![1–15 Exercises
Homework Exercises for Chapter 1
Overview
EXERCISE 1.1
[A:15] Do Archimedes’ problem using a circumscribed regular polygon, with n = 1, 2, . . . 256. Does the
sequence converge any faster?
EXERCISE 1.2
[D:20] Select one of the following vehicles: truck, car, motorcycle, or bicycle. Draw a two level decomposition
of the structure into substructures, and of selected components of some substructures.
1–15](https://image.slidesharecdn.com/ifem-151105131147-lva1-app6891/85/Ifem-ch01-15-320.jpg)
Ifem.ch01
- 2. Chapter 1: OVERVIEW 1–2 TABLE OF CONTENTS Page §1.1. WHERE THIS MATERIAL FITS 1–3 §1.1.1. Computational Mechanics . . . . . . . . . . . . . 1–3 §1.1.2. Statics vs. Dynamics . . . . . . . . . . . . . . 1–4 §1.1.3. Linear vs. Nonlinear . . . . . . . . . . . . . . . 1–5 §1.1.4. Discretization methods . . . . . . . . . . . . . . 1–5 §1.1.5. FEM Variants . . . . . . . . . . . . . . . . . 1–5 §1.2. WHAT DOES A FINITE ELEMENT LOOK LIKE? 1–6 §1.3. THE FEM ANALYSIS PROCESS 1–7 §1.3.1. The Mathematical FEM . . . . . . . . . . . . . 1–8 §1.3.2. The Physical FEM . . . . . . . . . . . . . . . . 1–9 §1.3.3. Synergy of Physical and Mathematical FEM . . . . . . 1–9 §1.4. INTERPRETATIONS OF THE FINITE ELEMENT METHOD 1–11 §1.4.1. Physical Interpretation . . . . . . . . . . . . . . 1–11 §1.4.2. Mathematical Interpretation . . . . . . . . . . . . 1–12 §1.5. KEEPING THE COURSE 1–13 §1.6. *WHAT IS NOT COVERED 1–13 EXERCISES . . . . . . . . . . . . . . . . . . . . . . 1–15 1–2
- 3. 1–3 §1.1 WHERE THIS MATERIAL FITS This book is an introduction to the analysis of linear elastic structures by the Finite Element Method (FEM). It embodies three Parts: I Finite Element Discretization: Chapters 2-11. This part provides an introduction to the discretization and analysis of skeletal structures by the Direct Stiffness Method. II Formulation of Finite Elements: Chapters 12-20. This part presents the formulation of displacement assumed elements in one and two dimensions. III Computer Implementation of FEM: Chapters 21-28. This part uses Mathematica as the implementation language. This Chapter presents an overview of where the book fits, and what finite elements are. §1.1. WHERE THIS MATERIAL FITS The field of Mechanics can be subdivided into three major areas: Mechanics Theoretical Applied Computational (1.1) Theoretical mechanics deals with fundamental laws and principles of mechanics studied for their intrinsic scientific value. Applied mechanics transfers this theoretical knowledge to scientific and engineering applications, especially as regards the construction of mathematical models of physical phenomena. Computational mechanics solves specific problems by simulation through numerical methods implemented on digital computers. REMARK 1.1 Paraphrasing an old joke about mathematicians, one may define a computational mechanician as a person who searches for solutions to given problems, an applied mechanician as a person who searches for problems that fit given solutions, and a theoretical mechanician as a person who can prove the existence of problems and solutions. §1.1.1. Computational Mechanics Several branches of computational mechanics can be distinguished according to the physical scale of the focus of attention: Computational Mechanics Nanomechanics and micromechanics Continuum mechanics Solids and Structures Fluids Multiphysics Systems (1.2) Nanomechanics deals with phenomena at the molecular and atomic levels of matter. As such it is closely interrelated with particle physics and chemistry. Micromechanics looks primarily at the crystallographic and granular levels of matter. Its main technological application is the design and fabrication of materials and microdevices. 1–3
- 4. Chapter 1: OVERVIEW 1–4 Continuum mechanics studies bodies at the macroscopic level, using continuum models in which the microstructure is homogenized by phenomenological averages. The two traditional areas of application are solid and fluid mechanics. The former includes structures which, for obvious reasons, are fabricated with solids. Computational solid mechanics takes a applied-sciences ap- proach, whereas computational structural mechanics emphasizes technological applications to the analysis and design of structures. Computational fluid mechanics deals with problems that involve the equilibrium and motion of liquid and gases. Well developed related areas are hydrodynamics, aerodynamics, atmospheric physics, and combustion. Multiphysics is a more recent newcomer. This area is meant to include mechanical systems that