作者：Carlos A. Felippa
Material assembled from Lecture Notes for the course Introduction to Finite Elements Methods (ASEN 5007) offered from 1986 to date at the Aerospace Engineering Sciences Department of the University of Colorado at Boulder.
材料来源于科罗拉多大学博尔德分校航空航天工程科学系1986年至今提供的有限元方法导论（ASEN 5007）课程讲义。
Preface前言
This textbook presents an Introduction to the computer-based simulation of linear structures by the Finite Element Method (FEM). It assembles the “converged” lecture notes of Introduction to Finite Element Methods or IFEM. This is a core graduate course offered in the Department of Aerospace Engineering Sciences of the University of Colorado at Boulder.
本教材介绍了有限元法（FEM）对线性结构的计算机模拟。它汇集了有限元方法导论或IFEM的“聚合”讲义。这是科罗拉多大学博尔德分校航空航天工程科学系提供的核心研究生课程。
IFEM was first taught on the Fall Semester 1986 and has been repeated every year since. It is taken by both first-year graduate students as part of their M.S. or M.E. requirements, and by senior undergraduates as technical elective. Selected material in Chapters 1 through 3 is used to teach a two week introduction of Matrix Structural Analysis and Finite Element concepts to junior undergraduate students who are taking their first Mechanics of Materials course.
IFEM于1986年秋季学期首次教授，此后每年都会重复。一年级研究生和高年级本科生都将其作为M.S.或M.E.要求的一部分，并将其作为技术选修课。第1章至第3章中的精选材料用于向正在参加第一门材料力学课程的初级本科生教授为期两周的矩阵结构分析和有限元概念介绍。
Prerequisites for the graduate-level course are multivariate calculus, linear algebra, a basic knowledge of structural mechanics at the Mechanics of Materials level, and some familiarity with programming concepts learnt in undergraduate courses.
研究生课程的先决条件是多元微积分、线性代数、材料力学水平的结构力学基础知识，以及对本科课程中学习的编程概念的一些熟悉。
The course originally used Fortran 77 as computer implementation language. This has been gradually changed to Mathematica since 1995. The changeover is now complete. No prior knowledge of Mathematica is required because that language, unlike Fortran or similar low-level programming languages, can be picked up while “going along.” Inasmuch as Mathematica supports both symbolic and numeric computation, as well as direct use of visualization tools, the use of the language is interspersed throughout the book.
该课程最初使用Fortran 77作为计算机实现语言。自1995年以来，它已逐渐改为Mathematica。转换现已完成。不需要事先了解Mathematica，因为与Fortran或类似的低级编程语言不同，该语言可以在“进行”时学习。由于Mathematica支持符号和数字计算，以及直接使用可视化工具，因此该语言的使用贯穿于整本书。
Book Objectives书籍目标
“In science there is only physics; all the rest is stamp collecting” (Lord Kelvin). The quote reflects the values of the mid-XIX century. Even now, at the dawn of the XXIth, progress and prestige in
the natural sciences favors fundamental knowledge. By contrast, engineering knowledge consists of three components:
“科学中只有物理学，其余的都是集邮”（开尔文勋爵）。这句话反映了十九世纪中期的价值观。即使是现在，在第二十九届大会的黎明，进步和声望自然科学偏爱基础知识。相比之下，工程知识由三个组成部分组成：
1. Conceptual knowledge: understanding the framework of the physical world.
1.概念知识：理解物理世界的框架。
2. Operational knowledge: methods and strategies for formulating, analyzing and solving problems,or “which buttons to push.”
2.操作知识：制定、分析和解决问题的方法和策略，或“按下哪些按钮”
3. Integral knowledge: the synthesis of conceptual and operational knowledge for technology development.
3.综合知识：为技术发展综合概念知识和操作知识。
The language that connects conceptual and operational knowledge is mathematics, and in particular the use of mathematical models. Most engineering programs in the USA correctly emphasize both conceptual and operational components. They differ, however, in how well the two are integrated.The most successful curricula are those that address the tendency to “horizontal disconnection” that bedevils engineering students suddenly exposed to a vast array of subjects.
