Keywords: mixed time integration, modal decomposition, non-linear dynamics, Legendre polynomials.

This paper presents and discusses a hybrid-mixed stress finite element model for the dynamic analysis of structures in the time domain. This model is based on a direct and independent approximation of both the stress and the displacement fields in the domain of each finite element. The displacements along the static boundary are also independently modelled. Complete sets of orthonormal Legendre polynomials are used to define all approximations required by this numerical model. The properties of the Legendre polynomials lead to the definition of closed form solutions for the integrations involved in the computation of all structural linear operators.

The time integration procedure introduced by Freitas [1,2] is implemented with the hybrid-mixed stress model. The displacement and the velocity fields are approximated independently in time using hierarchical bases. The time approximation criterion preserves hyperbolicity in the sense that it replaces the solution of hyperbolic problems by the solution of uncoupled elliptic problems, which can be subsequently solved using the alternative methods currently used for static analysis.

The time integration algorithm discussed in this work corresponds to a modal decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion. This is achieved by solving an eigenvalue problem in the time domain that only depends on the approximation basis being implemented. The dimension of this problem is in general very small, as it depends on the time approximation basis being used. Due to its characteristics, this method is well suited to parallel processing and to large time stepping [1,2].

In the numerical model reported in this paper, orthonormal Legendre polynomials are used to define the approximations required by the time integration procedure. To assess the accuracy and the efficiency of the model being discussed, a set of numerical tests is presented. The competitiveness of the proposed approach is illustrated by comparing its performance with the one presented by the classical Newmark method in the analysis of both linear [3] and nonlinear [4] dynamic problems.

1

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

2

J.A.T. Freitas, "Mixed finite element solution of timed-dependent problems", Computer Methods in Applied Mechanics and Engineering, 197, 3657-3678, 2008. doi:10.1016/j.cma.2008.02.014

3

N.M. Newmark, "A method of computation for structural dynamics", J. Engng Mech. Div., ASCE, 85, EM3, 67-94, 1959.

4

I. Miranda, R.M. Ferencz, T.J.R. Hughes, "An improved Implicit-explicit time integration method of structural dynamics", Earthquake Engineering and Structural Dynamics, 18, 643-653, 1989. doi:10.1002/eqe.4290180505

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