Isogeometric Collocation Method for Random Field Discretization Based on Adaptive Moment Abscissae

Abstract

The discretization of random fields is the first and most important step in the stochastic analysis of engineering structures with spatially dependent random parameters. The essential step of discretization is solving the Fredholm integral equation to obtain the eigenvalues and eigenfunctions of the covariance functions of the random fields. The collocation method, which has fewer integral operations, is more efficient in accomplishing the task than the time-consuming Galerkin method, and it is more suitable for engineering applications with complex geometries and a large number of elements. With the help of isogeometric analysis that preserves accurate geometry in analysis, the isogeometric collocation method can efficiently achieve the results with sufficient accuracy. An adaptive moment abscissa is proposed to calculate the coordinates of the collocation points to further improve the accuracy of the collocation method. The adaptive moment abscissae led to more accurate results than the classical Greville abscissae when using the moment parameter optimized with intelligent algorithms. Numerical and engineering examples illustrate the advantages of the proposed isogeometric collocation method based on the adaptive moment abscissae over existing methods in terms of accuracy and efficiency.

Introduction

Realistic engineering structures exhibit stochastic characteristics because of random variations in their material properties and external loads [1]. Traditional deterministic analyses that disregard such uncertainties cannot accurately capture the actual structural responses of engineering structures [2]. Consequently, stochastic structural analysis, which considers the inherent randomness within engineering structures, is necessary for further developing structural analysis. Spatially dependent randomness in engineering structures is modelled as a random field. Representing a continuous-parameter random field using a finite set of random variables is essential for the stochastic analysis of structures with random-field uncertainties. This process, known as random field discretization, aims to achieve a balanced compromise between accuracy and computational efficiency.

The Karhunen–Loève (K–L) expansion is one of the most commonly used methods for discretizing the random field. The K–L expansion can hold both homogeneous and inhomogeneous fields, minimize the mean-square error committed by a finite-term K–L approximation, and converge the sequence of K–L approximations in mean square to the correct limit [1]. Given these advantages, the K–L expansion requires the solution of a Fredholm integral equation that cannot be solved easily. An analytical solution to the integral equation exists only when the physical domain and covariance function of the random field are simple. Numerical solutions are the only options for more complex practical engineering applications.

In the field of finite element analysis (FEA), the Galerkin method is among the most popular methods for numerically solving the Fredholm integral equation [3, 4, 5–6]. However, the Galerkin method requires multiple integral operations when assembling system matrices [7, 8–9]. In practical engineering applications, the geometry of the physical domain can be very large and complex, and the number of elements can be large, which can lead to low efficiency of the Galerkin method. As an alternative method for the problem of random field discretization, the collocation method [10] has an obvious advantage over the Galerkin method in terms of efficiency. The collocation method has considerable potential for engineering applications with complex geometries and numerous elements with fewer integral operations and points to traverse. In 2005, Hughes proposed a new computer-aided analysis method referred to as isogeometric analysis (IGA) [11]. This method applies spline functions such as the non-uniform rational B-spline (NURBS) [12] and T-spline [13] as the shape functions, preserving the geometric consistency of engineering products between computer-aided design (CAD) and computer-aided engineering (CAE) phases [14, 15, 16–17]. This innovative approach can help reduce model errors in the early stages of stochastic analysis, including in the discretization of the random field.

Thus far, researchers have conducted several studies integrating collocation and IGA techniques. For example, Cho et al. [18] proposed overlapping additive Schwarz preconditioners for the isogeometric collocation discretization of a system of linear elasticity in two- and three-dimensions. Ali and Ma [19] investigated the general formulation of the isogeometric collocation method for parameterizing computational domains for the IGA. Hou et al. [20] applied the isogeometric collocation method to simulate the movement of crack edges in an extended isogeometric analysis of strongly discontinuous problems. For collocation points, recent studies proposed alternatives that can achieve improved convergence rates in specific situations [21], such as Cauchy–Galerkin points [22], Gauss points [23], and superconvergent points [24, 25–26]. However, the classical Greville abscissa remains the first choice for calculating the coordinates of collocation points [27, 28–29].

Despite its popularity, the accuracy of the Greville abscissa remains unsatisfactory. With no parameters to adjust the abscissae, the same expressions are expected when applying the collocation method with Greville abscissae to discretize the random fields on different meshes. An adaptive moment abscissa was proposed in this paper to preserve the stable interpolation property and improve the accuracy of the Greville abscissa. The old expression of the Greville abscissae was changed into an adaptive moment abscissae by adding an adaptive moment parameter to the expression of the Greville abscissae. The moment parameter was optimized using an optimization algorithm, which led to an adjusted abscissa specialized for the knot vector of a certain IGA mesh. This adaptive parameter makes the collocation points more suitable for the IGA mesh, and therefore, it improves the accuracy of the isogeometric collocation method for the discretization of random fields.

The remainder of this paper is organized as follows. Section 2 describes the random field and its isogeometric representation based on K-L expansion. Section 3 describes the proposed isogeometric collocation method with an adaptive moment abscissa. Section 4 presents the three illustrative examples of 1D/2D/3D random fields used for assessing the performance of the proposed approach. Finally, Section 5 presents the conclusions of this study.

Random Field Discretization Based on Karhunen–Loève Expansion

Random Field

A random field is a series of random variables distributed in space that is primarily used to describe random parameters in engineering applications with spatial variability and correlation [9]. The spatial correlation of random variables at different locations was described using a covariance function.

A Gaussian random field is among the most widely used random fields for describing random parameters in engineering applications. A Gaussian random field $H_G(\mathit{x},\theta )$ can be expressed as

1

$$H_G(\mathit{x},\theta )={\overline{H}}_G(\mathit{x})+H_G^{△}(\mathit{x},\theta ),$$where $\theta \in \mathrm{\Omega }$ and $(\mathrm{\Omega },F,P)$ represent the probability space, in which $\mathrm{\Omega }$, F, and P represent the sample space, σ-algebra, and probability measure, which is a function on F such that $P:F\to [0,1]$, respectively [30]. $\mathit{D}\in {\mathit{R}}^d$ represents the physical domain of a geometrical object, where $d$ represents the dimension of the random field. Further, x represents the coordinates of the random field locations (1D: x = x, 2D: x = (x, y), 3D: x = (x, y, z)), ${\overline{H}}_G(\mathit{x})$ represents the mean of the random field $H_G(\mathit{x},\theta )$, and $H_G^{△}(\mathit{x},\theta )$ represents a zero-mean random field that shares the same covariance function as $H_G(\mathit{x},\theta )$.

