The geodesic dynamic relaxation method for problems of equilibrium with equality constraint conditions

This paper presents an extension to the existing dynamic relaxation method to include equality constraint conditions in the process. The existing dynamic relaxation method is presented as a general, gradient-based, minimization technique. This representation allows for the introduction of the projected gradient, discrete parallel transportation and pull back operators that enable the formulation of the geodesic dynamic relaxation method, a method that accounts for equality constraint conditions. The characteristics of both the existing and geodesic dynamic relaxation methods are discussed in terms of the system's conservation of energy, damping (viscous, kinetic, and drift), and geometry generation. Particular attention is drawn to the introduction of a novel damping approach named drift damping. This technique is essentially a combination of viscous and kinetic damping. It allows for a smooth and fast convergence rate in both the existing and geodesic dynamic relaxation processes. The case study was performed on the form-finding of an iconic, ridge-and-valley, pre-stressed membrane system, which is supported by masts. The study shows the potential of the proposed method to account for specified (total) length requirements. The geodesic dynamic relaxation technique is widely applicable to the form-finding of force-modeled systems (including mechanically and pressurized pre-stressed membranes) where equality constraint control is desired.