Identification of welding residual stress of buttwelded plate using the boundary element inverse analysis with Tikhonov's regularization

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1 IOP Conference Series: Materials Science and Engineering Identification of welding residual stress of buttwelded plate using the boundary element inverse analysis with Tikhonov's regularization To cite this article: Y Ohtake and S Kubo 2010 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancements. Related content - Distributions of temperature and stress fields on penetration assembly during multi-pass welding Hong Li and Li Li - Inverse analysis of inner surface temperature history from outer surface temperature measurement of a pipe S Kubo, S Ioka, S Onchi et al. - Identification of a microscopic randomness of a particle reinforced composite material with Monte-Carlo Simulation and inverse homogenization analysis S-I Sakata, F Ashida and Y Shimizu This content was downloaded from IP address on 08/04/2019 at 17:12

2 Identification of welding residual stress of butt-welded plate using the boundary element inverse analysis with Tikhonov s regularization Y Ohtake 1, S Kubo 2 1 Research Laboratory, IHI Corporation 2 Department of Mechanical Engineering, Osaka University Yasuhiro_ootake@ihi.co.jp Abstract. This paper deals with an inverse problem for estimating the magnitude of the residual stress in a butt-welded plate from measured data. The welding residual stresses in the butt-welded plate are caused by inherent strain. Thus, the welding residual stress in the buttwelded plate can be calculated if the inherent strains are estimated from measurement data. In previous works, a method was proposed to estimate the inherent strains in the butt-welded plate using thermo-elastic boundary element inverse analysis. It was found in the paper that the evaluated value of the inherent strains in the butt-welded plate depended on the location or the number of measurement points in the butt-welded plate. The accuracy of the evaluated inherent strains was low when the location of the measurement point was far from the welding line and the number of measurement points was small. Thus, this paper uses Tikhonov s regularization to improve the accuracy of the evaluated values of the inherent strains in the butt-welded plate. Numerical simulations were conducted for an analysis model which was the same with that used in the previous paper. In the simulations, measurement errors were included in stresses at measurement points. The maximum value of the error was 4.9 (MPa). The results of the calculations showed that the inherent strains in the butt-welded plate were evaluated in high accuracy from a few measured data by using Tikhonov s regularization. The discrepancy principle was applied for estimating a proper value of the regularization parameter. It was found that Tikhonov s method with the discrepancy principle was useful in evaluating the residual stress in the butt-welded plate. 1. Introduction Large structures like ship, crane and LNG tank are categorized into welded structures because they are assembled using welds. The weld generates the residual stresses that may cause buckling or brittle fractures of the welding structures [1][2]. Thus, the estimation of the welding residual stress is required to protect the welded structures from those fractures. The welding residual stress is caused by the shrinkage of the weld metal [3]-[6]. The shrinkage is expressed by thermal strains or equivalent inherent strains. If the inherent strains are estimated from 1 2 1, Shin-Nakahara-Cho, Isogo-ku, Yokohama Japan. 2-1, Yamadaoka, Suita, Osaka , Japan. c 2010 Published under licence by Ltd 1

3 measurement data, the welding residual stresses can be calculated from the inherent strains in the stress analysis. An inverse analysis using thermo-elastic boundary element method [7]-[9] was proposed in the previous papers [10]-[12] to estimate the inherent strains in the butt-welded plate. The method can be used for identifying the inherent strains from a few measurement data, which are measured at points away from the welding line of the butt-welded plate. The distribution and the highest value of the residual stress can be obtained from the boundary element analysis when the inherent strains are identified from measured data by using the proposed method. The accuracy of the evaluated inherent strain was reduced when the number of measurement points was small or the location of the measurement points was far from the welding line. Thus, the improvement of the accuracy of the evaluated inherent strain was a problem in the method proposed in the previous paper. This paper uses Tikhonov s regularization [13] to improve the accuracy of the evaluated values of the inherent strains in the butt-welded plate. Discrepancy principle is also applied to determine a proper value of the regularization parameter. Numerical simulations are conducted for an analysis model which is the same with that in the previous paper [11]. Numerical examples with and without errors in the measured stresses are made in the simulations considering that errors are included in measured stresses. A trapezoidal distribution of the inherent strain in the butt-weld plate is used in the simulation. The inherent strains are evaluated from the measured stresses using the boundary element inverse analysis with Tikhonov s regularization. 2. Procedure of boundary element inverse analysis 2.1. Tikhonov s regularization for evaluating inherent strains from measured stresses Tikhonov s regularization [6][13] is one of methods to stabilize the solution. The method is defined by the following functional Π that is formulated in terms of solution z, measurement value u (m), operator A, regularization parameter α and stabilizing functional Λ(z). ( m) T ( m ( z) = [ Az u ] [ Az u ] + αλ( z) Π ) (1) Tikhonov s regularization is applied to improve the accuracy of the evaluated values of the inherent strains in a butt-welded plate. The stresses {σ} at the measurement point are calculated from the inherent strains {ε * }. The following equation indicates the relationship between the stresses {σ} and the inherent strains {ε * }. {} = [ ] { ε * } σ A (2) The functional Π in equation (1) is formulated by the stresses {σ} in equation (2), measurement stresses {σ (m) }, regularization parameter α and inherent strains {ε * }. The functional Π is then shown as, Π = [{} { ( m ) T }] [{ } { ( m ) }] { * T σ σ σ σ + α ε } { ε * } (3) By substituting equation (2) into equation (3), the functional Π in equation (3) is replaced by the following equation. Π = [ ]{ * } { ( m ) T }] [ ]{ * } { ( m ) }] { * T ε σ A ε σ α ε } { ε * } A (4) The extreme value of equation (4) is calculated from the following equation. 2

