A numerical study on eddy current signal characteristics of imitative stress corrosion cracks

Abstract

This paper presents numerical simulation results of eddy current signals to determine the appropriated dimensions of imitative stress-corrosion cracks (SCCs) in viewpoint of similar eddy current testing signals. The imitative SCCs were modelled as multiple tiny slits in three dimensions. A Finite Element and Boundary Element hybrid method (FEM-BEM) was utilized to simulate the eddy current signals in an impedance form of an absolute pancake coil. Equivalent conductivity ( $\sigma_{E}$ ) of the imitative SCCs was estimated by using a uniform notch model at a fixed width of 0.2 mm, and adjusting the conductivity in the notch region. According to the numerical results, the impedance amplitude could be significantly used to quantify the $\sigma_{E}$ of imitative SCCs. In addition, the variation of test frequency affecting to $\sigma_{E}$ value was investigated. Finally, the appropriated imitative SCCs dimensions for the eddy current testing were quantitatively predicted by using a statistical methodology.

Keywords

Stress corrosion cracks imitative SCC eddy current testing FEM-BEM

1. Introduction

Stress corrosion crack (SCC) is an essential service failure in engineering materials of the components and structures. The SCC occurs by environmentally induced crack propagation [1, 2, 3]. It can form as only a single crack or a group of colonies depended on many factors such as material, shape, type of stress, pH environment and type of corrosion.

Evaluation of SCC by using the eddy current testing (ECT) is an important issue in nuclear power plants, petrochemical and aerospace industries [4, 5, 6]. Although the ECT has high performance to treat the surface breaking SCC in length, it is still difficult to quantify the actual profiles with the measured signals. At the presents, SCCs have been researched and simulated to be conductive cracks in the viewpoint of ECT. There were some reports to explore the equivalent conductivity inside the SCC regions by numerical simulation of the uniform conductive cracks [7, 8, 9]. The results indicated that there was no correlation between conductivity and SCC width, but the SCC size related to its equivalent electrical resistance. The equivalent conductivity of SCC was applied to reconstruct the crack profile using inversion methods by authors [10, 11]. There is also a report of the local conductivity determination in the SCC region by using tested signals of the direct current potential drop (DCPD) [12]. The SCC conductivity showed as a linear distribution along the crack depth. In addition, imitative SCCs in perspective of ECT using multiple slits fabricated by different techniques were presented [13, 1, 4, 15]. They could be corresponding to the conductive distribution of the actual SCCs by introducing the conductive bridge between tiny slits.

As in literatures, the imitative SCCs are not exactly identical form to the actual SCCs, but they provide the same response to ECT signal. In order to determine the appropriated dimensions of imitative SCCs for simulating practical SCCs, this paper studied on the characteristics of the ECT signals due to different sizes of the imitative SCCs through numerical analyses. The specimens of imitative SCCs were modelled as a plate with tiny rectangular slits by varying the dimensions of gap between the tiny slits, length of the tiny slits and width of the tiny slits respectively. An absolute pancake coil was utilized in the numerical simulation as an ECT probe with different testing frequencies. Based on the numerical results, the equivalent conductivity ( $\sigma_{E}$ ) of the imitative SCCs was estimated, and the appropriated dimensions of the imitative SCC were predicted by using a response surface method. The results of this work are helpful to improve the reconstruction of SCC profile in inversion analysis. Furthermore, it can contribute to design the SCC reference specimen for practical SCC sizing, and for personnel training of ECT.

2. Numerical simulations

2.1 Basic scheme of FEM-BEM

To predict impedance of ECT probe, the solution of the magnetic vector potential ( $A$ ) field is necessary. The numerical algorithm of the Finite Element and Boundary Element (FEM-BEM) hybrid methods developed by the authors [16, 17] were utilized. The FEM-BEM code based on the $A$ - $\Phi$ formulation needs only conductor being meshed area, which can significantly reduce the number of the finite elements and the computational resources. In addition, as the probe is not necessary to be subdivided into FEM mesh, a probe of complicated geometry can be easily treated. The governing equations of the ECT problem in this study are described mathematically in the conductive region as the Eqs (1) and (2), where $\Phi$ , $\sigma$ and $\mu$ are the magnetic scalar potential, conductivity and magnetic permeability respectively.

