Analysis and adaptive modeling of highly heterogeneous elastic structures

In this paper, we develop a theory and methodology for obtaining approximate solutions to boundary value problems describing the deformation of highly heterogeneous linearly-elastic structures. The method, which represents a significant departure from traditional homogenization methods, provides a systematic and rigorous approach towards resolving the effects of microstructure of different scales on the macroscopic response of complex heterogeneous structures. An early variant of this method was first introduced in [10]. The method, referred to as HDPM (Homogenized Dirichlet Projection Method), proceeds by first solving an auxiliary homogenized boundary value problem, describing the deformation of a structure with the same exterior geometry, but with a selected set of uniform material properties. The adequacy of this homogenized solution is then determined using an a posteriori error estimate that provides a measure of the error of the homogenized solution in subdomains of the structure compared to the solution of the fine-scale heterogeneous problem, without specific knowledge of this fine-scale solution. In those subdomains where the homogenized solution is deemed acceptable, it is retained. However, in subdomains where the homogenized solution is inadequate, as is determined when the estimated error exceeds a preset tolerance, a local boundary value problem is constructed by projecting the homogenized displacements onto a partition designed to isolate these subdomains. These local boundary value problems are then solved in those subdomains where the homogenized solution is inaccurate using the exact microstructure with the approximate local boundary conditions. A posteriori error estimates are made to ascertain the quality of the resulting solution. If the quality of the solution remains inadequate, above a preset error tolerance, a two stage adaptive procedure is implemented. Stage-I ('material adaptivity') corresponds to modifying the homogenized structure's material properties. If, after Stage I, the solution quality is still inadequate, the subdomains of local solution are enlarged, thereby modeling in greater detail the actual microstructure, and the local solution process is repeated (Stage-II, 'subdomain unrefinement'). The main feature of this method is that only in subdomains where the error in the usual homogenized solution is above a preset tolerance is the microstructure taken into account. The cost of this method is shown to be orders of magnitude cheaper than direct large-scale computational simulations of micromechanical events. The results of several numerical experiments are provided to demonstrate the method and verify theoretical estimates.