| A formal methodology for reducing the size of models used in the analysis of elastic systems is presented. This involves an explicit representation of the model at various resolutions and is accomplished using a projection generated numerical homogenization procedure. The framework for this analysis is derived from the multi-resolution analysis associated with the construction of wavelet bases. This is applied to elasticity operators to average fine scale properties and behavior while limiting loss of information. In discretized form, the method produces equivalent smaller stiffness matrices that can be used as building blocks (super-elements) to construct reduced models of larger systems. The principal application envisioned is in design problems involving complex structural systems, such as in crash-worthiness design, where very intensive computations demand computational efficiency. This model reduction scheme is applied to the problem of layout optimization of structures involving multi-scale heterogeneities of sizes that may be comparable to the size of the structure. Numerical examples in which the heterogeneities are in the form of perforations of various sizes are presented. |
