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 initial formulations result in the same principal finite element equation. The different formulations will be elaborated in detail based on the following descriptions:
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matrix methods,
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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.
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Notes
- 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 first-order derivative \(\tfrac{\mathrm{d}}{\mathrm{d}x}\).
- 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.
References
Kuhn G, Winter W(1993) Skriptum Festigkeitslehre Universität Erlangen-Nürnberg
Brebbia CA, Telles JCF, Wrobel LC (1984) Boundary element techniques: theory and applications. Springer, Berlin
Zienkiewicz OC, Taylor RL (2000) The finite element method volume 1: the basis. Butterworth-Heinemann, Oxford
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Öchsner, A., Merkel, M. (2018). Motivation for the Finite Element Method. In: One-Dimensional Finite Elements. Springer, Cham. https://doi.org/10.1007/978-3-319-75145-0_2
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DOI: https://doi.org/10.1007/978-3-319-75145-0_2
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