various computer-aided design (CAD) tools, such as solid modeling. To directly blend the FEM into CAD design tools, T.J.R. Hughes, and his co-workers such as J.A. Cottrell and Y. Bazilevs [171], [172] developed the isogeometric analysis (IGA) FEM, which established the Galerkin variational weak formulation in the control mesh and uses the non-uniform rational basis spline (NURBS) functions as the FEM shape function to solve the problem at design stage. IGA-FEM method successfully integrates FEM into conventional NURBS-based CAD tools, without converting data between CAD and FEA packages in analyzing new designs during development stage.
如前所述，FEM 取得巨大成功的主要原因是它们对跨学科工程分析和设计的广泛适用性。另一方面，大多数机械工程设计都是通过使用各种计算机辅助设计（CAD）工具（例如实体建模）来执行的。为了将有限元直接融合到 CAD 设计工具中，T.J.R. Hughes 和他的同事，如 J.A. Cottrell 和 Y. Bazilevs 171 开发了等几何分析（IGA） FEM，它在控制网格中建立了 Galerkin 变分弱公式，并使用非均匀有理基样条（NURBS）函数作为 FEM 形状函数来解决设计阶段的问题。IGA-FEM 方法成功地将 FEM 集成到基于 NURBS 的传统 CAD 工具中，在开发阶段分析新设计时，无需在 CAD 和 FEA 软件包之间转换数据。

Due to the emergence of nanotechnology, various multiscale methods have been developed to couple atomistic methods such as molecular dynamics and density functional theory (DFT) and other ab initio methods with continuum scale FEMs. The most notable contributions in this area are hand-shake method [173] quasi-continuum FEM [174], and the bridging scale method developed by Wagner and Liu [175]. In 2007, Gavini, Bhattacharya, and Ortiz developed quasi-continuum orbital-free density-functional theory (DFT) FEM for multi-million atom DFT calculations (see Gavini et al. [176]).
由于纳米技术的出现，已经开发了各种多尺度方法，将原子方法（如分子动力学和密度泛函理论（DFT））和其他从头计算方法与连续尺度 FEM 耦合。该领域最显著的贡献是握手法 [173]、准连续体有限元法 [174] 以及 Wagner 和 Liu [175] 开发的桥接标度法。2007 年，Gavini、Bhattacharya 和 Ortiz 开发了准连续体无轨道密度泛函理论（DFT） FEM，用于数百万个原子的 DFT 计算（参见 Gavini 等 [176]）。

Broadly speaking, there has been a rapid development in computational homogenization methods since the late 1990s, which is an alternate approach to obtaining continuum-scale properties based on smaller scale microstructures. The computational homogenization method or the FEM homogenization methods for composite materials may be divided into two main categories: (1) Computational asymptotic homogenization method or multiscale computational homogenization, which is aimed for modeling composite materials with periodic microstructure. The pioneer of multiscale homogenization FEM may be credited to T.Y. Hou and his co-workers [177], and other earlier contributors are N. Kikuchi [101], Ghosh et al. [178], Fish et al. [179, 180]) and M. S. Shephard, (2) Computational micromechanics method, which is mainly aimed for composite materials with random microstructure, even though it may also be applied to materials with periodic microstructure, where the main contributors are: P. Suquet et al. at the French National Centre for Scientific Research (CNRS), Dvorak et al.[181] and J. Fish at Rensselaer Polytechnic Institute and C. Miehe and his co-workers (e.g. [182]) at the University of Stuttgart. During the same period, many novel FEM fluid–structure interaction solvers have been developed, for instance, the immersed FEM developed by Zhang et al. [183], which was motivated by the immersed boundary method pioneered by C. Peskin of Courant Institute of Mathematical Sciences, New York University.
从广义上讲，自 1990 年代后期以来，计算均质化方法得到了快速发展，这是基于较小尺度微观结构获得连续尺度特性的另一种方法。复合材料的计算均质法或 FEM 均质法可分为两大类：（1）计算渐近均质法或多尺度计算均质法，旨在对具有周期性微观结构的复合材料进行建模。多尺度均质有限元法的先驱可能要归功于 T.Y. Hou 和他的同事 [177]，其他早期的贡献者是 N. Kikuchi [101]、Ghosh 等 [178]、Fish 等 [179， 180]）和 M. S. Shephard，（2）计算微力学方法，主要针对具有随机微观结构的复合材料，尽管它也可以应用于具有周期性微观结构的材料。其中主要贡献者是：法国国家科学研究中心（CNRS）的 P. Suquet 等人，伦斯勒理工学院的 Dvorak 等人 [181] 和 J. Fish 以及斯图加特大学的 C. Miehe 和他的同事（例如 [182]）。在同一时期，已经开发了许多新颖的有限元流固耦合求解器，例如，Zhang 等人 [183] 开发的浸没式有限元求解器，其灵感来自纽约大学 Courant 数学科学研究所的 C. Peskin 开创的浸没边界方法。