Random fields are often assumed to be homogeneous or stationary, implying that their finite-dimensional probability distributions are invariant under arbitrary translations. This indicates that the covariance function is a function of argument difference $\mathit{x}-{\mathit{x}}^{\mathbf{\prime }}$. Further, random fields are sometimes assumed to be isotropic, i.e., invariant under orthogonal transformations. In this case, the covariance function is a function of distance $∥\mathit{x}-{\mathit{x}}^{\mathbf{\prime }}∥$[1].

Karhunen–Loève Expansion

Discretization, which transforms a continuous random field into a finite number of discrete random variables, is an essential step for establishing the representation of a random field. The K–L expansion method is the most widely used discretization method for random fields. K–L expansion can be used to discretize both Gaussian and non-Gaussian random fields and has the advantages of high accuracy and good convergence [31]. The problem of dealing with random fields can be simplified to the problem of dealing with random variables by using K-L expansion to discretize the random field. After calculating the eigenvalue and eigenfunction of the covariance function of the random field, which are unknown parameters in the discretization process, the random field can be successfully discretized, and the value of any location in the random field can be calculated.

K–L expansion is based on the spectral decomposition of the covariance function $C_{\mathit{xx}}(\mathit{x},{\mathit{x}}^{\mathbf{\prime }})$ of the random field $H(\mathit{x},\theta )$.

2

$$C_{\mathit{xx}}(\mathit{x},{\mathit{x}}^{\mathbf{\prime }})=\sum _{i=0}^{\infty }{\lambda }_i{\varphi }_i(\mathit{x}){\varphi }_i({\mathit{x}}^{\mathbf{\prime }}),$$where ${\lambda }_i$ and ${\varphi }_i(\mathit{x})$ represent the eigenvalues and corresponding eigenfunctions of the covariance function, respectively.

Given an eigenpair $\left\{{\lambda }_i,{\varphi }_i(\mathit{x})\right\}$, a random field can be expanded using a set of uncorrelated random variables ${\left\{{\zeta }_i(\theta )\right\}}_{i\in \mathbb{N}}$ as

3

$$H(\mathit{x},\theta )=\overline{H}(\mathit{x})+\sum _{i=1}^{\infty }\sqrt{{\lambda }_i}{\varphi }_i(\mathit{x}){\zeta }_i(\theta ).$$

For a Gaussian random field $H_G(\mathit{x},\theta )$, ${\left\{{\zeta }_i(\theta )\right\}}_{i\in \mathbb{N}}$ is a set of independent standard Gaussian random variables.

In practical engineering applications, the K–L expansion of a random field is always approximated by a finite number of terms M. For a Gaussian random field, its discretization form obtained using K-L expansion is given as

4

$$H_G(\mathit{x},\theta )={\overline{H}}_G(\mathit{x})+\sum _{i=1}^M\sqrt{{\lambda }_i}{\varphi }_i(\mathit{x}){\zeta }_i(\theta ).$$

Isogeometric Representation

This study applies IGA to construct CAD models and random field representations of all examples, considering the advantages of the complete preservation of geometric information and high-order continuity. The basis function $R_I(\mathbf{\xi })$ of the IGA is the complete basis of the Hilbert space $L^2(\mathit{D})$[32], and therefore, the eigenfunction of the random field’s covariance function can be expressed using the IGA basis functions in the subspace ${\mathbb{W}}_s=\text{span}\left\{R_I(\mathbf{\xi })\right\}$ expanded by the IGA basis functions as

5

$${\varphi }_i(\mathit{x})\to {\varphi }_i^s(\mathit{x})=\sum _{I=1}^{N_c}{f}_I^i{R}_I(\mathbf{\xi }),$$where $f_I^i$ represents the unknown coefficient corresponding to the Ith IGA basis function $R_I(\mathbf{\xi })$, and $N_c$ represents the number of control points of an IGA model. The IGA model is already completed with mesh refinement, which satisfies the accuracy requirements for structural analysis. For detailed information on the IGA model and its basis functions, readers are referred to Ref. [12].

Substituting ${\varphi }_i^s(\mathit{x})$ into Eq. (4), the discretization form of the random field can be rewritten as

6

$$H_G(\mathit{x},\theta )={\overline{H}}_G(\mathit{x})+\sum _{i=1}^M\sqrt{{\lambda }_i}{\zeta }_i(\theta )\left(\sum _{I=1}^{N_c}{f}_I^i{R}_I(\mathbf{\xi })\right),$$where the mean of the random field ${\overline{H}}_G(\mathit{x})$ can be represented by the IGA basis function as

7

$${\overline{H}}_G(\mathit{x})=\sum _{I=1}^{N_c}{\mu }_I{R}_I(\mathbf{\xi }).$$

Finally, a Gaussian random field can be represented by IGA basis functions as

8

$$H_G(\mathit{x},\theta )=\sum _{I=1}^{N_c}{\mu }_I{R}_I(\mathbf{\xi })+\sum _{i=1}^M\sqrt{{\lambda }_i}{\zeta }_i(\theta )\left(\sum _{I=1}^{N_c}{f}_I^i{R}_I(\mathbf{\xi })\right),$$where ${\mu }_I$ represents the mean value of the random field at the Ith control point. For a homogeneous Gaussian random field, the mean values of the random field at all control points are the same, i.e., $\forall {\mu }_I=\mu$ and ${\overline{H}}_G(\mathit{x})=\mu$.

Isogeometric Collocation Method with Adaptive Moment Abscissae

The two unknown parameters in Eq. (8), namely, the eigenvalue ${\lambda }_i$ and coefficient $f_I^i$, must be calculated by solving the Fredholm integral equation

9

$${\int }_{\mathit{D}}{C}_{\mathit{xx}}(\mathit{x},{\mathit{x}}^{\mathbf{\prime }}){\varphi }_i({\mathit{x}}^{\mathbf{\prime }})\text{d}{\mathit{x}}^{\mathbf{\prime }}={\lambda }_i{\varphi }_i(\mathit{x}).$$

The analytical solutions to Eq. (9) can be obtained only for a few problems with simple CAD models. In most cases, a numerical solution is the only choice in most cases [33]. Currently, the commonly used numerical methods for solving the Fredholm integral equations are the Galerkin method [34, 35], Nystrom method [36, 37–38], and collocation method [27, 28].

