The nonlinear kinematic hardening (NLKH) model was first proposed by Armstrong and Frederick in 1966 using a non-linear backstress. An extension, superimposing multiple non-linear backstresses, was proposed by Chaboche, which allowed a better description of cyclic stress-strain curves [1]. Other hardening models like the Ohno-Wang model were also proposed to describe the ratcheting phenomenon under multi-axial loading conditions [2][3][4]. Such endeavors have been continued to date, to improve the accuracy of phenomenological modeling of cyclic deformation, at the expense of increasing the model complexity. However, obtaining kinematic hardening parameters in the NLKH models from stress-strain data also brings challenges, as the kinematic hardening function involves multiple hardening parameters, yield strength, and elastic modulus.
When the computational power was limited, the identification of kinematic hardening models typically used a multi-step linear regression to extract the optimal set of parameters [5,6]. Chaboche proposed three backstresses to describe the stabilized hysteresis loops within 0.01-4.0 % strain ranges, and the model includes two NLKH terms and one linear term [1]. The linear term represents the backstress that passes through the origin and could be first calculated from the linear portion of the backstress curve [4]. The parameters of the two nonlinear terms could be first estimated by appropriately segmenting of the backstress curve and manually calibrated to achieve a good fitness [6,7]. However, this segmentation method often requires manual calibration of hardening parameters to reach a target objective function, which is time-consuming and restricts the number of backstress terms that could be optimized.
With the development of modern computers, computation no longer becomes a limiting factor for the calculating process, so more sophisticated optimization algorithms can easily be implemented to address the identification of hardening parameters within a reasonable amount of time. The NLKH parameters of multiple backstress terms could be optimized using global optimization algorithms without curve segmentations and manual calibrations [5]. The built-in function in the finite element (FE) software like Abaqus also allows the extraction of hardening parameters from the half-cycle hysteresis loop [6]. One of the modern optimization algorithms used to optimize the kinematic hardening parameters is the genetic algorithm (GA). Different from the conventional approach which may require manual calibration, GA is an algorithm based on the theory of evolution that can completely automate the optimization process with one or multiple objective functions. Mahmoudi et al. employed a multi-objective genetic algorithm (MOGA) to optimize the nonlinear kinematic hardening parameters of three and four backstresses [8]. They used two fitness functions, viz., stress-strain curves and ratcheting curves, and solved the constrained optimization of hardening parameters. Optimizing the kinematic hardening parameters of more than three backstresses has also been achieved in other works [9,10]. In addition to GA, particle swarm optimization (PSO) has also been used in the literature for the optimization of hardening parameters with constrained conditions [11,12]. The modern algorithms such as GA and PSO allow more than three backstresses to be used in the kinematic hardening model so that a better fitting of experimental data can be achieved. However, these optimization processes do not involve the physical significance of parameters in the elastic-plastic data and thus may require an extensive number of trial-and-errors to reach convergence. Many literatures adopted the objective function that is calculated based on the fitness of the simulated stress and strain curve to the experimental data [8,10,12]. The number of objective function calculation that includes both stress and strain simulation and fitness calculation is proportional to the number of trial-and-errors in the optimization algorithm. Therefore, the robustness of modern algorithms such as GA and PSO may come with the cost of extended computing time and model complexity. In some cases, more than three backstresses may not be necessary, which tends to result in the overfitting of the experimental data, as the stress-strain data could vary by cycles and sampling rate.
The conventional segmentation method typically separates the elastic and plastic range of experimental stress-strain data by determining an appropriate elastic limit [6]. The concept of applying the elastic limit/offset to extract the linear portion of the hysteresis loop was first proposed by Cottrell [13] and improved by Fournier et al. to accommodate the scattering of experimental data [14]. However, the value of elastic limit is uncertain and will significantly influence the subsequent optimization process by changing the corresponding yield strength and fitting data [10,12]. To address this concern in the segmentation method, this paper presents a novel approach to optimize hardening parameters from the cyclic stress-strain data using a combination of a sequential search of the elastic range and a gradient-based, non-linear programming (NLP) solver. The incremental elastic-limit approach adopts a stepwise segmentation process with an appropriate step size to obtain the backstress data. A nonlinear optimization algorithm is then used to calculate the hardening parameters. The optimized hardening parameters at all the elastic limits are evaluated based on the fitness of the simulated stress-strain data using a numerical simulating algorithm. The proposed method removes the elastic parameters from the optimization process and thus reduces the time and complexity of the optimization process.
