Journal of Computational Science and Technology

Journal of Computational Science and Technology

Vol. 2, No. 4, 2008

A Stabilization Method for the Hydrogen Diffusion Model in Materials* Stephane NDONG-MEFANE**, Hiroshi KANAYAMA***, Masao OGINO*** and M. F. EL-AMIN***,**** **Graduate School of Engineering 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan E-mail: kanayama@ mech.kyushu-u.ac.jp ***Department of Mechanical Engineering, Kyushu University 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan **** Mathematics Department, Aswan Faculty of Science, Egypt

Abstract In this paper, we highlight the existence of some instability in finite element method appearing for high values of the Peclet number in the model of hydrogen diffusion in materials. A stabilization technique is used to overcome the instability problem and therefore improve this scheme. We manage to improve the scheme and decrease the instability, and we highlight the strong influence of the stabilization parameter in this particular case. Key words: Hydrogen Embrittlement, Trapping, Crack, Stabilization Technique, Stabilization Parameter

1.

Finite

Element,

Introduction

In spite of decades of research effort, hydrogen assisted cracking is still the most serious problem in steel welding. Hydrogen embrittlement is still not clear, and a coupled diffusion elasto-plastic stress analysis is needed to model the effects of hydrostatic stress and hydrogen trapping due to plasticity on the hydrogen distribution in a body. The finite element method is a very useful tool for this purpose. Sofronis and McMeeking conducted several researches on hydrogen diffusion in materials. In 1989, Sofronis and McMeeking established a Finite Element Method (FEM) model for the effect of hydrostatic stress and trapping on the hydrogen distribution in plastically deformed steel (1). Like this, they could investigate the hydrogen concentration near a blunting crack tip under small-scale yielding conditions. Unfortunately, their hydrogen transport model did not provide the correct balance of hydrogen concentration into the considered material. Based on their work, another model that provides the correct balance of hydrogen and also shows a strong dependence of the hydrogen concentration in lattice sites on the strain rate has been developed by Krom, Koers and Bakker(2). In our previous study (3), we managed to get Krom et al. results, using a non-symmetric coefficient matrix for each time step, but we have also noticed the appearance of some numerical complications for high Peclet numbers. In the present study, using the work of Franca et al. (4), we establish a stabilized scheme and apply it to the advection diffusion equation in order to correct the instabilities created by the high values of the Peclet number. We also encounter some troubles with the stabilization parameter and show a way to deal with it.

*Received 17 June, 2008 (No. 08-0423) [DOI: 10.1299/jcst.2.447]

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Vol. 2, No. 4, 2008

2.

The hydrogen transport equation

Fig. 1 Description of the boundary and initial conditions for the one side coupled diffusion and elasto-plastic problem under small scale yielding conditions (1). Using the model presented in Fig. 1, a body with a volume Ω and a surface ∂Ω = Γ is considered. In the previous work (3), the hydrogen transport equation mainly based on Krom et al. (2), which is itself based on Sofronis and McMeeking’s work (1) has been introduced. The weak form is

    1 + N L K T N T  ∂C L , C L*  + DL ∇C L , ∇C L* 2     (K T C L + N L )  ∂t Ω

(

)

Ω

(1)

 C D V  KT dN T ∂ε p *  −  L L H ∇σ h , ∇C L*  +  C L , C L = 0,    RT  Ω  K T C L + N L dε p ∂t Ω where ( A, B) Ω = A ⋅ BdΩ . ∫ Ω

The time-implicit scheme for the model equation presented in Krom et al.’s work is

  N L KT NT (ε pn )  C n+1 − C n   L  1 + , CL*  + DL ∇CLn+1 , ∇CL*  L 2    ∆t KT CLn + N L    Ω

(

(

)

)

Ω

(2)

 −ε  C DLVH   KT −5.5ε n ε −  ∇σ hn , ∇CL*  +  CLn 29.508NT (ε np )e p , CL*  = 0. n   ∆t  RT  Ω  KT CL + N L Ω n+1 p

n L

n p

On the other hand, the time-implicit scheme for the advection diffusion equation presented in our previous work is

