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Research Papers: Materials Technology

# Comparison of Various Surrogate Models to Predict Stress Intensity Factor of a Crack Propagating in Offshore PipingOPEN ACCESS

[+] Author and Article Information
Arvind Keprate

Department of Mechanical and Structural
Engineering and Material Science,
University of Stavanger,
Stavanger 4036, Norway
e-mail: arvind.keprate@uis.no

R. M. Chandima Ratnayake

Department of Mechanical and Structural
Engineering and Material Science,
University of Stavanger,
Stavanger 4036, Norway
e-mail: chandima.ratnayake@uis.no

Shankar Sankararaman

NASA Ames Research Center,
SGT Inc.,
Moffett Field, CA 94035
e-mail: shankar.sankararaman@nasa.gov

1Corresponding author.

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received April 10, 2017; final manuscript received July 13, 2017; published online August 16, 2017. Assoc. Editor: Hagbart S. Alsos.The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States Government purposes.

J. Offshore Mech. Arct. Eng 139(6), 061401 (Aug 16, 2017) (10 pages) Paper No: OMAE-17-1055; doi: 10.1115/1.4037290 History: Received April 10, 2017; Revised July 13, 2017

## Abstract

This paper examines the applicability of the different surrogate-models (SMs) to predict the stress intensity factor (SIF) of a crack propagating in topside piping, as an inexpensive alternative to the finite element methods (FEM). Six different SMs, namely, multilinear regression (MLR), polynomial regression (PR) of order two, three, and four (with interaction), Gaussian process regression (GPR), neural networks (NN), relevance vector regression (RVR), and support vector regression (SVR) have been tested. Seventy data points (consisting of load (L), crack depth (a), half crack length (c) and SIF values obtained by FEM) are used to train the aforementioned SMs, while 30 data points are used for testing. In order to compare the accuracy of the SMs, four metrics, namely, root-mean-square error (RMSE), average absolute error (AAE), maximum absolute error (MAE), and coefficient of determination (R2) are used. A case study illustrating the comparison of the prediction capability of various SMs is presented. python and matlab are used to train and test the SMs. Although PR emerged as the best fit, GPR was selected as the best SM for SIF determination due to its capability of calculating the uncertainty related to the prediction values. The aforementioned uncertainty representation is quite valuable, as it is used to adaptively train the GPR model (GPRM), which further improves its prediction accuracy and makes it an accurate, faster, and alternative method to FEM for predicting SIF.

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## Introduction

###### Background.

During its service life, topside piping on offshore oil and gas (O&G) platforms is continuously subjected to vibrations emanating from various excitation sources such as flow-induced turbulence, flow-induced pulsation, mechanical excitation, and acoustic-induced vibrations. [1]. Due to the aforementioned vibrations, dynamic stresses are induced in the pipework, which if above threshold level, may lead to fatigue crack initiation and propagation in the vibrating piping [2]. If the propagating crack reaches the critical level then it may cause concerns such as hydrocarbon release (HCR) and production down-time [2]. Although all HCRs emanating from the failed pipework are of concern, the main danger comes from gaseous releases, as the gas cloud produced has the potential to quickly spread across the O&G platform. For instance, a gas release from a 6-mm diameter hole in a piping system operating at about 150 bar (i.e., 15 MPa) has the potential to turn into a major HCR (i.e., greater than 300 kg) in approximately 8 min [3]. If ignited, HCR can escalate into a major accident, resulting in significant economic losses and environmental damage, while causing a serious threat to the lives of personnel [4].

In order to prevent HCRs from topside piping and to enhance offshore process safety, it is vital to perform accurate fatigue life assessment [5]. Currently, linear-elastic fracture mechanics (LEFM) approach (coupled with probabilistic analysis) serves as a basis for evaluating fatigue life for planning in-service inspection of fatigue cracks in offshore structural and mechanical items [6]. The basis of the LEFM approach lies in performing a cycle-by-cycle crack propagation analysis, using a suitable crack growth law on an initial crack size, to determine the number of cycles required for an initial crack size to reach the critical crack size [7]. Thus, the remaining fatigue life (RFL) assessment using LEFM approach is mainly dependent upon two factors namely: initial crack size and crack growth law. In regards to the first factor, DNV [7] recommends that the value of initial crack size should be determined by the detection threshold limit of the nondestructive examination method (for instance initial crack size based on detection limit of ultrasonic testing is 1.5 mm).

The second important factor for LEFM-based RFL assessment is a suitable crack growth law which calculates the crack growth increment per cycle ($i.e., da/dn)$. Most commonly used crack growth law in LEFM domain is the Paris equation, according to which $da/dn$ is directly proportional to the stress intensity factor (SIF), at the crack tip of the propagating crack [8,9]. Thus, once $da/dn$ is evaluated for the first cycle, it is added to the initial crack size in order to obtain crack size at first cycle. In this way, the entire process is iterated until initial crack size reaches critical crack size, and the number of cycles required to do so represent the RFL, which may then be used to decide inspection interval frequency for topside piping. Consequently, based on the aforementioned discussion, it is affirmed that the accuracy of the RFL estimate (and in turn of the inspection interval frequency) depends upon accurate evaluation of the SIF.

At present SIF evaluation is possible by several methods such as handbook solutions [1012], analytical methods (e.g., conformal mapping, body force), experimental methods (fatigue crack growth, interferometry, etc.) and numerical methods (finite element methods (FEM), and boundary collocation techniques, etc.). Usually, handbook solutions give accurate SIF results for simple crack geometries [13]. However, as the intricacy of crack geometry and complexity of the loading augment, the handbook solutions for SIF are not the best way to evaluate SIF [13]. Consequently, for such scenarios, FEM is used to predict SIF of a propagating crack with high accuracy [13]. Nevertheless, FEM techniques are afflicted with the following constraints:

1. (1)The accuracy and computing time required to solve a FEM simulation is dependent upon the finite element size (mesh density). This implies that simulations of finite element models constructed with fine mesh size deliver accurate results, nevertheless the simulations generally take longer computing time and vice versa [14].
2. (2)Generally, three types of error are attributed to FEM, namely: user error (which emanates due to inexperience of the analyst), modeling error (which arises due to erroneous representation of the real world phenomenon), and discretization error (originating due to inadequate mesh density which is unable to capture the solution appropriately) [15]. While the former two errors are in the hands of the analyst, the latter is inherent to FEM and must be separately quantified by the analyst for an accurate solution. Thus, quantification of discretization error increases the accuracy of FEM results; however, it further adds to the time required to run a FEM simulation.

