As modelling of incompressibility remains a sensitive issue, a study was launched to assess the analysis of hydrated soft tissue using the alternative stress and displacement models of the Trefftz variant of the hybrid finite element formulation, the latter of which is addressed here [3].

As is the usual practice, this formulation develops from the discretization in time of the parabolic system of differential equations. However, a high-order time integration procedure is used to enhance the application of the Trefftz concept [4]. This spectral-type decomposition method can be applied to both frequency and time domain analyses and leads, in either case, to a system of uncoupled elliptic equations.

This system is discretized next in space domain approximating directly the displacement field in the domain of the element using a naturally hierarchical basis that satisfies the incompressibility condition. This basis and the associated strain field are used to enforce on (Galerkin) average the equilibrium condition and the constitutive relations, respectively.

The solving system is symmetric, sparse and well-suited to parallel processing and adaptive refinement. The Trefftz constraint is enforced next. It consists in further constraining the displacement approximation basis to satisfy locally all domain conditions of the problem. As is typical of methods based on free-field solutions of the governing differential equations, all coefficients of the solving system have boundary integral expressions. However, the main advantage of this variant of the hybrid finite element formulation is the stability and the improved rates of convergence induced by a basis that embodies the physics of the problem.

The performance of the finite element model is illustrated with the unconfined indentation and the confined and unconfined compression tests of cartilage specimens. The boundary conditions are chosen to model one- and two-dimensional responses, as well as creep and stress relaxation processes.

1

V.C. Mow, S.C. Kuei, W.M. Lai, C.G. Armstrong, "Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments", J Biomechanics Engineering, 102, 73-83, 1980. doi:10.1115/1.3138202

2

E.S. Almeida, R.L. Spilker, "Mixed and penalty finite element models for the nonlinear behaviour of biphasic soft tissues in finite deformation: Part I - Alternative formulations", CMBBE, 1, 25-46, 1997. doi:10.1080/01495739708936693

3

J.A.T. Freitas, "Hybrid finite element formulations for elastodynamic analysis in the frequency domain", Int J Solids and Structures, 36, 1883-1923, 1999. doi:10.1016/S0020-7683(98)00064-X

4

J.A.T. Freitas, "Mixed finite element formulation for the solution of parabolic problems", Computer Methods in Applied Mechanics and Engineering, 191, 3425-3457, 2002. doi:10.1016/S0045-7825(02)00244-X

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