$\displaystyle\frac{1}{\mu}\,\nabla^{2}A\;=\;\sigma\left({\frac{\partial A}{% \partial t}{\kern 1.0pt}{\kern 1.0pt}+{\kern 1.0pt}\,\frac{\partial\Phi}{% \partial t}}\right)$ (1) $\displaystyle\nabla{\kern 1.0pt}\cdot\,\sigma\,\left({\frac{\partial A}{% \partial t}{\kern 1.0pt}{\kern 1.0pt}+{\kern 1.0pt}\,\frac{\partial\Phi}{% \partial t}}\right)\;=\,\;0$ (2)

The air region can be governed as in Eq. (3), where $\mu_{0}$ is the magnetic permeability of air, and $J_{0}$ is the current density of the excitation coil.

$\frac{1}{\mu_{0}}\,\nabla^{2}A\;=\;-J_{0}$ (3)

Based on the Galerkin’s discretization procedure, the following system of linear equations can be deduced to solve the solution of vector potential $A$ . The deduction system equations of the FEM-BEM hybrid approach is shown as follows

$\left({\,\left[P\right]\,+\,\left[K\right]\,+\,j\omega\left[Q\right]\,+\,j% \omega\left[S\right]\,}\right)\,\left\{{{\begin{array}[]{*{20}c}A\hfill\\ \Phi\hfill\\ \end{array}}}\right\}=\;\left\{{{\begin{array}[]{*{20}c}{\left[M\right]\,\left% [G\right]^{-1}\,\left\{{F_{0}}\right\}}\hfill\\ 0\hfill\\ \end{array}}}\right\}$ (4)

where, $F_{0}$ is the abbreviated function of equation in term of $J_{0}$ , The matrix $P$ , $Q$ , and $S$ are the coefficient metrics of FEM, while the matrix $K$ , $M$ and $G$ are the coefficient metrics discretized by the BEM. Details of expressions can be found in [18].

2.2 Modelling of imitative SCCs and uniform conductive notch

As shown in Fig. 1a, the imitative SCCs fabricated using the diffusion bounding technique were modelled using the multiple tiny slits separated from each other to allow the eddy current pass through the crack region. An austenitic stainless steel 316 L plate was chosen as the specimen for 3D simulation with size of 20 $\times$ 40 $\times$ 4 mm in width, length and depth respectively, and with conductivity of 1.35 MS/m. The imitative SCCs were symmetrically introduced at the center of the plate with a total length of 20 mm with varying gap between tiny slits, and the length and width of tiny slits. Concerning the spatial resolution of the ECT probe, the width and length of tiny slits were set smaller than the ECT probe diameter since the spatial resolution of ECT is almost to the diameter of the probe utilized [8]. The details of the imitative SCC dimensions for simulations are shown in Table 1.

Figure 1.

Models for simulations. (a) Imitative SCC model for simulation and definition of variable, (b) the model of uniform conductive notch with its dimensions.

Table 1

Details of the imitative SCC dimensions

Variables	Dimensions (mm)
Gap between tiny slits ( $G$ )	0.2, 0.4, 0.6, 0.8, 1.0
Length of tiny slit ( $L$ )	0.4, 0.8, 1.2, 1.6, 2.0
Width of tiny slit ( $W$ )	0.2, 0.4, 0.6, 0.8, 1.0
Depth of tiny slit ( $D$ )	3 for all cases
Total length	20 for all cases

Referring to works of Yusa et al. [8, 13], it is appropriate to simulate the ECT signal of the SCC using a notch with uniform conductive region. Thus, the equivalent modeling utilized in this paper was chosen as a uniform conductive notch of width, length and depth being 0.2 mm, 20 mm and 3 mm respectively. This model was used to estimate the equivalent conductivity ( $\sigma_{E}$ ) for all imitative SCCs. The details of the model are shown in Fig. 1b. The uniform conductivity was varied inside the notch region from 0% to 30% of the base material conductivity (1.35 MS/m).

For FEM mesh of the models, both of the imitative SCCs and the uniform conductive notch were subdivided into a unity mesh. FEM and BEM nodes and elements were set covering the volume and the surface of the model. Dense elements were specified in the flaw zone with 0.2 mm side length. The numbers of elements of the FEM and BEM were 5,512 and 3,708 respectively. Figure 2 shows details of meshes for numerical simulation.