连接概念知识和操作知识的语言是数学，尤其是数学模型的使用。美国的大多数工程项目都正确地强调了概念和操作两个方面。然而，它们的不同之处在于两者的融合程度。最成功的课程是那些解决了困扰工程专业学生突然接触到大量科目的“横向脱节”趋势的课程。
Integral knowledge is unique to the engineering profession. Synthesis ability is a personal attribute that cannot be coerced, only encouraged and cultivated, the same as the best music programs do not automatically produce Mozarts. Studies indicate no correlation between good engineers and good students.2 The best that can be done is to provide an adequate (and integrated) base of conceptual and operational knowledge to potentially good engineers.
综合知识是工程专业所独有的。综合能力是一种个人属性，不能被胁迫，只能被鼓励和培养，就像最好的音乐节目不会自动产生莫扎特一样。研究表明，优秀的工程师和优秀的学生之间没有相关性。2最好的办法是为潜在的优秀工程师提供充足的（综合的）概念和操作知识基础。
Where does the Finite Element Method (FEM) fit in this framework?
FEM was developed initially, and prospered, as a computer-based simulation method for the analysis of aerospace structures. Then it found its way into both design and analysis of complex structural systems, not only in Aerospace but in Civil and Mechanical Engineering. In the late 1960s it expanded to the simulation of non-structural problems in fluids, thermomechanics and electromagnetics. This “Physical FEM” is an operational tool, which fits primarily the operational knowledge component of engineering, and draws from the mathematical models of the real world. It is the form emphasized in the first part of this book.
有限元法作为一种用于航空航天结构分析的计算机模拟方法，最初得到了发展和繁荣。然后，它进入了复杂结构系统的设计和分析，不仅在航空航天领域，而且在土木和机械工程领域。20世纪60年代末，它扩展到模拟流体、热力学和电磁学中的非结构问题。这种“物理有限元”是一种运算工具，主要适用于工程的运算知识部分，并从现实世界的数学模型中提取。这是本书第一部分强调的形式。
The success of FEM as a general-purpose simulation method attracted attention in the 1970s from two quarters beyond engineering: mathematicians and software entrepreneurs. The world of FEM eventually split into applications, mathematics, and commercial software products. The former two are largely housed in the comfortable obscurity of academia. There is little cross-talk between these communities. They have different perpectives. They have separate constituencies, conferences and publication media, which slows down technology transfer. As of this writing, the three-way split seems likely to continue, as long as there is no incentive to do otherwise.
20世纪70年代，有限元法作为一种通用模拟方法的成功引起了工程以外两个方面的关注：数学家和软件企业家。FEM的世界最终分裂为应用程序、数学和商业软件产品。前两者在很大程度上处于学术界的默默无闻之中。这些社区之间几乎没有相互交流。它们有不同的垂直。它们有独立的选区、会议和出版媒体，这减缓了技术转让。截至本文撰写之时，只要没有其他动机，三方分裂似乎可能会继续。
This book aims to keep a presentation balance: the physical and mathematical interpretations of FEM are used eclectically, with none overshadowing the other. Key steps of the computer implementation are presented in sufficient detail so that a student can understand what goes on behind the scenes of a “black box” commercial product. The goal is that students navigating this material can eventually feel comfortable with any of the three “FEM communities” they come in contact during their professional life, whether as engineers, managers, researchers or teachers.
这本书旨在保持表述的平衡：FEM的物理和数学解释是折衷使用的，没有一种解释会掩盖另一种解释。充分详细地介绍了计算机实现的关键步骤，以便学生能够了解“黑匣子”商业产品的幕后情况。目标是，浏览这些材料的学生最终可以在职业生涯中接触到的三个“FEM社区”中的任何一个感到舒适，无论是作为工程师、经理、研究人员还是教师。
Book Organization
The book is divided into four Parts. The first three are of roughly similar length.
