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Combined High-Dimensional and Reduced-Order Modeling Based on an Iterative Domain Decomposition Method
Taiji Saito, Tomoshi Miyamura, Yasunori Yusa
TL;DR
Repeated structural analyses of large, finely discretized models require acceleration without abandoning detailed modeling where it matters. The paper combines multiple ROMs and an HDM within an iterative DDM, adds snapshot-generation and preconditioning methods, and evaluates hybrid MPI-CUDA implementations on nuclear power plant models. The solver converges successfully, with exact or near-exact reproduction in several configurations, while some ROM choices produce notable errors and GPU execution reduces computation time.
Problem
Large high-dimensional structural models are feasible but costly when many analyses are required, motivating acceleration while retaining detailed modeling for selected regions.
Method
The method combines multiple projection-based ROM subdomains with HDM subdomains through an iterative DDM, using GPU ROM computations and proposed snapshot-generation and preconditioning procedures.
Results
The solver converged successfully; some HDM–ROM configurations reproduced HDM solutions exactly or nearly exactly, while the HDM–ROM_cg_V20 model reached maximum relative errors of 2.24% in HDM and 10.30% in ROM regions.
Takeaways & Limitations
The combined method supports detailed HDM treatment of engineering-interest or strongly nonlinear components while modeling remaining components with ROMs in parallel analyses.
Takeaways & Limitations
The snapshot-generation study assumes ROM subdomains have only interface and fixed Dirichlet boundaries, with no Neumann loads, nonzero prescribed displacements, or body forces.
Abstract
from arXiv · showhide
A method is developed for structural analysis in which parts of an analysis domain are modeled by reduced-order models (ROMs) and the remaining part is modeled by a high-dimensional model (HDM). The multiple ROMs and HDM are combined based on an iterative domain decomposition method (DDM) with an arbitrary number of non-overlapping subdomains. A diagonal preconditioner is derived for the iterative DDM. The method is implemented based on a framework of the existing DDM-based analysis code. The ROM is implemented on GPU. Snapshot generation methods for the ROM are also proposed. Static structural analyses of nuclear power plant models are conducted using the combined multiple ROMs and HDM with multiple-subdomains model. The analyses are performed in parallel by the hybrid MPI-CUDA implementation. The convergence property of the iterative DDM is improved by the preconditioner.
1. Introduction
The paper addresses the need to accelerate repeated large-scale structural analyses by combining multiple ROMs with an HDM through an iterative DDM. Unlike prior two-subdomain approaches, the method supports arbitrary non-overlapping subdomains and proposes systematic snapshot generation and preconditioning.
- Large, finely discretized FEM models improve geometric fidelity and accuracy but require computational acceleration when many analyses are needed.
- Projection-based ROMs accelerate computation by reducing model dimension while minimizing analysis-accuracy loss, using HDM-generated snapshots and reduced-order bases.
- Earlier domain-decomposition approaches constructed local ROMs on overlapping subdomains or coupled one ROM subdomain with one HDM subdomain.
- The proposed method combines multiple ROMs with an HDM using iterative DDM and arbitrary numbers of non-overlapping subdomains.
- Snapshot-generation methods address ROM subdomain problems with Dirichlet boundary conditions, while preconditioning is proposed for the CG solver.
2. Overview of Galerkin Projection-Based ROM for Structural Analysis
The paper constructs a Galerkin projection-based ROM by approximating HDM displacements in an affine subspace spanned by a reduced-order basis. The basis is extracted from HDM snapshots using SVD and truncated according to cumulative singular-value energy.
- Static linear structural equilibrium is represented as K(μ)u(μ)=f(μ), where μ may include material properties, geometry, and boundary conditions.
- The ROM approximates the nodal displacement as u(μ)≈u_ref+Vy(μ), with V the reduced-order basis and y(μ) reduced coordinates.
- HDM solutions computed for m parameter sets are collected as snapshots for constructing the reduced-order basis.
- SVD supplies the basis vectors, with V formed from the first r columns of U.
