A Compact Residual-Screened Local Reduced-Order Model for Diffuse-Domain Shape Optimization

We generalize the compact residual-screened local reduced-order model (CL-ROM) from Brinkman-penalized flow to shape optimization with the diffuse domain method (DDM). As with Brinkman penalization, DDM extends evolving-domain problems to a fixed rectangular domain, allowing one grid for discretization and directly compatible snapshots from different designs. Unlike Brinkman penalization, DDM uses a phase field to weight operators and sources and treats Dirichlet, Neumann and Robin conditions through diffuse boundary terms. The CL-ROM mechanism carries over: recent full-order and corrected solutions determine local reduced spaces, while residual screening, correction with retained factors and objective verification decide when a full-order solve is needed. We derive reduced operators, residuals and sensitivities for each diffuse model, using Galerkin projection for the symmetric problems and least-squares Petrov--Galerkin projection for nonlinear flow. For the symmetric problems we prove a local approximation estimate that admits inexact, corrected snapshots. In numerical tests of shape identification, drag minimization in Navier--Stokes channel flow and compliance minimization of a cantilever, CL-ROM uses reduced solutions for nearly all design updates, reproduces the full-order objectives with relative differences below $5\times10^{-6}$, and is $3.85$ to $7.65$ times faster than a full-order baseline that reuses factorizations as preconditioners.