Journal of Engineering

Volume 2018, Article ID 4125765, 10 pages

https://doi.org/10.1155/2018/4125765

## Application of the Extended Isogeometric Analysis (X-IGA) to Evaluate a Pipeline Structure Containing an External Crack

^{1}Mechanical Engineering Laboratory, Faculty of Science and Technology Fez, Morocco^{2}Laboratory of Systems Engineering and Applications (LISA), National School of Applied Sciences of Fez, Morocco^{3}Department of Mechanical Engineering, Imperial College London, Exhibition Rd., SW7 2AZ London, UK^{4}Department of Mechanical Engineering, School of Engineering, University of Birmingham, B15 2TT, UK

Correspondence should be addressed to C. I. Pruncu; ku.ca.lairepmi@ucnurp.c

Received 7 August 2018; Revised 6 September 2018; Accepted 18 September 2018; Published 17 October 2018

Academic Editor: Yaowen Yang

Copyright © 2018 S. El Fakkoussi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work proposes a novel strategy for a two-dimensional problem that includes the approach of extended isogeometric analysis (X-IGA) in order to detect the behavior of a crack in pipeline structures. The nonrational B-Spline uniform function (NURBS) was used for the approximation of the solution fields (displacements) taking into account its geometry constrains. The modeling of the X-IGA was implemented under Abaqus/Standard software via subroutine (UEL) where the Stress Intensity Factor (KI) was extracted. The results permit detecting with accuracy the fracture toughness of a pipeline structure containing an external crack that can be submitted to critical pressures. To validate the performances of the novel strategy a careful comparison with existing literature and analytical and numerical computation methods was performed.

#### 1. Introduction

The mechanical evaluation of a metal structure that contains cracks still represents an important research topic in the aeronautical, nuclear, and naval fields. Further preventive detection/prediction of a crack remains a major concern for researchers and engineers. An early detection makes it possible to minimize the cost of the corrective maintenance and to avoid important material and human damage. Because of this challenge, many researchers have devoted their efforts to study the phenomenon of crack propagation in metal structures. They proposed modern and reliable tools to quantify the damage rate near the crack. The literature highlights the existence of some methods based on the Stress Intensity Factor (KI) concept that permits evaluating with accuracy the fracture of the structures. This factor can be determined analytically, semianalytically, and by numerical computation methods. The numerical methods are based on the classical finite element method (FEM) and the extended finite element method (X-FEM) [1]. The X-FEM method is considered a powerful tool in terms of its precision to calculate the KI factor; it makes it possible to model the crack without taking into consideration the mesh near the crack tip. To enhance its robustness, in the X-FEM an enrichment function is added with respect to the conventional finite element formulation. However, this method has some limitations when an elliptical crack is simulated and is unable to model the semielliptic or circular cracks. The KI factor generates instabilities on the mentioned above cases [2, 3]. To avoid this problem that generates errors in the modeling of displacement and its geometry, Hughes et al. [4] proposed a Computer Aided Design (CAD) approach based on the finite elements procedure. In this routine the Lagrange form functions with the nonrational B-Spline uniform function (NURBS) is replaced. This approach is known as isogeometric analysis (IGA). These functions permit checking the interpolation conditions; however, the NURBS functions are always positive.

The new IGA approach has been well implemented in several fields such as the heat transfer [5, 6], vibration analysis [7, 8], fluid flow analysis [9, 10], and wave propagation [11]. Besides, the IGA method showed promising results in the composite structures [12, 13], in thin-shell structures [14], and for crack propagation simulations [15]. In the fracture mechanics, the IGA approach was validated by comparison to the extended finite elements method (X-FEM) and the classical finite elements method (FEM). Sanches et al. [16] proposed a new function named i-Spline while imposing some modifications to the B-Spline function. The proposed method was implemented in two-dimensional problems and was integrated in the finite elements calculation using Matlab software.

Bornemann and Cirak [17] identified a new technique that can be implemented in the B-Spline function by using the finite element method. The new techniques are applied to obtain convergence on the linear and geometrically nonlinear, two- and three-dimensional problems, using Matlab and Abaqus/Standard software. Choi and Cho [18] calculated the KI factor for a semielliptic crack using the NURBS function in the finite element method (MEF) formulation. Several researchers started to implement the NURBS function in the Abaqus/Standard software via subroutine (UEL) in the problems case without crack [19–21].