Based on the previously mentioned two points, it is deduced that the higher accuracy in the SIF evaluated through FEM is achieved at the expense of the greater time required for simulations to run, which makes FEM computationally expensive and time-consuming. The aforementioned shortcoming of FEM causes hindrance while performing LEFM-based RFL assessment of offshore piping. This is because in real life the number of FEM simulations required to predict the SIF for cycle-by-cycle crack propagation analysis is of the order of $105$ or more. Hence, if for one FEM simulation 30 s are required to evaluate the SIF, then approximately a month will be required to predict SIF for $105$ cycles, thus making RFL assessment using FEM quite laborious and uneconomical.

In order to overcome the aforementioned shortcomings of FEM, in evaluating the SIF for RFL assessment, it seems more efficient and rational to design and run a limited number of FEM simulations (with small element size), that act as the training and testing points for a surrogate model, which in turn is used to predict the SIF value accurately and instantly at other data points (i.e., load (L), crack depth (a), half-crack length (c)), thus making RFL assessment less labor intensive and economical.

###### Surrogate Model.

Surrogate models (SMs) are data-driven models that try to predict the complex input/output (I/O) behavior of an underlying system, by using a limited set of computationally expensive simulations (CES) [16]. The two basic steps in constructing a SM are training and testing. The first corresponds to fitting a model to the intelligently chosen training points, while the second step involves comparing the predictions of the SM to the actual response.2 The SMs must not be mistaken as a simplified version (with low reliability) of the CES; conversely, SMs emulate the behavior of the CES as accurately as possible, coupled with low-computational cost [17]. Another interesting feature of the SMs is the fact that it is not essential to understand the physics of the SM's code, as the analyst is interested in establishing an accurate I/O relationship.2 Up until now, SMs have been employed in a range of fields, including design automation, parametric studies, design space exploration, optimization, and sensitivity analysis [16]. For all the aforementioned applications, SMs are commonly used to replicate the results of a single expensive simulation code that needs to be run time and again in order to generate the desired results [18]. Thus, SMs act as a “curve fit” to the training data (generated by an expensive simulation code) and thereafter may be used to estimate the quantity of interest without running the expensive simulation code. Readers interested in gaining more insight into the complete process of constructing a SM are referred to Ref. [18].

The main idea of using SMs as a replacement to the original simulation code or program is based on the fact that, once built, the SMs will be faster than the main simulation code, while still being usefully accurate [18]. In the aforementioned context, SMs may be used to predict the SIF of a propagating crack. Up until now, SMs have been developed for predicting crack growth in aluminum in the aeronautical and aerospace industries [17,19,20]. Yuvraj et al. [21] used a support vector machine to predict the critical SI F for concrete beams. However, based on the literature review, the authors discovered that, up until now, the commonly used standards for RFL assessment in the offshore industry, i.e., BS 7910 [22], API 579 [23], and DNVGL-RP-C210 [7], rely on either closed-form solutions given in various SIF handbooks or FEM to predict the SIF. Moreover, other than Appendix C of API 579 [23], which uses a third-order polynomial equation (which is a type of SM) along with an influence coefficient to predict SIF, the authors have not come across any other scientific paper or report demonstrating the application of SMs to predict SIF for assessing fatigue degradation of topside piping in the offshore industry.

###### Scopes and Objectives.

Building on the literature review, which highlighted the lack of application of SMs to predict the SIF of a crack propagating in offshore piping, the main goal of this paper is to examine the applicability of the different SMs to predict the SIF of a crack propagating in topside piping, as an inexpensive alternative to the FEM. Six different SMs, namely, MLR, polynomial regression (PR) of order two, three and four (with interaction), Gaussian process regression (GPR), neural networks (NN), relevance vector regression (RVR), and support vector regression (SVR) have been tested. Seventy data points (consisting of L, a, c, and SIF values obtained by FEM) are used to train the aforementioned SMs, while 30 data points are used as the testing points. In order to compare the accuracy of the SMs, four metrics, namely, root-mean-square error (RMSE), average absolute error (AAE), maximum absolute error (MAE), and coefficient of determination ($R2)$ are used.

The remainder of the paper is structured as follows: In Sec. 2, the paper briefly discusses various methods of evaluating SIF. Afterward, in Sec. 3, the paper discusses the theory and mathematical background of different SMs. This is followed by the discussion of an illustrative case study in Sec. 4. Finally, suitable conclusions are provided in Sec. 5.

## Stress Intensity Factor Evaluation Methods

At present large number of methods are available to evaluate SIF at the crack tip. These methods can be broadly classified into three stages, as shown in Fig. 1. Stage 1 methods include the SIF solutions that are included in various handbooks such as Refs. [1012] and in the industry standards such as Refs. [7], [22], and [23]. The handbook solutions are usually in the format of closed form equations, for instance as per BS7910 [22], the mode-I SIF at the crack tip of a semi-elliptic surface crack propagating in a flat plate is given by the following equation [24]:

Display Formula

(1)$KI=σ0Yπa$

In Eq. (1), $σ0$ is the nominal uniform stress field as it appears in the plate without the crack, Y is the geometric function, and a is the crack depth as shown in Fig. 2. Generally, the stage 1 methods give exact solutions for simple geometries; nevertheless, it is also possible to use these solutions to predict fairly accurate SIF values for complex geometries when used in conjunction with stage 2 methods [13].

Stage 2 methods are mainly used when the stage 1 methods are not adequate to evaluate the SIF. As depicted by Fig. 1, the evaluation of SIF by using stage 2 methods generally consume a few man-hours [25]. Some of the stage 2 methods for SIF evaluation are superposition principle, compounding analysis, stress concentrations method, stress distribution technique, Green's function technique, and weight function technique.

For certain applications where a particular value of SIF is frequently required and when the stage 1 and 2 methods are unable to predict the SIF with sufficient certainty, analyst normally resort to stage 3 methods, which includes analytical, experimental, and numerical methods. Generally, if the crack has initiated and is growing in a finite size body, then, the boundary conditions do not allow the usage of closed-form equations [13]. Under such circumstances, numerical methods such as FEM, boundary collocation technique, or the boundary integral method are the popular methods for SIF determination.