The Galerkin method is widely used along with the traditional FEA method because of its high accuracy. However, this method requires multiple integral operations and thus has a low computational efficiency on a mesh with a large number of elements. The Nyström method has high computational efficiency because it does not require integration; however, it requires a large number of uniformly distributed points to achieve sufficient accuracy, which makes it unsuitable for cases of complex geometry with a large number of elements. The computational efficiency of the collocation method is higher than that of the Galerkin method. Further, the high-order continuity of the IGA basis function ensures the accuracy of the isogeometric collocation method in solving partial differential equations. Therefore, with the help of the IGA, the collocation method can outperform other methods with an obvious advantage in terms of efficiency while ensuring accuracy.

Isogeometric Collocation Method

The isogeometric collocation method is used to find an appropriate pair of eigen parameters $\left\{{\lambda }^s,{\varphi }^s(\mathit{x})\right\}$ in the isogeometric basis function subspace such that the error of the isogeometric discretization is equal to 0 at specific points, and $\left\{\lambda ,\varphi (\mathit{x})\right\}$ is replaced by the approximation $\left\{{\lambda }^s,{\varphi }^s(\mathit{x})\right\}$. These points are referred to as collocation points.

To solve the unknown coefficients $f_I^i$ in Eq. (5), the number of collocation points should be equal to or more than the number of unknown coefficients. The number of collocation points can be greater than those of the basis functions, especially when the eigenfunction is not expressed by all IGA basis functions. Although a relatively small number of basis functions is sufficient to achieve a good approximation of the eigenfunction $\varphi (\mathit{x})$, many collocation points are required to minimize the global error of the isogeometric discretization of the random field $e(\mathit{x})$ in Eq. (10). The study was conducted under the condition that the number of collocation points was equal to the number of basis functions. The inequality is calculated in Ref. [33].

The error of the isogeometric discretization of a random field is given by

10

$$\begin{array}{c}e(\mathit{x})={\int }_{\mathit{D}}{C}_{\mathit{xx}}(\mathit{x},{\mathit{x}}^{\prime }){\varphi }^s({\mathit{x}}^{\prime })\text{d}{\mathit{x}}^{\prime }-{\lambda }^s{\varphi }^s(\mathit{x})\\ =\sum _{I=1}^{N_c}{f}_I^i\left[{\int }_{\mathit{D}}{C}_{\mathit{xx}}(\mathit{x},{\mathit{x}}^{\prime })R_I(\mathbf{\xi })\text{d}{\mathit{x}}^{\prime }-{\lambda }^s{R}_I(\mathbf{\xi })\right].\\ \end{array}$$

If the error of Eq. (10) on the collocation points is zero,

11

$$\sum _{I=1}^{N_c}{f}_I^i\left[{\int }_{\mathit{D}}{C}_{\mathit{xx}}(\mathit{x},{\mathit{x}}^{\mathbf{\prime }})R_I(\mathbf{\xi })\text{d}{\mathit{x}}^{\mathbf{\prime }}-{\lambda }_i^s{R}_I(\mathbf{\xi })\right]=0,i,I=1,\dots ,N_c.$$

The solution of Eq. (11) can be considered an approximation of the solution of Eq. (9).

For a convergence analysis, the mesh size of the refined parametric domain is defined as h, and the Hilbert–Schmidt operator $G_C:L^2(\mathit{D})\to L^2(\mathit{D})$ is defined as $(G_C\varphi )(\mathit{x}):=\lambda \varphi (\mathit{x})$. Let $f_s(\mathit{D})$ represent the space of a continuous function on $\mathit{D}$ with a supremum that can be represented by $W_s=\text{span}\left\{R_i(\xi )\right\}$, which is a sequence of finite-dimensional approximating subspaces. Further, let $G_C$ represent the compact operator from $f_s(\mathit{D})$ to $f_s(\mathit{D})$.

We introduced an interpolatory projection operator called $P_h:L^2(\mathit{D})\to W_s$. In other words, given $\varphi (\mathit{x})\in f_s(\mathit{D})$,$P_h\varphi (\mathit{x})$ represents the element of $W_s$ interpolates $\varphi (\mathit{x})$ at collocation points $\left\{{\mathit{x}}_{\mathbf{1}},...,{\mathit{x}}_{N_c}\right\}$. As explained in Ref. [39], $P_h$ is a bounded linear operator of $f_s(\mathbf{D})$ with the norm

12

$$∥P_h∥=\underset{\mathit{x}\in \mathit{D}}{sup}\sum _{j=1}^{N_c}\left|R_j(\mathit{x})\right|.$$

Then, the collocation in Eq. (11) is equivalent to

13

$$(P_h{G}_C{\varphi }^s)(\mathit{x})={\lambda }^s{\varphi }^s(\mathit{x}).$$

To ensure convergence, Eq. (14) should be satisfied.

14

$$\underset{h\to 0}{lim}∥P_h{G}_C-G_C∥=0.$$

Because $G_C$ is compact on $f_s(\mathit{D})$, a sufficient condition for Eq. (14) is

15

$$\underset{h\to 0}{lim}{∥P_h\varphi (\mathit{x})-\varphi (\mathit{x})∥}_{L^{\infty }}=0,\varphi (\mathit{x})\in f_s(\mathit{D}).$$

From Eq. (15), it follows that ${∥\varphi (\mathit{x})-{\varphi }^s(\mathit{x})∥}_{\infty }\to 0$ when $h\to 0$, which demonstrates the convergence of collocation solutions in the supremum norm. The rate at which ${\varphi }^s(\mathit{x})$ converges to $\varphi (\mathit{x})$ was identical to the rate at which $P_h\varphi (\mathit{x})$ converges to $\varphi (\mathit{x})$. Detailed analyses of the convergence of the isogeometric collocation method can be found in Refs. [1, 40].