Part I: The Direct Stiffness Method. This part comprises Chapters 1 through 11. It covers major aspects of the Direct Stiffness Method (DSM). This is the most important realization of FEM, and the one implemented in general-purpose commercial finite element codes used by practicing engineers. Following a introductory first chapter, Chapters 2-4 present the fundamental steps of the DSM as a matrix method of structural analysis. A plane truss structure is used as motivating example. This is followed by Chapters 5-10 on programming, element formulation, modeling issues, and techniques for application of boundary conditions. Chapter 11 deals with relatively advanced topics including condensation and global-local analysis. Throughout these chapters the physical interpretation is emphasized for pedagogical convenience, as unifying vision of this “horizontal” framework.
第一部分：直接刚度法。本部分包括第1章至第11章。它涵盖了直接刚度法（DSM）的主要方面。这是FEM最重要的实现，也是在执业工程师使用的通用商业有限元代码中实现的。在介绍性的第一章之后，第2-4章介绍了DSM作为结构分析矩阵方法的基本步骤。以一个平面特拉斯结构为例。接下来是关于编程、元素公式、建模问题和边界条件应用技术的第5-10章。第11章涉及相对高级的主题，包括凝聚和全局局部分析。在这些章节中，为了教学的方便，物理解释被强调为这个“水平”框架的统一愿景。
Part II: Formulation of Finite Elements. This part extends from Chapters 12 through 19. It is more focused than Part I. It covers the development of elements from the more general viewpoint of the variational (energy) formulation. The presentation is inductive, always focusing on specific elements and progressing from the simplest to more complex cases. Thus Chapter 12 rederives the plane truss (bar) element from a variational formulation, while Chapter 13 presents the plane beam element. Chapter 14 introduces the plane stress problem, which serves as a testbed for the derivation of two-dimensional isoparametric elements in Chapter 15 through 18. This part concludes with an overview of requirements for convergence.
第二部分：有限元的形式化。本部分从第12章延伸到第19章。它比第一部分更集中。它从变分（能量）公式的更普遍的角度介绍了元素的发展。演讲是归纳性的，总是关注特定的元素，从最简单的案例发展到更复杂的案例。因此，第12章从变分公式中重新推导了平面特拉斯（杆）单元，而第13章给出了平面梁单元。第14章介绍了平面应力问题，该问题在第15章至第18章中用作推导二维等参单元的试验台。本部分最后概述了收敛的要求。
Part III: Computer Implementation. Chapters 20 through 29 deal with the computer implementation of the finite element method. Experience has indicated that students profit from doing computer homework early. This begins with Chapter 5, which contains an Introduction to Mathematica, and continues with homework assignments in Parts I and II. The emphasis changes in Part III to a systematic description of components of FEM programs, and the integration of those components to do problem solving.
第三部分：计算机实现。第20章至第29章介绍有限元法的计算机实现。经验表明，学生早做电脑作业是有益的。这从第5章开始，其中包含Mathematica导论，并继续第一部分和第二部分的家庭作业。第三部分的重点改为系统描述有限元程序的组成部分，并将这些组成部分集成起来解决问题。
Part IV: Structural Dynamics. This part, which starts at Chapter 30, is under preparation. It is intended as a brief introduction to the use of FEM in structural dynamics and vibration analysis, and is by nature more advanced than the other Parts.
