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International audience; Isolated singularities typically occur at self-intersection points of planar algebraic curves, curve offsets, intersections between spatial curves and surfaces, and so on. The information characterizing the singularity can be captured in a local dual basis, expressing combinations of vanishing derivatives at the singular point. Macaulay's algorithm is a classic algorithm for computing such a basis, for a point in an algebraic set. The integration method of Mourrain constructs much smaller matrices than Macaulay's approach, by performing integration on previously computed elements. In this work we are interested in the efficiency of dual basis computation, as well as its relation to orthogonal projection. First, we introduce an easy to implement criterion that avoids redundant computations during the computation of the dual basis, by deleting certain columns from the matrices in the integration method. In doing so, we explore general (non-monomial) bases for the associated primal quotient ring. Experiments show the efficient behaviour of the improved method. Second, we introduce the notion of directional multi-plicity, which expresses the multiplicity structure with respect to an axis, and is useful in understanding the geometry behind projection. We use this notion to shed light on the gap between the degree of the generator of the elimination ideal and the corresponding factor in the resultant.

Subdivision surfaces offer great flexibility in capturing irregular topologies combined with higher order smoothness. For instance, Loop and Catmull–Clark subdivision schemes provide C 2 smoothness everywhere except at extraordinary vertices, where the generated surfaces are C 1 smooth. The combination of flexibility and smoothness leads to their frequent use in geometric modeling and makes them an ideal candidate for performing isogeometric analysis of higher order problems with irregular topologies, e.g. thin shell problems. In the neighborhood of extraordinary vertices, however, the resulting surfaces (and the functions defined on them) are non-polynomial. Thus numerical integration based on quadrature has to be performed carefully to preserve the expected higher order consistency. This paper presents a detailed case study of different quadrature schemes for isogeometric discretizations of partial differential equations on closed surfaces with Loop’s subdivision scheme. As model problems we consider elliptic equations with the Laplace–Beltrami and the surface bi-Laplacian operator as well as the associated eigenvalue problem for the Laplace–Beltrami operator. A particular emphasis is on the robustness of the approach in the vicinity of extraordinary vertices. Based on a series of numerical experiments, different quadrature schemes are compared. Mid-edge quadrature, which can easily be implemented via a lookup table, turns out to be a preferable choice due to its robustness and efficiency.

This paper is devoted to techniques for adaptive spline projection via quasi-interpolation, enabling the efficient approximation of given functions. We employ local least-squares fitting in restricted hierarchical spline spaces to establish novel projection operators for hierarchical splines of degree p. This leads to efficient spline projectors that require O ( p d ) floating point operations and O ( 1 ) evaluations of the given function per degree of freedom, while providing essentially the same accuracy as global approximation. Our spline projectors are based on a unifying framework for quasi-interpolation in hierarchical spline spaces. We present a detailed comparison with the scheme of Speleers and Manni (2016) .

Even though isogeometric analysis has a clear advantage regarding the number of degrees of freedom needed to achieve a certain level of accuracy, the time needed for matrix assembly (by means of numerical integration of stiffness or mass matrix entries) constitutes a severe bottleneck in the process. In order to overcome this difficulty, we propose the new approach of Integration by Interpolation and Lookup (IIL). Firstly, applying spline interpolation to the common factors in the occurring integrals approximately transforms them into integrals of piecewise polynomial functions, whose integrands are expressed in tensor-product B-spline form. The common factors represent the influence of the geometry mapping (i.e., the NURBS domain parameterization) and the contributions of possibly non-constant material coefficients. Secondly, these integrals are evaluated exactly using pre-computed look-up tables for integrals of tri-products of univariate B-splines and their derivatives. For the model case of elliptic problems, we perform a theoretical analysis to demonstrate that the IIL method maintains the overall approximation order of the Galerkin discretization, provided that the spline interpolation is sufficiently accurate. Moreover, we provide a comparison of the computational complexity of our method with that of a standard Gauss quadrature method. Finally, we present experimental results to illustrate the performance of the IIL method and to support our theoretical results.