Dynamic relaxation method with finite rotations and automatic mass-tuning for the static analysis of lightweight structures

This paper presents a Dynamic Relaxation Method framework for the nonlinear static analysis of lightweight structures predominantly composed of tensile cables and membranes. To accommodate a broader class of lightweight structures and include flexible beams, this study proposes an extension of the Dynamic Relaxation Method to address finite rotations, required in the behavior of geometrically exact beams. The framework utilizes the Rodrigues pseudovector to parameterize the rotation tensor with only three degrees of freedom, thereby eliminating the need for trigonometric functions and allowing direct updates of rotations in pseudovector form. This approach also integrates the kinetic damping technique with the allocation of fictitious masses, here denoted as mass-tuning, applied to each element in the finite element mesh to achieve a consistent critical time step across all elements, thus circumventing the need for assembling the global stiffness matrix. A special mass-tuning technique is developed for beam elements to evaluate a common parameter for both mass and inertia. The robustness of the proposed algorithm is evaluated using benchmark tests. Subsequently, a more complex structure, integrating beam, membrane, and cable elements, is tested to evaluate the feasibility and robustness of the proposed method, with the results being compared with independent structural analysis software.