An important FEM application area emerged with the developments of computational plasticity. The early finite element computational plasticity formulation is based on hypoelastic–plastic rate formulation. To satisfy the objectivity requirement, T.J, R. Hughes and J. Winget first proposed the so-called incremental objectivity (see Hughes and Winget [184]), which was probably in the first time the esoteric continuum mechanics theory was applied to practical FEM computational formulations and computations, and it in turn promoted the development of nonlinear continuum mechanics in 1980s and 1990s. Soon afterwards. Simo and Hughes then extended Hughes-Winget incremental objectivity algorithm to the finite deformation case in computational plasticity (see Simo and Hughes [185]). The notion of consistency between the tangent stiffness matrix and the integration algorithm employed in the solution of the incremental problem was introduced by Nagtegaal [186] and Simo and Taylor [78]. Consistent formulations have been subsequently developed for finite deformation plasticity by Simo and Oritz [187] (1988) within the framework of multiplicative decomposition of the deformation gradient and hyperelasticity. Also in 1980s, based on the Gurson model, Tvergaard and Needleman [188] developed the FEM based Gurson-Tvergaard-Needleman model, which is probably the most widely used FEM computational plasticity constitutive model used in material modeling, though recently attention has turned to developing machine-learning based or data-driven computational plasticity models e.g., F. Chinesta, et al. [189], and the unsupervised machine learning data-driven finite element methods, called as the Self-consistent Clustering Analysis (SCA) (see Z. Liu et al. [190, 191]).
随着计算塑性的发展，出现了一个重要的有限元应用领域。早期的有限元计算塑性公式基于低弹性 - 塑性速率公式。为了满足客观性要求，T.J、R. Hughes 和 J. Winget 首先提出了所谓的增量客观性（参见 Hughes 和 Winget [184]），这可能是深奥的连续介质力学理论首次应用于实际的有限元计算公式和计算，反过来又促进了 1980 年代和 1990 年代非线性连续介质力学的发展。然后，Simo 和 Hughes 将 Hughes-Winget 增量客观性算法扩展到计算塑性中的有限变形情况（参见 Simo 和 Hughes [185]）。在求解增量问题时采用的切线刚度矩阵和积分算法之间的一致性概念由 Nagtegaal [186] 和 Simo 和 Taylor [78] 提出。随后，Simo 和 Oritz [187] （1988）在变形梯度和超弹性的乘法分解框架内为有限变形塑性开发了一致的公式。同样在 1980 年代，基于 Gurson 模型，Tvergaard 和 Needleman [188] 开发了基于有限元的 Gurson-Tvergaard-Needleman 模型，这可能是材料建模中使用最广泛的有限元计算塑性本构模型，尽管最近注意力转向开发基于机器学习或数据驱动的计算塑性模型，例如 F. Chinesta 等 [189]，以及无监督机器学习数据驱动的有限元方法。称为自洽聚类分析（SCA）（参见 Z. Liu 等人 [190， 191]）。

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.
有限元法的一个重要进步是晶体塑性有限元方法（CPFEM）的发展，该方法首次在 Pierce 等 [192] 的一篇具有里程碑意义的论文中提出。在过去的近四十年中，有许多研究人员对这一主题做出了重大贡献，例如，麻省理工学院的 A. Arsenlis 和 DM. Parks 以及劳伦斯利弗莫尔国家实验室 [193]、[194]、[195]、康奈尔大学的 Dawson 等人（Quey 等人 [196];Mathur 和 Dawson [197];Max-Planck-Institute für Eisenforschung 的 Raabe et al. [196,197,198,201] 等。基于晶体滑移，CPFEM 可以计算位错、晶体取向等织构信息，以考虑计算过程中的晶体各向异性，并已应用于模拟晶体塑性变形、表面粗糙度、断裂等。最近，S. Li 和他的同事开发了一种基于 FEM 的多尺度位错模式动力学来模拟单晶中的晶体塑性 [202， 203]。Yu et al.， [204] 重新制定了有限变形下一般弹粘塑性材料的自洽聚类分析（SCA）。与传统的全场有限元器件进行比较，证明了预测多晶材料整体机械响应的准确性和效率。

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.
2013 年，由 L. Beirão da Veiga 和 F. Brezzi 领导的一组意大利科学家和工程师提出了一种所谓的虚拟元法（VEM）（参见 Beirao 等人 [205， 206]）。虚拟单元法是传统 FEM 的扩展，适用于任意单元几何形状。它允许多面离散化（二维多边形或三维多面体），甚至可能是高度不规则和非凸的单元域。名称 virtual 源于这样一个事实，即不需要了解局部形状函数基，实际上从未显式计算过。VEM 具有使其在某些特殊问题上优于传统 FEM 的功能，例如难以获得高质量网格的复杂几何形状问题、需要非常局部细化的解决方案等。在这些特殊情况下，当网格变形时，VEM 在数值计算中表现出稳健性和准确性。

As early as 1957, R. Clough introduced the first graduate FEM course at UC-Berkeley, and since then FEM courses at both graduate and undergraduate levels have been added into engineering higher education curriculums in all the major engineering schools and Universities all over the world. As J.T. Oden recalled in his 1963 paper, “I went on to return to academia in 1964 and among my first chores was to develop a graduate course on finite element methods. At the same time, I taught mathematics and continuum mechanics, and it became clear to me that finite elements and digital computing offered hope of transforming nonlinear continuum mechanics from a qualitative and academic subject into something useful in modern scientific computing and engineering.”.
早在 1957 年，R. Clough 就在加州大学伯克利分校开设了第一门研究生 FEM 课程，从那时起，研究生和本科阶段的 FEM 课程已被添加到世界各地所有主要工程学院和大学的工程高等教育课程中。正如 J.T. Oden 在他 1963 年的论文中回忆的那样，“我于 1964 年重返学术界，我的第一项工作是开发一门关于有限元方法的研究生课程。与此同时，我教授数学和连续介质力学，我清楚地意识到，有限元和数字计算为将非线性连续介质力学从定性和学术学科转变为现代科学计算和工程中有用的学科提供了希望。

