A manifold learning approach to nonlinear model order reduction of quasi-static problems in solid mechanics: proof of concept and parameter study

Abstract The proper orthogonal decomposition (POD)—a popular model order reduction (MOR) method—may require significant model dimensionalities to successfully capture a nonlinear solution manifold resulting from a parameterised quasi-static solid-mechanical problem. The local basis method by Amsallem et al. (Int J Numer Meth Eng 92(10):891–916, 2012) addresses this deficiency by introducing a locally, rather than globally, linear approximation of the solution manifold. However, this generally successful approach comes with some limitations, especially in the data-poor setting. In this investigation, we instead propose a graph-based manifold learning approach to nonlinear projection-based MOR which uses a global, continuously nonlinear approximation of the solution manifold. Approximations of local tangents to the solution manifold, which are necessary for a Galerkin scheme, are computed in the online phase. As an example application for the resulting nonlinear MOR algorithms, we consider simple representative volume element computations. On this example, the manifold learning approach Pareto-dominates the POD and local basis method in terms of the error achieved using a range of model dimensionalities. This contribution discusses the broad framework of the presented manifold learning approach in view of an application within MOR schemes; hyperreduction is beyond the scope of this article.