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To improve the conditioning of SGFEM, two stability techniques have been employed, namely, (a) a cubic polynomial has been used as the PU (partition of unity), instead of the standard FE hat-functions, to address the possible almost linear dependence between the PU functions and the enrichments, and (b) a local principal component analysis (LPCA) has been implemented to address the local bad conditioning produced by multi-fold enrichments at a node. The scaled condition number for the proposed SGFEM is shown to be <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(h^{-2})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>h<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (same as that of a standard Finite Element Method), for various relative positions of crack surface and mesh. The robustness of the scaled condition number for the proposed SGFEM, with respect to the relative positions of the crack-surface and the element boundaries, has been observed numerically. The numerical experiments for both the planar and fully 3D planar crack problems are presented to show the efficiency of the proposed SGFEM.<\/jats:p>","DOI":"10.1007\/s00211-022-01312-0","type":"journal-article","created":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T21:04:15Z","timestamp":1661807055000},"page":"475-509","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Stable generalized finite element method (SGFEM) for three-dimensional crack problems"],"prefix":"10.1007","volume":"152","author":[{"given":"Cu","family":"Cui","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qinghui","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Uday","family":"Banerjee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ivo","family":"Babu\u0161ka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,8,29]]},"reference":[{"key":"1312_CR1","doi-asserted-by":"publisher","first-page":"1051","DOI":"10.1016\/j.cma.2018.08.007","volume":"346","author":"K Agathos","year":"2019","unstructured":"Agathos, K., Bordas, S.P.A., Chatzi, E.: Improving the conditioning of XFEM\/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization. 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