Locally based construction of analysis-suitable $G^1$ multi-patch spline surfaces

2023-04-17

15:00 — 16:00

FAMNIT-MP1

Aljaž Kosmač (UP FAMNIT, Slovenia)

Isogeometric analysis was first introduced as an approach for solving partial differential equations (PDE) that aims towards integration of worlds of Computer aided design (CAD) and Finite element analysis (FEA) by employing the same spaces of functions in domain representation as well as describing the solution of the PDE on that domain. In comparison to traditional finite element functions, it has been observed that functions of higher smoothness have positive effect on stability and convergence properties of numerical solution. Using isogeometric analysis for solving PDEs on multi-patch spline surfaces is an active area of research. But not every multi-patch spline surface is suitable for such task. Analysis-suitable $G^1$ (AS-$G^1$) multi-patch spline surfaces [1, 2] are particular $G^1$-smooth multi-patch spline surfaces, which are needed to ensure the construction of $C^1$-smooth multi-patch spline spaces with optimal polynomial reproduction properties. In [3], a global method to construct AS-$G^1$ planar multi-patch parameterisations has been developed. In this talk we present a locally based approach for the design of AS-$G^1$ multi-patch spline surfaces. The approach is based on a Lagrange multiplier method and generates AS-$G1$ multi-patch spline surfaces by approximating a given $G^1$-smooth but non-AS-$G^1$ multi- patch surface. Several numerical examples demonstrate the potential of the presented technique for the construction of AS-$G^1$ multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, over them.

Joint work with: A. Farahat, M. Kapl, V. Vitrih.

References:

[1] A. Collin, G. Sangalli, and T. Takacs: Analysis-suitable G1 multi-patch parametriza- tions for C1 isogeometric spaces, Comput. Aided Geom. Des., 47 (2016), 93–113.

[2] A. Farahat, B. Jüttler, M. Kapl, and T. Takacs: Isogeometric analysis with C1- smooth functions over multi-patch surfaces, Comput. Methods Appl. Mech. Engrg. 403 (2023), 115706.

[3] M. Kapl, G. Sangalli, and T. Takacs: Construction of analysis-suitable G1 planar multi-patch parameterizations, Comput. Aided Des., 97 (2018), 41–55.

We look forward to sharing the passion for math with you!