Discussion paper: Has renaming the high fidelity generalized method of cells been justified?

Abstract

The high fidelity generalized method of cells (HFGMC) has been originally developed by Aboudi (2001) and Aboudi et al. (2001) as a micromechanical method for periodic multi-phase composite media. A computational implementation of the HFGMC equations has been proposed by Bansal and Pindera (2004) to enhance numerical efficiency, still with direct reference to the HFGMC formulation. Later, the same computational implementation is recast as a new method called “finite volume direct averaging micromechanics” (FVDAM), starting by Bansal and Pindera (2006). The current discussion paper has two aims. The first is to show that the FVDAM is not a new method and that it has the same assumptions and identical governing equations as those originally derived by the HFGMC. The only difference is in the solution procedure where intermediate dependent variables, in the form of average displacements at the interfaces, are used instead of directly solving for the unknown micro-variables; the coefficients of the displacement polynomials. Thus, renaming the HFGMC micromodel to FVDAM has not been justified. In fact, (Haj-Ali and Aboudi, 2009) have shown that the same reduction of variables can be achieved by a simple static condensation carried out at the global system of equations instead of introducing intermediate variables. The second aim of this paper is to address misrepresentations in a recent discussion paper by the FVDAM authors claiming, in part, that the HFGMC method using parametric geometry of the subcells should follow their formulation (termed parametric FVDAM). We show that the latter is limited to an incomplete quadratic expansion of the displacement and an approximation in the form of a priori constant Jacobian of the parametric mapping. However, the HFGMC with arbitrary cell geometry, (Haj-Ali and Aboudi, 2010), has been formulated in a direct and general manner, i.e. retaining the full quadratic expansion of the displacement together with the complete Jacobian. Thus, the parametric FVDAM is a special case of the parametric HFGMC, i.e. when the Jacobian is sampled and evaluated only at one point, namely the origin of the parametric coordinate system. The intended new contribution of Haj-Ali and Aboudi (2010) to refined micromechanics and progressive damage has been completely ignored by the FVDAM-discussion paper. Therefore, in order to maintain scientific clarity, it is strongly advocated to preserve the original name of the HFGMC method, regardless of the different computational implementations used for solving the governing equations for both orthogonal and parametric geometries of the subcells.