4 Π = { } {} ε 0 * (5) Thus, the relationship between the inherent strains and the measured stresses after Tikhonov s regularization is formulated as follows by substituting equation (4) into equation (5). 1 T ( m) [ + α] [ A] σ * T { ε } = [ A] [ A] { } (6) Matrix [A] for the present problem estimating inherent strains is deduced as follows. A method was proposed to estimate the inherent strains in the butt-welded plate from the measurement stress in previous works [10]-[12]. The proposed equation was formulated by using the following two equations [7]-[9] that were based on the theory of thermo-elastic boundary element method that is composed in the displacements {u} and the loads {t} of the boundary surface in the analysis model, the stresses at the measured points and the inherent strains {ε * }. [ H ]{} u [ G]{} t + [ B]{ ε * } = (7) { σ } [ S ]{} u + [ D]{ t} + [ R]{ ε * } = (8) Here [H], [G], [B], [S], [D] and [R] are the matrixes that are defined in the boundary element analysis [7]- [9]. Equations (7) and (8) are replaced by the following equations when the displacements {u} and the loads {t} at the boundary surface in the analysis model are divided into each values of unknown and known [11]. The unknown values of the displacements and the loads are {u u } and {t u } and the known values of that are {u k } and {t k }. where H mm H mn uu Gmm Gmn t B k mq = + { ε * } H nm H nn u k Gnm Gnn tu Bnq (9) uu t k { σ } = [ S ] [ ] [ ]{ ε * pm S pn + D pm D pn + R pq } u k tu (10) { u } { } T u = uu1, uu2, uu3, L LL, uum 1, uum { u } { } T k = uk1, uk 2, uk 3, L LL, ukn 1, ukn { t } { } T u = tu1, tu2, tu3, LL, tum 1, tum { t } = { t t, t, L LL, t, t } T k L (11) k1, k 2 k 3 kn 1 * * * * * * { ε } = { ε } T 1, ε 2, ε 3, L LL, ε q 1, ε q { σ } = { σ σ, σ, L LL,, } T 1, kn σ p σ p where m is the number of the unknown values of displacements and loads, n is of the known values of that, q is of the inherent strains and p is of the measured stresses, respectively. This paper introduces Tikhonov s regularization to the proposed method in previous work [11] to improve the accuracy of the evaluated values of the inherent strains in the butt-welded plate. As a result, the matrix [A] in equation (6) is derived from equations (9) and (10) as follows. 3