Among the aforementioned numerical methods, FEM is most commonly used for SIF determination in the O&G sector. Generally, various FEM-based softwares such as abaqus and ansys are currently utilized to evaluate SIF in the O&G industry [26]. However, as argued in Sec. 1 of the paper that repeated FEM simulations to evaluate SIF for cycle-by-cycle crack propagation analysis is computationally expensive and time-consuming, thus making RFL assessment labor intensive. Hence, in the author's opinion it is possible to make RFL assessment of topside piping less labor intensive by replacing FEM with an inexpensive SM which has been trained and tested using input (L, a, c) and output (SIF) variables. In Sec. 3, the theory and the mathematical background of various SMs are discussed.

## Surrogate Models

###### General.

As discussed in Sec. 1, the main purpose of SMs is to act as a replacement to the computationally expensive and/or time-consuming simulations, without compromising the accuracy of the output. The most commonly used SMs in the engineering domain are conventional response surface (linear and polynomial regression), GPR, NN, RVR, and SVR [27]. The mathematical background and theory of the various SMs used in this paper are discussed briefly in Secs. 3.23.7.

###### Multiple Linear Regression.

Multiple linear regression (MLR) attempts to model the relationship between two or more independent variables and a response variable by fitting a linear equation to observed data [28]. Every value of the independent variable x is associated with a value of the dependent variable y [28]. The model for MLR is written as Display Formula

(2)$ŷ=β0+β1x1+β2x2+⋯+βnxn+ε$

The aim is to find the value of exponents $β0,β1,…,βn$ such that the MLR fitting line best fits the training data and may then be used to predict the values of $ŷ$ (response or dependent), for the future values of x.

###### Polynomial Regression.

In essence, polynomial regression (PR) is very similar to a linear regression (LR), and the only difference being that the linear relationship among the independent and dependent variables in the case of the latter model is replaced by a polynomial of $nth$ degree in the former model [28]. Usually, the PR of degree n is stated as [29] Display Formula

(3)$ŷ=β0+β1x+β2x2+⋯+βnxn+ε$

For lower degrees, the relationship has a specific name (i.e., n = 2 is called quadratic, n = 3 is called cubic, etc.) [28]. Like MLR, the aim is to find the value of exponents $β0,β1,…,βn$, such that the PR curve best fits the training data and may then be used to predict the values of $ŷ$, for the future values of x.

###### Gaussian Process Regression.

A Gaussian process regression (GPR) is a nonparametric, kernel-based probabilistic model, which employs a set of observed inputs and outputs to construct an approximation to the underlying relationship [30]. Suppose the training points consist of a d-dimensional input variable vector stated as $x1,x2,x3,…xm$, and the output random vector $Y(x1),Y(x2),Y(x3),…Y(xm)$, it is also possible to write the training points as $xT$ versus $yT$, where $xT$ is a $m×d$ matrix and $yT$ is a $m×1$ vector. Now, if the analyst wants to predict the output values $(yp$) conforming to the input ($xp$), where $xp$ is a $p×d$ matrix, then, the joint density output values $yp$ are evaluated as [27] Display Formula

(4)$p(yp|xp,xT,yT,Θ)∼N(m⋅S)$

In Eq. (4), $Θ$ denotes the hyper-parameters of the GPR, the value of which is evaluated by the training data. The prediction mean and covariance matrix (m and S, respectively) are given by [27] Display Formula

(5)$m=KPT(KTT+σn2I)−1yTS=KPP−KPT(KTT+σn2I)−1KTP$

In Eq. (5), $KTT$ is the covariance function matrix (size $m×m$) among the input training points ($xT$), and $KPT$ is the covariance function matrix (size $p×m)$ between the input prediction point ($xp)$ and the input training points ($xT)$. These covariance matrices are composed of squared exponential terms, with each element of the matrix being computed as [27] Display Formula

(6)$Kij=K(xi,xj;Θ)=−θ2[∑q=1d(xi,j−xj,q)2lq]$

It must be noted here that all of the above computations require the estimate of the hyper-parameters $Θ$; the multiplicative term ($θ)$, the length scale in all dimensions ( and the noise standard deviation ($σn)$ constituting these hyper-parameters are estimated based on the training data, by maximizing the following log-likelihood function [27] Display Formula

(7)$logp(yT|xT;Θ)=−yTT2(KTT+σn2I)−1yT−12log |KTT+σn2I|+d2log(2π)$

Once the hyper-parameters are estimated, then, the GPR model (GPRM) can be used for predictions, utilizing Eq. (5).

###### Neural Network.

A neural network (NN) is defined as “a computing system made up of a number of simple, highly interconnected processing elements, which process information by their dynamic state response to external inputs” [31]. A NN typically comprises a large number of layers of neurons, as shown in Fig. 3 [32]. As can be seen, there are interconnecting lines between different neurons; these represent the trail of information flow [32]. Furthermore, each interconnecting line has a weight associated with it, which regulates the signal amid two connecting neurons [32].

When a NN is used in fitting problems, the analyst wants the NN to map between a data set of numeric inputs (independent variables) and a set of numeric targets (dependent variables) [33]. Generally, a NN is a good fitting function, and a fairly simple NN can fit any particular data [33]. However, the large amount of trial and error associated with a NN limits its use as a regression tool [29].

###### Support Vector Regression.

The primary feature of SVR is that it permits the analyst to define or calculate a margin (ε), within which he/she accepts errors in the sample data without influencing the prediction of the SM [34]. The aforementioned feature of SVR turns out to be very helpful if the sample is afflicted by a random error due to some constraints, for example, finite mesh size [34]. SVR employs the structural risk minimization (SRM) principle, which is deemed to be superior to the conventional empirical risk minimization (ERM) [35]. Suppose we have a linear function f(x), which is written as [21] Display Formula

(8)$f(x)=w⋅x+b$

In Eq. (8), w and b are the parameters of the function f(x), and x is the normalized test pattern [21]. SRM principle depends on minimalizing the empirical risk which is expressed by the error (ε)-insensitive loss function [21] Display Formula

(9)

In Eq. (9), $Lε$ is the ε-insensitive loss function, $(yi)$ is the target output, f(x) is predicted output, and $xi$ is the training data set [21]. The analyst wants to find the values of parameters w and b such that the minimal empirical risk with respect to ε-insensitive loss function is obtained. The aforementioned problem is equal to the convex optimization problem that minimizes the margin (w) and slack variables ($ξi,ξi*)$ and is written as [21] Display Formula

(10)

subjected to Display Formula

(11)

In Eq. (10), $(1/2)w⋅w$ is the margin, while the parameter C (>0) governs the trade-off between the value of w and the extent up to which aberrations higher than ε are allowed [35]. Figure 4 depicts the aforementioned situation graphically. As can be seen from Fig. 4, the data points that lie within the ±ε boundary are ignored, while the points which lie on or outside the ±ε band define the predictor [36].