Assembling the Solution Matrices

According to matrix algebra, Eq. (11) can be rewritten as

16

$$\mathit{Af}=\mathbf{\Lambda }\mathit{B}\mathit{f},$$where $\mathit{A}\in {\mathbb{R}}^{N_c\times N_c}$, $\mathit{B}\in {\mathbb{R}}^{N_c\times N_c}$, $\mathbf{\Lambda }\in {\mathbb{R}}^{N_c\times N_c}$, and $\mathit{f}\in {\mathbb{R}}^{N_c\times N_c}$ are symmetric matrices.

17

$$\begin{array}{c}{A}_{\mathit{ij}}={\int }_{\mathit{D}}{C}_{\mathit{xx}}({\mathit{x}}_i,{\mathit{x}}^{\mathbf{\prime }})R_j(\mathbf{\xi })\text{d}{\mathit{x}}^{\mathbf{\prime }},i,j=1,\dots ,N_c\\ B_{\mathit{ij}}=R_j({\mathbf{\xi }}_i),i,j=1,\dots ,N_c\\ {\mathrm{\Lambda }}_{\mathit{ij}}={\delta }_{\mathit{ij}}{\lambda }_j^s,i,j=1,\dots ,N_c\\ {\mathit{f}}_{\mathit{ij}}=f_i^j,i,j=1,\dots ,N_c\\ \end{array}$$where ${\delta }_{\mathit{ij}}$ represents the Kronecker delta. Therefore, the isogeometric collocation method transforms the problem of calculating the eigenparameter pairs of the random field covariance function into that of calculating the eigenvalues of the matrix. By solving Eq. (16), a pair of characteristic parameters $\left\{\lambda ,\varphi (\mathit{x})\right\}$ of the covariance function can be approximately replaced by a pair of characteristic parameters of the matrix $\left\{{\lambda }^s,{\varphi }^s(\mathit{x})\right\}$. The random field expression based on the isogeometric basis function can be obtained by substituting the calculated $\mathit{f}\in {\mathbb{R}}^{N_c\times N_c}$ and ${\lambda }^s$ into Eq. (8).

The process of assembling the matrix via the Galerkin method is similar to that with the collocation method when solving the Fredholm equation. However, according to the conclusion reported by Jahanbin and Rahman [1], in d-dimensional space, the calculation of the elements of matrix A required a d-fold integration operation in the collocation method. The same calculation required a 2d-fold integration operation in the Galerkin method; this operation is more expensive than that of the collocation method. In addition, calculating the elements of matrix B in the Galerkin method requires more d-fold integration operations than that in the collocation method. Thus, it can be concluded that the collocation method has obvious advantages in terms of computational efficiency compared to that of the Galerkin method.

Adaptive Moment Abscissae

The accuracy and stability of the collocation method depend on the choice of the collocation points. Currently, the most commonly used collocation point is the Greville abscissae.

In the k-th direction of the d-dimensional parameter space $\widehat{\mathit{D}}={[0,1]}^d,k=1,\dots ,d$, the Greville abscissa ${\overline{\xi }}_{i_k}^k$ is defined as

18

$${\overline{\xi }}_{i_k}^k=\frac{1}{p_k}\sum _{q=1}^{p_k}{\xi }_{i_k+q}^k\left(i_k=1,\dots ,N_{c,k}\right),$$where $N_{c,k}$ represents the number of control points in the k-th direction, and $\prod _{k=1}^d{N}_{c,k}=N_c$.

Eq. (18) shows that there is an ${\overline{\xi }}_1^k=0,{\overline{\xi }}_{N_{c,k}}^k=1,{\overline{\xi }}_{i_k}^k\in [0,1]$ for any k. Then, based on isogeometric mapping, the collocation points can be obtained from the Greville abscissae as

19

$${\mathit{x}}_i=\stackrel{\circ }{\mathit{X}}({\overline{\mathit{\xi }}}_i),{\overline{\mathit{\xi }}}_i=({\overline{\xi }}_{i_1}^1,\dots ,{\overline{\xi }}_{i_d}^d)\in \widehat{\mathit{D}}={[0,1]}^d,\forall i_k=1,\dots N_{c,k}.$$

Greville abscissae are usually the first choice for researchers applying the collocation method because of their stable interpolation properties. However, the collocation method using the Greville abscissae suffers from an unsatisfactory calculation accuracy. An adaptive moment abscissa is proposed to improve the calculation accuracy of the isogeometric collocation method. The moment abscissa can be defined as

20

$${\overline{\xi }}_{i_k}^k={\left(\frac{1}{p_k}\sum _{q=1}^{p_k}{({\xi }_{i_k+q}^k)}^r\right)}^{1/r},$$where $r\in {\mathbb{R}}^{+}$. When $r=1$, the moment abscissa degenerates into the Greville abscissa; i.e., the Greville abscissa is a special form of the moment abscissa.

According to Jator and Sinkala [27], the error in the spline function collocation method can be reduced by reducing the value of

21

$$f(r)=\underset{i}{max}\left\{\frac{min\left\{{\xi }_{i_k+p_k}^k-{\xi }_{i_k}^k:\left\{\left({\xi }_{i_k}^k,{\xi }_{i_k+p_k}^k\right)\cap \left({\overline{\xi }}_{i_k}^k(r),{\overline{\xi }}_{i_k+1}^k(r)\right)\ne 0\right\}\right\}}{{\overline{\xi }}_{i_k+1}^k(r)-{\overline{\xi }}_{i_k}^k(r)}\right\}.$$

Eq. (21) shows that the error in the collocation method increases when the abscissae gradually approaches the interval boundary of the knot vector of the spline. Therefore, parameter r in the moment abscissae can be automatically adjusted by an intelligent optimization algorithm, and the collocation points can be adjusted to different isogeometric meshes. The value of Eq. (21) can be sufficiently small on different isogeometric meshes for reducing the error of the isogeometric collocation method; therefore, it is called the adaptive moment abscissae for the collocation points.

This study adopts a derivative-free genetic algorithm for determining parameter r, which can be implemented using built-in functions in MATLAB (Mathworks). Both the population size $N_p^{\mathit{GA}}$ and maximum iteration number $N_t^{\mathit{GA}}$ must be set to a sufficiently large value to ensure the reliability of genetic algorithm for different examples.

