Investigation of existing and new approaches to step size control in a continuation framework

This paper presents a comprehensive study on step-size control in continuation methods. It reviews various well-known approaches, identifies suitable control parameters and introduces two schemes combining the parameters. Additionally a novel event-based control approach is presented. The effectiveness of these methods is evaluated through the use of simple examples and a nonlinearly coupled beam that is subject to harmonic excitation. The performance of each approach is compared in terms of the number of steps required, path quality, and computational cost. Results show that one of the proposed combination methods offers superior performance compared to existing approaches, especially in challenging scenarios such as highly nonlinear systems. These findings have practical implications for the development of more efficient and robust control strategies for continuation applications.