By the end of 2015, there have been more than several hundred monographs and textbooks on the FEM published in dozens of languages worldwide. An exposition of FEM mathematical theory by Strang and Fix [207] was among the earliest of those FEM books. J.T. Oden and his collaborators such as GF Carey and JN Reddy and others wrote a five-volume FEM monograph in 1980s (see [167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,213, 214]). Other influential finite element books or monographs are those by Zienkiewicz and Cheung [215], and Zienkiewicz and Taylor [216], ora later Zienkiewicz, Taylor and Zhu [217], TJR. Hughes [218], Cook et al. [219] or later Cook [220]), Bathe [221], and the Nonlinear FEM monographs by Belytschko et al. [222] and by De Borst et al. [223], among others, all have made major impacts on FEM educations and applications. Among all these FEM monographs and textbooks, the book by Zienkiewicz and Taylor [216] or Zienkiewicz et al. [217] probably have had most impacts on FEM technology popularization, which may be because Taylor wrote a FEM research computer program code named FEAP, which was placed in the appendix of that book, providing an immediate guidance and instruction on how to implement FEM in computers.
截至 2015 年底，全球已以数十种语言出版了数百本关于 FEM 的专著和教科书。Strang 和 Fix [207] 对 FEM 数学理论的阐述是最早的 FEM 书籍之一。J.T. Oden 和他的合作者，如 GF Carey 和 JN Reddy 等人在 1980 年代写了一本五卷本的 FEM 专著（见 [167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,213,214]）。其他有影响力的有限元书籍或专著是 Zienkiewicz 和 Cheung [215]、Zienkiewicz 和 Taylor [216]，或后来的 Zienkiewicz、Taylor 和 Zhu [217]、TJR 的书籍或专著。Hughes [218]、Cook 等 [219] 或后来的 Cook [220]）、Bathe [221] 以及 Belytschko 等 [222] 和 De Borst 等 [223] 的非线性有限元专著都对有限元数学的教育和应用产生了重大影响。在所有这些有限元研究专著和教科书中，Zienkiewicz 和 Taylor [216] 或 Zienkiewicz 等人 [217] 的书可能对有限元技术的普及影响最大，这可能是因为泰勒编写了一个名为 FEAP 的有限元研究计算机程序代码，该代码被放置在该书的附录中，为如何在计算机中实施有限元提供了直接的指导和说明。

The ready availability of FEM textbooks and tutorials, along with the large FEM software market, have made the FEM accessible to users across academia and industry. The development of FEM software technology started in the early1960s. In 1963, E.L. Wilson and R. Clough developed a structural mechanics computing code called Symbolic Matrix Interpretive System, SMIS, which was initially intended to fill the gap between the matrix method and hand calculating in structure mechanics. It turns out the development of SMIS led to the birth of FEM software. Then, based on SMIS, Wilson initiated and developed a general-purpose static and dynamic Structural Analysis Program, SAP. In late 1960s and the early 1970s, K, J. Bathe developed the nonlinear FEM code ADINA based on SAP IV and NONSAP. Today, the brand name SAP2000 has become synonymous with the state-of-the-art FEM structural analysis and design methods since its introduction over 55 years ago.
FEM 教科书和教程的现成可用性，以及庞大的 FEM 软件市场，使学术界和工业界的用户都可以使用 FEM。FEM 软件技术的发展始于 1960 年代初期。1963 年，E.L. Wilson 和 R. Clough 开发了一种称为符号矩阵解释系统（SMIS）的结构力学计算代码，该代码最初旨在填补结构力学中矩阵方法和手动计算之间的空白。事实证明，SMIS 的发展导致了 FEM 软件的诞生。然后，基于 SMIS，Wilson 发起并开发了一个通用的静态和动态结构分析程序 SAP。在 1960 年代末和 1970 年代初，K， J. Bathe 基于 SAP IV 和 NONSAP 开发了非线性 FEM 代码 ADINA。自 55 年前问世以来，SAP2000 品牌已成为最先进的 FEM 结构分析和设计方法的代名词。

At the same period, to compete with the Soviet Union’s space program, NASA developed its own FEM code called NASTRAN (NASA STRuctural ANalysis Program). The first version of NASTRAN was called COSMIC Nastran, which debuted in 1969, with a key figure in its development being R. H. MacNeal. As early as 1963, R. H. MacNeal founded the MacNeal-Schwendler Software Corporation (MSC) along with R. Schwendler. Under his leadership, MSC developed its first structural analysis software called SADSAM (Structural Analysis by Digital Simulation of Analog Methods), which showed the early form of FEM analysis software technology. In response to NASA’s request for proposals in 1965 for a general purpose structural analysis program, Dr. MacNeal contributed significantly to the early efforts of the aerospace industry by successfully simulating on-the-ground physical testing through computing to deliver the right answers and physics needed to take humans to the moon. In 1971, MSC Software released a commercial version of Nastran, named MSC Nastran.
与此同时，为了与苏联的太空计划竞争，NASA 开发了自己的 FEM 代码，称为 NASTRAN（NASA STRuctural ANalysis Program）。NASTRAN 的第一个版本称为 COSMIC Nastran，于 1969 年首次亮相，其开发的关键人物是 RH MacNeal。早在 1963 年，R. H. MacNeal 就与 R. Schwendler 一起创立了 MacNeal-Schwendler 软件公司（MSC）。在他的领导下，MSC 开发了第一款名为 SADSAM（模拟方法数字仿真结构分析）的结构分析软件，该软件展示了 FEM 分析软件技术的早期形式。1965 年，为了响应 NASA 对通用结构分析计划的提案请求，MacNeal 博士通过计算成功模拟地面物理测试，以提供将人类送上月球所需的正确答案和物理学，为航空航天业的早期工作做出了重大贡献。1971 年，MSC 软件公司发布了 Nastran 的商业版本，名为 MSC Nastran。