In terms of computational costs, the calculation of the moment abscissae and objective function in Eqs. (20) and (21) is related to the number of collocation points and order of the IGA basis function. The time complexity in a single dimension can be easily obtained as $\mathrm{O}(p_k{N}_{c,k}+(N_{c,k}-1)(N_{c,k}-p_k))$. For a d-dimensional space, the computation of Eqs. (20) and (21) relies only on the knot vector in the corresponding single dimension. Therefore, parameter $r$ in each direction needs to be calculated and optimized independently, which implies that the computational cost of the d-dimensional space linearly increases with dimension $d$, and the total time complexity is $\mathrm{O}\left(N_t^{\mathit{GA}},N_p^{\mathit{GA}},\sum _{k=1}^d,\left(p_k,N_{c,k},+,(N_{c,k}-1),(N_{c,k}-p_k)\right)\right)$. The physical domain coordinates of the collocation points need to be calculated according to Eq. (19) after obtaining coordinates in the adaptive collocation parameter domain. The computational cost of this step is exactly the same as the Greville collocation method, which shows an exponential relationship with the dimension.

A flowchart of the isogeometric collocation method with the proposed adaptive moment abscissae is presented in Figure 1.

[See PDF for image]

Figure 1

Flowchart of the isogeometric collocation method with the proposed adaptive moment abscissae

The implementation steps of the isogeometric collocation method are listed below:

• Step 1: Locate the optimal parameter value $\overline{r}$ by minimizing $f(r)$ in Eq. (21) using the genetic algorithm.

• Step 2: Substitute the optimized parameter $\overline{r}$ into Eq. (20) to calculate the adaptive moment abscissa.

• Step 3: Substitute the adaptive moment abscissa ${\overline{\xi }}_{i_k}^k$ into Eq. (19) to calculate the coordinates of the collocation points.

The eigenvalues and eigenfunctions of the covariance function can be solved using the determined collocation points, thereby achieving the discretization of the random field.

Experimental Results

1D Line Segment Experiment

Consider the 1D Gaussian random field $H_G(\mathbf{x},\theta )$ defined in $\mathit{D}=[0,1]$. Five one-dimensional IGA meshes of different orders were used in this example. The knot vector and control points for the coarsest mesh are listed in Table 1, and the weights corresponding to the control points are set to 1. A series of refined meshes with more elements was obtained from the coarsest mesh with one element by adding new knots and control points, which are also known as h-refinement for the IGA mesh.

Table 1. Knot vector and control points for the coarsest mesh

Order	Knot Vector	Control Point
1	[0,0,1,1]	(0,0) (1,0)
2	[0,0,0,1,1,1]	(0,0) (1/2,0) (1,0)
3	[0,0,0,0,1,1,1,1]	(0,0) (1/3,0) (2/3,0) (1,0)
4	[0,0,0,0,0,1,1,1,1,1]	(0,0) (1/4,0) (1/2,0) (3/4,0) (1,0)

Isogeometric collocation methods with the proposed adaptive moment abscissae and classical Greville abscissae were applied to calculate the coordinates of the collocation points. The search field of $\overline{r}$ was set to [0.05, 10], and the population size and maximum iteration number for the genetic algorithm were set to 10 and 100, respectively.

Figure 2 shows isogeometric meshes of the 1st to the 4th order with eight elements, knots of different orders, and collocation points obtained by the two different collocation methods marked by different colors. The positions of the collocation points in the 1st order mesh are exactly the same, while the positions of the collocation points obtained from the adaptive moment abscissae are slightly different from those obtained from the classical Greville abscissae.

[See PDF for image]

Figure 2

Distributions of collocation points obtained by the Greville and adaptive moment abscissae and knots for the meshes of different orders

The covariance function of the random field was set to the following two common types:

22

$$C_{\mathit{xx}}(x,x^{\prime })=\left\{\begin{array}{c}{\sigma }^{2}exp\left(-\frac{{\left|x-x^{\prime }\right|}^2}{{\left(\mathit{bL}\right)}^2}\right),(\text{Gaussian}),\hfill \\ {\sigma }^{2}exp\left(-\frac{\left|x-x^{\prime }\right|}{\mathit{bL}}\right),(\text{exponential}),\hfill \end{array}\right)$$where ${\sigma }^2=1,L=1,$ and $b=1$.

The eigenvalues of the Gaussian and exponential covariance functions of the random field were calculated on the IGA meshes of the 1st order to the 4th order with different times of h-refinement. The fifth eigenvalue was selected to compare the advantages and disadvantages of different collocation abscissae. The exact values of the fifth eigenvalue of the Gaussian and exponential covariance functions were 1.173953119186 × 10⁻⁵ and 1.227891385452×10⁻², respectively. Table 2 lists the moment result of the adaptive moment abscissae. The moment values on different meshes show considerable deviation from 1, which is the implicit moment value of the Greville abscissae. Figure 3 shows the convergence curves of the moment parameter optimization corresponding to meshes with different numbers of elements in the order of four. All mean fitness values of the populations corresponding to different meshes converge before reaching the prescribed maximum iteration number of 100, which indicates that the genetic algorithm is robust as long as a sufficiently large population size and maximum iteration number are utilized.

Table 2. The moment result of the adaptive moment abscissae on different meshes

Order of mesh	Number of elements
	4	8	16	32	64	128
1	0.077	6.076	6.555	4.094	7.520	6.984
2	1.200	1.100	0.900	0.900	0.900	0.900
3	0.470	0.292	0.610	0.885	0.875	0.846
4	1.000	0.438	0.900	0.500	0.430	0.430

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Figure 3

Convergence curves of the moment parameter optimization of the 1D line segment experiment.

Figures 4 and 5 show the relative errors of the results obtained using the collocation methods with Greville and adaptive moment abscissae, respectively. The Gaussian and exponential covariance functions are plotted in two subplots respectively. The convergence results of the collocation methods with two different abscissae are very similar, which confirms that the adaptive moment abscissa is a general form of Greville abscissa. In addition, the relative errors of the results for the adaptive moment abscissae were lower than those for the Greville abscissae. Figure 6 shows the error reduction of the collocation method with the adaptive moment abscissae compared with the collocation method with the Greville abscissae, thereby demonstrating the advantage of the adaptive moment abscissae in accuracy over that of the Greville abscissae.

