Approximation Method for Stress–Strain Using Metamodel Parameter Updating

Round 1

Reviewer 1 Report

In what follows some considerations and suggestions to improve the quality of the work:

• The abstract and introduction sections must be written in a much simpler way. It is very hard to understand the objective of the paper.
• The authors refere to the “typical method”, but there is no description of it
• A workflow should be added to understand the analysis logic
• The statement “When a necking phenomenon occurs in the specimen, the stress and strain distributions appear nonlinearly in the necking area, and the assumption of uniaxial stress is no longer valid” is not clear.
• The statement “FE simulation is a numerical analysis software application that can derive the engi-

neering stress–strain value of the FE model using the true stress–strain metamodel properties” should be written in a better form

• Figure 7 and 10 should be extensively commented in order to make them understandable
• In general, it is difficult for someone not working in that specific research area to deeply understand the quality of the work

Author Response

Response to Reviewer 1 Comments

Dear Reviewer of Applied Sciences:

 

We agree with your valuable comments and suggestions. we have revised the manuscript accordingly. Additionally, in this letter, we have included our point-by-point response to your comments, which are listed as follows.

 

Point 1: The abstract and introduction sections must be written in a much simpler way. It is very hard to understand the objective of the paper.

 

Response 1: Thank you to have taken in account all advices/comments. The abstraction in this paper has been completely rewritten, and the research needs and methods of implementation have been significantly revised.

 

In the Abstract (page 1, line 12-30)

Abstract:  The properties of the material applied to the FE simulation can be expressed by constitutive models, and simple constitutive models and complex constitutive models can be used to show the actual phenomenon. And the technology to improve the accuracy of the constitutive model applied to FE simulation is the inverse method. Inverse method is a method to curve fit the FE simulation result to test data by utilizing Finite element model updating(FEMU). Inverse methods are generally approaches to update of material properties. The inverse method can iteratively run many FE simulations for constitutive model optimization and consider metamodel-based simulation optimization (MBSO) to reduce this resource waste. With MBSO, you can get significant results with less resources. However, the MBSO algorithm has the problem that the optimization performance deteriorates as the number of parameters increases. The typical method of the inverse method is to adjust these factor values ​​individually. If there are many factors in the constitutive model, the optimization result may deteriorate due to the performance limit of MBSO when the structural method is used. This paper proposes a method of fitting a stress-strain constitutive model with a scaling factor in order to improve the efficiency of the inversion method using MBSO. For this purpose, a process was performed to determine the curve characteristics during the pretreatment stage. The results show that the proposed method significantly improved the prediction efficiency of the combination function. Thus, we conclude that initializing the combination function and setting the parameters of the inverse method by applying the proposed approach improves the efficiency of large deformation analyses.

 

 

In the INTRODUCTION (page 2, line 69-72)

If, Given simple boundary conditions with uni-axial loads, inverse methods are available as an alternative to professional equipment and software. The inverse method is based on continuously updating the material parameters of the FE simulation, and it requires two curve-fitting processes [30-32].

 

 

In the INTRODUCTION (page 2, line 76-99)

The simulation results are compared with the tensile test data, and the load–displacement curve is the measurement data required to determine the conformity between the simulation and actual test results [10]. The methods for updating the factors of the Inverse method can be classified into the direct search method and the metamodel based simulation optimization (MBSO). Both methods aim to improve the accuracy of the FE simulation by updating the factors of the constitutive equation. Of these, MBSO is a method that uses design of experiment (DOE), response surface (RS), and global optimization techniques, and can reduce the number of trial and error and derive statistically significant factor values. Leveraging MBSO with the Inverse method can take advantage of the ability to solve optimization problems with less resources. However, MBSO has a problem that the performance of the search algorithm deteriorates as the number of factors in the constitutive equation increases. DOE, RS, and global optimization techniques can be modified to improve the performance of navigation algorithms, but filtering is more important to minimize the number of factors that do not affect results. increase. The inverse method that utilizes MBSO also has such a problem. Inappropriate constitutive equation use, excessive factor number setting, improper variable initialization method, etc. consume a considerable amount of resources in the inverse method utilizing MBSO. In this paper, we proposed a method of adjusting the constitutive model coefficient to a scale factor in order to reduce the resources consumed by the inversion method. In order to verify the validity of the inverse method presented in this paper, we performed a typical method that did not utilize the scale factor and a proposed method that used the scaling factor. Then, the performance of each method was compared and analyzed in three stages: Pre-process, FE simulation, and inverse method.