Figure 2.

Meshing details of the imitative SCCs and the uniform conductive notch models.

2.3 Simulation of ECT probe and excitation frequencies

Relations between the imitative SCC dimensions and the equivalent conductivity were investigated by simulation of the absolute pancake coil with a diameter of 3.2 mm and height of 0.8 mm as the ECT probe. The turn number of the coil was set to 140 turns. Total excitation current was set to 1 A, and the probe was discretized to 10 $\times$ 10 elements in its cross-section, i.e., was approximated by 100 ring currents. The simulation frequencies were 50 kHz, 100 kHz and 300 kHz respectively to consider the effects of frequency, i.e., a frequency (50 kHz) with skin depth deeper than the thickness of the specimen, a frequency (100 kHz) with skin depth deeper than the flaw but still in the conductor area, and a frequency (300 kHz) with skin depth less than the flaw depth. The simulation signals were acquired by scanning ECT probe along the length direction of the flaw (in Y direction referred from Fig. 2). A liftoff setting of ECT probe was constant at 0.5 mm.

2.4 Validation of the imitative SCC model

An ECT experiment was conducted to validate the simulation results by using the tiny slit dimension of 0.4 mm, 0.8 mm and 0.2 mm in width length and gap between tiny slit respectively. They were fabricated to a thickness of 10 mm of SS316L plate by using the diffusion bonding technique as illustrated in Fig. 3a. The tolerances of tiny slit sizes were $-$ 10% to $+$ 20%. The ECT instrument used in the experiment was the Ectane model by EDDYfi ${}^{\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}\textregistered}$ (Canada). An ECT probe with the same specification in the simulation was employed by excitation frequencies at 50 kHz, 100 kHz and 300 kHz to compare the results with the numerical data. The system used in the experiment is illustrated in Fig. 3b.

Figure 3.

(a) The imitative SCC dimensions of tiny slit width $=$ 0.4 mm, tiny slit length $=$ 0.8 mm and gap between tiny slits $=$ 0.2 mm, (b) The eddy current testing for validation of numerical results.

The average impedance signals obtained from the experimental data are compared to the simulation results as shown in Fig. 4. The impedances were plotted between resistance (R) and inductive reactance (XL) of the utilized ECT probe with error bars of standard deviation. The correlation between the experiment and the numerical results were in good agreement for all tested frequencies. The phases of signals could be similarly observed from both of the experimental and simulation results. The errors of the peak signal amplitudes were less than 3% of all cases.

Figure 4.

Comparison of ECT signals between numerical results and experiment results of the imitative SCC dimension of the tiny slit width $=$ 0.4 mm, tiny slit length $=$ 0.8 mm and gap between tiny slits $=$ 0.2 mm.

3. Simulation results

The simulation results of ECT signals corresponding to imitative SCCs are categorized to the impedance signals of varying gap between tiny slits, varying length of tiny slits and varying width of tiny slits. Then, the equivalent conductivities are estimated for different imitative SCCs. In summary, the prediction results of appropriated imitative SCC dimensions are established based on the specified equivalent conductivity for the ECT probe utilized.

3.1 Numerical results of varying gap between tiny slits

The ECT impedance amplitudes and phases affected by the varying gap between tiny slits at the constant tiny slit width $=$ 0.4 mm and length $=$ 0.8 mm are revealed in Fig. 5. Results of the impedance amplitude in logarithm scale gave a linear tendency for different frequencies at 50 kHz, 100 kHz and 300 kHz. The signal amplitudes decreased when the gap between tiny slits was increased as the basis of increasing conductive bridge between tiny slits. The impedance phases seemed to be little changes for the gap between tiny slits of 0.2 mm, 0.4 mm and 0.6 mm. However, the impedance phases for the gap between tiny slit at 0.8 mm and 1.0 mm were significant upward trend. The simulation results of frequency at 300 kHz could be clearly observed the difference than others. The results of tiny slits with further widths and lengths (not show in the above figure) gave the similar tendency.

Figure 5.

ECT impedance amplitudes and phases of varying gap between tiny slits at constant width of tiny slit 0.4 mm and length of tiny slit 0.8 mm for testing frequencies at 50 kHz, 100 kHz and 300 kHz.