transcendtheclassicalboundariesofsolidandfluidmechanics, asininteractingfluidsandstructures. Phase change problems such as ice melting and metal solidification fit into this category, as do the interaction of control, mechanical and electromagnetic systems. Finally, system identifies mechanical objects, whether natural or artificial, that perform a distin- guishable function. Examples of man-made systems are airplanes, buildings, bridges, engines, cars, microchips, radio telescopes, robots, roller skates and garden sprinklers. Biological systems, such as a whale, amoeba or pine tree are included if studied from the viewpoint of biomechanics. Ecological, astronomical and cosmological entities also form systems.1 In this progression of (1.2) the system is the most general concept. A system is studied by de- composition: its behavior is that of its components plus the interaction between the components. Components are broken down into subcomponents and so on. As this hierarchical process continues the individual components become simple enough to be treated by individual disciplines, but their interactions may get more complex. Consequently there is a tradeoff art in deciding where to stop.2 §1.1.2. Statics vs. Dynamics Continuum mechanics problems may be subdivided according to whether inertial effects are taken into account or not: Continuum mechanics Statics Dynamics (1.3) In dynamics the time dependence is explicitly considered because the calculation of inertial (and/or damping) forces requires derivatives respect to actual time to be taken. Problems in statics may also be time dependent but the inertial forces are ignored or neglected. Static problems may be classified into strictly static and quasi-static. For the former time need not be considered explicitly; any historical time-like response-ordering parameter (if one is needed) will do. In quasi-static problems such as foundation settlement, creep deformation, rate-dependent 1 Except that their function may not be clear to us. “The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?” (Stephen Hawking). 2 Thus in breaking down a car engine, say, the decomposition does not usually proceed beyond the components you can buy at a parts shop. 1–4
- 5. 1–5 §1.1 WHERE THIS MATERIAL FITS plasticity or fatigue cycling, a more realistic estimation of time is required but inertial forces are still neglected. §1.1.3. Linear vs. Nonlinear A classification of static problems that is particularly relevant to this book is Statics Linear Nonlinear Linear static analysis deals with static problems in which the response is linear in the cause-and- effect sense. For example: if the applied forces are doubled, the displacements and internal stresses also double. Problems outside this domain are classified as nonlinear. §1.1.4. Discretization methods A final classification of CSM static analysis is based on the discretization method by which the continuum mathematical model is discretized in space, i.e., converted to a discrete model of finite number of degrees of freedom: Spatial discretization method Finite Element Method (FEM) Boundary Element Method (BEM) Finite Difference Method (FDM) Finite Volume Method (FVM) Spectral Method Mesh-Free Method (1.4) For linear problems finite element methods currently dominate the scene, with boundary element methods posting a strong second choice in specific application areas. For nonlinear problems the dominance of finite element methods is overwhelming. Classical finite difference methods in solid and structural mechanics have virtually disappeared from practical use. This statement is not true, however, for fluid mechanics, where finite difference discretization methods are still important. Finite-volume methods, which address finite volume method conservation laws, are important in highly nonlinear problems of fluid mechanics. Spectral methods are based on transforms that map space and/or time dimensions to spaces where the problem is easier to solve. A recent newcomer to the scene are the mesh-free methods. These are finite different methods on arbitrary grids constructed through a subset of finite element techniques and tools. §1.1.5. FEM Variants The term Finite Element Method actually identifies a broad spectrum of techniques that share common features outlined in §1.3 and §1.4. Two subclassifications that fit well applications to structural mechanics are FEM Formulation Displacement Equilibrium Mixed Hybrid FEM Solution Stiffness Flexibility Mixed (a.k.a. Combined) (1.5) 1–5
