Modelling error estimation and adaptive modelling of perforated materials

Traditional computational analyses of perforated materials, much like traditional computational analyses of heterogeneous materials, involve the use of homogenization, using either the assumption of periodicity or the assumption of statistical homogeneity. Recently, Oden and Vemaganti (Journal of Computational Physics 2000; 164:22–47) have proposed a goal-oriented approach to the modelling of heterogeneous multi-phase linearly elastic materials that does not rely on such idealized situations. In this approach, the mathematical model is adaptively selected based on estimates of the modelling error: the error resulting from the smearing out of rapidly varying material moduli. The approach is said to be goal-oriented because the adaptive process is driven by local estimates of modelling error in quantities of interest to the analyst, instead of global estimates. We extend this goal-oriented adaptive approach to the case of perforated domains composed of linearly elastic materials. Toward this end, new global and local bounds on modelling errors resulting from homogenization of perforated materials are developed. A representative numerical experiment is presented