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Reduced order computational continua. (English) Zbl 1253.74085
Summary: We present a new multiscale approach, hereafter referred to as reduced order computational continua $$(RC^{2})$$, that possesses computational efficiency of phenomenological models for heterogeneous media with accuracy inherent to generalized and nonlocal continua models. The $$RC^{2}$$ approach introduces no scale separation, makes no assumption about infinitesimality of the fine-scale structure, does not require higher order continuity, introduces no new degrees-of-freedom, is free of higher order boundary conditions and exploits a pre-computed material database to effectively solve a unit cell (representative volume) problem. It features three building blocks: (i) the nonlocal quadrature scheme, (ii) the coarse-scale stress function and (iii) the residual-free fields. The nonlocal quadrature scheme permits nonlocal interactions to extend over finite neighborhoods and thus introduces nonlocality into the two-scale integrals employed in the multiple-scale asymptotic expansion methods, or alternatively, into the Hill-Mandel macrohomogeneity condition. The coarse-scale stress function, which replaces the classical notion of coarse-scale stress being the average of fine-scale stresses, is constructed to express the governing equations in terms of coarse-scale fields only. Finally, the residual-free fields are constructed to avoid costly discrete equilibrium solution of the unit cell problems, which is known to be the bottleneck of multiscale computations.

##### MSC:
 74Q05 Homogenization in equilibrium problems of solid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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