  N L KT NT (ε pn )   C n+1 − C n   L  1 + , CL*  + DL∇CLn+1 , ∇CL*  L     K C n + N 2   ∆t  Ω  T L L 

(

)

(

)

Ω

(3)

 −ε  C DLVH   KT −5.5ε n ε −  ∇σ hn , ∇CL*  +  CLn+1 29.508NT (ε pn )e p , CL*  = 0. n   ∆t KT CL + N L  RT Ω  Ω n+1 L

n+1 p

n p

In the above equation, C L and CT are respectively the number of hydrogen atoms per unit volume in lattice sites and in trap sites. R is the gas constant, KT is the equilibrium constant 448


Vol. 2, No. 4, 2008

such that K T = e − ∆ET

RT

with ∆ET as the trap binding energy, T is the absolute 3

temperature, and σ h is defined as the hydrostatic stress such that σ h = 1 ∑ σ ii where the 3

i =1

component σ ij is the stress tensor component. The subscripts i and j are usually denoted x, y, z instead of 1, 2, 3. VH is the partial molar volume of hydrogen, ε P is the plastic strain,

N L and N T are the lattice site and the trap site densities, and DL is the diffusion constant. More details about these expressions can be found in the previous paper (3).

3.

Overview of previous results

The model is a half disc with a radius of 1.5 × 10-1[m] and a thickness of 8.25 × 10-4[m]. The crack can be considered as a horizontal band going from the center to the left until the circumference of the half disc with an initial crack tip opening of 1.0 × 10-5[m]. The mesh consists of 10,656 nodes and 3,706 10-node tetrahedral elements, and is refined near the crack tip(5). A recapitulation of the model dimensions (Table1 and Fig. 2), numerical parameters (Table 2), and constant values (Table 3) can be found in the following three tables. Table 1 Model’s dimensions.

−5

Initial crack tip opening

b0

1.0 × 10

Model radius

R

1.5 × 10

Model thickness

h

8.25 × 10 −4 [m]

[m]

−1 [m]

Fig. 2 The mesh has been made using 3D tetrahedral elements.

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Table 2 Numerical parameters.

Table 3 Constant values.

The hydrogen distribution in lattice sites at 130[s] has been compared with Krom et al. results in Fig. 3. The results with the strain rate factor and without the strain rate factor are computed for a loading time of 130[s]. C L 0 is the initial hydrogen concentration on the

(

)

symmetry line θ = 0 D . Effects of the strain rate factor can be observed in both results.

Fig. 3  C L 

 ahead of the crack tip by our scheme and by Krom et al. scheme at 130[s] C L 0 

(end of loading). 450


Vol. 2, No. 4, 2008

4.

An instability problem The Peclet number is a dimensionless number relating the rate of advection of a flow to

its rate of diffusion and defined as Velocity × Chareacteristic Length . In the previous

Diffusion Constant

study

(3)

, considering the crack opening

b0 as the characteristic length, a Peclet number of

the order 1.0 × 10+01 is computed. If this number increases one-hundred times, the appearance of instabilities with significant oscillations in the hydrogen concentration happens and negative hydrogen concentration can be observed. Figure 4 and Fig. 5 illustrate the above explanation.

Fig. 4 Hydrogen distribution near the crack tip for a high value of the Peclet number (100 times higher).

Fig. 5 Change of the hydrogen concentration distribution near the crack tip. (a) Original value of the Peclet number (Pe), (b) 10 times Pe, (c) 100 times Pe.

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5.

A stabilized scheme

In order to solve the instability problem presented above, a stabilization method is applied to the non-symmetric scheme related to Eq. (3). To do so, the work of Franca et al. on stabilized schemes for advection diffusion equations (4) has been used. Let us consider the following stationary advection diffusion problem where the unknown variable is a scalar field u ; (4)

p ⋅ ∇u − k∆u = f ,

in a bounded domain Ω ⊂ ℜ 3 with zero value on the polyhedral boundary Γ . Here, p is the given flow velocity with ∇ ⋅ p = 0 in prescribed

source

function.

