A fast and accurate dynamic relaxation approach for form-finding and analysis of bending-active structures

Abstract

Active bending is an increasingly popular construction technique that uses elastically bent structural members to form complex curved shapes. The design and analysis of bending-active structures requires an accurate simulation of the bending process, which is often complicated by the occurrence of large displacements. In this article, we propose to combine a previously developed implicit dynamic relaxation method with co-rotational beam elements to obtain a fast and accurate method for form-finding and analysis of bending-active structures. This approach is applied to four test cases. Implicit dynamic relaxation is compared to the classic Newton–Raphson method and conventional dynamic relaxation. The results show that the proposed implicit dynamic relaxation approach can be stabilized intuitively by changing the time step and damping ratio, making it more stable than the classic Newton–Raphson method. Moreover, the proposed approach converges fast compared to the conventional dynamic relaxation: the total computation time is considerably lower, even though the computation time per iteration is higher. Finally, a high accuracy is achieved due to the use of co-rotational beam elements. The combination of high accuracy and low computation time makes this approach well-suited for both form-finding and analysis of bending-active structures.

A novel shape optimization approach for strained gridshells: Design and construction of a simply supported gridshell

“…A prototype of a simply supported, cantilevering strained gridshell was designed using the proposed shape optimization approach, with the method of moving asymptotes [33] as the optimization algorithm, and implicit dynamic relaxation [15] to solve the nonlinear equilibrium equations. The prototype shows that the proposed optimization approach is capable of increasing the overall stiffness of the gridshell with respect to multiple load cases, as compared to a gridshell of similar structural height designed by means of a classical form-finding procedure.…”

“…The nonlinear equilibrium equations in the formfinding, relaxation, and analysis stage (equations ( 1), ( 2), ( 3)) are solved using implicit dynamic relaxation. For more details on implicit dynamic relaxation, the reader is referred to [15]. Co-rotational beam elements [29] are used to model the grid laths.…”

“…To validate the proposed optimization approach, a cantilevering, simply supported gridshell is designed and built. The method of moving asymptotes [33] is chosen as the optimization algorithm, and implicit dynamic relaxation [15] is used to solve the nonlinear equilibrium equations.…”

“…More specifically, if the originally explicit dynamic relaxation solver is made implicit by selecting a mass matrix proportional to the stiffness matrix, the two solvers become equivalent. Moreover, it has been shown that the resulting implicit dynamic relaxation is faster than classic explicit dynamic relaxation for bending-active beam structures such as gridshells [15].…”

“…In an effort to improve the accuracy of dynamic relaxation in dealing with bendingactive structures, researchers have worked on the development of bending elements for the existing dynamic relaxation framework. Barnes [17] and Van Mele et al [18] proposed a beam model with three degrees of freedom per node, Du Peloux et al [19,20] and D'Amico et al [21] presented a model with 4 degrees of freedom per node, and Wakefield [22], D'Amico et al, [7], Bessini et al [23] and Rombouts et al [15] use beam elements with 6 degrees of freedom per node. The latter are generally more accurate [21], and better suited to deal with torsion and anisotropic cross sections [7].…”