 

 

In the Conclusion (page 18, line 407-425)

 

The proposed method is a simple idea of adding a scale factor to the constitutive model that represents the material characteristics. Using the scale factor as a parameter in the inverse method, as in the proposed method, greatly reduces the number of cases that need to be tried, thus improving optimization efficiency. To validate the proposed method, nonlinear prediction characteristics were compared by applying three constitutive models, two pre-processing methods, and two inverse methods. The combination function that applied the proposed method could accurately predict the nonlinear characteristics with fewer iterations. However, when the stress-strain curve characteristics of constitutive model are inadequate, the inverse method does not converge or leads to inaccurate results. Therefore, if you want to solve MBSO incompatibility in a multifactorial environment with scaling factors, it may be better to set the constitutive model to the combination function. It is predicted that initializing the combination constitutive model and setting the inverse method parameters by applying the proposed method will improve the efficiency and accuracy of nonlinear analyses performed for large deformations. If you want to get more accurate results, it is recommended to acquire the test data as true stress-strain. The reference data of this study was engineering stress-stain, but it is considered that more accurate and accurate results than this study can be obtained by acquiring True stress-strain using DIC technique and performing inverse method.

 

 

 

 

Point 2: The authors refere to the “typical method”, but there is no description of it.

 

Response 2: We sincerely appreciate your insightful comments and feedback. We added the definition of typical method in abstract.

 

In the Abstract (page 1, line 20-22)

However, the MBSO algorithm has the problem that the optimization performance deteriorates as the number of parameters increases. The typical method of the inverse method is to adjust these factor values ​​individually.

 

 

 

 

Point 3: A workflow should be added to understand the analysis logic.

 

Response 3: Thank you for your thoughtful and constructive feedback. We briefly described the preprocess and inverse method procedures in the modified Figures 7 and 10. Furthermore, the general procedure of the reverse method was described with reference to Figure 10 (c). Through these graphic elements, I tried to help the reader's understanding.

 

In the Figure 7 (page 8, line 227-228)

 

[Link] [Link] (a) (b) Figure 7. FE simulation procedures: (a) method that initializes the parameter once; (b) method that applies CFT.

 

 

In the Figure 10 (page 12-13)

 

[Link] (a) [Link] (b) [Link] (c) Figure 10. Inverse method procedure: (a) Typical method (metamodel-based simulation optimization with one-time initialization of variables); (b) Proposed method (metamodel-based simulation optimization using CFT); (c) Concept diagram of inverse method procedure.

 

Point 4: The statement “When a necking phenomenon occurs in the specimen, the stress and strain distributions appear nonlinearly in the necking area, and the assumption of uniaxial stress is no longer valid” is not clear.

 

Response 4: We sincerely appreciate your insightful comments and feedback. The content pointed out by the reviewers was a sentence with many misunderstandings. So I rewrote the text to reflect his point.

 

In the INTRODUCTION (page 2, line 47-49)

 

When a necking phenomenon occurs in a material, the stress-strain characteristics change non-linearly. Then, different stress-strain curve characteristics can be observed for each material [11-14].

 

 

 

 

Point 5: The statement “FE simulation is a numerical analysis software application that can derive the engineering stress–strain value of the FE model using the true stress–strain metamodel properties” should be written in a better form.

 

Response 5: We sincerely appreciate your insightful comments and feedback. The content pointed out by the reviewers was a sentence with many misunderstandings. So I rewrote the text to reflect his point.

 

In the [3. FE simulation] (page 8, line 223-225)

 

FE simulations can define material properties with the constitutive model of true stress-strain. Then, the engineering stress-strain  can be derived as the FE simulation result..

 

 

 

 

Point 6: Figure 7 and 10 should be extensively commented in order to make them understandable.

 

Response 6: We thank the reviewer for pointing out our description mistake. This answer was presented at point 3. It seems that the answer to Point 6 and the answer to Point 3 are duplicated. I completely repainted the picture and tried to avoid using complex symbols as much as possible. I would like to explain the internal algorithm more briefly than to explain it in a little more detail.