5 [ A ] [ S ][ F] + [ D ][ C][ E] + [ R ] = (12) pm pn pq where 1 [ F ] = [ H ] [ G ][ C][ E] + [ B ] mm mn mq 1 [ ] 1 [ C ] [ H ][ H ] [ G ] [ G ] = nm mm mn nn 1 [ E] = [ B] [ H ][ H ] [ B ] nq nm mm mq The inherent strains in the butt-welded plate are evaluated from the measured stresses by using those equations Discrepancy principle for determining a proper value of the regularization parameter The discrepancy principle [13] is applied to determine a proper value of the regularization parameter α. The residual R σ is evaluated between the measured stresses and those corresponding to the inherent strains ε * estimated using a regularization parameter α. R σ = 1 p p ( m ) ( σ n _ α σ ) n= 1 2 (13) The discrepancy principle predicts that the regularization parameter α is optimum when the residual R σ is equal to that coincident with the measurement errors. 3. Numerical examinations Numerical simulations are made to identify the inherent strains in the butt-welded plate model from the measurement data. Figure 1 shows the model for the simulation of the boundary element analysis. Two-dimensional plane stress condition is assumed in the calculation because the thickness of the specimen is small. Constant-type 57 boundary elements are used in the analysis model shown in figure 1. Linear-type 42 cells are placed in the domain of specimen. Young modulus of the analysis model is (GPa) and Poisson s ratio is 0.3. For the sake of symmetry of the analysis model the displacement in the x- direction on y-axis and that in the y-direction on x-axis are constrained. The distribution of the inherent strain is assumed to be trapezoid and the maximum value of the magnitude is The inherent strains of the trapezoid distribution are shown in figure 2. Table 1 indicates the location of the measurement points. The y-coordinate of all measurement points is 12.5mm and the location is away from the welding line. The stresses σ x and σ y at the measurement points are used for the calculations in the numerical simulations. The number of the measurement points N is 6 or 11. The measurement points 1, 3, 5, 7, 9 and 11 in table 1 are used when N=6 and the measurement points 1 through 11 are used when N=11. The measurement errors are included in the stresses at the location of the measurement points. The maximum value of the error is 4.9 (MPa). The errors in the measurement stresses are shown in figure 3. The errors in the measurement stresses are distributed from maximum value δ to minimum value δ by uniform random numbers. The value of δ is 4.9 (MPa). The expected value of R σ due to the distributed errors is then δ 2 /3 = (MPa 2 ). Figure 4 shows the flowchart of the calculation procedure to identify the distribution of the inherent strain from the measured data. The actual inherent strains in the simulation are given in the domain cells of the analysis model in figure 1. The stresses at the measurement points are calculated by the 4

6 thermo-elastic boundary element analysis. The stresses are used to identify the inherent strain in the domain cells of the analysis model from the stresses at the measurement points. The results of the direct analysis are used for the calculation in inverse analysis. The calculation of the inverse analysis is conducted by using equation (6). The inherent strains are evaluated from a few measured values of the residual stress away from the welding line. Then, the magnitude and the distribution of the residual stress in the butt-welded plate were calculated from the identified inherent strains. y 55 x 100 Figure 1. Model of boundary element method to evaluate inherent strain in butt-welded plate Inherent strain ε* x axis position (mm) Figure 2. Actual inherent strain in cell of boundary analysis model of butt-welded plate Table 1. Location of measurement points y x-coordinate of measurement point p' (mm) (mm)

7 Measurement error (MPa) : Measurement error in σx : Measurement error in σy Measurement points Figure 3. Errors with maximum value of 4.9 (MPa) introduced in measured stresses at measurement points Assumption of inherent strains of trapezoid distribution in butt-weld plate Boundary element analysis Direct analysis Calculation of stresses at measurement points Modeling to evaluate inherent strains from measurement stresses Boundary element inverse analysis Inverse analysis Comparison between assumed inherent strains and evaluated ones Figure 4. Flowchart of the calculation procedure in numerical simulation 6

8 4. Calculation results 4.1. Estimated inherent strains using stresses without errors The simulations of the inverse analysis are conducted to identify the distribution of the inherent strains in the butt-welded plate shown in figure 1. The inherent strains are estimated from the stresses at the location of the measurement points shown in table 1 by the proposed boundary element inverse analysis method. For preliminary examination of the applicability of the proposed method, constant inherent strain of the magnitude of is assumed and no measurement errors are introduced. Table 2 shows the inherent strains estimated by the proposed boundary element inverse analysis method when the stresses at the measurement points include no measurement errors. Calculations are made for two cases that the number of the measurement points N located on the line of y=12.5mm is 6 and 11. From table 2 the evaluated inherent strains agree well with the actual ones when no errors are included in the measured stresses. The results show that the proposed boundary element inverse analysis method is useful in the evaluation of the inherent strains in the butt-welded plate from the measured stresses. Table 2. Inherent strains estimated by inverse analysis Cell number Assumed Evaluated values q values (N =6) (N =11) Evaluated results of inherent strains using stresses with errors Numerical simulations were made to examine the effectiveness of the proposed method with Tikhonov s regularization. For evaluating the results the residual R ε between the actual and estimated inherent strains is defined as, R ε = q * * ( ε n _ α ε actual ) n= 1 2 (14) where ε * n_α is the estimated inherent strains after the regularization and ε * actual is the actual inherent strain. Figures 5 (a) and (b) show the residual in inherent strain R ε evaluated by equation (14) from the stresses at 6 and 11 measurement points. The residual in stress R σ is also shown in the figure. The circle mark on the line of R σ in figure 5 corresponds to the point where R σ is equal to its expectation value in accordance with the discrepancy principle. The triangle mark on the line of R σ indicates its 7

