{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T21:05:50Z","timestamp":1792789550920,"version":"4.5.0"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,21]],"date-time":"2021-04-21T00:00:00Z","timestamp":1618963200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11702087"],"award-info":[{"award-number":["11702087"]}]},{"name":"National Natural Science Foundation of China","award":["U1704158"],"award-info":[{"award-number":["U1704158"]}]},{"name":"Key Research Project of Higher School of Henan Province","award":["21A520020"],"award-info":[{"award-number":["21A520020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Compared with continuous elements, discontinuous elements advance in processing the discontinuity of physical variables at corner points and discretized models with complex boundaries. However, the computational accuracy of discontinuous elements is sensitive to the positions of element nodes. To reduce the side effect of the node position on the results, this paper proposes employing partially discontinuous elements to compute the time-domain boundary integral equation of 3D elastodynamics. Using the partially discontinuous element, the nodes located at the corner points will be shrunk into the element, whereas the nodes at the non-corner points remain unchanged. As such, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. First, we present a numerical integration scheme of the partially discontinuous element. For the singular integral, an improved element subdivision method is proposed to reduce the side effect of the time step on the integral accuracy. Then, the effectiveness of the proposed method is verified by two numerical examples. Meanwhile, we study the influence of the positions of the nodes on the stability and accuracy of the computation results by cases. Finally, the recommended value range of the inward shrink ratio of the element nodes is provided.<\/jats:p>","DOI":"10.3390\/a14050129","type":"journal-article","created":{"date-parts":[[2021,4,21]],"date-time":"2021-04-21T12:47:07Z","timestamp":1619009227000},"page":"129","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3524-2662","authenticated-orcid":false,"given":"Yuan","family":"Li","sequence":"first","affiliation":[{"name":"School of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China"},{"name":"Key Laboratory of Artificial Intelligence, Personalized Learning in Education of Henan Province, Xinxiang 453007, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ni","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China"},{"name":"Key Laboratory of Artificial Intelligence, Personalized Learning in Education of Henan Province, Xinxiang 453007, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuejiao","family":"Gong","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5335-9517","authenticated-orcid":false,"given":"Wentao","family":"Mao","sequence":"additional","affiliation":[{"name":"School of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China"},{"name":"Key Laboratory of Artificial Intelligence, Personalized Learning in Education of Henan Province, Xinxiang 453007, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shiguang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China"},{"name":"Key Laboratory of Artificial Intelligence, Personalized Learning in Education of Henan Province, Xinxiang 453007, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,21]]},"reference":[{"key":"ref_1","unstructured":"Brebbia, C.A. (1978). The Boundary Element Method for Engineers, Pentech Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"106044","DOI":"10.1016\/j.ijfatigue.2020.106044","article-title":"Stochastic approach to fatigue crack-growth simulation for a railway axle under input data variability","volume":"144","author":"Giannella","year":"2021","journal-title":"Int. J. Fatigue"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.engfracmech.2007.03.039","article-title":"Comparison of crack growth simulation by DBEM and FEM for SEN-specimens undergoing torsion or bending loading","volume":"75","author":"Citarella","year":"2008","journal-title":"Eng. Fract. Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1730","DOI":"10.1016\/j.engfracmech.2010.03.012","article-title":"Comparison of DBEM and FEM Crack Path Predictions in a notched Shaft under Torsion","volume":"77","author":"Citarella","year":"2010","journal-title":"Eng. Fract. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Citarella, R., and Federico, L. (2018). Advances in Vibroacoustics and Aeroacustics of Aerospace and Automotive Systems. Appl. Sci., 8.","DOI":"10.3390\/app8030366"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Citarella, R., Federico, L., and Barbarino, M. (2020). Aeroacustic and Vibroacoustic Advancement in Aerospace and Automotive Systems. Appl. Sci., 10.","DOI":"10.3390\/app10113853"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"926","DOI":"10.1016\/j.cma.2019.06.012","article-title":"Structural shape optimization of three dimensional acoustic problems with isogeometric boundary element methods","volume":"355","author":"Chen","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"112806","DOI":"10.1016\/j.cma.2019.112806","article-title":"Acoustic topology optimization of sound absorbing materials directly from subdivision surfaces with isogeometric boundary element methods","volume":"362","author":"Chen","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_9","unstructured":"Hughes, T.J.R. (2000). