I³MS - Dedé Seminar

Prof. Dr. Luca Dedé - Isogeometric Analysis of High Order, Surface, and Geometric Partial Differential Equations with Applications

Department of Mathematics, Politecnico di Melano, Italy

Abstract

In this talk, we consider the spatial approximation of high order, surface, and geometric Partial Differential Equations (PDEs) by means of Isogeometric Analysis (IGA). Specifically, we discretize the PDEs by means of NURBS-based IGA in the framework of the Galerkin method and we show that IGA is particularly suitable for approximating these classes of PDEs. Indeed, NURBS-basis IGA straightforwardly encapsulates the exact representation of the computational domain (the geometry) in the numerical approximation of the PDE, thus significantly enhancing the accuracy of the solution of surface and geometric PDEs. Moreover, we consider trial spaces of NURBS basis functions with high degree of continuity in the computational domain, a feature that is particularly efficient for approximating high order PDEs. We show the efficiency and accuracy of the method by solving high order PDEs on both open and closed surfaces and phase field problems driven by the Cahn-Hilliard and crystal growth equations. Moreover, we approximate by means of NURBS-based IGA some benchmark geometric PDEs, specifically the mean curvature and Willmore flow problems and the Canham-Helfrich curvature model. The latter is indeed suited to model the shape of biomembranes and vescicles as red blood cells. Finally, we present and discuss a dynamical model for the simulation of the interaction of the biomembranes with the fluid.