Hyper-reduction of generalized continua.

*(English)*Zbl 1398.74337Summary: This paper deals with the reduced order modeling of micromorphic continua. The reduced basis model relies on the proper orthogonal decomposition and the hyper-reduction. Two variants of creation of reduced bases using the proper orthogonal decomposition are explored from the perspective of additional micromorphic degrees of freedom. In the first approach, one snapshot matrix including displacement as well as micromorphic degrees of freedom is assembled. In the second approach, snapshots matrices are assembled separately for displacement and micromorphic fields and the singular value decomposition is performed on each system separately. Thereafter, the formulation is extended to the hyper-reduction method. It is shown that the formulation has the same structure as for the classical continua. The relation of higher order stresses introduced in micromorphic balance equations to creation of the reduced integration domain is examined. Finally, the method is applied to examples of microdilatation
extension and clamped tension and to a size-dependent stress concentration in Cosserat elasticity. It is shown that the proposed approach leads to a good level of accuracy with significant reduction of computational time.

##### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74A60 | Micromechanical theories |

74A99 | Generalities, axiomatics, foundations of continuum mechanics of solids |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

##### Keywords:

reduced order modeling; POD; hyper-reduction; reduced integration domain; micromorphic continua; generalized continua; Cosserat continuum##### Software:

OOFEM
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\textit{M. Horák} et al., Comput. Mech. 59, No. 5, 753--778 (2017; Zbl 1398.74337)

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