European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   Isogeometric Analysis and Fluid Structure Interaction (Part 1)
Chair: Anotida Madzvamuse
25 mins
A direct projection to low-order level for p-multigrid methods in Isogeometric Analysis
Roel Tielen, Matthias Möller, Kees Vuik
Abstract: Isogeometric Analysis (IgA) can be considered as the natural extension of the Finite Element Method (FEM) to high-order B-spline basis functions. The development of efficient solvers for discretizations arising in IgA is a challenging task, as most (standard) iterative solvers have a detoriating performance for increasing values of the approximation order p of the basis functions. Recently, p-multigrid methods have been developed as an alternative solution strategy. With p-multigrid methods, a multigrid hierarchy is constructed based on the approximation order p instead of the mesh width h (i.e. h-multigrid). The coarse grid correction is then obtained at level p=1, where B-spline basis functions coincide with standard Lagrangian P_1 basis functions, enabling the use of well known solution strategies developed for the Finite Element Method to solve the residual equation. Different projection schemes can be adopted to go from the high-order level to level p=1. In this paper, we compare a direct projection to level p=1 with a projection between each level 1 <= k <= p in terms of iteration numbers and CPU times. Numerical results, including a spectral analysis, show that a direct projection leads to the most efficient method for both single patch and multipatch geometries.
25 mins
Equilibrium Path Analysis Including Bifurcations with an Arc-Length Method Avoiding A Priori Perturbations
H. M. Verhelst, M. Möller, J.H. Den Besten, F.J. Vermolen, M.L. Kaminski
Abstract: Wrinkling or pattern formation of thin (floating) membranes is a phenomenon governed by buckling instabilities of the membrane. For (post-) buckling analysis, arc-length or continuation methods are often used with a priori applied perturbations in order to avoid passing bifurcation points when traversing the equilibrium paths. The shape and magnitude of the perturbations, however, should not affect the post-buckling response and hence should be chosen with care. In this paper, our primary focus is to develop a robust arc-length method that is able to traverse equilibrium paths and post-bifurcation branches without the need for a priori applied perturbations. We do this by combining existing methods for continuation, solution methods for complex roots in the constraint equation, as well as methods for bifurcation point indication and branch switching. The method has been benchmarked on the post-buckling behaviour of a column, using geometrically non-linear isogeometric Kirchhoff-Love shell element formulations. Excellent results have been obtained in comparison to the reference results, from both bifurcation point and equilibrium path perspective.
25 mins
An Error-estimate-based Adaptive Integration Scheme for Immersed Isogeometric Analysis
S.C. Divi
Abstract: Finite Cell Method (FCM) -- an immersed finite element method introduced by Rank and co-workers -- together with Isogeometric analysis (IGA) -- a spline-based finite element framework proposed by Hughes et al. -- has been applied successfully in various problems in solid mechanics, in image-based analysis, fluid-structure interaction and in many other applications. A challenging aspect of the isogeometric finite cell method is the integration of cut cells. In particular in three dimensional simulations the computational effort associated with integration can be the critical component of a simulation. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems (particularly in 3D) is not yet available. In this contribution we will investigate the accuracy and computational effort of the octtree integration scheme, which in this study is supplemented with a triangulation procedure at the lowest level of bisectioning (proposed by C.V.Verhoosel et al.) to construct an explicit approximation of the geometry. We study the contribution of the integration error using the theoretical basis provided by Strang's first lemma. Based on this study we propose an error-estimate-based adaptive integration scheme for immersed isogeometric analysis. Additionally, we will apply and investigate the proposed integration technique to flow problems.
25 mins
On finite element approximation of aeroelastic problems with consideration of laminar-turbulence transition
Petr Sváček
Abstract: The mutual interactions of fluid flow and structure motion is imporant in many technical applications. Usually, the methods focus on determination of the critical velocity, i.e. the velocity for which the aeroelastic system looses its stability. In such a situation the structure responds by undamped vibrations to the aerodynamics forces, which can lead to complete destruction of the elastic structure. In order to solve this problem the linearized aerodynamical models are being used based on several simplifications. This allows to determine the critical velocity, but the post-flutter behaviour or other non-linear phenomena can not be captured. In this paper the problem of numerical approximation of a two dimensional fluid-structure interaction problem is addressed. The fully coupled formulation of fluid flow over a structure is considered. For the flow model the incompressible system of Navier-Stokes equations is used written in the Arbitrary Lagrangian form in order to allow the treatment of the time dependent computational domain. The motion of the structure is governed using a simplified model, where bending, torsion and possibly torsion of the control section is considered. The main attention is paid to the finite element approximations of the turbulent incompressible viscous flow over a flexibly supported airfoil, where a suitable model of transition is used to model the laminar-turbulence transition. The numerical results shall be presented.