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This book covers the fundamentals of the finite element method in the context of numerical simulation with specific reference to the simulation of the response of structural and mechanical components to mechanical and thermal loads.

We begin with the question: what is the meaning of the term “simulation”? By its dictionary definition, simulation is the imitative representation of the functioning of one system or process by means of the functioning of another. For instance, the membrane analogy introduced by Prandtl¹ in 1903 made it possible to find the shearing stresses in bars of arbitrary cross‐section, loaded by a twisting moment, through mapping the deflected shape of a thin elastic membrane. In other words, the distribution and magnitude of shearing stress in a twisted bar can be simulated by the deflected shape of an elastic membrane.

The membrane analogy exists because two unrelated phenomena can be modeled by the same partial differential equation. The physical meaning associated with the coefficients of the differential equation depends on which problem is being solved. However, the solution of one is proportional to the solution of the other: At corresponding points the shearing stress in a bar, subjected to a twisting moment, is oriented in the direction of the tangent to the contour lines of a deflected thin membrane and its magnitude is proportional to the slope of the membrane. Furthermore, the volume enclosed by the deflected membrane is proportional to the twisting moment.

In the pre‐computer years the membrane analogy provided practical means for estimating shearing stresses in prismatic bars. This involved cutting the shape of the cross‐section out of sheet metal or a wood panel, covering the hole with a thin elastic membrane, applying pressure to the membrane and mapping the contours of the deflected membrane. In present‐day practice both problems would be formulated as mathematical problems which would then be solved by a numerical method, most likely by the finite element method.

There are many other useful analogies. For example, the same differential equations simulate the response of assemblies of mechanical components, such as linear spring‐mass‐viscous damper systems and assemblies of electrical components, such as capacitors, inductors and resistors. This has been exploited by the use of analogue computers. Obviously, it is much easier to build and manipulate electrical circuitry than mechanical assemblies. In present‐day practice both simulation problems would be formulated as mathematical problems which would be solved by a numerical method.

At the heart of simulation of aspects of physical reality is a mathematical problem cast in a generalized form². The solution of the mathematical problem is approximated by a numerical method, such as the finite element method, which is the subject of this book. The quantities of interest (QoI) are extracted from the approximate solution. The errors of approximation in the QoI depend on how the mathematical problem was discretized³ and how the QoI were extracted from the numerical solution. When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by action of the analyst.

Estimation and control of numerical errors are fundamentally important in numerical simulation. Consider, for example, the problem of design certification. Design rules are typically stated in the form

(1.1)

where (resp. ) is the maximum (resp. allowable) value of a quantity of interest, for example the first principal stress. Since in numerical simulation only an approximation to Fmax is available, denoted by , it is necessary to know the size of the numerical error τ:

(1.2)

In design and design certification the worst case scenario has to be considered, which is underestimation of Fmax, that is,

(1.3)

Therefore it has to be shown that

(1.4)

Without a reliable estimate of the size of the numerical error it is not possible to certify design and, furthermore, numerical errors penalize design by lowering the allowable value, as indicated by eq. (1.4). Generally speaking, it is far more economical to ensure that τ is small than to accept the consequences of decreased allowable values.

We distinguish between finite element modeling and numerical simulation. As explained in greater detail in Chapter 5, finite element modeling evolved well before the theoretical basis of numerical simulation was developed. In finite element modeling a numerical problem is formulated by assembling elements from a library of finite elements that contains intuitively constructed beam, plate, shell, solid elements of various description. The numerical problem so created may not correspond to a well defined mathematical problem and therefore a solution may not even exist. For that reason it is not possible to speak of errors of approximation. Nevertheless, finite element modeling is widely practiced with success in some cases but with disappointing results in others. Such practice should be regarded as a practice of art, guided by intuition and experience, rather than a scientific activity. This is because practitioners of finite element modeling have to balance two kinds of very large errors: (a) conceptual errors in the formulation and (b) approximation errors in the numerical solution of an improperly posed mathematical problem.

In numerical simulation, on the other hand, the formulation of mathematical models is treated separately from their numerical solution. A mathematical model should be understood to be a precise statement of an idea of physical reality that permits the prediction of the occurrence, or probability of occurrence, of physical events, given certain data. The intuitive aspects of simulation are confined to the formulation of mathematical models whereas their numerical solution involves the application of well established procedures of applied mathematics. Separation of mathematical models from their numerical solution makes separate treatment of errors associated with the formulation of mathematical models and their numerical approximation possible. Errors associated with the formulation of mathematical models are called model form errors. Errors associated with the numerical solution of mathematical problems are called errors of approximation or errors of discretization. In the early papers and books on the finite element method no such distinction was made.

In this chapter we introduce the finite element method as a method by which the exact solution of a mathematical problem, cast in a generalized form, can be approximated. We also introduce the relevant mathematical concepts, terminology and notation in the simplest possible setting. Generalization of these concepts to two‐ and three‐dimensional problems will be discussed in subsequent chapters.

We first consider the formulation of a second order ordinary differential equation without reference to any physical interpretation. This is to underline that once a mathematical problem was formulated, the approximation process is independent from why the mathematical problem was formulated. This important point is often missed by engineering users of legacy finite element codes because the formulation and approximation of mathematical problems is mixed in finite element libraries.

We show that the exact solution of the generalized formulation is unique. Approximation of the exact solution by the finite element method is described and various discretization strategies are explored. Efficient methods for the computation of QoIs and a posteriori error estimation are described. This chapter serves as a foundation for subsequent chapters.

We would like to assure engineering students who are not yet familiar with the concepts and notation of that branch of applied mathematics on which the finite element method is based that their investment of time and effort to master the contents of this chapter will prove to be highly rewarding.