A comparison of smooth basis constructions for isogeometric analysis

H. M. Verhelst*, P. Weinmüller, A. Mantzaflaris, T. Takacs, D. Toshniwal

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost-C1, Analysis-Suitable G1 and the Approximate C1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C1 and Analysis-Suitable G1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost-C1 and D-Patch provide relatively easy construction on complex geometries. The Almost-C1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C1 and Analysis-Suitable G1 applicable to more complex geometries.

Original languageEnglish
Article number116659
Number of pages27
JournalComputer Methods in Applied Mechanics and Engineering
Publication statusPublished - 2024


Funding Information:
HMV is grateful to Delft University of Technology , specifically the faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) and the faculty of Mechanical, Maritime and Materials Engineering (3mE) for the financial support to conduct this research. AM acknowledges support from EU’s Horizon 2020 research under the Marie Sklodowska-Curie grant 860843 and DT is grateful to ANSYS Inc. for their financial support. Furthermore, the authors thank Wei Jun Wong from Delft University of Technology for delivering the ABAQUS reference results, and the community of the Geometry + Simulation Modules (a.k.a. gismo) for their continuous effort on improving the code.


  • Biharmonic equation
  • Isogeometric analysis
  • Kirchhoff–Love shell
  • Unstructured splines


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