Isogeometric Shape Optimization in Fluid-Structure Interaction

The objective of this work is to examine the potential of isogeometric methods in the context of multidisciplinary shape optimization. We introduce a shape optimization problem based on a coupled fluid-structure system, whose geometry is defined by NURBS (Non-Uniform Rational B-Spline) curves. This shape optimization problem is then solved by using either an isogeometric approach, or a classical grid-based approach. In spite of the fact that optimization results do not show any major differences, conceptional advantages of the new isogeometric method become apparent. In particular, control points of the spline can be directly handled as design variables without the need of a spline-fit and consequently geometry errors can be excluded at every stages of the optimization loop. Key-words: Shape Optimization, NURBS, Isogeometric Analysis, FluidStructure Interaction ∗ Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, 85748 Garching, Germany † INRIA Opale Project-Team in ria -0 05 98 36 7, v er si on 2 17 J un 2 01 1 Optimisation de forme isogéométrique en interaction fluide-structure Résumé : Le but de ce travail est d’examiner le potentiel des méthodes isogeometriques dans le contexte de l’optimisation de forme multidisciplinaire. On introduit un problème d’optimisation de forme basé sur un système couplé fluide-structure, dont la géométrie est définie par des courbes NURBS (NonUniform Rational B-Spline). Ce problème d’optimisation de forme est ensuite résolu à l’aide soit d’une approche isogéométrique, soit d’une approche classique s’appuyant sur un maillage. Bien que les résultats de l’optimisation ne présentent pas de différences majeures, les avantages conceptuels de l’approche isogéométrique apparaissent. Notamment, les points de contrôle des splines peuvent être directement manipulées en tant que variables de conception, sans devoir employer une approximation de la géométrie. Par conséquent, les erreurs géométriques sont exclues à toutes les étapes de la procédure d’optimisation. Mots-clés : Optimisation de forme, NURBS, analyse isogéométrique, interaction fluide-structure in ria -0 05 98 36 7, v er si on 2 17 J un 2 01 1 Isogeometric Shape Optimization 3 Introduction For many years the interplay of Computer Aided Design (CAD) and numerical analysis suffered from the bottleneck of different representations of geometry. Whereas polynomials were predominant in the latter discipline, function classes like B-Splines and Non-Uniform Rational B-Splines (NURBS) were employed by CAD systems. With the advent of Isogeometric Analysis (IA) [7] in 2005, a methodology has been proposed to bridge this gap. In particular, IA describes both the geometry and the numerical solution in terms of NURBS basis functions and consequently can be seen as an isoparametric Finite Element Method (FEM). Especially in shape optimization the benefits of a unified geometry representation become apparent. On the one hand geometries are represented exactly in the PDE (partial differential equation) solver which is called several times in the optimization loop. On the other hand NURBS data, i.e. control points (and weights), can be directly used to define design variables. In contrast to classical methods no spline fit has to be performed to reduce the number of unknowns on the boundary of interest. This contribution discusses a multidisciplinary shape optimization problem in the isogeometric setting. In particular we consider a fluid-structure interaction (FSI) problem with the shape of a bent pipe and a flexible part of the boundary. Another part of the boundary is subject to optimization. The single field problems, fluid, structure and fluid mesh, are introduced and their numerical treatment is discussed. References for isogeometric FSI are [3, 2]. This article is organized as follows: The first section introduces NURBS in a nutshell. In Section 2 the fluid-structure interaction problem is presented. Therein we address the numerical solution of the single fields as well as a solution algorithm for the coupled FSI problem. Isogeometric shape optimization and its benefits are presented in Section 3. We introduce the test case of a bent pipe together with its linear approximation to be able to compare the results with the classical case in Section 4. The article closes with conclusions in Section 5. RR n° 7639 in ria -0 05 98 36 7, v er si on 2 17 J un 2 01 1 Isogeometric Shape Optimization 4 1 Non-Uniform Rational B-Splines (NURBS) In this contribution we concentrate on computational domains defined by a NURBS geometry function. Therefore we introduce in very short the basics on NURBS and refer to the standard work [9]. A NURBS is a function F : Ω0 → Ω, F(ξ) = x(ξ), (1) which maps a parametric domain Ω0 to a physical domain Ω. For the remainder of this article we restrict ourselves to the planar case such that the corresponding coordinates read x = (x, y) and ξ = (ξ, η) , cf. Fig. 1. Figure 1: Mapping of the parametric domain to the physical space. A control volume Ω0 in parametric space and its image Ω = F ( Ω0 ) are highlighted in grey. Let Ξ = (ξ0, . . . , ξl) ∈ R be the knot vector of a NURBS of degree p consisting of nondecreasing real numbers. We assume open knot vectors, i.e. the first and the last knot have multiplicity p+ 1. Therefore the endpoints are interpolatory, which is important for representing the computational domain exactly. The ith B-spline basis function of p-degree Ni,p is defined recursively as Ni,0(ξ) = { 1 if ξi ≤ ξ < ξi+1, 0 otherwise; (2) Ni,p(ξ) = ξ − ξi ξi+p − ξi Ni,p−1(ξ) + ξi+p+1 − ξ ξi+p+1 − ξi+1 Ni+1,p−1(ξ). (3) Note that the quotient 0/0 is assumed to be zero. In one dimension, a NURBS of degree p is then given by Ri,p(ξ) = wiNi,p(ξ) ∑ j∈J wjNj,p(ξ) (4) with B-splines Ni,p, weights wi ∈ R, and an index set J = {0, . . . , l − p− 1}. Bivariate NURBS are constructed via (suppressing the degrees pξ and pη) Rk`(ξ, η) = wk`Nk(ξ)N`(η) ∑ i∈I ∑ j∈J wijNi(ξ)Nj(η) . (5) This representation requires knot vectors Ξξ = (ξ0, . . . , ξl1) and Ξη = (η0, . . . , ηl2) for each parameter direction. The index sets are I = {0, . . . , l1 − pξ − 1} and J = {0, . . . , l2 − pη − 1}. RR n° 7639 in ria -0 05 98 36 7, v er si on 2 17 J un 2 01 1 Isogeometric Shape Optimization 5 A single patch NURBS parameterization of the physical domain consists of the geometry function F(ξ, η) = ∑