不确定性损伤识别的改进贝叶斯理论及模态区间分析法

Uncertainties may highly restrain the practicability of deterministic damage identification methods in solving real-world problems. Therefore, this project attempts to develop probabilistic and interval methods that can deal with uncertainty effects caused by parameter variability and measurement noises. An improved Bayesian theory is investigated, which incorporates the merits of the approximate Bayesian computation, the Markov Chain Monte Carlo sampling and the stochastic response surface. By this means, the solution of likelihood functions is avoided and the probabilistic properties of responses can be fast calculated. Moreover, the solution process for estimating parameter posterior distributions is considerably simplified with an improvement in computational efficiency. On the other hand, modal interval analysis is combined with an interval inverse optimization algorithm seeking for the interval envelopes of parameters and responses. Interval overestimation can be avoided leading to an improvement in estimation precision of parameter intervals. Simultaneously, damage thresholds are defined with the aid of statistical hypothesis testing and the theory of confidence interval. After that, robust damage indices are established under both probabilistic and interval frameworks. Lastly, the feasibility and reliability of our methods are validated using numerical examples, laboratory-scale models and data from real-world structures. In a word, the new theories proposed in this project may enhance the applicability of damage identification techniques to practical problems, and thus have their particular values and contributions.

实际工程问题中的不确定性因素极大限制了确定性损伤识别方法的实用性。为此，本项目拟针对结构参数变异性及测量噪声不确定性，研究适用的概率和区间损伤识别方法。拟结合近似贝叶斯计算、马尔科夫链蒙特卡罗抽样和随机响应面提出一种改进的贝叶斯理论，避免了求解似然函数，同时快速计算响应概率统计特征值，以大幅简化参数后验概率分布的求解过程并提高计算效率。利用具有逻辑分析功能的模态区间分析法，通过构建区间优化反演算法求解参数和响应的区间包络线，以抑制区间扩张现象并提高参数区间估计精度。结合统计假设检验和置信区间分别研究概率和区间损伤临界值，并构建鲁棒性损伤指标。最后，基于数值算例、模型试验和实际结构数据，验证理论方法的可行性和可靠性。项目研究成果可以增强损伤识别技术在工程实践中的实用性，具有一定的理论与应用价值。

实际工程结构中总存在着不确定性，限制了确定性损伤识别方法的实用性。为此，本项目考虑了结构几何参数、材料特性及测量噪声的不确定性，并根据不确定性因素是否具有统计特性，提出了基于贝叶斯推断和区间分析的损伤识别方法。首先，结合近似贝叶斯计算、马尔科夫链蒙特卡罗抽样和随机响应面，提出了一种改进近似贝叶斯计算方法，避免求解似然函数的同时还能快速计算结构响应的统计特征值，以此提高计算效率并大幅简化参数后验概率分布估计的求解过程，而后进一步构建和对比损伤前后结构的概率损伤指标，进而预测损伤的位置和程度；接着，为避免陷入低概率区抽样并提高抽样效率，改进了群体蒙特卡洛抽样算法，利用每个迭代步的样本方差来搅动粒子群并求取自适应权重系数，再结合近似贝叶斯计算和随机响应面提出了一种概率损伤识别方法；然后，针对难以统计量化的不确定性因素，提出将结构静力学方程表述成区间形式，再利用模态区间逻辑重新释义该方程，进而通过响应测量值是否超出由区间方程计算得到的区间包络线来判断损伤的发生；最后，针对结构不确定性损伤识别中经典区间算法容易发生扩张的问题，用约束条件方程组表示结构的静力平衡方程，并提出一种改进解析冗余度约简方法，以去除方程组中难以测量的转动变量，同时把与假定损伤单元相关的方程归于同一子集，再代入区间变量将约束条件方程拓展成区间形式，通过模态区间运算判断各约束条件的满足情况，以此为损伤指标来实现损伤定位。项目研究成果可以增强损伤识别技术在工程实践中的实用性，具有一定的理论意义与应用价值。