Novel Implicit Integration Algorithms With Identical Second-Order Accuracy and Flexible Dissipation Control for First-Order Transient Problems

Abstract

For the solutions of first-order transient problems with optimal efficiency, conventional single-step implicit methodologies, unaided by additional post-processing techniques, encounter limitations in concurrently attaining identical second-order accuracy and controllable numerical dissipation. Addressing this challenge, the present study contributes not only a comprehensive analytical framework for formulating implicit integration algorithms but also leverages the auxiliary variable and the composite sub-step technique to propose two distinct types of implicit integration algorithms. Each type is characterized by self-initiation, unconditional stability, identical second-order accuracy, controllable numerical dissipation, and zero-order overshoots. Recognizing that single-step implicit methods can be conceptualized as composite single-sub-step ones, both algorithm types achieve identical effective stiffness matrices, thereby reducing the computational effort for solving linear problems. The utilization of auxiliary variables endows the proposed single-step method with numerical attributes akin to established counterparts, while achieving identical second-order accuracy. Conversely, the adoption of the composite sub-step technique in the proposed two-sub-step methods surpasses the published algorithms by significantly reducing the error constants. This superiority persists when enforcing the same sub-step sizes and sub-step dissipation levels. Numerical simulations affirm that the proposed methods consistently outperform existing alternatives without incurring additional computational expenses.