Eduard Marenić

Ivica Skozrit

Zdenko Tonković

On the Calculation of Stress Intensity Factors and J-Integrals Using the Submodeling Technique

Keywords: finite element analysis, submodeling technique, stress intensity factor, Jintegral, semi-elliptical surface crack, reference stress method

In the present paper, calculations of the stress intensity factor (SIF) in the linear elastic range and the J-integral in the elastoplastic domain of cracked structural components are performed by using the shell-to-solid submodeling technique to improve both the computational efficiency and accuracy. In order to validate the submodeling technique, several numerical examples are analyzed. The influence of the choice of the submodel size on the SIF and the J-integral results is investigated. Detailed finite element (FE) solutions for elastic and fully plastic J-integral values are obtained for an axially cracked thick-walled pipe under internal pressure. These values are then combined, using the General Electric/Electric Power Research Institute (GE/EPRI) method and the reference stress method (RSM), to obtain approximate values of the J-integral at all load levels up to the limit load. The newly developed analytical approximation of the reference pressure for thick-walled pipes with external axial surface cracks is applicable to a wide range of crack dimensions.

Introduction

Assessment of the structural integrity of cracked pressure vessel components fabricated from ductile materials can be performed by using elastoplastic fracture mechanics (EPFM) parameters and the limit load analysis. The most commonly used fracture parameter in EPFM is the J-integral. The calculation of the J-integral and the plastic limit load by using the incremental theory of plasticity requires a high computational effort [START_REF] Simha | Jintegral and crack driving force in elastic-plastic materials[END_REF][START_REF] Kim | Plastic Limit Pressures for Cracked Pipes Using Finite Element Limit Analyses[END_REF]. The J-integral estimation method based on the reference stress method [START_REF] Ainsworth | The Assessment of Defects in Structures of Strain Hardening Material[END_REF] could reduce this effort since it only requires three basic data to be applied: the linear elastic fracture mechanics (LEFM) parameter (called the stress intensity factor) of the cracked component, the uniaxial true stress-strain curve of the material and the limit load able to produce its plastic collapse [START_REF] R6 | Assessment of the Integrity of Structures Containing Defects[END_REF][START_REF] Kim | Elastic-Plastic Fracture Mechanics Method for Finite Internal Axial Surface Cracks in Cylinders[END_REF][START_REF] Moreno | Assessment of the Reference Stress Method for J-Integral Estimation of Cracked Riveted Beams of an Old Wrought Iron[END_REF].

Due to fabrication defects, or damage in service, cracks exist generally in the areas of geometrical discontinuity of pressure vessel components where the stress concentration occurs. Therefore, cracks can be found near tubular joints, around weld toes. For a cracked tubular joint, the junction curve is a complicated space curve and the 3D meshing is very difficult and is even more demanding when the crack geometry has to be included. In order to calculate SIF one needs to use singular elements along the crack front, which generally leads to a 3D model. The number of elements required for a three dimensional FE model of the whole vessel volume, for example, would be very high, leading to extremely time-consuming solutions; therefore, the methodology would not be appropriate for design purposes. Furthermore, the accuracy of the numerical SIF results depends on the element type, mesh quality, mesh refinement, integration scheme and weld shape modeling. Elements near the crack front often have a high aspect ratio and, occasionally, some elements near the crack front are found to be badly distorted. This will influence SIF results significantly, and hence, in [START_REF] Chiew | Stress Intensity Factors for Surface Crack in a Tubular T-Joint[END_REF][START_REF] Lie | Model and Mesh Generation of Cracked Tubular Y-Joints[END_REF][START_REF] Lie | Mesh Modelling and Analysis of Cracked Uni-Planar Tubular K-Joints[END_REF] a new method is proposed to model the surface crack and to generate the FE mesh for cracked tubular joints. The tubular joint is divided into different, distinct zones which can greatly simplify the mesh generation process. For each zone, an FE mesh is generated separately and then merged in order to perform an analysis. The mesh quality can be controlled and improved, thus producing accurate and stable numerical results.

The line spring model introduced by Rice and Levy [START_REF] Rice | The part-through surface crack in an elastic plate[END_REF] reduces a 3D surface crack problem into a two-dimensional plate or shell theory problem. This approach is computationally inexpensive compared to fully 3D models and, within certain restrictions, provides acceptable accuracy [START_REF] Parks | The inelastic line-spring: estimates of elastic-plastic fracture mechanics parameters for surface cracked plates and shells[END_REF][START_REF] Berg | Two-parameter fracture mechanics and circumferential crack growth in surface cracked pipelines using line-spring elements[END_REF]. The main disadvantage of the model is that it does not take into account the curvature of the crack front and is not a good approximation near the ends where the crack intersects the free surface and its depth varies rapidly, due to the three-dimensional nature of the solution in such areas.

Recently, the submodeling technique has often been used in the FE numerical analysis to study in detail an area of interest in a model [START_REF] Haryadi | Global/local analysis of composite plates with cracks[END_REF][START_REF] Bakuckas | Comparison of boundary correction factor solutions for two symmetric cracks in a straight-shank hole[END_REF][START_REF] Diamantoudis | Stress Intensity Factors of Semi-Elliptical Surface Cracks in Pressure Vessels by Global-Local Finite Element Methodology[END_REF][START_REF] Levy | The Combined Stress Intensity Factors of Multiple Longitudinally Coplanar Cracks in Autofrettaged Pressurized Tubes Influenced by the Bauschinger Effect[END_REF][START_REF] Liu | Fatigue crack initiation life prediction of railroad wheels[END_REF][START_REF] Mikhaluk | Experimental observations and finite element modelling of damage initiation and evolution in carbon/epoxy non-crimp fabric composites[END_REF]. Herein, the area of interest is the region of high stress caused by geometric discontinuities. The main idea of the submodeling technique is to perform a global (shell)-local (3D) transition. This approach gives an opportunity to make a local mesh refinement which brings considerable advantages. In the submodeling technique, the results from the global model are interpolated onto the nodes on the appropriate parts of the submodel boundary. A basic condition of the technique is that in the analysis the global model must define the submodel boundary response with sufficient accuracy. That means, when one is defining the submodel boundaries, in that area on the shell global model the displacement results must be accurate to a certain extent. In order to accomplish accuracy in the global model with a relatively coarse shell mesh, the submodel boundary must lie far enough from the intensive stress concentration. As the submodel region has a finer mesh, the submodel can provide an accurate, detailed solution.

Besides better accuracy, another advantage is that one can model a real problem more accurately by using a 3D submodel whose boundaries are driven by the shell submodel. This gain in accuracy is obtained because the 3D submodel can model cracks, welds, fillets, chamfers, etc. Unlike a large number of papers where the submodeling capability is used to obtain linear elastic SIF [START_REF] Haryadi | Global/local analysis of composite plates with cracks[END_REF][START_REF] Bakuckas | Comparison of boundary correction factor solutions for two symmetric cracks in a straight-shank hole[END_REF][START_REF] Diamantoudis | Stress Intensity Factors of Semi-Elliptical Surface Cracks in Pressure Vessels by Global-Local Finite Element Methodology[END_REF][START_REF] Levy | The Combined Stress Intensity Factors of Multiple Longitudinally Coplanar Cracks in Autofrettaged Pressurized Tubes Influenced by the Bauschinger Effect[END_REF][START_REF] Liu | Fatigue crack initiation life prediction of railroad wheels[END_REF][START_REF] Mikhaluk | Experimental observations and finite element modelling of damage initiation and evolution in carbon/epoxy non-crimp fabric composites[END_REF], a very limited number of studies have been reported in the area dealing with the submodeling techniques in nonlinear EPFM. In that case, the loads are usually chosen so that the plastic zone around the crack tip is sufficiently small [START_REF] Simha | Jintegral and crack driving force in elastic-plastic materials[END_REF].

In the present investigation the submodeling capability is used to obtain a more accurate stress field at the crack tip in the linear elastic and nonlinear elastoplastic fracture mechanics problems. Efficiency and accuracy of the submodeling technique are demonstrated by several numerical examples. Thereby, the following is considered: a membrane with a semi-elliptical crack, a pipe with an external axial crack and a welded vessel-nozzle junction with a semi elliptical crack in the weld toe. The objective of this work is to determine the submodel size where nodal displacements for driven nodes of the submodel will be correctly interpolated from the results of the global model. Additionally, on the basis of detailed FE results obtained from the submodel analysis and using the reference stress approach [START_REF] Ainsworth | The Assessment of Defects in Structures of Strain Hardening Material[END_REF], new reference pressure and J estimation equations are proposed for thick-walled pipes with external axial surface cracks under internal pressure. All computations have been performed within the FE software ABAQUS/Standard [START_REF] Hibbitt | ABAQUS/Standard. User's Guide and Theoretical Manual[END_REF].

Elastic fracture mechanics analysis using submodeling

Three examples are used to check and verify submodeling in the SIF calculation with the results from the literature. The first example is a membrane with a semi-elliptical crack. In the second, a pipe with an external axial crack is analyzed. Finally, a semielliptical crack is included into the submodel of the vessel-nozzle junction.