- 6. Chapter 1: OVERVIEW 1–6 (The distinction between these subclasses require advanced technical concepts, and will not be covered here.) Using the foregoing classification, we can state the topic of this book more precisely: the computa- tional analysis of linear static structural problems by the Finite Element Method. Of the variants listed in (1.5), emphasis is placed on the displacement formulation and stiffness solution. This combination is called the Direct Stiffness Method or DSM. §1.2. WHAT DOES A FINITE ELEMENT LOOK LIKE? The subject of this book is FEM. But what is a finite element? The concept will be partly illustrated through a truly ancient problem: find the perimeter L of a circle of diameter d. Since L = π d, this is equivalent to obtaining a numerical value for π. Draw a circle of radius r and diameter d = 2r as in Figure 1.1(a). Inscribe a regular polygon of n sides, where n = 8 in Figure 1.1(b). Rename polygon sides as elements and vertices as nodal points or nodes. Label nodes with integers 1, . . . 8. Extract a typical element, say that joining nodes 4–5 as shown in Figure 1.1(c). This is an instance of the generic element i– j shown in Figure 1.1(d). The element length is Li j = 2r sin(π/n). Since all elements have the same length, the polygon perimeter is Ln = nLi j , whence the approximation to π is πn = Ln/d = n sin(π/n). 1 2 3 4 5 6 7 8 r 4 5 i j (a) (b) (c) (d) d r 2r sin(π/n) 2π/n Figure 1.1. The “find π” problem treated with FEM concepts: (a) continuum object, (b) a discrete approximation (inscribed regular polygon), (c) disconnected element, (d), generic element. Values of πn obtained for n = 1, 2, 4, . . . 256 are listed in the second column of Table 1.1. As can be seen the convergence to π is fairly slow. However, the sequence can be transformed by Wynn’s algorithm3 into that shown in the third column. The last value displays 15-place accuracy. Some of the key ideas behind the FEM can be identified in this simple example. The circle, viewed as a source mathematical object, is replaced by polygons. These are discrete approximations to the circle. The sides, renamed as elements, are specified by their end nodes. Elements can be separated by disconnecting the nodes, a process called disassembly in the FEM. Upon disassembly 3 A widely used extrapolation algorithm that speeds up the convergence of many sequences. See, e.g, J. Wimp, Sequence Transformations and Their Applications, Academic Press, New York, 1981. 1–6
- 7. 1–7 §1.3 THE FEM ANALYSIS PROCESS Table 1.1. Rectification of Circle by Inscribed Polygons (“Archimedes FEM”) n πn = n sin(π/n) Extrapolated by Wynn- Exact π to 16 places 1 0.000000000000000 2 2.000000000000000 4 2.828427124746190 3.414213562373096 8 3.061467458920718 16 3.121445152258052 3.141418327933211 32 3.136548490545939 64 3.140331156954753 3.141592658918053 128 3.141277250932773 256 3.141513801144301 3.141592653589786 3.141592653589793 a generic element can be defined, independently of the original circle, by the segment that connects two nodes i and j. The relevant element property: length Li j , can be computed in the generic element independently of the others, a property called local support in the FEM. Finally, the desired property: the polygon perimeter, is obtained by reconnecting n elements and adding up their length; the corresponding steps in the FEM being assembly and solution, respectively. There is of course nothing magic about the circle; the same technique can be be used to rectify any smooth plane curve.4 This example has been offered in the FEM literature to aduce that finite element ideas can be traced to Egyptian mathematicians from circa 1800 B.C., as well as Archimedes’ famous studies on circle rectification by 250 B.C. But comparison with the modern FEM, as covered in Chapters 2–3, shows this to be a stretch. The example does not illustrate the concept of degrees of freedom, conjugate quantities and local-global coordinates. It is guilty of circular reasoning: the compact formula π = limn→∞ n sin(π/n) uses the unknown π in the right hand side.5 Reasonable people would argue that a circle is a simpler object than, say, a 128-sided polygon. Despite these flaws the example is useful in one respect: showing a fielder’s choice in the replacement of one mathematical object by another. This is at the root of the simulation process described in the next section. §1.3. THE FEM ANALYSIS PROCESS A model-based simulation process using FEM involves doing a sequence of steps. This sequence takes two canonical configurations depending on the environment in which FEM is used. These are reviewed next to introduce terminology. 4 A similar limit process, however, may fail in three or more dimensions. 5 This objection is bypassed if n is advanced as a power of two, as in Table 1.1, by using the half-angle recursion √ 2 sin α = 1 − 1 − sin2 2α, started from 2α = π for which sin π = −1. 1–7