[See PDF for image]

Figure 4

Relative errors of the fifth eigenvalue obtained by the collocation method with Greville abscissae: (a) Covariance function of the random field is a Gaussian function, (b) Covariance function of the random field is an exponential function

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Figure 5

Relative errors of the fifth eigenvalue obtained by the collocation method with adaptive moment abscissae: (a) Gaussian covariance function, (b) Exponential covariance function

[See PDF for image]

Figure 6

Error reduction of the collocation method with adaptive moment abscissae compared to that of the collocation method with Greville abscissae: (a) Gaussian covariance function, (b) Exponential covariance function

Figure 6 shows that, on the 1st order mesh, the eigenvalues obtained by the two collocation methods with different abscissae are the same regardless of the number of elements, type of covariance function, and type of collocation abscissae. This is because the positions of the collocation points achieved by these two collocation methods are the same on the 1st order mesh as shown in Figure 2, which results in the same calculations of the eigenvalues. Except for the 1st order mesh, the eigenvalue results of the proposed adaptive moment abscissa are more accurate than those of the classical Greville abscissa. However, the error reduction ranges of the adaptive moment abscissae are different, and there is no obvious positive correlation between the error reduction range and number of elements or order of mesh. This phenomenon can be attributed to the random search properties of the genetic algorithm. The parameter r calculated by the adaptive moment abscissa is only the local optimum. The accuracy of the collocation method with the proposed adaptive moment abscissae was obviously better compared to that of the collocation method with the classical Greville abscissae. For example, the accuracy of the eigenvalue result can be improved by 98% on the 4th order mesh when the covariance function of a random field is Gaussian.

The advantage of the proposed isogeometric collocation method with adaptive moment abscissae in terms of accuracy was demonstrated by calculating the parameters of the random field discretization.

2D Surface Experiment

Consider a 2D Gaussian random field $H_G(\mathit{x},\theta )$ defined as a quarter of a square with a hole in the middle. A square with a hole is shown in Figure 7a, with sizes L = 20 and R = 1. The 2nd order IGA meshes are applied in this example. The search field of $\overline{\mathbf{r}}$ is set to [0.05, 10], and the population size and maximum iteration number for the genetic algorithm are set to 10 and 100, respectively. The knot vectors of the coarsest mesh are ${\mathrm{\Xi }}_1=[0,0,0,0.5,1,1,1]$ and ${\mathrm{\Xi }}_2=[0,0,0,1,1,1]$, and their control points and weights are listed in Table 3. As in the first example, five refined meshes with 8–2048 elements were obtained from the coarsest mesh with two elements through h-refinement. The coarsest mesh and mesh refined through h-refinement performed five times are illustrated in Figure 7b, c, respectively.

[See PDF for image]

Figure 7

Quarter of a square with a hole in the middle (Control points are marked by green): (a) Shape, (b) Coarsest IGA mesh with two elements, (c) Refined IGA mesh obtained through h-refinement performed five times with 2048 elements

Table 3. Control points and weights for the coarsest mesh

Control Point	weight	Control Point	weight	Control Point	weight
(-1,0)	1	($-\frac{15}{4}$,$\frac{21}{2}$)	1	(− $\frac{21}{2}$,0)	1
(-1,$\sqrt{2}$-1)	$\frac{1+\sqrt{2}}{2\sqrt{2}}$	(0,$\frac{21}{2}$)	1	($-\frac{21}{2}$,$\frac{15}{4}$)	1
($-\sqrt{2}$+1,1)	$\frac{1+\sqrt{2}}{2\sqrt{2}}$	(− 20,0)	1	(− 20,20)	1
(0,1)	1	(− 20,20)	1	(0,20)	1

The covariance function of the random field is

23

$$C_{\mathit{xx}}(\mathit{x},{\mathit{x}}^{\mathbf{\prime }})={\sigma }^{2}exp\left(-\frac{∥\mathit{x}-{\mathit{x}}^{\mathbf{\prime }}∥}{\mathit{bL}}\right),\mathit{x},{\mathit{x}}^{\mathbf{\prime }}\in {\mathbb{R}}^2,$$where ${\sigma }^2=0.01$ and $b=0.5$.

Isogeometric collocation methods with the proposed adaptive moment and classical Greville abscissae are applied to calculate the eigenvalues of the covariance function. There were no analytical results for the eigenvalues, and therefore, the isogeometric Galerkin method was applied as reference. Random field discretization was demonstrated on six meshes, including the coarsest mesh and the meshes refined through h-refinement performed one, two, three, four, and five times.

The third eigenvalues and eigenfunctions were selected to compare the results of the three methods. Table 4 presents the results of the three methods on the six meshes, while Figure 8 shows eigenfunctions obtained by the three methods on the mesh with 2048 elements. Table 4 indicates that eigenvalues calculated by the three methods are almost the same for the last three meshes with a large number of elements. Further, Figure 8 shows that eigenfunctions calculated using the three methods were almost the same and indiscernible to the naked eye. Thus, the accuracy of the isogeometric collocation method with the proposed adaptive moment abscissa was verified. However, Table 4 also shows that the collocation method with the proposed adaptive moment abscissae converged faster than that with the classical Greville abscissae.

