A novel inverse method for distributed dynamic loads reconstruction with adaptively determining hyper-parameters

Distributed dynamic loads, which is regarded as an essential component of the dynamic loads, occupy a very important role in practical engineering. This paper explores a novel and effective methodology, utilizing the Gaussian prior model and the orthogonal polynomials, to reconstruct the distributed dynamic load loads varying with the distribution of spatial and temporal history in time domain. The unknown orthogonal polynomial coefficients and the measurement are considered as the random vectors to determine the corresponding probability density distribution functions using Bayesian framework. Thus, the posterior density function of the unknown parameters is obtained through the Bayesian formulation. Then, the unknown parameters are identified using the Maximum A Posteriori estimator and the numerical iteration to further reconstruct the distributed loads. Especially, the hyper-parameters of the load identification model based on the Gaussian prior model are also regarded as the random vectors to decrease the impact of imprecise estimation of these hyper-parameters on the load reconstruction. From this perspective, the prior probability distributions of the hyper-parameters are determined to further obtain the joint posterior probability density function of the distributed dynamic load identification problem. The innovation of this methodology is that the Bayesian framework on the basis of Gaussian prior is first applied to the time-domain distributed dynamic loads reconstruction from the perspective of randomness, which decreases the ill-posedness and adaptively determines the hyper parameters, overcoming the disadvantage in selecting optimal regularization parameters of traditional regularization. In addition, concerning the discretization representation of distributed dynamic loads, the method to determine the coupling truncation orders of time-spatial domain is also explored. Additionally, an array of numerical examples are discussed to reveal the reasonability, accuracy and the operation efficiency in the case of various loading conditions. Simultaneously, the noise resistance is discussed in terms of different noise levels contrasted with the Tikhonov method. The results highlight the effectiveness of the discussed approach in different structures and loading conditions.