Comparing Different Metamodelling Approaches to ...

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Jun 25, 2017 - ABSTRACT. This paper examines the applicability of the different meta-models (MMs) to predict the Stress Intensity Factor (SIF).
Proceedings of the ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering OMAE2017 June 25-30, 2017, Trondheim, Norway

OMAE2017-62333

COMPARING DIFFERENT METAMODELLING APPROACHES TO PREDICT STRESS INTENSITY FACTOR OF A SEMI-ELLIPTIC CRACK Arvind Keprate Department of Mechanical and Structural Engineering and Material Science, University of Stavanger, Norway

R.M. Chandima Ratnayake Department of Mechanical and Structural Engineering and Material Science, University of Stavanger, Norway

Shankar Sankararaman SGT Inc., NASA Ames Research Center Mailstop 269-3, Bldg. T35B, Room 102, PO Box 1 Moffett Field, CA 94035, USA ABSTRACT This paper examines the applicability of the different meta-models (MMs) to predict the Stress Intensity Factor (SIF) of a semi-elliptic crack propagating in topside piping, as an inexpensive alternative to the Finite Element Methods (FEM). Five different MMs, namely, multi-linear regression (MLR), second order polynomial regression (PR-2) (with interaction), Gaussian process regression (GPR), neural networks (NN) and support vector regression (SVR) have been tested. Seventy data points (SIF values obtained by FEM) are used to train the aforementioned MMs, while thirty data points are used as the testing points. In order to compare the accuracy of the MMs, four metrics, namely, Root Mean Square Error (RMSE), Average Absolute Error (AAE), Maximum Absolute Error (AAE), and Coefficient of Determination (R! ) are used. Although PR-2 emerged as the best fit, GPR was selected as the best MM 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, which further improves its prediction accuracy.

predictions of the MM to the actual response [2]. Readers interested in gaining more insight into the complete process of constructing a MM are referred to [3]. However, for practical purposes, the Surrogate Modelling (SUMO) Matlab toolbox, available in [1], may be used to generate the desired MM. A schematic illustrating the construction of MMs using SUMO is shown in Figure 1.

Figure 1. Schematic showing creation of MMs using SUMO [1] The MMs must not be mistaken as a simplified version (with low reliability) of the CES; conversely, MMs emulate the behavior of the CES as accurately as possible, coupled with low computational cost [4]. Another interesting feature of the MMs is the fact that it is not essential to understand the physics of the MM’s code, as the analyst is interested in establishing an accurate I/O relationship [2]. Up until now, MMs have been employed in a range of fields, including design automation, parametric studies, design space exploration, optimization and sensitivity analysis [1]. For all the aforementioned applications, MMs are commonly used to replicate the results of a single expensive simulation code that needs to be run time and again

1.

INTRODUCTION Meta-models (MMs) (also known as surrogate models or response surface models) 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) [1]. The two basic steps in constructing a MM are training and testing. The first corresponds to fitting a model to the intelligently chosen training points, while the second step involves comparing the

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The stress intensity factor (SIF) defines the state of the stress field at the crack tip [8]. The term SIF was coined by George R Irwin in 1957, to define the stress field at the crack front [11]. The fundamental basis of SIF were the Westergaard’s solution (given by Equation 1.) which used the Airy stress function to describe the local stress field at the crack front [12]. The Westergaard’s solution are stated as:

in order to generate the desired results [3]. Thus, MMs 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. The main idea of using MMs as a replacement to the original simulation code or program is based on the fact that, once built, the MMs will be faster than the main simulation code, while still being usefully accurate [3]. In the aforementioned context, MMs may be used to predict the stress intensity factor (SIF) of a propagating crack. Up until now, MMs have been developed for predicting crack growth in aluminum in the aeronautical and aerospace industries [4, 5, and 6]. Yuvraj et al. [7] used a support vector machine (SVM) to predict the critical SIF for concrete beams. However, based on the literature review, authors have not come across a scientific paper or report demonstrating the application of MMs to predict SIF in the offshore industry. The commonly used standards for fatigue life assessment in the offshore industry, i.e. BS-7910 [8] and DNVGL-RP-C210 [9], rely on either closed form solutions or Finite Element Methods (FEM) to predict the SIF in metallic structures. Thus, the main objective of this manuscript is to assess the applicability of various MMs to predict the SIF of a semi-elliptic crack in an offshore mechanical component.