###### Relevance Vector Regression.

A relevance vector machine (RVM) provides a regression method in a Bayesian framework [37]. When used for regression, the basis functions are given by kernels, with one kernel associated with each of the data points from the training set [28]. The general form of a RVR is [28] Display Formula

(12)$y(x)=∑n=1Nwnk(x,xn)+b$

where b is a bias parameter. The number of parameters in this case is M = N + 1, and y(x) has the same form as the predictive model for the support vector machine. Suppose we are given a set of N observations of the input vector x, which we denote collectively by a data matrix X. The corresponding target values are given by T. Thus, the likelihood function is given by [28] Display Formula

(13)$p(T|X,w,β)=∏n=1Np(tn|xn,w,β−1)$

Next, we introduce a prior distribution (a zero-mean Gaussian prior) over the parameter vector w. The weight prior takes the form [28] Display Formula

(14)$p(w|α)=∏n=1MN(wi|0,αi−1)$

where $αi$ represents the precision of the corresponding parameter $wi$, and $α$ denotes $(α1,…αM)T$. The basis function associated with these parameters plays no role in the predictions made by the model and so are effectively pruned out, resulting in a sparse model [28]. Furthermore, the posterior distribution for the weights is again Gaussian and takes the form [28] Display Formula

(15)

where the mean and covariance are given by [28] Display Formula

(16)$m=β∑φTT∑=(A+βφTφ)−1$

where φ is the N × M design matrix with the elements $φni=∅i(xn)$, and A = diag ($αi)$. The values of α and β are determined using type-2 maximum likelihood, also known as the evidence approximation, the details of which can be found in Ref. [28].

A case study demonstrating the applicability of the various SMs is performed in Sec. 4.

## Illustrative Case Study

###### General.

The offshore piping material considered for numerical analysis is in accordance with the industry practice and is assumed to be API5L-Grade B. In order to ease the reader's understanding, instead of a semi-elliptical crack in a pipe, we have considered a semi-elliptic surface flaw at the center of the flat plate (as depicted in Fig. 5) and evaluated the SIF by using three different methods. The aforementioned approximation is supported by the fact that most of the large diameter pipelines used in the O&G sector are manufactured from flat plates using the UOE forming process [38]. Thus, the SIF solutions for plates may be utilized to approximate the solution for pipes by introducing an appropriate bulging factor [23]. The schematic of the plate and crack geometry used in the case study is shown in Fig. 5, while the details of the plate geometry along with its material properties are given in Table 1.

###### Stress Intensity Factor Calculation.

In this paper, the SIF calculation is performed using three different methods. The first is analytical, using the formula given in BS7910 [22]; the second uses FEM, and the third employs SMs. These methods are expounded in Secs. 4.2.14.2.3.

###### BS7910.

As per BS7910, the SIF can be simply expressed as a function of crack size and loading conditions, using a closed form solution given in Eq. (1). In Eq. (1), the parameter Y depends upon the geometry of the component and the crack, so it is a complex function of the crack size [22]. The geometric function for the calculation of the SIF is given in Annex M of BS-7910 and is stated as Display Formula

(17)$Y=M*fw*Mm$

where M is the bulging factor and equals 1 for the plate, $fw$ is the finite-width correction factor, and $Mm$ is the factor for membrane loading. $fw$ and $Mm$ are given by Eqs. (18) and (19), respectively, Display Formula

(18)$fw=(sec(πcLe*at) for2cLe≤0.8$
Display Formula
(19)$Mm=[M1+M2(at)2+M3(at)4]*g*fθ/ϕ$

The detailed calculation for evaluating Y is given in Ref. [5]. Once the value of Y has been calculated, the next step involves the calculation of $SIF$ (using Eq. (1)) for 30 testing points shown in Table 2. The resulting values of SIF for different data points obtained using BS7910 solution are shown in Table 3. It must be mentioned here that the aforementioned SIF solution presented in BS-7910 is adopted from the work of Newman and Raju [39].

###### Finite Element Method.

A finite element model is constructed using ansys 17 [40]. The finite element model of the plate with a semi-elliptical surface crack is shown in Fig. 6. As can be seen, two different mesh sizes have been used in the analysis, with the mesh around the crack location (at the crack front and surrounding areas) being more refined than the rest of the plate geometry. The reason for a finer mesh at the crack location is to obtain a more accurate SIF solution and to avoid convergence problems. Singular elements were used to model the crack tip region. The reason being that stresses and strains are singular at the crack tip. Thus, to produce singularity in the stresses and strains, the elements around the crack tip were modeled using singular elements. Other relevant information related to the FEM is given in Table 4. The reason for using hex dominant element was because when compared to tetrahedron elements, the hex dominant meshing uses significantly less number of elements thus making FEM computations faster.

A uniaxial tensile load is applied on the smaller sides of the rectangular plate. The damage under consideration is a central fatigue crack in mode-I opening; its length runs perpendicular to the loading axis. The crack depth is a, while crack length is 2c. The finite element model is run for data (combination of L, a, c) obtained using Latin hypercube sampling (LHS). For the analysis, the range of L varies from 100 MPa to 200 MPa, while the range of a lies between 1 mm and 8 mm. Likewise, the range of c varies from 2 mm to 22 mm. The resulting value of SIF for the aforementioned data set is shown in Table 2.

###### Surrogate Models.

Since, SMs are data-driven models, therefore, before using SMs to predict SIF, it is vital to understand how the various parameters (L, a, c and SIF) of the 100 data points (training and testing) are related to each other. For this reason, a plot is shown in Fig. 7.

From Fig. 7, it can be seen that a and c are linearly related to each other. This is because these variables are dependent upon each other, and the relationship between the two is termed as aspect ratio ( = a/2c). Furthermore, from Fig. 7, it can be inferred that L variable is independent of a and c variables. However, SIF (i.e., response variable) seems to share an almost linear relationship with the input variables, i.e., L, a and c, indicating that the value of the response variable (SIF) increases as the values of either of the input variables is increased.

