We propose and analyze discretizations of the de Rham complex built from Tchebycheffian B-splines (TB-splines). In particular, we present a unified theoretical framework for approximating scalar and vector fields in mixed isogeometric methods using TB-spline spaces whose local pieces lie in the nullspaces of suitable constant-coefficient linear differential operators. This construction enriches the classical polynomial spline spaces by incorporating trigonometric and exponential shape functions while preserving key features of polynomial B-splines (e.g., local support, partition of unity, stable bases) and avoiding the limitations of rational (NURBS) models. For one-, two- and three-dimensional settings, we construct TB-spline de Rham complexes together with bounded commuting projections. The spaces in the resulting discrete complexes enable robust discretizations of mixed scalar and vector Laplacians, the Maxwell eigenvalue problem, and others. We also provide numerical experiments illustrating optimal approximation rates, spectrally-correct solutions to the Maxwell eigenvalue problem, and an application to incompressible flows. These tests validate the theory and demonstrate the practical advantages of TB-splines over purely polynomial constructions.

In this paper, we describe a general class of C¹ smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids — some of the most important primitives for CAD and CAE. The univariate rational splines are assembled by transforming multiple sets of NURBS basis functions via so-called design-through-analysis compatible extraction matrices; different sets of NURBS are allowed to have different polynomial degrees and weight functions. Tensor products of the univariate splines yield multivariate splines. In the bivariate setting, we describe how similar design-through-analysis compatible transformations of the tensor-product splines enable the construction of smooth surfaces containing one or two polar singularities. The material is self-contained, and is presented such that all tools can be easily implemented by CAD or CAE practitioners within existing software that support NURBS. To this end, we explicitly present the matrices (a) that describe our splines in terms of NURBS, and (b) that help refine the splines by performing (local) degree elevation and knot insertion. Finally, all C¹ spline constructions yield spline basis functions that are locally supported and form a convex partition of unity.

This paper addresses theoretical considerations behind the algorithmic computation of polynomial multi-degree spline basis functions as presented in Toshniwal et al. (2017). The approach in Toshniwal et al. (2017) breaks from the reliance on computation of integrals recursively for building B-spline-like basis functions that span a given multi-degree spline space. The gains in efficiency are indisputable; however, the theoretical robustness needs to be examined. In this paper, we show that the construction of Toshniwal et al. (2017) yields linearly independent functions with the minimal support property that span the entire multi-degree spline space and form a convex partition of unity.

In this paper, we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described.