A proposed computer method for structural analysis matched a full high-dimensional calculation in some tests, but the results varied with the construction of the reduced model. In a simplified nuclear power plant model, the point-based version reproduced the reference solution exactly, while the version based on planar interface motion lacked sufficient accuracy.
The study examined a rectangular solid and simplified nuclear power plant configurations with one or four small modular reactors, comparing high-dimensional subdomains with reduced-order subdomains inside the same solver.
A solver built from full and compact models
The method places projection-based reduced-order models alongside high-dimensional subdomains in an iterative, non-overlapping domain-decomposition solver. In practical terms, the structure is split into regions, with each region handled by one of the two model types. The solver uses a conjugate-gradient method, which repeatedly updates the calculation until the regions agree at their interfaces.
The researchers derived a diagonal preconditioner for the iterative domain-decomposition method. They also implemented reduced-order-model calculations on GPUs using CUDA.
The reduced models were built from three types of snapshots, meaning stored examples of structural response: a unit displacement imposed at each interface degree of freedom, two-direction planar tilting, and intermediate solutions recorded during a full high-dimensional conjugate-gradient calculation.
Different snapshots, different levels of agreement
The first example was a rectangular solid represented by a 3 by 3 by 12 hexahedral mesh, with 208 nodes and 108 elements. Its bottom region formed one reduced-order subdomain, while the remaining high-dimensional region was divided into six subdomains.
The reduced-order region had an interface of 16 nodes. The three snapshot approaches generated 48 unit-displacement snapshots, four plane-tilt snapshots and 53 conjugate-gradient snapshots. The reduced bases contained either 20 or 24 vectors, selected with energy thresholds of 0.9999 and 0.99999.
The point-snapshot model exactly matched the full high-dimensional result, while the plane-snapshot model was approximate. The 24-vector conjugate-gradient model was almost identical to the reference. With 20 vectors, the maximum relative displacement errors reached 2.24% in the high-dimensional regions and 10.30% in the reduced-order region.
The study also compared snapshots collected under different load directions. Conjugate-gradient snapshots collected for a load in the x direction did not reproduce the full-model solution under the tested load in the y direction. The point and plane constructions also behaved differently when the load conditions changed.
The plant models delivered mixed results
The second set of calculations used a simplified nuclear power plant model with one-SMR and four-SMR configurations, named SMR1 and SMR4. SMR1 used five subdomains and SMR4 used eight, with four high-dimensional subdomains in each model.
The solver converged in the simplified plant calculations. The point-snapshot HDM-ROM model reproduced the exact solution, whereas the plane-snapshot model lacked sufficient accuracy.
The NPP evaluation reported maximum displacement errors of 66.69% in high-dimensional regions and 55.28% in reduced-order regions. The supplied analysis does not identify the precise configuration associated with those maximum values, so the figures should not be treated as one overall error rate for the plant models.
Iteration counts fell, but timings were uneven
The listed comparisons recorded fewer conjugate-gradient iterations with diagonal scaling. For the SMR4 high-dimensional case, the count was 5,426 with scaling and 11,111 without it. For the SMR1 high-dimensional case, it was 3,481 versus 5,755. In the SMR4 hybrid case running on a CPU, the count was 1,435 with scaling and 1,795 without.
The reported computation times were uneven across the listed runs. The paper gives 76.7 seconds for the SMR4 high-dimensional calculation and 4.6 seconds for its hybrid CPU calculation. For SMR1, it reports 48.5 seconds for the high-dimensional case, 404.1 seconds for the hybrid CPU case and 13 seconds for the hybrid GPU case.
The timing figures require careful reading. The supplied analysis says the timing columns do not reconcile arithmetically with the listed iteration counts and per-iteration values and that the table requires verification of its units or column alignment. The results are therefore best read as reported figures for the listed runs, not as a settled performance ranking.
The evidence remains limited to the tested setups
These findings come from one rectangular solid and simplified nuclear power plant constructions. They show exact or insufficient agreement under the tested model constructions, but do not establish the same outcome for other structural problems.
The tested change in load direction left open whether snapshots would generalize across loads. The plane-based construction also lacked sufficient accuracy in the simplified plant case, while conjugate-gradient snapshots failed to reproduce the solution under the tested different load direction. Results for broader load and interface cases remain unresolved.
The implementation used open-source ADVENTURE modules, but the parallel conjugate-gradient solver underlying the hybrid method is not publicly available. The work was supported by JSPS KAKENHI Grant Number 24K14984 and partly by the Kajima Foundation’s General Research Grants. Fugaku resources were used for snapshot computations under Project ID hp250298.
Paper data and sources
Original title: Combined High-Dimensional and Reduced-Order Modeling Based on an Iterative Domain Decomposition Method
Authors: Taiji Saito, Tomoshi Miyamura, Yasunori Yusa
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text