Isogeometric Analysis of high order Partial Differential Equations on surfaces

Abstract

We consider the numerical approximation of high order Partial Differential Equations (PDEs) defined on surfaces in the three dimensional space, with particular emphasis on closed surfaces. We consider computational domains that can be represented by B-splines or NURBS, as for example the sphere, and we spatially discretize the PDEs by means of NURBS-based Isogeometric Analysis in the framework of the standard Galerkin method. We numerically solve benchmark Laplace–Beltrami problems of the fourth and sixth order, as well as the corresponding eigenvalue problems, with the goal of analyzing the role of the continuity of the NURBS basis functions on closed surfaces. In this respect, we show that the use of globally high order continuous basis functions, as allowed by the construction of periodic NURBS, leads to the efficient solution of the high order PDEs. Finally, we consider the numerical solution of high order phase field problems on closed surfaces, namely the Cahn–Hilliard and crystal equations

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

“…To confirm this, we solve the monodomain problem on the LA by means of IGA with the spatial discretizations indicated in Table 3. We remark that, for all the meshes T h,2 -T h, 6 , we obtain results which are qualitatively and quantitatively very similar to the ones of Fig. 15.…”

“…We highlight this aspect by comparing in Fig. 16 the front locations of the dimensionless transmebrane potential computed for the discretizations associated to the meshes T h,1 , T h,2 , T h, 4 , and T h, 6 . The results associated to the meshes T h, 5 and T h,6 are in practice coincident also for "large" values of the time t. Therefore, we remark that accurate results can be obtained with relatively coarse discretizations if smooth NURBS basis functions are used for the IGA spatial approximation.…”

“…This assumption at first sight ignores the inherent transmural anisotropy that is present in the fiber structure of the atrial walls, but recent works [13,21] have shown that a proper homogenization treatment of the transmural anisotropy allows the definition of an effective conductivity tensor that recovers the mid-wall activation pattern with sufficient accuracy. In this respect, IGA provides a natural framework for the spatial approximation of elecrophysiology models defined on surfaces, specifically when these are defined by NURBS, as highlighted in [6] and [28]. While sophisticated ionic models for the electrophysiology of the atria exist -e.g.…”

Construction of analysis-suitableG1planar multi-patch parameterizations

“…Let ω be the interval [0,1] or the unit square [0, 1] 2 . The space S p,r k (ω) is the (tensor-product) spline space of degree p (in each direction) and continuity C r at all inner knots, which is defined on ω by choosing (in each direction) k uniform inner knots of multiplicity p − r, where k ∈ Z j , i, j = 0, .…”

“…The high smoothness of the isogeometric spaces allows to solve high order partial differential equations directly via their weak form using a standard Galerkin discretization, see e.g. [1,18,19,34].…”