On the mathematical foundations of the self-consistent clustering analysis for non-linear materials at small strains

• Published in: Computer methods in applied mechanics and engineering 2019-09, Vol.354, p.783-801
• Main Author: Schneider, Matti
• Format: Article
• Language: English
• Subjects: Cluster analysis; Clustering; Composites; Computational micromechanics; Construction materials; Discretization; Hashin–Shtrikman variational principle; Lippmann–Schwinger equation; Market segmentation; Nonlinear analysis; Process planning; Self-consistent clustering analysis; Transformation field analysis; Uniqueness
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2019.06.003

We investigate, both mathematically and numerically, the self-consistent clustering analysis recently introduced by Liu–Bessa–Liu and, independently, by Wulfinghoff–Cavaliere–Reese. We establish, in the small strain setting and non-softening material behavior, existence and uniqueness of the solution to the discretized equations for fixed (possibly anisotropic) reference material, by a constructive method. Furthermore, we establish convergence of the solution to the discretized equation to the continuous solution upon refinement of the clusters. Thus we generalize the work of Tang–Zhang–Liu to spatial dimensions larger than 1. We explore the specific structure of the Lippmann–Schwinger equation governing clustering analysis, proving strict equivalence to a problem of Eyre–Milton type. For the latter formulation, existence and uniqueness are easily established based on recent progress in the understanding of polarization schemes in FFT-based computational homogenization methods. Last but not least, for elasto-viscoplastic constituents we clarify the relationship of the self-consistent clustering analysis to the transformation field analysis of Dvorak–Benveniste. Our theoretical considerations are confirmed by pertinent numerical experiments. •Stream-lined derivation of the clustered Lippmann–Schwinger equation.•Reformulation of the clustered Lippmann–Schwinger equation as an Eyre–Milton equation.•Well-posedness of discretized equations for small strain hardening materials.•Proof of convergence of discrete solutions upon cluster refinement.•Numerical investigations of the convergence rate.