!!"

𝜎! =

!!"

!!

𝑐𝑜𝑠

!

𝑐𝑜𝑠

!

! !

!

!!

!

!

!

!!

!

!

1 − 𝑠𝑖𝑛 𝑠𝑖𝑛 1 + 𝑠𝑖𝑛 𝑠𝑖𝑛

(1) 𝜏!" =

!! !!"

!

!

!!

!

!

!

𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠

where r and 𝜃 are cylindrical coordinates with origin at the crack tip. 𝐾! is referred as the SIF and is given by the equation: 𝐾! = 𝜎! 𝑌 𝜋𝑎

(2)

In the above equation, 𝜎! 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.1. Weld Toe - Hot Spot Base Metal

t

As an alternative to computationally expensive FEM (i.e. ANSYS software), this manuscript uses five different MMs, namely multi-linear regression (MLR), second order polynomial regression (PR-2) (with interaction), Gaussian Process Regression (GPR), neural networks (NN) and support vector regression (SVR) to predict the SIF of a crack. Firstly, the aforementioned MMs are trained and then, during the testing process, they are used to predict the SIF. In order to compare the accuracy of the MMs, four metrics, namely, Root Mean Square Error (RMSE), Average Absolute Error (AAE), Maximum Absolute Error (MAE), and Coefficient of Determination (R! ) are used. Although PR-2 emerged as the best fit, GPR was selected as the best MM for SIF determination, due to its capability of calculating the uncertainty related to the prediction values. The SIF predicting capability of GPR, along with the existing methods of estimating SIF in the offshore industry, is presented in [10].

2c (crack length) a (crack depth)

Weld Root

Figure 1. Schematic of crack geometry on offshore piping A crack in a solid body is usually stressed in three different modes as depicted in Figure 2. Mode I is the opening mode, mode II the sliding mode and mode III is the tearing mode [13]. Since, in practical situations for a piping component undergoing fatigue degradation, mode–I fracture is a main concern [14]; therefore, in this manuscript SIF for a crack growing under mode-I fracture is considered.

The remainder of the paper is structured as follows: In Section 2, the manuscript briefly discusses the concept of SIF. Afterwards, in Section 3, the manuscript briefly discusses the theory and mathematical background of different MMs. Thereafter, in Section 4, an illustrative case study is presented, wherein the five MMs are compared based on their accuracy to predict SIF. Finally, a suitable conclusion and recommendations are provided in Section 4. 2.

!!

𝜎! =

Figure 2. Three modes of fracture [13] At present large number of methods such as handbook solutions, weight function technique, superposition principle, experimental methods and FEM are available to evaluate SIF at the crack tip [15]. Nevertheless, due to enhanced computational abilities and due to availability of commercial softwares (such as ANSYS and ABAQUS), FEM is currently the most

STRESS INTENSITY FACTOR

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Since the PR of degree 2 has been used in this manuscript, so a more general form of 2!" order polynomial, which considers the interaction of terms and errors (ε) in the prediction, is expressed as [17]:

commonly used method for SIF determination in the Oil and Gas (O&G) industry. Although, FEM provides accurate SIF results for intricate crack geometries and complex loading conditions, however it is computationally expensive and time consuming. Especially, when repeated FEM simulations are required to evaluate SIF (for cycle-by-cycle crack growth analysis) during fracture mechanics based RFL assessment of topside piping, the aforementioned method (i.e. FEM) becomes prohibitively expensive and time-consuming. Hence, in the author’s opinion, in order to make RFL assessment of topside piping less labor intensive, FEM need to be replaced with an inexpensive MM which has been trained and tested using input (loading, crack depth and half crack length) and output (SIF) variables. 3.

𝑦 = 𝛽! +

𝑥! +

! !!! 𝛽!!