In the modeling of some discontinuities problem like cracks, the IGA method has been coupled with the X-FEM method to generate a novel approach known as X-IGA. The new approach is proposed to understand the crack propagation phenomenon and fatigue of metal structures. Ghorashi et al. [22] investigated stationary and propagation of cracks using the combination of IGA approach and X-FEM method. They validated the results for a two-dimensional plane crack on a plate. Later, they compared the Stress Intensity Factor (KI) obtained with the analytical values and with the values of X-FEM method. Jung and Taciroglu [23] have used the natural cubic spline function in the X-FEM formulation to evaluate a semielliptic crack propagation on a two-dimensional plate. Bhardwaj et al. [24] studied the benefits of using the NURBS functions by applying the X-FEM method with the main objective of validating the X-IGA method. Based on the X-IGA approach, Singh et al. [25] used Bézier extraction configured on the T-Spline function to treat cracks in the two-dimensional specimen. On the other hand, Tinh Quoc Bui [26] used an extended isogeometric analysis (X-IGA) for the simulation of two-dimensional fracture mechanics problems in electromagnetic piezoelectric materials. A robust research on understanding of the thermal buckling analysis of functionally graded plates with internal defects was considered later by [27]. Bui et al. [28] have developed an innovative dynamic extended isogeometric analysis (X-IGA) for transient fracture of magnetoelectroelastic crack (MEE) solids under coupled electro-magneto-mechanical loading. The X-IGA approach permits obtaining a greater accuracy with respect to X-FEM methods [29] and conventional FEM approach [30], in terms of errors convergence and modeling strategy. Therefore, the novel X-IGA approach can be a robust tool to evaluate the fracture of metal structures.

It is noticed that the IGA and X-IGA approaches are implemented only in the Matlab software. So far, from our knowledge, this was partially implemented in Abaqus/Standard software and this is done only for structures without cracks. In that reason, we propose to extend the novel X-IGA approach 0068 in the evaluation of the structure containing a crack and presenting a particular case to calculate the Stress Intensity Factor (KI) using the X-IGA approach for a pipeline containing an external crack. To validate our model a comparison with the Stress Intensity Factor (KI) results obtained via an analytical method and numerical methods (FEM and X-FEM) was implemented. The proposed method implemented under the Abaqus/Standard software offers a modern approach to investigate the fracture mechanisms for industrial site where the complex problems are encountered. The coupling between the two powerful methods IGA and X-FEM provides a new strategy capable of modeling cracked structures without remeshing and to model complex geometries. This approach will allow reducing considerable the computing time with direct benefit of improving the cost.

#### 2. Theoretical Background

In this section, IGA and X-IGA as well as the analytical solution for SIF finite plate and an annular tube were briefly reviewed. More details of IGA and X-IGA and the LEFM are referred to the literature [4, 24, 33].

##### 2.1. Isogeometric Analysis (IGA)

###### 2.1.1. B-Spline Function

The B-Spline functions are written in the form of Cox-de Boor recursion formula [34]:1.For 2.For The derivative of the B-Spline function is written as follows:

###### 2.1.2. Nonuniform Function Rational B-Spline (NURBS)

NURBS are a generalization of B-Splines based not on polynomials but on rational functions [34]. The advantage of using the NURBS function and not the B-Spline function is that the former is able to represent complex shapes such as conic sections circles and elliptical.

and are the basic functions, B-Splines with p and q are the order of the interpolations function, n and m are the number of control points and are the weights corresponding to the control points, and and are knots vectors (nodes).

###### 2.1.3. Surface NURBS

The surface NURBS functions are represented by the following equation:

is the control points and is the bivariate NURBS basic functions.

##### 2.2. Extended Isogeometric Analysis (X-IGA)

The advantage of using the X-FEM method [1] together with IGA approach [4] is that the first method is able to model the crack in the structures using the enrichment functions and the second one is able to model geometries. The coupling between two methods is made by using NURBS functions.

The approximation of displacement for a typical point can be written in generalized form

is NURBS basic function, is the nodal shift in the classical method, is the Heaviside function, and are the enrichment functions near the crack tip and are additional degree of freedom for Heaviside function and enrichment function in of the crack tip. and are, respectively, the number of the points of the enriched controls for the Heaviside functions and the functions of crack tip.

The Heaviside function is defined byThe functions of enrichments in the crack tip are defined by

##### 2.3. IGA/X-FEM Implementation in Abaqus/Standard Software

Duval et al. [35] implemented the IGA method, in the Abaqus/standard software. The elastic domain was considered for the crack case. They used a subroutine UELMAT written in Fortran language as a main code programming. There the NURBS functions were added as secondary file in the form ( NB). This file contains all the necessary input information of the NURBS functions such as the vector knot, the weight functions, and the order of the function. The present work will apply this strategy to implement the IGA approach coupled with the X-FEM method in the Abaqus/Standard software. Subsequently, the functions NURBS are implemented in Abaqus/Standard software via the UEL subroutine (UEL) that allows modifying the formulation of the finite elements. Some other parameters of the method (X-FEM) were added in this code, such as the functions of enrichment at the crack tip and the Heaviside function. The main code (UEL) is detailed in Algorithm 1 at the end of this document. To facilitate the addition of extra information to the main code, we have used the subroutine uexternaldb. This later on allows adding the necessary information of the crack and information of the NURBS function. These parameters are used later in the main program (UEL). The details of program implantation are summarized in Figure 1.