Once the relationship between input and response variables of our dataset is understood, the next step involves dividing the data set into training and testing points. Seventy data points were considered to be training points, while the rest 30 points are considered to be the testing points which are shown in Table 2. Thereafter, using the flowchart shown in Fig. 8, different SMs are constructed to predict the value of the SIF. The first step while building a SM is to train it using the training data points. For instance, the equation for MLR generated after fitting them to the training data, is given by the below equation, respectively,

Display Formula

(20)$SIF(Prediction)=−445.36+3.85(Load)+31.55(a)+32.65(c)$

Once the SM has been trained, the next step is to test the SM by comparing the values of the SIF obtained from ansys and those predicted by the SM. The values of the SIF predicted by different SMs are shown in Table 3.

###### Result Discussion.

The values of the SIF obtained by three methods (BS7910, ANSYS, and SMs) have been plotted and presented in Fig. 9 and Table 3. Following inferences can be made based on the data presented in Fig. 9 and Table 3:

1. (1)The value of SIF obtained by using the solution given in BS7910 is always higher than that obtained from ansys. The aforementioned results indicate that the solution given in BS7910 overestimates the value of SIF. The reason for the aforementioned overestimation is because higher value of SIF implies lower value of RFL of an operating asset, which in turn implies smaller inspection intervals and enhanced process safety [4]. Thus, by overestimating the SIF value, BS7910 provides a conservative estimate of RFL and inspection intervals which is in accordance with the industry practice.
2. (2)The three methods of SIF determination are compared on various attributes in Table 5. The comparison for the time consumed to determine the value of K is done on the basis of prediction for 30 data points. The value of error for BS7910 (i.e., handbook solution) has been taken from Ref. [25], while the value of error for FEM is based on author's experience.

Based on the results presented in Table 5, the authors affirm that SIF prediction using the SMs seems to be fairly accurate and less time-consuming than the FEM. Furthermore, in the present case study, the authors have considered the prediction of the SIF for 30 data points only, therefore the time savings (i.e., 30 min required for the SIF evaluation using FEM and 2–10 s using the SMs) accrued by using the SMs highlighted in the case study may not appear to be substantial. However, in real-life, the number of FEM simulations required to predict the SIF for cycle-by-cycle CPA is of the order of $105$ or more. Hence, if for one FEM simulation, 30 s are required to predict the SIF, then approximately a month will be required to evaluate SIF for $105$ cycles, thus making RFL assessment using FEM quite laborious and uneconomical. On the contrary, the SMs has the capability to evaluate the SIF for $105$ cycles in 30–60 s, thus saving both time and money for the engineering companies and also reducing the toil of the analyst performing the RFL assessment.

It is also important to compare the accuracy of the various SMs used in the case study. In order to compare the accuracy of the SMs, four metrics, namely, RMSE, average absolute error (AAE), maximum absolute error (MAE), and coefficient-of-determination ($R2)$ are used. Mathematically, these are written as Display Formula

(21)

The fundamental basis behind all four accuracy measuring metrics lies in comparing the values of predictions with the true response, which in our case, is the SIF value obtained by FEM (i.e., ansys). On using Eq. (20), the values of the aforementioned metrics for the various SMs used in the case study are shown in Table 6.

In order to discover the most accurate of the competing SMs, it is vital to comprehend the results of Table 6. Generally, the lower the value of RMSE, AAE, and MAE, the higher is the accuracy of the predicting model. Furthermore, a model with the value of $R2$ closer to 1 depicts high level of prediction accuracy. Based on the aforementioned premise, it is inferred that PR (second, third. and fourth-order with interaction terms) is the most accurate SM out of the six contestants, while GPR is the second most accurate SM. It must be mentioned here that, although PR outperforms GPR on accuracy, however, GPR unlike PR, has the capability of calculating uncertainty related to the predicted values of the SIF, due to which GPR can be adaptively trained to further increase its accuracy. As a result, GPR has been chosen by the authors as the best SM (out of six competing SMs) to predict the SIF of a propagating crack. The detailed procedure for establishing an adaptive GPR model is discussed by authors in Ref. [41]. In order to investigate the influence of the number of training points on the prediction quality of the GPR model, it is trained first by the first 50 data points and thereafter by all 100 data points. It is shown that by increasing the number of the training points the prediction accuracy of the GPRM increased [41].

At the end of the case study, the authors wish to express the following points:

1. (1)In the given case study, the SIF values of a semi-elliptic surface crack in a flat plate (for constant amplitude loading) have been determined using the SMs. However, the proposed SMs have the capability to predict the SIF values for any crack shape (in complex geometries) and variable loading conditions which can be modeled in ansys (FEM). The reason is because the only input of the SM is the output from the FEM simulations. Thus, the usage of the proposed SM is not constrained by the crack shape, geometrical complexity, or the loading conditions.
2. (2)Building on the argument given in point 1, authors state that the main limitation of using the SM for SIF determination is that its prediction accuracy depends upon the accuracy of the FEM used to train the SM. Thus, an analyst must ensure high degree of accuracy while constructing the FEM, which in turn would ensure higher prediction accuracy of the SM.

## Conclusion

The paper proposes the use of SMs as a replacement to computationally expensive FEM to predict the SIF of a crack propagating in offshore piping. The viability of six different SMs, namely, MLR, polynomial regression with order two, three, and four (with interaction), GPR, NN, RVR, and SVR have been tested in the paper. Seventy data points were used to train the aforementioned meta-models, while 30 data points were used as testing points.

In order to compare the accuracy of the aforementioned SMs, four metrics, namely, RMSE, AAE, MAE, and coefficient-of-determination ($R2)$ were used. The value of $R2$ for MLR, PR-2, PR-3, PR-4, GPR, NN, RVR, and SVR were found to be 0.979, 0.996, 0.998, 0.992, 0.991, 0.982, 0.978, and 0.981, respectively. Generally, a model with the value of $R2$ closer to 1 depicts high level of prediction accuracy. Based on the aforementioned premise, it is inferred that PR, is the most accurate SM out of the six contestants, while GPR is the second most accurate SM. It must be mentioned here that, although PR outperforms GPR on accuracy, unlike PR, GPR has the capability of calculating uncertainty related to the predicted values of the SIF, due to which, GPR can be adaptively trained to further increase its accuracy. As a result, GPR has been chosen as the best SM (out of six competing SMs) to predict the SIF of a propagating crack, the details of which are given in Ref. [41].