A membrane with a semi-elliptical crack

The first geometry to be analyzed is a membrane with a symmetric, centrally located, semi-elliptical crack loaded in Mode I by uniform tension applied to its top and bottom surfaces, as shown in Fig. 1. Employing symmetry, only one-quarter of the membrane is analyzed. The dimensions of the membrane relative to the membrane thickness t are as follows: the half-width / 10 Wt  , the half-length / 25 Ht  , the crack depth / 0.6 at 

, and the crack half-length / 0.8 ct  . A numerical solution for an uncracked membrane (global model) is obtained initially, where the membrane has been modeled without any crack. For shell-to-solid submodeling, the global model consists of 110 S8R shell elements, as shown in Fig. 2a. The results from the global analysis are interpolated on the boundary of the submodel with a crack included. The submodel consists of about 990 three-dimensional C3D20R elements (Fig. 2b). To avoid problems associated with incompressibility, the reduced integration is applied. In order to model strain singularity at the crack tip correctly, collapsed wedge-shaped elements are used. In order to be able to carry out the global analysis without a crack, and drive the cracked submodel with results from the global analysis, the submodel boundaries have to be chosen with good care. The submodel boundaries should be taken as close as possible to the crack tip, but sufficiently far so that the stress field in the boundaries is completely unaffected by the crack. Herein, the submodel size 2s is chosen to be 2.5

times the size of the semi-elliptical crack, (i.e. the length 2 = (2.5...3) (2 ) sc  in Fig. 1)

according to [START_REF] Diamantoudis | Stress Intensity Factors of Semi-Elliptical Surface Cracks in Pressure Vessels by Global-Local Finite Element Methodology[END_REF]. This example is simple and is used for the verification purpose. The assumption of a typical submodel size 2s is checked by modeling the whole membrane with three dimensional elements and with a crack included (Fig. 3). Numerical results

show that stress distribution is altered only near the defect and it is enough to choose the submodel boundary such as 2 = 2.5 [START_REF] Kim | Plastic Limit Pressures for Cracked Pipes Using Finite Element Limit Analyses[END_REF] sc  . Compendium of stress intensity factor solutions [START_REF]Recommended practice for fitness-for-service[END_REF]. According to the quoted reference, SIF is estimated by using the following equation:

I () m m b b a K M M Q    , (1)
where M m and M b are membrane and bending correction factors, respectively, and Q is the elliptical integral approximated by

1.65 1 1.464 a Q c     . ( 2
)
A comparison of results for the SIF obtained numerically by using the submodeling technique and those obtained by using Eq. ( 1) is shown in Table 1. As one can see from the table, the results are in very good agreement. Good agreement shows that it is possible to perform the linear fracture mechanics analysis using the global-local technique. Although there are two steps that have to be carried out, this approach allows only the local mesh refinement with 3D elements together with singularity elements along the crack front. This kind of analysis where the refined 3D submodel is driven by a coarse global model enables a cost-effective analysis.

A pipe with an external axial surface crack

The next step is to check the submodeling technique when a crack is introduced into a model with a more complex geometry, namely pressurized pipe. The surface-cracked pipe geometry and loading are shown in Fig. 4. R i and R o are the inner and the outer radius, while t denotes the wall thickness. The external surface crack is assumed to have a semi-elliptical shape described by a length 2c and a maximum depth a. An axial surface-cracked pipe is characterized by three non-dimensional parameters, i.e. R i /t, c/a and a/t. Besides that, solutions for the limit pressure in literature are usually expressed in terms of a non-dimensional parameter ρ associated with the crack length

t R c i /   .
As evident from Fig. 4a, the internal pressure p is applied as a distributed load to the inner surface of the computational model. Since a pipe capped at both ends is considered, the corresponding axial tension force p R F ). The overlay plot of the global model and the submodel is shown in Fig. 6. PVT -09-1091, Tonković 11 In addition, a full three-dimensional C3D20R solid element mesh is considered for use as a reference solution (Fig. 7). The full solid element mesh in the crack-tip area is very similar to that used in the submodeling analyses. Moreover, both models have the same general characteristics. A difference exists in the modeling of pressure loading using the shell and continuum elements. The pressure loading is applied to the midsurface of the shell elements in contrast to the solid elements, where the pressure is accurately applied along the inside surface of the pipe. Fig. 7 A solid reference model of the cracked pipe (R i /t = 4, c/a = 10, a/t = 0.8)

A parametric study is performed, in which an inner radius-to-thickness ratio R i /t takes the values of 4 and 10. Thereat, the wall thickness of the pipe t is held constant at a value of 1.625 mm. These values are chosen to represent a standard thick-walled pipes geometry (inner radius-to-thickness ratio R i /t typically less than 10). Four different half crack length-to-crack depth ratios of c/a = 5, 10, 15 and 20, and crack depth-tothickness ratios of a/t = 0.2, 0.4, 0.6 and 0.8 are considered. The actual dimension of the crack depth ranges from 0.325 mm to 1.3 mm and the crack length varies from 3.25 to 52 mm. Thus, a total of 32 cases are considered in the present investigation, as summarized in Table 2. In all analyses, the total length 2L of the pipe is chosen large enough so that the length would have a negligible effect on results (L/c ≥ 10). The material properties chosen are: E = 200 GPa, ν = 0.3, σ y = 250 MPa where E is Young's modulus, ν Poisson's ratio and σ y is the yield stress. The FE SIF solutions will be properly normalized in such a manner that the above mentioned choices do not affect the present results. The linear-elastic FE computations yield the stress intensity factor K for an external axial surface crack in a pipe under internal pressure, which may be expressed by the following relation [START_REF] Raju | Stress-Intensity Factors for Internal and External Surface Cracks in Cylindrical Vessels[END_REF]:

) , , , ( m m    t a t R F a t R p K  (3) where ) , , / , / ( m   t a t R F
is a dimensionless function depending on the pipe and crack geometry and φ is the angle on an inscribed circle for locating a point on the crack, as shown in Fig. The variations of the difference in stress intensity factor with the increasing submodel size 2s from 2 to 12 times the size of the crack length 2c in the submodeled analysis are compared to the full solid (3D) element analysis in Fig. 8. The comparison is expressed as a percent difference that is defined as

3D 3D 100 ( ) / K K K 
. It can be concluded from the results that for the pipe with an inner radius/thickness ratio of 10 a submodel size 2s equal to 2.5 times the size of the crack length 2c is sufficiently large for the linear elastic analysis (Fig. 8a). However, for very thick pipe with R i /t = 4 the K value is stabilized for the ratio of 2s/(2c) equal to 5 (Fig. 8b). The error thereby introduced in the SIF value is up to around 1%.

Fig. 8 Dependence of the stress intensity factor on the submodel size: As mentioned before, the same pressure is applied on the inner radius of the 3D solid FE model as on the mean radius of the shell model (Fig. 13a). For very thin pipes this doesn't make a practical difference, but for pipes studied here the membrane displacement for pipes with ratio i / 10 Rt  and i /4 Rt  is too high in shell model for The values of the dimensionless function F for the stress intensity factor obtained by the submodeled analysis are also compared with the solutions obtained by [START_REF] Raju | Stress-Intensity Factors for Internal and External Surface Cracks in Cylindrical Vessels[END_REF] for the selected values of t/R i = 0.1, a/c = 0.2, 0.4, 1.0, a/t = 0.2, 0.5, 0.8 and s/c = 3 (Fig. 15).

a) i /
The results compare satisfactorily, while only slight differences in less than 6% of the F values are exhibited. These differences could arise from the fact that the solutions obtained by Raju and Newman are given for the model with two diametrically located cracks. Additionally, the differences occur due to the numerical inaccuracy of the results in [START_REF] Raju | Stress-Intensity Factors for Internal and External Surface Cracks in Cylindrical Vessels[END_REF], as also shown in [START_REF] Kim | Elastic-Plastic Fracture Mechanics Method for Finite Internal Axial Surface Cracks in Cylinders[END_REF].

A welded vessel-nozzle junction with a semi-elliptical crack in weld toe

In the above examples, submodeling has been verified as an efficient, cost-effective approach in the calculation of fracture mechanics parameters in simpler examples; it is justified to use the same procedure for a real welded vessel-nozzle junction. The accurate calculation of stress intensity factors for semi-elliptical weld toe cracks (Fig. 16) is a rather difficult task because the geometry of the joint with crack is extremely complex and, hence, difficult to model.

