A Compact Residual-Screened Local Reduced-Order Model for Diffuse-Domain Shape Optimization
Abstract
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 5×10−6, and is $3.85$ to $7.65$ times faster than a full-order baseline that reuses factorizations as preconditioners.
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BibTeX
@misc{zhang2026compact,
title = {{A Compact Residual-Screened Local Reduced-Order Model for Diffuse-Domain Shape Optimization}},
author = {Xiaoping Zhang and Anusooya Vasudevan and Lili Ju and Zhu Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04864},
eprint = {2611.04864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04864},
}
RIS
TY - UNPB
TI - A Compact Residual-Screened Local Reduced-Order Model for Diffuse-Domain Shape Optimization
AU - Xiaoping Zhang
AU - Anusooya Vasudevan
AU - Lili Ju
AU - Zhu Wang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04864
UR - https://arxiv.org/abs/2611.04864
AB - 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 5×10^−6, and is $3.85$ to $7.65$ times faster than a full-order baseline that reuses factorizations as preconditioners.
ER -