Isogeometric analysis of diffusion problems on random surfaces

TL;DR

This paper presents an isogeometric method for solving diffusion equations on random surfaces, employing low rank approximations and numerical validation on complex geometries derived from surface triangulations.

Contribution

The paper introduces a novel isogeometric approach for diffusion on random surfaces, integrating low rank approximation algorithms and handling complex geometries.

Findings

Effective modeling of diffusion on random surfaces.

Validation of the approach on complex geometries.

Successful application of low rank approximation in this context.

Abstract

In this article, we discuss the numerical solution of diffusion equations on random surfaces within the isogeometric framework. We describe in detail, how diffusion problems on random surfaces can be modelled and how quantities of interest may be derived. In particular, we employ a low rank approximation algorithm for the high-dimensional space-time correlation of the random solution based on an online singular value decomposition, cp. [7]. Extensive numerical studies are performed to validate the approach. In particular, we consider complex computational geometries originating from surface triangulations. The latter can be recast into the isogeometric context by transforming them into quadrangulations using the procedure from [41] and a subsequent approximation by NURBS surfaces.