Corneliu Cismasiu

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 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.