- 8. Chapter 1: OVERVIEW 1–8 Discretization + solution error REALIZATION IDEALIZATION solution error Discrete model Discrete solution VERIFICATION VERIFICATION FEM IDEALIZATION & DISCRETIZATION SOLUTION Ideal physical system Mathematical model generally irrelevant Figure 1.2. The Mathematical FEM. The mathematical model (top) is the source of the simulation process. Discrete model and solution follow from it. The ideal physical system (should one go to the trouble of exhibiting it) is inessential. §1.3.1. The Mathematical FEM The process steps are illustrated in Figure 1.2. The process centerpiece, from which everything emanates, is the mathematical model. This is often an ordinary or partial differential equation in space and time. A discrete finite element model is generated from a variational or weak form of the mathematical model.6 This is the discretization step. The FEM equations are processed by an equation solver, which delivers a discrete solution (or solutions). On the left Figure 1.2 shows an ideal physical system. This may be presented as a realization of the mathematical model; conversely, the mathematical model is said to be an idealization of this system. For example, if the mathematical model is the Poisson’s equation, realizations may be a heat conduction or a electrostatic charge distribution problem. This step is inessential and may be left out. Indeed FEM discretizations may be constructed without any reference to physics. The concept of error arises when the discrete solution is substituted in the “model” boxes. This replacement is generically called verification. The solution error is the amount by which the discrete solution fails to satisfy the discrete equations. This error is relatively unimportant when using computers, and in particular direct linear equation solvers, for the solution step. More relevant is the discretization error, which is the amount by which the discrete solution fails to satisfy the mathematical model.7 Replacing into the ideal physical system would in principle quantify modeling errors. In the mathematical FEM this is largely irrelevant, however, because the ideal physical system is merely that: a figment of the imagination. 6 The distinction between strong, weak and variational forms is discussed in advanced FEM courses. In the present course such forms will be stated as recipes. 7 This error can be computed in several ways, the details of which are of no importance here. 1–8
- 9. 1–9 §1.3 THE FEM ANALYSIS PROCESS Physical system simulation error= modeling + solution error solution error Discrete model Discrete solution VALIDATION VERIFICATION FEM CONTINUIFICATION Ideal Mathematical model IDEALIZATION & DISCRETIZATION SOLUTION generally irrelevant Figure 1.3. The Physical FEM. The physical system (left) is the source of the simulation process. The ideal mathematical model (should one go to the trouble of constructing it) is inessential. §1.3.2. The Physical FEM The second way of using FEM is the process illustrated in Figure 1.3. The centerpiece is now the physical system to be modeled. Accordingly, this sequence is called the Physical FEM. The processes of idealization and discretization are carried out concurrently to produce the discrete model. The solution is computed as before. Just like Figure 1.2 shows an ideal physical system, 1.3 depicts an ideal mathematical model. This may be presented as a continuum limit or “continuification” of the discrete model. For some physical systems, notably those well modeled by continuum fields, this step is useful. For others, such as complex engineering systems, it makes no sense. Indeed FEM discretizations may be constructed and adjusted without reference to mathematical models, simply from experimental measurements. The concept of error arises in the Physical FEM in two ways, known as verification and validation, respectively. Verification is the same as in the Mathematical FEM: the discrete solution is replaced into the discrete model to get the solution error. As noted above this error is not generally important. Substitution in the ideal mathematical model in principle provides the discretization error. This is rarely useful in complex engineering systems, however, because there is no reason to