{

The

}

Wh = v ∈ H 01 (Ω) v |K ∈ P2 (K ), K ∈ ℑh , where

Ω , k > 0 is the diffusion constant, and f is a scalar

u

field

is

approximated

in

ℑh is a partition of Ω into quadratic tetrahedral

elements (P2 (K )) and H 01 (Ω ) is the Sobolev space of functions with square-integrable values and derivatives in Ω , and with zero value on the polyhedral boundary Γ .Then, one stabilized finite element scheme is: Find uh ∈Wh such that

B(uh , vh ) = F (vh ), vh ∈Wh ,

(5)

where B(u , v ) and F (v ) are defined as B (u , v ) = ( p ⋅ ∇u , v )Ω + (k∇u , ∇v )Ω +

∑τ ( p ⋅ ∇u − k∆u, p ⋅ ∇v − k∆v )

K,

(6)

K ∈ℑh

F (v ) = ( f , v )Ω +

∑ τ ( f , p ⋅ ∇v − k ∆ v )

(7)

K,

K ∈ℑh

and τ is the stability parameter. Applying the same principle on our advection diffusion equation, we have n+1 n n    1 + N L KT NT (ε p )  C Lh − C Lh 2   ∆t K T C Lnh + N L  

(

)

(

+ DL ∇C Lnh+1 , ∇C L*h

)

Ω

 *    DV , C L −  C Lnh+1 L H ∇σ hn , ∇CL*h   h  RT Ω  Ω 

n +1  − ε pn *  KT −5.5ε np ε p n +  CLnh+1 N e 29 . 508 ( ) , C Lh  ε T p   ∆t KT C Lnh + N L  Ω

n +1 n n     1 + N L KT N T (ε p )  C Lh − CLh  + ∇ ⋅  C n+1 DLVH ∇σ n   h  2  Lh RT   ∆t KT C Lnh + N L       − ∇ ⋅ D ∇C n+1  L Lh   n + ∑τ  n +1 n  = 0. −εp KT −5.5ε np ε p n K ∈ℑh  + CLn+1  29.508 NT (ε p )e , h ∆t KT C Lnh + N L      ∇ ⋅  C * DLVH ∇σ n  − ∇ ⋅ D ∇C *  h  L Lh   Lh RT     K

( (

)

)

(

(8)

)

It is important to notice that in our case we are using the conservative advection diffusion equation. On the other hand, Franca et al. use in their paper(4) the non-conservative advection diffusion equation. The stabilization parameter is first defined by considering the 452


Vol. 2, No. 4, 2008

non-stationary case and using a previous work conducted by Tabata and Suzuki (6) as

   2  hk h   ∆t τ n = min  , , k ,  2 2 D LVH ∇σ n 12 D L  h   RT  

(9)

and we also set

(

)

a = τ n ∇ ⋅ D L ∇C L*h ,

(10)

where C L*h is the test function and hk is the maximum length of the tetrahedral element. It is noted that ∇σ hn is computed by the linear interpolation based on vertex nodal values over the quadratic tetrahedral element. It will later be shown that the value of τ n needs to be adjusted.

6.

Results

At first, the stabilized scheme is applied to the advection diffusion equation by using the previously mentionned stability parameter. In both cases of a = 0 and a ≠ 0 , no good result is obtained. In the case of a = 0 , the stabilized scheme gives the same result as the Galerkin method for the initial Peclet number. However, an increace in the value of this number will stop the computation. In the case of a ≠ 0 , the computation does not run at all and displays a “segmentation fault” error message even for the initial Peclet number. From this point, decreasing the stability parameter appears to be the only way . It has been noticed that for lower values of the parameter, no interuption occurs in the computation anymore. For example, if the value τ*=1.0 × 10-03τn is used as the stability parameter, the computation runs correctly and some results are obtained in both cases of a* = 0 and

(

)

a* ≠ 0 with a* = τ * ∇ ⋅ DL ∇C*Lh . This value was not chosen randomly and later in this paper, the process used to choose a satisfying stabilization parameter will be explained in detail. For now, the results obtained with this new stabilization parameter are presented. (i) Case a*

=0

In this case, we can observe the results in Table 4. The results obtained in this case give not only results close to the Galerkin-stable cases (cases 1 and 2) but also well-improved results in case 3. It can be seen in Fig. 6 how the stabilized scheme influences and improves the results provided by the Galerkin method.