 

 

 

 

Point 7: In general, it is difficult for someone not working in that specific research area to deeply understand the quality of the work.

 

Response 7: We appreciate your meticulous review. With the feedback that the content was difficult, I took the following measures.

 

■ Word correction

○ Fixed words in Metamodel → Constitutive model.

○ Metamodel was used only for the purpose of pointing to the response surface in Metamodel based simulation optimization (MBSO).

○ Fixed in Preprocess → Pre-process.

 

■ Model simplification

○ Reduced the number of Constitutive models. These can be seen in Table 2. As you can see from the previous version, the performance difference between Polynomial functions was not large, and the performance difference between Exponential functions was not large. The constitutive equation was clearly different, but the reader found that it was not significant to the view.

○ The core items of the Proposed method have been corrected so that they can be understood at a glance by the formula. (Equation (5)-(7)) These are also shown in Figure 10.

 

In the [2.2. Initialization of constitutive model parameters for true stress–strain curve] (page 5, line 154-155)

 

[Link]

 

In the [4.2 Inverse method using metamodel-based simulation optimization] (page 10, line 298-301)

 

[Link]

 

■ Quantification

○ Entered the values ​​for all factors. The results of the pretreatment are shown in Table 3, and the results of the inversion method are shown in Tables 4 and 5.

 

In the [4.3. Results of the inverse method] (page 18, line 402-405)

 

[Link]

 

Again, we thank you for your thoughtful feedback and suggestions. Your favorable decision will be greatly appreciated.

 

Sincerely,

 

Euysik Jeon, Ph.D.

Professor

Department of Mechanical and Automotive Engineering & Future Convergence Engineering

Kongju National University

Author Response File: Author Response.docx

Reviewer 2 Report

The article describes an inverse method for identifying the stress-strain curve of a material starting from the data of a tensile test. Different constitutive models are used, and the result of the optimization with the method proposed by the authors is compared with other analysis strategies.

The English form is generally acceptable, and the bibliography is adequate.

However, the work has some obscure parts, some confusing or perhaps even wrong. The innovativeness lies only in the slightly customized or modified minimization strategy compared to normal existing strategies. It also appears that constitutive models are used in the wrong way, that is, constitutive models (i.e. analytical laws containing true stresses and strains) are used to describe the engineering experimental data.

Moreover, even if the curve of the material considered is a little different from the usual (only a little, in truth) the problem faced appears rather simple; I do not see much progress for an engineer/researcher, and it seems strange to me that with well-established existing strategies one could not be able to fit the given curve well.

 

Detailed comments:

• The first thing that bothers and catches the eye is that the term "metamodel" is used to indicate what in the mechanical engineering community is referred to as the "constitutive model" that links sigma (stress) to epsilon (strain). This creates a lot of confusion.
• What in table 2 are called "metamodels" are actually "constitutive models". Even the use of expressions such as "pre-processing" and "post-processing" in an unusual way creates a lot of confusion
• In the introduction it is said "The first curve fitting is performed to initialize the parameters of the metamodel" but this is completely trivial. Just as the use of the Matlab toolbox is acceptable but does not represent an advancement in the world of research.
• The fact of using 2 scaling factors (one for the load and one for the displacement) has no physical sense and I do not understand its usefulness, except as a pure mathematical stratagem, applied however to a rather simple problem. The usage of these scale parameters is quite obscure. You cannot deform, stretch, increase or reduce the experimental data at will !
• At row 127 it is said "When the plastic deformation of the tensile specimen is initiated, the specimen is deformed unevenly, resulting in nonlinear data". In tensile tests, the deformation is even, or uniform, until the necking initiation.
• The second of equations (1) seems wrong, there is an error in the symbols.
• The models shown in table 2 are wrong: in the models of Hollomon, Ludwick, Gosh and Voce, the strain must be the true one, not the engineering one. This is an absolutely fundamental fact.
• At row 164 it is said "This result implies that the parameters must be initialized differently depending on the metamodel". I didn’t understand if the authors in this section implemented a fitting of the models on the experimental curve; if not, it is obvious that a simple initialization on engineering data cannot provide a good fitting with models that are intended to represent the true stress-strain data.
• Figure 4 makes no sense; it is not a result to be seen. It is clear that if constitutive models, that were born to describe true curves, are used on engineering data, the matching will not be good. Moreover, here the coefficients of the models have just been initialized and not optimized; this comparison is pointless.
• Figure 5 also makes no sense. Here at least an optimization of the coefficients has been made, and the curves are more similar; however, the experimental engineering stress is again being compared with constitutive models which are intended to give true stress. On the y-axis there is written "true stress" but curve 1, which seems to be the one you want to fit, is clearly engineering. In fact, in the text it is not very clear which is the curve that the authors believe they have to fit. It almost seems like the authors don't know that the engineering-to-true conversion is valid up to the necking or maximum load.
• The authors used LS-Dyna, which is acceptable but rather unusual given that quasi-static tests were simulated. An implicit solver would have been more suitable.
• On line 233 it is said “The maximum displacement of the experimental data was applied as the termination condition for the FE simulation and was performed from zeros until the termination condition was satisfied. In this process, the termination condition was satisfied differently depending on the properties of the metamodel, and it was difficult to predict the termination time. " Perhaps the authors refer to convergence problems due to the high localized deformation? Otherwise it is not clear what is meant by "termination condition", since the displacement law has been imposed equal to the experimental one
• The first 5 lines of paragraph 3.3 are obscure. What is meant by "normal analysis"?
• In conclusion, in reality, the whole first part up to section 3.3 is practically wrong and useless.
• Finally, in paragraph 4.1 the term metamodel is used correctly
• In equation 2, what does the upper bar symbol mean? How many rows and columns does the system of equations have? Why is there that "0" on the diagonal?
• The diagram of figure 10a is the very well-known FEMU scheme; the novelty of the whole article appears to reside in diagram 10b. But it really seems like a very minimal change of strategy.
• In figure 12a, what are models 1 and 2? It was said they were engineering data and converted to true, respectively, so how do they get that shape?
• Figure 12c is the same as 12a.
• Figure 12d shows the result of the classic FEMU, but it is unrealistic that a better fit than this cannot be obtained.
• For the whole article, and also for figure 13, the coefficients of the various constitutive models are not reported.

Author Response

Response to Reviewer 2 Comments

 

Dear Reviewer of Applied Sciences:

 

We agree with your valuable comments and suggestions. we have revised the manuscript accordingly. Additionally, in this letter, we have included our point-by-point response to your comments, which are listed as follows.

※ This post cannot represent detailed pictures or tables. Please see the attached file.

Point 1: The first thing that bothers and catches the eye is that the term "metamodel" is used to indicate what in the mechanical engineering community is referred to as the "constitutive model" that links sigma (stress) to epsilon (strain). This creates a lot of confusion.

 

Response 1: We appreciate your meticulous review. With the feedback that the content was difficult, I took the following measures.

 

 

Point 2: What in table 2 are called "metamodels" are actually "constitutive models". Even the use of expressions such as "pre-processing" and "post-processing" in an unusual way creates a lot of confusion.

 

Response 2: Thank you for pointing out the difficulty of the content, mixed words, confusion of sentences, etc. at multiple points. I wanted to maximize the comments of the reviewers. The answer to Point 2 was made at Point 1. These feedbacks were applied throughout the paper. Today, we are Pass the revised paper to the editor. It would be really nice if the reviewers would consider it again.

 

 

Point 3: In the introduction it is said "The first curve fitting is performed to initialize the parameters of the metamodel" but this is completely trivial. Just as the use of the Matlab toolbox is acceptable but does not represent an advancement in the world of research.

 

Response 3: We sincerely appreciate your insightful comments and feedback. I realized that the dissertation flow is concentrated in MATLAB CFT. Corrected many sentences, including summaries, prolegomenon, and conclusions.

 

 

Point 4: The fact of using 2 scaling factors (one for the load and one for the displacement) has no physical sense and I do not understand its usefulness, except as a pure mathematical stratagem, applied however to a rather simple problem. The usage of these scale parameters is quite obscure. You cannot deform, stretch, increase or reduce the experimental data at will !

