Tchebycheffian B-splines in isogeometric Galerkin methods

TL;DR

This paper explores the use of Tchebycheffian B-splines, which generalize polynomial splines, as a flexible and robust alternative to NURBS in isogeometric Galerkin methods, enabling enhanced geometric and analytical control.

Contribution

It introduces the application of TB-splines with null-space pieces in isogeometric Galerkin methods, expanding the flexibility beyond traditional polynomial and NURBS approaches.

Findings

TB-splines provide a versatile basis for isogeometric analysis.

They allow combining polynomials with exponential and trigonometric functions.

The approach offers a robust environment beyond NURBS limits.

Abstract

Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. In this paper, we consider the use of TB-splines with pieces belonging to null-spaces of constant-coefficient linear differential operators as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods. We discuss how to exploit the large flexibility of the geometrical and analytical features of the underlying Tchebycheff spaces according to problem-driven selection strategies. TB-splines offer a wide and robust environment for the isogeometric paradigm beyond the limits of the rational NURBS model.