Abstract
For one-dimensional domain the Lagrange interpolation, with the polynomial structure, furnishes an excellent approximation. For two non-zero data points, the Lagrange interpolant is a straight line. Piecewise parabolic approximations are also quite popular.
The linear interpolation over a triangulated mesh is the two-dimensional counterpart of the Lagrange interpolation with two data points.
Courant illustrated two-dimensional spatial discretization for the structural mechanics problem of torsion pertaining to a non-circular prismatic bar (Courant, Bull Am Math Soc 49(1):1–29, 1943) (In his first sentence of the last paragraph, Courant wrote: “Of course, one must not expect good local results from a method using so few elements.” Obviously, he was talking about a finite number of elements. We can therefore credit Courant to be initiator of finite elements.). He demonstrated the intimate connection between triangulation and linear interpolants.
A collection of scholarly papers appears in Michal Křížek and Stenberg (eds) (Finite element methods: fifty years of the Courant element. CRC Press, Boca Raton, 1994 (Marcel Dekker, Jyvaskyla, 1993)). This chapter prepares readers to undertake such in-depth studies.
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Notes
- 1.
This corresponds to the unit virtual displacement \(\left \{\delta r^{(i)}\right \}\) of Sect. 2.1.6.
- 2.
\(\left \{1,x,y\right \}\), the independent polynomial terms, are the Rayleigh modes, since they satisfy equilibrium, for generating nodal shape functions \(\mathfrak{n}_{i}(x,y).\) This is elaborated in Sect. 3.2.3.
- 3.
We can utilize the List structure of Mathematica to evaluate the denominator only once for all three shape functions.
- 4.
The word is derived from Latin tessella that is a small piece to make mosaics.
- 5.
Change of u to ϕ in Sect. 3.3.1.2 avoids conflicts with displacements u i (x, y, z).
- 6.
- 7.
This gives us an opportunity to familiarize ourselves with the divergence theorem.
References
Courant R (1943) Variational methods for the solution of problems of equilibrium and vibration. Bull Am Math Soc 49(1):1–29
Felippa CA (1994) 50 year classic reprint: an appreciation of R. Courant’s ‘variational methods for the solution of problems of equilibrium and vibrations,’ 1943. Int J Numer Methods Eng 37:2159–2187
Fenner RT (1975) Finite element methods for engineers. Imperial College Press/Macmillan, London/New York
Gupta KK, Meek JL (1996) A brief history of the beginning of the finite element method. Int J Numer Methods Eng 39:3761–3774
Love AEH (1944) A treatise on the mathematical theory of elasticity. Dover, New York
Michal Křížek PN, Stenberg R (eds) (1994) Finite element methods: fifty years of the Courant element. CRC Press, Boca Raton (Marcel Dekker, Jyvaskyla, 1993)
Wolfram S (2015) An elementary introduction to the Wolfram Language. Wolfram Media, New York
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Dasgupta, G. (2018). Courant’s Triangular Elements with Linear Interpolants. In: Finite Element Concepts. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7423-8_3
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