The authors affirm that SIF prediction using the SMs seems to be fairly accurate and less time-consuming than the FEM which is currently used in the O&G industry. Thus, based on the aforementioned discussions and the results presented in the paper, the authors propose that, by replacing computationally expensive FEM with the SM for SIF prediction, operators may accrue substantial savings in terms of both time and money without compromising on the accuracy of RFL estimate of topside piping undergoing fatigue degradation.

## Acknowledgements

This work has been carried out as part of a Ph.D. research project, performed at the University of Stavanger. The research is fully funded by the Norwegian Ministry of Education. The research reported in this paper was partly supported by NASA under Award No. NNX12AK33A. The support is gratefully acknowledged.

## Nomenclature

• $a$ =

crack depth

• b =

parameter of SVR and RVR

• $da/dn$ =

crack growth increment per cycle

• $fw$ =

finite-width correction factor

• $fθ$ =

function of crack depth, half crack length, and crack orientation

• g =

function of crack depth and orientation

• $KI$ =

stress intensity factor of mode-I crack

• $KPT$ =

covariance function matrix of size $p×m$

• $KTT$ =

covariance function matrix of size $m×m$

• l =

length scale of covariance function of the Gaussian process

• $Lε$ =

insensitive loss function

• m =

prediction mean

• M =

bulging factor

• $Mm$ =

• S =

covariance matrix

• s2 =

prediction variance

• $t$ =

plate thickness

• w =

margin of SVR

• $xp$ =

$p×d$ matrix of input prediction points

• $xT$ =

$m×d$ matrix of input training points

• $x1,…,xn$ =

independent variable in SMs

• $y$ =

response variable in SMs

• Y =

geometric function

• $yp$ =

$p×1$ vector of output prediction points

• $yT$ =

$m×1$ vector of output training points

• 2c =

crack length

• α =

precision of parameter in RVR

• ε =

error insensitive loss function

• $θ$ =

multiplicative term of covariance function of the Gaussian process

• $Θ$ =

hyper-parameters of the Gaussian process

• $ξ$ =

slack variable of SVR

• $σn$ =

noise standard deviation of covariance function of the Gaussian process

• $σ0$ =

nominal uniform stress

• $ϕ$ =

elliptic integral of the second kind

## References

Keprate, A. , and Ratnayake, R. M. C. , 2017, “ Enhancing Offshore Process Safety by Selecting Fatigue Critical Piping Locations for Inspection Using Fuzzy-AHP Based Approach,” Process Saf. Environ. Prot., 102, pp. 71–84.
EI, 2007, “ Guidelines for the Avoidance of Vibration Induced Fatigue Failure in Process Pipework,” The Energy Institute, London.
EI, 2013, “ Guidelines for the Design, Installation and Management of Small Bore Tubing Assemblies,” The Energy Institute, London.
Keprate, A. , and Ratnayake, R. M. C. , 2016, “ Handling Uncertainty in the Remnant Fatigue Life Assessment of Offshore Process Pipework,” ASME Paper No. IMECE2016-65504.
Keprate, A. , Ratnayake, R. M. C. , and Sankararaman, S. , 2017, “ Minimizing Hydrocarbon Release From Offshore Piping by Performing Probabilistic Fatigue Life Assessment,” Process Saf. Environ., 106, pp. 34–51.
Naess, A. A. , 2009, Fatigue Handbook: Offshore Steel Structures, Tapir Publisher, Trondheim, Norway, Chap. 3.
DNV, 2015, “ Probabilistic Methods for Planning of Inspection for Fatigue Cracks in Offshore Structures,” Det Norske Veritas AS, Høvik, Norway, Standard No. DNV-RP-C210.
Antaki, G. A. , 2003, Piping and Pipeline Engineering: Design, Construction, Maintenance, Integrity, and Repair, CRC Press, Boca Raton, FL, Chap. 7.
Lassen, T. , and Recho, N. , 2006, Fatigue Life Analyses of Welded Structures, ISTE, London, Chap. 6.
Tada, H. P. , Paris, P. C. , and Irwin, G. R. , 1973, The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, PA.
Sih, G. C. , 1973, Handbook of Stress Intensity Factors: Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, PA.
Rooke, D. P. , and Cartwright, D. J. , 1976, Compendium of Stress Intensity Factors, HMSO, London.
Miedlar, P. C. , Berens, A. P. , Gunderson, A. , and Gallagher, J. P. , 2002, Handbook for Damage Tolerant Design, AFGROW, U.S. Air Force, Dayton, OH, pp. 11.2.1–11.2.5.
More, S. T. , and Bindu, R. S. , 2015, “ Effect of Mesh Size on Finite Element Analysis of Plate Structure,” Int. J. Eng. Sci. Innovative Technol., 4(3), pp. 181–185.