On the Calculation of Stress Intensity Factors and J-Integrals Using the Submodeling Technique PVT -09-1091, Tonković 20 In general, a major problem in using solid elements lies in generating an adequate mesh for accurate solutions (for example to create a good 3D, hexahedral mesh). This problem is partially solved when the global-local approach is used because just part of interest is modeled as a solid. Usually, the geometry of this relatively small submodel (with respect to the whole global model) can be divided into considerably simpler cells which can be meshed easily. A similar approach is used in [START_REF] Chiew | Stress Intensity Factors for Surface Crack in a Tubular T-Joint[END_REF][START_REF] Lie | Model and Mesh Generation of Cracked Tubular Y-Joints[END_REF][START_REF] Lie | Mesh Modelling and Analysis of Cracked Uni-Planar Tubular K-Joints[END_REF], as described in the introduction section of the paper, where the joint is divided into zones, meshed zone by zone, and afterwards merged together for analysis.

The computational model of the considered junction is shown in Fig. 16. Weld joints are marked with numerals from 1 to 3. In the present analysis only weld 1 is taken into consideration. The semi-elliptical surface crack is introduced into the model near the weld toe of the vessel-nozzle junction, where a high stress concentration exists. In [START_REF]Recommended practice for fitness-for-service[END_REF], this type of crack which is located on the outer surface is labeled as type D. In order to have accurate results from the global (shell) model, the optimal submodel size for the vessel-nozzle junction (Fig. 17c) is selected on the basis of the error analysis on the extracted submodel displacement boundary conditions. In Fig. 18 variations of the difference in radial displacements distribution of mid surface along a longitudinal and circumferential cross-section of the pressure vessel obtained from the shell model are presented and compared to the full 3D solid FEA. Thereby, the crack and the weld are not taken into account in the shell model (Fig. 17a andb). On the other hand, the 3D model mesh models the crack and the ring of welded material accurately to include the welded joint (Fig. 17d). The comparison is expressed as a percent difference that is defined as

,shell ,3D ,3D 100 ( ) / r r r u u u 
. It can be concluded from the results that a submodel size s equal to 10 times the size of the wall thickness t 1 is sufficiently large for the SIF analysis. The error thereby introduced in the radial displacement values is up to around 1.5%. It should be noted that this is true when the crack in the weld toe is small. Additionally it can be concluded from Fig. 18 that the submodel size in circumferential direction may be somewhat smaller than in the longitudinal. The numerical results and submodeling technique are verified with the results from [START_REF]Recommended practice for fitness-for-service[END_REF]. Using API code [START_REF]Recommended practice for fitness-for-service[END_REF], K I can be calculated by means of the following equation:

I 0 ( , ) ( ) a K h x a x dx    , (4)
where h(x, a) is the weight function, and σ(x) is the stress normal to the crack plane but when the junction is in uncracked state. Coordinate x is the through-thickness distance measured from the outer surface of the vessel. A comparison between numerical results obtained by using the submodeling technique and the results calculated by means of Eq. ( 4) is shown in The results are in good agreement, so it is obvious that the submodeling technique is an efficient technique, especially for complicated geometries like in this case.

Table 4 Comparison of the SIF results between the present work and [20]

K I (MPa mm ) φ = 0° φ = 90° φ = 180° API

Elastoplastic fracture mechanics analysis using submodeling

In the previous section, the SIF calculation is performed by using the FE submodeling technique. In this section, the intention is to verify the submodeling technique in the elastoplastic J ep -integral evaluation for a pipe with an external axial surface crack considered in the section 2.2.

J estimation by the GE/EPRI method

In the structural integrity assessment procedures the values of the elastoplastic J epintegral are obtained by using the deformation theory of plasticity, where the stressstrain response in the FE analyses is described by the well-known Ramberg-Osgood model in the following form:

0 y y n             . ( 5
)
In Eq. ( 5), 0  denotes the associated reference strain 0y / E   while the values  and n represent the parameters fitting the experimentally obtained curve. With the method developed by the General Electric/ Electric Power Research Institute (GE/EPRI) , the total crack driving force J ep can be divided into elastic, J el , and plastic, J pl , parts, as

ep el pl J J J  . (6)
The elastic component of the J ep -integral, J el , can be estimated from the SIF with a plasticity correction [START_REF] R6 | Assessment of the Integrity of Structures Containing Defects[END_REF] as follows:

  2 e el , Ka J E    , (7)
where

              . ( 8
)
In Eq. ( 8), p L is the plastic limit pressure that produces the plastic collapse of the surface cracked pipe for a rigid-plastic, non-hardening material with the yield stress y  [START_REF] Miller | Review of Limit Loads of Structures Containing Defects[END_REF][START_REF] Tonković | Influence of Flow Stress Choice on the Plastic Collapse Estimation of Axially Cracked Steam Generator Tubes[END_REF]. The empirical expression for the estimation of the plastic limit pressure for the considered pipe and crack geometry in terms of the non-dimensional crack configuration parameters a/t and ρ was proposed by the authors [START_REF] Tonković | Influence of Flow Stress Choice on the Plastic Collapse Estimation of Axially Cracked Steam Generator Tubes[END_REF][START_REF] Tonković | Numerical Modelling of Deformation Responses of Cracked Tubes[END_REF]. The basic equations are compiled in Table 5. In the previously mentioned references, extensive

FEAs have been performed to validate the results of the plastic limit pressure by comparing the computed values with the existing solutions [START_REF] Kim | Plastic Limit Pressures for Cracked Pipes Using Finite Element Limit Analyses[END_REF][START_REF] Carter | A Library of Limit Loads for FRACTURE TWO[END_REF][START_REF] Staat | Local and Global Collapse Pressure of Longitudinally Flawed Pipes and Cylindrical Vessels[END_REF]. As shown, the analytical approximation of the plastic limit pressure for thick-walled pipes is applicable to a wide range of crack dimensions.

Table 5 Basic equations for the estimation of the plastic limit pressure [START_REF] Tonković | Influence of Flow Stress Choice on the Plastic Collapse Estimation of Axially Cracked Steam Generator Tubes[END_REF] Plastic limit pressure: Dimensionless parameter which is a function of crack configuration parameters a/t and ρ:

) , / ( 0 L  t a P p p   . ( 9
2 2 1 1               
          P . ( 11
)
Nonlinear FE analysis has been performed to evaluate the J-integral. The values of the J-integral are computed around 5 contours surrounding the crack tip. The result from the 1st contour closest to the crack tip is discarded, and the J-integral value is the average of all the values obtained on the 2nd to 5th contour. The submodeling capability is now used to analyze the crack-tip region when the material behavior is elastoplastic. In the present investigation, the FE solutions for elastic and fully plastic J-integral values are first obtained for an axially cracked pipe, and are then combined, using GE/EPRI and the reference stress method [START_REF] Ainsworth | The Assessment of Defects in Structures of Strain Hardening Material[END_REF], to obtain approximate values of the J ep -integral at all load levels up to the limit load. In order to validate the model, the mesh convergence is investigated to describe the fully plastic flow field accurately. A total of 160 cases are considered. Herein, values of the strain hardening index n are systematically varied: n = 3, 5, 7 and 10, while parameter  is fixed to  = 1.

Additionally, in order to investigate the dependence of the J ep -integral results on the submodel size, ten different submodel size-to-crack length ratios of 2s/(2c) = 2, 3, 4, 6, 8, 10, 12, 14, 16 and 18 are considered. The results for the J ep -integral values at the deepest crack front location (φ = π/2), as functions of applied pressure, from the submodeled analyses are compared to the full solid (3D) element mesh. Figure 19 shows the relative error that is defined as

3D 3D 100 ( ) / J J J 
. As well known, if the plastic zone size is less than about 10% of the crack length, small-scale yielding conditions exist around the crack tip. In that case, the far-field elastic region is not affected by the plastic zone. In our case, this will be true for the relatively small pressures that cause the sufficiently small plastic zones. As may be observed from Fig. 19, with a further pressure increase, larger plastic strains occur on the model and a submodel size 2s equal to 2.5 times the size of the crack length 2c is not sufficiently large for analysis. For the load levels of up to 180% of limit load, the J value is stabilized for the ratio of 2s/(2c) equal to 4. It is valid only for the pipe with i / 10 Rt  (Fig. 19a). However, for very thick pipe with R i /t = 4 there is no significant stabilization of the J value (Fig. 19a). Since the differences in results are generally lower for submodel size of s/c = 4 then for other submodels, the submodel size of s/c = 4 can be taken as relevant for submodeled elastoplastic analysis of cracked pipes. For this reason, submodeled analyses for calculation of the J-integral are performed by using the submodel size-to-crack length ratio of 4.

From Fig. 19 it can also be concluded that the difference between the results obtained by submodeled and full solid element analysis becomes larger with increasing pressure. Thus, for the larger submodels the maximum differences in results are 1.3%

for pipe with R i /t = 10 and 4% for R i /t = 4. These differences are connected with the modeling of the spread of the plastic zone through the thickness of the pipe and different finite element formulations of shell and solid elements applied to perform submodeled and full solid element analysis. In the present example, the global model for the submodeling analysis is meshed with S8R thick shell elements (Fig. 5a) with five integration points through the thickness. On the contrary, the full solid element analysis is performed by using a single C3D20R element with reduced integration (2x2x2) through the thickness (Fig. 7) further away from the crack.

