On the application of high-performance model reduction techniques to homogenization of heterogeneous materials

J. HERNANDEZ; J. OLIVER,; A E. HUESPE; M. CAICEDO; J.C.CANTE

Barcelona

Congreso; COMPLAS XII, Int. Conf. on Comp. Plasticity, Fundamentals and Applic.,; 2013

The present work deals with the fast solution of the fine-scale boundary value problem (BVP)arising in homogenization of heterogeneous materials such as composites and polycristalline metals.We propose to solve such a BVP using a high-performance, model reduction strategy. The mainingredients of this strategy are: 1) A Galerkin approximation that uses a low-dimensional set ofglobally-supported, displacement basis functions, computed offline from a larger set of finite element(FE) solutions. 2) An interpolatory integration scheme, constructed by replacing the stress tensor inthe weak formulation of the BVP by a low-dimensional interpolant, and that therefore only requiresevaluation of stresses at a few, strategically pre-selected FE gauss points. We demonstrate that, inthe case of the cell BVP, the strategy widely adopted in the reduced-order modeling literature ofdetermining the low-dimensional space in which the interpolant of the nonaffine term resides onlyfrom snapshots of such a term evaluated at the solution leads invariably to ill-posed, discreteformulations. The main contribution of the present work is a method that overcomes this type of ill-posednessby introducing a hybrid interpolant, constructed using the dominant modes obtained fromstress FE snapshots together with the gradient of the displacement basis functions. Theperformance of the developed method is critically assessed for two types of heterogeneous material(a composite and a porous material), both displaying highly complex, statistically homogeneousmicro-structures. Results obtained exhibits gains in performance with respect to classical finiteelement analyses of above three orders of magnitude.