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.

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.

We extend the recently introduced divergence-conforming immersed boundary (DCIB) method [1] to fluid-structure interaction (FSI) problems involving closed co-dimension one solids. We focus on capsules and vesicles, whose discretization is particularly challenging due to the higher-order derivatives that appear in their formulations. In two-dimensional settings, we employ cubic B-splines with periodic knot vectors to obtain discretizations of closed curves with C² inter-element continuity. In three-dimensional settings, we use analysis-suitable bi-cubic T-splines to obtain discretizations of closed surfaces with at least C¹ inter-element continuity. Large spurious changes of the fluid volume inside closed co-dimension one solids are a well-known issue for IB methods. The DCIB method results in volume changes orders of magnitude lower than conventional IB methods. This is a byproduct of discretizing the velocity-pressure pair with divergence-conforming B-splines, which lead to negligible incompressibility errors at the Eulerian level. The higher inter-element continuity of divergence-conforming B-splines is also crucial to avoid the quadrature/interpolation errors of IB methods becoming the dominant discretization error. Benchmark and application problems of vesicle and capsule dynamics are solved, including mesh-independence studies and comparisons with other numerical methods.

In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure that the smoothness across a subset of the mesh edges can be reduced while maintaining stability of the dimension. The conditions have a simple geometric interpretation. Examples are presented to show the applicability of the results on both hierarchical and non-hierarchical T-meshes. For hierarchical T-meshes it is shown that mixed smoothness spline spaces that contain the space of PHT-splines (Deng et al., 2008) always have stable dimension.

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.

Analysis-suitable T-splines (ASTS) including both extraordinary points and T-junctions are used to solve Kirchhoff–Love shell problems. Extraordinary points are required to represent surfaces with arbitrary topological genus. T-junctions enable local refinement of regions where increased resolution is needed. The benefits of using ASTS to define shell geometries are at least two-fold: (1) The manual and time-consuming task of building a new mesh from scratch using the CAD geometry as an input is avoided and (2) C¹ or higher inter-element continuity enables the discretization of shell formulations in primal form defined by fourth-order partial differential equations. A complete and state-of-the-art description of the development of ASTS, including extraordinary points and T-junctions, is presented. In particular, we improve the construction of C¹-continuous non-negative spline basis functions near extraordinary points to obtain optimal convergence rates with respect to the square root of the number of degrees of freedom when solving linear elliptic problems. The applicability of the proposed technology to shell analysis is exemplified by performing geometrically nonlinear Kirchhoff–Love shell simulations of a pinched hemisphere, an oil sump of a car, a pipe junction, and a B-pillar of a car with 15 holes. Building ASTS for these examples involves using T-junctions and extraordinary points with valences 3, 5, and 6, which often suffice for the design of free-form surfaces. Our analysis results are compared with data from the literature using either a seven-parameter shell formulation or Kirchhoff–Love shells. We have also imported both finite element meshes and ASTS meshes into the commercial software LS-DYNA, used Reissner–Mindlin shells, and compared the result with our Kirchhoff–Love shell results. Excellent agreement is found in all cases. The complexity of the shell geometries considered in this paper shows that ASTS are applicable to real-world industrial problems.

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.

For T a planar triangulation, let R_m ^r(T) denote the space of bivariate splines on T such that f∈R_m ^r(T) is C^r(τ) smooth across an interior edge τ and, for triangle σ in T, f|_σ is a polynomial of total degree at most m(σ)∈Z_≥0. The map m:σ↦Z_≥0 is called a non-uniform degree distribution on the triangles in T, and we consider the problem of computing (or estimating) the dimension of R_m ^r(T) in this paper. Using homological techniques, developed in the context of splines by Billera (1988), we provide combinatorial lower and upper bounds on the dimension of R_m ^r(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in R_m ^r(T) can be determined exactly. In the special case of a constant map m, the lower and upper bounds are equal to those provided by Mourrain and Villamizar (2013).