 

Response 4: We thank the reviewer for pointing out our description mistake. The revised paper distinguished test data from constitutive models. And we used only the configuration model for the inverse method. And I reduced the number of models so as not to recover the configuration model with the same effect. These are shown in Table 2.

 

 

Point 5: At row 127 it is said "When the plastic deformation of the tensile specimen is initiated, the specimen is deformed unevenly, resulting in nonlinear data". In tensile tests, the deformation is even, or uniform, until the necking initiation.

 

Response 5: We sincerely appreciate your insightful comments and feedback. Fixed a potentially misleading statement.

 

 

Point 6: The second of equations (1) seems wrong, there is an error in the symbols.

 

Response 6: We thank the reviewer for pointing out our description mistake. Added equations and explanations for engineering stress-strain and true stress-strain to help understand equation (1).

 

 

Point 7: The models shown in table 2 are wrong: in the models of Hollomon, Ludwick, Gosh and Voce, the strain must be the true one, not the engineering one. This is an absolutely fundamental fact.

 

Response 7: We thank the reviewer for pointing out our description mistake. Corrected the notation in Table 2.

 

 

Point 8: At row 164 it is said "This result implies that the parameters must be initialized differently depending on the metamodel". I didn’t understand if the authors in this section implemented a fitting of the models on the experimental curve; if not, it is obvious that a simple initialization on engineering data cannot provide a good fitting with models that are intended to represent the true stress-strain data.

 

Response 8: Thanks to the reviewers for their comments that it is not the right fitting method. It is true that the reliability of the reference data(engineering stress-strain data of tensile test) is a little unreliable. However, the force-displacement data of the tensile test to be compared is the actual measured value. And the result of FE simulation was also obtained by force-displacement data. In this way, the RMSE was calculated using the two force-displacement data. I realize that the content of the previous document was difficult to understand. Therefore, through this revision(Figures 7 and Figure 10 (c)), it is simply and clearly expressed that the RMSE was calculated by the nominal stress-deformation. true stress-strain was used only to define the physical properties of the FE simulation. If we have test data on true stress-deformation, we can get more meaningful results and conclusions. There is no disagreement with that feedback. However, we conducted tensile tests with limited resources and focused on ways to improve the efficiency of the reversal method for this result. We do not overlook the fact that true stress-strain data can be used to increase the reliability of the study. These matters were added as future research content of the conclusion.

 

 

Point 9: Figure 4 makes no sense; it is not a result to be seen. It is clear that if constitutive models, that were born to describe true curves, are used on engineering data, the matching will not be good. Moreover, here the coefficients of the models have just been initialized and not optimized; this comparison is pointless.

 

Response 9: We appreciate your comment. I agree with the bar you pointed out. Anyone can easily predict the improper true stress-strain in Figure 4. What I wanted to show in Figure 4 is the appearance between the configuration models that are quite far apart. Results of factor initialization dependent on ASTM and experience did not show low RMSE and the curve characteristics were not similar. The results of these initializations were also reflected in the inversion method, which affected inaccurate results. These are the same as Figure 12 and Figure 14.

 

 

 

 

Point 10: Figure 5 also makes no sense. Here at least an optimization of the coefficients has been made, and the curves are more similar; however, the experimental engineering stress is again being compared with constitutive models which are intended to give true stress. On the y-axis there is written "true stress" but curve 1, which seems to be the one you want to fit, is clearly engineering. In fact, in the text it is not very clear which is the curve that the authors believe they have to fit. It almost seems like the authors don't know that the engineering-to-true conversion is valid up to the necking or maximum load.

 

Response 10: We appreciate your meticulous review. As pointed out, it is correct to regard Figure 4 and Figure 5 as factor initialization results rather than optimization results. The title of the picture was modified to match this. And the result of the initialization of these factors played an important role in determining the range of factors used in the inversion method. Model 1 in Figure 14 (a) shows that inadequate factor ranges cause high RMSE interruptions. And there was a correction of the graph style due to the influence of other reviewers. In general, all graphs from Figure 4 onwards have been modified.

 

 

Point 11: The authors used LS-Dyna, which is acceptable but rather unusual given that quasi-static tests were simulated. An implicit solver would have been more suitable.

