{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T09:20:16Z","timestamp":1783070416827,"version":"3.54.6"},"reference-count":53,"publisher":"SAGE Publications","issue":"7","license":[{"start":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T00:00:00Z","timestamp":1772150400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Damage Mechanics"],"published-print":{"date-parts":[[2026,7]]},"abstract":"<jats:p>The presented case study benchmarks a novel design approach for evaluating the survival function of multidimensional dynamic systems subjected to stochastic, nonstationary environmental loading, with particular focus on naval architecture. The proposed design methodology combines a novel log-integral concept of the Integrated Cumulative Distribution Function (ICDF) for accurate modeling of failure probabilities with the Smoothed Particle Hydrodynamics (SPH) Computational Fluid Dynamics (CFD) method to simulate slamming forces on the vessel hull mid-section. The proposed design approach offers a robust tool for reliability and safety assessment of vessels and offshore structures, particularly in complex, nonlinear, adverse marine environments. A traditional four-parameter Weibull parametric fit is used to cross-validate the predicted design values. The combination of ICDF and SPH simulations may provide naval architects with a robust framework for enhancing the reliability analysis of marine structures under dynamic, rapidly changing loading conditions.<\/jats:p>\n                  <jats:p>The major novelty of this study lies in combining an SPH-based CFD approach with a successfully benchmarked novel probabilistic integral ICDF extrapolation scheme, which is particularly suitable for design when the underlying dataset is representative but limited in size. System performance or limit-state function depends on multiple random variables (e.g. load, resistance, and environmental factors). This method enables efficient estimation of extreme impact loads, thereby providing practical support for reliability-based design and operational safety assessment of high-speed craft undergoing underwater\/above-water entry and exit processes under nonstationary sea conditions.<\/jats:p>\n                  <jats:p>\n                    Engineering relevance: assessing the reliability of high-reliability structures where failure probabilities\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"normal\">Failure<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:mspace width=\"0.25em\"\/>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    are low (e.g.\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mo>\u2264<\/mml:mo>\n                        <mml:msup>\n                          <mml:mn>10<\/mml:mn>\n                          <mml:mrow>\n                            <mml:mo>\u2212<\/mml:mo>\n                            <mml:mn>6<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    ). Fundamental design concepts, such as the Most Probable Maximum (MPM) for non-Gaussian processes with clustering effects, are expressed in terms of a memory-modified mean up-crossing rate in a practical engineering context. The presented ICDF design scheme is shown to provide enhanced accuracy to the design values and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:msub>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"normal\">Failure<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    estimates, when the underlying data sample is of limited size.\n                  <\/jats:p>","DOI":"10.1177\/10567895261423606","type":"journal-article","created":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T08:41:14Z","timestamp":1772181674000},"page":"1045-1070","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["Cross-validation of vessel hull mid-section slamming damage assessment by novel log-integral reliability scheme"],"prefix":"10.1177","volume":"35","author":[{"given":"Zhihao","family":"Zhang","sequence":"first","affiliation":[{"name":"Jiangsu University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3196-8562","authenticated-orcid":false,"given":"Oleg","family":"Gaidai","sequence":"additional","affiliation":[{"name":"Shanghai Ocean University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiayao","family":"Sun","sequence":"additional","affiliation":[{"name":"Jiangsu University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Changjun","family":"Gu","sequence":"additional","affiliation":[{"name":"Changzhou FRP boatbuilding Co. 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