An efficient and unconditionally stable numerical algorithm for nonlinear structural dynamics

Abstract

Summary This article proposes an algorithm for express solutions in nonlinear structural dynamics. Our strategy is to adopt a typical time integrator and accept the solution after a constant number of iterations using a constant Jacobian matrix. Its success may not be initially obvious, but we demonstrate that the proposed algorithm not only is fully operational but also inherits the advantages of the host time integrators such as the unconditional stability, the order of accuracy, and the numerical dissipation that helps suppress the spurious higher mode oscillation. The use of a constant Jacobian matrix plays the key role in minimizing the computational expense associated with matrix operations. We first study the optimization of the number of iterations, then present the consistency and stability analysis followed by some examples verifying these features, and conclude by showing the exponential efficiency improvement in a response history analysis of a high‐rise building fully equipped with nonlinearities.

A convergence‐enhanced Timoshenko beam element for the stochastic nonlinear analysis of reinforced concrete components

“…To avoid constructing the iterative matrix for each iteration, the Modified Newton method employs the initial iterative matrix and its corresponding inverse matrix to approximate the solutions and thus the method has a linear rate of convergence. This idea serves as the foundation for the efficient and unconditionally stable numerical algorithm presented by Xu et al, 11 based on the assumption that the acceleration remains constant during the discretized time interval. The quasi-Newton method, [12][13][14] which has ultra-linear convergence and improved robustness, is a promising solution scheme for governing equations of systems involving non-convex energy functions.…”