The optimization method was applied to calculate the combined hardening parameters of coke drum materials and Ni-based weld metals. The cyclic stress-strain data were obtained from the fully reversed lowcycle fatigue tests at 250 • C. The severe heating and quenching cycles of coke drums result in the low-cycle fatigue damage of the vessel body and attachments, further exacerbated by material degradation due to thermal ageing [15][16][17][18]. Ni-based alloys such as Alloy 625 and Alloy 182 have been extensively used to restore the excavated damage or provide additional support to the damaged region of coke drums. The discrepancy of cyclic behavior between weld and base metal determines the strain distribution at the dissimilar metal joint under high cyclic plastic deformation [19,20]. The optimization process was implemented using the MATLAB™ software to extract the hardening parameters from cyclic stress-strain data for the fitness-for-service evaluation of coke drum repair welds.
Isothermal low-cycle fatigue (ILCF) tests are performed at high strain amplitudes to obtain the cyclic behavior of coke drum materials and welds. The tested coke drum base materials are a C-0.5Mo steel (SA 204 Grade C), three heats of as-received 1.25Cr-0.5Mo steels (SA 387 Grade 11) and a service-aged 1.25Cr-0.5Mo steel. The composition and condition of the base materials (BMs) are summarized in Table 1. Two types of common Ni-based alloys, Alloy 182 and Alloy 625 shown in Table are selected to fabricate repair weld mockups. Three different welding processes are adopted for Alloy 625 mockups. The composition and welding processes of the tested weld metals (WMs) are summarized in Table 2. Fig. 1 schematically illustrates the sample locations in a weld cross-section and the sample geometry. The BM and WM samples are extracted transverse to the welding direction and machined into the dogbone shape. The gauge section of the sample is 10 mm in length and mm × 6 mm in cross-section. The ILCF tests are performed using a Gleeble™ thermo-mechanical simulator based on the ASTM E606 standard [21]. In these experiments, the force is measured by the load cell in the Gleeble™ machine, and the strain is measured by an extensometer mounted to the surface of the gauge section [22].
The ILCF test temperature is controlled at 250 • C, which represents the temperature at which coke drums typically experience the maximum deformation during the "quenching" step of the coke drum operation [23]. The strain-controlled tests are performed in a fully reversed condition (strain ratio = -1) at ±2.0 % strain amplitude (=Δε/2). A strain rate of 0.2 %/s or 2x10 -3 /s is adopted, and the data sampling rate is Hz. Cyclic stress and strain data from the first 50 cycles of fully reversed loading are adopted to optimize the NLKH and IH parameters. In particular, the NLKH parameters are determined from the 50th cycle, assuming that by that time the material behavior (cyclic hardening or softening) has been saturated. The IH are then determined using all cycles of stress and strain data.
The total strain, ε, consists of the elastic strain (ε e ) and plastic strain (ε p ) components as ε = ε e + ε p .
(1)
For one-dimensional uniaxial loading, the Hooke's law is defined as
where E is the elastic modulus. According to a yielding criterion, the yield function is defined as
where α is the back stress term, and σ Y is the yield strength defined by the sum of the isotropic hardening (IH) term (R) and the initial yield strength (σ 0 ). As is shown in Fig. 2, the isotropic hardening describes the change of the yield surface size, which is equivalent to the yield strength.
The current radius of the yield surface is described by the IH term using an evolutional equation as
in which Q and b are material constants, and dp is the increment of the accumulative plastic strain, which is given by
The integration of Equation ( 4) gives the non-linear isotropic hardening term as
The isotropic hardening term can be used to describe the cyclic hardening/softening due to plastic strain accumulation, but it cannot be used to reveal the translation of the yield surface under a fully reversed cyclic loading.
Fig. 3 (a) and (b) show a fully reversed stress-plastic strain curve with the Bauschinger effect and the kinematic hardening in the twodimensional stress space. Compared with isotropic hardening, kinematic hardening describes the changes in yield surface position. The position of a yield surface in the stress space is determined by the backstress term, α. The backstress term in kinematic hardening can describe the Bauschinger effect, which refers to a lowering yield point when the stress is switched to a reversed direction [24][25][26]. The mechanism of the Bauschinger effect is related to the irreversible dislocation motions during cyclic loading, which is caused by impermeable dispersed particles, and defects such as grain boundaries [27].