Table 4. The third eigenvalue of the random field’s covariance function calculated by three different methods

Number of element	Eigenvalue
	Isogeometric collocation method		Isogeometric Galerkin method
	Adaptive moment abscissae	Greville abscissae
2	0.35709	0.33114	0.42677
8	0.42456	0.41410	0.43658
32	0.43715	0.43542	0.43805
128	0.43799	0.43776	0.43800
512	0.43799	0.43796	0.43798
2048	0.43798	0.43798	0.43798

[See PDF for image]

Figure 8

Third eigenfunction of the covariance function of the random field calculated by three methods on the mesh with 2048 elements: (a) Isogeometric collocation method with adaptive moment abscissae, (b) Isogeometric collocation method with Greville abscissae, (c) Isogeometric Galerkin method

Further, the running time in different steps is recorded for a comparison analysis of the computational efficiency of the three methods. Table 5 lists the running time of three methods in calculating the coordinates of collocation points, assembling the matrix, and the total time of calculating eigenvalues. Figure 9 shows the ratio of the total running time of the isogeometric collocation method with Greville abscissae and the isogeometric Galerkin method when the total running time of the isogeometric collocation method with adaptive moment abscissae is 1. Table 5 and Figure 9 show that the absolute running time of the isogeometric collocation methods in this step is not long, although the calculation time of the isogeometric collocation method with adaptive moment abscissae is longer than that of the isogeometric collocation method with Greville abscissae. The longest running time on the mesh with 2048 elements is only ~3.5 s. The difference in running time between the two collocation methods can be considered the time for optimizing parameter $\overline{\mathbf{r}}$ using the genetic algorithm. The running time of these two collocation methods in the step of assembling matrices is similar, and it is only about one tenth of that of the Galerkin method. Further, with an increase in the number of elements, the total time of the isogeometric collocation method with adaptive moment abscissae decreases compared to that of the isogeometric collocation method with Greville abscissae. In addition, compared with the Galerkin method, the obvious time advantage of the collocation method in the step of the assembling matrix leads to the obvious advantage of the total time of eigenvalue calculation.

Table 5. The running time of three methods when calculating the eigenvalues in different steps

	Number of elements			2	8	32	128	512	2048
Running time (s)	Calculating the coordinates of collocation points	Isogeometric collocation method	Adaptive moment abscissae	0.2	0.21	0.3	0.48	0.97	3.49
			Greville abscissae	0.01	0.01	0.02	0.03	0.17	1.74
		Isogeometric Galerkin method		0	0	0	0	0	0
	Assembling matrices	Isogeometric collocation method	Adaptive moment abscissae	0.02	0.04	0.36	4.37	68.25	1304.69
			Greville abscissae	0.02	0.04	0.47	4.21	77.27	1337.53
		Isogeometric Galerkin method		0.02	0.12	3.29	40.55	785.88	11532.87
	Total running time	Isogeometric collocation method	Adaptive moment abscissae	0.22	0.28	0.66	4.86	69.24	1308.58
			Greville abscissae	0.16	0.13	0.5	4.26	77.49	1339.67
		Isogeometric Galerkin method		0.2	0.32	3.29	40.66	786.55	11534.58

[See PDF for image]

Figure 9

Ratio of the total running time of isogeometric collocation method with Greville abscissae and isogeometric Galerkin method, compared with the proposed isogeometric collocation method with adaptive moment abscissae.

Table 5 and Figure 9 also indicate that the ratio of the running time of the Galerkin method compared to that of the collocation method increases with an increase in the number of elements and reaches the maximum value at the mesh with 512 elements; however, it decreases on the mesh with 1024 elements. This phenomenon was investigated by Jahanbin and Rahman [1], who reported that this could be attributed to the local support of the NURBS functions. When the collocation and Galerkin methods were applied to calculate the elements of matrices A and B in Eq. (16), the integral operations that must traverse the entire physical domain were realized through the local calculation of the NURBS basis functions ${\overline{R}}_i(\mathbf{x})$ and ${\overline{R}}_j({\mathbf{x}}^{\prime })$. This local support made the actual integral domain considerably smaller than that of the entire physical domain. This effect increases gradually with an increase in the number of elements, which weakens the advantage of the collocation method in terms of computational efficiency over the Galerkin method because of the need for fewer integration operations. Consequently, the computational efficiency advantage of the collocation method on a mesh with more elements is smaller than that on a mesh with fewer elements.

Based on all the abovementioned investigations, the accuracy of the proposed isogeometric collocation method with adaptive moment abscissae was verified, and its advantage in terms of efficiency over existing methods was demonstrated.

Reflector of Antenna Experiment

The Cassegrain antenna shown in Figure 10a is widely used in satellite communications, radio telescopes, and other fields. Gravity and wind loads significantly influence the surface precision of the Cassegrain antenna because of its relatively large dimensions. The electrical performance of an antenna is affected when a reflector deviates from its original shape. Considering the spatially dependent random uncertainties of material properties, the modelling and discretization of the random field of the material properties are essential when calculating the deformation and evaluating the reliability of the reflector of the antenna.

[See PDF for image]

Figure 10

Main reflector of the antenna and its IGA model (Control points are marked by blue): (a) Shape of the reflector, (b) Mesh 1 with 8 elements, (c) Mesh 2 with 32 elements, (d) Mesh 3 with 128 elements, (e) Mesh 4 with 512 elements, (f) Mesh 5 with 2048 elements

Considering the random field of the Young’s modulus $H_G(\mathbf{x},\theta )$ defined on the main reflector of the antenna, the covariance function of the random field was

24

$$C_{\mathit{xx}}(\mathit{x},{\mathit{x}}^{\mathbf{\prime }})={\sigma }^{2}exp\left(-\frac{∥\mathit{x}-{\mathit{x}}^{\mathbf{\prime }}∥}{L}\right),\mathit{x},{\mathit{x}}^{\mathbf{\prime }}\in {\mathbb{R}}^3,$$where ${\sigma }^2=0.02$ and $L=5$.

The proposed isogeometric collocation method with an adaptive moment abscissae was applied to six NURBS meshes. The coarsest mesh with eight elements is shown in Figure 10b, whose knot vectors are ${\mathrm{\Xi }}_1=[0,0,0,0\text{.25,0}.25,0\text{.5,0}.5,0\text{.75,0}.75,1,1,1]$ and ${\mathrm{\Xi }}_2=[0,0,0,0.5,1,1,1]$, and their control points and weights are listed in Table 6. Further, through h-refinement, four refined meshes with 32–2048 elements were obtained from the coarsest mesh. The meshes refined through h-refinement performed 1–4 times are shown in Figure 10c–f, respectively.