𝑥!! +

!

! 𝛽!" 𝑥! 𝑥!



(5)

Like MLR, the aim is to find the value of exponents 𝛽! , 𝛽! , . . , 𝛽! , such that the PR curve best fits the training data and may then be used to predict the values of 𝑦, for the future values of x. 3.3

Gaussian Process Regression A Gaussian Process (GP) is a generalization of the Gaussian probability distribution [19]. Whereas a probability distribution describes random variables, which are scalars or vectors (for multivariate distributions), a stochastic process governs the properties of functions [19]. Gaussian process regression (GPR) is a non-parametric, kernel-based probabilistic model, which employs a set of observed inputs and outputs to construct an approximation to the underlying relationship [20].

METAMODELS

As discussed in Section 1, the main purpose of MMs 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 MMs in the engineering domain are conventional response surface (linear and polynomial regression), SVR, NN, and GPR [16]. The mathematical background and theory of the various MMs used in this manuscript are discussed briefly in the following subsections.

Suppose the training points consist of a d-dimensional input variable vector (the input variables being the crack size and loading conditions, in our case), stated as 𝑥! , 𝑥! , 𝑥! , … . 𝑥! , and the output random vector 𝑌(𝑥! ), 𝑌(𝑥! ), 𝑌(𝑥! ), … . 𝑌(𝑥! ) (the SIF, in this case). It is also possible to write the training points as 𝑥 ! vs. 𝑦! , where 𝑥 ! is a 𝑚 𝑋 𝑑 matrix and 𝑦! is a 𝑚 𝑋 1 vector. Now, if the analyst wants to predict the output values (𝑦! ) conforming to the input (𝑥! ), where 𝑥! is a 𝑝 𝑋 𝑑 matrix, then, the joint density output values 𝑦! are evaluated as [16]:

3.1

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 [17]. Every value of the independent variable x is associated with a value of the dependent variable y [17]. The model for MLR is written as: 𝑦 = 𝛽! + 𝛽! 𝑥! + 𝛽! 𝑥! + ⋯ + 𝛽! 𝑥! + ε

! !!! 𝛽!

𝑝 𝑦! 𝑥! , 𝑥 ! , 𝑦! , 𝛩 ̴ 𝑁(𝑚. 𝑆)

(3)

(6)

The aim is to find the value of exponents 𝛽! , 𝛽! , . . , 𝛽! such that the MLR fitting line best fits the training data and may then be used to predict the values of 𝑦 (response or dependent), for the future values of x.

In Equation (6), 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 [16]:

3.2

𝑚 = 𝐾!" 𝐾!! + 𝜎!! 𝐼 !! 𝑦! 𝑆 = 𝐾!! − 𝐾!" 𝐾!! + 𝜎!! 𝐼 (7)

Polynomial Regression In essence, Polynomial Regression (PR) is very similar to a linear regression (LR), the only difference being that the linear relationship amongst the independent and dependent variables in the case of the latter model is replaced by a polynomial of 𝑛!! degree in the former model [17]. Until now, a plethora of researchers have utilized PR to design complex engineering systems [17]. Usually, the PR of degree n is stated as [18]: y = β! + β! x + β! x ! + ⋯ + β! x ! + ε

!!

𝐾!"

In Equation (7), 𝐾!! is the covariance function matrix (size 𝑚 𝑋 𝑚) amongst the input training points (𝑥 ! ), and 𝐾!" is the covariance function matrix (size 𝑝 𝑋 𝑚) between the input prediction point (𝑥! ) and the input training points (𝑥 ! ). These covariance matrices are composed of squared exponential terms, with each element of the matrix being computed as [16]:

(4)

!

𝐾!" = 𝐾 𝑥! , 𝑥! ; 𝛩 = − [

For lower degrees, the relationship has a specific name (i.e. n= 2 is called quadratic, n=3 is called cubic, etc.) [17].

!

3

(!!,! !!!,! ) ! !!! ! !

!

