Hybrid-Trefftz stress and displacement elements for axisymmetric incompressible biphasic media J.A. Teixeira de Freitas * , M. Toma Departamento de Engenharia Civil e Arquitectura, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal article info Article history: Received 16 March 2008 Received in revised form 15 February 2009 Accepted 18 February 2009 Available online 25 February 2009 Keywords: Trefftz elements Incompressibility Spectral analysis Time domain analysis Saturated porous media abstract The stress and displacement models of the hybrid-Trefftz finite element formulation are applied to the solution of incompressible axisymmetric saturated porous media. The use of a Trefftz approximation basis ensures that all domain conditions of the problem are satisfied in a strong form, namely the equi- librium, the constitutive and the strain–displacement relations and the mixture incompressibility condi- tion. The alternative stress and displacement models are fully complementary in terms of approximation criteria. The stress (displacement) model is derived from the direct approximation of the stress and pres- sure fields (the displacements in the solid and fluid phases) in the domain of the element. The displace- ments of the solid phase and the normal displacement of the fluid phase are approximated independently on the boundary of the stress element and used to enforce in a weak form the inter-element and bound- ary equilibrium conditions on the forces in the solid phase and on the fluid pressure. The boundary approximation used in the displacement element is on the solid surface forces and the fluid pressure, and is used to enforce on average the inter-element and boundary displacement continuity conditions. The resulting finite element governing systems are sparse, well-suited to adaptive refinement and paral- lel processing, and their coefficients are defined by boundary integral expressions. The energy statements associated with the formulation are recovered and sufficient conditions for the uniqueness of the finite element solutions are stated. Benchmark tests on hydrated soft tissue modelling are used to assess the performance of the alternative stress and displacement models of the hybrid-Trefftz formulation. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction This paper closes the report on a study on the modelling of the response of hydrated soft tissues with hybrid-Trefftz finite ele- ments [1,2]. It extends to axisymmetric problems the formulation and implementation of the elements developed for the analysis of two-dimensional problems [3,5,6], and is used to establish a di- rect comparative analysis of the alternative stress and displace- ment models. The parabolic model proposed in Mow et al. [7] for incompress- ible biphasic media is adopted here. This model is first discretized in the time dimension in a format that can accommodate alterna- tive time integration procedures, namely the trapezoidal rules fre- quently used in the finite element modelling of the response of soft tissues, as illustrated in the work reported by Spilker and his co- workers, e.g., [8] and the spectral decomposition method that is used here to assess the response of the hybrid-Trefftz elements in both frequency and time domains [9]. The resulting boundary value problem is discretized next using a technique in every aspect similar to that applied in the develop- ment of hybrid and hybrid-Trefftz elements for two-dimensional problems. Thus, and to avoid unnecessary duplication, the finite element equations for the alternative displacement and stress models of the hybrid formulation are stated and their specific roles are recalled. The hybrid-Trefftz variants are obtained by constrain- ing the approximation bases to the solution set of the governing system of differential equations, the distinguishing feature of the Trefftz approach [10]. The first part of the paper closes with the presentation of the finite element governing systems. The main aspects of its numerical implementation are briefly recalled. The associated energy statements and the sufficient conditions that ensure the uniqueness of the finite element solutions presented for two- dimensional problems are here adapted to axisymmetric problems. The second part of the paper addresses the assessment of the performance of the hybrid-Trefftz elements when applied to the frequency and time domain analyses. The results that are pre- sented show that the Trefftz elements for hydrated soft tissue modelling preserve the high performance features that have been consistently reported on their application to progressively wider classes of modelling problems since the pioneering work of Jirou- sek and Leon [11,12]. 0045-7825/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2009.02.023 * Corresponding author. Tel.: +351 218418236. E-mail address: freitas@civil.ist.utl.pt (J.A.T. de Freitas). Comput. Methods Appl. Mech. Engrg. 198 (2009) 2368–2390 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma