{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T19:02:12Z","timestamp":1768071732930,"version":"3.49.0"},"reference-count":21,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2020,3,20]],"date-time":"2020-03-20T00:00:00Z","timestamp":1584662400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"name":"State Key Laboratory of Scientific and Engineering Computing (LSEC), and National Center for Mathematics and Interdisciplinary Sciences of Chinese Academy of Sciences"},{"name":"National Magnetic Confinement Fusion Science Program of China","award":["2015GB110003"],"award-info":[{"award-number":["2015GB110003"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["91430215, 91530323, 11725106, 11831016, and 11771440"],"award-info":[{"award-number":["91430215, 91530323, 11725106, 11831016, and 11771440"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"National Key Research and Development Program of China","award":["2016YFB0201304"],"award-info":[{"award-number":["2016YFB0201304"]}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2020,3,31]]},"abstract":"<jats:p>\n            Given a shape regular tetrahedron and a curved surface that is defined implicitly by a nonlinear level set function and divides the tetrahedron into two sub-domains, a general-purpose, robust, and high-order numerical algorithm is proposed in this article for computing both volume integrals in the sub-domains and surface integrals on their common boundary. The algorithm uses a direct approach that decomposes 3D volume integrals or 2D surface integrals into multiple 1D integrals and computes the 1D integrals with Gaussian quadratures. It only requires finding roots of univariate nonlinear functions in given intervals and evaluating the integrand, the level set function, and the gradient of the level set function at given points. It can achieve arbitrarily high accuracy by increasing the orders of Gaussian quadratures, and it does not need extra\n            <jats:italic>a priori<\/jats:italic>\n            knowledge about the integrand and the level set function. The code for the algorithm is freely available in the open-source finite element toolbox Parallel Hierarchical Grid (PHG) and can serve as a basic building block for implementing 3D high-order numerical algorithms involving implicit interfaces or boundaries.\n          <\/jats:p>","DOI":"10.1145\/3372144","type":"journal-article","created":{"date-parts":[[2020,3,20]],"date-time":"2020-03-20T10:23:08Z","timestamp":1584699788000},"page":"1-18","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":16,"title":["High-order Numerical Quadratures in a Tetrahedron with an Implicitly Defined Curved Interface"],"prefix":"10.1145","volume":"46","author":[{"given":"Tao","family":"Cui","sequence":"first","affiliation":[{"name":"Chinese Academy of Sciences, China and University of Chinese Academy of Sciences, Yuquan Road, Beijing, China"}]},{"given":"Wei","family":"Leng","sequence":"additional","affiliation":[{"name":"Chinese Academy of Sciences, China and University of Chinese Academy of Sciences, Yuquan Road, Beijing, China"}]},{"given":"Huaqing","family":"Liu","sequence":"additional","affiliation":[{"name":"Chinese Academy of Sciences, China and University of Chinese Academy of Sciences, Yuquan Road, Beijing, China"}]},{"given":"Linbo","family":"Zhang","sequence":"additional","affiliation":[{"name":"Chinese Academy of Sciences, China and University of Chinese Academy of Sciences, Yuquan Road, Beijing, China"}]},{"given":"Weiying","family":"Zheng","sequence":"additional","affiliation":[{"name":"Chinese Academy of Sciences, China and University of Chinese Academy of Sciences, Yuquan Road, Beijing, China"}]}],"member":"320","published-online":{"date-parts":[[2020,3,20]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/060666482"},{"key":"e_1_2_1_2_1","first-page":"567","article-title":"An adaptive immersed finite element method with arbitrary Lagrangian-Eulerian scheme for parabolic equations in time variable domains","volume":"12","author":"Chen Z.","year":"2015","journal-title":"Int. J. Numer. Anal. Model."},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00524-8"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1137\/090763093"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2009.03.044"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.161114.021015a"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/140966290"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.12.005"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.4569"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3282"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/16M1102227"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.5121"},{"key":"e_1_2_1_13_1","unstructured":"Haijun Wu and Yuanming Xiao. 2010. An unfitted hp-interface penalty finite element method for elliptic interface problems arXiv preprint arXiv:1007.2893.  Haijun Wu and Yuanming Xiao. 2010. An unfitted hp -interface penalty finite element method for elliptic interface problems arXiv preprint arXiv:1007.2893."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.06.004"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/0721042"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02238487"},{"key":"e_1_2_1_17_1","first-page":"89","article-title":"A set of symmetric quadrature rules on triangles and tetrahedra","volume":"27","author":"Zhang L. B.","year":"2009","journal-title":"J. Comput. Math."},{"key":"e_1_2_1_18_1","first-page":"65","article-title":"A parallel algorithm for adaptive local refinement of tetrahedral meshes using bisection","volume":"2","author":"Zhang L. B.","year":"2009","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"e_1_2_1_19_1","unstructured":"The toolbox Parallel Hierarchical Grid (PHG). 2019. Retrieved from http:\/\/lsec.cc.ac.cn\/phg.  The toolbox Parallel Hierarchical Grid (PHG). 2019. Retrieved from http:\/\/lsec.cc.ac.cn\/phg."},{"key":"e_1_2_1_20_1","volume-title":"A parallel sparse direct solver","author":"MUMPS","year":"2017"},{"key":"e_1_2_1_21_1","volume-title":"Iterative Methods for Sparse Linear Systems","author":"Saad Y.","edition":"2"}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3372144","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3372144","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T23:45:06Z","timestamp":1750203906000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3372144"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,20]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,3,31]]}},"alternative-id":["10.1145\/3372144"],"URL":"https:\/\/doi.org\/10.1145\/3372144","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"value":"0098-3500","type":"print"},{"value":"1557-7295","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,20]]},"assertion":[{"value":"2018-09-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-11-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-03-20","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}