- The basis size r is the smallest integer meeting the prescribed cumulative singular-value energy threshold η.
- Galerkin projection yields the reduced system V^T K(μ)V y(μ)=V^T f(μ), whose solution determines the approximate displacement.
3. Domain Decomposition Method
The iterative DDM partitions the structural domain into non-overlapping subdomains, assembles their contributions, and separates interior from interface equilibrium. Substructuring condenses interior degrees of freedom into Schur-complement systems, enabling reduced global systems for iterative solution.
- 3.1 Non-Overlapping Domain Decomposition Method: The analysis domain is divided into N non-overlapping subdomains whose local stiffness, displacement, and load quantities are assembled into the global equilibrium equation.
- 3.1 Non-Overlapping Domain Decomposition Method: Boolean matrices map subdomain-local degrees of freedom to global degrees of freedom and represent shared interface nodes.
- 3.2 Substructuring-Based Domain Decomposition Method: Substructuring partitions local quantities into interior and interface degrees of freedom and eliminates the interior displacement using a local Schur complement.
- 3.1 Non-Overlapping Domain Decomposition Method: Interior equilibrium is solved independently within each subdomain, while interface equilibrium couples subdomains through assembled contributions.
- 3.2 Substructuring-Based Domain Decomposition Method: Assembling local Schur complements produces a global interface system, while the corresponding right-hand-side vectors are assembled from local condensed terms.
- 3.3 Combination of Non-Substructuring and Substructuring-Based DDMs: Combining non-substructuring and substructuring DDMs condenses selected subdomains, reducing global degrees of freedom by their eliminated interior degrees of freedom.
4. Solution Procedure of the CG Method within the Framework of the DDM
The DDM solution procedure uses preconditioned conjugate gradients, with subdomain reaction forces computed through local Schur-complement operations or equivalent local Dirichlet problems.
- CG iteration: The coefficient system is solved with the preconditioned CG method, using M^-1 as the preconditioning matrix.Algorithm 1 describes the corresponding CG procedure.
- Interface reaction evaluation: The interface reaction for subdomain k is computed as r_B^(k) = S^(k)p_B^(k), then assembled with reactions from all subdomains.The vector p_B^(k) contains the search-direction entries associated with the subdomain's interface degrees of freedom.
- Local subdomain solves: Local Dirichlet problems are more efficient than explicitly constructing each local Schur complement for interface reaction evaluation.Interface degrees of freedom are imposed as Dirichlet boundary conditions alongside the original prescribed-displacement conditions.
- Iteration loop: The subdomain interior system is solved for u_I^(k), after which the reaction force r_B^(k) is obtained from the corresponding interface relation.The procedure is repeated inside the CG iteration loop.
5. Combined High-Dimensional and Reduced-Order Modeling Based on Iterative DDM
The HDM–ROM method couples multiple ROM subdomains with HDM subdomains through an iterative DDM, using ROM approximations of local Schur complements and diagonal preconditioning.
- HDM–ROM coupling: The HDM–ROM model uses an HDM for the region of interest and multiple ROMs for the remaining regions within an iterative DDM framework.The ROMs are incorporated into the CG solution procedure, and the subdomains are non-overlapping.
- ROM local problem: The ROM treats varying interface search directions p_B^(k) as parameter vectors μ, while assuming zero Neumann load in ROM regions.Under this assumption, the local interior problem becomes K_II^(k)u_I^(k) = -K_IB^(k)(μ)f_B^(k).
- Reduced solve: The ROM equation projects the local interior problem onto the reduced basis V^(k), producing the reduced variable y^(k).The reduced system is solved with a linear solver before reconstructing the interior displacement and reaction force.
- Schur-complement approximation: The ROM approximation replaces each local Schur complement S^(k) with S_hat^(k) = V^(k)T K^(k)(μ)V^(k) in the global interface equilibrium equation.The resulting coefficient matrix combines contributions from all ROM and HDM subdomains.