3.2 Numerical results of varying length of tiny slits

The results of ECT impedance signals by varying length of tiny slits are shown in Fig. 6 by keeping constant width of tiny slit and gap between tiny slits at 0.4 mm and 0.2 mm respectively. The impedance amplitudes rose up owing to the increasing slit length from 0.4 mm to 1.2 mm for all of test frequencies in logarithm scale. Afterward, only small amplitude changes could be indicated for varying the tiny slit length from 1.6 mm to 2.0 mm. These results were caused from the spatial resolution of the ECT probe utilized. For the impedance phase results, there were small phase angle changes of all of test frequencies. However, shifting in phase of simulated frequency at 300 kHz was higher than those other frequencies. The results of further tiny slit widths and gap between tiny slits showed the similar tendency.

Figure 6.

ECT impedance amplitudes and phases of varying length of tiny slit at constant width of tiny slit $=$ 0.4 mm and gap between tiny slit $=$ 0.2 mm for tested frequencies at 50 kHz, 100 kHz and 300 kHz.

3.3 Numerical results of varying width of tiny slits

As shown in Fig. 7, the amplitudes of ECT impedance signals linearly increased in logarithm scale when the tiny slit widths were enlarged from 0.2 mm to 1.0 mm for all of test frequencies. These results were obtained from fixing the length of tiny slit at 0.8 mm and the gap between tiny slits at 0.2 mm. On the other hand, the impedance phases tended to decrease for all of the test frequencies. There was a significant phase decline in testing frequency of 300 kHz. The simulation results of other tiny slit lengths and the gap between tiny slits showed in the same trend.

Figure 7.

ECT impedance amplitudes and phases of varying width of tiny slit at constant length of tiny slit $=$ 0.8 mm and gap between tiny slit $=$ 0.2 mm for tested frequencies at 50 kHz, 100 kHz and 300 kHz.

3.4 Estimation results of equivalent conductivity in crack region

Correlations between the absolute values of impedance ( $Z$ ) and the equivalent conductivity for different frequencies obtained by varying uniform conductivity in the notch width of 0.2 mm are shown in Fig. 8a. The correlation followed the fitting relation of the Eq. (5) with 95% confidence interval. From the regression analysis results, the R-square of data variation were higher than 0.99 for all of test frequencies. The residual of fitting between the mathematical model and the data less than 0.8% with random distribution were observed in Fig. 8b. In addition, the Root Mean Square Errors (RMSE), which estimates the standard deviation in the data were less than 0.5%. Consequently, the estimating conductivity for imitative SCCs could be a value from 0% to 58% $\pm$ 0.5% of the based material conductivity. The details of approximated coefficients are presented in Table 2.

$\sigma_{E\;}=\;a\,\cdot\,e^{bZ}\,$ (5)

Table 2

Approximated coefficients of correlation between impedance (Z) and the equivalent conductivity

Frequency	Coefficients at 95% confidence interval
50 kHz	a $=$ 58.08 $\pm$ 2.43, b $=$ $-$ 10.10 $\pm$ 0.42
100 kKz	a $=$ 57.92 $\pm$ 2.53, b $=$ $-$ 3.18 $\pm$ 0.14
300 kHz	a $=$ 56.83 $\pm$ 2.71, b $=$ $-$ 0.62 $\pm$ 0.03

Figure 8.

(a) Curve fitting between the impedance (Z) and the equivalent conductivity from the uniform notch at width of 0.2 mm, (b) Residual from fitting.

Figure 9 shows the calculated $\sigma_{E}$ to represent the response of changing imitative SCCs dimensions for frequencies of 50 kHz, 100 kHz and 300 kHz respectively. In Fig. 9a, the $\sigma_{E}$ values are obtained by varying gap between tiny slits but fixing the width of tiny slit = 0.4 mm and length of tiny slit $=$ 0.8 mm. It could be indicated that the value of $\sigma_{E}$ significantly increased for all of test frequencies according to varying the gap size at small value such as 0.2 mm, 0.4 mm and 0.6 mm. However, the $\sigma_{E}$ value at frequency of 300 kHz rose up separately when the gap between tiny slits was changed to 0.8 mm and 1.0 mm.

Figure 9.

The equivalent conductivity obtained from (a) varying gap between tiny slits, (b) varying length of tiny slit, (c) varying width of tiny slit.

