Investigation of the interactions between the numerical and the modeling errors in the Homogenized Dirichlet Projection Method

Nicolas Moës (1) , John Tinsley Oden (1) , Tarek Zohdi (1)

1 University of Texas at Austin [Austin]

The Homogenized Dirichlet Projection Method, HDPM, was developed in [1,2] as a systematic technique for analyzing highly-heterogeneous elastic structures. The method can provide an analysis of structures composed of composite materials with very complex microstructure at a fraction of the cost of solving the full fine-scale model. In the present investigation, the HDPM is revisited to take into account the unavoidable numerical errors produced in finite element approximations of the associated boundary value problems. The total error of the HDPM, which now takes into account both for the modeling and numerical errors, is split into several terms, each accounting for a parameter in the method. The parameters are: the choice of the homogenized material property, the partition into subdomains, the coarse finite element mesh used to solve the homogenized problem and the fine meshes used to solve the subdomain problems. Numerical experiments are carried out on 1-D problems for which the exact solutions are easily calculated. The experiments reveal that the influence of the coarse and fine meshes are very different. When free of numerical error the HDPM is based on four main results. We rewrite these in the framework of the error in the constitutive law. This leads to a clear mechanical interpretation of the results. Moreover, it allows us to extend the results to nonlinear constitutive laws and to find new properties of the HDPM. Some of the theoretical results are validated for specific cases involving known numerical error. Finally, explicit a posteriori upper bounds are derived for the total and numerical error. A simple adaptive strategy is presented for choosing the fine mesh and the subdomain partition. The strategy is tested on a 1-D model problem.

Domaines: Mécanique [physics.med-ph]; Matériaux

Nicolas Moës, John Tinsley Oden, Tarek Zohdi. Investigation of the interactions between the numerical and the modeling errors in the Homogenized Dirichlet Projection Method. Computer Methods in Applied Mechanics and Engineering, 1998, 159 (1-2), pp.79-101. ⟨10.1016/S0045-7825(98)80104-7⟩. ⟨hal-01007001⟩