Hyper-reduction in nonlinear mechanics of solid materials
Talk by Prof. Dr. David Ryckelynck
Speaker invited by: Prof. Dr. Rémi Abgrall
Date: 10.12.14 Time: 16.15 - 17.15 Room: Y27H25
Abstract: Reduced-order models reveal common features between solutions of parametric partial differential equations (PDE). They aim to save computational time when modelling complex mechanical phenomena, as setting-up convenient nonlinear constitutive equations or boundary conditions for instance. The Garlerkin weak form of nonlinear reduced equations do not provide sufficient speed-up, mainly because the computation of the reduced-residual can’t be performed offline. Hence, this computation remains affected by the complexity of the original model. Hyper-reduction methods aim to generate reduced-order model whose complexity does not depend on the complexity of the original model by introducing a reduced integration domain (RID). This domain contains only few elements of the original mesh. We propose both an explicit hyper-reduced scheme and implicit hyper-reduced schemes applied to nonlinear mechanics of solid materials. The explicit approach aims to interpolate missing boundary conditions on the boundary of the RID. The implicit approach predicts the reduced coordinate of the displacement field by using balance conditions restricted to the RID. The RID is generated offline by aggregating the magic points of various reduced bases, depending on the physics involved in the original model. These magic points depend on the shape of the empirical modes by following the empirical interpolation method. The reduced-bases are obtained by using either: an a posteriori approach such as the snapshot POD method and the derivative extended POD method; or an a priori approach such as the APHR (A priori Hyper Reduction) method. In case of standard materials, an a posteriori error estimation is proposed by following the constitutive-relation-error method. We also develop a snapshotless approach to adapt the hyper-reduced approximation, without additional FE solution of the DPE.