expect that the mathematical model exists, and if it does, that it is more physically relevant than the discrete model. Validation tries to compare the discrete solution against observation by computing the simulation error, which combines modeling and solution errors. As the latter is typically insignificant, the simulation error in practice can be identified with the modeling error. One way to adjust the discrete model so that it represents the physics better is called model updating. The discrete model is given free parameters. These are determined by comparing the discrete solution against experiments, as illustrated in Figure 1.4. Inasmuch as the minimization conditions are generally nonlinear (even if the model is linear) the updating process is inherently iterative. 1–9
- 10. Chapter 1: OVERVIEW 1–10 Physical system simulation error Parametrized discrete model Experimental database Discrete solution FEM EXPERIMENTS Figure 1.4. Model updating process in the Physical FEM. §1.3.3. Synergy of Physical and Mathematical FEM The foregoing physical and mathematical sequences are not exclusive but complementary. This synergy8 is one of the reasons behind the power and acceptance of the method. Historically the Physical FEM was the first one to be developed to model very complex systems such as aircraft, as narrated in Appendix H. The Mathematical FEM came later and, among other things, provided the necessary theoretical underpinnings to extend FEM beyond structural analysis. A glance at the schematics of a commercial jet aircraft makes obvious the reasons behind the physical FEM. There is no differential equation that captures, at a continuum mechanics level,9 the structure, avionics, fuel, propulsion, cargo, and passengers eating dinner. There is no reason for despair, however. The time honored divide and conquer strategy, coupled with abstraction, comes to the rescue. First, separate the structure and view the rest as masses and forces, most of which are time-varying and nondeterministic. Second, consider the aircraft structure as built of substructures:10 wings, fuselage, stabilizers, engines, landing gears, and so on. Take each substructure, and continue to decompose it into components: rings, ribs, spars, cover plates, actuators, etc, continuing through as many levels as necessary. Eventually those components become sufficiently simple in geometry and connectivity that they can be reasonably well described by the continuum mathematical models provided, for instance, by Mechanics of Materials or the Theory of Elasticity. At that point, stop. The component level discrete equations are obtained from a FEM library based on the mathematical model. The system model is obtained by going through the reverse process: from component equations to substructure equations, and from those to the equations of the complete aircraft. This system assembly process is governed by the classical principles of Newtonian mechanics expressed in conservation form. This multilevel decomposition process is diagramed in Figure 1.5, in which the intermediate sub- structure level is omitted for simplicity. 8 This interplay is not exactly a new idea: “The men of experiment are like the ant, they only collect and use; the reasoners resemble spiders, who make cobwebs out of their own substance. But the bee takes the middle course: it gathers its material from the flowers of the garden and field, but transforms and digests it by a power of its own.” (Francis Bacon, 1620). 9 Of course at the atomic and subatomic level quantum mechanics works for everything, from landing gears to passengers. But it would be slightly impractical to model the aircraft by 1036 interacting particles. 10 A substructure is a part of a structure devoted to a specific function. 1–10
- 11. 1–11 §1.4 INTERPRETATIONS OF THE FINITE ELEMENT METHOD FEM Library Component discrete model Component equations Physical system System discrete model Complete solution Mathematical model SYSTEM LEVEL COMPONENT LEVEL Figure 1.5. Combining physical and mathematical modeling through multilevel FEM. Only two levels (system and component) are shown for simplicity; intermediate substructure levels are omitted. REMARK 1.2 More intermediate decomposition levels are used in some systems, such as offshore and ship structures, which are characterized by a modular fabrication process. In that case the decomposition mimics the way the system is actually constructed. The general technique, called superelements, is discussed in Chapter 11. REMARK 1.3 There is no point in practice in