Towards Helicopter Simulation with an Index-1 Differential-Algebraic Equations System - Efficient Time Integration Methods

Abstract

Comprehensive helicopter simulation has been an important subject of research ever since the rise of the first helicopters. Numerous modeling approaches and software frameworks emerged, each with its individual advantages and drawbacks. However, a comprehensive approach that yields an efficient and effective simulation adaptable for research is still missing. Comprehensive here means that all relevant aspects are taken into account, e. g. vortex dynamics, rotor behavior, vibrations, etc. The German Aerospace Center (DLR) is working on a new solution using general state-space models for submodels of the helicopter which are coupled to a comprehensive model. This coupling yields an index-1 differential algebraic equation (DAE) system. In this thesis, we analyze existing numerical algorithms for the solution of index-1 DAE systems and define a new familiy of methods that suits the challenges of helicopter simulation best. The challenges here consist in the interaction of large systems that are complex and individual in their behavior on their own. We focus our analysis on half-explicit Runge-Kutta HERK) methods. Their explicit approach delivers the efficiency we seek. However, these methods are not able to handle stiff systems like the highly vibratory helicopter well. Stiff ordinary differential equation (ODE) problems can be solved by explicit exponential Runge-Kutta (EERK) methods. In order to apply them to index-1 DAE systems, we derive the new half-explicit exponential Runge-Kutta (HEERK) methods. We test our HEERK methods in comparison to HERK methods on a mechanical model of the main rotor. The results show that HEERK methods need substantially fewer time steps than HERK methods for the same approximation quality. We also see that a good choice of submodels, here the stiff variables are solely placed in the state function, boosts this effect. So, HEERK methods constitute an essential step towards an effective real-time simulation in comprehensive rotorcraft simulation.