Prager's kinematic hardening model is the first model to simulate the evolution of backstress using a linear form. However, this model is not capable of describing the non-linear hysteresis loop during cyclic loading. A modification of the linear kinematic hardening was proposed by Armstrong and Frederick to add a non-linear term as
Chemical composition and conditions of tested coke drum base materials [15].
Materials C Mn Si Mo Cr Ni Cu Fe P S Condition C-0.5Mo 0.2 0.7 0.21 0.48 0.09 0.11 0.16 Bal. 0.01 0.01 Normalized only 1.25Cr-0.5Mo Heat A 0.14 0.44 0.54 0.55 1.35 0.14 0.14 Bal. 0.01 0.01 Normalized & Tempered Heat B 0.13 0.61 0.55 0.6 1.5 0.18 0.11 Bal. 0.01 0.01 Quenched & Tempered Heat C 0.13 0.6 0.58 0.56 1.43 0.2 Bal. 20 Years Service-aged where C and γ are the kinematic hardening constants and dp is the equivalent plastic strain defined in Equation ( 7). A superposition of several A-F non-linear kinematic hardening models was proposed by Chaboche [1] to better represent the hysteresis loops:
where C i and γ i are the kinematic hardening coefficients. Typically, three to four terms are sufficient to describe the behavior of structural metals [28][29][30]. In the present work, two non-linear terms and one linear term are adopted in the Chaboche model, to simulate cyclic hysteresis curves. Under uniaxial loading conditions, the stress can be calculated from the sum of the initial yield strength, the IH term and the backstress term as
An implicit stress integration method is created based on the radial return algorithm of nonlinear kinematic hardening published by Simo and Taylor [31]. The stress integration method does not consider the change of the IH term and thus is only applied to the calculation of stress and strain data when the cyclic hardening/softening is negligible. The radial return algorithm is implemented first by calculating an elastic trial stress, σ T n+1 , at the Δε n+1 . For instance, using Eq. ( 2), the stress, σ n+1 , is given as (11) where E, Δε n+1 , and Δε p n+1 are the elastic modulus, total strain increment, and plastic strain increment, respectively. Assuming a trial stress as
Eq. ( 12) then gives
where φ = ± 1 is the direction of plastic strain change. Assuming that the one-dimensional yield function, F n+1 , is
where σ Y is the yield strength without the isotropic hardening term. If the consistency condition (F n+1 ≤ 0) is satisfied with the given trial stress, then each i-th component of the NLKH term is written as
This equation is valid for uniaxial loading only. It can be reorganized as
where
Therefore,
The differential of the yield function is given by
If the trial stress falls outside the yield surface (F n+1 > 0), an iterative process of calculation will be initiated, using the strain corrector,
n+1 , until the trial stress satisfies the consistency condition. The update of strain corrector is given as
where k is the current iteration. The iterative process to calculate the equivalent plastic strain increment, Δp n+1 , based on the consistency condition and Newton-Raphson is described in Table 3.
The elastic range defines the linear portion of a cyclic stress-strain curve that follows Hooke's law. Similar to the common offset value, 0.2 %, a certain elastic limit (d) can be used to calculate the yield stress (σ Y ) and elastic modulus (E) in a stabilized hysteresis loop.
As is shown in the shadowed region in Fig. 4 (a), the elastic range is defined by the plastic strain (ε p ) as
where Δε p /2 is the plastic strain amplitude of a stress-strain curve. The stabilized yield strength (σ s Y ) can be estimated using the maximum stress (σ e max ) and minimum stress (σ e min ) in the elastic range as
Then the ith backstress (α i ) can be calculated by
The plastic strain, ε p , and backstress, α, outside the elastic range of the upper stress-strain curve is used to extract the nonlinear kinematic hardening parameters.
The re-generation of the experimental data serves two purposes: to reduce the impact of experimental factors on parameter optimization and to generate fitting data for the optimization process with minimum loss of the data integrity. The first step of data re-generation is to separate the upper and lower stress and strain data of a stabilized hysteresis cycle and to fit the data using the 'fit' function in the MATLAB™ software with the 'smoothingspline' algorithm. The 'smoothingspline' algorithm provides a non-parametric regression model typically used to fit noisy signal data. A smoothing spline function, s(ε), is generated in the algorithm to minimize the objective function with a specified smoothing parameter, h, and weights w i (taken here to be 1):
The smoothing parameter h is selected here to be 0.995, which produces approximately a cubic spline interpolant, with an intention to reflect the curve of experimental data with a high fidelity. Using the generated smoothing spline function, a new set of stress data can be created using a logarithmically distributed strain array. The comparison between experimental and re-generated stress-strain data in Fig. 5 shows that the data proportion in the elastic range increases significantly, and data points in the plastic range become sparse in the plastic range. The benefits of such distribution are presented in the discussion section.
