European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   Computational Fluid and Solid Mechanics: (Visco-)elasticity
Chair: Marc Gerritsma
25 mins
Modal analysis of elastic vibrations of incompressible materials based on a variational multiscale finite element method
Ramon Codina, Önder Türk
Abstract: In this study, we extend the standard modal analysis technique that is used to approximate vibration problems of elastic materials to incompressible elasticity. In modal analysis, the second order time derivative of the displacements in the inertia term is utilized, and the problem is transformed into an eigenvalue problem in which the eigenfunctions are precisely the amplitudes, and the eigenvalues are the squares of the frequencies. While this approach is applied directly to compressible materials in different structural models, incompressible media pose the difficulty associated to the need of introducing the pressure (or mean stress) as a variable and to interpolate it in an adequate manner. In particular, when the problem is approximated using finite elements, the standard Galerkin formulation requires the use of interpolations for the displacement and the pressure that satisfy the classical inf-sup condition, often called Ladyzhenskaya-Babuska-Brezzi condition in this context. The alternative to use finite element interpolations satisfying the inf-sup condition is to resort to stabilized finite element formulations. However, particular care is needed when dealing with the eigenvalue problem since in general, stabilization techniques yield a quadratic eigenvalue problem even if the original one is linear. We introduce a finite element formulation we have developed for the Stokes eigenvalue problem that is based on the variational multiscale (VMS) concept, preserving the linearity, and accommodating arbitrary interpolations for the displacement and the pressure, into the modal analysis of incompressible elastic materials, using displacements and pressures as variables. We show that each mode of the modal analysis (amplitude and frequency) can be obtained from an eigenvalue problem that can be split into the finite element scale and the subgrid scale. The latter needs to be approximated, and we show that this approximation should depend on the frequency of the mode being considered. Since this frequency is unknown, an iterative procedure must be devised. The result is a problem for the finite element component of the displacement amplitude and the pressure which allows for any spatial interpolation. Several eigenvalues and eigenfunctions of the Stokes eigenvalue problem need to be computed to perform the modal analysis. The time approximation to the continuous solution is obtained taking a few modes of the whole set, those with higher energy. We present an example of the vibration of a linear incompressible elastic material showing how our approach is able to approximate the problem. It is shown how the energy of the modes associated to higher frequencies rapidly decreases, allowing one to get good approximate solution with only a few modes.
25 mins
Simulating two-dimensional viscoelastic fluid flows by means of the "Tensor Diffusion" approach
Patrick Westervoß, Stefan Turek
Abstract: In this work, the novel "Tensor Diffusion" approach for simulating viscoelastic fluids is proposed, which is based on the idea, that the extra-stress tensor in the momentum equation of the flow model is replaced by a product of the strain-rate tensor and a tensor-valued viscosity. At least for simple flows, this approach offers the possibility to reduce the full nonlinear viscoelastic model to a generalized "Tensor Stokes" problem, avoiding the need of considering a separate stress tensor in the solution process. Besides fully developed channel flows, the "Tensor Diffusion" approach is evaluated as well in the context of general two-dimensional flow configurations, which are simulated by a suitable four-field formulation of the viscoelastic model respecting the "Tensor Diffusion".
25 mins
ALE space-time discontinuous Galerkin method for the interaction of compressible flow with linear and nonlinear dynamic elasticity with applications to vocal fold vibrations
Miloslav Feistauer, Monika Balazsova, Jaromir Horacek, Adam Kosik
Abstract: The paper deals with the space-time discontinuous Galerkin method (STDGM) for the solution of compressible Navier-Stokes equations in the ALE form in time dependent domains combined with the solution of linear and nonlinear dynamic elasticity. The discretization of the dynamic system is based on splitting the system into two systems of first order in time. The developed numerical schemes are analyzed from the point of view of the stability, accuracy and robustness with the aid of several test problems. The applications are oriented to fluid-structure interaction (FSI), particularly to the simulation of air flow in a time dependent domain representing vocal tract and vocal folds vibrations. We compare results obtained with the aid of linear and nonlinear elasticity models. The results show that it is more adequate to use the nonlinear elasticity St. Venant-Kirchhoff or Neo-Hookean model in contrast to linear elasticity model.
25 mins
Conservation laws in viscoelastic flows
Sébastien Boyaval
Abstract: Many viscoelastic fluids have been proposed to capture a wealth of rheological phenomenas and model real flows. But standard viscoelastic fluids inspired by Maxwell's 1867 seminal proposition have not yet delivered their promise. Standard viscoelastic equations do nor allow for useful numerical results in practical flow situations, neither define good mathematical models. We generalize Maxwell's one-dimensional proposition to multi-dimensional flows using conservation laws, quite differently from standard viscoelastic equations.