The results of varying length of tiny slit are shown in Fig. 9b by setting the constant width of tiny slits and the gap between tiny slits as 0.4 mm and 0.2 mm respectively. The value of $\sigma_{E}$ tended to decrease for the lengths of tiny slit as 0.2 mm and as 1.2 mm all of test frequencies. After that, the $\sigma_{E}$ seemed to be constant for increasing tiny slit length between 1.6 and 2.0 mm. As shown in Fig. 9c, the results by varying the width of tiny slit are obtained by fixing length of tiny slit to 0.8 mm and the gap between tiny slits to 0.2 mm. The values of $\sigma_{E}$ were significant declines in exponential form for all of frequencies, but the results of 300 kHz had little difference with those 50 kHz and 100 kHz.

From the estimation results, it could be indicated that there were small variations in the equivalent conductivity for frequencies of 50 kHz, 100 kHz and 300 kHz due to varying gap between tiny slits $\leqslant$ 0.6 mm, length of tiny slits $\leqslant$ 1.2 mm, and width of tiny slits $\leqslant$ 0.6 mm. These dimensions were much less than the diameter of the probe utilized (3.2 mm). By adjusting the slit dimensions in this range, the equivalent conductivity of the imitative SCCs could be controlled from 2.6% to 38.3% of the base material, and were almost the same for different frequencies.

4. Design strategy of imitative SCC dimensions at specified equivalent conductivity

In order to design the imitative SCC dimensions, statistical methodology was utilized to treat the significant variables affecting the equivalent conductivity including frequency and dimensions of slits. All variables were primarily analyzed by using the analysis of variance (ANOVA). The results could be indicated that there was no significant effect to the response of $\sigma_{E}$ by changing frequency. On the other hand, the dimensions of the gap between tiny slits $\leqslant$ 0.6 mm, the length of tiny slit $\leqslant$ 1.2 mm and the width of tiny slit $\leqslant$ 0.6 mm affected to the value of $\sigma_{E}$ significantly. Therefore, response surface method (RSM) was used to analyze the relationship between the dimensions of imitative SCC and the target of equivalent conductivity.

Table 3
Estimated parameters of response surface at specified different width

Slit width	Estimated coefficient parameters						Box-Cox coefficient
(mm)	$\beta_{1}$	$\beta_{2}$	$\beta_{3}$	$\beta_{4}$	$\beta_{5}$	$\beta_{6}$	( $\lambda$ )
0.2	4.199	1.180	$-$ 3.464	$-$ 1.368	0.938	1.885	0.124
0.4	3.788	2.502	$-$ 4.196	$-$ 2.434	1.237	2.111	$-$ 0.010
0.6	3.465	3.347	$-$ 4.755	$-$ 2.969	1.489	2.199	$-$ 0.009

Figure 10.

Contour plots for prediction the length of tiny slit and gap between tiny slit for the imitative SCC at fixed width of tiny slit to (a) 0.2 mm, (b) 0.4 mm, (c) 0.6 mm.

As RSM scheme, the equivalent conductivity was fitted by a full quadratic model in terms of the gap between tiny slits ( $G$ ) and the length of tiny slit ( $L$ ) as shown in Eq. (6). The responses of $\sigma_{E}$ from simulated frequencies at 50 kHz, 100 kHz and 300 kHz were employed in the model. Box-Cox transformation was also utilized to improve the data distribution by the logarithmic function [19]. The results of fitting could be estimated by RSM coefficients as shown in Table 3 under the confidence interval at 95%, and the R-square higher than 0.99. Finally, the predicted $\sigma_{E}$ values of imitative SCCs represented in contour plot between tiny slit gap and length by setting different widths of tiny slit as shown in Fig. 10. The RSM results of imitative SCC dimensions covered the $\sigma_{E}$ in ranges of 9% to 39%, 5% to 30%, and 3% to 25% for setting width to 0.2 mm, 0.4 mm, and 0.6 mm respectively.

