Toward Optimality of Proper Generalised Decomposition Bases

Shadi Alameddin,Udo Nackenhorst,Amélie Fau,David Néron,Pierre Ladevèze

doi:10.3390/mca24010030

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.

Highlights

Numerical simulations appeal as an attractive augmentation to experiments to design and analyse mechanical structures
These techniques and their effect on the Proper Generalized Decomposition (PGD) greedy algorithm are illustrated throughout examples with a varying number of degrees of freedom
It is found that a randomised singular value decomposition (SVD) algorithm is a promising scheme to ensure the optimality of PGD expansions

Summary

Introduction

Numerical simulations appeal as an attractive augmentation to experiments to design and analyse mechanical structures. A posteriori model reduction techniques such as the Proper Orthogonal Decomposition (POD) is based on an offline training computations which extract a reduced order basis (ROB) from the solution of a high fidelity model. This optimal basis is practically built through a singular value decomposition (SVD) of a snapshot matrix. An issue may be caused by the rapid growth of the ROB basis, whereas the primary interest of MOR is to benefit from a small sized ROB which provides a nondemanding temporal updating step This step is equivalent to a POD step where the spatial modes are fixed and only the temporal ones are updated. In [5], some advanced strategies have been proposed to use an optimal parametric path allowing for controlling the basis expansion optimally

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