arXiv subjects

Naoto Mitsume

Publications and source records attributed to Naoto Mitsume.

9 recordsLinked to original sources

arXiv2026

Domain-decomposed parallelization of B-spline based s-version of the finite element method via generalized graph abstraction

The s-version of the finite element method (SFEM) enables locally high-resolution analysis by superimposing independently defined finite element meshes. However, domain-decomposed parallelization is nontrivial because complex interactions arise among degrees of freedom distributed over multiple meshes. In this study, we propose a method for constructing a graph structure that uniformly represents interactions among computational points, including both intra- and inter-mesh interactions, based on the overlap of basis-function supports. We apply the proposed graph representation to the B-spline based SFEM (BSFEM), a high-accuracy SFEM formulation previously proposed by the authors. The resulting graph partition enables the consistent assignment of degrees of freedom and elements to processes and the construction of the MPI communication structure, thereby realizing domain-decomposition-based distributed-memory parallelization of BSFEM. To the best of the authors' knowledge, this BSFEM implementation constitutes the first domain-decomposition-based distributed-memory parallelization of an SFEM-based method. Furthermore, as an example demonstrating the utility of the proposed graph representation, we apply cost-weighted graph partitioning in which the matrix-generation costs specific to BSFEM are incorporated into node weights, and demonstrate effective static load balancing that accounts for the nonuniform matrix-generation workload.

math.NA↗

arXiv2026

A hybrid s-version isogeometric strategy for dynamic crack propagation in 2D and 3D problems

A hybrid s-version of isogeometric analysis (hS-IGA) strategy is proposed for accurate and efficient evaluation of near-crack fracture quantities in dynamic crack propagation analysis. The strategy retains the global-local superposition framework of the conventional s-method, while introducing B-spline basis functions only into the global discretisation and preserving a Lagrange-based local mesh in the crack domain. This hybrid formulation is motivated by the continuity-related bottleneck in the global-local coupling integration of the conventional Lagrange-based s-method, and by the need to retain a Lagrange-based local mesh for crack representation and post-processing of the dynamic stress intensity factor (DSIF) and local stress. The resulting formulation removes discontinuities in the coupling integrands caused by the global approximation and enables accurate coupling integration by standard Gauss quadrature without recursive subdivision. The proposed strategy is verified using two-dimensional stationary and dynamic straight-crack problems against the standard finite element method and the conventional s-method, and is further assessed using three-dimensional stationary and dynamically propagating circular-crack problems against the conventional s-method. Results show that the proposed hS-IGA strategy accurately evaluates the DSIF and local stress while retaining the global-local modelling advantages of the s-method. It also substantially reduces the number of integration points required for coupling integration, by approximately 81% in the two-dimensional dynamic benchmark and 95.6% in the three-dimensional dynamic benchmark relative to the conventional s-method. These results demonstrate that the proposed hS-IGA framework provides an accurate and efficient global-local strategy for dynamic crack propagation analyses requiring reliable evaluation of near-crack fracture quantities.

cs.CE↗

arXiv2026

A generalized vertical coordinate transformation based on SPH(2) for efficient free surface flow simulations

We propose three new particle methods that improve computational efficiency by introducing a generalized Vertical Coordinate Transformation (VCT) for free surface flow problems with complex bottom boundaries. The first method is a bottom boundary-fitted particle method (BF-SPH). The BF-SPH is simply an arrangement of the body-fitted-coordinate system in the finite difference method to the particle method. The BF-SPH can accurately impose the bottom boundary conditions, while a simple procedure is performed by transforming the complex bottom into a flat one. The second method is the bottom boundary-fitted ellipsoidal particle method (BFE-SPH), which combines the BF-SPH with the ellipsoidal particle model proposed by Shibata et al. The BFE-SPH can speed up the particle simulation by choosing a reasonable aspect ratio of ellipsoidal particles. The last method is the $σ$-SPH method, which automatically selects the aspect ratios of ellipsoidal particles concerning water depth using the $σ$-coordinate system. The $σ$-coordinate is often employed in numerical simulations of oceanographic fields, such as in the Princeton Ocean Model. However, this is the first attempt to apply the $σ$-coordinate to a particle method. Vertical resolution is required from offshore to the coastal region in oceanographic problems such as tsunamis, especially when conducting detailed analysis using a 3-D particle method. Using the $σ$-coordinate allows for a stepwise transition to a naturally efficient coordinate system by referencing water depth. In this paper, we have shown that the above three methods can be generalized as Vertical Coordinate Transformations (VCTs), and the VCTs are successfully achieved by employing SPH(2) with the second-order accuracy of the second-order derivatives, including cross derivatives.

math.NA↗

arXiv2026

Metagraph-Based Domain-Decomposed Galerkin Reduced-Order Model

This study proposes a metagraph-based domain-decomposed Galerkin reduced-order model (MBDD-G-ROM) for distributed-memory parallel reduced-order analysis of large-scale problems. The method represents domain-decomposed Galerkin reduced-order models over arbitrary domain decompositions using two graph levels: calculation-point graphs for interactions among discretization points and metagraphs for connectivity among local approximation-space subdomains. In the proper orthogonal decomposition (POD)-based implementation, POD computation subdomains are represented as metanodes, while metaedges encode the block-sparsity induced by overlaps between local POD basis supports. Partitioning the metagraph enables the POD computation subdomains to be decoupled from the parallel computation subdomains, allowing distributed-memory parallelization of both offline and online phases, including reduced-system assembly and iterative linear solution, without requiring the two decompositions to coincide. The metagraph also supports static load balancing through metanode weights that approximate computational costs. The method is evaluated for an unsteady diffusion equation and incompressible Navier-Stokes flow around a three-dimensional cylinder. The results show that MBDD-G-ROM preserves reduced-order solution accuracy while achieving high online parallel efficiency. A load-balancing test further demonstrates that cost-based metanode weights can improve computational efficiency.

