From the perspective of the test algorithm, it is not necessary for the model to exactly replicate an actual structure. However, it must accurately capture the negative stiffness characteristics that are inherent in real structures. Additionally, by adjusting parameters, the algorithm can simulate a wide range of structural behaviors and features. In practice, the negative stiffness behavior of real structures tends to be simpler than what is often tested in such simulations.
The multi-dimensional virtual spring method offers an effective approach to addressing structural negative stiffness problems. As demonstrated in the example above, this method has several advantages. First, under various parameter settings, the multi-dimensional additional spring technique allows for precise control over the load ratio throughout the entire process, and the algorithm exhibits excellent convergence properties. Second, regarding displacement response, certain displacements may initially increase before decreasing. This suggests that not only does the applied load follow a similar pattern, but also some localized deformations in the softened structure—derived from these displacements—will experience a similar trend. Third, in this particular example, each degree of freedom is subjected to an external load, and there is coupling between any two degrees of freedom, resulting in a full-rank global stiffness matrix. However, the complexity of real-world problems typically does not reach this level, which makes this method highly applicable in practical engineering scenarios, offering faster and more stable convergence.
Regarding the algorithm’s convergence, beyond the examples provided, a qualitative description can be given. Looking at the iterative process, the key to achieving convergence lies in whether the redundant load vector {F_r} decreases with each iteration. This is indeed the case. The reason is that the excess load calculated at the end of one iteration is reversed and applied back to the structure. The influence of this reverse load on the internal forces of other degrees of freedom diminishes with distance, leading to a gradual reduction in the new excess loads generated in subsequent iterations. This behavior closely resembles the decay of node imbalance torque seen in the traditional torque distribution method of structural mechanics. Overall, the algorithm demonstrates robust stability and reliability, making it a valuable tool for analyzing complex structural systems with negative stiffness characteristics.
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