going beyond a certain component level while considering the complete model, since the level of detail can become overwhelming without adding significant information. Further refinement or particular components is done by the so-called global-local analysis technique outlined in Chapter 11. This technique is an instance of multiscale analysis. For sufficiently simple structures, passing to a discrete model is carried out in a single idealization and discretization step, as illustrated for the truss roof structure shown in Figure 1.6. Multiple levels are unnecessary here. Of course the truss may be viewed as a substructure of the roof, and the roof as a a substructure of a building. §1.4. INTERPRETATIONS OF THE FINITE ELEMENT METHOD Just like there are two complementary ways of using the FEM, there are two complementary interpretations for teaching it. One interpretation stresses the physical significance and is aligned with the Physical FEM. The other focuses on the mathematical context, and is aligned with the Mathematical FEM. §1.4.1. Physical Interpretation The physical interpretation focuses on the view of Figure 1.3. This interpretation has been shaped by the discovery and extensive use of the method in the field of structural mechanics. This relationship 1–11
- 12. Chapter 1: OVERVIEW 1–12 joint Physical System support member IDEALIZATION ;;;;;; ;; ;;;; Figure 1.6. The idealization process for a simple structure. The physical system, here a roof truss, is directly idealized by the mathematical model: a pin-jointed bar assembly. For this particular structure, the idealization coalesces with the discrete model. is reflected in the use of structural terms such as “stiffness matrix”, “force vector” and “degrees of freedom.” This terminology carries over to non-structural applications. The basic concept in the physical interpretation is the breakdown (≡ disassembly, tearing, partition, separation, decomposition) of a complex mechanical system into simpler, disjoint components called finite elements, or simply elements. The mechanical response of an element is characterized in terms of a finite number of degrees of freedom. These degrees of freedoms are represented as the values of the unknown functions as a set of node points. The element response is defined by algebraic equations constructed from mathematical or experimental arguments. The response of the original system is considered to be approximated by that of the discrete model constructed by connecting or assembling the collection of all elements. The breakdown-assembly concept occurs naturally when an engineer considers many artificial and natural systems. For example, it is easy and natural to visualize an engine, bridge, aircraft or skeleton as being fabricated from simpler parts. As discussed in §1.3, the underlying theme is divide and conquer. If the behavior of a system is too complex, the recipe is to divide it into more manageable subsystems. If these subsystems are still too complex the subdivision process is continued until the behavior of each subsystem is simple enough to fit a mathematical model that represents well the knowledge level the analyst is interested in. In the finite element method such “primitive pieces” are called elements. The behavior of the total system is that of the individual elements plus their interaction. A key factor in the initial acceptance of the FEM was that the element interaction can be physically interpreted and understood in terms that were eminently familiar to structural engineers. §1.4.2. Mathematical Interpretation This interpretation is closely aligned with the configuration of Figure 1.2. The FEM is viewed as a procedure for obtaining numerical approximations to the solution of boundary value problems 1–12
- 13. 1–13 §1.6 *WHAT IS NOT COVERED (BVPs) posed over a domain . This domain is replaced by the union ∪ of disjoint subdomains (e) called finite elements. In general the geometry of is only approximated by that of ∪ (e) . The unknown function (or functions) is locally approximated over each element by an interpolation formula expressed in terms of values taken by the function(s), and possibly their derivatives, at a set of node points generally located on the element boundaries. The states of the assumed unknown function(s) determined by unit node values are called shape functions. The union of shape functions “patched” over adjacent elements form a trial function basis for which the node values represent the generalized coordinates. The trial function space may be inserted into the