A fast and stable first-order method for simulation of flexible beams and cables

Abstract

Gradient deficient beam elements (commonly known as cable elements in the literature) based on the absolute nodal coordinate formulation (ANCF) have great potential for large-scale and real-time simulation applications. In this report, a novel and stable first-order integration method for the gradient deficient ANCF element is developed in a simple linear form. The extension of the integration method to the fully parameterized cable element is also discussed. The method involves splitting the elastic potential using the quadrature points involved in the numerical integration of the continuous body and treating the forces developed by the potential at each quadrature point as though it has arisen from a relaxed constraint in the compliant constraints formalism. The integration method exhibits excellent stability properties, scales well, and can be solved efficiently, with only a single linear solve. Efficient methods for computing the required Jacobians and deformation functions and solving the resulting linear equation are discussed. The integration method is tested in three dimensions, on a cable segment formed from multiple elements and compared to other first-order integration methods in terms of the speed and accuracy. The method is demonstrated via simulating various systems, including a pendulum, a cantilever bar, and a flexible beam dropped onto cylindrical supports. Lastly, the utility and use cases of the proposed integrator are discussed.