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. This approach allows for the creation of various structural behaviors through parameter tuning. In reality, the negative stiffness features tend to be less complex compared to what is simulated in these tests.
The multi-dimensional virtual spring method offers a promising solution to the challenge of structural negative stiffness. As demonstrated in the previous example, this method has several advantages. First, under different parameter settings, the multi-dimensional additional spring approach effectively controls the load ratio throughout the process, and the algorithm exhibits excellent convergence properties. Second, regarding displacement response, certain displacements initially increase and then decrease. This indicates that not only does the load follow a similar pattern, but local deformations in the softened structure—derived from displacement—also experience an initial rise followed by a decline. Third, in this example, each degree of freedom is subjected to a load, and there is coupling between any two degrees of freedom (the overall stiffness matrix is full rank). In practical scenarios, the complexity of the actual problem typically does not reach such a level, making this method highly applicable and capable of achieving faster and more stable convergence.
In terms of algorithm convergence, beyond the examples provided, we can also describe the behavior qualitatively. Observing the iterative process, the key to convergence at each stage lies in whether the redundant load vector {F_r} is decreasing. This is indeed the case. The reason is that the excess load calculated at the end of each iteration is applied in reverse to the structure. The influence of this reversed load on internal forces at other degrees of freedom diminishes with distance, leading to a gradual attenuation of internal force changes. These attenuated internal force adjustments result in new excess loads that are smaller than those from the previous iteration. This behavior closely resembles the decay of node imbalance torque seen in the torque distribution method of structural mechanics.
Overall, the proposed method provides a robust and flexible way to simulate and analyze structures with negative stiffness characteristics, offering both accuracy and efficiency in practical applications.
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