IGS: an IsoGeometric approach for Smoothing on surfaces

TL;DR

This paper introduces an IsoGeometric Smoothing (IGS) method utilizing Isogeometric Analysis to estimate functions on NURBS surfaces from noisy data, effectively solving a 4th-order PDE for applications like aerodynamic pressure estimation.

Contribution

The paper presents a novel IGS method combining IGA and penalized least squares for smoothing on NURBS surfaces, enabling exact geometry representation and solving high-order PDEs efficiently.

Findings

Effective smoothing on NURBS surfaces demonstrated through numerical simulations.

Successful application to aerodynamic pressure estimation on a space shuttle winglet.

Method leverages $C^1$-continuous NURBS basis functions for PDE solution.

Abstract

We propose an Isogeometric approach for smoothing on surfaces, namely estimating a function starting from noisy and discrete measurements. More precisely, we aim at estimating functions lying on a surface represented by NURBS, which are geometrical representations commonly used in industrial applications. The estimation is based on the minimization of a penalized least-square functional. The latter is equivalent to solve a 4th-order Partial Differential Equation (PDE). In this context, we use Isogeometric Analysis (IGA) for the numerical approximation of such surface PDE, leading to an IsoGeometric Smoothing (IGS) method for fitting data spatially distributed on a surface. Indeed, IGA facilitates encapsulating the exact geometrical representation of the surface in the analysis and also allows the use of at least globally $C^1-$continuous NURBS basis functions for which the 4th-order PDE can be solved using the standard Galerkin method. We show the performance of the proposed IGS method by means of numerical simulations and we apply it to the estimation of the pressure coefficient, and associated aerodynamic force on a winglet of the SOAR space shuttle.