The incremental searching is implemented at a pre-determined step size, Δd, and iteration number, n. The adopted step size and the iteration number are 10 -4 and 20. The elastic limit at the ith iteration, Δd⋅i, is used to separate the elastic and plastic data as is shown in Fig. 4. The elastic data provides the yield stress and elastic modulus, and the plastic data is used for the kinematic hardening parameters optimization. The NLKH model used in this work includes two nonlinear terms and one linear term (N2L1), which are sufficient to describe the cyclic behavior of steels within the 0.04 strain range according to Chaboche [1]. The NLKH equation used to fit the stabilized back stress versus plastic strain is provided by [1].
where α i 0 is the initial back stress value and ε p0 is the initial plastic strain ( -Δεp 2 ). The coefficient of the linear term (C 3 ) is calculated from the linear portion of the fitting data in the high plastic range, to reduce the number of optimization variables. The MultiStart solver built in MAT-LAB™ Global Optimization Toolbox is used to search for the non-linear hardening parameters in the NLKH model. The following initial values and limits in Table 4 are used for the optimization of Eq. ( 26). Δσ/2 is the stress amplitude of the fitting hysteresis loop.
The MultiStart Algorithm allows parallel computing of local minima of the objective function from multiple start points, which enables a thorough searching of the global minimum within the given constraints [32]. The 'lsqcurvefit' solver of MATLAB™ is applied to fit the NLKH Table 3 Equivalent plastic strain calculation through iterative process.
) ,
Fig. 4. Elastic region defined by a given elastic limit, d, in a stabilized cycle represented by the plastic strain and stress data.
equations. The number of starting points to run the solver algorithm is set to be 50. Fig. 6 (a) presents the fitting results against the experimental data extracted at the elastic limit = 2.25 x 10 -4 and with three backstresses. Then a fully reversed stabilized cycle is simulated using the stress integration method in Table 3 and the optimized hardening parameters. Fig. 6 (b) shows the comparison of a stabilized cycle of stress-strain data and the simulated curve using these optimized parameters.
The optimization of the NLKH parameters is implemented at each elastic limit, Δd • i, and then the fitness (ϕ i ) is calculated based on the root mean square error (RMSE) between the simulated and experimental stress and strain data [10], which is given as
The fitness calculation requires that the strain values are consistent between simulated and experimental data, so the experimental stress data is regenerated using the smoothing spline function in the data regeneration step. Fig. 7 presents the variation of objective function and yield strength with IEL in an example of hardening parameters optimization. The fitness decreases and stabilizes at 2.25 x 10 -4 . Fig. 8 presents the simulated stress-strain data using the NLKH parameters optimized at four different elastic limits including 5 x 10 -5 , 7.5 x 10 -5 , 2.25 x 10 -4 , and 9.75 x 10 -4 .
Fig. 9 summarized the optimization flow chart of the combined hardening parameters that includes the optimization of NLKH parameters and IH parameters. The optimization of NLKH parameters follows the procedures presented in Section 4.3, which generates a list of NLKH parameters and yield strengths at various elastic limits. Although the set of NLKH parameters with the lowest fitness is always prioritized to use for the IH optimization, it may not work well for the other cycles due to the corresponding yield strength. In such cases, the fitness result needs to be balanced with the yield strength value to select the optimal NLKH parameters. The classical isotropic hardening with one term shown in equation ( 9) is used. Although the fitting of yield strength can be easily performed using any nonlinear fitting algorithm, it is important to obtain high-quality yield strengths from the cyclic stress and strain data using the method shown in Fig. 4. Once again, the IEL approach was applied to calibrate the yield strength of loading cycles using the NLKH parameters obtained from the stabilized cycle to ensure consistency of the optimization.