About the same time in 1960s, J. A. Swanson worked at Westinghouse Astronuclear Laboratory in Pittsburgh, and he was responsible for stress analysis of the components in NERVA nuclear reactor rockets. While there he then developed 3D FEM model and computer codes to analyze and predict transient stresses and displacements of the reactor system. To integrate different computer codes and streamline the processing, Swanson asked his employer Westinghouse to develop a general-purpose FEM computer code, but his suggestion was rejected, and then he left the company and developed the initial ANSYS FEM code. Today, ANSYS has become one of the major FEM commercial software worldwide.
大约在 1960 年代的同一时间，JA Swanson 在匹兹堡的西屋天体核实验室工作，他负责 NERVA 核反应堆火箭部件的应力分析。在那里，他开发了 3D FEM 模型和计算机代码，以分析和预测反应堆系统的瞬态应力和位移。为了集成不同的计算机代码并简化处理，Swanson 要求他的雇主 Westinghouse 开发一种通用的 FEM 计算机代码，但他的建议被拒绝了，随后他离开了公司并开发了最初的 ANSYS FEM 代码。如今，ANSYS 已成为全球主要的 FEM 商业软件之一。

Several years later, J.O. Hallquist at Lawrence Livermore National Laboratory also developed a 3D nonlinear FEM code called DYNA3D, which was extensively used impact, dynamic contact, and failure analysis of structures, which later evolved to LS-DYNA. LS-DYNA is the major FEM software used in automobile design and crashworthiness analyses. In 2018, Livermore Software Technology was purchased by ANSYS, and LS-DYNA became a part of ANSYS as ANSYS LS-DYNA.
几年后，劳伦斯利弗莫尔国家实验室的 J.O. Hallquist 还开发了一种名为 DYNA3D 的 3D 非线性有限元代码，该代码被广泛用于结构的冲击、动态接触和失效分析，后来演变为 LS-DYNA。LS-DYNA 是用于汽车设计和耐撞性分析的主要 FEM 软件。2018 年，Livermore Software Technology 被 ANSYS 收购，LS-DYNA 成为 ANSYS 的一部分，更名为 ANSYS LS-DYNA。

In the early development of nonlinear FEM software, there were two early pioneers: P. V. Marcal and D. Hibbitt, who was Marcal’s student at Brown University. In the early 1970s, Marcal founded the MARC corporation to develop the first general purpose nonlinear FEM program, which is still widely used today in industry and academia for analysis of complex structures, such as nuclear reactors, car crashworthiness and manufacturing processes. While Hibbitt together with B. Karlsson and P. Sorenson formed a FEM company called HKS in 1978, and they released a large-scale commercial FEM software called ABAQUS. One of the major features of ABAQUS is that it allows the user defined subroutines, which greatly facilitates the researchers to conduct their researches by using the standard FEM solvers with reliability and efficiency. Together with ANSYS, ABAQUS is one of the two commercial FEM software that dominate the market.
在非线性有限元软件的早期开发中，有两位早期的先驱：P. V. Marcal 和 D. Hibbitt，后者是 Marcal 在布朗大学的学生。1970 年代初，Marcal 创立了 MARC 公司，开发了第一个通用非线性 FEM 程序，该程序至今仍广泛用于工业和学术界，用于分析复杂结构，例如核反应堆、汽车耐撞性和制造工艺。Hibbitt 与 B. Karlsson 和 P. Sorenson 于 1978 年成立了一家名为 HKS 的 FEM 公司，并发布了一款名为 ABAQUS 的大型商业 FEM 软件。ABAQUS 的主要特点之一是它允许用户定义的子程序，这极大地方便了研究人员通过使用可靠和高效的标准 FEM 求解器进行研究。ABAQUS 与 ANSYS 一起成为主导市场的两款商用 FEM 软件之一。

By the late 1990s to early 2000s, the FEM software industry had become a multi-billion-dollar business. There were several household FEM software company names such as ANSYS, ABAQUS, ADINA, LS-DYNA, NASTRAN, COMSOL Multiphysics, CSI, among others. Today, there are also a plethora of open-source FEM software available online, such as FreeFEM, OpenSees, Elmer, FEBio, FEniCS Project, DUNE, among some others.
到 1990 年代末到 2000 年代初，FEM 软件行业已成为一项价值数十亿美元的业务。有几家家喻户晓的 FEM 软件公司名称，例如 ANSYS、ABAQUS、ADINA、LS-DYNA、NASTRAN、COMSOL Multiphysics、CSI 等。今天，网上还有大量的开源 FEM 软件，例如 FreeFEM、OpenSees、Elmer、FEBio、FEniCS Project、DUNE 等。

4 Present and Future 4 现在与未来

The modern form of the FEM can routinely solve many large and complex industrial problems. It enables developing a fundamental understanding and allows for the predictive analysis for product design. For new scientific discoveries and engineering innovations, the development of new scientific principles often trails the pace of new inventions with the sheer volume of data that are generated across multiple spatial, temporal, and design parameter (spatial–temporal-parameter) spaces. For this reason, FE researchers are studying various forms of machine and deep learning methods, of which this class of methods covers the largest class of interpolations. According to the universal approximation theorem, a neural network (NN) can be designed and trained to approximate any given continuous function with any desired accuracy [224, 225, 226, 227] which is believed to drive new discoveries and enable future computational discretization technologies. In this context, Mechanistic Data Science (MDS) FEMs, which combine known scientific principles with newly collected data, will provide the critically needed research that can be a boon for new inventions.
FEM 的现代形式可以解决许多大型和复杂的工业问题。它有助于形成基本理解，并允许对产品设计进行预测分析。对于新的科学发现和工程创新，新科学原理的发展往往落后于新发明的步伐，因为在多个空间、时间和设计参数（空间 - 时间 - 参数）空间中生成的数据量很大。出于这个原因，FE 研究人员正在研究各种形式的机器和深度学习方法，其中这类方法涵盖了最大的插值类别。根据通用近似定理，可以设计和训练神经网络（NN）以任何所需的精度逼近任何给定的连续函数 [224， 225， 226， 227]，这被认为可以推动新发现并支持未来的计算离散化技术。在这种情况下，机械数据科学（MDS） FEM 将已知的科学原理与新收集的数据相结合，将提供急需的研究，为新发明带来福音。

