Uncertainties are ubiquitous in engineering applications that may arise from different sources such as inherent material randomness, geometric dimensions, manufacturing errors, and dynamic loading. Because deterministic analysis fails to characterize randomness field, stochastic analysis techniques have been extensively studied to strengthen the credibility of computational prediction of uncertainty problems [1]. There are three main variants of stochastic analysis: perturbation based techniques [2], stochastic spectral approaches [3,4], and Monte Carlo simulation (MCs) [5–7]. Among them, the MCs is regarded as the most versatile and simplest approach, and often used as the reference solution to verify the results of Perturbation method and Spectral method, but the exhaustive sampling in MCs leads to a heavy computational burden arising from both solving physical problems and constructing analysis-suitable geometric models that must be addressed carefully.

Combining Proper Orthogonal Decomposition (POD) and Radial Basis Functions (RBF) [8–11] is an effective technique of model order reduction [12–14]. The POD represents the solution field with the special ordered orthogonal functions in a low-dimensional subspace, which is constructed based on a discrete number of system responses obtained from the evaluation of the full order models (FOM) [15]. POD reduces the degrees of freedom by capturing the dominant components of high-dimensional processes because it offers the optimal basis in the sense that the approximation error is minimal in L₂ norm. On the other hand, the RBF builds a surrogate model through interpolating the data in the reduced space, whereby it admits continuous approximation of system responses for any arbitrary combination of input parameters [16] and thus does not need to solve partial differential equation for each sample.

The pre-processing time of MCs in constructing geometric models can be reduced with isogeometric analysis (IGA) [17]. The key idea of IGA is employing spline functions used to construct geometric models in Computer Aided Design (CAD), for example Non-Uniform Rational B-splines (NURBS), T-splines [18], PHT-splines [19,20], and subdivision surfaces [21], as the basis functions to discretize physical fields. Compared to traditional Lagrange polynomial based methods, the main advantage of IGA lies in its ability of integrating numerical analysis and CAD. IGA enables one to perform numerical analysis directly from CAD models without meshing, which is particularly beneficial to uncertainty qualification since it requires fast generation of a large number of models. IGA also offers the benefits of geometric exactness, flexible refinement scheme and high order continuity that are amenable to numerical analysis.

Fracture mechanics is crucial in structural integrity assessment and damage tolerance analyses, but simulation of fracture behaviors poses significant challenges to Finite Element Methods (FEM) for the following reasons: (1) the mesh in proximity to cracks should be several orders of magnitude finer than that used for stress analysis; (2) a remeshing procedure is inevitable when cracks extend; (3) stress singularity or high stress gradients need to be captured. The extended finite element method (XFEM) incorporates enrichment functions to solution space and thus allows for crack propagation in a fixed mesh, but it still relies on a fine and good-quality mesh and necessitates special techniques to represent crack surfaces like Level Set Method. In comparison, Boundary Element Method (BEM) [22–26] has proven a useful tool for fracture simulation. Since BEM only discretizes the boundary of the domain, it not only reduces the degrees of freedom and facilities mesh generation, but more importantly, extends the crack by simply adding new elements at the front of cracks. In addition, as a semi-analytical method, BEM evaluates stress more accurately which is critical in extracting stress intensity factors. Furthermore, isogeometric analysis within the context of boundary element method (IGABEM) inherits the advantage of IGA in integration of CAD and numerical analysis and that of BEM in dimension reduction. IGA with BEM is also natural because both of them are boundary-represented. Since its inception, IGABEM has been successfully applied to potential [27–31], linear elasticity [32–36], acoustics [37–44], electromagnetics [45], structural optimization [40,46–49], etc., Peng et al. [50,51] applied IGABEM to two dimensional and three dimensional linear elasticity fracture mechanics, and demonstrates the accuracy and efficiency of IGABEM in this area.

This paper presents a novel procedure for solving uncertainty problems of fracture mechanics. In this method, the IGABEM is employed for fracture analysis, and MCs for addressing multiple random parameters. The computational cost is reduced by combination of POD and RBF. The remaining of this paper is structured as follows. Section 2 introduces the fundamentals of MCs in stochastic analysis. Section 3 illustrates how to apply POD and RBF to MCs. Section 4 formulates IGABEM in linear elasticity fracture mechanics. Several numerical examples are given in Section 5 to test the reliability, accuracy and efficiency of the proposed method, followed by conclusions in Section 6.

Monte Carlo simulation (MCs) directly characterizes uncertainties by calculating expectations and variances from a large number of samples. For a random variable X associated with the probability density function p(x), the two probabilistic moments are defined as

(1) E[X]=∫-∞+∞xp(x)dxD[X]=∫-∞+∞(x-E[X])2p(x)dx

where E[X] is the expectation of the random variable X, and D(X) represents its variance, whose square root is the standard deviation.

According to the law of large numbers, the average of the results obtained from a number of samples should converge to the expectation as more sampling points are selected, which is the theoretical basis of MCs. Suppose g(X) is an arbitrary function of the random variable X. The expectation and variance of g(X) can be approximated by

(2) E[g(X)]≈1N∑i=1Ng(xi)

(3) D[g(X)]≈1N-1[∑i=1N(g(xi)-E[g(x)])]2

where N is the sample size, and the order of convergence rate is O(N^−1/2).

