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A Node-Based Smoothed Finite Element Method (NS-FEM) for Free and Forced Vibration Analysis of Three-Dimensional (3D) Structures

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INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS 2025年第4期22卷

作者： Zhao, Jingui Liu, Guirong Huo, Shuhao Wang, Gang Sun, Chao Li, Zirui Hebei Univ Technol Sch Mech Engn Tianjin 300401 Peoples R China Hebei Univ Technol State Key Lab Reliabil & Intelligence Elect Equipm Tianjin 300130 Peoples R China Univ Cincinnati Dept Aerosp Engn & Engn Mech Cincinnati OH 45221 USA Taiyuan Univ Technol Coll Aeronaut & Astronaut Taiyuan 550001 Peoples R China

The smoothed finite element method (S-FEM) has been found to be an effective solution method for solid mechanics problems. This paper proposes an effective approach to compute the lower bound solution of free vibration and the upper bound solution of the forced vibration of solid structures, by making use of the important softening effects of the node-based smoothed finite element method (NS-FEM). Through the gradient smoothing technique, the strain-displacement matrix is obtained in the smoothing domain based on the element mesh nodes. Subsequently, the stiffness matrix is computed in a manner consistent with the standard finite element method (FEM). Here, the practical lanczos algorithm and the modal superposition technique are employed to obtain the frequencies, modes, and transient responses of a given homogeneous structure. For three-dimensional (3D) solid structures, the automatically generated four-node tetrahedron (T4) element meshes are utilized. The results obtained from the NS-FEM are compared with the standard FEM in terms of accuracy, convergence and computational efficiency.

关键词： Node-based smoothed finite element method free vibration forced vibration lanczos algorithm mode superposition method

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Rethinking eigenpairs: Community detection with orthogonality

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KNOWLEDGE-BASED SYSTEMS 2025年 311卷

作者： Yan, Liping Cao, Jie Yu, Weiren Wang, Yuyao Vukovic, Darko B. Nanjing Univ Sci & Technol Sch Comp Sci & Engn Nanjing Peoples R China Hefei Univ Technol Sch Management Hefei Peoples R China Univ Warwick Dept Comp Sci Coventry England Nanjing Univ Posts & Telecommun Sch Comp Sci Nanjing Peoples R China St Petersburg State Univ Grad Sch Management St Petersburg Russia

Spectral clustering and matrix factorization are two widely utilized algorithms for community detection. On the one hand, most existing spectral clustering algorithms focus on learning node representations from a well-designed similarity matrix that thoroughly incorporates data attribute information. However, they often overlook the intrinsic attribute information embedded within the algorithm itself. On the other hand, the node representations generated by most existing matrix factorization algorithms often exhibit a lack of linear independence, leading to the presence of redundant information. Motivated by them, we propose an algorithm, SOCD (Selective Orthogonalization for Community Detection), which leverages the orthogonality loss of the lanczos algorithm to monitor whether eigenvalues converge or not, and saves the already converged eigenpairs, then the corresponding converged eigenvectors are employed for community structure detection. Meanwhile, the already converged eigenvectors continue to be orthogonalized against the lanczos vector to preserve the orthogonality of the lanczos algorithm. A large number of experiments conducted on real datasets with community labels demonstrate the superiority of our algorithm over its competitors. The experimental code is available for download at https://***/AnonSimRank/SOCD/.

关键词： Community detection lanczos algorithm Spectral clustering Selective orthogonalization Dimensionality reduction

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Using an uncontracted inter-molecular basis to assess the convergence of contracted inter-molecular bases when computing the spectrum of H2O-CO

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MOLECULAR PHYSICS 2025年

作者： Wang, Xiao-Gang Carrington, Tucker Queens Univ Chem Dept Kingston ON K7L 3N6 Canada

We report energy levels of H $ _2 $ 2O-CO computed with a large product contracted (PC) basis. Intra-molecular levels are obtained using a large basis of products of contracted intra-molecular functions and contracted inter-molecular functions. To determine the size of the contracted inter-molecular basis required to achieve convergence, we compute inter-molecular levels without contracting the inter-molecular basis. We call this a $ \vert v, L \rangle $ |v,L > basis. We find that a large contracted inter-molecular basis is necessary. Previous calculations with a smaller inter-molecular basis have convergence errors similar to those caused by using the rigid-monomer approximation. For example, the largest relative splitting error is 37%. Owing to the size of the inter-molecular basis, a quadrature-point intermediate matrix $ \boldsymbol {F} $ F, rather than a DVR-point intermediate matrix $ \boldsymbol {F} $ F, and MPI parallelisation are important for reducing the calculation time.

