Model reduction methods for dynamic analyses of large structures

A Generic Bilevel Formalism for Unifying and Extending Model Reduction Methods

“…For first or second order (J = 1 or 2) cases, this resembles the method of Häggblad and Eriksson [29]. The span of the columns of $ f /, for $ f i given by (4.27), will later be shown to reduce to a conventional Krylov subspace under special circumstances.…”

“…The Lanczos method is usually discussed in terms of a second-order formulation, but it also has a first-order formulation [26] for nonproportional damping, as well as block [27], modal-hybrid [28], and other variants. Noting the analogy of their method to those based on Krylov subspaces (in particular, those of Arnoldi and Lanczos), Häggblad and Eriksson [29] recognized that efficient, "generalized Krylov" subspace descriptions can be generated as recursive relations from series solutions of the governing equa-tions. They take a physical approach and derive advanced methods of this type from series in which "each component is successively computed by balancing the inertia forces from the previous component in the series."…”

A Static condensation Reduced Basis Element method : approximation anda posteriorierror estimation

“…First, it is clear that if we consider "quasi-one-dimensional" systems -components with significant internal complexity but simple port structure [4] -then we may arguably choose m COM small in our truth discretization. 3 We can be more rigorous: in each reference component we prescribe m COM large as always (which implies associated n G-P large) -in order to err on the good side of the truth; we then retain (Online) for any system port G-P ∈ P SYS only the n G-P < n G-P basis functions Φ k,G-P (µ), 1 ≤ k ≤ n G-P , associated with the lowest modes -in effect, a "port-reduced" space X PSYS ⊂ X PSYS ; we construct a port-reduced stiffness matrix and load vector, A and F , respectively, in terms of the retained modes -the higher modes are removed; we obtain a port-reduced U from (44) with A and F replaced by A and F , respectively; we find a port-reducedλ min from the definition (37) with A replaced by A ; we bound the error due to port reduction in U by our error bound (73) (note the residual term will now be non-zero), and the error due to port reduction inλ min by the eigenproblem residual norm [15,18]. We thus save significantly in Online 2 since even for the error bounds (residual evaluation) we need only compute the columns of A associated with the retained modes.…”

“…In short, our approach provides greater flexibility in the inexpensive Online stage: interchangeability of components is extended to include parametric variations which arise in geometry, constitutive laws, and sources and loads. Furthermore, in the static condensation RBE approach, a posteriori error bounds for the RB approximations at the component level permit us to develop a posteriori error bounds at the system level without recourse to the truth FE residual over the full domain [15]. We should note that in this paper we consider only elliptic coercive partial differential equations and not the more difficult eigenproblems or dynamic problems to which CMS approaches are typically applied.…”

“…As in the static condensation RBE method, the CMS approach combines static condensation at the interdomain level with model order reduction at the intradomain level. In the earlier work [10,16] the CMS model order reduction is typically of the intradomain eigenfunction modal truncation variety, however more recently Krylov spaces are also considered [15].…”