Geometrically or physically non-linear problems are often characterized by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to ine ciency of standard numerical solution techniques. Special numerical procedures are therefore required to pass critical points. This paper presents a solution technique which is based on a constraint equation that is deÿned on a subplane of the degrees-of-freedom (dof 's) hyperspace or a hyperspace constructed from speciÿc functions of the degrees-of-freedom. This uniÿed approach includes many existing methods which have been proposed by various authors. The entire computational process is driven from only one control function which is either a function of a number of degrees-of-freedom (local subplane method) or a single automatically weighted function that incorporates all dof 's directly or indirectly (weighted subplane method). The control function is generally computed in many points of the structure, which can be related to the ÿnite element discretization. Each point corresponds to one subplane. In the local subplane method, the subplane with the control function that drives the load adaptation is selected automatically during the deformation process. Part I of this two-part series of papers fully elaborates the proposed solution strategy, including a fully automatic load control, i.e. load estimation, adaptation and correction. Part II presents a comparative analysis in which several choices for the control function in the subplane method are confronted with classical update algorithms. The comparison is carried out by means of a number of geometrically and physically non-linear examples. General conclusions are drawn with respect to the e ciency and applicability of the subplane solution control method for the numerical analysis of engineering problems.

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In this paper, the transient computational homogenization scheme is extended to allow for nonlinear elastodynamic phenomena. The framework is used to analyze wave propagation in a locally resonant metamaterial containing hyperelastic rubber-coated inclusions. The ability to properly simulate realistic nonlinearities in elasto-acoustic metamaterials constitutes a step forward in metamaterial design as, so far, the literature has focused only on academic nonlinear material models and simple lattice structures. The accuracy and efficiency of the framework are assessed by comparing the results with direct numerical simulations for transient dynamic analysis. It is found that the band gap features are adequately captured. The ability of the framework to perform accurate nonlinear transient dynamic analyses of finite-size structures is also demonstrated, along with the significant computational time savings achieved.

In this paper, based on a representative volume element (RVE) and Slattery's averaging theorem, parameters of Biot's poroelastic equations for homogenous isotropic porous materials are obtained. According to Slattery's averaging theorem, the coupling terms, which describe the inertial effects and the viscous effects, are represented by an integral of the solid-fluid interaction force. This relation provides a new approach to obtain the parameters required in Biot's equations through a direct numerical simulation of the RVE. An example of a 2D RVE is given and simulations of sound propagation in an impedance tube with a foam are conducted using Biot's equations. It is shown that the numerical coupling mass obtained from this new approach behaves qualitatively the same as an associated phenomenological model.

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