Chandresh, S. , 2002, “ Mesh Discretization Error and Criteria for Accuracy of Finite Element Solutions,” ANSYS Users Conference, Pittsburgh, PA, Apr. 22–24, pp. 45–56.
Deschrijver, D. , and Dhaene, T. , 2012, “ Surrogate Modelling Lab,” University of Ghent, Ghent, Belgium, accessed Mar. 3, 2017,
Hombal, V. K. , and Mahadevan, S. , 2013, “ Surrogate Modelling of 3D Crack Growth,” Int. J. Fatigue, 47, pp. 90–99.
Forrester, A. I. J. , Sobester, A. , and Keane, A. J. , 2008, Engineering Design Via Surrogate Modelling, Wiley, Chichester, UK, Chap. 1.
Sankararaman, S. , Ling, Y. , Shantz, C. , and Mahadevan, S. , 2011, “ Uncertainty Quantification and Model Validation of Fatigue Crack Growth Prediction,” Eng. Fract. Mech., 78(7), pp. 1487–1504.
Leser, P. E. , Hochhalter, J. D. , Warner, J. E. , Newman, J. A. , Leser, W. P. , Wawrzynek, P. A. , and Yuan, F. G. , 2016, “ Probabilistic Fatigue Damage Prognosis Using Surrogate Models Trained Via Three-Dimensional Finite Element Analysis,” Struct. Health Monit., 16(3), pp. 291–308.
Yuvraj, P. , Murthy, A. R. , Iyer, N. R. , Samui, P. , and Sekar, S. K. , 2014, “ Prediction of Critical Stress Intensity Factor for High Strength and Ultra High Strength Concrete Beams Using Support Vector Regression,” J. Struct. Eng., 40(3), pp. 224–233.
BS, 2013, “ Guide to Methods for Assessing Acceptability of Flaws in Metallic Structures,” British Standards Institute, London, Standard No. BS 7910.
API, 2007, “ Recommended Practice for Fitness-for-Service,” API Publishing Services, Washington, DC, API Recommended Practice 579.
Irwin, G. R. , 1957, “ Analysis of Stresses and Strains Near the End of a Crack Traversing in a Plate,” ASME J. Appl. Mech., 24(3), pp. 361–364.
Rooke, D. P. , Baratta, F. I. , and Cartwright, D. J. , 1981, “ Simple Methods of Determining Stress Intensity Factors,” Eng. Fract. Mech., 14(2), pp. 397–426.
Ali, Z. , Meysam, K. E. S. , Iman, A. , Aydin, B. , and Yashar, B. , 2014, “ Finite Element Method Analysis of Stress Intensity Factor in Different Edge Crack Positions and Predicting Their Correlation Using Neural Network Method,” Res. J. Recent Sci., 3(2), pp. 69–73.
Sankararaman, S. , 2012, “ Uncertainty Quantification and Integration in Engineering Systems,” Ph.D. dissertation, Vanderbilt University, Nashville, TN.
Bishop, C. M. , 2006, Pattern Recognition and Machine Learning, Springer, New York, Chap. 12.
Jin, R. , Chen, W. , and Simpson, T. W. , 2001, “ Comparative Studies of Metamodelling Techniques Under Multiple Modelling Criteria,” Struct. Multidiscip., 23(1), pp. 1–13.
McFarland, J. M. , 2008, “ Uncertainty Analysis for Computer Simulations Through Validation and Calibration,” Ph.D. dissertation, Vanderbilt University, Nashville, TN.
Maureen, C. , 1987, “ Neural Network Primer—Part I,” AI Expert, 2(12), pp. 46–52.
Siffman, D. , 2012, The Nature of Code, Self Published, Mountain View, CA, Chap. 1.
MathWorks, 1994, “ Deep Learning,” The MathWorks, Inc., Natick, MA, accessed Mar. 14, 2017,
Forrester, A. I. J. , and Keane, A. J. , 2009, “ Recent Advances in Surrogate Based Optimization,” Prog. Aerosp. Sci., 45(1–3), pp. 50–79.
Gunn, S. R. , 1998, “ Support Vector Machines for Classification and Regression,” University of Southampton, Southampton, UK, Technical Report No. ISIS-1-98.
Smola, A. J. , and Scholkopf, B. , 2004, “ A Tutorial on Support Vector Regression,” Stat. Comput., 14(3), pp. 199–222.
Tipping, M. E. , 2001, “ Sparse Bayesian Learning and the Relevance Vector Machine,” J. Mach. Learn. Res., 1, pp. 211–244.
Ren, Q. , Zou, T. , Li, D. , Tang, D. , and Peng, Y. , 2015, “ Numerical Study on the X80 UOE Pipe Forming Process,” J. Mater. Process. Technol., 215, pp. 264–277.
Newman, J. C. , and Raju, I. S. , 1979, “ Stress Intensity Factors for a Wide Range of Semi-Elliptical Surface Cracks in Finite Thickness Plates,” Eng. Fract. Mech., 11(4), pp. 817–829.
ANSYS, 2017, “ Product Data Sheet: ANSYS Student Data Sheet,” ANSYS, Inc., Canonsburg, PA, accessed Mar. 14, 2017,
Keprate, A. , Ratnayake, R. M. C. , and Sankararaman, S. , 2017, “ Adaptive Gaussian Process Regression as an Alternative to FEM for Prediction of Stress Intensity Factor to Assess Fatigue Degradation in Offshore Piping,” Int. J. Pressure Vessels Piping, 153, pp. 45–58.
View article in PDF format.

