Dynamic Relaxation Using Continuous Kinetic Damping—Part I: Basic Algorithm

Abstract

Dynamic relaxation (DR) is the most widely used approach for static equilibrium analyses. Specifically, DR compels dynamic systems to converge to a static equilibrium through the addition of fictitious damping. DR methods are classified by the method in which fictitious damping is applied. Conventional DR methods use a fictitious mass matrix to increase the fictitious damping while maintaining numerical stability. There are many calculation methods for the fictitious mass matrix; however, it is difficult to select the appropriate method. In addition, these methods require a stiffness matrix of a model, which makes it difficult to apply nonlinear models. To resolve these problems, a new DR method that uses continuous kinetic damping (CKDR) is proposed in this study. The proposed method does not require the fictitious mass matrix and any tuning coefficients, and it possesses a second-order convergence rate. The aforementioned advantages are unique and significant when compared to those of conventional methods. The stability and convergence rate were analyzed by using an eigenvalue analysis and demonstrated by simulating nonlinear models of a pendulum and cable. Simple but representative models were used to clearly demonstrate the features of the proposed DR method and to enable the reproducibility of the verification results.