Bridging proper orthogonal decomposition methods and augmented Newton–Krylov algorithms: An adaptive model order reduction for highly nonlinear mechanical problems

Abstract

This article describes a bridge between POD-based model order reduction techniques and the classical Newton/Krylov solvers. This bridge is used to derive an efficient algorithm to correct, "on-the-fly", the reduced order modelling of highly nonlinear problems undergoing strong topological changes. Damage initiation problems are addressed and tackle via a corrected hyperreduction method. It is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm, even when strong topological changes are involved.

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

“…We believe that the hybrid SmXFEM/LSM developed here is an effective numerical tool for nano-scale material analysis and designs, and can be used to study nano-scale inhomogeneous materials and structures with complex morphological changes. Future works includes extension to realistic three-dimensional cases and the coupling with model reduction techniques for highly non-linear problems [58].…”

A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics

“…In fact, the solution to parametric problems involving the evolution of topological changes cannot, in general, be obtained efficiently using a method based on the separation of variables (unless one manages to map the physical space to a reference space were correlation in the data can be retrieved [26]). One systematic way to circumvent the problem would be to enrich the snapshot "online" [25,56], but this leads to difficulties related to the cost of evaluating the projection error.…”

“…Europe PMC Funders Author Manuscripts (25) or in the (Petrov-) Galerkin framework (26) The strategies proposed in the literature for this two subset of techniques differ in the way of building operator , which requires a critical trade-off between optimality, stability and tractability. In [34], is constructed such that the condition number of operator is minimised.…”

“…Nonlinear problem (7) can be solved by a classical Newton algorithm. The linearisation of reduced problem (7) at iteration i + 1 of a Newton solver (see for instance [25] for more details) leads to the following problem: (8) where Δα i+1 = α i+1 -α i is the unknown quantity of the linear prediction and . Linearised system (8) is a Galerkin reduction (or a least-square reduction as these two approaches are equivalent for the linearised problem when using a -norm) of linearised equation (3) (9) providing the reduced linearised operator (of very small size n c ) is invertible.…”

“…The theory and applicability of various projection-based model order reduction methods such as component mode synthesis [13,7], the reduced basis method [14,15,16], the proper orthogonal decomposition [17,18,2] which will be used in this work, the a priori hyperreduction method [19,20] or the proper generalised decomposition [21,22,23] are now well-established in the linear to mildly nonlinear cases. Some attempts have been proposed to extend this concept to strong nonlinearities, in particular in structural mechanics [24,19,25,26]. This background makes it conceivable to use such methods in complex engineering problems such as fracture mechanics.…”