As is illustrated in Fig. 2, isotropic hardening refers to the homogenous expansion and shrinkage of the yield surface and thus describes the hardening behavior under cyclic loading. The yield strengths are extracted from the 50th cycle of stress and strain data since it is assumed that cyclic hardening or softening is saturated (stabilized) by that time. On the other hand, the IH parameters are extracted considering all cycles. At the nth cycle, the accumulative plastic strain, p n , is given by
where Δεp 2 is the plastic strain amplitude. The yield strength at the nth cycle, σ eq n , can be calculated using Eq. ( 17). Then the objective becomes searching for the cyclic yield strengths to describe the cyclic stress-strain data using the optimized NLKH parameters. The calculation is implemented from the first loading cycle. At the i th cycle, the IEL approach is applied to produce yield strength candidates at different elastic limits, and the fitness terms are calculated defined in Eq. (27). In addition to the fitness term, an additional convergence term is introduced to restrict the scattering of IH data. The converge term is the absolute difference between the yield strength and the average of the yield strengths from three previous cycles. The modified fitness, φj n , is then given by: where ϕ j n and σj n are the fitness term and calculated yield strength at the elastic limit of Δd • j, and k is the coefficient of consistency term. The stress value with the lowest fitness is selected as the IH stress at the nth cycle. Once the iterative calculation of IH terms is completed for the 50 cycles of stress-strain data, the isotropic hardening parameters, Q and b, are calculated based on the fitting of yield strengths and accumulated plastic strain.
As described above, the optimization process is divided into two steps: NLKH parameters optimization and IH parameters optimization. The NLKH parameters are calculated from a stabilized cycle (here, the 50th cycle), and used to calculate the optimal equivalent stress per cycle. Then the equivalent stress data from each cycle are used to fit the IH parameters.
Table 5 summarizes the IH and NLKH parameters of the low-alloy steels and weld metals of this study, as determined with the optimization process shown in Fig. 9. The k-values (Equation ( 29)) adjusted for the IH parameter optimization of base and weld metals are summarized in Table 6. Figs. 10 and 11 show the optimized yield strength and accumulative plastic strain of the weld and base metals, and the IH fitting of the hardening/softening behavior. Among the pressure vessel steels, C-0.5Mo steel and 1.25Cr-0.5Mo steel N&T exhibit a notable hardening before stabilizing; whereas the 1.25Cr-0.5Mo steel serviceaged and Q&T exhibit a mild softening before stabilizing. On the other hand, the Alloy 625 weld metals show a similar stress level after stabilizing, but the GTAW weld metal exhibits a higher initial stress than the weld metals fabricated using SMAW and GMAW processes. In terms of the IH parameters, all the Alloy 625 weld metals exhibit a similar exponential coefficient, b, but the coefficient, Q, varies with the welding process. In comparison, the Alloy 182 weld metal fabricated using the SMAW process exhibits a lower stabilized stress level than the Alloy 625 weld metals. The IH parameters of Alloy 182 are smaller than the Alloy 625 weld metal using the SMAW process.
In the final step, FE analysis of the cyclic hardening behavior are implemented using the optimized parameters included in Table 5. Figs. 12 and 13 show the comparison of experimental and simulated data at the 1st and 50th cycle. The optimized hardening parameters susccessfuly describe the cyclic hardening/softening and the stabilized plastic flow of the base and weld metals. However, the deviation was observed at the monontonic and 1st cycle, which will be elaborated in the discussion.
A novel optimization approach is proposed in this study to generate the combined hardening parameters from fully reversed loading stress and strain data. The IEL approach generates a set of yield strengths and optimized NLKH parameters at each elastic limit using the nonlinear optimization package available in the MATLAB™ software. The IEL approach proposed in this study adopts a stepwise segmentation process and a robust global optimization algorithm. At each elastic limit, the elastic modulus and yield strength can be determined from the stress and strain data within the elastic range, which reduces the computational time by removing two parameters to be optimized. In contrast, advanced optimization algorithms such as GA and PSO could be implemented without segmenting the elastic range from the cyclic data, because the optimizers are based on evolution or other statistical theories instead of the physical meaning of the parameters. However, a large population of parameter set may need to be generated during the optimization process. Since the calculation of each objective function requires both cyclic data simulation and fitness calculation, the computation time required to implement advanced optimization algorithms is expected to be long. The combination of the IEL approach and nonlinear optimization could reduce the extensive objective function calculation and obtain good fitting parameters efficiently. In addition, the IEL approach is considered straightforward to apply without requiring extensive optimization knowledge, as it is a conventional approach to determining the elastic range using a certain elastic limit [6].
