Toward Optimality of Proper Generalised Decomposition Bases

Abstract

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram–Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method.

On the Benefits of a Multiscale Domain Decomposition Method to Model-Order Reduction for Frictional Contact Problems

“…This problem can be palliated, for example, by systematically performing a Gram-Schmidt orthonormalization for the space modes. However, even after orthonormalization of space modes, redundancy may still occur on time modes [58]. An approach that can be adopted is to perform a full SVD computation of the solution after each enrichment step, and to keep the most significant modes as basis for the next iteration.…”

“…An approach that can be adopted is to perform a full SVD computation of the solution after each enrichment step, and to keep the most significant modes as basis for the next iteration. Less expensive methods could make use of SVD updating techniques [59,60] or randomized SVD [58]. However, these methods aim at computing precisely the SVD of the solution throughout the iterations of the LATIN, with an effort which may be not worthwhile knowing that the solution may be far from convergence.…”

PGD reduced-order modeling for structural dynamics applications

“…One idea to improve the convergence of the PGD approximation is to update the temporal modes in a global manner [4,33,37,38]. In other words, after a new mode is found, the updating algorithm reevaluates the modes (λ k ) 1 k m in order to obtain a better combination of the spatial modes (µ k ) 1 k m .…”

“…In other words, after a new mode is found, the updating algorithm reevaluates the modes (λ k ) 1 k m in order to obtain a better combination of the spatial modes (µ k ) 1 k m . This procedure will significantly improve the convergence of the PGD at a fairly low computational cost, which only depends on N t and m [33]. However, this procedure requires the computation of some matrices that can become ill-conditioned with the increase of the number of modes.…”

“…Only stable, energy conservative discretization schemes are considered here, namely the Crank-Nicolson method [29,30,31] (also known as the implicit trapezoidal rule), and the Newmark method [32] with γ = 1/2 and β = 1/4. Moreover, we apply two post-processing procedures that aim at improving the convergence of the Galerkin-based version of the PGD [4,33], namely 1) the orthogonalization of the spatial modes via a modified Gram-Schmidt algorithm, and 2) the update procedure of the temporal modes. These procedures are applied to both the L-PGD and H-PGD.…”