Sensitivity analysis for problems exhibiting geometric nonlinearities and follower loads using the complex-variable finite element method

This study presents an enhanced approach for conducting sensitivity analysis of nonlinear problems involving a combination of geometric nonlinearities and follower loads, particularly those involving displacement-dependent forces. The method utilizes the complex-variable finite element method (ZFEM), incorporating complex algebra into the conventional finite element incremental-iterative procedure to achieve highly accurate derivative calculations. A crucial task in this process is computing a complex-valued, non-constant external force that depends on a complex-valued displacement. The key innovation lies in overcoming challenges associated with sensitivity computation for geometric nonlinearities and follower loads through a streamlined and computationally efficient methodology that can be integrated with commercial finite element software. The method enhances implementation efficiency by avoiding the need for intricate analytical derivations and not depending on unstable numerical approximations, such as the Finite Difference Method (FDM). ZFEM’s versatility and robustness were verified against sensitivity analytical solutions for cantilever beam problems undergoing large elastic rotations and displacements under static and dynamic loading conditions. The numerical examples demonstrated excellent agreement with analytical solutions and finite differencing results, maintaining accuracy and stability across all cases. This research demonstrates that ZFEM significantly increases accessibility for computing sensitivities in complex solid mechanics problems, providing a user-friendly and efficient method for both static and dynamic scenarios involving geometric and follower loads.