arXiv subjects

Shigeki Kaneko

Publications and source records attributed to Shigeki Kaneko.

3 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.

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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.

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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.

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