{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,29]],"date-time":"2026-08-29T19:16:06Z","timestamp":1788030966935,"version":"build-2784847793"},"reference-count":54,"publisher":"SAGE Publications","issue":"9","license":[{"start":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T00:00:00Z","timestamp":1775692800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"funder":[{"name":"Blue Project of Jiangsu Province"},{"name":"Youth Funding of Rocket Force University of Engineering","award":["2023QN-S003"],"award-info":[{"award-number":["2023QN-S003"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12072104"],"award-info":[{"award-number":["12072104"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12202136"],"award-info":[{"award-number":["12202136"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Damage Mechanics"],"published-print":{"date-parts":[[2026,9]]},"abstract":"<jats:p>\n                    The computational cost of peridynamics has historically limited its use to small-scale problems, despite its conceptual appeal for fracture modeling and failure analysis. This work introduces a revised peridynamic integration paradigm that enables high-fidelity simulations using substantially fewer integration points while maintaining accuracy. The key observation is that conventional peridynamic volume integration can be reformulated as two separable components, radial and angular, allowing independent numerical treatment of each. Building on this decoupling, we propose a peridynamic integration framework that couples Gauss\u2013 quadrature in the radial direction with Lebedev spherical quadrature for angular integration (PD-GL). This reformulation yields three main advances: (i) systematic reduction of integration points via structured projection algorithms, (ii) removal of repeated neighbor-search overhead through spatial hashing, and (iii) retention of high accuracy even for small horizons\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi>\u03b4<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                        <mml:mspace width=\".1em\"\/>\n                        <mml:mrow>\n                          <mml:mi mathvariant=\"normal\">\u0394<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mi>x<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    , where conventional peridynamics typically deteriorates. Across a range of horizon sizes (\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi>\u03b4<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                        <mml:mspace width=\".1em\"\/>\n                        <mml:mrow>\n                          <mml:mi mathvariant=\"normal\">\u0394<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mi>x<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    to\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mn>4<\/mml:mn>\n                        <mml:mspace width=\".1em\"\/>\n                        <mml:mrow>\n                          <mml:mi mathvariant=\"normal\">\u0394<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mi>x<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    ), PD-GL achieves more than a sevenfold speedup relative to standard implementations, with the advantage increasing as the horizon grows. By integrating multiple established acceleration strategies into a single unified scheme, PD-GL addresses the primary computational bottlenecks of peridynamic simulations and facilitates practical, high-fidelity fracture and failure modeling in engineering applications.\n                  <\/jats:p>","DOI":"10.1177\/10567895261436512","type":"journal-article","created":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T12:56:02Z","timestamp":1775739362000},"page":"1455-1481","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["An efficient peridynamic model for damage and fracture analysis: the PD-GL framework with Gauss\u2013Legendre\u2013Lebedev quadrature"],"prefix":"10.1177","volume":"35","author":[{"given":"Han","family":"Wang","sequence":"first","affiliation":[{"name":"Rocket Force University of Engineering"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5789-2044","authenticated-orcid":false,"given":"Liwei","family":"Wu","sequence":"additional","affiliation":[{"name":"Nanjing Tech University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dan","family":"Huang","sequence":"additional","affiliation":[{"name":"Hohai University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chuanqiang","family":"Yu","sequence":"additional","affiliation":[{"name":"Rocket Force University of Engineering"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2026,4,9]]},"reference":[{"key":"e_1_3_3_2_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-246X.2007.03214.x"},{"key":"e_1_3_3_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2022.114142"},{"key":"e_1_3_3_4_1","doi-asserted-by":"publisher","DOI":"10.1201\/9781315373331"},{"key":"e_1_3_3_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10704-015-0056-8"},{"key":"e_1_3_3_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2015.02.015"},{"key":"e_1_3_3_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2018.11.025"},{"key":"e_1_3_3_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijimpeng.2021.104004"},{"key":"e_1_3_3_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112592"},{"key":"e_1_3_3_10_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.5596"},{"key":"e_1_3_3_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tafmec.2024.104840"},{"key":"e_1_3_3_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2016.05.017"},{"key":"e_1_3_3_13_1","volume-title":"Introduction to Numerical Analysis","author":"Hildebrand FB","year":"1987","unstructured":"Hildebrand FB (1987) Introduction to Numerical Analysis. 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