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We propose a novel semi\u2010analytical energy model for boundary handling that unifies fluid simulation and boundary interactions within a variational framework. The model comprises two key components: a semi\u2010analytical bulk energy formulation that mitigates particle deficiency issues in the evaluation of bulk energy, and a nonlocal contact potential that effectively prevents particle penetration into boundaries. Both energy terms are naturally compatible with the Semi\u2010Implicit SPH (SISPH), and a unified Hessian\u2010free solver combined with reduced\u2010order collision detection enables an efficient and stable GPU\u2010based implementation for both fluid dynamics and nonlinear fluid\u2010solid interactions. Furthermore, the unified treatment of fluid bulk energy and boundary energy via the semi\u2010analytical formulation robustly corrects penetrations in practice, even under severe compression scenarios involving complex moving boundaries. Compared with existing semi\u2010analytical boundary treatments, our method is more robust under fast boundary motion and strong compression. Across challenging benchmarks with sharp features, narrow gaps, and moving meshes, it remains stable and penetration\u2010free where prior methods often fail.<\/jats:p>","DOI":"10.1111\/cgf.70333","type":"journal-article","created":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T13:18:07Z","timestamp":1775827087000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Semi\u2010Analytical Energy Model for Particle\u2010Based Fluid Simulation Involving Complex Moving Boundaries"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-1829-9914","authenticated-orcid":false,"given":"Junyuan","family":"Liu","sequence":"first","affiliation":[{"name":"University of Chinese Academy of Sciences  Beijing China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7416-5110","authenticated-orcid":false,"given":"Shusen","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Software Chinese Academy of Sciences  Beijing China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-4578-9701","authenticated-orcid":false,"given":"Yuzhong","family":"Guo","sequence":"additional","affiliation":[{"name":"Institute of Software Chinese Academy of Sciences  Beijing China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruikai","family":"Liang","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences  Beijing China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Li","sequence":"additional","affiliation":[{"name":"University of Chinese Academy of Sciences  Beijing China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8870-2482","authenticated-orcid":false,"given":"Xiaowei","family":"He","sequence":"additional","affiliation":[{"name":"Institute of Software Chinese Academy of Sciences  Beijing China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2026,4,10]]},"reference":[{"key":"e_1_2_10_2_2","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-fluid-121108-145519"},{"key":"e_1_2_10_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2012.05.005"},{"key":"e_1_2_10_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185558"},{"key":"e_1_2_10_5_2","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185592"},{"key":"e_1_2_10_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/31804862"},{"key":"e_1_2_10_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cag.2018.08.001"},{"key":"e_1_2_10_8_2","doi-asserted-by":"crossref","unstructured":"Bender JanandKoschier Dan. \u201cDivergence-Free Smoothed Particle Hydrodynamics\u201d.Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation.2015 147\u2013155. doi:10.1145\/2786784.27867962.","DOI":"10.1145\/2786784.2786796"},{"key":"e_1_2_10_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cag.2014.07.004"},{"key":"e_1_2_10_10_2","doi-asserted-by":"crossref","unstructured":"Bender Jan Kugelstadt Tassilo Weiler Marcel andKoschier Dan. \u201cVolume maps: An implicit boundary representation for SPH\u201d.Proceedings of the 12th ACM SIGGRAPH Conference on Motion Interaction and Games.2019 1\u2013103.","DOI":"10.1145\/3359566.3360077"},{"key":"e_1_2_10_11_2","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2020.3004245"},{"key":"e_1_2_10_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(99)00051-1"},{"key":"e_1_2_10_13_2","doi-asserted-by":"publisher","DOI":"10.1145\/2601097.2601116"},{"key":"e_1_2_10_14_2","unstructured":"Becker MarkusandTwschner Matthias. \u201cWeakly Compressible SPH for Free Surface Flows\u201d.Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation.2007 209\u20132172."},{"key":"e_1_2_10_15_2","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.fluid.30.1.329"},{"key":"e_1_2_10_16_2","doi-asserted-by":"publisher","DOI":"10.1145\/3414685.3417804"},{"key":"e_1_2_10_17_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14132"},{"key":"e_1_2_10_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7091-7486-9_5"},{"key":"e_1_2_10_19_2","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459678"},{"key":"e_1_2_10_20_2","doi-asserted-by":"publisher","DOI":"10.1002\/fld.3666"},{"key":"e_1_2_10_21_2","volume-title":"Classification and Overview of Meshfree Methods","author":"Fries Thomas-Peter","year":"2003"},{"key":"e_1_2_10_22_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12754"},{"key":"e_1_2_10_23_2","doi-asserted-by":"publisher","DOI":"10.1145\/3284980"},{"key":"e_1_2_10_24_2","doi-asserted-by":"crossref","unstructured":"Huerta Antonio Belytschko Ted Fern\u00e1ndez-M\u00e9ndez Sonia andRabczuk Timon. \u201cMeshfree methods\u201d. 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