9 minimum value which corresponds to the optimum value of α and the optimum solution among all the solutions obtained using Tikhonov s regularization. Figure 5 shows that the value of the regularization parameter α estimated using the discrepancy principle shown by the circle is close to its optimum value shown by the triangle. Thus, it was found that discrepancy principle is useful in estimating a proper value of the regularization parameter. Triangular symbols in figure 6 show the inherent strains estimated by the proposed boundary element inverse analysis with Tikhonov s regularization and the regularization parameter determined by the discrepancy principle. The estimated inherent strains are compared with the actual ones. For comparison purpose the inherent strains estimated without the regularization is shown by circular symbols. The inherent strains estimated by the regularization are in good agreement with the actual ones even when the number of the measurement points is 6. The results show that Tikhonov s regularization with the discrepancy principle can be used to improve the accuracy of the estimated inherent strain in the butt-welded plate. Residual in stress Rσ, Residual in inherent strain Rε 1.E+04 1.E+02 1.E+00 1.E-02 R σ (Equation (13)) R ε (Equation (14)) 1.E-04 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Regularization parameter α 1.E-04 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 (a) Measurement points N=6 (b) Measurement points N=11 Figure 5. Relationships regularization parameter α and residual in stress R σ, residual in inherent strain R ε from measurement stresses with maximum error of 4.9 (MPa) at measurement points N=6 and 11 Residual in stress Rσ, Residual in inherent strain Rε 1.E+04 1.E+02 1.E+00 1.E-02 R σ (Equation (13)) R ε (Equation (14) Regularization parameter α Evaluated inherent strain ε* : Actual inherent strain : Before regularization : After regularization Cell number Cell number (a) Measurement points N=6 (b) Measurement points N=11 Figure 6. Evaluated inherent strains before and after Tikhonov s regularization from measurement stresses with maximum error of 4.9 (MPa) at measurement points N=6 and 11 Evaluated inherent strain ε* : Actual inherent strain : Before regularization : After regularization

10 5. Conclusions Tikhonov s regularization was used to improve the accuracy of the evaluated values of the inherent strains in the butt-welded plate. Numerical simulations were conducted by using thermo-elastic boundary element analysis for an analysis model which was the same with that used in the previous paper. In the simulations, measurement errors were included in measurement stresses. The maximum value of the error was 4.9 (MPa). The results of the calculations showed that the inherent strains of the butt-welded plate were evaluated in high accuracy from a few measured data by using Tikhonov s regularization. A proper value of the regularization parameter was estimated using the discrepancy principle. It was found that Tikhonov s regularization with the discrepancy principle was useful in evaluating the residual stresses in the butt-welded plate. It was found that the proposed method could effectively predict the inherent strains of the butt-welded plate. References [1] Watanabe M and Sato K 1965 Welding Mechanics and Application, Asakura shoten, Tokyo. [2] Yonetani S 1983 Occurrence of Residual Stress and Plan, Yokendo, Tokyo [3] Ueda Y, Fukuda K, Nakacho K and Endo S 1975 J. Soc. Naval Arch. of Japan Vol 138 pp [4] Ueda Y and, Yuan M G 1993 Journal of Engineering Materials and Technology Vol 115 pp [5] Koguchi H, Tomishima T and Yada T 1990 International Journal Pressure Vessels and Piping Vol 44 pp [6] Kubo S 1992 Inverse problem Baifukan Tokyo [7] Brebbia C A, Telles J C F and Wrobel L C 1984 Boundary Element Techniques Springer- Verlag [8] Yuuku R and Kisu H 1987 Elastic Analysis by Boundary Element Analysis Baifukan Tokyo [9] Tanaka M and Tanaka M 1984 Fundamentals of Boundary Element Analysis Baifukan Tokyo [10] Ohtake Y 1995 Transactions of the Japan Society of Mechanical Engineers, Series A Vol pp [11] Ohtake Y 2009 Transactions of the Japan Society of Mechanical Engineers, Series A Vol pp [12] Ohtake Y and Kubo S 2009 Proceedings of 12 th International Conference on Fracture Vol 10 [13] Tikhonov A N and Arsenin V Y 1997 Solutions of ill-posed problems (John Wiley & Sons) 9