Keywords: p-FEM, high-order finite elements, nonlinear finite elements.

This paper provides a survey of the basic concepts of the p-version of the finite element method in structural mechanics, discusses achievements in research over the past two decades, addresses recent results related to nonlinear problems and gives an outlook to consider open problems and future developments.

The first systematic investigations on the p-version of the finite element method were performed by Babûska, Szabó and their co-workers [1] in the early nineteen-eighties. It is a systematic extension process of the Ansatz spaces leaving the mesh unchanged and increasing the polynomial degree of the shape functions locally or globally [2]. It has turned out to be a very efficient discretization strategy for many linear elliptic problems, for example, the Poisson equation, the Lamé equations, the Reissner–Mindlin problem, shell discretizations etc. For this class of problems the p-version is in general superior to the classical h-version approach. Also, in the case of singularities, the p-version shows an exponential rate of convergence in the energy norm in the pre-asymptotic range, when combined with a proper mesh design [3].

In recent years, the p-version has been developed further to solve nonlinear problems such as elastoplasticity. It has been demonstrated that it can be applied efficiently to industrial problems arising, for example, in sheet metal forming. A very important advantage of the p-FEM is its ability to use elements with a very large aspect ratio. This feature permits the modelling of even very thin-walled structures in a strictly three-dimensional setting [2]. Moreover, the inherent independence of the geometric shape from the space of the Ansatz functions opens many possibilities to a tight connection between CAD-models and a p-FEM computation. It has also been demonstrated that it is possible to incorporate model adaptivity in a very natural way in order to approximate a given mechanical problem optimally.

Many of the advantages being observed from the above mentioned problems can be exploited as well if the p-FEM is used in multi-field problems such as fluid-structure interaction [4].

The most serious problem in the way of a broader application of high order finite elements in commercial software is the more demanding quality for a precise description of the geometry and mesh generation suited for the p-FEM. Many difficulties, however, can be overcome by using a mortar-type coupling of non-conforming finite element meshes [5].

[1]

I. Babûska, B.A. Szabó, I.N. Katz, "The p-Version of the Finite Element Method", SIAM Journal on Numerical Analysis, 18, 515-545, 1981. doi:10.1137/0718033

[2]

A. Düster, H. Bröker, E. Rank, "The p-version of the finite element method for three-dimensional curved thin walled structures", International Journal for Numerical Methods in Engineering, 52, 673-703, 2001.doi:10.1002/nme.222

[3]

B. Szabó, A. Düster, E. Rank, "The p-version of the Finite Element Method", in E. Stein, R. de Borst, T.J.R. Hughes, (Editors), Encyclopedia of Computational Mechanics, John Wiley & Sons, Vol. 1, 119-139, 2004. doi:10.1002/0470091355.ecm003g

[4]

S. Kollmannsberger, S. Geller, A. Düster, J. Tölke, C. Sorger, M. Krafczyk, E. Rank, "Fixed-grid Fluid-Structure interaction in two dimensions based on a partitioned Lattice Boltzmann and p-FEM approach", International Journal for Numerical Methods in Engineering, 79, 817-845, 2009. doi:10.1002/nme.2581

[5]

Z. Wassouf, "Die Mortar Methode für Finite Elemente hoher Ordung", Doctoral Thesis, Technische Universität München, Computation in Engineering.

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