Annabelle Collin

Analysis-suitable G1 multi-patch parametrizations for C1 isogeometric spaces

One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from p-degree splines (and extensions, such as NURBS), they enjoy up to C p - 1 continuity within each patch. However, global continuity beyond C 0 on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only C 0 at the patch interface. On such domains we study the h-refinement of C 1 -continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the C 1 -continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recently (Kapl et al., 2015b) has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) C 1 splines. This is the starting point of our study. We introduce the class of analysis-suitable G 1 geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of C 1 isogeometric spaces over analysis-suitable G 1 parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of C 1 isogeometric spaces is prevented. We study h-refinement for C 1 continuous isogeometric spaces over multi-patch domains.We introduce analysis-suitable G 1 (AS G 1 ) geometry parametrizations.AS G 1 parametrizations allow optimal approximation properties.For non-AS G 1 geometries the solution may be locked and convergence is prevented.

A SURFACE-BASED ELECTROPHYSIOLOGY MODEL RELYING ON ASYMPTOTIC ANALYSIS AND MOTIVATED BY CARDIAC ATRIA MODELING

Computational electrophysiology is a very active field with tremendous potential in medical applications, albeit it leads to highly intensive simulations. We here propose a surface-based electrophysiology formulation, motivated by the modeling of thin structures such as cardiac atria, which greatly reduces the size of the computational models. Moreover, our model is specifically devised to retain the key features associated with the anisotropy in the diffusion effects induced by the fiber architecture, with rapid variations across the thickness that cannot be adequately represented by naive averaging strategies. Our proposed model relies on a detailed asymptotic analysis in which we identify a limit model and establish strong convergence results. We also provide detailed numerical assessments that confirm an excellent accuracy of the surface-based model – compared with the reference 3D model – including in the representation of a complex phenomenon, namely, spiral waves.

Surface-based electrophysiology modeling and assessment of physiological simulations in atria

The objective of this paper is to assess a previously-proposed surface-based electrophysiology model with detailed atrial simulations. This model --- derived and substantiated by mathematical arguments --- is specifically designed to address thin structures such as atria, and to take into account strong anisotropy effects related to fiber directions with possibly rapid variations across the wall thickness. The simulation results are in excellent adequacy with previous studies, and confirm the importance of anisotropy effects and variations thereof. Furthermore, this surface-based model provides dramatic computational benefits over 3D models with preserved accuracy.

Sequential State Estimation for Electrophysiology Models with Front Level-Set Data Using Topological Gradient Derivations

We propose a new sequential estimation method for making an electrophysiology model patient-specific, with data in the form of level sets of the electrical potential. Our method incorporates a novel correction term based on topological gradients, in order to track solutions of complex patterns. Our assessments demonstrate the effectiveness of this approach, including in a realistic case of atrial fibrillation.