governing equations and the unknown node values determined by the Ritz method (if the solution extremizes a variational principle) or by the Galerkin, least-squares or other weighted-residual minimization methods if the problem cannot be expressed in a standard variational form. REMARK 1.4 In the mathematical interpretation the emphasis is on the concept of local (piecewise) approximation. The concept of element-by-element breakdown and assembly, while convenient in the computer implementation, is not theoretically necessary. The mathematical interpretation permits a general approach to the questions of convergence, error bounds, trial and shape function requirements, etc., which the physical approach leaves unanswered. It also facilitates the application of FEM to classes of problems that are not so readily amenable to physical visualization as structures; for example electromagnetics and thermal conduction. REMARK 1.5 It is interesting to note some similarities in the development of Heaviside’s operational methods, Dirac’s delta-function calculus, and the FEM. These three methods appeared as ad-hoc computational devices created by engineers and physicists to deal with problems posed by new science and technology (electricity, quantum mechanics, and delta-wing aircraft, respectively) with little help from the mathematical establishment. Only some time after the success of the new techniques became apparent were new branches of mathematics (operational calculus, distribution theory and piecewise-approximation theory, respectively) constructed to justify that success. In the case of the finite element method, the development of a formal mathematical theory started in the late 1960s, and much of it is still in the making. §1.5. KEEPING THE COURSE The first Part of this course, which is the subject of Chapters 2 through 11, stresses the physical interpretation in the framework of the Direct Stiffness Method (DSM) on account of its instructional advantages. Furthermore the computer implementation becomes more transparent because the sequence of computer operations can be placed in close correspondence with the DSM steps. Subsequent Chapters incorporate ingredients of the mathematical interpretation when it is felt convenient to do so. However, the exposition avoids excessive entanglement with the mathematical theory when it may obfuscate the physics. A historical outline of the evolution of Matrix Structural Analysis into the Finite Element Method is given in Appendix H, which provides appropriate references. In Chapters 2 through 6 the time is frozen at about 1965, and the DSM presented as an aerospace engineer of that time would have understood it. This is not done for sentimental reasons, although that happens to be the year in which the writer began his thesis work on FEM under Ray Clough. Virtually all finite element codes are now based on the DSM and the computer implementation has not essentially changed since the late 1960s. 1–13
- 14. Chapter 1: OVERVIEW 1–14 §1.6. *WHAT IS NOT COVERED The following topics are not covered in this book: 1. Elements based on equilibrium, mixed and hybrid variational formulations. 2. Flexibility and mixed solution methods of solution. 3. Kirchhoff-based plate and shell elements. 4. Continuum-based plate and shell elements. 5. Variational methods in mechanics. 6. General mathematical theory of finite elements. 7. Vibration analysis. 8. Buckling analysis. 9. General stability analysis. 10. General nonlinear response analysis. 11. Structural optimization. 12. Error estimates and problem-adaptive discretizations. 13. Non-structural and coupled-system applications of FEM. 14. Structural dynamics. 15. Shock and wave-propagation dynamics. 16. Designing and building production-level FEM software and use of special hardware (e.g. vector and parallel computers) Topics 1–7 pertain to what may be called “Advanced Linear FEM”, whereas 9–11 pertain to “Nonlinear FEM”. Topics 12-15 pertain to advanced applications, whereas 16 is an interdisciplinary topic that interweaves with computer science. For pre-1990 books on FEM see Appendix G: Oldies but Goodies. 1–14
- 15. 1–15 Exercises Homework Exercises for Chapter 1 Overview EXERCISE 1.1 [A:15] Do Archimedes’ problem using a circumscribed regular polygon, with n = 1, 2, . . . 256. Does the sequence converge any faster? EXERCISE 1.2 [D:20] Select one of the following vehicles: truck, car, motorcycle, or bicycle. Draw a two level decomposition of the structure into substructures, and of selected components of some substructures. 1–15