The elastodynamic response of saturated poroelastic media is modelled approximating independently the solid and seepage displacements in the domain and the force and pressure components on the boundary of the element. The domain and boundary approximation bases are used to enforce on average the dynamic equilibrium and the displacement continuity conditions, respectively. The resulting solving system is Hermitian, except for the damping term, and its coefficients are defined by boundary integral expressions as a Trefftz basis is used to set up the domain approximation. This basis is taken from the solution set of the governing differential equation and models the free-field elastodynamic response of the medium. This option justifies the relatively high levels of performance that are illustrated with the time domain analysis of unbounded domains.
The hybrid-{T}refftz displacement element is applied to the elastodynamic analysis of bounded and unbounded media in the frequency domain. The displacements are approximated in the domain of the element using local solutions of the wave equation, the Neumann conditions are enforced directly and the surface forces are approximated on the Dirichlet and inter-element boundaries of the finite element mesh. Two alternative elements are developed to model unbounded media, namely a finite element with absorbing boundaries and an unbounded element that satisfies explicitly the Sommerfeld condition. The finite element equations are derived from the fundamental relations of elastodynamics written in the frequency domain. The numerical implementation of these equations is discussed and numerical tests are presented to assess the performance of the formulation.
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.
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.
An adaptive p-refinement procedure for the implementation of the displacement model of the hybrid-{T}refftz finite element formulation is presented. The procedure is designed to select and implement automatically the degrees of freedom in the domain (displacements) and on the boundary (surface forces) of the element to attain a prescribed level of accuracy. This accuracy is measured on the strain energy of the system for a prescribed finite element mesh. Local measures of error can be easily accounted for. The performance of the adaptive procedure suggested is illustrated using two-dimensional potential problems.
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