Fig. 19

Dependence of the J ep -integral results on the submodel size:

a) i / 10, / 10, / 0.4 R t c a a t    , b) i / 4, / 5, / 0.4 R t c a a t   
Following reference [START_REF] Kumar | Elastic-Plastic Fracture Analysis of Through-Wall and Surface Flaws in Cylinders[END_REF], the plastic part of J ep -integral, J pl , for semi-elliptical surface cracked pipes is given by   [START_REF] Berg | Two-parameter fracture mechanics and circumferential crack growth in surface cracked pipelines using line-spring elements[END_REF] and pipe geometry. In order to obtain the values of the function   1i / , / , / , h R t a t c a  , the plastic part of J ep integral is determined by

1 2 y i pl 1 L , , , , , n p R a c J t a h n E p t t a           
pl_FE FE el J J J . (13)
It is to be noted that the value of h 1 depends on the load magnitude, as shown in Fig. 20.

Since the power-law part of Eq. ( 5) describing the material behavior during the loading process is dominant at sufficiently large loads, the values of h 1 (a/t, c) at high loads should be taken from appropriate diagrams [START_REF] Kim | Elastic-Plastic Fracture Mechanics Method for Finite Internal Axial Surface Cracks in Cylinders[END_REF]. The values of h 1 (a/t, c) obtained from the submodel analysis for the considered pipe geometry and ratio of 2s/(2c) equal to 4 at the deepest crack front location ( /2

 

) are presented in Table 6. Thus, according to the GE/EPRI method, the J ep -integral can be estimated by using Eqs ( 5) to ( 13) and the data given in Tables 3 and6.

Fig. 20 Dependence of the plastic influence h 1 -function on the load magnitude

( 5 / , 4 / i   a c t R ): a) 2 . 0 /  t a , b) 6 . 0 /  t a

J estimation by the reference stress method

As presented, the J estimation based on GE/EPRI method requires the Ramberg-Osgood coefficient and strain hardening index as basic input to represent material tensile data.

The disadvantage here is that Ramberg-Osgood fitting of the stress-strain curve can be seriously inaccurate, leading to inaccuracy in the estimated J [START_REF] Kim | Elastic-Plastic Fracture Mechanics Method for Finite Internal Axial Surface Cracks in Cylinders[END_REF][START_REF] Chattopadhyay | Improved J and COD estimation by GE/EPRI method in elastic to fully plastic transition zone[END_REF]. The limitation of this method is that plastic influence functions should be found, which requires detailed 3D elastoplastic FE analyses [START_REF] Kim | Elastic-Plastic Fracture Mechanics Method for Finite Internal Axial Surface Cracks in Cylinders[END_REF][START_REF] Kumar | Elastic-Plastic Fracture Analysis of Through-Wall and Surface Flaws in Cylinders[END_REF][START_REF] Chattopadhyay | Improved J and COD estimation by GE/EPRI method in elastic to fully plastic transition zone[END_REF][START_REF] Kim | Elastic-Plastic J and COD Estimates for Axial Through-Wall Cracked Pipes[END_REF]. Thereby, the solutions for plastic influence functions, available in fracture mechanics handbooks, are limited to a relatively simple geometrical configuration and dimensions of both the crack and the structural element under simple loading conditions. For more realistic and complex crack geometries and loading conditions, the GE/EPRI J estimation method is impractical for engineering application. Additionally, Kim et al. [START_REF] Kim | Elastic-Plastic Fracture Mechanics Method for Finite Internal Axial Surface Cracks in Cylinders[END_REF] and Chattopadhyay et al. [START_REF] Chattopadhyay | Improved J and COD estimation by GE/EPRI method in elastic to fully plastic transition zone[END_REF] showed that GE/EPRI method over-estimates the plastic J with respect to the incremental plasticity FE solutions. For application to general stress-strain laws, the J ep estimation equation is now re-formulated by using the reference stress method in the form [3, 4, , determined from the true-stress strain data, while p ref stands for the reference pressure. Unlike the conventional reference stress method [START_REF] Ainsworth | The Assessment of Defects in Structures of Strain Hardening Material[END_REF][START_REF] R6 | Assessment of the Integrity of Structures Containing Defects[END_REF] where the reference stress is defined by using the plastic limit load, the enhanced reference stress method (ERSM), proposed in [START_REF] Kim | Elastic-Plastic Fracture Mechanics Method for Finite Internal Axial Surface Cracks in Cylinders[END_REF], defines the reference stress with the optimized reference load. The optimized reference load is found to give the best estimates of J.

Detailed information on the proposed method can be found in Ref. [START_REF] Kim | Elastic-Plastic Fracture Mechanics Method for Finite Internal Axial Surface Cracks in Cylinders[END_REF][START_REF] Kim | Elastic-Plastic J and COD Estimates for Axial Through-Wall Cracked Pipes[END_REF]. Based on the FE results from the submodeled analyses and the mentioned enhanced reference stress concept, the empirical expression for the estimation of the optimized reference pressure p ref for pipes with external axial semi-elliptical surface cracks is derived as follows:

  1 1 1 ref 0 1 ( 1) /, () n L hn p p p S a t hn           , (15)
where p L and p 0 are defined by Eqs ( 9) and [START_REF] Rice | The part-through surface crack in an elastic plate[END_REF]. S is the dimensionless factor which is a function of non-dimensional crack configuration parameters a/t and ρ given by:

2 1 2 1 aa S S S tt                , (16)
where [START_REF] Tonković | Influence of Flow Stress Choice on the Plastic Collapse Estimation of Axially Cracked Steam Generator Tubes[END_REF]

S S                     (17)
The reference stress-based J ep estimation (Eq. 14) is compared with the GE/EPRI solutions as well as with FE results where the non-linear material behavior is modeled by using incremental plasticity with the von Mises yield function, together with the associated flow rule and isotropic hardening assuming small strains. The material data for the austenitic steel 08X18H10T that corresponds to the German grade 1.4541 (equivalent to AISI 321), used in the analysis and determined experimentally in a standard tensile test, are taken from [START_REF] Norms | Norms for the Calculation of Strength of the Equipment and the Pipelines of Nuclear Power Installations (PNAE G-7-002-86)[END_REF]. The basic material parameters are summarized in Table 8 for 300 C. Figure 21 displays the engineering stress-strain curve (σ e , ε e ). The true stress and strain values (σ t , ε t ) used in the incremental plasticity analysis are determined by the following relationships:

  e e t 1      , (18)
 

e t 1 ln     . ( 19
)
The values of both the engineering terms and true terms are shown in Table 9. The data is given up to the ultimate tensile stress level (25% of strain). After this specified point, the response is assumed to be perfectly plastic. The influence of the choice of the submodel size on the SIF results is investigated.

Together with the submodel size study, the effect of the error caused by the same pressure defined on the inner radius of the 3D solid FE model as on the mean radius of the shell model is analyzed. It is concluded that the choice of the submodel can compensate for pressure definition error and that there is no need for pressure correction. In accordance with literature results [START_REF] Diamantoudis | Stress Intensity Factors of Semi-Elliptical Surface Cracks in Pressure Vessels by Global-Local Finite Element Methodology[END_REF], the study also concluded that a submodel size equal to 2.5 times the size of the crack length is sufficiently large for the linear elastic analysis. Finally, the semi-elliptical surface crack is introduced into the real model near the weld toe of the vessel-nozzle junction, where a high stress concentration exists. The submodeling technique, based on the transition from the global shell model to the three-dimensional submodel discretized with a finer mesh of elements, is proven to be very efficient for this purpose.

As the last example, in order to verify the submodeling technique in the elastoplastic J-integral evaluation, a thick-walled pipe with an external axial surface crack under internal pressure is considered. The dependence of the J-integral results on the submodel size is investigated for the load levels up to the limit load. In addition, the submodel mesh convergence is analyzed to describe the fully plastic flow field accurately. It is shown that, unlike the LEFM problems, the submodel size must be minimum 4 times the size of the crack length. Based on the FE results from the submodel analyses and by using the enhanced reference stress concept, an empirical expression for the estimation of the optimized reference pressure for cracked pipes is proposed. The proposed SIFs, J-integral and the analytical approximation of reference pressure for the considered pipe and crack geometries are very useful tools for assessing the integrity of pressurized pipes. [START_REF]Recommended practice for fitness-for-service[END_REF] Table 5 Basic equations for the estimation of the plastic limit pressure [START_REF] Tonković | Influence of Flow Stress Choice on the Plastic Collapse Estimation of Axially Cracked Steam Generator Tubes[END_REF] Table 6 Values of the plastic influence h 1 -functions Table 7 Summary of computational requirements for the elastoplastic analysis of the pipe with an external axial surface crack Table 8 Mechanical properties of the austenitic steel 08X18H10T at 300°C [START_REF] Norms | Norms for the Calculation of Strength of the Equipment and the Pipelines of Nuclear Power Installations (PNAE G-7-002-86)[END_REF] Table 9 Material data of the austenitic steel 08X18H10T at 300 C [START_REF] Norms | Norms for the Calculation of Strength of the Equipment and the Pipelines of Nuclear Power Installations (PNAE G-7-002-86)[END_REF]

