Deepesh Toshniwal

Deepesh Toshniwal, Bernard Mourrain, Thomas Hughes

Polynomial splines are ubiquitous in the fields of computer aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation requires the construction of a suitable spanning set of linearly independent splines, and a theoretical understanding of the spline space dimension can be a useful tool when assessing possible approaches for building such splines. Here, we provide such a tool. Focusing on T-meshes, we study the dimension of the space of bivariate polynomial splines, and we discuss the general setting where local mesh adaptivity is combined with local polynomial degree adaptivity. The latter allows for the flexibility of choosing non-uniform bi-degrees for the splines, i.e., different bi-degrees on different faces of the T-mesh. In particular, approaching the problem using tools from homo-logical algebra, we generalize the framework and the discourse presented by Mourrain (2014) for uniform bi-degree splines. We derive combinatorial lower and upper bounds on the spline space dimension and subsequently outline sufficient conditions for the bounds to coincide.

Diogo C. Cabanas, Kendrick M. Shepherd, Deepesh Toshniwal et al.

The de Rham complex arises naturally when studying problems in electromagnetism and fluid mechanics. Stable numerical methods to solve these problems can be obtained by using a discrete de Rham complex that preserves the structure of the continuous one. This property is not necessarily guaranteed when the discrete function spaces are hierarchical B-splines, and research shows that an arbitrary choice of refinement domains may give rise to spurious harmonic fields that ruin the accuracy of the solution. We will focus on the two-dimensional de Rham complex over the unit square $Ω\subseteq \mathbb{R}^2$, and provide theoretical results and a constructive algorithm to ensure that the structure of the complex is preserved: when a pair of functions are in conflict some additional functions, forming an L-chain between the pair, are also refined. Another crucial aspect to consider in the hierarchical setting is the notion of admissibility, as it is possible to obtain optimal convergence rates of numerical solutions and improved stability by limiting the multi-level interaction of basis functions. We show that, under a common restriction, the admissibility class of the first space of the discrete complex persists throughout the remaining spaces. As such, admissible refinement can be combined with our new algorithm to obtain admissible meshes that also respect the structure of the de Rham complex. Moreover, we detail how our algorithm can be easily included in standard adaptive mesh refinement schemes. Finally, we include numerical results that motivate the importance of the previous concerns for the vector Laplace and Maxwell eigenvalue problems.

Christopher Zimmermann, Deepesh Toshniwal, Chad M. Landis et al.

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation. Both PDEs can be efficiently discretized using $C^1$-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise $C^1$-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-$α$ scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.