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.
科学和工程问题通常分为三类：（1）数据丰富但科学原理未开发或不可用的问题，（2）数据和科学知识有限的问题，以及（3）已知科学原理但参数不确定的问题，可能具有很高的计算负荷 [228]。从本质上讲，机械数据科学（MDS）模仿了人类文明从一开始就为困难和无法解决的问题寻找解决方案的方式。MDS 使用主动深度学习和分层神经网络等机器学习方法，而不是启发式方法来处理输入数据、提取机理特征、减少维度、通过回归和分类学习隐藏的关系，并提供知识数据库。由此产生的简化订单可用于设计和优化新的科学和工程系统（参见 Liu 等 [229]）。因此，FEM 研究的新重点已转向开发基于机器学习的 FEM 和降阶模型。

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]).
随着机器学习和深度学习方法的最新发展，通过构建深度神经网络来解决 FEM 问题已成为一项最先进的技术。早期的研究集中在遵循 FEM 结构构建浅层神经网络来解决边值问题。Takeuchi 和 Kosugi [230] 提出了一种有限元的神经网络表示来解决泊松方程问题。Yagawa 和 Aoki [231] 用互连神经网络（NN）的网络能量取代了 FEM 泛函，以解决热传导问题。由于浅层神经网络计算能力的限制和缓慢的收敛速度，早期的应用只能解决简单的 PDE 问题。2010 年代之后，随着深度学习技术的快速发展和更复杂的神经网络结构（如卷积神经网络（CNN）、生成对抗网络（GAN）和残差神经网络（ResNet））的发展，用于解决计算力学问题的神经网络越来越受欢迎。由于其高维回归能力，一些研究人员，例如 Ghavamiana 和 Simone [232] 使用深度神经网络作为回归模型来学习材料行为或微观结构响应。其他工作侧重于使用深度学习神经网络解决偏微分方程。G. Karniadakis 和他的同事（参见 Raissi 等 [233,234]，Karniadakis 等 [235]）提出了一种物理信息神经网络（PINN）来求解强形式的高维偏微分方程，并具有约束条件以适应自然和基本边界条件。使用高级神经网络方法再次研究了遵循 FEM 结构构建深度神经网络的想法。Weinan E 和他的同事 [236] 以及 B. Yu [237] 提出了一种解决变分问题的 Deep Ritz 方法。Sirignano 和 Spiliopoulos [238] 提出了所谓的深度加辽金方法（DGM）来求解高维偏微分方程。Zabaras 和他的同事提出了一个基于 CNN 的物理约束深度学习框架，用于高维代理建模和不确定性量化（参见 Zhu 等人 [239]）。Rabczuk [240] 通过求解偏微分方程的能量格式，系统地探索了将神经网络用于计算力学的潜力（参见 Samaniego 等人 [241]）。Lee [242] 提出了一个单位网络分区，用于偏微分方程的深度 hp 近似，并广泛地提出了训练和初始化策略，以加速求解过程的收敛（参见 Lee 等人 [242]）。J. Opschoor 和他的同事已经研究了激活函数构建元素形状函数（参见 [243， 244]）。

Inspired by the universal approximation of deep neural networks (DNN), Zhang et al. [245] published the first paper on the construction of the FEM shape functions-based on the hierarchical nature of the DNN, called Hierarchical Deep-learning Neural Networks (HiDeNN). Specifically, the authors demonstrated the construction of a few classes of deep learning interpolation functions, such as the reproducing kernel particle method (RKPM), non-uniform rational B-spline (NURBS), and isogeometric analysis (IGA), among other approximation techniques. Saha et al. [228] generalized HiDeNN to a unified Artificial intelligence (AI)-framework, called HiDeNN-AI. HiDeNN-AI can assimilate many data-driven tools in an appropriate way, which provides a general approach to solve challenging science and engineering problems with little or no available physics as well as with extreme computational demand.
受到深度神经网络（DNN）的通用近似的启发，Zhang 等 [245] 发表了第一篇基于 DNN 的分层性质构建 FEM 形状函数的论文，称为分层深度学习神经网络（HiDeNN）。具体来说，作者演示了几类深度学习插值函数的构造，例如再现核粒子法（RKPM）、非均匀有理 B 样条法（NURBS）和等几何分析（IGA）等近似技术。Saha 等 [228] 将 HiDeNN 推广到一个统一的人工智能（AI）框架，称为 HiDeNN-AI。HiDeNN-AI 可以以适当的方式吸收许多数据驱动的工具，这提供了一种通用的方法来解决具有挑战性的科学和工程问题，这些问题几乎没有或没有可用的物理学以及极端的计算需求。