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice Hall."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Armentani, E., Giannella, V., Citarella, R., Parente, A., and Pirelli, M. (2019). Substructuring of a Petrol Engine: Dynamic Characterization and Experimental Validation. Appl. Sci., 9.","DOI":"10.3390\/app9224969"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Perrella, M., Gerbino, S., and Citarella, R. (2017). BEM in Biomechanics: Modelling Advances and Limitations, Book Chapter in Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes, Academic Press.","DOI":"10.1016\/B978-0-12-811718-7.00008-3"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/S0021-9991(95)90238-4","article-title":"Orthogonal collocation in the nonconforming boundary element method","volume":"121","author":"Karur","year":"1995","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/0955-7997(92)90150-6","article-title":"Use of discontinuous boundary elements for fracture mechanics analysis","volume":"10","author":"Trevelyan","year":"1992","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/j.soildyn.2007.07.007","article-title":"Elastodynamic boundary element formulation employing discontinuous curved elements","volume":"28","author":"Yerli","year":"2008","journal-title":"Soil Dyn. Earthq. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.enganabound.2019.04.011","article-title":"Discontinuous isogeometric boundary element (IGABEM) formulations in 3D automotive acoustics","volume":"105","author":"Sun","year":"2019","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Xu, J.M., and Brebbia, C.A. (1986, January 8). Optimum positions for the nodes in discontinuous boundary elements. In Proceeding of the 8th Conference on Boundary Elements, Tokyo, Japan.","DOI":"10.1007\/978-3-662-22335-2_22"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/0264-682X(88)90018-4","article-title":"On the accuracy of continuous and discontinuous boundary elements","volume":"5","author":"Parreira","year":"1988","journal-title":"Eng. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1142\/S0218396X03001985","article-title":"Influence of Element Types on Numeric Error for Acoustic Boundary Elements","volume":"11","author":"Marburg","year":"2008","journal-title":"J. Comput. Acoust"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1885","DOI":"10.1002\/nme.1620231008","article-title":"Conforming versus non-conforming boundary elements in three-dimensional elastostatics","volume":"23","author":"Manolis","year":"1986","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"ref_20","unstructured":"Patterson, C., and Elsebai, N.A.S. (1983, January 8\u201311). A family of partially discontinuous boundary element for three-dimensional analysis. In Proceeding of the 5th Conference on Boundary Elements, Hiroshima, Japan."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0955-7997(95)00003-7","article-title":"A comparison of the semidiscontinuous element and multiple node with auxiliary boundary collocation approaches for the boundary element method","volume":"15","author":"Subia","year":"1995","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"920","DOI":"10.1016\/j.oceaneng.2008.01.011","article-title":"Removing irregular frequencies by a partial discontinuous higher order boundary element method","volume":"35","author":"Sun","year":"2008","journal-title":"Ocean Eng."},{"key":"ref_23","unstructured":"Manolis, G.D., and Beskos, D.E. (1988). Boundary Element Method in Elastodynamics, Spon Press."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/S0045-7825(98)00294-1","article-title":"On the numerical stability of time-domain elastodynamic analyses by BEM","volume":"173","author":"Frangi","year":"1999","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0955-7997(02)00082-6","article-title":"Numerical behavior of time domain BEM for three-dimensional transient elastodynamic problem","volume":"27","author":"Marrero","year":"2003","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.enganabound.2018.08.001","article-title":"On the stability of direct time-domain boundary element methods for elastodynamics","volume":"96","author":"Bai","year":"2018","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.enganabound.2016.02.002","article-title":"Element-subdivision method for evaluation of singular integrals over narrow strip boundary elements of super thin and slender structures","volume":"66","author":"Gao","year":"2016","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1016\/j.apm.2019.11.053","article-title":"Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach","volume":"80","author":"Xie","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"934","DOI":"10.1016\/j.enganabound.2010.04.009","article-title":"An element implementation of the boundary face method for 3D potential problems","volume":"34","author":"Qin","year":"2010","journal-title":"Eng. Anal. Bound. 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