Isogeometric Analysis: a high-order method for PDEs

Abstract: Isogeometric Analysis was proposed in the seminal work of Hughes, Cottrell, and Bazilevs in 2005, and be seen as a generalisation of the finite element method that replaces classical $C^0$ finite elements with smooth splines. Doing so, IGA aims to be easily compatible with computer-aided geometric design systems, where smooth splines are used to create computational geometric models. In this framework, there has been a successful creation of novel, robust, high-order accurate numerical methods for solving PDEs.

The concept of k-refinement (or K-method) was proposed as one of the key features of isogeometric analysis, "a new, more efficient, higher-order concept", in the original isogeometric article by Hughes and co-workers. The idea of using high-degree and continuity splines (or NURBS, etc.) as a basis for a new high-order method appeared very promising from the beginning. The k-refinement leads to several advantages: higher accuracy per degree-of-freedom, improved spectral accuracy, the possibility of structure-preserving smooth discretizations are the most interesting features that have been studied actively in the community. At the same time, the k-refinement brings significant challenges at the computational level: using standard finite element routines, its computational cost grows with respect to the degree, making degree raising computationally expensive. After a brief introduction of Isogeometric Analysis, I will discuss ideas from [Sangalli and Tani, CMAME, 2018, arXiv:1712.08565] and following works, that allow a computationally efficient k-refinement.

computational engineering, finance, and sciencenumerical analysis

Audience: researchers in the topic

Australian Seminar on Computational Mathematics

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