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1 Introduction to the finite element method

      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

      (1.2)

      In design and design certification the worst

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