To reduce the FE computational cost, the so-called two-stage data-driven methods have been proposed of which during the offline stage, where a database generated by the FEM is first developed, and the final solutions are computed during the online stage. C. Farhat and his group at Stanford University have developed several dimensional reductions of nonlinear FEM dynamic models, including mesh sampling and weighting for the hyper reduction of nonlinear Petrov‐Galerkin reduced‐order models (see Farhat et al. [246], and Grimberg et al. [247]). To further reduce the FEM computational burden in multiscale analysis, Liu et al. [190] applied the unsupervised machine learning techniques, such as the k-mean clustering method to group the material points during the offline stage and obtain the final solutions by solving the reduced-order Lippmann–Schwinger micromechanics equations. This class of data-driven approaches circumvents the computational burden of the well-established FE square method, the offline-online database approach to solve the concurrent FEM problems. They named the method the Finite Element-self-consistent clustering analysis (FE-SCA) of which the computational cost of the microscale analysis is reduced tremendously in multiple orders of magnitude speed up (see Li et al. [248]). Gao et al. proposed an alternative (FE-SCAxSCA…xSCA) clustering analysis, of which the continuum FEM scale is concurrently solved with the (n-1) coupled-scale Lippmann–Schwinger micromechanics equations [249, 250]. It has been recently extended to (FE-SCA**2) by He et al. [251, 252].
为了降低有限元的计算成本，人们提出了所谓的两阶段数据驱动方法，其中在离线阶段，首先开发由有限元生成的数据库，然后在在线阶段计算最终解决方案。斯坦福大学的 C. Farhat 和他的团队开发了非线性有限元动力学模型的几种降维方法，包括用于非线性 Petrov-Galerkin 降阶模型超降维的网格采样和加权（参见 Farhat 等 [246] 和 Grimberg 等 [247]）。为了进一步减轻多尺度分析中的有限元计算负担，Liu 等 [190] 应用了无监督机器学习技术，如 k 均值聚类方法，在离线阶段对材料点进行分组，并通过求解降阶 Lippmann-Schwinger 微观力学方程获得最终解。这类数据驱动的方法规避了成熟的 FE 平方法的计算负担，即解决并发 FEM 问题的离线 - 在线数据库方法。他们将该方法命名为有限元自洽聚类分析（FE-SCA），其中微尺度分析的计算成本大大降低了多个数量级（参见 Li 等 [248]）。Gao 等人提出了一种替代方案（FE-SCAxSCA…xSCA）聚类分析，其中连续体 FEM 尺度与（n-1）耦合尺度 Lippmann-Schwinger 微观力学方程 [249， 250] 同时求解。He 等 [251， 252] 最近将其扩展到（FE-SCA**2）。

Ortiz and his co-workers at Caltech developed data driven FEMs for dynamics and noisy data (see [253, 254]). Chen and his co-workers have developed a physics-constrained data-driven RKPM method based on locally convex reconstruction for noisy databases (see He and Chen [255]). S. Li and his group at UC-Berkeley utilized FEM solution generated data to develop a machine learning based inverse solution to predict pre-crash data of car collision (see Chen et al. [256]). Bessa et al. [257] proposed a data-driven framework to address the longstanding challenge of a two-scale analysis and design of materials under uncertainty applicable to problems that involve unacceptable computational expense when solved by standard FEM analysis of representative volume elements. The paper defined a framework that incorporates the SCA method to build large databases suitable for machine learning. The authors believe that this will open new avenues to finding innovative materials with new capabilities in an era of high-throughput computing (“big-data”).
Ortiz 和他在加州理工学院的同事为动力学和噪声数据开发了数据驱动的 FEM（参见 [253， 254]）。Chen 和他的同事开发了一种基于噪声数据库局部凸重建的物理约束数据驱动的 RKPM 方法（参见 He 和 Chen [255]）。加州大学伯克利分校的 S. Li 和他的团队利用有限元解决方案生成的数据开发了一种基于机器学习的逆解决方案来预测汽车碰撞的前数据（参见 Chen 等人 [256]）。Bessa 等 [257] 提出了一个数据驱动的框架，以解决在不确定性下进行双尺度分析和材料设计的长期挑战，该框架适用于通过代表性体积元素的标准有限元分析解决时涉及不可接受的计算费用的问题。该论文定义了一个框架，该框架结合了 SCA 方法，以构建适合机器学习的大型数据库。作者认为，这将为在高吞吐量计算（“大数据”）时代寻找具有新功能的创新材料开辟新途径。

Reduced order modeling has been an active research field over the last decades. Early research works focused on the proper orthogonal decomposition (POD) method (also known as Karhunen-Loève transform, or principal component analysis) with the purpose of reducing the degrees of freedom of the discretized equations. The POD based model reduction has shown great success in computational fluid dynamics, see e.g., the works of Berkooz et al. [258]. For further accelerating the simulations, K. Willcox and her coworkers (see [259, 260]) proposed a missing point estimation method, which is known later as a hyper reduction method. Other notable works related to POD and hyper reduction methods are the Gauss–Newton with approximated tensors (GNAT) method, Grassmann manifold based reduced basis adaptation, thanks to C. Farhat and his coworkers (see Carlberg and Farhat [261], Carlberg et al. [262], Grimberg et al. [247], Amsallem and Farhat [263], and Farhat et al. [246]). For solid mechanics, Ryckelynck et al. [264] proposed a hyper reduction method based on FEM for dealing with nonlinear problems. Lu et al. [265] proposed an adaptive hyper reduction for coupled thermal-fluid analysis. Another type of model reduction method, which is based on mathematics and has a rigorous error bound estimate, is called reduced basis method, as proposed by Maday and Rønquist [266]. The proper generalized decomposition (PGD) based model reduction, as an extension of POD, can be
在过去的几十年里，降阶建模一直是一个活跃的研究领域。早期的研究工作集中在正确的正交分解（POD）方法（也称为 Karhunen-Loève 变换或主成分分析），目的是减少离散方程的自由度。基于 POD 的模型简化在计算流体动力学方面取得了巨大成功，参见 Berkooz 等人 [258] 的工作。为了进一步加速模拟，K. Willcox 和她的同事（参见 [259， 260]）提出了一种缺失点估计方法，后来被称为超还原方法。其他与 POD 和超还原方法相关的著名工作是近似张量的高斯 - 牛顿（GNAT）方法，以及基于格拉斯曼流形的约化基适应，这要归功于 C. Farhat 和他的同事（参见 Carlberg 和 Farhat [261]、Carlberg 等 [262]、Grimberg 等 [247]、Amsallem 和 Farhat [263] 以及 Farhat 等 [246]）。对于固体力学，Ryckelynck 等 [264] 提出了一种基于有限元的超还原方法，用于处理非线性问题。Lu et al. [265] 提出了一种用于耦合热 - 流体分析的自适应超还原。另一种基于数学的模型约简方法，具有严格的误差限估计，称为约简基法，由 Maday 和 Rønquist [266] 提出。作为 POD 的扩展，基于适当广义分解（PGD）的模型缩减可以是

