Process-zone condensation: a discretization-exact reduced cell model for FE$^2$ analysis of composites with fibre-matrix debonding and matrix plasticity

Periodic composites with cohesive interfaces and matrix plasticity require nonlinear cell solutions in multiscale analysis. Process-zone condensation eliminates the linear elastic remainder of the cell by a Schur complement, retaining all interfacial and process-zone internal variables. Influence matrices recover macroscopic stress and consistent tangent. One back-substitution per increment checks that the matrix outside the retained zone remains elastic. While this holds, the reduction is exact relative to the full discretisation, also with a plastic matrix, and requires no training. In an E-glass/epoxy cell, eight of nine two-dimensional paths agree with the full cell to 1e-11 in stress and tangent; the check flags the ninth. Three paths on each of two three-dimensional meshes need zone expansion. Excluding setup and checks, two-dimensional path timings show an aggregate speed-up of 3.3; certified three-dimensional zones show none. In adaptive finite element squared (FE$^2$) simulations of a perforated plate, the reduced solver reproduces the full-cell response within solver tolerances, 1.51 times faster for glass/polypropylene and slower for glass/epoxy. Independent Ansys models agree to below 1e-3 in cell stress and about 0.6% in plate reactions, although local damage and plasticity onsets differ. The advantage depends on zone size, active set and adaptation costs.