]

(8)

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The fundamental basis of support vector regression (SVR) stems from the theory of support vector machines (SVM), developed at AT&T Bell Laboratories in the 1990s [24]. 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 MM [25]. 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 [25]. SVR employs the structural risk minimization (SRM) principle, which is deemed to be superior to the conventional Empirical Risk Minimization (ERM) [26]. Suppose we have a linear function f(x), which is written as [7]:

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 (𝑙! , 𝑞 = 1 𝑡𝑜 𝑑), and the noise standard deviation (𝜎! ) constituting these hyper-parameters are estimated based on the training data, by maximizing the following log-likelihood function [16]: 𝑙𝑜𝑔𝑝 𝑦! │𝑥 ! ; 𝛩 = − !

!!! !

𝐾!! + 𝜎!! 𝐼

!!

!

𝑦! − log 𝐾!! + !

𝜎!! 𝐼 + log (2𝜋) !

(9)

Once the hyper-parameters are estimated, then the GPR model (GPRM) can be used for predictions, utilizing Equation (7). A vital issue while constructing the GPRM is how an analyst selects the training points. Usually, the training points emanate either from the field experiments or from using a computer algorithm. In this manuscript, the authors have considered the latter case for generation of training points, as a result of which the data is noise free, consequently leading to elimination of 𝜎! from Equations (7) and (9).

𝑓 𝑥 = 𝑤. 𝑥 + 𝑏 (10) In Equation (10), w and b are the parameters of the function f(x), and x is the normalized test pattern [7]. SRM principle depend on minimalizing the empirical risk which is expressed by the error (ε)-insensitive loss function [7]: 𝐿! 𝑦! , 𝑓 𝑥

3.4

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” [21]. A NN typically comprises a large number of layers of neurons, as shown in Figure 2 [22]. As can be seen, there are interconnecting lines between different neurons; these represent the trail of information flow [22]. Furthermore, each interconnecting line has a weight associated with it, which regulates the signal amid two connecting neurons [22].

=

(11)

ε, if 𝑦! − 𝑓 𝑥 ≤ ε 𝑦! − 𝑓 𝑥 − ε, otherwise

In Equation (11), 𝐿! is the ε-insensitive loss function, (𝑦! ) is the target output, f(x) is predicted output, and 𝑥! is the training data set [7]. 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 (𝜉! , 𝜉!∗ ), and is written as [7]: lim!,!,!!,!!∗ (12)

! !

𝑤. 𝑤 + 𝐶

! ∗ !!! 𝜉!

+

! !!! 𝜉!

subjected to 𝑦! − 𝑤. 𝑥! − 𝑏 ≤ ε + 𝜉! 𝑤. 𝑥! + 𝑏−𝑦! ≤ ε + 𝜉! , 𝑖 = 1, … , 𝑛 𝜉! , 𝜉!∗ ≥ 0 (13)

Figure 2. Schematic of a Neural [23] 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) [23]. Generally, a NN is a good fitting function, and a fairly simple NN can fit any particular data [23]. However, the large amount of trial and error associated with a NN limits its use as a regression tool [18]. 3.5

!

In Equation (12), 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 [24]. Figure 3 depicts the aforementioned situation graphically. As can be seen from Figure 3, 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 [24].

Support Vector Regression

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In Equation (14), the parameter Y depends upon the geometry of the component and the crack, so it is a complex function of the crack size [8]. The geometric function for the calculation of the SIF is given in Annex M of BS-7910 and is stated as: 𝑌 = 𝑀 ∗ 𝑓! ∗ 𝑀! (15) where M is the bulging factor and equal to 1 for the flat plate [8]. The detailed calculation for evaluating Y is given in [28]. Once the value of Y has been calculated, the next step involves the calculation of 𝐾, using Equation (14). The resulting values of 𝐾 for different combinations of loading and crack depth are shown in Table 2 in Annex A. It must be mentioned here that the aforementioned SIF solution presented in BS-7910 is adopted from the work of Newman and Raju [29]. 4.2.2 Finite Element Modelling A finite element model is constructed using the commercially available software ANSYS 17 [30]. The finite element model of the plate with a semi-elliptical surface crack is shown in Figure 5. 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.