$\displaystyle\ln\,\left({\sigma_{E}}\right)=\;\beta_{1}\,+\,\beta_{2}G+\,\beta% _{3}L+\,\beta_{4}G^{2}+\,\beta_{5}L^{2}+\,\beta_{6}GL$ (6)

In practice, the width of tiny slit was specified to design the imitative SCCs dimensions as given equivalent conductivity. Four typical observations of the values, for example cases, 20%, 15%, 10% and 5% of the base material, were chosen as target values. By setting the specified widths of tiny slit, the optimized RSM results of the gap between tiny slits and length of tiny slits could be determined and presented as in Table 4. These dimensions also gave the accurate prediction of the $\sigma_{E}$ values nearby the targets (less than 1% of error). However, high error of the prediction was indicated for the target $\sigma_{E}$ value at 5% through setting width of tiny slit 0.2 mm. This was due to out of suitable range for determining low values of $\sigma_{E}<$ 7.6%. From the prediction results, this strategy revealed the proper imitative SCC dimensions for given the equivalent conductivity. The importance imitative SCC dimensions containing gap between tiny slits, length of slit and width of slit were obtained, which could be used as the design value for fabrication of the imitative SCC specimens.

Table 4

Prediction results of the imitative SCC dimensions at specified the equivalent conductivity

Specified width of	Targets of	Prediction of	% Error	Prediction of dimensions (mm)
tiny slit (mm)	(%)	(%)		Gap of tiny slit	Length of tiny slit
0.2	20	20.00	$-$	0.20	0.54
0.4		20.05	0.25	0.40	0.52
0.6		19.89	0.55	0.39	0.84
0.2	15	15.03	0.20	0.20	0.68
0.4		15.00	$-$	0.40	0.67
0.6		15.06	0.40	0.26	0.40
0.2	10	10.01	0.10	0.30	1.20
0.4		10.00	$-$	0.20	0.63
0.6		10.02	0.20	0.40	0.72
0.2	5	7.60	52.0	0.20	1.20
0.4		5.00	$-$	0.20	1.04
0.6		5.00	$-$	0.20	0.75

5. Conclusion

This work studied the characteristics of ECT signals due to imitative SCCs of different dimensions through numerical simulation by using an FEM-BEM code. The SS316L specimens were modelled as a plate with multiple tiny slits. An absolute pancake ECT probe was utilized in the numerical simulation. The ECT signals of different excitation frequencies were simulated by scanning the probe along the length of the crack region with constant lift off, and scanning steps. The simulation results from the conductive notch model width of 0.2 mm were used to estimate the equivalent conductivity. The dependences of the $\sigma_{E}$ on the parameters of the imitative SCCs were then, analyzed. The design strategy of the imitative SCC to control its conductivity was presented. Based on the numerical results, the following conclusions were obtained:

Increasing gap between tiny slits decreased the impedance amplitude of the ECT signals. On the other hand, increasing tiny slit length and width led to increase the impedance amplitude of the ECT signals.

Phase of the ECT signals changes significantly at a higher frequency. Therefore, the test frequency of the pancake probe should be set to produce the eddy current penetration deeper than the crack depth to avoid the effect of phase shifting.

The significant dimensions of the imitative SCC should be smaller than the utilized ECT probe to prevent the effects from the phase shifting and spatial resolution of the probe. Following the simulation results obtained by an ECT probe with a diameter of 3.2 mm and height of 0.8 mm, the gap between tiny slits and width of tiny slit should be set $\leqslant$ 0.6 mm, and the length of tiny slit should be set $\leqslant$ 1.2 mm.

The maximum amplitude of ECT impedance signals could be used effectively to estimate the equivalent conductivity. In addition, there was no significant effect from changing the frequencies on the value.

The statistical methodology is efficient to predict the appropriated dimensions of imitative SCCs, which is important to design an imitative SCC specimens, and helpful to SCC profile reconstruction through inverse analysis of ECT signals.

Footnotes

Acknowledgments

The authors would like to thank the National Magnetic Confinement Fusion Program of China (2013GB113005), National Science Foundation of China (No. 51577139 and 51407132) for funding this study.

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A numerical study on eddy current signal characteristics of imitative stress corrosion cracks

Abstract

Keywords

1. Introduction

2. Numerical simulations

2.1 Basic scheme of FEM-BEM

2.4 Validation of the imitative SCC model

3.1 Numerical results of varying gap between tiny slits

Table 3 Estimated parameters of response surface at specified different width

Footnotes

Acknowledgments

References

Table 3
Estimated parameters of response surface at specified different width