On artifacts in approximate solutions of nonlinear dynamic systems

• Applicant: Professor Dr.-Ing. Utz von Wagner, since 1/2023
• Subject Area: Mechanics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 455173332

The increasing demands on the predictive capacity of simulation results has lead to a frequent use of nonlinear models in various fields of engineering. Since those models are often described by nonlinear differential equations, for which an exact solution can only be found in a limited number of cases, the analysis often incorporates the use of approximation methods, as e. g. the Harmonic Balance method. As our own preliminary work has shown, it is possible that artifacts occur in the computation of approximate solutions for the duffing-oscillator and similar systems. Herein the term artifacts is used to describe results of an approximation method which satisfy the algebraic equations that arise during the computation but do not solve the underlying differential equation. Therefore the evaluation of artifacts may lead to erroneous results in regards to the system behavior which poses a significant challenge in the utilization of approximation methods. During the currently ongoing project, several approximation methods were applied for the computation of approximate solutions for nolinear dynamical systems with different excitation mechanisms. The obained results show, that artifacts occur frequently and that they can be identified through the utilization of an error criterion, which is derived from the residuum that is associated with the solution. However, it became also clear thatthe computational cost of the currently used method for the identification of artifacts is inconveniently high. The fundamental objective of the proposed renewal of the project is therefore the development and implementation of a more efficient method for the identification of artifacts, which should be based on a preceding investigation into the causes that lead to the occurence of artifacts. To achieve this objective, the following aims are pursued:1) The variety of systems that are analyzed in regard to the occurence of artifacts should be enlarged. Therefore the analysis will be extended to systems with a piecewise polymial nonlinearity as well as systems with a larger number of degrees of freedom.2) An investigation of the convergence behavior of the solutions should be conducted to determine whether the occurence of artfacts is restricted to low approximation orders.3) An analysis of the underlying algebraic equations and the performance of the numerical methods, that are used for their solution, should be used to derive additional characteristics for the identification of artifacts.4) An addaptive algorithm for the implementation of the harmonic balance method under consideration of the occurence of artifacts should be developed.5) All computer programms that have been written in the course of the project should be published as open-source.

• DFG Programme: Research Grants
• Ehemaliger Antragsteller: Dr.-Ing. Lukas Lentz, until 12/2022