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Freitas, J. A. T., and C. Cismaşiu. "Developments with hybrid-{T}refftz stress and displacement elements." Computer Assisted Mechanics and Engineering Sciences. 8 (2001): 289-311. Abstract

The paper reports on the work on hybrid-{T}refftz finite elements developed by the Structural Analysis Research Group, ICIST, Technical University of Lisbon. A dynamic elastoplastic problem is used to describe the technique used to establish the alternative stress and displacement models of the hybrid-{T}refftz finite element formulations. They are derived using independent time, space and finite element bases, so that the resulting solving systems are symmetric, sparse, naturally $p$-adaptive and particularly well suited to parallel processing. The performance of the hybrid-{T}refftz stress and displacement models is illustrated with a number of representative static and dynamic applications of elastic and elastoplastic structural problems.

Cismaşiu, C. The hybrid-{T}refftz Displacement Element for Static and Dynamic Structural Analysis Problems. Lisboa, Portugal: Instituto Superior Técnico, 2000. Abstract

The displacement model of the hybrid-{T}refftz finite element formulation is applied to the solution of geometrically and physically linear static and dynamic problems. As the approximation bases solve locally the governing system of differential equations, the errors in the approximation affect only the implementation of the boundary conditions. Potential and elastostatic problems are used to illustrate the enforcement of the boundary conditions and the convergence of the solutions in energy, stresses and displacements, under both p- and h-refinement sequences and their insensitivity to mesh distortion, incompressibility and positioning of the coordinate system of the approximation basis. Also illustrated is the use of elements with arbitrary geometry and the efficiency that can be reached by including in the bases the solutions associated with dominant local effects, in particular those associated with singular stress fields. An adaptive p-refinement algorithm that exploits the naturally hierarchical nature of the approximation bases is presented and assessed. The formulation is generalised for elastodynamic analysis in the frequency domain of both bounded and unbounded domains, which are modelled either with absorbing boundary conditions or with semi-infinite elements that satisfy the Sommerfeld condition. The performance of the formulation is illustrated with tests on the convergence of the solutions in energy, stresses and displacements and on their insensitivity to mesh distortion, wave length and position of the absorbing boundary, for a wide spectrum of forcing frequencies and under both p- and h-refinement sequences.

Freitas, J. A. T., C. Cismasiu, and Z. M. Wang. "Comparative analysis of hybrid-Trefftz stress and displacement elements." Archives of Computational Methods in Engineering. State of the art reviews. 6.1 (1999): 35-39.
Freitas, J. A. T., and C. Cismaşiu. "Numerical implementation of hybrid-{T}refftz displacement elements." Computers & Structures. 73 (1999): 207-225. Abstract

The numerical implementation of the displacement model of the hybrid-{T}refftz finite element formulation is presented. The geometry of the supporting element is not constrained a priori. Unbounded, non-convex and multiply connected elements can be used. The approximation basis is naturally hierarchical and very rich. It is constructed on polynomial solutions of the governing differential equation, and extended to include the particular terms known to model accurately important local effects, namely the singular stress patterns due to cracks or point loads. Numerical and semi-analytical methods are used to compute the finite element matrices and vectors, all of which present boundary integral expressions. Appropriate procedures to store, manipulate and solve symmetric highly sparse systems are used. The characteristics of the finite element solving system in terms of sparsity and conditioning are analysed, as well as its sensitivity to the effects of mesh distortion, incompressibility and rotation of the local reference systems. Benchmark tests are used also to illustrate the performance of the element in the estimation of displacements, stresses and stress intensity factors.