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Art"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The construction of complex volumetric parametric models has long been a bottleneck in achieving integrated design and simulation modeling. To enhance model quality and simplify the modeling process, this paper proposes an innovative method for improving the continuity of complex volumetric parametric models. First, depending on whether the input consists of design parameters or surface models, a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$G^0\/C^0$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\/<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>C<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -continuous volumetric parametric model is generated using either a creation or recreation approach. A new data structure is also developed to store the control-point indices and their topological relationships. Next, based on the actual connections among different patches in the volumetric models generated using the two aforementioned methods, the continuity conditions for three different scenarios are formulated. For each scenario, the corresponding systems of equations are established to determine the control points. Subsequently, an algorithm is developed to automatically sort, store, and adjust the relevant control points so that the volumetric parametric model satisfies the\n                    <jats:italic>G<\/jats:italic>\n                    <jats:sup>1<\/jats:sup>\n                    continuity condition. The generated examples demonstrate that the volumetric parametric modeling method proposed in this study is effective for constructing complex models, significantly improving the model quality and rendering them more suitable for subsequent analysis and processing.\n                  <\/jats:p>","DOI":"10.1186\/s42492-026-00217-1","type":"journal-article","created":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T07:00:23Z","timestamp":1775718023000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Construction of complex non-uniform rational B-spline volume parametric models with G1 continuity"],"prefix":"10.1186","volume":"9","author":[{"given":"Dan","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4914-514X","authenticated-orcid":false,"given":"Long","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiahong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,9]]},"reference":[{"issue":"39\u201341","key":"217_CR1","doi-asserted-by":"publisher","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","volume":"194","author":"TJR Hughes","year":"2005","unstructured":"Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. 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