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Table 4 Results for different values of the Peclet number with the stabilization parameter τ*=1.0 × 10-03τn. Peclet number

Galerkin

Stabilization parameter a*=0

Case 1 32.6 Case 2 326 Case 3 3260 : Good agreement with Galerkin results, : The computation gives negative results or stops, : Improved results with some negative values. In the case of a* = 0 and in Fig. 7 , the stabilization term reduces the oscillations and the negative values appearing in the Galerkin method. One can also notice that in cases 2 and 3, the maximum concentration is decreased. Using a smaller stability parameter allows to run the computation, and especially in the case of a* = 0 , to get some satisfying results. However, changing the stabilization parameter can change the concentration distribution of hydrogen in some unexpected ways. It can improve the results presented above, but it can also make the instability worse. The stabilization parameter cannot be increased or decreased at will, a specific procedure is needed in order to choose the best sabilization parameter in the case of a* = 0 .

Fig. 6 Results of hydrogen distribution near the crack tip with a stabilization parameter of τ*=1.0 × 10-03τn. (a): case 1, (b): case 2, (c): case 3.

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Fig. 7 Detailed view of the hydrogen distribution near the crack tip in case 3, for τ*=1.0 × 10-03τn.

(ii ) Case a*≠0

For this case the results can be seen in Table 5. In this case, we applied the same

Table 5 Results for different values of the Peclet number with the stabilization parameter τ*=1.0 × 10-05τn. Peclet number

Galerkin

Stabilization method a*≠0

Case 1 32.6 Case 2 326 Case 3 3260 : Good agreement with Galerkin results, : No full agreement with Galerkin’s results, : Improved results with only few negative values. stabilization parameter as the case of a* = 0 , but the results obtained do not work (“segmentation fault” error message) for the Galerkin-stable case and results are obtained only for the Galerkin-unstable case. In the Galerkin-unstable case, the results are very well improved, but to get some results in the Galerkin-stable case, a smaller stabilization parameter is needed. After decreasing 100 times the stabilization parameter, the computation runs normally and we still have a good improvement in case 3, but we don’t have full aggrement with the original Galerkin results in case 1 and case 2. Decreasing more the stabilization parameter would mean decreasing the influence of the stabilization parameter and having non-satisfying results in the attempt to solve the instability appearing in the Galerkin method for case 3 (if the stabilization parameter is too small, the stabilization term has no effect on the advection diffusion equation anymore). For these reasons, the case a* ≠ 0 could not be validated. In the next chapter, only the case of

a* = 0 is considered.

7.

Choice of the stabilization parameter

As it is said in the previous section, the stabilization parameter cannot be chosen randomly. If it is too small, the action of the stabilization term is cancelled and the results in the unstable case (case 3) are just the same as those provided by the Galerkin method. 455


Vol. 2, No. 4, 2008

Increasing the stabilization parameter too much will first lead to an important degradation of the quality of the results in cases 1 and 2: the results are not the same as those provided by the Galerkin method anymore (see Fig. 8), but the results are still being improved in case 3 (see Fig. 9; the stabilized result is exactly the same for both stabilization parameters). If the stabilization parameter is increased more, the computation does not run anymore.

Fig. 8 Influence on the stabilization parameter for hydrogen diffusion near the crack tip in the case 1.

Fig. 9 Improvement of the results in case 3 for both stabilization parameters in hydrogen diffusion near the crack tip.