dated back to 1980s, and it was introduced by P. Ladevèze (See Ladeveze [267], Ladeveze and Rougee [268]) under the name of radial time–space approximation. F. Chinesta et al. [269, 270] developed a PGD method to account for the parameter space, aiming at building offline computational vademecum for fast online predictions. It is noted that PGD methods are based on the idea of separation of variables and in particular a canonical tensor decompostion (TD).
它的历史可以追溯到 1980 年代，由 P. Ladevèze （参见 Ladeveze [267]、Ladeveze 和 Rougee [268]）以径向时空近似的名义提出。F. Chinesta et al. [269， 270] 开发了一种 PGD 方法来解释参数空间，旨在构建用于快速在线预测的离线计算平台。值得注意的是，PGD 方法基于变量分离的思想，特别是规范张量分解（TD）。

Recent works have been conducted to combine deep machine learning methods with reduced order modeling methods. Zhang et al. [271] consolidated the various attributes of TD and PGD methods with HiDeNN and proposed the so-called HiDeNN-TD and HiDeNN-PGD methods. The comparison of FEM, TD/PGD, HiDeNN-TD/PGD, HiDeNN, and DNN has been conducted in terms of accuracy and speed. It is shown that the HiDeNN-TD/PGD outperforms other methods with a good balance between accuracy and speed. The proposed HiDeNN-TD/PGD method is expected to provide novel powful tools for solving large scale high dimensional problems while maintaining a high accuracy. Various applications, including multiphysics coupled additive manufacturing, multiscale composite modeling, and structural topology optimization, have been discussed in a generalized reduced order machine learning finite element framework [272]. In particular, this reduced order machine learning framework is expected to enable ultra large-scale high resolution topology design that is currently challenging for the FEM based topology optimization. In this regard, Lu et al. [273] recently developed a convolution HiDeNN-TD formulation with a built-in density filter for high resolution topology design. This convolution formulation incorporates the concept of meshfree approximation into the finite element function approximation and allows smoother solutions and automatic length-scale control in topology design. It is shown that the convolution HiDeNN-TD leads to better design with smoother and fine structures. This general convolution formulation opens new perspectives to resolve the length scale effect and can be applied to many orther problems, such as additive manufacturing and microstructure modeling.
最近的工作是将深度机器学习方法与降阶建模方法相结合。Zhang 等 [271] 将 TD 和 PGD 方法的各种属性与 HiDeNN 合并，并提出了所谓的 HiDeNN-TD 和 HiDeNN-PGD 方法。FEM、TD/PGD、HiDeNN-TD/PGD、HiDeNN 和 DNN 在准确性和速度方面进行了比较。结果表明，HiDeNN-TD/PGD 在精度和速度之间取得了良好的平衡，优于其他方法。所提出的 HiDeNN-TD/PGD 方法有望为解决大规模高维问题提供新颖的工具，同时保持高精度。广义降阶机器学习有限元框架中讨论了各种应用，包括多物理场耦合增材制造、多尺度复合材料建模和结构拓扑优化 [272]。特别是，这种降阶机器学习框架有望实现超大规模高分辨率拓扑设计，而目前基于 FEM 的拓扑优化具有挑战性。在这方面，Lu et al. [273] 最近开发了一种内置密度滤波器的卷积 HiDeNN-TD 公式，用于高分辨率拓扑设计。这种卷积公式将无网格近似的概念合并到有限元函数近似中，并允许在拓扑设计中实现更平滑的解和自动长度尺度控制。结果表明，卷积 HiDeNN-TD 可以带来更好的设计，具有更平滑和精细的结构。这种通用卷积公式为解决长度尺度效应开辟了新的视角，并且可以应用于许多其他问题，例如增材制造和微观结构建模。

