结构损伤识别中的若干正则化问题研究

STUDIES ON SOME OF REGULARIZATION PROBLEMS IN STRUCTURAL DAMAGE IDENTIFICATION

针对结构损伤识别计算中由于矩阵奇异带来的识别结果不稳定问题,利用列主元QR分解、截断奇异值分解这两种正则化技术研究识别计算中的迭代方法,给出了这两种技术改善识别结果稳定性的证明;同时研究了对得到合理识别结果有较大影响的两个计算参数:实施正则化技术的阀值以及限制迭代增量的步长限值。该文以IASC-ASCE的BENCHMARK结构为算例,分别采用列主元QR分解、截断奇异值分解,识别出BENCHMARK结构第I阶段损伤问题第一种工况下第二种损伤模式的位置,识别的损伤程度最大误差分别为7.33%、6.36%,验证了研究的正则化技术的有效性。通过算例对阀值与步长限值进行研究和讨论,发现列主元QR分解宜应用在步长限值α较大(α≥0.5)的情况下,截断奇异值分解则既可应用在步长限值α较大、也可应用在α较小(α≥0.25)的情况下。

Two of regularization techniques, i.e., QRD （QR Decomposition） with column pivoting and TSVD （Truncated Singular Value Decomposition）, are studied respectively in order to solve ill-posed optimization problems transformed from structural damage identification. The improvement on the stability of identification results is demonstrated. Then, two parameters affecting identification results, i.e., valve for implementing regularization techniques and step-size limit α for limiting iterative increments in solving identification problems, are investigated using the example of the BENCHMARK structure proposed by IASC （International Association for Structural Control） and ASCE （American Society of Civil Engineers）. The case study of the BENCHMARK structure illustrates how to implement the two regularization techniques, as well as how to determine the proper values of the two parameters, valve and α. The results show that, the location of damage pattern Ⅱ in the 1^st case of the phase Ⅰ problem of the BENCHMARK structure are identified successfully, and the errors of damage severity calculated using TSVD and QRD with column pivoting are 6.36% and 7.33% respectively. The discussion about the two parameters, valve and α, shows that longer step-size limit α （≥0.5） is suitable for QRD with column pivoting, but the step-size limit can be longer or shorter （ α≥0.25） for TSVD.