An Energy Stable Discontinuous Galerkin Discretization for the Damped Intrinsic Beam Equations

Kurzfassung

The German Aerospace Center is developing MAECOsim, a software framework for simulating rotary-wing aircraft. A major challenge is efficiently simulating the dynamic behavior of rotor blades. To address this, we investigate the Geometrically Exact Intrinsic Beam Model [1], focusing on a variant incorporating Kelvin-Voigt structural damping, as derived in [2]. The undamped intrinsic beam model is a one-dimensional, geometrically exact model suitable for simulating beams with large deformations. Its equations classify as hyperbolic advection equations with quadratic source terms [3]. In [2], Kelvin-Voigt damping is introduced, transforming this system into a nonlinear advection-diffusion system. With the help of a carefully chosen auxiliary variable, we reformulate this second-order PDE system into a first-order PDE system. The resulting formulation can be understood as a parabolic extension of the hyperbolic advection equation of the undamped model. This simplifies both the analysis and discretization. Energy stability of the solution was shown analytically in [2]. We derive an analogous energy stability statement for the solution in the semi-discrete context after discretising the system in space using a local discontinuous Galerkin approach for the system of first-order PDEs. Our theoretical considerations are validated by numerical experiments using the simulation framework Trixi [4]. The experiments show that the numerical solution of a dynamic simulation converges to the exact solution at an expected rate. Furthermore, we investigate the energy of the numerical solution and the relationship between the damping coefficients in the Kelvin-Voigt model and the energy decay rate. We show that in a dynamic simulation, the solution converges to the expected steady state solution for relevant mechanical setups. Future investigations will address incorporating the damped intrinsic beam model into the multibody system of MAECOsim. Our aim is to validate the model for rotor blades with rather complex characteristics, e.g., kinks or jumps in material parameters. References [1] D. H. Hodges, “Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams,” AIAA Journal, vol. 41, no. 6, pp. 1131-1137, 2003. [2] M. Artola, A. Wynn, and R. Palacios, “Generalized kelvin–voigt damping for geometrically nonlinear beams,” AIAA Journal, vol. 59, no. 1, pp. 356-365, 2021. [3] C. Rodriguez and G. Leugering, “Boundary feedback stabilization for the intrinsic geometrically exact beam model,” SIAM Journal on Control and Optimization, vol. 58, no. 6, pp. 3533-3558,2020. [4] M. Schlottke-Lakemper, G. J. Gassner, H. Ranocha, A. R. Winters, and J. Chan, “Trixi.jl: Adaptive high-order numerical simulations of hyperbolic PDEs in Julia.” https://github.com/trixi-framework/Trixi.jl, 2021.