关键词： Vibrational spectroscopy Van der Waals complexes contracted bases lanczos algorithm

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Numerical Solution of Partial Symmetric Generalized Eigenvalue Problems in Piezo Device Modal Analysis

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COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION 2025年第3期7卷 1002-1015页

作者： Muratova, Galina V. Martynova, Tatiana S. Oganesyan, Pavel A. Southern Fed Univ Vorovich II Inst Math Mech & Comp Sci Rostov Na Donu 344090 Russia

We demonstrate how different computational approaches affect the performance of solving the generalized eigenvalue problem (GEP). The layered piezo device is studied for resonance frequencies using different meshes, sparse matrix representations, and numerical methods in COMSOL Multiphysics and ACELAN-COMPOS packages. Specifically, the matrix-vector and matrix-matrix product implementation for large sparse matrices is discussed. The shift-and-invert lanczos method is used to solve the partial symmetric GEP numerically. Different solvers are compared in terms of the efficiency. The results of numerical experiments are presented.

关键词： Partial symmetric generalized eigenvalue problem (GEP) Modal analysis lanczos algorithm

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Quantum dynamics in Krylov space: Methods and applications

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Physics Reports 2025年 1125-1128卷 1-82页

作者： Nandy, Pratik Matsoukas-Roubeas, Apollonas S. Martínez-Azcona, Pablo Dymarsky, Anatoly del Campo, Adolfo Center for Gravitational Physics and Quantum Information Yukawa Institute for Theoretical Physics Kyoto University Kitashirakawa Oiwakecho Sakyo-ku Kyoto 606-8502 Japan RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) 2-1 Hirosawa Wako Saitama 351-0198 Japan Cavendish Laboratory University of Cambridge J.J. Thomson Avenue Cambridge CB3 0HE United Kingdom Department of Physics and Materials Science University of Luxembourg L-1511 Luxembourg Department of Physics and Astronomy University of Kentucky Lexington 40506 KY United States Donostia International Physics Center E-20018 San Sebastián Spain

The dynamics of quantum systems unfolds within a subspace of the state space or operator space, known as the Krylov space. This review presents the use of Krylov subspace methods to provide an efficient description of quantum evolution and quantum chaos, with emphasis on nonequilibrium phenomena of many-body systems with a large Hilbert space. It provides a comprehensive update of recent developments, focused on the quantum evolution of operators in the Heisenberg picture as well as pure and mixed states. It further explores the notion of Krylov complexity and associated metrics as tools for quantifying operator growth, their bounds by generalized quantum speed limits, the universal operator growth hypothesis, and its relation to quantum chaos, scrambling, and generalized coherent states. A comparison of several generalizations of the Krylov construction for open quantum systems is presented. A closing discussion addresses the application of Krylov subspace methods in quantum field theory, holography, integrability, quantum control, and quantum computing, as well as current open problems. © 2025 Elsevier B.V.

关键词： Krylov complexity lanczos algorithm Operator growth Quantum chaos

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Explicitly restarted lanczos algorithms in an MPP environment

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PARALLEL COMPUTING 1999年第5期25卷 613-631页

作者： Szularz, M Weston, J Clint, M Univ Ulster Sch Informat & Software Engn Coleraine BT52 1SA Londonderry North Ireland Queens Univ Belfast Dept Comp Sci Belfast BT7 1NN Antrim North Ireland Univ Edinburgh Parallel Comp Ctr Edinburgh EH8 9YL Midlothian Scotland

