On the equivalence of dynamic relaxation and the Newton-Raphson method: application to the design and analysis of bending-active structures

摘要

Dynamic relaxation is a form-finding and analysis method that has proven its effectiveness in the context of tension/compression structures such as cable nets, membranes and tensegrity structures. Recently, however, an increasing interest in bending-active structures has stimulated researchers to include the effects of bending and torsion in the dynamic relaxation process, using either Three-Degree-of-Freedom (3-DoF) beam elements (Barnes et al. [4]), or 4-DoF beam elements (Du Peloux et al. [12], D’Amico et al. [10]), or 6-DoF beam elements (Li and Knippers [13]). The stability and convergence speed of the dynamic relaxation solver depends on the choice of the fictitious masses for all degrees of freedom. Although 6-DoF beam elements are in principle preferred over 3-DoF and 4-DoF beam elements because of their higher accuracy (D’Amico et al. [10]), their use in dynamic relaxation is hindered by the fact that no heuristic rules are available for the choice of the fictitious masses.In this paper the numerical stability of the dynamic relaxation solver is investigated for 6-DoF beam elements using modal analysis. A fictitious mass matrix proportional to the stiffness matrix is put forward as the most suitable choice considering numerical stability and convergence, as it causes all eigenfrequencies of the structure to coincide. Moreover, it is shown that for this choice of mass matrix and a specific choice for the damping ratio and the time step, the dynamic relaxation method becomes identical to the Newton-Raphson method, which is well known for its fast convergence. For numerically challenging problems, the stability of the classic Newton-Raphson method can be improved by increasing the damping ratio and/or decreasing the time step.We applied the proposed approach to three test cases involving bending-active structures in order to verify its accuracy and convergence speed. The results show that the dynamic relaxation routine indeed converges in a very small number of iterations, while still maintaining the accuracy of 6-DoF beam elements. The combination of high accuracy and low computation time makes this approach well-suited for both the form finding and the analysis of spatial structures undergoing large displacements.