 

Response 11: Thank you for your feedback on the FE simulation solver. In some cases, the Implicit solver provided a more accurate and faster solution. However, as a result of the case study, the effect was not so great. Below are the results of a case study I tried with the LS-Dyna solver. Since the shape of the test piece is different, it was not included in the paper.

 

And the factor was updated and sometimes it fell into an infinite loop. In the case of the explicit solver, the interpretation was interrupted immediately, so it was possible to switch between different cases quickly. When an experimental point that falls into an infinite loop occurs, the amount of resources consumed for one experimental point differs greatly. We did not want the amount of resources consumed by one experimental point to be different. I wanted them to be similar, if not the same.

However, not all inverse methods consistently apply the above case study results. In the future, we will always prepare for what you have pointed out. Thank you for your comment.

 

 

 

Point 12: On line 233 it is said “The maximum displacement of the experimental data was applied as the termination condition for the FE simulation and was performed from zeros until the termination condition was satisfied. In this process, the termination condition was satisfied differently depending on the properties of the metamodel, and it was difficult to predict the termination time. " Perhaps the authors refer to convergence problems due to the high localized deformation? Otherwise it is not clear what is meant by "termination condition", since the displacement law has been imposed equal to the experimental one.

 

Response 12: Thank you for your thoughtful and constructive feedback. The results of the FE simulation were derived from the force-displacement data. And the displacement generated by FE simulation does not need to exceed the maximum value of the test data(maximum displacement of tensile test). Therefore, the termination condition of the analysis is set to the maximum displacement of the tensile test. These are reflected in Figure 8. I wish I could understand it more easily

 

 

Point 13: The first 5 lines of paragraph 3.3 are obscure. What is meant by "normal analysis"?

 

Response 13: We thank the reviewer for pointing out our description mistake. “Normal analysis” means that the maximum displacement of the analysis and the maximum displacement of the test are equal. We lacked a lot of explanations. So I modified the text and the picture like Response 12.

 

 

 

 

 

 

 

 

 

 

Point 14: In conclusion, in reality, the whole first part up to section 3.3 is practically wrong and useless.

 

Response 14: We appreciate your comment. For the readers who read the paper, I added an explanation of the meaning of each figure and table. I briefly described the expected results so that the reader could read the content and think about the future implications.

 

 

 

Point 15: Finally, in paragraph 4.1 the term metamodel is used correctly.

 

Response 15: We appreciate your meticulous review. Metamodel was used only for the purpose of pointing to the response surface in Metamodel based simulation optimization (MBSO).

 

 

 

Point 16: In equation 2, what does the upper bar symbol mean? How many rows and columns does the system of equations have? Why is there that "0" on the diagonal?

 

Response 16: We appreciate your meticulous review. With the feedback that the content was difficult, I took the following measures. first, The core items of the Proposed method have been corrected so that they can be understood at a glance by the formula. (Equation (5)-(7)) These are also shown in Figure 10. And, We briefly described the preprocess and inverse method procedures in the modified Figures 7 and 10. Furthermore, the general procedure of the reverse method was described with reference to Figure 10 (c). Through these graphic elements, I tried to help the reader's understanding.

 

 

Point 17: The diagram of figure 10a is the very well-known FEMU scheme; the novelty of the whole article appears to reside in diagram 10b. But it really seems like a very minimal change of strategy.

 

Response 17: Thank you for your valuable comment. our method is a method that slightly modifies the typical method. It's a very easy way to access. The effect of the method is shown in Fig. 14. The performance of the proposed method was not good for all configuration models. The contents of such future research are described in the conclusion.

 

 

 

Point 18: Figure 12c is the same as 12a.

 

Response 18: We thank the reviewer for pointing out our description mistake. Fixed all images related to the result of the inverse method. As the number of constitutive models decreased, so did the figure showing the results.

 

 

 

Point 19: Figure 12d shows the result of the classic FEMU, but it is unrealistic that a better fit than this cannot be obtained.

 

Response 19: We appreciate your meticulous review. We have reflected these matters in future research results regarding research results.

 

 

 

Point 20: For the whole article, and also for figure 13, the coefficients of the various constitutive models are not reported.

 

Response 20: We sincerely appreciate your insightful comments and feedback.