Figure 3. Schematic showing loss function and slack variable in SVR [24] 4. ILLUSTRATIVE CASE STUDY 4.1 General The offshore piping material considered for numerical analysis is in accordance with industry practice and is assumed to be API5L-Grade B. Since most of the large-diameter pipelines used in the oil and gas sector are manufactured from flat plates using the UOE forming process [27], to ease the reader’s understanding, we have considered a semi-elliptic surface crack at the center of the flat plate, as depicted in Figure. 4. The component is subjected to uniaxial tensile loading, perpendicular to the crack front. The details of the geometry and material properties are given in Table 1 in Annex A. 2c θ

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 1 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 load, a and c) obtained using Latin Hypercube Sampling (LHS). For the analysis, the range of load (L) varies from 100MPa to 200MPa, while the range of crack depth (a) lies between 1mm and 8mm. Likewise, the range of half crack length (c) varies from 2mm to 22mm.The resulting value of K for the aforementioned data set is shown in Table 2 in Annex A.

a

t w Figure 4. Schematic of plate and crack geometry 4.2

SIF Calculation In this manuscript, the SIF calculation is performed using three different methods. The first is analytical, using the formula given in BS7910 [8]; the second uses FEM, and the third employs different MMs, namely, MLR, PR-2, GPR, NN, and SVR. These methods are expounded in the following subsections.

4.2.1

BS7910 Solution As per BS7910, the SIF can be simply expressed as a function of crack size and loading conditions, using a closed form solution given as:

𝐾! = 𝜎! 𝑌 𝜋𝑎 (14) Figure 5. FEM model of plate and crack geometry

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MLR PR(2) GPR NN SVR

4.2.3

Meta-models Using the flowchart shown in Figure 6 (given in Annex A), a MM is constructed to predict the value of the SIF. Seventy different combinations of crack size and loading levels were considered, and the SIF was calculated through finite element analyses. These seventy values of the SIF obtained by FEM were then used to train each meta-model separately. For instance, the equations for MLR and PR (2), generated after fitting them to the training data, are given in Equations (16) and (17) respectively:

SIF Prediction = −4.84𝑒 !! 𝐿 ∗ 𝐿 + 2.01𝑒 !! 𝐿 ∗ 𝑎 + 2.13𝑒 !! 𝐿 ∗ 𝑐 + 3.58 𝑎 ∗ 𝑎 − 1.84 𝑎 ∗ 𝑐 + 8.8𝑒 !! 𝑐 ∗ 𝑐 9 + +1.08 𝐿 − 3.65 𝑎 − 12.94 𝑐 + 38.67 (17)



In Equations (16) and (17), L, a, and c stand for loading, crack depth and half crack length, respectively, as shown in Figure 4. Once the MM has been trained, the next step is to test the MM by comparing the values of the SIF obtained from ANSYS and those predicted by the MM. The values of the SIF predicted by different MMs are shown in Table 2 in Annex A.

𝐴𝐴𝐸 =

− 𝑦! )!

In order to compare the accuracy of the aforementioned MMs, four metrics, namely, Root Mean Square Error (RMSE), Average Absolute Error (AAE), Maximum Absolute Error (MAE), and coefficient-of-determination (R! ) were used. The value of R! for MLR, PR-2, GPR, NN and SVR was found to be 0.970, 0.995, 0.90, 0.982 and 0.963 respectively. Generally, a model with the value of R! closer to 1 depicts high level of prediction accuracy. Based on the aforementioned premise, it is inferred that PR (2), i.e. second order polynomial regression with interaction terms, is the most accurate MM out of the five contestants, while GPR is the second most accurate MM. It must be mentioned here that, although PR (2) outperforms GPR on accuracy, unlike PR (2), 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

𝑦𝑖 −𝑦𝑖 !