Fig. 1 AFig. 2

 12 Fig.1A membrane with a semi-elliptical crack subjected to tension: a) geometry,

Fig. 3 A

 3 Fig. 3 A solid reference model of the cracked membrane

Fig. 4 A

 4 Fig.[START_REF] R6 | Assessment of the Integrity of Structures Containing Defects[END_REF] A pipe with an external axial surface crack subjected to internal pressure: a)

  5b

Fig. 5 Fig. 6

 56 Fig. 5 Typical FE meshes for a pipe: a) global (shell) model mesh without a crack, b)

Fig. 9 Fig. 10 Fig. 11 Fig. 12

 9101112 Fig. 9 Radial displacements distribution of inner surface of the pipe along a

Fig. 13

 13 Fig.[START_REF] Haryadi | Global/local analysis of composite plates with cracks[END_REF] Submodel and full 3D model without crack under the same pressure:

Fig. 15 A

 15 Fig. 15 A comparison of the dimensionless function F for the stress intensity factor

Fig. 16 Fig. 17

 1617 Fig. 16 Computational model of the vessel-nozzle junction with a crack in the weld toe

Fig. 18

 18 Fig.[START_REF] Mikhaluk | Experimental observations and finite element modelling of damage initiation and evolution in carbon/epoxy non-crimp fabric composites[END_REF] Variations of the difference in radial displacements distribution of mid surface

  ) Limit pressure of an un-cracked thick-walled pipe based on the Von Mises yield criterion:

Fig. 21

 21 Fig.[START_REF] Raju | Stress-Intensity Factors for Internal and External Surface Cracks in Cylindrical Vessels[END_REF] Engineering stress-strain curves of the 08X18H10T steel at 300 C[START_REF] Norms | Norms for the Calculation of Strength of the Equipment and the Pipelines of Nuclear Power Installations (PNAE G-7-002-86)[END_REF]

Figure captions:Fig. 1 AFig. 2

 12 Figure captions:

Fig. 3 AFig. 4 AFig. 5 Fig. 6 Fig. 7 AFig. 8 Fig. 9 Fig. 10 Fig. 11 Fig. 12 Fig. 13 Fig. 15 A 21 ]Fig. 16 Fig. 17 Fig. 18 Fig. 19 Fig. 20 Fig. 21

 3456789101112131521161718192021 Fig. 3 A solid reference model of the cracked membrane

Fig. 22

 22 Fig.[START_REF] Miller | Review of Limit Loads of Structures Containing Defects[END_REF] Comparison of FE and GE/EPRI J results with the proposed reference

Table 1

 1 Checking the validity of the submodeling technique

	K I (MPa mm )	φ = 0°	φ = 90°
	API 579 [20]	127.8	156.4
	submodeled FE analysis	123.4	152.8

Table 2

 2 FE models considered in the present analyses

	FEM model no.	R i /t	c/a	a/t	FEM model no.	R i /t	c/a	a/t
	1	4	5	0.2	17	10	5	0.2
	2			0.4	18			0.4
	3			0.6	19			0.6
	4			0.8	20			0.8
	5		10	0.2	21		10	0.2
	6			0.4	22			0.4
	7			0.6	23			0.6
	8			0.8	24			0.8
	9		15	0.2	25		15	0.2
	10			0.4	26			0.4
	11			0.6	27			0.6
	12			0.8	28			0.8
	13		20	0.2	29		20	0.2
	14			0.4	30			0.4
	15			0.6	31			0.6
	16			0.8	32			0.8

  the considered pipe geometry and the ratio of 2s/(2c) equal to 2.5 are tabulated in Table3. As the SIF values reach the largest values at the deepest crack front location ( = π/2), only the results at that location are given.

	submodeled analysis for
	4. The values of function	F	( m R	/	t	,	a	/	t	,		,		)	obtained from the

Table 3

 3 Dimensionless function F for the stress intensity factor

	R m /t			4					10
	a/t	5	10	c/a	15	20	5	10	c/a	15	20
	0.2 0.921 0.993 1.019 1.030 1.031 1.117 1.151 1.168
	0.4 1.171 1.350 1.413 1.443 1.280 1.516 1.615 1.667
	0.6 1.593 1.966 2.098 2.165 1.693 2.235 2.486 2.614
	0.8 2.187 2.895 3.138 3.256 2.200 3.346 3.911 4.230

  In order to validate this assumption, the error analysis on the extracted submodel displacement boundary conditions is performed. In Figs 9 and 10 radial displacements distribution of inner surface of the pipe along a longitudinal and circumferential crosssection obtained from the FE submodeling approach is presented and compared to the full 3D solid and shell models. Responses in hoop stresses are plotted in the Figs11 and 12. The results given in this graphs refer to i / 10

						Rt  , / ca  , / 0.4 10 at	, t = 1.625
	mm, /2 sc  and 3. As may be observed from Figs 9 to 12, the results obtained by the
	submodel ratio of 2s/(2c) equal to 2 deviate significantly. On the other hand, for the
	submodel ratio of 2s/(2c) equal to 3 the displacements and hoop stresses show very little
	difference between the two analysis.
	R t		10, / c a		10, / 0.4 a t 	, b) i / 4, / R t c a 		5, / 0.4 a t 

  obtained by submodeling analysis (driven with higher global displacements) are not overestimated compared with cracked full 3D model. Described effect of displacement mismatch and submodel bending and it's influence to hoop stress is more obvious in thicker pipes. For pipe with radius-to-thickness ratio R i /t = 4, in order to have equivalent submodeling analysis to full 3D, one have to build bigger submodel (Fig.14), say s/c = 4. In addition it can be concluded from Figs 9. to 12. that the submodel size in circumferential direction may be smaller than in the longitudinal. So for the pipe with ratio of R i /t = 10 optimum submodel size in circumferential direction is

		circ /2 sc  ,
	and in longitudinal	long /3 sc  .

3.4% and 8.9% respectively. For the hoop stress this difference is 4.8% and 11.4% for ratio 10 and 4 respectively. In order to study this effect, in Figs 9 to 12 the results for the radial displacements and hoop stresses are also presented for the submodel without crack. Internal pressure acting on the internal surface of the 3D submodel results in smaller load than in the shell's mid surface. Due to smaller load on inner surface of 3D submodel without crack, displacement distribution (Fig.

9

) underestimate global, shell membrane displacement which is used as a submodel boundary condition. This displacement mismatch causes submodel bending shown in Figs

13a and b

. Besides shell and submodel without crack, in Fig 13a deformed configuration of full 3D model without crack is shown. Even though the radial displacement of the global model are too high and causes submodel bending, when crack is introduced in submodel firstly unbending and then opening happens. That is why the displacement and hoop stress and finally SIF

Table 4 .

 4 In this table

		  0	represents the right point (far from the

Table 6

 6 Values of the plastic influence h 1 -functions

	R i /t c/a a/t	3	5	n	7	10	R i /t c/a a/t	3	5	n	7	10
	4	5 0.2 1.358 1.737 2.020 2.370 10 5 0.2 1.521 1.872 2.161 2.540
		0.4 5.563 7.228 8.357 9.482	0.4 6.424 8.254 9.739 11.386
		0.6 15.943 18.571 19.581 20.009	0.6 19.903 23.354 25.421 26.766
		0.8 37.093 38.727 38.132 34.822	0.8 51.712 57.183 60.255 59.333
		10 0.2 1.570 2.075 2.449 2.868	10 0.2 1.803 2.299 2.688 3.174
		0.4 6.556 8.604 9.708 10.356	0.4 8.395 11.071 12.818 14.223
		0.6 16.592 17.815 17.681 17.112	0.6 26.156 29.434 31.093 32.007
		0.8 29.855 27.973 25.438 21.583	0.8 57.742 58.603 60.943 60.396
		15 0.2 1.581 2.102 2.483 2.879	15 0.2 1.865 2.405 2.819 3.324
		0.4 6.312 8.256 9.146 9.504	0.4 8.678 11.579 13.262 14.492
		0.6 14.925 15.394 14.875 14.002	0.6 25.213 27.009 27.327 27.030
		0.8 27.514 27.749 27.309 25.622	0.8 47.198 45.301 45.762 43.527
		20 0.2 1.536 2.033 2.392 2.743	20 0.2 1.850 2.388 2.792 3.272
		0.4 5.958 7.772 8.554 8.803	0.4 8.387 11.241 12.750 13.676
		0.6 14.133 14.476 14.087 13.490	0.6 23.587 24.393 24.021 23.255
		0.8 22.731 22.968 22.615 21.040	0.8 43.821 42.614 44.701 45.104

A summary of the typical computational requirements for the full 3D solid and submodeled elastoplastic FE analysis of the pipe with an external axial surface crack is given in

Table 7 .

