Underlying microstructural morphology and physical phenomena have oftentimes a strong effect on an apparent behaviour and mechanical properties of a considered material at a much larger (engineering) scale. This especially holds true in the regime of nonlinear microstructural deformations, plasticity, or damage, when effective properties significantly depend on changes in the deformation state of the underlying microstructure. For practical engineering applications, the specimen length scale is usually much larger than that of the typical microstructural features, making it difficult to use a fully resolved model capturing both scales at the same time. This calls for efficient computational tools, within which first- and second-order computational homogenization are well known and established. These methods allow, under certain assumptions, to transfer the key information from the microscale to the macroscale through homogenized quantities, by solving microstructural response on a Representative Volume Element (RVE), to establish an effective homogeneous continuum. This continuum may, in turn, be discretized at the macroscopic level and solved in a computationally efficient way. Such two-scale simulations may, however, still be computationally involved mainly when considered RVEs include a significant number of degrees of freedom and features.
The goal of this talk is to address this issue by introducing Reduced Basis Method  and Empirical Cubature Method  to build a Reduced Order Model (ROM) for the microscopic problem solved within RVEs. This ROM methodology is designed to treat geometrical parametrization of the microstructure, which allows for two-scale shape optimization of the microstructure to yield optimal performance at a higher scale. During this talk, overall theory will be first introduced in the case of first-order computational homogenization. Several examples will be considered to demonstrate the methodology and its performance. Extensions and preliminary results towards second-order computational homogenization will be considered as well.
 Quarteroni,A.,Manzoni,A.,Negri,F.(2015). Reduced basis methods for partial differential equations: an introduction (Vol. 92). Springer.
 Hernandez, J. A., Caicedo, M. A., Ferrer, A. (2017). Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Computer Methods in Applied Mechanics and Engineering, 313, 687-722.
17.05.2023, 14.45 - 15.30, Room B-366 @ Thákurova 7, 166 29 Prague 6