 

 

 

Author Response File: Author Response.docx

Reviewer 3 Report

In my opinion authors should correct/modify the following issues:

#1) Page 4. Line 123. Fig. 3. For the sake of clarity, please identify each curve with a legend with colored lines inside the figure. In addition, please use kN instead of N in the y-axis.

#2) Page 4. Line 129. Please use italics for “n-th”

#3) Page 5. Line 155. Please include the values of the parameters of the metamodels used for obtained results plotted in Fig. 4 and describe properly how these values were selected.

#4) Page6. Line 164. Fig. 4. Curves corresponding to models 4 and 5 are not represented completely. Please modify the limits of the y-axis to see those curves completely. In addition, please explain why the results corresponding to model 7 are linear.

#5) Page 6. Line 182. “The difference between the experimental data and the metamodel can be observed qualitatively in Figures 4 and 5” For the sake of clarity, which metamodel?. Please be more precise and discuss this properly. In addition, please indicate that the curve corresponding to model 1 is the experimental data in plots.

#6) Page 9. Equation (2). Something seems to be missed in this equation. Please include “….”.

#7) Page 13. Line 347-349. The discussion of results plotted is quite brief. Please include a deep discussion of the obtained results comparing each model for Figs. 12 and 13. Please do not use the same symbol for different curves as it is done in Fig. 12 d and Fig. 13d.

#8) Page 13. Line 358. Authors state “the polynomial model exhibit significant differences”. Please discuss this properly.

#9) Lines 354-356. The discussion is not complete. Please discuss the results properly.

#10) Line 364-366. Fig. 13d. Please discuss why main differences in the combined model appears at the beginning of plasticity (material yield strength). In addition, for the sake of clarity please use different symbols for models nº 8 and nº 9 and select another symbol (not a star) for nº10 for Fig, 13d. Please do the same in Fig. 13c. In addition, please include in the legend the name of the model as it was done in the legend of Fig. 12d.

#11) Reference section. Please revise references according to journal template for books. Do not use capital letters for book title. References #23, #40, #41 and #45.

Author Response

Response to Reviewer 3 Comments

 

Dear Reviewer of Applied Sciences:

 

We agree with your valuable comments and suggestions. we have revised the manuscript accordingly. Additionally, in this letter, we have included our point-by-point response to your comments, which are listed as follows.

 

Point 1: Page 4. Line 123. Fig. 3. For the sake of clarity, please identify each curve with a legend with colored lines inside the figure. In addition, please use kN instead of N in the y-axis.

 

Response 1: Thank you very much for your comment. I changed the unit of y-axis according to the matter pointed out. ([N] → [kN]) And added a legend. We also changed the line color and style to distinguish the graph.

 

In the [2.Pre-processing of the constitutive model] (page 4, line 126-127)

 

[Link] Figure 3. Results of the tensile test (ASTM E8/E8M).

 

 

 

 

 

Point 2: Page 4. Line 129. Please use italics for “n-th”.

 

Response 2: Thank you for your comment. The position of the word pointed out was deleted by the correction of the sentence. However, italics were reflected in the order notation of the constitutive model throughout the paper. The answers below are just a few examples. Throughout the dissertation, the order reflects italics.

 

In the [2. Pre-processing of the constitutive model] (page 5, line 153-155)

 

Table 2. Various constitutive model details for FE simulation. [Link]

 

In the [2. Pre-processing of the constitutive model] (page 5, line 169-170)

 

Table 3. Definition of parameter for constitutive model. [Link]

 

 

Point 3: Page 5. Line 155. Please include the values of the parameters of the metamodels used for obtained results plotted in Fig. 4 and describe properly how these values were selected.

 

Response 3: We sincerely appreciate your insightful comments and feedback.

 

■ Quantification

○ Entered the values ​​for all factors. The parameter values in the configuration model are shown in Tables 3 and 4 as quantitative values. The true stress-strain curves obtained from these parameters can be seen in Figures 3, 4, 12, and 13.