 7 The results given in this table refer to i / 10

	Rt  , / ca  , / 0.4 10 at	,

Table 7

 7 Summary of computational requirements for the elastoplastic analysis of the pipe with an external axial surface crack

		Measures of computational effort
	Model	Total number of degrees of freedom	CPU s	Size of data library, (megabytes)
	Full 3D solid model	76461	686	38.0
	Global shell model + 3D submodel	55052	398	22.7

Table 8

 8 Mechanical properties of the austenitic steel 08X18H10T at 300°C[START_REF] Norms | Norms for the Calculation of Strength of the Equipment and the Pipelines of Nuclear Power Installations (PNAE G-7-002-86)[END_REF]

	Temperature, T (C)	Modulus, E (GPa)	Yield stress, σ y (MPa)	Ultimate stress, σ u (MPa)		n
	300	184	160	420	1.375	4.01

Table 9

 9 Material data of the austenitic steel 08X18H10T at 300 C[START_REF] Norms | Norms for the Calculation of Strength of the Equipment and the Pipelines of Nuclear Power Installations (PNAE G-7-002-86)[END_REF] ConclusionsIn the present investigation the shell-to-solid submodeling capability is used to obtain a more accurate crack tip stress field in the linear elastic and nonlinear elastoplastic fracture mechanics problems. Computational efficiency of the submodeling technique On the Calculation of Stress Intensity Factors and J-Integrals Using the Submodeling Technique PVT -09-1091, Tonković 35 for SIF calculation is demonstrated by three numerical examples. The first one is a membrane with a semi-elliptical crack under tension, and the other one is a pressurized thick-walled pipe with a finite external axial surface crack. These two examples are used to verify submodeling in the SIF calculation with the results from the literature.

	Engineering stress,	Engineering plastic	True stress	True plastic strain,
	σ e (MPa)	strain, ε e,pl	σ t (MPa)	ε t,pl
	160	0.000	160	0.000
	184	0.006	186	0.006
	223	0.016	226	0.016
	251	0.026	258	0.026
	283	0.042	295	0.041
	312	0.059	330	0.057
	344	0.079	371	0.076
	365	0.098	401	0.094
	413	0.198	495	0.181
	420	0.248	524	0.221
	As may be observed from Fig. 22, excellent agreement of the two RSM and FE
	solutions is exhibited. On the other hand, the results obtained by the GE/EPRI deviate
	significantly.

Table captions :Table 1

 captions1 Checking the validity of the submodeling technique

Table 2

 2 FE models considered in the present analyses

Table 3

 3 Dimensionless function F for the stress intensity factor

Table 4

 4 Comparison of the SIF results between the present work and

Acknowledgements

The authors express their gratitude to the Ministry of Science, Education and Sports of the Republic of Croatia for their financial support.


Quentin Wu
email: quentin.wu@zodiacaerospace.com

Sophie Boudau
email: sophie.boudau@zodiacaerospace.com

David Gouyon
email: david.gouyon@univ-lorraine.fr

Eric Levrat
email: eric.levrat@univ-lorraine.fr

Éric Levrat

A review of Know-How reuse with patterns in Model-Based Systems Engineering

The increasing complexity of systems to be developed requires engineers to review their practices in order to improve the efficiency of engineering and meet the needs of a competitive market. That is why models supported by formal or semi-formal languages are preferred to avoid the understanding variability of natural languages. In this context, Model-Based Systems Engineering (MBSE) made it possible to change the engineering paradigm by putting forward a unique, shared system model. To promote its adoption, a solution would be to allow reuse of knowledge and know-how, to encourage engineers seizing and adapting MBSE to their needs. This paper aims to review and evaluate the concept of patterns towards reuse in engineering, especially in a MBSE approach.

Introduction

The design of increasingly complex systems is implicating longer engineering phases and greater costs during the design lifecycle of a project. Those negative impacts are accentuated by the current document-centred application of Systems Engineering (SE) processes inside companies. Indeed, system development teams are working on standalone models, communicating with other teams through documents written in natural language. This implies a huge work concerning consistency and comprehension, as information shared through those documents has to be comprehensive and unique, to avoid rework and non-conformity to customer expectations. So, there is a challenge concerning the engineering efficiency (how to enhance productivity, quality, communications, and reduce risk) needed in a highly competitive environment, where the need is to shorten engineering cycle period.

In order to manage complexity, maintain consistency, and ensure traceability during systems engineering, the SE community has turned to the Model-Based Systems Engineering (MBSE) [START_REF] Estefan | Survey of Model-Based Systems Engineering ( MBSE ) Methodologies[END_REF]. Popularized by the International Council on Systems Engineering (INCOSE) with the MBSE Initiative1 , MBSE is defined as "the formalized application of modelling to support system requirements, design, analysis, verification and validation activities beginning in the conceptual design phase and continuing throughout development and later life cycle phases". However, adoption of MBSE takes time, as many inhibitors remain such as cultural and general resistance to change, lack of related Knowledge and Know-How (K&KH), or the need to higher degree of guidance and reuse.

For a wider MBSE adoption, several advances seem to be necessary concerning organizational, methodological, and tools perspectives. In particular, from a methodological point of view, reuse seems to be promising. Reusing engineer's Knowledge and Know-How is an act of capitalization on previous experiences or projects, whether it is on the System Of Interest (SOI) or on the Systems Engineering Activities (SEA). But, often, those data are kept in their mind, and works have to be done to formalize them, with the goal of sharing them so they can be reused. The expected benefits make the assumption that reused modelling artefacts satisfy some maturity criteria to grant that they have reached a level of quality compatible with reuse objectives. This article reviews engineering practices which intent to capitalize on K&KH, and to facilitate information sharing and reuse. A focus is made on reusing K&KH through the concept of "pattern". In this way, the second section presents reuse challenges in engineering and related works, the third a short history on patterns, the fourth a literature review of pattern for SE, and the last section discusses on the interest of using patterns in MBSE.

Challenges and related works

The fundamental difference between knowledge and know-how is that knowledge provides only solutions and answers to problems and questions, whereas knowhow provides solutions but also a manner to construct these solutions [START_REF] Gzara | Product Information Systems Engineering: An Approach for Building Product Models by Reuse of Patterns[END_REF]. Thereby, engineer's know-how is built from their experience, allowing them to reuse information gathered in order to be more efficient in their tasks. However, those "archives" are stuck in engineers' mind, making it difficult to share them to someone else to foster reuse [START_REF] Mourtzis | An Inference-Based Knowledge Reuse Framework for Historical Product and Production Information Retrieval[END_REF][START_REF] Demian | An Ethnographic Study of Design Knowledge Reuse in the Architecture, Engineering, and Construction Industry[END_REF]). Yet, dynamic information flowing among engineering teams is a critical challenge for many companies who need to manage complex systems as information must be shared, comprehensive, and coherent among the project [START_REF] Miled | Reusing Knowledge Based on Ontology and Organizational Model[END_REF]). This aspect is very important as it allows a better comprehension of the SOI and SEA. For example in requirement engineering, [START_REF] Darimont | Deploying a Template and Pattern Library for Improved Reuse of Requirements Across Projects[END_REF] presents the results of a survey where 55-60% of engineers use "copy & paste" and "duplication" techniques, and only 13% use "requirements patterns catalogue". As the complexity, the quantity of K&KH and also engineering artefacts are exploding, those practices are no longer sufficient to answer challenges of nowadays complex system development. That is why there is a need to promote efficient way to transfer K&KH, in order to facilitate their circulation and reuse, and this is why current expectations are to promote models over natural language and its variability of understanding.

Research works have already been done for reuse K&KH in SE [START_REF] Bollinger | Facilitating Model Reuse and Integration in an Urban Energy Simulation Platform[END_REF][START_REF] Barter | A Systems Engineering Pattern Language[END_REF]Cloutier 2008;[START_REF] Cook | Utilizing Mbse Patterns To Accelerate System Verification[END_REF][START_REF] Gautam | Design Reuse Framework : A Perspective for Lean Development[END_REF][START_REF] Gzara | Product Information Systems Engineering: An Approach for Building Product Models by Reuse of Patterns[END_REF]Haskins 2005;[START_REF] Korff | Re-Using Sysml System Architectures[END_REF][START_REF] Wang | Reuse in Systems Engineering[END_REF], but as there are many different ways to capitalize K&KH, it is important to define the targeted perimeter or the engineering artefacts before considering reusing. Indeed, "reuse" activities in SE can be distinguished in three different approaches: Opportunistic reuse: when the first project was not developed with reusable capacity; Planned reuse: when the first project was developed with reusable capacity; Variance: on a product line, common core model but different options. Those approaches belong to the process of "knowledge transfer" which consists of two sub-processes defined by [START_REF] Majchrzak | Knowledge Reuse for Innovation[END_REF]). On the one hand, the process by which an entity's K&KH is captured, called "knowledge sharing", and on the other hand, the process by which an entity is able to locate and to use K&KH, called "knowledge reuse". It is important to ensure that the engineering artefact on which a reuse solution is applied may be the SOI or SEA [START_REF] Pfister | A Proposed Meta-Model for Formalizing Systems Engineering Knowledge, Based on Functional Architecture Patterns[END_REF].

