Local Goal-Oriented Estimation of Modeling Error for Multi-Scale Modeling of Heterogeneous Elastic Materials
Albert Romkes, Tristan C. Moody · International Journal for Computational Methods in Engineering Science and Mechanics · 2007
We consider the multi-scale analysis of the mechanics of multiphase composites with complex microstructure, based on the Goal-Oriented Adaptive Modeling method [1 Zohdi, T. I., Oden, J. T. and Rodin, G. J. 1996. Hierarchical Modeling of Heterogeneous Bodies. Comput. Methods Appl. Mech. Eng., 138: 273–298. [Crossref], [Web of Science ®] , [Google Scholar], 2 Oden, J. T. and Zohdi, T. I. 1997. Analysis and Adaptive Modeling of Highly Heterogeneous Elastic Structures. Computer Methods in Applied Mechanics and Engineering, 148: 367–391. [Crossref], [Web of Science ®] , [Google Scholar], 3 Oden, J. T., Vemaganti, K. and Moës, N. 1998. Hierarchical Modeling of Heterogeneous Solids. Computer Methods in Applied Mechanics and Engineering, 172: 3–25. [Crossref], [Web of Science ®] , [Google Scholar], 4 Oden, J. T. and Vemaganti, K. S. 2000. Adaptive Modeling of Composite Structures: Modeling Error Estimation. Int. J. Comput. Civil Str. Eng., 1: 1–16. [Google Scholar], 5 Oden, J. T. and Vemaganti, K. 2000. Estimation of Local Modeling Error and Goal-Oriented Adaptive Modeling of Heterogeneous Materials. Part I: Error Estimates and Adaptive Algorithms. Comp. J. Phys., 164: 22–47. [Crossref], [Web of Science ®] , [Google Scholar], 6 Vemaganti, K. and Oden, J. T. 2001. Estimation of Local Modeling Error and Goal-Oriented Adaptive Modeling of Heterogeneous Materials. Part II: A Computational Environment for Adaptive Modeling of Heterogeneous Elastic Solids. Comput. Methods Appl. Mech. Eng., 190: 6089–6124. [Crossref], [Web of Science ®] , [Google Scholar], 7 Romkes, A. and Oden, J. T. 2004. Adaptive Modeling of Wave Propagation in Heterogeneous Elastic Solids. Comput. Methods Appl. Mech. Eng., 193: 539–559. [Crossref], [Web of Science ®] , [Google Scholar], 8 Romkes, A., Oden, J. T. and Vemaganti, K. 2006. Multi-scale Goal-Oriented Adaptive Modeling of Random Heterogeneous Materials. Mechanics of Materials, 38: 859–872. [Crossref], [Web of Science ®] , [Google Scholar], 9 Oden, J. T., Prudhomme, S., Romkes, A. and Bauman, P. 2006. Multi-scale Modeling of Physical Phenomena: Adaptive Control of Models. SIAM Journal of Scientific Computing, 28: 2359–2389. [Crossref], [Web of Science ®] , [Google Scholar], 10 Oden, J. T., Prudhomme, S., Romkes, A. and Bauman, P. 2005. Multi-scale Modeling of Physical Phenomena: Adaptive Control of Models, The University of Texas at Austin. ICES Report 05-13 and 05-20 [Google Scholar]]. The underlying approach of this method is to perform an initial analysis using homogenized, or effective, properties and sequentially improve these by adding only enough of the actual microstructure to control the modeling error in a user-specified quantity of interest. The quantification of the modeling error is established by providing residual-based a posteriori error estimates [11 Oden, J. T. and Prudhomme, S. 2002. Estimation of Modeling Error in Computational Mechanics. Comp J. Phys., 182: 496–515. [Crossref], [Web of Science ®] , [Google Scholar]]. However, this involves solving an additional global dual problem and computing global integrals of governing residual functionals. In the case of multiphase composite materials this estimation process can be computationally prohibitive. We therefore propose a technique for local, a posteriori estimation of the modeling error. It requires solving a local dual problem, of computationally small size, and computing local residual integrals. We introduce this new approach for the analysis of linear elastostatics problems of multiphase composites and show two-dimensional numerical verifications.