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A symmetric fully optimized second‐order method for nonlinear homogenization
Author(s) -
Furer Joshua,
Ponte Castañeda Pedro
Publication year - 2018
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201700065
Subject(s) - homogenization (climate) , nonlinear system , legendre polynomials , composite number , mathematics , mathematical analysis , mathematical optimization , physics , algorithm , biodiversity , ecology , quantum mechanics , biology
We consider an alternate formulation of the recently developed ‘fully optimized second‐order’ (FOSO) nonlinear homogenization method, which is based on a stationary variational principle for the macroscopic energy function. In this method, the trial fields are the properties of a suitably designed linear comparison composite (LCC), thus allowing for the estimation of the effective response and field statistics of the nonlinear composite in terms of available estimates for the corresponding quantities in the LCC. The formulation considered in this paper makes use of an alternative choice for the linear comparison composite, leading to homogenization estimates that are more symmetric with respect to Legendre duality than earlier FOSO estimates. The new ‘symmetric’ FOSO method is applied to a class of two‐phase power‐law composites with fibrous microstructures subjected to plane strain loading. The resulting estimates for the effective response and field statistics are found to improve on earlier estimates, and to be in good agreement with full‐field numerical simulations for nonlinear composite cylinder assemblages, as well as with available results for sequentially layered composites.