Within existing reuse approaches for the SOI, the use of Components Off The Shelves (COTS) consists in breaking down a problem into solvable sub-problems by already existing components. However, the advantages of COTS are accompanied by integration issues, early identified by [START_REF] Boehm | COTS Integration: Plug and Pray?[END_REF] which are: functionality and performance (what it is expected to do), interoperability (no standard exists), product evolution (risk of no longer meeting the need) and vendor behaviour (false promises). For the reuse in SEA which aim to produce the SOI, [START_REF] Darimont | Deploying a Template and Pattern Library for Improved Reuse of Requirements Across Projects[END_REF]) deployed a local reuse library for each engineers and a shared reuse library for improving reuse of requirements across projects, and [START_REF] Majchrzak | Knowledge Reuse for Innovation[END_REF]) identifies a six-stage reusefor-innovation process, with the capacity to quickly capture and present information on potentially reusable ideas. Other works both addressed the SOI and SEA, such as the extension of the Constructive Systems Engineering Cost Model (COSYSMO) by [START_REF] Wang | Reuse in Systems Engineering[END_REF] that consists in defining reuse categories and weights for each of the category. While COSYSMO considers the whole, the PABRE approach focuses on requirements management [START_REF] Palomares | Requirements Reuse with the PABRE Framework[END_REF] and is based on a metamodel of Software Requirements Patterns (SRP), a method of reuse, a catalogue of 111 SRP, and a software tool that supports the management and the use of the catalogue.

As shown, many research works are looking to reuse K&KH to improve engineering efficiency. One way that looks particularly promising is achieved through the adoption of patterns, for both SEA and SOI, to systematize complex systems engineering [START_REF] Cochard | Contribution à La Génération de Séquences Pour La Conduite de Systèmes Complexes Critiques[END_REF]. As they can be used in all stages of the development cycle [START_REF] Gzara | Product Information Systems Engineering: An Approach for Building Product Models by Reuse of Patterns[END_REF], reuse of patterns seems to be a very suitable form of reuse [START_REF] Schindel | Requirements Statements Are Transfer Functions: An Insight from Model-Based Systems Engineering[END_REF].

A little history of patterns

Most people in the pattern community attribute the first promoter of the value of "pattern" to [START_REF] Alexander | A Pattern Language[END_REF] in a book on architecture, urban design and community liveability. They formalized a "pattern language", made of a myriad of patterns that helped them to express design in terms of relationships between the parts of a house, and the rules to transform those relationships [START_REF] Coplien | Idioms and Patterns as Architectural Literature[END_REF]. They began to identify patterns with the idea that "Each pattern describes a problem which occurs over and over again in our environment, and then describes the core of the solution to that problem, in such a way that you can use this solution a million times over, without ever doing it the same way twice" [START_REF] Alexander | A Pattern Language[END_REF]. The same way engineers reuse their knowledge based on their previous experience, [START_REF] Cloutier | Applicability of Patterns to Architecting Complex Systems[END_REF] point out that Alexander and his co-authors "did not invent these patterns, they came from observation and testing; and only then were they documented as patterns".

Since these pioneer works, the pattern approach has been introduced in various engineering fields such as Software, Requirements, Telecommunications and Control Systems Engineering [START_REF] Cloutier | Applicability of Patterns to Architecting Complex Systems[END_REF]. [START_REF] Beck | Using Pattern Languages for Object-Oriented Programs[END_REF] were the first to propose object-oriented patterns in the Software community. The goal was to improve quality and to facilitate code writing by adopting good practices. [START_REF] Gamma | Design Patterns: Elements of Reusable Object-Oriented Software[END_REF], also known as the "Gang of Four", wrote an authoritative book describing 23 Software Design Patterns such as Composite, Iterator, Command… A Design Pattern is a general, reusable solution to a recurring problem in the design of object-oriented applications; it describes a proven solution for solving software architecture problems. As Design Patterns are not a finished design (concrete algorithm), but a structured description of computer programming, it means they are independent from programming languages. Design Patterns have been widely accepted, and encouraged other domains to write patterns to capture their experience.

In the field of SE, the value of patterns appears towards the growing complexity of systems and the difficulty to capture large body of knowledge. That is why [START_REF] Barter | A Systems Engineering Pattern Language[END_REF] proposes the creation of a Systems Engineering Pattern Language, which is a collection of patterns that, when combined, address problems larger than the problems that an individual pattern can address. In the same way, [START_REF] Haskins | 1.1.2 Using Patterns to Share Best Results -A Proposal to Codify the SEBOK[END_REF] proposes the use of SE patterns to capture the information in the Systems Engineering Body of Knowledge (SEBOK). Other works have used the concept of pattern in SE, especially in the Product Information System field, where [START_REF] Cauvet | Ingénierie Des Systèmes d'information Produit : Une Approche Méthodologique Centrée Réutilisation de Patrons[END_REF][START_REF] Gzara | Product Information Systems Engineering: An Approach for Building Product Models by Reuse of Patterns[END_REF] propose a methodological framework based on the reuse of patterns during all the lifecycle, or [START_REF] Conte | P-Sigma : Un Formalisme Pour Une Représentation Unifiée de Patrons[END_REF]) who proposed patterns libraries to support a methodological framework for the conception of product information system.

After this short history of patterns, the next section aims at improving the comprehension of what is a pattern in SE.

Patterns for Systems Engineering

It happens that similar designs are made independently by different engineers [START_REF] Gaffar | Semantics of a Pattern System[END_REF]. This phenomenon acknowledges the fact that the same design elements exist in multiple designs, and the study and documentation of such designs foster reuse among projects. Indeed, it prevents from "reinventing the wheel" and provides a vocabulary for the design concepts that projects can share. This is consistent with the notion that patterns "are not created from a blank page; they are mined" [START_REF] Hanmer | Documenting Architectures with Patterns[END_REF]). It appears that SE patterns are embedded in existing designs, and that it is necessary to find a mechanism to identify them. Those patterns are called "buried patterns" by [START_REF] Pfister | A Proposed Meta-Model for Formalizing Systems Engineering Knowledge, Based on Functional Architecture Patterns[END_REF]) and represent a scientific issue. As the process of "Mining" appears to be essential for creating Pattern Languages, various approaches have been identified to write patterns from the element extracted from pattern mining. According to (DeLano 1998) classification, it is possible to classify mining's processes into three categories: Individual contributions where writers of the pattern used their own experiences or ones from their colleagues; Second-hand contributions where patterns are written based on interviews with experts or by guiding another person in the writing of patterns, it can also come from borrowing patterns from the literature or from companies in the same domain; Workshops/Meeting contributions that consists of groups of around ten people who brainstorm the elements of a patterns, along with a moderator and a facilitator.

When mining a pattern, depending on the language used (textual or modelling), it appears that a minimal set of information is always provided, as a pattern seems to possess an inherent triptych composed of {Context, Problem, Solution}. [START_REF] Gaffar | Semantics of a Pattern System[END_REF]) define a "Minimal Triangle" that defines the core meaning of a pattern (Fig. 1). It summarizes the idea that a pattern provides a solution to a recurring problem in a particular context. However, a general consensus enlarges the minimal elements needed in a pattern, [START_REF] Barter | A Systems Engineering Pattern Language[END_REF]) describe a generic pattern with the minimal elements needed to be written (Fig. 2). [START_REF] Cloutier | Applying the Concept of Patterns to Systems Architecture[END_REF] conduct a survey that allow them to list a recommended Systems Pattern Form. They also underline the fact that concepts used in Systems Engineering represent higher levels of complexity and abstraction that the prevailing notions of Alexander in architecture. For instance, the architecture of the underlying concepts of control-command requires a more complex notation than the sketch used in [START_REF] Alexander | A Pattern Language[END_REF], thus [START_REF] Pfister | A Proposed Meta-Model for Formalizing Systems Engineering Knowledge, Based on Functional Architecture Patterns[END_REF]) used the Enhanced Functional Flow Block Diagram (eFFBD) to represent the model of their control-command and rely on formal conceptual foundations in the form of a meta-model. [START_REF] Barter | A Systems Engineering Pattern Language[END_REF] Like models, patterns are abstractions or a set of abstractions of the reality and not a magical solution. They allow people to solve complex problems by leveraging experience, K&KH from their predecessors. The results of a study conducted, in the Open Source Software community, by [START_REF] Hahsler | A Quantitative Study of the Adoption of Design Patterns by Open Source Software Developers[END_REF] show that the larger the team size was, the greater the use of patterns was for documenting changes: from 11.4% for a unique developer to 82.2% in a team of ten or more developer. The capacity of patterns to deliver at each level of the development the correct amount of information for the stage it is applied, allow its quick adoption and most importantly its active use as Hahsler concludes in his study: "design patterns are adopted for documenting changes and thus for communication in practice by many of the most active open source developers". Patterns offer the possibility to create a common lexicon between systems architects that foster a common understanding of systems architecture, validated by experts. In this way, the experience acquired by the software community on pattern will be valuable, and help systems engineers to walk in their footsteps in order to develop patterns that will foster reuse, as well as helping control the complexity of a system.

