Gaussian quadrature for C1C^1 cubic Clough-Tocher macro-triangles

A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud [14]. The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C1 cubic Clough-Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results into a factor of three reduction in the number of quadrature points needed to integrate the Clough-Tocher spline space exactly

Scalable multicomponent spectral analysis for high-throughput data annotation

Orchestrating parametric fitting of multicomponent spectra at scale is an essential yet underappreciated task in high-throughput quantification of materials and chemical composition. We present a systematic approach compatible with high-performance computing infrastructures using the MapReduce model and task-based parallelization. Our approach is realized in a software, pesfit, to enable efficient generation of high-quality data annotation and online spectral analysis as demonstrated using experimental materials characterization datasets

Transient wall shear stress estimation in coronary bifurcations using convolutional neural networks

Background and Objective: Haemodynamic metrics, such as blood flow induced shear stresses at the inner vessel lumen, are associated with the development and progression of coronary artery disease. Understanding these metrics may therefore improve the assessment of an individual's coronary disease risk. However, the calculation of such luminal Wall Shear Stress (WSS) using traditional Computational Fluid Dynamics (CFD) methods is relatively slow and computationally expensive. As a result, CFD based haemodynamic computation is not suitable for integrated and large-scale use in clinical settings. Methods: In this work, deep learning techniques are proposed as an alternative method to CFD, whereby luminal WSS magnitude can be predicted in coronary bifurcations throughout the cardiac cycle based on the steady state solution (which takes <120 seconds to calculate including preprocessing), vessel geometry and additional global features. The deep learning model is trained on a dataset of 101 patient-specific and 2626 synthetic left main bifurcation models with 26 separate patient-specific cases used as the test set. Results: The model showed high fidelity predictions with <5% (normalised against mean WSS magnitude) deviation to CFD derived values as the gold-standard method, while being orders of magnitude faster with on average <2 minutes versus 3 hours computation for transient CFD. Conclusions: This method therefore offers a new approach to substantially reduce the computational cost involved in, for example, large-scale population studies of coronary haemodynamic metrics, and may therefore open the pathway for future clinical integration

On dimension and existence of local bases for multivariate spline spaces

AbstractWe consider spaces of splines in k variables of smoothness r and degree d defined on a polytope in Rk which has been divided into simplices. Bernstein-Bézier methods are used to develop a framework for analyzing dimension and basis questions. Dimension formulae and local bases are found for the case r = 0 and general k. The main result of the paper shows the existence of local bases for spaces of trivariate splines (where k = 3) whenever d > 8r

Nonlinear elasticity complex and a finite element diagram chase

In this paper, we present a nonlinear version of the linear elasticity (Calabi, Kr\"oner, Riemannian deformation) complex which encodes isometric embedding, metric, curvature and the Bianchi identity. We reformulate the rigidity theorem and a fundamental theorem of Riemannian geometry as the exactness of this complex. Then we generalize an algebraic approach for constructing finite elements for the Bernstein-Gelfand-Gelfand (BGG) complexes. In particular, we discuss the reduction of degrees of freedom with injective connecting maps in the BGG diagrams. We derive a strain complex in two space dimensions with a diagram chase.Comment: Manuscript prepared for proceedings of the INdAM conference "Approximation Theory and Numerical Analysis meet Algebra, Geometry, Topology'', which was held in September 2022 at Cortona, Ital

Splines in geometry and topology

This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.Comment: 18 page