Due to the experience of natural disasters, oxidation, corrosion, material degradation and continuous loading, structures in the process of long-term service may gradually develop inconspicuous internal flaws, such as micro-cracks and voids. These flaws often degrade the local strength and stability of the structure, or even result in catastrophic failures. As a pivotal link in ensuring structural safety and quality, identification of flaws and damage in critical structure components has long been a research focus in the field of structural health monitoring. In recent years, nondestructive testing technologies for structures have been widely applied in various fields, including civil engineering, water conservancy, transportation, aerospace, and mechanical engineering. The main non-destructive testing methods include ultrasonic testing [1,2], ground-penetrating radar [3,4], infrared imaging [5,6], and acoustic emission testing [7,8]. Among these methods, internal flaws are identified by analyzing the responses of materials or structures to external signals based on their physical properties, which can be regarded as direct detection of damaged regions within the structure. Additionally, inverse analysis of internal flaws in structures can be conducted based on the external responses of the structure [9,10].

With the advancement of computer technology, many scholars have conducted relevant research and achieved promising results. Ma et al. [11] proposed a flaw inversion model for detection of multiple complex flaw clusters based on the extended finite element method (XFEM) and the artificial bee colony algorithm. The hierarchical clustering analysis was conducted to capture potential subdomains containing flaws and the positions of complex flaw clusters in structures can be located effectively. Subsequently, Zhao et al. [12] employed XFEM to solve the forward problems of flaw detection and used the K-means clustering algorithm to estimate the number of flaws. The proposed approach not only detected the positions of complex-shaped flaws in structures, but also significantly improved the convergence speed of flaw inversion. A couple of method of the level set method with the finite element method was proposed by Zhang et al. [13] for identifying two-dimensional and three-dimensional holes in structures. To a certain extent, the aforementioned studies have preliminarily addressed the issues of quantitative inversion of flaw information in the field of nondestructive testing. However, when the traditional finite element methods are employed in those numerical models, the meshes or integration regions usually need to be regenerated or adjusted for fitting well with geometric boundaries. What’s worse, the flaw detection iterative process usually contains hundreds and thousands of forward analyses to obtain defect parameters within the allowable error range. The computational workload is huge and it would be heavier due to the complex shapes and the unknown number of flaws.

To reduce computational costs, several studies combined artificial intelligence algorithms with the computational mechanics model to identify flaws [14,15]. Dong et al. [16] utilized Convolutional Neural Networks (CNNs) to propose a localizing and classifying model of anomalies in the industrial component inspection task. The performance was capable both at the pixel level (localizing flawed regions) and the image level (identifying whether an image contains flaws). Yang et al. [17] integrated a deep learning feature extraction method with the Extreme Learning Machine (ELM) classification approach to establish a deep extreme learning machine model for flaw detection in wood images. This model achieved a flaw recognition accuracy of 96.72% for wood, with a testing time of only 187 ms. Nguyen-Ngoc et al. [18] proposed a new Structural Health Monitoring (SHM) damage detection method for truss bridges by coupling a DNN model with the evolved Artificial Rabbit Optimization (EVARO) algorithm, which demonstrated excellent performance in damage localization and quantification. Huang et al. [19] employed artificial neural networks to determine the severity of damage. This method can accurately identify damage in beam structures using only a small amount of displacement data. The average error between the obtained damage severity and the true value does not exceed 4%. In conclusion, it’s feasible to establish the flaw detection model by combining computational mechanics and artificial intelligence algorithms. The nonlinear mapping relationships are revealed between structural responses and flaw parameters.

Therefore, we take the advantages of artificial intelligence algorithms and SBFEM to set up the deep learning model for identifying internal flaws. The scaled boundary finite element method (SBFEM) is an alternative method with a semi-analytical accuracy in calculation [20], which exhibits outstanding works on automatic flaw/damage simulations [21,22]. Combined with the quadtree algorithm and SBFEM, only 6 types of unique elements are needed to cover all mesh morphologies and all the shapes are square [23–25]. The handing node is common with the regular node and doesn’t affect the calculation accuracy. It overcomes the difficulties of low data generation efficiency and complex interface processing to a certain extent. On the other hand, we utilize DNN as the learning framework and employ a Bayesian regularization algorithm to suppress the risk of local convergence of the objective function. A deep learning model for square plates with flaws is established to accurately locate and quantify the size of flaws in the structure. In addition, we analyze the impact of the number of training set samples and flaw size on the inversion results, providing insights for the research on structural disaster prevention and mitigation technologies that combine computational mechanics methods with DNNs.

SBFEM is an alternative method that is analogous to the finite element method. In the solution process, the computational domain is discretized into multiple S-elements. Subsequently, the stiffness matrices of these individual S-elements are assembled to solve for the unknowns. When performing calculations using SBFEM, only the boundaries of the structural domain require discretization. Hanging nodes can be directly treated as regular nodes, which endows it with unique advantages in addressing complex material interfaces and problems involving singularities [26].