MCs in structural analysis can be conducted in the following steps [15]: (1) Identify the random variables that are the source of the uncertainties in the system. (2) Determine the probability density functions of the random variables. (3) Use a random number generator to produce a set of samples which are adopted as the input parameters. (4) Employ a numerical method in deterministic analysis to evaluate the solution for each sample. (5) Based on the outputs of numerical analysis for all of the samples, we calculate the expectation and variance of the system using Eqs. (2) and (3) [52]. From the above, it can be seen that MCs is easy to implement because the existing numerical simulation codes can be directly used without modification. In addition, MCs is versatile and is suitable for complex uncertainty problems. However, Step 3 of MCs is very time consuming because the numerical simulation needs to be conducted as many times as the number of sampling points. The large sample size can enhance the accuracy but may lead to higher computational cost.

As mentioned above, MCs is prohibitively expensive because it needs to solve the physical problems at many samples. This procedure can be accelerated by the reduced-order moeling based on POD and RBF. Let α be an input random variable that has an influence on the structural responses. At the first step, we solve the FOM for a series of samples αi of the random variable. The system responses λ(αi) corresponding to the i-th sample point is called snapshots, which can be collected to form the following snapshot matrix Λ:

(4) Λ=[λ(α1),λ(α2),…,λ(αm)]=[λ11λ12…λ1mλ21λ22…λ2m⋮⋮⋱⋮λn1λn2…λnm ]

where Λ∈Rn×m, n indicates the number of responses for any input variable, and m the number of samples. λi,j is the i-th response of the structure for the j-th sample. Decomposing the matrix Λ through Singular Value Decomposition (SVD), we arrive at

(5) Λ=UΣVT=∑j=1rujσjvjT

where r= min(m,n). U∈Rn×n and V∈Rm×m are orthogonal matrices whose entries are denoted by u_ij and v_ij, respectively. uj and vj are the eigenvectors of ΛΛT and ΛTΛ, respectively. The two eigenvectors are also called the left and right singular matrices of matrix Λ, respectively. Σ∈Rn×m is a diagonal matrix in which the diagonal elements σj (also called singular values) are arranged in descending order.

By defining φj=uj and aj(αi)=σjvij, Eq. (5) can be rewritten as

(6) λ(αi)=∑j=1rφjaj(αi)

where φj is defined as the orthogonal basis, and aj(αi) is the corresponding amplitude. Using Eq. (6), the system responses at the selected samples can be expressed by a linear combination of φj and aj(αi), which constitutes a reduced-order model (ROM) with less degrees of freedom than the FOM.

Eq. (6) only approximates a discrete number of system responses that are already computed using the FOM. To achieve a continuous approximation of system responses for any arbitrary input parameters, the radial basis functions (RBF) are used to interpolate the amplitudes in the reduced subspace

(7) a(α)≈â(α)=∑i=1Nηiϕi(α)

where N is the number of samples, ϕi is the i-th RBF and ηi the corresponding coefficient. ϕ(α) takes the form of Gaussian kernel function

(8) ϕi(α)=e-(1/γi2)∥α-αi∥

in which the symbol ∥⋅∥ represents the Euclidean norm, and the coefficient γi determines the width of the basis functions.

By letting a(αj)=â(αj), the values of the coefficient η can be determined by solving Eq. (7). Substituting Eq. (7) into Eq. (6) gives rise to

(9) λ(α)=∑j=1rφjâj(α)

Hence, the system response for any sample of the stochastic variable can be obtained straightforwardly by Eq. (9) without needing to solve the partial differential equations repeatedly. The algorithm mentioned above is illustrated in the Fig. 1. The POD-RBF enable us to conduct MCs without needing to perform FOM simulation for all of the sample points. However, the FOM is still essential for getting snapshots which is solved by IGABEM in our work as detailed in the following section. Therefore, the number of the samples or snapshots should be selected carefully to strike a balance between the accuracy and efficiency. In the future, we will introduce error estimation technique to improve the performance of our method. In addition, it is highlighted that in order to further enhance the computational efficiency, the number of basis functions in Eqs. (5) and (9) can be decreased by selecting the basis functions corresponding to the elements with larger values in Σ.

In the following examples, the input random variables are supposed to follow the Gaussian distribution with the expectation being 0.5 and the standard deviation being 0.033. The sampling method for Gaussian distribution function used in this paper is to select sample points in the interval [E-3D,E+3D]. It is noted that only the problems with a single random input variable are considered in the present paper.The application of the method for the problems with multidimensional random input variables will be investigated in the future.

The paper presents a novel framework for Monte Carlo simulation of multi-dimensional uncertainties of two-dimensional linear fracture mechanics. We use the NURBS to build the geometric model and discretize boundary integral equations. The IGABEM eliminates the repeatedly meshing procedure in uncertainty quantification, and retains the geometric exactness. The combination of POD and RBF accelerates the stochastic analysis and maintains good accuracy at low frequencies. In this work, the POD-RBF is used to approximate the structural response of a single random input variable. The method can also be extended to the problems with multidimensional random input variables, which will be investigated in the future. In addition, the present method will be applied to three dimensional problems and multi-physics coupling problems.