Abstract
The approach to the finite element method can be derived from different motivations. Essentially there are three ways:
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a rather descriptive way, which has its roots in the engineering working method,
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a physical or
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mathematically motivated approach.
Depending on the perspective, different formulations result, which however all result in a common principal equation of the finite element method. The different formulations will be elaborated in detail based on the following descriptions:
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matrix methods,
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physically based working and energy methods and
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weighted residual method.
The finite element method is used to solve different physical problems. Here solely finite element formulations related to structural mechanics are considered [1, 5–7, 9–12].
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- 1.
The additional index ‘e’ is to be dropped at displacements since the nodal displacement is identical for each linked element in the displacement method.
- 2.
In the one-dimensional case the differential operator simplifies to the derivative \(\tfrac{\text{d}}{\text{d} x}\).
- 3.
The index ‘e’ of the element coordinate is neglected in the following—in the case it does not affect the understanding.
- 4.
Since the static boundary conditions are implicitly integrated in the overall potential, the shape functions do not have to fulfill those. However, if the shape functions fulfill the static boundary conditions additionally, an even more precise approximation can be achieved.
- 5.
Usually a separate local coordinate system \(0 \le x^\text{e} \le L^\text{e}\) is introduced for each element ‘e’. The coordinate in Eq. (2.88) is then referred to as global coordinate and receives the symbol \(X\).
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Öchsner, A., Merkel, M. (2013). Motivation for the Finite Element Method. In: One-Dimensional Finite Elements. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31797-2_2
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DOI: https://doi.org/10.1007/978-3-642-31797-2_2
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