第四部分：结构动力学。本部分从第30章开始，目前正在编写中。它旨在简要介绍有限元法在结构动力学和振动分析中的应用，本质上比其他部分更先进。
目录
1 Overview . . . . . . . . . . . . . . . . 1- 1
概述
2 The Direct Stiffness Method: Breakdown . . . . . . . . . . 2- 1
直接刚度方法：分解
3 The Direct Stiffness Method: Assembly and Solution . . . . . . . . . 3- 1
直接刚度法：聚合与求解
4 The Direct Stiffness Method: Miscellaneous Topics . . . . . . . . 4- 1
直接刚度方法：其他主题
5 Analysis of Example Truss by a CAS . . . . . . . . . . . . 5- 1
CAS对特拉斯实例的分析
6 Constructing MOM Members . . . . . . . . . . . . 6- 1
构造MOM成员
7 Finite Element Modeling: Introduction . . . . . . . . . . . 7- 1
有限元建模：简介
8 Finite Element Modeling: Mesh, Loads, BCs . . . . . . . . . . 8- 1
9 Multifreedom Constraints I . . . . . . . . . . . . . 9- 1
10 Multifreedom Constraints II . . . . . . . . . . . . . 10- 1
11 Superelements and Global-Local Analysis . . . . . . . . . . . 11- 1
12 The Bar Element . . . . . . . . . . . . . . . 12- 1
13 The Beam Element . . . . . . . . . . . . . . . 13- 1
14 The Plane Stress Problem . . . . . . . . . . . . . 14- 1
15 The Linear Triangle . . . . . . . . . . . . . . . 15- 1
16 The Isoparametric Representation . . . . . . . . . . . . 16- 1
17 Isoparametric Quadrilaterals . . . . . . . . . . . . . 17- 1
18 Shape Function Magic . . . . . . . . . . . . . . 18- 1
19 FEM Convergence Requirements . . . . . . . . . . . . 19- 1
20 (Moved to AFEM) . . . . . . . . . . . . . . 20- 1
21 Implementation of One-Dimensional Elements . . . . . . . . . . 21- 1
22 FEM Programs for Plane Trusses and Frames . . . . . . . . . . 22- 1
23 Implementation of iso-P Quadrilateral Elements . . . . . . . . . . 23- 1
24 Implementation of iso-P Triangular Elements . . . . . . . . . . 24- 1
23 The Assembly Procedure . . . . . . . . . . . . . . 23- 1
24 FE Model Definition . . . . . . . . . . . . . . 24- 1
25 Solving FEM Equations . . . . . . . . . . . . . . 25- 1
26 (under revision) . . . . . . . . . . . . . . . 26- 1
27 (under revision) . . . . . . . . . . . . . . . 27- 1
28 Stress Recovery . . . . . . . . . . . . . . . 28- 1
29 (placeholder) . . . . . . . . . . . . . . . . 29- 1
30 (under preparation) . . . . . . . . . . . . . . 30- 1
31 (under preparation) . . . . . . . . . . . . . . . 31- 1
第一章：概述
§1.1. Book Scope 1–3
§1.2. Where the Material Fits 1–3
§1.2.1. Top Level Classification . . . . . . . . . . . . . . 1–3
§1.2.2. Computational Mechanics . . . . . . . . . . . . . 1–3
§1.2.3. Statics versus Dynamics . . . . . . . . . . . . . . 1–5
§1.2.4. Linear versus Nonlinear . . . . . . . . . . . . . 1–5
§1.2.5. Discretization Methods . . . . . . . . . . . . . . 1–5
§1.2.6. FEM Formulation Levels . . . . . . . . . . . . . 1–6
§1.2.7. FEM Choices . . . . . . . . . . . . . . . . . 1–7
§1.2.8. Finally: What The Book Is About . . . . . . . . . . 1–7
§1.3. What Does a Finite Element Look Like? 1–7
§1.4. The FEM Analysis Process 1–9
§1.4.1. The Physical FEM . . . . . . . . . . . . . . . . 1–9
§1.4.2. The Mathematical FEM . . . . . . . . . . . . . 1–11
§1.4.3. Synergy of Physical and Mathematical FEM . . . . . . . 1–11
§1.4.4. Streamlined Idealization and Discretization . . . . . . . 1–13
§1.5. Method Interpretations 1–13
§1.5.1. Physical Interpretation . . . . . . . . . . . . . . 1–13
§1.5.2. Mathematical Interpretation . . . . . . . . . . . . 1–14
§1.6. Keeping the Course 1–15
§1.7. *What is Not Covered 1–15
§1.8. The Origins of the Finite Element Method 1–16
§1.9. Recommended Books for Linear FEM 1–16
§1.9.1. Hasta la Vista, Fortran . . . . . . . . . . . . . . 1–16
§1. Notes and Bibliography . . . . . . . . . . . . . . . . . . . . . . 1–17
§1. References . . . . . . . . . . . . . . . . . . . . . . 1–18
§1. Exercises . . . . . . . . . . . . . . . . . . . . . . 1–19