In the local coordinate system ξ Oη, ξ denotes the radial coordinate, where ξ = 0 is at the scaling center and ξ = 1 at the polygon boundary, and η is the circumferential coordinate along the polygon boundary. Within a two-dimensional finite domain, for a Cartesian coordinate system xOy, the coordinate values x and y of any arbitrary point inside an element can be expressed in the local coordinate system: (1)x=ξxb(η)=ξN(η)xb (2)y=ξyb(η)=ξN(η)ybwhere N(η) is the shape functions with N(η)=[N1(η),N2(η),⋯], x_b and y_b denote the nodal coordinates vectors of the two-node line element on the boundary with xb=[x1,x2]T, yb=[y1,y2]T.

With the coordinate transformation to the scaled boundary coordinate, the displacements at point inside the subdomain are formed as (3)u(ξ,η)=N(η)u(ξ)where u(ξ) is the analytical displacements on the radial direction.

Based on the principle of virtual work, the two-dimensional linear elastic static equilibrium equation for SBFEM can be derived. (4)E0ξ2u(ξ),ξξ+(E0+E1T−E1)ξu(ξ),ξ−E2u(ξ)=0where E₀, E₁ and E₂ are coefficient matrices, which are similar to the static stiffness matrices in finite element method (FEM) [26].

When solving using SBFEM, the coefficient matrices are determined solely by the boundary conditions and material parameters of the computational domain. Eq. (4) is transformed into a first-order differential equation. (5)ξX(ξ),ξ=−Z{u(ξ)q(ξ)}where q(ξ) is the equivalent internal force vector and the Hamiltonian coefficient matrix Z is (6)Z=[−E0−1E1TE0−1E2−E1E0−1E1TE1E0−1]

The displacements u(ξ) can be obtained by solving the eigenvalues of the Hamiltonian matrix. (7)u(ξ)=Ψnξ−Sncwhere Ψ_n is the modal displacement vector. S_n is the positive real part of the eigenvalue vector of the modal displacement. c is the integral constant vector.

The quadtree mesh generation algorithm is a method that enables rapid transition between coarse and fine meshes in different computational regions by controlling thresholds. High-quality square meshes are generated by the quadtree algorithm and it can be directly converted into SBFEM subdomains. The resulting hanging nodes are treated as regular nodes without affecting solution accuracy. Considering the rotation and symmetry of square meshes, the quadtree mesh generation process only involves six types of node arrangements. When performing linear elastic analysis using SBFEM, only these six element patterns need to be calculated, with no special treatment required [27,28].

A DNN for plate-like structures with circular holes is established based on SBFEM combined with an image quadtree approach. Training the model requires generating a large number of training samples in a short period. Image quadtree SBFEM offers unique advantages when solving problems related to plate-like structures with circular holes [29,30]. It allows direct mesh generation on the image structure, thereby reducing the modeling time for irregular structural models. During mesh generation, only local refinement of the mesh at material interfaces is necessary, which reduces the number of mesh elements and significantly lowers computational costs [31].

Using the image quadtree SBFEM approach, quadtree mesh generation is performed on the image of a heterogeneous concrete aggregate model with double holes. The left end of the model is fixed, and a uniformly distributed load of P = 1 MPa is applied at the right end. The color inside the holes is represented by 255, while that of the matrix is represented by 0. The maximum element size is set to 32, the minimum to 1, and the threshold to 0.05. The completed quadtree mesh, as shown in Fig. 1a, consists of 102,921 elements in total. All calculations in this study were conducted on a desktop PC equipped with an Intel(R) Core (TM) i7-8700 CPU @ 3.20 GHz and 64 GB of RAM. The contour plot of the structural horizontal displacement obtained from static calculations is illustrated in Fig. 1b, with a computational time of only 5.372 s.

In DNNs, mathematical transformations are achieved by stacking different layers and adjusting the weights of each layer through functions, thereby describing data patterns. In this chapter, the image quadtree SBFEM is used to calculate the displacement responses of structural observation points under different loads. Based on a DNN with a backpropagation algorithm, an inversion model for internal flaws or damaged areas of the structure is constructed.

(1)

An inversion model suitable for internal structural flaws was established based on the image quadtree SBFEM and DNN algorithm. The sample collection process and data preprocessing methods are presented. Detailed explanations are provided regarding the observation point arrangement scheme, relevant parameters of the DNN, and the establishment process of the flaw inversion model. Based on the model evaluation method, it was determined that when the proposed inversion model uses the tanh activation function and 25 neurons, it can accurately predict structural flaw parameters while controlling time costs.