European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   Isogeometric Analysis and Fluid Structure Interaction (Part 2)
Chair: Anotida Madzvamuse
25 mins
General order Virtual Elements for Magnetostatic problems
Lourenco Beirao da Veiga, Franco Brezzi, Franco Dassi, Donatella L. Marini, Alessandro Russo
Abstract: We consider the variational formulation of magnetostatic problems roposed by Kikuchi, where the magnetic field, H, is the unknown vectorial function. We depart from the standard Finite Element Method (FEM) and we apply for the first time the Virtual Element Method (VEM) to solve such problems. The Virtual Element Method is a novel way to discretize a partial differential equation. It avoids the explicit integration of shape functions and introduces an innovative construction of the stiffness matrix so that it acquires very interesting properties and advantages. One among them is the possibility to apply the VEM to general polygonal/polyhedral domain decomposition, also characterized by non-conforming and non-convex elements. In this talk we focus on the definition/construction of the lowest degree elements to better appreciate the novelty of VEM. Then, we give an hint of the general order case. Finally, we show some numerical examples to numerically validate the theoretical analysis of the method.
25 mins
A Low-rank Approach for Nonlinear Parameter-dependent Fluid-structure Interaction Problems
Roman Weinhandl, Peter Benner, Thomas Richter
Abstract: Parameter-dependent discretizations of linear fluid-structure interaction problems can be approached with low-rank methods. When discretizing with respect to a set of parameters, the resulting equations can be translated to a matrix equation since all operators involved are linear. If nonlinear FSI problems are considered, a direct translation to a matrix equation is not possible. We present a method that splits the parameter set into disjoint subsets and, on each subset, computes an approximation of the problem related to the upper median parameter by means of the Newton iteration. This approximation is then used as initial guess for one Newton step on a subset of problems.
25 mins
Numerical approximation of fluid-structure interaction problem near the stability boundary
Jan Valášek, Petr Sváček, Jaromír Horáček
Abstract: This contribution deals with the numerical simulation of a fluid-structure interaction problem. The elastic body is modelled with the aid of linear elasticity. The fluid flow is described by the incompressible Navier-Stokes equations and the arbitrary Lagrangian-Eulerian (ALE) method is used in order to incorporate the time dependence of the fluid domain into the flow description. Both models are coupled using dynamic and kinematic interface conditions. The whole problem is solved by the finite element method solver applied both to the elastic part as well as to the fluid flow approximation. For the flow part a residual based stabilization is used, see \cite{CFD_Health2018}. The coupled problem is solved by a strongly coupled scheme. A special attention is paid to the method how to calculate the aerodynamic forces exerted by fluid flow on the interface of the elastic body. The three different possibilities are investigated -- one is based on the extrapolation of stress tensor from the interior of the domain, the second one based on a reconstruction technique. Third one is derived from a weak reformulation of the acting forces. Moreover, the newly proposed penalization boundary condition is used at the inlet, see \cite{Svacek2017profily}. This approach allows to relax an exact value of the inlet velocity on the boundary during channel closing phase, which is the configuration of our interest. The behaviour of the penalization boundary condition for open channel is very similar to the Dirichlet boundary condition, \cite{APPL2019}. Numerical results of an example of flow-induced vibrations near the stability boundary are presented, e.g. the critical velocity of flutter instability is determined. Moreover, the comparison of different evaluation of aerodynamic forces is given.
25 mins
Isogeometric Analysis of a Reaction-Diffusion Model for Human Brain Development
Jochen Hinz, Joost van Zwieten, Fred Vermolen, Matthias Moller
Abstract: Neural development has become a topic of growing interest in the past decades. On the one hand healthy adult individuals exhibit qualitatively similar neural structures, on the other hand neural development exhibits a substantial degree of randomness, which is largely confirmed by the observation that even monozygotic twins exhibit significant anatomical differences. Among other factors, this neural ‘fingerprint’ manifests itself mainly through the patterns formed in the neural folding and buckling process occurring naturally after the twentieth week of fetal development. This suggests that environmental factors can have a profound influence on the course of neural development, which in turn suggests that the underlying biological process, mathematically, exhibits a high degree of sensitivity toward perturbations in the initial condition. On the other hand, a proficient model for human brain development should be capable of producing qualitatively similar outcomes for similar setups and explain neural pathologies like lissencephaly and polymicrogyria by quantitatively different starting conditions. The derivation of proficient models for human brain development is greatly hindered by the unethicalness of experimentation on human fetuses. We propose a numerical scheme based on Isogeometric Analysis (IgA) for the development of the geometry of a brain. The development is modelled by the use of the Gray-Scott equations for pattern formation in combination with an equation for the displacement of the brain surface. The method forms an alternative to the classical finite-element method. Our method is based on a partitioning of a sphere into six patches, which are mapped onto the six faces of a cube. Major advantages of the new formalism are the use of a smooth reconstruction of the surface based on the third-order basis functions used for the computation of the concentrations. These features give a smooth representation of the brain surface. Though the third order basis functions outperform lower order basis functions in terms of accuracy, a drawback remains its higher cost of assembly. This drawback is compensated by the need of a lower resolution in case of higher order basis functions.