{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:53:05Z","timestamp":1776847985973,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"215","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n \n T<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {T}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a tetrahedral mesh. We present a 3-D local refinement algorithm for\n \n \n \n \n T<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {T}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which is mainly based on an 8-\n subtetrahedron subdivision<\/italic>\n procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron\n \n \n \n \n \n T<\/mml:mi>\n <\/mml:mrow>\n \n \u2208\n \n <\/mml:mo>\n \n T<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \\mathbf {T} \\in \\mathcal {T}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore,\n \n \n \n \n \n \u03b7\n \n <\/mml:mi>\n (<\/mml:mo>\n \n \n T<\/mml:mi>\n <\/mml:mrow>\n i<\/mml:mi>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n )<\/mml:mo>\n \n \u2265\n \n <\/mml:mo>\n c<\/mml:mi>\n \n \u03b7\n \n <\/mml:mi>\n (<\/mml:mo>\n \n T<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \\eta (\\mathbf {T}_i^{n}) \\geq c \\eta (\\mathbf {T})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \n T<\/mml:mi>\n <\/mml:mrow>\n \n \u2208\n \n <\/mml:mo>\n \n T<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \\mathbf {T} \\in \\mathcal {T}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n c<\/mml:mi>\n c<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a positive constant independent of\n \n \n \n \n T<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {T}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and the number of refinement levels,\n \n \n \n \n \n T<\/mml:mi>\n <\/mml:mrow>\n i<\/mml:mi>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n \\mathbf {T}_i^{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is any refined tetrahedron of\n \n \n \n \n T<\/mml:mi>\n <\/mml:mrow>\n \\mathbf {T}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n \n \u03b7\n \n <\/mml:mi>\n \\eta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a tetrahedron shape measure. It is also proved that local refinements on tetrahedra can be smoothly extended to their neighbors to maintain a conforming mesh. Experimental results show that the ratio of the number of tetrahedra actually refined to the number of tetrahedra chosen for refinement is bounded above by a small constant.\n <\/p>","DOI":"10.1090\/s0025-5718-96-00748-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1183-1200","source":"Crossref","is-referenced-by-count":67,"title":["Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision"],"prefix":"10.1090","volume":"65","author":[{"given":"Anwei","family":"Liu","sequence":"first","affiliation":[]},{"given":"Barry","family":"Joe","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","unstructured":"R. E. Bank (1990), PLTMG: A Software Package for Solving Elliptic Partial Differential Equations: Users\u2019 Guide 6.0, SIAM, Philadelphia."},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/BF02241777","article-title":"An adaptive, multilevel method for elliptic boundary value problems","volume":"26","author":"Bank, R. E.","year":"1981","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"3","series-title":"Johns Hopkins Series in the Mathematical Sciences","isbn-type":"print","volume-title":"Matrix computations","volume":"3","author":"Golub, Gene H.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0801837723","edition":"2"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"718","DOI":"10.1137\/0910044","article-title":"Three-dimensional triangulations from local transformations","volume":"10","author":"Joe, Barry","year":"1989","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"key":"5","doi-asserted-by":"crossref","unstructured":"B. Joe (1991), Delaunay versus max-min solid angle triangulations for three-dimensional mesh generation, Intern. J. Num. Meth. Eng., 31, pp. 987-997.","DOI":"10.1002\/nme.1620310511"},{"key":"6","doi-asserted-by":"crossref","unstructured":"B. Joe (1994), Tetrahedral mesh generation in polyhedral regions based on convex polyhedron decompositions, Intern. J. Num. Meth. Eng., 37, pp. 693-713.","DOI":"10.1002\/nme.1620370409"},{"key":"7","doi-asserted-by":"crossref","unstructured":"B. Joe (1995), Construction of three-dimensional improved-quality triangulations using local transformations, SIAM J. Sci. Comput., 16, 1292\u20131307.","DOI":"10.1137\/0916075"},{"issue":"144","key":"8","doi-asserted-by":"publisher","first-page":"1147","DOI":"10.2307\/2006341","article-title":"A proof of convergence and an error bound for the method of bisection in \ud835\udc45\u207f","volume":"32","author":"Kearfott, Baker","year":"1978","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"9","unstructured":"A. Liu (1994), Quality local refinement of tetrahedral meshes, Ph.D. Thesis, Department of Computing Science, University of Alberta."},{"issue":"207","key":"10","doi-asserted-by":"publisher","first-page":"141","DOI":"10.2307\/2153566","article-title":"On the shape of tetrahedra from bisection","volume":"63","author":"Liu, Anwei","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"A. Liu and B. Joe (1994), Relationship between tetrahedron shape measures, BIT, 34, pp. 268-287.","DOI":"10.1007\/BF01955874"},{"key":"12","doi-asserted-by":"crossref","unstructured":"A. Liu and B. Joe (1995), Quality local refinement of tetrahedral meshes based on bisection, SIAM J. Sci. Comput., 16, 1269\u20131291.","DOI":"10.1137\/0916074"},{"key":"13","unstructured":"R. L\u00f6hner, K. Morgan, and O. C. Zienkiewicz (1986), Adaptive grid refinement for the compressible Euler equations, in I. Babu\u0161ka et al., Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley, New York."},{"key":"14","doi-asserted-by":"crossref","unstructured":"W. F. Mitchell (1989), A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. on Mathematical Software, 15, pp. 326-347.","DOI":"10.1145\/76909.76912"},{"key":"15","unstructured":"D. Moore and J. Warren (1990), Adaptive mesh generation I: packing space, Technical Report, Department of Computer Science, Rice University."},{"key":"16","doi-asserted-by":"crossref","unstructured":"T. W. Nehl and D. A. Field (1991), Adaptive refinement of first order tetrahedral meshes for magnetostatics using local Delaunay subdivisions, IEEE Trans. on Magnetics, 27, pp. 4193-4196.","DOI":"10.1109\/20.105026"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1002\/nme.1620200412","article-title":"Algorithms for refining triangular grids suitable for adaptive and multigrid techniques","volume":"20","author":"Rivara, M.-Cecilia","year":"1984","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"18","doi-asserted-by":"crossref","unstructured":"M.-C. Rivara (1987), A grid generator based on 4-triangles conforming mesh-refinement algorithms, Intern. J. Num. Meth. 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Zhang (1988), Multi-level iterative techniques, Ph.D. Thesis, Department of Mathematics, Pennsylvania State University."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00748-X\/S0025-5718-96-00748-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00748-X\/S0025-5718-96-00748-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:12:36Z","timestamp":1776719556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00748-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":21,"journal-issue":{"issue":"215","published-print":{"date-parts":[[1996,7]]}},"alternative-id":["S0025-5718-96-00748-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00748-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1996]]}}}