 

In the [2. Pre-processing of the constitutive model] (page 5, line 169-170)

 

Table 3. Definition of parameter for constitutive model.   [Link]

 

 

In the [4.3. Results of the inverse method] (page 18, line 402-405)

 

Table 4. Definition of parameter for constitutive model. [Link]

 

 

 

 

Point 4: Page6. Line 164. Fig. 4. Curves corresponding to models 4 and 5 are not represented completely. Please modify the limits of the y-axis to see those curves completely. In addition, please explain why the results corresponding to model 7 are linear.

 

Response 4: Thank you for your thoughtful and constructive feedback. Most of the paintings have been redrawn through the reviewers' comments. In the process, I removed the y-axis restriction that hides the curve.

 

In the [4.3. Results of the inverse method] (page 18, line 402-405)

 

[Link]  Figure 4. The constitutive model properties initialization using the ASTM E646 standard

 

 

 

 

Point 5: Page 6. Line 182. “The difference between the experimental data and the metamodel can be observed qualitatively in Figures 4 and 5” For the sake of clarity, which metamodel?. Please be more precise and discuss this properly. In addition, please indicate that the curve corresponding to model 1 is the experimental data in plots.

 

Response 5: We sincerely appreciate your insightful comments and feedback.

 

■ Corrected word

Corrected the word metamodel to a construct model. Metamodel was used only for the purpose of pointing to the response surface in Metamodel based simulation optimization (MBSO).

○ Fixed words in Metamodel → Constitutive model.

 

Then, following the comments of other reviewers, I modified the material model settings.

 

■ Model simplification

○ Reduced the number of Constitutive models. These can be seen in Table 2. As you can see from the previous version, the performance difference between Polynomial functions was not large, and the performance difference between Exponential functions was not large. The constitutive equation was clearly different, but the reader found that it was not significant to the view.

○ The core items of the Proposed method have been corrected so that they can be understood at a glance by the formula. (Equation (5)-(7)) These are also shown in Figure 10.

 

In the [2.2. Initialization of constitutive model parameters for true stress–strain curve] (page 5, line 154-155)

 

Table 2. Various constitutive model details for FE simulation. [Link]

 

 

 

Point 6: Page 9. Equation (2). Something seems to be missed in this equation. Please include “….”..

 

Response 6: We appreciate your meticulous review. With the feedback that the content was difficult, I took the following measures. We appreciate your meticulous review. With the feedback that the content was difficult, I took the following measures. first, The core items of the Proposed method have been corrected so that they can be understood at a glance by the formula. (Equation (5)-(7)) These are also shown in Figure 10. And, We briefly described the preprocess and inverse method procedures in the modified Figures 7 and 10. Furthermore, the general procedure of the reverse method was described with reference to Figure 10 (c). Through these graphic elements, I tried to help the reader's understanding.

 

In the [4.2 Inverse method using metamodel-based simulation optimization] (page 10, line 298-301)

 

[Link]

 

In the Figure 7 (page 8, line 227-228)

 

[Link]  [Link] (a) (b) Figure 7. FE simulation procedures: (a) method that initializes the parameter once; (b) method that applies CFT.

 

 

 

In the Figure 10 (page 12-13)

 

[Link] (a) [Link] (b) [Link] (c) Figure 10. Inverse method procedure: (a) Typical method (metamodel-based simulation optimization with one-time initialization of variables); (b) Proposed method (metamodel-based simulation optimization using CFT); (c) Concept diagram of inverse method procedure.

 

>> Review figure : [Link]

 

Again, we thank you for your thoughtful feedback and suggestions. Your favorable decision will be greatly appreciated.

 

Sincerely,

 

Euysik Jeon, Ph.D.

Professor

Department of Mechanical and Automotive Engineering & Future Convergence Engineering

Kongju National University

Author Response File: Author Response.docx

Round 2

Reviewer 3 Report

  

Author Response

Dear Reviewer of Applied Sciences:

 

We agree with your valuable comments and suggestions. we have revised the manuscript accordingly. Additionally, in this letter, we have included our point-by-point response to your comments, which are listed as follows.

 

Point 1: English language and style are fine/minor spell check required

 

Response 1: Thank you very much for your comment. I have corrected all the English in my dissertation with a proofreading institution. The overall content of the paper was not revised.

 

 

Again, we thank you for your thoughtful feedback and suggestions. Your favorable decision will be greatly appreciated

Author Response File: Author Response.docx