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PROCEEDINGS OF THE FIFTEENTH UK CONFERENCE OF THE ASSOCIATION OF COMPUTATIONAL MECHANICS IN ENGINEERING
Edited by: B.H.V. Topping
Mixed and Hybrid Stress Elements for Biphasic Media
J.A. Teixeira De Freitas, I.D. Moldovan and M. Toma
Department of Civil Engineering and Architecture, Instituto Superior Técnico, Technical University of Lisbon, Portugal
J.A. Teixeira De Freitas, I.D. Moldovan, M. Toma, "Mixed and Hybrid Stress Elements for Biphasic Media", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 4, 2007. doi:10.4203/ccp.85.4
Keywords: saturated porous media, incompressible soft tissues, hybrid elements.
The stress model of the hybrid-mixed, hybrid and hybrid-Trefftz finite element formulations is applied to the analysis of biphasic media. The objective is to relate the alternative finite element formulations of the dynamic response of mixtures and to address the modelling difficulties reported in the literature on incompressible soft tissues and unbounded saturated porous media.
The governing equations are first discretized in the time dimension using a finite element approach designed to replace the parabolic/hyperbolic problem by an equivalent elliptic problem, which can be implemented either in the frequency domain or in the time domain. The time domain applications are implemented on a wavelet approximation basis.
The resulting elliptic problem is discretized using first a mixed finite element formulation. It is based on the direct approximation of the stress and pressure fields in the domain of the element and of the displacements in the solid and fluid phases on the boundary of the element. The hybrid formulation is derived by particularization, by constraining the domain approximation to satisfy locally the equilibrium condition independently in the solid and in the fluid phases. The hybrid-Trefftz variant is stated next by constraining further the domain approximation basis to satisfy also the compatibility condition and the constitutive relations of the mixture. As the alternative formulations are derived from first principles, the associated energy statements are recalled and sufficient conditions for uniqueness of the finite element solutions are stated.
The paper closes with the illustration of the application of the least known formulation, the hybrid-Trefftz formulation, to well established test problems. The main purpose is to illustrate how the hybrid-Trefftz stress element performs in terms of sensitivity to mesh distortion and mesh sensitivity to wavelength, under full incompressibility conditions and in the modelling of both near- and far-field responses of unbounded media.
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