- Preconditioning: Diagonal scaling preconditioning uses the diagonal of the assembled interface matrix, approximating each Schur-complement diagonal with the corresponding K_BB^(k) diagonal.The resulting M_Diag-KBB^-1 is employed as the CG preconditioning matrix.
6. Generation of ROM Snapshots for the HDM–ROM Method
ROM snapshots are generated by prescribing displacements on subdomain interfaces, with three application strategies considered for representing local Dirichlet boundary conditions.
- Boundary assumptions: The ROM subdomain assumes interfaces and fixed zero-displacement boundaries, with no Neumann boundaries, nonzero prescribed displacements, or body forces.These assumptions define the boundary-condition scope of the snapshot-generation procedure.
- Method (1): One snapshot strategy applies unit displacement to each interface degree of freedom while setting all other degrees of freedom to zero.Unity is sufficient because the structural problem is linear.
- Method (2): A second strategy computes prescribed interface displacements from displacement fields, and is expected to help when an assembly component is replaced by a ROM.The passage specifically discusses planar interfaces and components connected through supporting structures.
- Snapshot representation: Without singular-value truncation, the unit-displacement snapshots can represent any interface displacement field as their linear combination and reproduce the HDM exactly.This exactness statement applies before reducing the ROB vectors by singular value decomposition.
7. Implementation
The implementation extends the ADVENTURE framework with non-overlapping domain decomposition, GPU-accelerated ROM operations, and CPU alternatives for dense computations.
- Software framework: The implementation uses ADVENTURE modules for finite-element computation, input/output, and graph-based domain decomposition.The parallel CG solver uses libfem and ADVENTURE_IO, while ADVENTURE_Metis uses Metis and ParMetis capabilities.
- Domain decomposition: ROM regions are kept as single subdomains, while the remaining HDM region is partitioned automatically into an appropriate number of subdomains.The HDM partitioning is performed using ADVENTURE_Metis.
- Dense computations: ROM construction, ROM solves, and HDM-solution recovery require dense matrix operations.These operations include singular value decomposition and dense linear-system solution.
- GPU and CPU execution: GPU implementations use cuSOLVER and cuBLAS in MPI processes handling ROMs, while CPU implementations use LAPACKE in Intel oneAPI MKL.The GPU and CPU implementations provide alternative execution paths for the dense computations.
8. Numerical Examples
The numerical examples evaluate HDM–ROM models on a rectangular solid and nuclear power plant models, comparing snapshot-generation strategies, solution accuracy, convergence, and computational performance.
- Rectangular solid: The rectangular solid uses one ROM subdomain for the bottom cubic region and six HDM subdomains for the remaining region.The bottom surface is fixed, and horizontal prescribed displacement is applied to the top surface.
- Snapshot generation: 48 interface snapshots are generated by applying unit displacements to each interface degree of freedom in the rectangular-solid ROM region.The ROM region interface has 16 nodes, with prescribed displacements applied in three coordinate directions.
- Snapshot generation: ROM_plane uses four snapshots from tilted planar interfaces, whereas ROM_cg collects 53 snapshots from nodal displacement vectors during CG iterations.ROM_cg_V20 and ROM_cg_V24 retain 20 and 24 ROB vectors, respectively, based on singular-value criteria.
- Accuracy: HDM–ROM_point exactly reproduces the HDM solution, while HDM–ROM_plane only approximately reproduces it when the deformed interface is approximately planar.The point model reproduces the HDM solution for various load conditions except when the load is applied to the ROM region.
- Nuclear power plant model: The nuclear power plant model contains a building and four SMRs, with each SMR separately represented by a ROM because their meshes differ slightly.Each SMR is supported on three surfaces that serve as interfaces between subdomains.
- Computational performance: The CG solver converges for the HDM–ROM nuclear power plant models, diagonal scaling reduces convergence iterations, and GPU implementation significantly reduces computation time.The analyses use parallel hybrid MPI-CUDA execution on a CPU–GPU workstation.