The lanczos algorithm is one of the principal methods for the computation of a small part of the eigenspectrum of large, sparse, real symmetric matrices. A single-vector, explicitly restarted variant of the lanczos method is proposed in this paper. The algorithm finds only one eigenpair at a time using a deflation technique in which the lanczos factorization for the current eigenpair is generated in the null space of all previously computed eigenvectors. This approach yields a fixed k-step restarting scheme which permits short lanczos factorizations and almost completely eliminates the reorthogonalization among the lanczos vectors. The orthogonalization strategy developed falls naturally into the class of selective orthogonalization strategies as classified by Simon. 'Reverse communication' software for the implementation of the proposed variant on a Connection Machine CM-200 with 8K processors and on a Gray T3D with 32 processors is discussed. Test results on the CM-200 using examples from the Harvell-Boeing collection of sparse matrices show the method to be very effective when compared with Sorensen's state-of-the-art routine taken from the ARPACK library. The method has fixed, small storage requirements, copes easily with genuinely multiple eigenvalues and is guaranteed to converge to the desired eigenvalues. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： lanczos algorithm restarting deflation reorthogonalization MPP

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SOME HISTORY OF THE CONJUGATE-GRADIENT AND lanczos algorithmS - 1948-1976

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SIAM REVIEW 1989年第1期31卷 50-102页

作者： GOLUB, GH OLEARY, DP UNIV MARYLAND DEPT COMP SCI COLLEGE PK MD 20742 USA UNIV MARYLAND INST ADV COMP STUD COLLEGE PK MD 20742 USA

This paper gives some of the history of the conjugate gradient and lanczos algorithms and an annotated bibliography for the period 1948-1976

关键词： 65F10 65F15 65H10 65F03 conjugate gradient algorithm lanczos algorithm variable metric algorithms

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A CONVERGENCE ANALYSIS FOR NONSYMMETRIC lanczos algorithmS

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MATHEMATICS OF COMPUTATION 1991年第194期56卷 677-691页

作者： YE, Q Department of Mathematics and Statistics University of Calgary

A convergence analysis for the nonsymmetric lanczos algorithm is presented. By using a tridiagonal structure of the algorithm, some identities concerning Ritz values and Ritz vectors are established and used to derive approximation bounds. In particular, the analysis implies the classical results for the symmetric lanczos algorithm.

关键词： lanczos algorithm CONVERGENCE BOUND TRIDIAGONAL MATRICES

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An application of the discrete-time Toda lattice to the progressive algorithm by lanczos and related problems

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JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 2018年第17期51卷 174001-174001页

作者： Nakamura, Yoshimasa Sekido, Hiroto Kyoto Univ Grad Sch Informat Dept Appl Math & Phys Kyoto 6068501 Japan

The finite or the semi-infinite discrete-time Toda lattice has many applications to various areas in applied mathematics. The purpose of this paper is to review how the Toda lattice appears in the lanczos algorithm through the quotient-difference algorithm and its progressive form (pqd). Then a multistep progressive algorithm (MPA) for solving linear systems is presented. The extended lanczos parameters can be given not by computing inner products of the extended lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost. The asymptotic behavior of the pqd algorithm brings us some applications of MPA related to eigenvectors.

关键词： discrete-time Toda lattice qd algorithm lanczos algorithm multistep progressive algorithm

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Fast Approximate kNN Graph Construction for High Dimensional Data via Recursive lanczos Bisection

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JOURNAL OF MACHINE LEARNING RESEARCH 2009年第9期10卷 1989-2012页

作者： Chen, Jie Fang, Haw-ren Saad, Yousef Univ Minnesota Dept Comp Sci & Engn Minneapolis MN 55455 USA Argonne Natl Lab Div Math & Comp Sci Argonne IL 60439 USA

Nearest neighbor graphs are widely used in data mining and machine learning. A brute-force method to compute the exact kNN graph takes Theta(dn(2)) time for n data points in the d dimensional Euclidean space. We propose two divide and conquer methods for computing an approximate kNN graph in Theta(dn(t)) time for high dimensional data (large d). The exponent t is an element of (1, 2) is an increasing function of an internal parameter a which governs the size of the common region in the divide step. Experiments show that a high quality graph can usually be obtained with small overlaps, that is, for small values of t. A few of the practical details of the algorithms are as follows. First, the divide step uses an inexpensive lanczos procedure to perform recursive spectral bisection. After each conquer step, an additional refinement step is performed to improve the accuracy of the graph. Finally, a hash table is used to avoid repeating distance calculations during the divide and conquer process. The combination of these techniques is shown to yield quite effective algorithms for building kNN graphs.

关键词： nearest neighbors graph high dimensional data divide and conquer lanczos algorithm spectral method

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