An Energy Stable Discontinuous Galerkin Approach for the Geometrically Exact Intrinsic Beam Model

Kurzfassung

With the Versatile Aeromechanic Simulation Tool (VAST), the German Aerospace Center is developing a software framework for the simulation of rotary wing aircraft. One challenge consists of simulating the dynamic behaviour of rotor blades. For this purpose we are investigating the Geometrically Exact Intrinsic Beam Model [1] and its discretization. In [2] we derive an energy stable discretization for the governing equations of the model using Discontinuous Galerkin techniques. As a theoretical basis for our approach we use the Geometrically Exact Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams (Intrinsic Beam Model), which was developed by Hodges in 2003 [1]. It allows for the modelling of anisotropic and initially curved beams, making it well suited for rotor blades. Furthermore, in contrast to other well-known beam models like Euler-Bernoulli or Timoschenko, the intrinsic beam model contains non-linearities and is geometrically exact. Considered beams are idealized by a one dimensional reference line so that the governing equations of the model can be formulated as system of partial differential equations (PDE) in one space dimension and a time dimension. The solution of the PDE consists of internal forces and moments as well as linear and angular velocities each in three dimensions so that there are 12 unknowns in total. If they are known, these intrinsic variables can be integrated to obtain displacements and rotations of the modelled beam. We use a practical representation of the PDE as a system of linear hyperbolic balance laws to derive boundary conditions that describe the mechanical setup of a clamped-free beam. From these boundary conditions, we also show that they are sensible from a mathematical point of view. Further, we use the energy method to show that the model is energy conserving in general and energy stable when applying the boundary conditions and potential external forces and moments along the modelled beam. In particular, we derive an estimation for the solution's energy which shows that for bounded external forces and moments and zero boundary data, the energy can not grow faster than quadratically in time. As the underlying PDE can be understood as a linear hyperbolic system of balance laws, an appropriate choice for the spatial discretization of the underlying problem is a Discontinuous Galerkin (DG) Approach. The DG approach not only has the advantage that it is very efficient and helps minimizing the degrees of freedom in the computationally intensive process of simulating helicopters. It is also able to depict discontinuities, which may enable us to take jumps in material parameters into account which are not unusual within helicopter rotor blades. As numerical flux we use a slightly modified version of the well-known Lax-Friedrichs flux which contains a so called upwind parameter. With the help of this DG discretization, we derive a semi-discrete formulation of the problem that still continuously depends on the time variable. We further analyse the energy of this semi-discrete problem and derive an estimation for it, that mimics the estimation for the energy of the original problem's solution. That is, also the semi-discrete energy can not increase faster than quadratically in time and the DG discretization is in fact energy stable. The energy of the numerical solution emerging from the semi-discrete problem has, however, additional numerical dissipation whose amount can be controlled by the upwind parameter of the numerical flux.4. Numerical Experiments To implement our theoretical considerations, we use the simulation framework Trixi [3]. To obtain a fully discrete problem, we use an explicit fourth order Runge-Kutta scheme for the discretization of the semi-discrete formulation which is represented by an ordinary differential equation in time. In our numerical experiments, we investigate an exemplary case of the intrinsic beam model and use the techniques of manufactured solutions to verify the convergence of the numerical solution resulting from our discretization scheme. The results show an optimal convergence rate, i.e. the empirical order of convergence reaches the degree of the polynomials that are used in the DG approach for the spatial discretization. Moreover, we experimentally verify our predictions concerning the energy of the semi discrete solution. That is, for example that a simulated beam that is not exposed to any external influences like external boundary data or external forces and moments along the beam has a non-increasing energy. Additionally, we verify that the amount of numerical dissipation, i.e. the amount by which the discrete energy decreases in that case is determined by the upwind parameter of the numerical flux. Another point that is investigated in our numerical experiments is the post processing that is needed in order to obtain the displacements and rotations of the one dimensional reference line representing the beam. Our results confirm that the DG approach suits the problem of the intrinsic beam model well. Our theoretical investigations as well as our numerical experiments show that the discretization scheme is numerically stable. Moreover, we could verify the convergence of the resulting numerical solution to the exact solution at an optimal convergence rate. What remains to be investigated is the correctness of our simulation results by comparing them to experimental data and other existing results. Often the available data consists of steady state cases, which leads to another question that will be investigated in the future: Can the Intrinsic Beam Model together with the DG discretization be united with a damping model? That will not only be interesting for the comparison of simulation data with experimental data and exact steady state solutions but also for the actual application on helicopter rotor blades. An example of how damping models can be integrated into the intrinsic beam model is to find in [4], where Artola, Wynn and Palacios derive a damped version of the intrinsic beam model using Generalised Kelvin-Voigt damping. Currently, we are investigating the compatibility of the damped model with our considerations on the DG discretization so far. References [1] D. Hodges. Geometrically Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams, AIAA Journal, 41 (2003) 1131-1137 [2] C. Bleffert. An Energy Stable Discontinuous Galerkin Discretization Approach for the Geometrically Exact Intrinsic Beam Model, Master's Thesis, University of Cologne, 2022. [3] M. Schlottke-Lakemper et al. Trixi.jl: Adaptive high-order numerical simulations of hyperbolic PDEs in Julia, https://github.com/trixi-framework/Trixi.jl, 2021. [4] M. Artola, A. Wynn, R. Palacios. A Generalised Kelvin-Voigt Damping Model for Geometrically-Nonlinear Beams, AIAA Journal, 59 (2021) 356-365