Another novelty of this approach is using the stress integration method to calculate the objective function of the hardening parameters at different elastic limits. Many literatures have reported using the fitness of the experimental and simulated data as an objective function [5,8,9], but the optimization time could increase significantly without an efficient simulation algorithm of the hardening parameters. Previous studies have adopted the coupling of optimization algorithms and FE software to evaluate the similarities between simulated and experimental data [10]. However, such efforts may be time-costly and require good knowledge of FE software subroutine scripting. Based on the fundamental plasticity theory, the stress integration method in this paper is used to simulate the stress and strain under fully reversed loading with great accuracy and efficiency. This allows a prompt calculation of the objective function and contributes to the reduction of total optimization time.
On the other hand, the re-generation of stress and strain data is considered critical to implement the IEL approach, for the following reasons. At a constant strain rate, the stress-strain data from a cyclic test are mainly distributed within the plastic range when the strain amplitude is large, while the data amount in the elastic range tends to be low. However, such distribution is not favorable for an incremental search of the elastic limit, because the low amount of data provides few elastic range options for the optimization process, and the fitting result can be subject to variation due to different test sampling rates. On the other hand, the high end of the fitting data is mostly determined by the linear coefficient, C 3 , whereas the other NLKH parameters are mostly determined by the initial segment of the fitting data [10]. As is shown in Fig. 5, the data at the high end take a large proportion of the fitting data, which becomes redundant and could reduce the curve fitness represented by the NLKH parameters. These questions were resolved here by regenerating the fitting data with a logarithmically distributed strain, as shown in Fig. 5. This data re-generation enables the optimization process to be sampling-rate agnostic and provides sufficient data inside and close to the elastic range for the IEL approach.
The study uses the strain-controlled low-cycle fatigue test data for the optimization of IH and NLKH parameters. The IEL approach is first applied to segment the elastic and plastic range from the stabilized hysteresis loop and then used to calculate the yield strengths for isotropic hardening. However, the assumption that the NLKH parameters from the stabilized cycle works for all the cycle may not necessarily .25Cr-0.5Mo Service-aged 361 145,761 -45 0.93 37,340 1,341 17,838 252 2,717 1.25Cr-0.5Mo N&T 264 162,974 53 1.51 89,059 1,375 24,784 250 3,906 1.25Cr-0.5Mo Q&T 400 167,451 -71 0.83 82,985 1,074 16,393 234 2,508 Weld Metal Alloy 182 SMAW 182 137,885 142 1.94 177,065 2,227 35,309 309 3,072 Alloy 625 GTAW 227 168,798 138 1.92 106,105 2,490 59,900 279 3,776 Alloy 625 GMAW 161 201,131 257 4.02 139,632 1,760 76,771 337 4,996 Alloy 625 SMAW 165 198,987 182 3.25 121,379 2,153 69,452 272 4,167 be accurate. Figs. 12 and 13 show that the simulated stress and strain curves using optimized parameters agrees well with the experimental data at the stabilized cycle, but some discrepancies exist at the 1st cycle. The reason could be related to the accumulative plastic strain dependence of kinematic hardening parameters. Okorokov et al. addressed the parameters dependence of accumulative strain by assuming saturation functions, which were calibrated using multiple cycles including the monotonic curve [33]. For simplicity, this study adopts consistent kinematic hardening parameters optimized from the stabilized cycles, thus some discrepancies are expected at the initial cycles.
The IEL concept was developed in this work to optimize the isotropic and kinematic hardening parameters from fully reversed cyclic stressstrain data of pressure vessel steels and Ni-base weld metals. At a given elastic limit, back stress, and strain data were extracted from the stabilized stress-strain curve and fitted using nonlinear fitting algorithms in MATLAB™. The data re-generation procedure was introduced to regenerate the stress and strain data with a logarithmically distributed strain and a nonparametric regression function. The regenerated stress-strain data exhibited a concentrated distribution within the elastic range, which was suitable for the IEL optimization and enabled the Fig. 12. Comparison between experimental results and finite element analysis of coke drum base materials. Stress-strain data are extracted from the 1st cycle and the 50th cycle of 2.0 % strain experimental data. The hollow circles represent experimental results, and the solid lines represent numerical simulations. fitting data to be sampling-rate agnostic. In addition, an implicit stress integration method was developed to simulate the stress-strain curves, which were then used to calculate the objective functions of NLKH and IH parameters optimization. Workflows of NLKH and IH optimization were established and programmed in the MATLAB™ software. The combined hardening parameters of pressure vessel steels and Ni-base weld metals were obtained using the established workflows and validated through 1-element FE simulations of the experiments.