## References

Keprate, A. , and Ratnayake, R. M. C. , 2017, “ Enhancing Offshore Process Safety by Selecting Fatigue Critical Piping Locations for Inspection Using Fuzzy-AHP Based Approach,” Process Saf. Environ. Prot., 102, pp. 71–84.
EI, 2007, “ Guidelines for the Avoidance of Vibration Induced Fatigue Failure in Process Pipework,” The Energy Institute, London.
EI, 2013, “ Guidelines for the Design, Installation and Management of Small Bore Tubing Assemblies,” The Energy Institute, London.
Keprate, A. , and Ratnayake, R. M. C. , 2016, “ Handling Uncertainty in the Remnant Fatigue Life Assessment of Offshore Process Pipework,” ASME Paper No. IMECE2016-65504.
Keprate, A. , Ratnayake, R. M. C. , and Sankararaman, S. , 2017, “ Minimizing Hydrocarbon Release From Offshore Piping by Performing Probabilistic Fatigue Life Assessment,” Process Saf. Environ., 106, pp. 34–51.
Naess, A. A. , 2009, Fatigue Handbook: Offshore Steel Structures, Tapir Publisher, Trondheim, Norway, Chap. 3.
DNV, 2015, “ Probabilistic Methods for Planning of Inspection for Fatigue Cracks in Offshore Structures,” Det Norske Veritas AS, Høvik, Norway, Standard No. DNV-RP-C210.
Antaki, G. A. , 2003, Piping and Pipeline Engineering: Design, Construction, Maintenance, Integrity, and Repair, CRC Press, Boca Raton, FL, Chap. 7.
Lassen, T. , and Recho, N. , 2006, Fatigue Life Analyses of Welded Structures, ISTE, London, Chap. 6.
Tada, H. P. , Paris, P. C. , and Irwin, G. R. , 1973, The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, PA.
Sih, G. C. , 1973, Handbook of Stress Intensity Factors: Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, PA.
Rooke, D. P. , and Cartwright, D. J. , 1976, Compendium of Stress Intensity Factors, HMSO, London.
Miedlar, P. C. , Berens, A. P. , Gunderson, A. , and Gallagher, J. P. , 2002, Handbook for Damage Tolerant Design, AFGROW, U.S. Air Force, Dayton, OH, pp. 11.2.1–11.2.5.
More, S. T. , and Bindu, R. S. , 2015, “ Effect of Mesh Size on Finite Element Analysis of Plate Structure,” Int. J. Eng. Sci. Innovative Technol., 4(3), pp. 181–185.
Chandresh, S. , 2002, “ Mesh Discretization Error and Criteria for Accuracy of Finite Element Solutions,” ANSYS Users Conference, Pittsburgh, PA, Apr. 22–24, pp. 45–56.
Deschrijver, D. , and Dhaene, T. , 2012, “ Surrogate Modelling Lab,” University of Ghent, Ghent, Belgium, accessed Mar. 3, 2017,
Hombal, V. K. , and Mahadevan, S. , 2013, “ Surrogate Modelling of 3D Crack Growth,” Int. J. Fatigue, 47, pp. 90–99.
Forrester, A. I. J. , Sobester, A. , and Keane, A. J. , 2008, Engineering Design Via Surrogate Modelling, Wiley, Chichester, UK, Chap. 1.
Sankararaman, S. , Ling, Y. , Shantz, C. , and Mahadevan, S. , 2011, “ Uncertainty Quantification and Model Validation of Fatigue Crack Growth Prediction,” Eng. Fract. Mech., 78(7), pp. 1487–1504.
Leser, P. E. , Hochhalter, J. D. , Warner, J. E. , Newman, J. A. , Leser, W. P. , Wawrzynek, P. A. , and Yuan, F. G. , 2016, “ Probabilistic Fatigue Damage Prognosis Using Surrogate Models Trained Via Three-Dimensional Finite Element Analysis,” Struct. Health Monit., 16(3), pp. 291–308.
Yuvraj, P. , Murthy, A. R. , Iyer, N. R. , Samui, P. , and Sekar, S. K. , 2014, “ Prediction of Critical Stress Intensity Factor for High Strength and Ultra High Strength Concrete Beams Using Support Vector Regression,” J. Struct. Eng., 40(3), pp. 224–233.
BS, 2013, “ Guide to Methods for Assessing Acceptability of Flaws in Metallic Structures,” British Standards Institute, London, Standard No. BS 7910.
API, 2007, “ Recommended Practice for Fitness-for-Service,” API Publishing Services, Washington, DC, API Recommended Practice 579.
Irwin, G. R. , 1957, “ Analysis of Stresses and Strains Near the End of a Crack Traversing in a Plate,” ASME J. Appl. Mech., 24(3), pp. 361–364.
Rooke, D. P. , Baratta, F. I. , and Cartwright, D. J. , 1981, “ Simple Methods of Determining Stress Intensity Factors,” Eng. Fract. Mech., 14(2), pp. 397–426.
Ali, Z. , Meysam, K. E. S. , Iman, A. , Aydin, B. , and Yashar, B. , 2014, “ Finite Element Method Analysis of Stress Intensity Factor in Different Edge Crack Positions and Predicting Their Correlation Using Neural Network Method,” Res. J. Recent Sci., 3(2), pp. 69–73.
Sankararaman, S. , 2012, “ Uncertainty Quantification and Integration in Engineering Systems,” Ph.D. dissertation, Vanderbilt University, Nashville, TN.
Bishop, C. M. , 2006, Pattern Recognition and Machine Learning, Springer, New York, Chap. 12.
Jin, R. , Chen, W. , and Simpson, T. W. , 2001, “ Comparative Studies of Metamodelling Techniques Under Multiple Modelling Criteria,” Struct. Multidiscip., 23(1), pp. 1–13.
McFarland, J. M. , 2008, “ Uncertainty Analysis for Computer Simulations Through Validation and Calibration,” Ph.D. dissertation, Vanderbilt University, Nashville, TN.
Maureen, C. , 1987, “ Neural Network Primer—Part I,” AI Expert, 2(12), pp. 46–52.
Siffman, D. , 2012, The Nature of Code, Self Published, Mountain View, CA, Chap. 1.
MathWorks, 1994, “ Deep Learning,” The MathWorks, Inc., Natick, MA, accessed Mar. 14, 2017,
Forrester, A. I. J. , and Keane, A. J. , 2009, “ Recent Advances in Surrogate Based Optimization,” Prog. Aerosp. Sci., 45(1–3), pp. 50–79.
Gunn, S. R. , 1998, “ Support Vector Machines for Classification and Regression,” University of Southampton, Southampton, UK, Technical Report No. ISIS-1-98.
Smola, A. J. , and Scholkopf, B. , 2004, “ A Tutorial on Support Vector Regression,” Stat. Comput., 14(3), pp. 199–222.
Tipping, M. E. , 2001, “ Sparse Bayesian Learning and the Relevance Vector Machine,” J. Mach. Learn. Res., 1, pp. 211–244.
Ren, Q. , Zou, T. , Li, D. , Tang, D. , and Peng, Y. , 2015, “ Numerical Study on the X80 UOE Pipe Forming Process,” J. Mater. Process. Technol., 215, pp. 264–277.
Newman, J. C. , and Raju, I. S. , 1979, “ Stress Intensity Factors for a Wide Range of Semi-Elliptical Surface Cracks in Finite Thickness Plates,” Eng. Fract. Mech., 11(4), pp. 817–829.
ANSYS, 2017, “ Product Data Sheet: ANSYS Student Data Sheet,” ANSYS, Inc., Canonsburg, PA, accessed Mar. 14, 2017,
Keprate, A. , Ratnayake, R. M. C. , and Sankararaman, S. , 2017, “ Adaptive Gaussian Process Regression as an Alternative to FEM for Prediction of Stress Intensity Factor to Assess Fatigue Degradation in Offshore Piping,” Int. J. Pressure Vessels Piping, 153, pp. 45–58.

## Figures

Fig. 1

Methods for determining SIF (Adapted from Ref. [25])

Fig. 2

Schematic of crack geometry on offshore piping

Fig. 3

Schematic of a neural network. Adapted from Ref. [33].

Fig. 4

Schematic showing loss function and slack variable in SVR. Adapted from Ref. [36].

Fig. 5

Schematic of plate and crack geometry used in the case study

Fig. 6

FEM model of plate and crack geometry used in the case study

Fig. 7

Plots indicating relation between load, a, c, and SIF

Fig. 8

Flowchart to build SM for SIF prediction

Fig. 9

Comparison of SIF values obtained from ANSYS, BS7910, and different SMs

## Tables

Table 1 Material and geometry properties of API5L-Grade B
Table 2 Testing data points used in the case study
Table 3 SIF value obtained by different methods for 30 testing points
Table 4 Details of FEM
Table 5 Comparison of SIF determination methods
Table 6 Comparison of different meta-models

## Discussions

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