𝑀𝐴𝐸 = max 𝑦! − 𝑦! , 𝑦! − 𝑦! , . . , 𝑦! − 𝑦! R! = 1 −

(18)

!!!""#" !!!"#

%$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. On using Equation (18), the values of the aforementioned metrics for the various MMs used in the case study are shown in Table 3. Table 3. Comparison of different meta-models RMSE

0.970 0.995 0.990 0.982 0.963

CONCLUSION The manuscript proposes the use of MMs as a replacement to computationally expensive FEM to predict the SIF of a crack propagating in offshore piping. The viability of five different MMs, namely, multi-linear regression (MLR), second order polynomial regression (PR-2) (with interaction), Gaussian Process Regression (GPR), neural network (NN) and support vector regression (SVR) have been tested in the manuscript. Seventy data points (SIF values obtained by FEM) were used to train the aforementioned meta-models, while thirty data points were used as testing points.

Result Discussion The values of the SIF obtained by the various MMs have been plotted and presented in Figure 7 and Table 2; see Annex A. In order to compare the accuracy of the MMs, four metrics, namely, Root Mean Square Error (RMSE), Average Absolute Error (AAE), Maximum Absolute Error (MAE), and are used [12]. coefficient-of-determination (R! ) Mathematically, these are written as:

! !!!

83.38 33.4 51.97 69.62 79.74

5.

4.3

! !!!(𝑦!

24.11 9.68 13.29 18.85 28.69

In order to discover the most accurate of the competing MMs, it is vital to comprehend the results of Table 3. 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 R! closer to 1 depicts high level of prediction accuracy. Based on the aforementioned premise, it is inferred that PR (2), i.e. second order polynomial regression with interaction terms, is the most accurate MM out of the five contestants, while GPR is the second most accurate MM. It must be mentioned here that, although PR (2) outperforms GPR on accuracy, GPR, unlike PR (2), 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 MM (out of five competing MMs) to predict the SIF of a propagating crack. The detailed procedure for establishing a GPR model is discussed by authors in [10].

SIF Prediction = −445.36 + 3.85 L + 31.55 a + 32.65 c (16)

𝑅𝑀𝑆𝐸 =

32.62 12.55 18.56 25.35 36.54

AAE

MAE

𝑹𝟐

6

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12. Westergaard, H. M., "Bearing Pressures and Cracks," Journal of Applied Mechanics, 6, pp. 49-53, 1939. 13. Naess, A. A., (2009). Fatigue Handbook: Offshore Steel Structures. ISBN 9788251906623. Tapir Publisher, Norway. 14. Antaki, G.A., (2003). Piping and Pipeline Engineering. Design, Construction, Maintenance, Integrity, and Repair. ISBN 9780824709648, CRC Press, USA. 15. Rooke, D. P., Baratta, F. I., and Cartwright, D. J., (1981). “Simple Methods of Determining Stress Intensity Factors”. Engineering Fracture Mechanics,14(2), pp. 397-426. 16. Sankararaman, S., (2012). “Uncertainty Quantification and Integration in Engineering Systems”. PhD Dissertation, Vanderbilt University, USA. 17. Bishop, C. M., (2006). Pattern Recognition and Machine Learning. ISBN 9780387310732. Springer, UK. 18. Jin, R., Chen, W., and Simpson, T. W., (2001). “Comparative Studies of Metamodelling Techniques under Mutliple Modelling Criteria,” Structural and Multidisciplinary Optimization, 23, pp. 1–13. 19. Rasmussen, C. E., and Williams, C. K. I., (2006). Gaussian Processes in Machine Learning. ISBN 026218253X, MIT Press, USA. 20. McFarland, J. M., (2008). “Uncertainty Analysis for Computer Simulations through Validation and Calibration”. PhD Dissertation, Vanderbilt University, USA. 21. Maureen, C., (1989). “Neural Network Primer: Part I”. AI Expert, 2 (12), 1987. 22. Siffman, D., (2012). The Nature of Code. ISBN 9780985930806, USA. 23. https://se.mathworks.com/products/neuralnetwork/feat ures.html#deep-learning 24. Smola, A. J., and Scholkopf, B., (1998). “A Tutorial on Support Vector Regression”. Neuro COLT2 Technical Report Series NC2-TR-1998-030. 25. Forrester, A. I. J., and Keane, A. J., (2009). “Recent Advances in Surrogate Based Optimzation”. Progress in Aerospace Sciences, 45 (1-3), pp. 50-79. 26. Gunn, S. R., (1998). “Support Vector Machines for Classification and Regression”. Technical Report ISIS-1-98, University of Southampton, UK. 27. Ren, Q., Zou, T., Li, D., Tang, D., and Peng, Y., (2015). “Numerical Study on the X80 UOE Pipe Forming Process”. Journal of Materials Processing Technology, 215, pp. 264-277. 28. Keprate, A., Ratnayake, R. M. C., and Sankararaman, S., (2017). “Minimizing Hydrocarbon Release from Offshore Piping by Performing Probabilistic Fatigue Life Assessment”. Process Safety and Environmental Protections, 106, pp. 34-51.

