A two‐scale approximation of the Schur complement and its use for non‐intrusive coupling

Abstract

SUMMARYThis paper presents a two-scale approximation of the Schur complement of a subdomain's stiffness matrix, obtained by combining local (i.e. element strips) and global (i.e. homogenized) contributions. This approximation is used in the context of a coupling strategy that is designed to embed local plasticity and geometric details into a small region of a large linear elastic structure; the strategy consists in creating a local model that contains the desired features of the concerned region and then substituting it into the global problem by the means of a non-intrusive solver coupling technique adapted from domain decomposition methods. Using the two-scale approximation of the Schur complement as a Robin condition on the local model enables to reach high efficiency. Examples include a large 3D problem provided by our industrial partner Snecma.

Analysis of three-dimensional fracture mechanics problems: A non-intrusive approach using a generalized finite element method

“…The reference values for K I are taken from Walters et al [62] and are used to compute e r ðK I Þ using Eq. (24). The relative error e r ðK I Þ of stress intensity factors extracted from local solutions, u L , computed on the domains shown in Figs.…”

“…Their implementation is based on the user element library (UEL) capabilities provided by Abaqus. Gendre et al [23,24] proposed a non-intrusive methodology with Abaqus based on approximations of the Schur complement of local degrees of freedom. Their methodology focus on problems with localized plasticity.…”

Non-invasive global–local coupling as a Schwarz domain decomposition method: acceleration and generalization

“…• Mixed approach: the condition Q F = ∞ is a poor approximation of the optimal choice. In [31] a two-scale approximation of S C was proposed for the global-local coupling.…”

A hierarchical non-intrusive algorithm for the generalized finite element method

“…For further details on the choice of spring stiffness, please refer to [24]. It is noted that an in-depth study of the choice of the spring value (or impedance) in mixed non-intrusive coupling is presented in [30]. The local solution u L defined in this section is the same solution provided by the classical global-local or sub-modeling analysis [31] with local solutions computed by a GFEM instead of the standard FEM.…”