Deep residual U-net with input of static structural responses for efficient U* load transfer path analysis
Introduction
Load transfer path analysis has been an essential procedure in structural design and optimization of engineering structures. It is crucial to learn how the internal force is transferred in a structure in order to strengthen the weak regions and remove the excessive materials. Several methods have been developed and studied by researchers to depict the load transmission in a structure, such as the principal stress method [1], [2], [3]. However, stress concentrations at holes or sharp internal corners can produce confusing information for load transfer path visualization by using these conventional methods [4], [5]. A common approach with clear theoretical background and mechanism is needed to accurately and efficiently depict the load trajectories in engineering structures with general boundary conditions.
Recently, U* index theory is drawing great attention due to the ease of load transfer path visualization by modern finite element analysis (FEA) software [6], [7]. U* index theory originated from the concept of relative rigidity that was defined by Takahashi et al. [8]. It represents the relative stiffness between the loading point and an arbitrary point of a structure, and the main load paths in the structure can be tracked from the U* contour graph. Design criteria based on U* index were also introduced to effectively improve the structural performance in the design stage [4]. U* index theory has been experimentally validated by Wang [9] and applied to crashworthiness design by the use of commercial FEA programs in automotive industries [10], [11], [12], [13]. Furthermore, in the field of structural health monitoring, U* index was found sensitive to structural damages in plate-type structures as well [14]. Despite the great benefits of U* index theory, the calculation time of U* for a large-scale structure is tremendous. For U* calculation of every single node, the global stiffness matrix of the finite element model is accordingly altered due to the repetitive changes of boundary conditions. To avoid iteratively rebuilding the FE model, a fast computation algorithm was presented by Sakurai et al. [15] to convert multiple geometry conditions into multiple load conditions. Nonetheless, it is still time-consuming compared to typical stress analysis due to the nature of U* index theory. To efficiently evaluate and enhance the structural performance and design, a real-time computing algorithm is required to output the U* distribution instantly.
With the recent advances in computer science and optimization techniques [16], [17], [18], [19], machine learning algorithms have become more accessible to researchers and engineers to solve the knowledge-intensive problems in engineering applications. In the field of structural analysis, deep learning (DL) has been used in computational mechanics [20], [21], [22], structural health monitoring [23], [24], [25], [26], and structural design optimization [27], [28], [29]. Wang et al. [30] introduced the first attempt of U* prediction by a multi-layer perceptron (MLP) model with input of geometric information. The U* distributions generated by the proposed method showed good agreements with the ground truth from FEA according to the case studies. However, apparent limitations of this method are noticed that the boundary and load conditions of the structure must be constant. It is cumbersome to perform real-world structural optimization with such limitations as the dimensions of the structure design are modified from time to time.
As opposed to the traditional MLP networks, convolutional neural networks (CNNs) can better extract features from spatial input data with fewer parameters. The state-of-the-art performance of CNN has shown in various computer vision tasks, including image recognition [31], [32], segmentation [33], [34], and reconstruction [35], [36]. Fully convolutional network (FCN) is one of the most used networks for semantic segmentation which is first introduced by Shelhamer et al. [37]. However, spatial information is lost in extracting deeper features leading to rough output results. Ronneberger et al. [38] presented an improved FCN called U-Net with a U-shape architecture. To recover the spatial information in the expanding path (decoding phase), features maps from the contracting path (encoding phase) are concatenated to each corresponding step in the expanding path. On the other hand, a deeper architecture may cause vanishing gradient problems. A common approach to overcome this problem is to adapt residual blocks with identity skip connections [39]. The combination of residual blocks and U-Net has been proven to be effective in road extraction [40] and medical image segmentation [41]. Inspired by these works, a CNN with static structural responses as inputs named Ustar-Net is developed to speed up the U* load transfer path analysis inside the plate-type structures with arbitrary dimensions, loading conditions, and boundary conditions. The network is based on the architecture of U-Net that consists of skip connections from encoding levels to the corresponding decoding levels. Also, residual blocks are used to further facilitate model training in a deep network. While only geometrical information was used as model input in Wang’s method [30], static structural responses are also exploited in the proposed approach. The effects of different input data combinations on model performance are analyzed and discussed. Case studies of homogeneous plates with different profiles demonstrate high accuracy and efficiency of the approach in U* estimation. Moreover, the mechanical analysis of structures composed of functionally graded materials (FGM) has been a trending research topic in recent years [42], [43], [44], [45], [46]. To further study the feasibility of the proposed method on analyzing these types of structures, four datasets of functionally graded (FG) plates with variable Young’s modulus are also studied. This research combines FEA and DL algorism for efficient and robust U* based load transfer path analysis. By utilizing FEA results in the DL model, this time-saving approach can significantly improve the efficiency of structural design optimization and structural health monitoring.
Section snippets
U* index theory [4,15]
The internal stiffness between an arbitrary point and the loading point of a structure can be presented by U* index. A linear elastic structure is shown in Fig. 1(a) with an external load at point A, a constraint at point B, and an arbitrary point C for U* calculation. The structure can be simplified by 3 linear springs between any two points in the structures without considering rotational degrees of freedom (DOFs). The relationship between load and displacement of in the original system can
Methodology of DL-based U* prediction
U* load transfer path analysis with the finite element method (FEM) is computationally expensive even with the inspection loading method because of the iterative U* calculation on each node of the finite element model. The main objective of this research is to explore the relationship between U* distributions and static structural responses such as stress and displacement fields by DL. The dataset generation, network architecture, model loss definition, and evaluation metrics are demonstrated
Case studies
To verify the accuracy and efficiency of Ustar-Net, homogenous plates and FG plates with linearly varying Young’s modulus are separately tested on Tensorflow. The model training is accelerated by an Nvidia RTX 2070 Super GPU with a batch size of 32. The network will converge in 1000 epochs by using Adam optimizer with an adaptive learning rate as shown in Fig. 8. The best model with the minimum validation loss of the 1000 epochs is saved to achieve the optimal performance of the network. While
Conclusions
This research introduces a novel DL-based approach by the use of static structural responses to efficiently generate precise U* distributions of general plate-type structures for load transfer path analysis. The proposed network is built with the architecture of U-Net and ResNet. In addition to the geometrical information, this network also exploits the stress and displacement data from static analysis with FEM as input data. While the previous MLP-based method can only estimate U* values on
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.
Acknowledgments
This research is supported in part by the University of Manitoba, Research Manitoba, and Natural Sciences and Engineering Research Council of Canada (NSERC).
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