Camp chain for inverse problems of structures
Introduction
Optimisation progress is usually described as a search for the optimal solution in the design space. It is also posed as an inverse problem under certain circumstances. The task of inversion in the hydraulic domain often aims to find specific physical parameters or boundary conditions of the medium corresponding to recorded observations [1]. Numerical modelling methods such as the finite element method (FEM) have been regarded as standard tools for modern engineering structures, and their simulation reliability has been significantly increased with recent advances [2], [3]. For brevity, different kinds of numerical modelling are represented by FEM in the paper. On the assumption that the FEM simulations make rigorous adherence to the practical system state, an inverse analysis is feasible by revisiting the numerical process [4], [5]. Therefore, inverse problems have been implemented more and more by constructing mapping networks between sampled sought-for parameters and corresponding FEM responses at the monitoring points [6], [7], [8], [9], [10]. However, the use of network models is found to be subject to a few limitations.
On the one hand, building network models demands a considerable number of initial network nodes to avoid bad local minima and uninvertible singular matrices. In general, a network node requires a pair of input–output samples, which corresponds to a direct FEM forward calculation. Unfortunately, FEM computation in the engineering domain is typically time-consuming because of the complex mesh and boundary conditions. Even network models of low density cannot avoid unnecessary sampling at points far away from the objectives [11], [12]. In addition, a substantial amount of monitoring data by recent technological advances [13], [14], [15], [16] subsequently increases the requisite number of training samples [17], [18].
On the other hand, a range of network models are subject to one-way computing features, and thereby often resort to different search algorithms to achieve fast inversion. The radial basis function (RBF) network coupled with particle swarm optimization has been used for parameter estimations of high dams and foundations [11], [19], [20]. Classical backpropagation neural networks (NNs) combined with various tools, such as harmony search [4] and Bayesian theory [21], have been utilised to carry out inverse analysis of the physical or mechanical parameters for dam body materials. A Kriging network integrated by a genetic algorithm has also been applied to optimise the hydroelectric flow [22].
In this light, this paper proposes a camp chain (CC) model to solve inverse problems of engineering structures. The CC aims to reduce the total usage of FEM simulations and seeks for the unknowns independently. The CC consists of a chain of lightweight networks named camps, and each camp is a pattern recognition model constructed from only a few pairs of input–output samples. The material property or physical variables generally play the role of the objective parameters to be inversely estimated. The observed or simulated system responses, such as temperature changes and external deformation, are often referred to as verifying references. A pair of sampled objective parameters and the corresponding verifying references constitutes a node of the camp. The initial camp has a comparatively lower predictive accuracy than a sufficiently sampled model. Thereafter, the camp progressively moves towards the target by adding new and discarding old training samples during each update. Simultaneously the camp shrinks itself quickly, and its predictive accuracy is subsequently improved. The moving process of the camp forms a chain structure, and the predictions of the former camp direct the construction of the latter camp.
The mapping function between the inputs and outputs of each camp is simulated by a Linear-Gaussian (L-G) kernel transformation, which combines the linear function and a Gaussian function auxiliary item. This framework increases the R-squared value [23] of the model without taking in more interfering variables and subsequently improves the fitting and predictive accuracy of each camp. The nonlinear relationship between physical parameters and system responses is emulated by the Gaussian function. Moreover, the Gaussian function contributes to smoothing the model and adds stability. In fact, the unsatisfying errors produced by rough modelling with few nodes are compensated for when the camp approaches the objective point and shrinks into its close vicinity; this is because a coarse prediction at this point still attains high accuracy.
Essential techniques and procedures for building the CC are illustrated in detail in Section 2, including the sampling method for the initial camp, the pattern recognition function used in the camps, and the inversion scheme. Section 3 first gives a walkthrough of implementing the CC on an example of estimating the concrete thermal conductivities of a 2-D dam model, through which the inversion effectiveness and accuracy have been validated. Subsequently, the optimal number of camp nodes, the ability to deal with large unknowns, possible ill-conditions, and sensitivity to hyperparameters are discussed. Comparisons to other network-based models, respectively involving multiple linear regression (MLR), RBF and NN are demonstrated in Section 4. Finally, conclusions are given in Section 5.
Section snippets
Initial camp
Similar to all other pattern recognition models [24], each camp of the CC attempts to identify the mapping relationship between the input and the output data. The input is extracted from the experienced value range of the unknown parameters. Suppose the numerical simulation highly approximates the practical engineering project, then the corresponding output at the verification points can be obtained by using the numerical model. In the CC, only the input of the initial camp involves
Problem set up
The Xiluodu double-curved arch dam is located on the upper Yangtze River, China. The simplified 2D model of the intermediate cross-section of the dam for studying heat transfer is used to test the feasibility and validity of the CC inversion [11], [39]. Its crest elevation, top thickness and bottom thickness in the middle are respectively 610 m, 14 m and 60 m [40], [41], [42]. As displayed in Fig. 4, the dam is designed to use 3 different kinds of concrete thereby corresponding to 3 thermal
Comparison
The case study of estimating the thermal conductivities of the concrete dam model will continue to be used in this section, to allow comparison with inverse methods respectively based on networks of MLR, RBF and NN. Also, Latinized stratified sampling will continue to be used as the initial camp sampling method, and will be used as the termination criterion.
Conclusion
To avoid unnecessary numerical computation during simulation-based optimisation and inversion, the camp chain (CC) method is proposed. The CC is comprised of a series of networks with a small number of training samples for each, called camps for short, where the predictions of the previous camp constitute part of the training samples of the next camp. One pair of training samples is associated with a network node. Each camp is a plainly equipped pattern recognition model, with the L-G kernel
Data accessibility
The code for complete CC inversion is written in MATLAB and the code for Fig. 1, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14 was compiled with Plotly and ggplot2 in R, which can be accessed for educational purposes on Mendeley Data [63].
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
A grateful acknowledgement is given to the State Key Laboratory of Hydro-science and Engineering, Tsinghua University. The author would also like to thank Daniel Leditschke for the English language editing.
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