TY - CHAP
T1 - Smooth multi-patch discretizations in Isogeometric Analysis
AU - Hughes, Thomas J.R.
AU - Sangalli, Giancarlo
AU - Takacs, Thomas
AU - Toshniwal, Deepesh
PY - 2021
Y1 - 2021
N2 - With the aim of a seamless integration with Computer-Aided Design, Isogeometric Analysis has been proposed by Hughes et al. (2005) as a numerical technique for the solution of partial differential equations. Indeed, isogeometric analysis is based on splines, the same functions that are adopted for geometry parametrizations in CAD. Smooth splines yield two important benefits when compared to C0 piecewise polynomial approximations: superior accuracy and stability, and the possibility to directly discretize high-order differential equations, such as the ones arising in thin-shell theory, in fracture models, in phase-field based multiphase flows, and in geometric flows on surfaces. In this chapter we review three different methods to construct C1 isogeometric spaces on multi-patch domains or unstructured quadrilateral meshes. The first, proposed in Collin et al. (2016), is based on the concept of geometric continuity, well-known in geometric design. The second, from Toshniwal et al. (2017c), uses a specific singular construction (the D-patch construction) at extraordinary points. The third is a polar construction from Toshniwal et al. (2017a). Such constructions possess properties that make them suitable for performing both computational analysis and geometric modeling.
AB - With the aim of a seamless integration with Computer-Aided Design, Isogeometric Analysis has been proposed by Hughes et al. (2005) as a numerical technique for the solution of partial differential equations. Indeed, isogeometric analysis is based on splines, the same functions that are adopted for geometry parametrizations in CAD. Smooth splines yield two important benefits when compared to C0 piecewise polynomial approximations: superior accuracy and stability, and the possibility to directly discretize high-order differential equations, such as the ones arising in thin-shell theory, in fracture models, in phase-field based multiphase flows, and in geometric flows on surfaces. In this chapter we review three different methods to construct C1 isogeometric spaces on multi-patch domains or unstructured quadrilateral meshes. The first, proposed in Collin et al. (2016), is based on the concept of geometric continuity, well-known in geometric design. The second, from Toshniwal et al. (2017c), uses a specific singular construction (the D-patch construction) at extraordinary points. The third is a polar construction from Toshniwal et al. (2017a). Such constructions possess properties that make them suitable for performing both computational analysis and geometric modeling.
KW - C approximation
KW - Computer-Aided Design
KW - extraordinary points
KW - geometric continuity
KW - Isogeometric Analysis
KW - multi-patch
KW - NURBS
KW - polar parametrization
KW - singular parametrization
KW - splines
UR - http://www.scopus.com/inward/record.url?scp=85094823651&partnerID=8YFLogxK
U2 - 10.1016/bs.hna.2020.09.002
DO - 10.1016/bs.hna.2020.09.002
M3 - Chapter
AN - SCOPUS:85094823651
VL - 22
T3 - Handbook of Numerical Analysis
SP - 467
EP - 543
BT - Geometric Partial Differential Equations - Part II
ER -