best MM (out of five competing MMs) to predict the SIF of a propagating crack, the details of which are given in [10].

ACKNOWELDGEMENT This work has been carried out as part of a PhD 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 NNX12AK33A. The support is gratefully acknowledged. REFERENCES 1. Surrogate Modelling Lab., University of Gent, Belgium. http://www.sumo.intec.ugent.be/research 2. https://en.wikipedia.org/wiki/Surrogate_model 3. Forrester, A. I. J., Sobester, A., and Keane, A. J., (2008). Engineering Design via Surrogate Modelling. ISBN 9780470060681. Wiley, UK. 4. Hombal, V. K., and Mahadevan, S., (2013). “Surrogate Modelling of 3 D Crack Growth”. International Journal of Fatigue, 47, pp. 90-99. 5. Sankararaman, S., Ling, Y., Shantz, C. and Mahadevan, S., (2011). “Uncertainty Quantification and Model Validation of Fatigue Crack Growth Prediction”. Engineering Fracture Mechanics, 78(7), pp. 1487-1504. 6. Leser, P. E., Hochhalter, J. D., Warner, J. E., Newman, J. A., Leser, W. P., Wawrzynek, P. A., and Yuan, F. G., (2015). “Probabilistic Fatigue Damage Prognosis Using Surrogate Models Trained via ThreeDimensional Finite Element Analysis”. International Workshop on Structural Health Monitoring, Stanford, California, USA. 7. 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”. Journal of Structural Engineering, 40(3), pp. 224-233. 8. BS-7910, (2005). “Guide on Methods for Assessing Acceptability of Flaws in Metallic Structures”. UK. 9. DNVGL-RP-C203 (2010). “Fatigue Design of Offshore Steel Structures”. Det Norske Veritas, Høvik, Norway. 10. Keprate, A., Ratnayake, R.M.C., and Sankararaman, S., (2017). “Surrogate Model for Predicting Stress Intensity Factor: A Novel Application to Oil and Gas Industry”. International Conference on Offshore Mechanics and Arctic Engineering, 2017, Trondheim, Norway. 11. Irwin, G. R., (1957). “Analysis of Stresses and Strains Near the End of a Crack Traversing in a Plate”. Journal of Applied Mechanics, 24, pp. 361–364.

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29. Newman, J. C., and Raju, I. S., (1979). “Stress Intensity Factors for a Wide Range of Semi-Elliptical Surface Cracks in Finite Thickness Plates”. Engineering Fracture Mechanics, 11, pp. 817-829. 30. ANSYSWebsite:http://www.ansys.com/Products/Acad emic/ANSYS-Student.

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ANNEX A Latin Hypercube Sampling to Generate Design Space (i.e. L, a and c Values)

Training Points (Combination of L, a, c )

Testing Points (Combination of L, a , c )

Run FEM to predict SIF for Training Points

Run FEM to predict SIF for Testing Points

Final Training Points (L, a, c, SIF)

Final Testing Points (L, a, c, SIF)