math.NA↗

arXiv2024

Graph Neural PDE Solvers with Conservation and Similarity-Equivariance

Utilizing machine learning to address partial differential equations (PDEs) presents significant challenges due to the diversity of spatial domains and their corresponding state configurations, which complicates the task of encompassing all potential scenarios through data-driven methodologies alone. Moreover, there are legitimate concerns regarding the generalization and reliability of such approaches, as they often overlook inherent physical constraints. In response to these challenges, this study introduces a novel machine-learning architecture that is highly generalizable and adheres to conservation laws and physical symmetries, thereby ensuring greater reliability. The foundation of this architecture is graph neural networks (GNNs), which are adept at accommodating a variety of shapes and forms. Additionally, we explore the parallels between GNNs and traditional numerical solvers, facilitating a seamless integration of conservative principles and symmetries into machine learning models. Our findings from experiments demonstrate that the model's inclusion of physical laws significantly enhances its generalizability, i.e., no significant accuracy degradation for unseen spatial domains while other models degrade. The code is available at https://github.com/yellowshippo/fluxgnn-icml2024.

cs.LG↗

arXiv2023

2D Slice-driven Physics-based 3D Motion Estimation Framework for Pancreatic Radiotherapy

Pancreatic diseases are difficult to treat with high doses of radiation, as they often present both periodic and aperiodic deformations. Nevertheless, we expect that these difficulties can be overcome, and treatment results may be improved with the practical use of a device that can capture 2D slices of organs during irradiation. However, since only a few 2D slices can be taken, the 3D motion needs to be estimated from partially observed information. In this study, we propose a physics-based framework for estimating the 3D motion of organs, regardless of periodicity, from motion information obtained by 2D slices in one or more directions and a regression model that estimates the accuracy of the proposed framework to select the optimal slice. Using information obtained by slice-to-slice registration and setting the surrounding organs as boundaries, the framework drives the physical models for estimating 3D motion. The R2 score of the proposed regression model was greater than 0.9, and the RMSE was 0.357 mm. The mean errors were 5.11 $\pm$ 1.09 mm using an axial slice and 2.13 $\pm$ 0.598 mm using concurrent axial, sagittal, and coronal slices. Our results suggest that the proposed framework is comparable to volume-to-volume registration, and is feasible.

physics.med-ph↗

arXiv2023

Higher-continuity s-version of finite element method with B-spline functions

This paper proposes a strategy to solve the problems of the conventional s-version of finite element method (SFEM) fundamentally. Because SFEM can reasonably model an analytical domain by superimposing meshes with different spatial resolutions, it has intrinsic advantages of local high accuracy, low computation time, and simple meshing procedure. However, it has disadvantages such as accuracy of numerical integration and matrix singularity. Although several additional techniques have been proposed to mitigate these limitations, they are computationally expensive or ad-hoc, and detract from its strengths. To solve these issues, we propose a novel strategy called B-spline based SFEM. To improve the accuracy of numerical integration, we employed cubic B-spline basis functions with $C^2$-continuity across element boundaries as the global basis functions. To avoid matrix singularity, we applied different basis functions to different meshes. Specifically, we employed the Lagrange basis functions as local basis functions. The numerical results indicate that using the proposed method, numerical integration can be calculated with sufficient accuracy without any additional techniques used in conventional SFEM. Furthermore, the proposed method avoids matrix singularity and is superior to conventional methods in terms of convergence for solving linear equations. Therefore, the proposed method has the potential to reduce computation time while maintaining a comparable accuracy to conventional SFEM.

math.NA↗

arXiv2023

Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions

Graph neural network (GNN) is a promising approach to learning and predicting physical phenomena described in boundary value problems, such as partial differential equations (PDEs) with boundary conditions. However, existing models inadequately treat boundary conditions essential for the reliable prediction of such problems. In addition, because of the locally connected nature of GNNs, it is difficult to accurately predict the state after a long time, where interaction between vertices tends to be global. We present our approach termed physics-embedded neural networks that considers boundary conditions and predicts the state after a long time using an implicit method. It is built based on an E(n)-equivariant GNN, resulting in high generalization performance on various shapes. We demonstrate that our model learns flow phenomena in complex shapes and outperforms a well-optimized classical solver and a state-of-the-art machine learning model in speed-accuracy trade-off. Therefore, our model can be a useful standard for realizing reliable, fast, and accurate GNN-based PDE solvers. The code is available at https://github.com/yellowshippo/penn-neurips2022.

cs.LG↗

arXiv2021

Isometric Transformation Invariant and Equivariant Graph Convolutional Networks

Graphs are one of the most important data structures for representing pairwise relations between objects. Specifically, a graph embedded in a Euclidean space is essential to solving real problems, such as physical simulations. A crucial requirement for applying graphs in Euclidean spaces to physical simulations is learning and inferring the isometric transformation invariant and equivariant features in a computationally efficient manner. In this paper, we propose a set of transformation invariant and equivariant models based on graph convolutional networks, called IsoGCNs. We demonstrate that the proposed model has a competitive performance compared to state-of-the-art methods on tasks related to geometrical and physical simulation data. Moreover, the proposed model can scale up to graphs with 1M vertices and conduct an inference faster than a conventional finite element analysis, which the existing equivariant models cannot achieve.

cs.LG↗