As the interest for MBSE increases, it is important to also examine the work done for integrating the concept of pattern in this framework. The integration of the OMG System Modelling Language (OMG SysML) and its consequences on how to define problems and describe solutions are particularly interesting and will be examined in the next section.

Patterns for Model-Based Systems Engineering (MBSE)

Although research works have been made to assess whether the concept of pattern can be applied in the Systems Engineering field such as [START_REF] Pfister | A Proposed Meta-Model for Formalizing Systems Engineering Knowledge, Based on Functional Architecture Patterns[END_REF][START_REF] Cloutier | Applicability of Patterns to Architecting Complex Systems[END_REF]Haskins 2005), the value of patterns in a MBSE framework has not been fully explored. Yet, it appears crucial to consider all the different needs, requirements and constraints of the different stakeholders in the early design stages. Perceived by many companies as a time loss, it appears that introducing or reinforcing reuse capacity in MBSE methodologies allows the design of a new project with much less human effort, benefiting from the reuse of the already existing system models [START_REF] Shani | Reuse in Model-Based Systems Engineering[END_REF]. In this way, the capitalization and reuse of system models through the concept of pattern can be implemented in MBSE, and thus, favour its adoption at a larger scale.

Models are abstraction or a set of abstractions of the reality (i.e. the reality can be represented under different consistent views), which means that it can be easy to reuse a model in a new project since no physical limitations get in the way. However, depending on the type of reuse to do, the complexity of the system under design, and also the heterogeneity of methodologies and tools, it appears that the adoption of MBSE is penalized. Indeed, reusing existing modelling artefacts (even if their designs have been made to be reusable) is harder than expected. As [START_REF] Korff | Re-Using Sysml System Architectures[END_REF]) stated, the "biggest problem is to transfer and manage the knowledge [of] what is actually available for re-use". He emphasizes on the fact that it is necessary for system engineers to be aware of system assets that can be defined and propagated among teams designing complex systems. However, the creation of assets library is not sufficient, as the purpose is to allow engineers to reuse those assets in their ongoing projects. Korff underlines the fact that users should have the possibility to search, publish, and reuse assets in defined libraries and catalogues, without any specific technical pre-requisite. Contrary to [START_REF] Korff | Re-Using Sysml System Architectures[END_REF], [START_REF] Paydar | A Semi-Automated Approach to Adapt Activity Diagrams for New Use Cases[END_REF] do not focus on the creation of assets but propose an approach concerning the adaptation of promising reusable assets during a model reuse process, especially on the adaptation of OMG Unified Modeling Language (OMG UML) activity diagrams to new use cases, in the context of web engineering. This work proposes to semi-automatically create an activity diagram from existing activity diagrams according to the input use case diagram. Even though this approach is not presented in a MBSE framework, the fact that between OMG UML and OMG SysML, use case diagrams are identical and that activity diagrams presents the same use, allows considering a transposition in the SE field.

In the case of variant modelling in MBSE, [START_REF] Oster | Applying Composable Architectures to the Design and Development of a Product Line of Complex Systems[END_REF] propose an approach for building and exploiting composable architectures to the design and development of a product line of complex systems in the aerospace and defence market. They choose OMG SysML as the core language to define descriptive models of the composable system reference architecture and extended it to define parametric models. This methodology allowed the product line to evolve more readily as the impact of information propagation of adding, updating or modifying new components was automatic. As their works consider physical layer, [START_REF] Maio | Challenges in the Modelling of SoS Design Alternatives with MBSE[END_REF] focus their attention on the development of a functional architectures that can accommodate to change due to decisions made in the logical layer for System of Systems (SoS). The results of their study are a MBSE process that consists in the integration of a system model before the consideration of the variants. It requires that the system model should contain both the original configuration and the variant one. This separation is important in case of a new technology is introduced but the older one are not abandoned yet. They also investigate the aspects of including variant modelling into the OMG SysML, with a focus on extending an existing and operating model to support a new variant in the case where a similar technology is used.

The introduction of a reuse capacity in MBSE frameworks has proven to improve engineering efficiency in engineers work. However, the steep learning curve induced for organizations to adopt MBSE methodologies, results in the need of helping the engineers to "quickly identify not only valid architectural solutions, but optimal value solutions for the mission need" [START_REF] Oster | Applying Composable Architectures to the Design and Development of a Product Line of Complex Systems[END_REF]). Thus, it appears that the concept of patterns could be an answer to this challenge. Indeed, works have been done to introduce patterns during various phases of the engineering cycles. [START_REF] Gasser | Structuring Activity Diagrams[END_REF]) described behavioural construct patterns (Fig. 3) to facilitate and systematize the modelling of system behaviour. Instead of thinking at the level of atomic graphical elements, he defined a structured way to represent elementary behavioural constructs. In this way, he advocates the use of an "insert policy", like in the construction of Functional Flow Block Diagram (FFBD) where the resizing of the diagram is automatic when new elements are inserted. The proposed behavioural construct patterns will allow engineers to work in an algorithmic way of thinking, which implies a higher modelling level that will permit to focus more on the expected behaviour than on the aesthetics of the diagrams.

Fig. 3 Loop Exit Construct, extracted from [START_REF] Gasser | Structuring Activity Diagrams[END_REF] In order to help engineers to focus on what is important, patterns should guide the development to avoid deviation. For example, [START_REF] Barbieri | A SysML Based Design Pattern for the High-Level Development of Mechatronic Systems to Enhance Re-Usability[END_REF] proposed a process for the development of mechatronic systems based on a SysML design pattern. Their intent is to demonstrate that adequate guidelines for modelling can benefits the development process by allowing an efficient traceability of all information within the system model to trace change influences more easily. This approach proves to be particularly helpful for facilitating the impact analysis in later lifecycle phases and for the reuse for future projects.

Pursuing the work of (Haskins 2005) on patterns, [START_REF] Schindel | Requirements Statements Are Transfer Functions: An Insight from Model-Based Systems Engineering[END_REF] proposed an engineering paradigm where patterns are re-usable models, that enables what he calls Pattern-Based Systems Engineering (PBSE), where patterns can be configured or specialized into product lines or into product systems. With the advent of MBSE, this modelling framework has led to the creation of an INCOSE working group called MBSE Patterns2 . In this context, [START_REF] Schindel | Introduction to Pattern-Based Systems Engineering (PBSE): Leveraging MBSE Techniques[END_REF] developed their approach, they see "patterns as re-usable models" and apply them to requirements and design. At a high-level, they constitute a generic system pattern model that can be customized according to enterprise needs, configuration, uses, so that engineers can benefit from the concepts of MBSE without being an expert of modelling methodologies. [START_REF] Cook | Utilizing Mbse Patterns To Accelerate System Verification[END_REF] applies it for the Verification and Validation processes, and [START_REF] Bradley | Optimizing Delivery of Global Pharmaceutical Packaging Solutions, Using Systems Engineering Patterns[END_REF] in the pharmaceutical market.

Conclusion

As presented in the introduction, a main issue for system engineers is to shorten engineering cycle period, and MBSE appears to be a great candidate to face this challenge. For a wider MBSE adoption, this paper highlights the strong

Fig. 1 Fig. 2

 12 Fig.1The Minimal Triangle, extracted from[START_REF] Gaffar | Semantics of a Pattern System[END_REF]

http://www.omgwiki.org/MBSE/doku.php (visited on 31/05/2018)

http://www.omgwiki.org/MBSE/doku.php?id=mbse:patterns:patterns (visited on

31/05/2018)

methodological need to capitalize on previous projects to reuse K&KH, and focuses on the concept of pattern, which offers the possibility to make information dynamic between stakeholders during the development of complex systems, in order to share it and foster its reuse for future MBSE projects.

From a methodological perspective, improvements from processes, methods and tools should be made. It appears that the act of capitalization is not self-evident, as patterns need to be mined, and imply the ability to detect and bring out K&KH. A first step is to evaluate the maturity of such capitalized patterns, as done in the automated production systems domain by [START_REF] Vogel-Heuser | Key Maturity Indicators for Module Libraries for PLC-Based Control Software in the Domain of Automated Production Systems[END_REF]) on the maturity on control modules in libraries. A second step is to improve the general maturity of reuse approaches as done in the software domain by [START_REF] Manzoni | Identifying Extensions Required by RUP (Rational Unified Process) to Comply with CMM (Capability Maturity Model) Levels 2 and 3[END_REF], using for example metrics inspired by Capability Maturity Model. A next step to improve engineering effectiveness concerns the development and the adoption of MBSE software tools that integrate patterns libraries supporting their capitalization, selection, reuse, and update.