Table 6. Control points and weights for the coarsest mesh

Control Point	Weight	Control Point	Weight	Control Point	Weight
(4.2459,-4.2459,1.9424)	$\frac{1}{\sqrt{2}}$	(2.2502,− 2.2502,0.4919)	$\frac{1}{\sqrt{2}}$	(− 1,1,0)	$\frac{1}{\sqrt{2}}$
(0,-4.2459,1.9424)	1	(0,− 2.2502,0.4919)	1	(0,1,0)	1
(-4.2459,-4.2459,1.9424)	$\frac{1}{\sqrt{2}}$	(− 2.2502,− 2.2502,0.4919)	$\frac{1}{\sqrt{2}}$	(1,1,0)	$\frac{1}{\sqrt{2}}$
(-4.2459,0,1.9424)	1	(− 2.2502,0,0.4919)	1	(1,0,0)	1
(-4.2459,4.2459,1.9424)	$\frac{1}{\sqrt{2}}$	(− 2.2502,2.2502,0.4919)	$\frac{1}{\sqrt{2}}$	(− 3.4605,3.4605,1.1109)	$\frac{1}{\sqrt{2}}$
(0,4.2459,1.9424)	1	(0,2.2502,0.4919)	1	(0,3.4605,1.1109)	1
(4.2459,4.2459,1.9424)	$\frac{1}{\sqrt{2}}$	(2.2502,2.2502,0.4919)	$\frac{1}{\sqrt{2}}$	(3.4605,3.4605,1.1109)	$\frac{1}{\sqrt{2}}$
(4.2459,0,1.9424)	1	(2.2502,0,0.4919)	1	(3.4605,0,1.1109)	1
(3.4605,-3.4605,1.1109)	$\frac{1}{\sqrt{2}}$	(1,− 1,0)	$\frac{1}{\sqrt{2}}$	(− 1,0,0)	1
(0,-3.4605,1.1109)	1	(0,− 1,0)	1	− 3.4605,0,1.1109)	1
(-3.4605,-3.4605,1.1109)	$\frac{1}{\sqrt{2}}$	(− 1,− 1,0)	$\frac{1}{\sqrt{2}}$

The eigenvalues obtained using the isogeometric collocation method with adaptive moment abscissa are listed in Table 7. The eigenvalues converged at mesh 4 with 512 elements. The final random field discretization is demonstrated on Mesh 4, and the three realizations of the random field are shown in Figure 11. Therefore, the feasibility of the proposed isogeometric collocation method with adaptive moment abscissae for the discretization of a random field in 3D space was verified.

Table 7. The eigenvalues obtained by the isogeometric collocation method with adaptive moment abscissae.

	Mesh	1	2	3	4	5
Eigenvalue	1	0.6294	0.6266	0.6262	0.6261	0.6261
	2	0.1497	0.1464	0.1459	0.1458	0.1458
	3	0.1497	0.1464	0.1459	0.1458	0.1458
	4	0.0541	0.0465	0.0461	0.0461	0.0461
	5	0.0394	0.0458	0.0461	0.0461	0.0461
	6	0.0302	0.0290	0.0286	0.0286	0.0286
	7	0.0190	0.0193	0.0197	0.0197	0.0197
	8	0.0190	0.0193	0.0197	0.0197	0.0197
	9	0.0150	0.0141	0.0138	0.0137	0.0137
	10	0.0150	0.0141	0.0138	0.0137	0.0137

[See PDF for image]

Figure 11

Three realizations of the random field

Discussion

The investigation of the above illustrative examples indicates that the Galerkin method hardly achieves a good balance between computational cost and accuracy of random field discretization when implemented on a mesh with a large number of elements, although with considerably more running time. The need for a tradeoff between accuracy and efficiency makes the Galerkin method less attractive than the collocation method. With fewer integral operations, the collocation method obtained the same results as the Galerkin method, with only approximately one-tenth of the running time. This significant advantage makes the collocation method a strong competitor for calculating eigenvalues in the discretization of random fields in practical applications.

In terms of collocation points, the proposed adaptive moment abscissae yielded more accurate results than the classical Greville abscissae. Using a genetic algorithm, the value of the moment parameter can differ when dealing with different meshes, which makes the positions of the collocation points more suitable for the mesh and improves the accuracy of the calculated eigenvalues. Besides a mesh with a few elements, the running time of the collocation method with the proposed adaptive moment abscissa is almost the same as that of the collocation method with the Greville method, which implies that the running of the genetic algorithm has negligible influence on the efficiency of the collocation method. With little and acceptable time increase and significant improvement in accuracy, the adaptive moment abscissae shows great potential as a better substitute for the Greville abscissae in the collocation method.

Conclusions

In this study, a new isogeometric collocation method based on the adaptive moment abscissae was proposed to efficiently and accurately discretize random fields. The conclusions are summarized as follows:

• (1) With isogeometric basis functions applied to the generation of geometric models and representation of random fields, the loss of geometric information between the CAD and CAE stages was avoided, and the error caused by meshing process was eliminated.

• (2) The collocation method has an obvious advantage in terms of efficiency over the Galerkin method because fewer integral operations are involved in solving the Fredholm integral equation, making it more suitable for practical engineering applications.

• (3) The adaptive moment abscissa was proposed to further improve the accuracy of the collocation method. With an automatically calculated moment parameter added to the Greville abscissa, the collocation points are adaptive to different IGA meshes, thereby leading to optimized eigenparameters for the discretization of random fields.

• (4) Illustrative examples in the 1D/2D/3D space demonstrate the advantage of the proposed adaptive moment abscissae in accuracy over the classical Greville abscissae and the advantage of the isogeometric collocation method over the isogeometric Galerkin method. The feasibility of the proposed isogeometric collocation method with adaptive moment abscissae for the discretization of a random field was verified. Future work will explore its application in stochastic structural analysis and robust design optimization since the computational performance and engineering practicability of the proposed method need to be further demonstrated.

Acknowledgements

Not applicable.

Author Contributions

ZL proposed the original idea and coordinated the study. DP and MY were in charge of the literature review, coding, results analysis, and writing. JC conceptualized the method and provided guidance on the experimental design, analysis of results, and writing. CQ and JT reviewed and revised the manuscript. All the authors have read and approved the final manuscript.

Funding

Supported by National Natural Science Foundation of China (Grant Nos. U22A6001 and 52375273), Major Project of Science and Technology Innovation 2030 (Grant No. 2021ZD0113100), and Zhejiang Provincial Natural Science Foundation of China (Grant No. LZ24E050005).

Data Availability

The data that support the findings of this study will be made available on reasonable request.

Declarations

Competing Interests

The authors declare no competing financial interests.

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