The extra-ordinary interpolating capability of neural network has resulted in numerous research in approximating and solving the ordinary and partial differential equations e.g. Chen et al. [274]. Currently, the researchers are looking into approximating the mathematical operators directly from universal approximation of operator theory (see Chen and Chen [275]). The goal of this endeavor is to solve the integral equations such as the Lippmann–Schwinger micromechanics equations (Z. Liu et. al. [190]). A major step forward towards this target is made by Li et al. [276] in their proposed Graph Kernel Network, in which they developed neural operators for solving partial differential equations. In this work, a kernel-based graph neural network is shown to be able to mimic the Green’s function method for solving partial differential equation. One drawback of this method is the associated computational cost and storage requirement increases with the size of the problem. A more general version of approximating the Green’s function operator is proposed by Lu et al. in their DeepONet [277] and by Gin et al. in their DeepGreen methods (see [278]). These approximation works are vital in computational mechanics as these can directly solve the micromechanics equation for multiscale analysis like FE-SCA. Moreover, these networks have suggested that the analytical calculus method such as differentiation and integration, and solution of differential and integration equations can be directly expressed as an approximation of neural network. In this regard, recurrent neural networks have shown promised to be identical in structure of wave equations [279]. These researches are directed towards a future when the research and education in the science, technology, engineering, and mathematics (STEM) sector with discrete calculus will be transformed with the aid of deep learning. We are envisioning this field as deep learning discrete calculus, a new perspective in teaching calculus by the integration of calculus definition, numerical analysis, and deep learning (see Liu et al. [280]).
神经网络的超常插值能力导致了大量关于逼近和求解常微分方程和偏微分方程的研究，例如 Chen 等 [274]。目前，研究人员正在研究直接从算子理论的通用近似中逼近数学算子（参见 Chen 和 Chen [275]）。这项工作的目标是求解积分方程，例如 Lippmann-Schwinger 微观力学方程（Z. Liu et al. [190]）。Li et al. [276] 在他们提出的 Graph Kernel Network 中向这一目标迈出了重要一步，其中他们开发了用于求解偏微分方程的神经运算符。在这项工作中，基于核的图神经网络被证明能够模拟格林函数方法来求解偏微分方程。这种方法的一个缺点是相关的计算成本和存储要求随着问题的大小而增加。Lu 等人在他们的 DeepONet [277] 和 Gin 等人的 DeepGreen 方法中提出了一种更通用的近似格林函数算子版本（见 [278]）。这些近似工作在计算力学中至关重要，因为它们可以直接求解 FE-SCA 等多尺度分析的微观力学方程。此外，这些网络表明，微分和积分、微分和积分方程的解等解析微积分方法可以直接表示为神经网络的近似值。在这方面，递归神经网络已经表明，在波动方程的结构上是相同的 [279]。这些研究面向未来，届时具有离散微积分的科学、技术、工程和数学（STEM）领域的研究和教育将在深度学习的帮助下发生转变。我们将这个领域设想为深度学习离散微积分，通过整合微积分定义、数值分析和深度学习，为微积分教学提供一种新的视角（参见 Liu 等人 [280]）。

The development of model reduction methods meets the urgent demand in the industry for fast and nearly real time simulations of engineering problems, such as online dynamic system control, structural health monitoring, vehicle health monitoring, on-line advanced manufacturing feedback control, automated driving controls and decisions, etc. Such applications usually require an intensive interaction between sensors, control algorithms, and simulation tools. Practical optimal control may require a reliable prediction within the range of milli- or sub-milli-second. Reducing the computation cost of simulations has been one of the major motivations for developing model reduction methods. Other reduced order modeling related topics are the feature engineering and data analytics, which constitute an extensive literature in field of machine learning. Thus, the reduced order modeling and the machine learning have intrinsic connections. Developing reduced order machine learning methods may enable physics-data combined models that can overcome the current bottleneck in model reduction methods and purely data-driven machine learning approaches.
模型归约方法的开发满足了行业对工程问题快速、近乎实时仿真的迫切需求，如在线动态系统控制、结构健康监测、车辆健康监测、在线先进制造反馈控制、自动驾驶控制和决策等。此类应用通常需要传感器、控制算法和仿真工具之间的密集交互。实际的最优控制可能需要毫秒或亚毫秒范围内的可靠预测。降低仿真的计算成本一直是开发模型归约方法的主要动机之一。其他与降阶建模相关的主题是特征工程和数据分析，它们构成了机器学习领域的广泛文献。因此，降阶建模和机器学习具有内在的联系。开发降阶机器学习方法可能会使物理 - 数据组合模型成为可能，从而克服当前模型降阶方法和纯数据驱动机器学习方法的瓶颈。

Acknowledgements

The authors would like to acknowledge the sources of some historic facts cited in this paper, which
are taken from the following finite element history articles and books:

Oden, J.T. (1990). Historical comments on finite elements. In A history of scientific computing , 152-166.
Lax, P. (1993). Feng Kang. SIAM News, 26(11).
Clough, R.W. and Wilson, E.L. (1999), August. Early finite element research at Berkeley.
In Fifth US National Conference on Computational Mechanics, 1-35.
Felippa, C.A. (2004). Introduction to finite element methods. University of Colorado, 885.
Owen, D.R.J. and Feng, Y.T. (2012). Fifty years of finite elements—a solid mechanics perspective. Theoretical and Applied Mechanics Letters, 2(5), p.051001.
Stein, E. (2014). History of the finite element method–mathematics meets mechanics–part I: Engineering developments. In The History of Theoretical, Material and Computationa Mechanics-Mathematics Meets Mechanics and Engineering (pp. 399-442). Springer, Berlin, Heidelberg.
Clough, R.W. (1980). The finite element method after twenty-five years: a personal view. Computers & Structures, 12(4), 361-370.
Wikipedia: Finite Element Method, https://en.wikipedia.org/wiki/Finite_element_method

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Change history

28 June 2022

A Correction to this paper has been published:

https://doi.org/10.1007/s11831-022-09784-x

via：

Eighty Years of the Finite Element Method: Birth, Evolution, and Future | Archives of Computational Methods in Engineering

https://link.springer.com/article/10.1007/s11831-022-09740-9