Final Testing Points (L, a, c) Values

Meta-model

Final Testing Points (SIF) Values

Trained Meta-model

Predict Values of SIF for Each Combination of L, a, and c

Compute RMSE, AAE, MAE and R2

Figure 6. Flowchart to build MM for SIF prediction

Table 1. Material and Geometry Properties of API5L-Grade B Material Properties

Value

Geometrical Properties

Value

Modulus of Elasticity

210 GPa

Length

273 mm

Poisson Ratio

0.3

Width

60 mm

Yield Stress/Tensile Stress

241 GPa/350 GPa

Thickness

10 mm

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Copyright © 2017 ASME

1200

1000

SIF(MPa-mm^0.5)

800

600

400

200

0 0

5

10

15

20

25

30

35

DATA POINTS (Combina:on of L, a and c) ANSYS

MLR

POL(2)

GPR

NN

SVR

BS7910

Figure 7. SIF Comparison for ANSYS, BS7910 and different MMs,

Table 2. SIF for 30 test points using ANSYS and different MMs SIF(MPa- mm) Loading S. No

(MPa)

Crack Depth in mm (a)

Half Crack Length in mm (c)

BS7910

ANSYS

MLR

POL -2

GPR

NN

SVR

807.7

722.2

722.5

733.7

731.2

758.4

696.9

558.5

503.1

524.5

496.3

504.9

517.4

526.3

569.9

519.9

538.9

518.5

526.6

535.8

537

448.1

403.4

403.1

400.9

396.9

407.7

411

1

176

6.1

9.1

2

126.1

4.8

10

3

166

3.92

6.7

4

116.4

4.45

7.9

10

Copyright © 2017 ASME

5

192.2

3.39

9.9

6

141.5

7.85

21

7

133.7

3.62

5.5

8

120.4

7.6

12

9

150.6

1

2.2

10

143.5

5.95

15

11

108.8

3.95

7.6

12

187.3

5.27

9.9

13

189.8

3.27

10

14

177.9

6.97

11

15

145.1

2.62

5.2

16

179.2

3.79

5.7

17

184.1

3.23

9.7

18

167.9

1.31

1.9

19

110.3

5.06

7.5

20

115.8

6.53

20

21

136.2

4.7

9.7

22

178.3

3.53

7.9

23

157.2

2.02

3.2

24

146.2

1.89

2.7

25

103.9

7.76

13

26

122.4

4.35

7.7

27

191.1

1.65

3

28

159.5

2.87

6.5

29

152.5

2.52

4.2

30

124.7

3.33

9.3

716.7 1138 409.3 696.8 249.3 828.7 394.2 853.9 704.9 970.4 400.3 562.4 667 275.2 428.2 807.4 586.4 631.3 340.7 292.6 649.7 460.4 386.9 488.1 384.5 453.4

11

675.7

726.5

689.8

704.4

707.2

722.1

1056

1025

1032

1027

1126

993.4

380.3

364.7

381

369

382.3

375.3

635.8

641.1

630.1

630.2

663.4

617.8

243.6

239.6

263.2

240.7

232

270.1

755.6

772.3

744.8

774.7

795.2

758

361.6

346.7

351.3

335.6

361.8

361.6

753.9

767.2

760.1

764.2

793.3

746.2

662.9

730.2

696.4

714.9

693.2

727.7

853.4

827.3

860

864.2

902.7

792.8

377.6

367.1

368.9

368.9

379.3

383.8

518.9

551.2

522.8

526.5

534.8

546.6

626.2

683.8

646.4

655.5

652.8

683.3

268.6

305,5

293,6

287

259,5

327,9

380.3

383.5

395.5

374

382.3

387.3

772.3

855.7

779.1

810

813.5

844.5

529.3

545.7

522.6

528.9

546.3

545.4

584.3

613.5

577.1

576.1

606.8

611.8

324.3

330.1

331

332.8

320.7

348.9

279.8

267.8

294.6

284.2

271.8

291

581.5

628.3

579.7

582.8

603.6

608.3

415.7

414.5

413.5

415.4

421.3

421.6

373.2

440.7

375.4

410

374.5

453

459.2

472.5

448.9

462.1

469.6

482.7

362.6

360.3

361.4

366.2

362.9

375.8

426.4

445.7

423.5

431.6

433.1

462

Copyright © 2017 ASME