Fig. 10 Choice of the initial stabilization parameter in hydrogen diffusion near the crack tip.

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Fig. 11 Average of negative concentrations for different stabilization parameters. (a) a first view of the stabilization parameter requires a zoom (in the red circled zone) to display the other points clearly. (b) This first zoom reveals lower values of the stabilization parameter, but is still not enough. (c) the best value of the stabilization parameter is found in this second and last zoom. The first step to choose a good stabilization parameter is to find, for the hydrogen distribution in case 1, the maximum value of the stabilization parameter that still provides results close to those provided by the Galerkin method. In Fig. 10, this value is set to τ*=2.0 × 10-03τn =3.91 × 10-07. If we go beyond τ*=2.0 × 10-03τn, the concentration distribution becomes too different from the Galerkin concentration distribution. Once this “initial value” has been determined, the second step consists in plotting the average of negative concentrations appearing for each value of the stabilization parameter with values of the stabilization parameter located between zero and the “initial value” set previously. 457


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The best value of the stabilization parameter as we can see in Fig. 11 is the one with the value of average concentration closest to zero, and corresponds to τ*=9.0 × 10-04τn = 1.77 × 10-07 .

8.

Conclusions

After re-constructing our own finite element scheme based on the Krom et al. scheme, and applying successfully the Galerkin method in the previous work, it has been shown this time that some numerical complications are encountered for great values of the Peclet number, and using a stabilization method, the Galerkin’s results for the high Peclet number can be improved. In this work, a way to choose correctly the stabilization parameter in this particular stabilization problem is shown. However, some negative values of the hydrogen concentration still remain, and further research is needed to remove completely them. Later, a fully coupled analysis might also be required; in order to perform a fully coupled analysis, the hydrostatic stress and the strain rate factor should fully depend on the total hydrogen concentration in lattice and trap sites (7)(8). Those are the next goals for this study on hydrogen diffusion problems in materials.

Acknowledgment This research has been conducted as part of “Fundamental Research Project on Advanced Hydrogen Science” funded by New Energy and Industrial Technology Development Organization (NEDO).

References (1) P. Sofronis, R. M. McMeeking, Numerical analysis of hydrogen transport near a blunting crack tip, Journal of the Mechanics and Physics, Vol. 37 (1989), pp. 317-350. (2) A.H.M. Krom, R.W.J Koers, A. Bakker, Hydrogen transport near a blunting crack, Journal of the Mechanics and Physics of Solids, Vol. 47 (1999), pp. 971-992. (3) H. Kanayama, T. Shingoh, S. Ndong-Mefane, M. Ogino, R. Shioya, H. Kawai, Numerical analysis of hydrogen diffusion problems using the finite element method, Theoretical and Applied Mechanics, Vol. 56 (2008), pp. 389-400. (4) L.P. Franca, S. L. Frey, T.J.R. Hughes, Stabilized finite element methods: I. Application to the advective-diffusive model, Computer Methods in Applied Mechanics and Engineering, Vol. 95 (1992), pp. 253-276. (5) R. Miresmaeili, M. Ogino, R.Shioya, H. Kawai, H. Kanayama, Modeling of the hydrostatic stress and equivalent plastic strain distributions around the blunting crack tip in impure iron, in preparation. (6) M. Tabata, A. Suzuki, A stabilized finite element method for the Rayleigh-Benard equations with finite Prandtl number in a spherical shell, Computer Methods in Applied Mechanics and Engineering, Vol. 190 (2000), pp. 387-402. (7) P. Sofronis, Y. Liang, N. Aravas, Hydrogen induced shear localization of the plastic flow in metals and alloys, European Journal of Mechanics - A/Solids, Vol. 20, issue 6, (2001), pp. 857-872. (8) H. Kotake, R. Matsumoto, S. Taketomi, N. Miyazaki, Transient hydrogen diffusion analyses coupled with crack-tip plasticity under cyclic loading, International Journal of Pressure Vessels and Piping, Vol. 85, No. 8 (2008), pp. 540-549.

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