Machine Learning Model for Gas–Liquid Interface Reconstruction in CFD Numerical Simulations^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The “Machine learning model for gas-liquid interface reconstruction” suggests a new method to reconstruct the gas-liquid interface based on GNNs. This is a very important area that can greatly benefit from innovative solutions.

General concept comments
This document is well-written, has a good English level and provides a solid background review. However, there are specific areas that require attention to enhance its clarity

Specific comments 

1)      A key work pointing to unstructured meshes would be highly recommended.

2)      Several statements throughout the document should be supported by appropriate references (for example the first one in the introduction as well as section 3)

3)      Referring to works solely by numbers (e.g., “[5] proposed”) decreases the readability. It would be more reader-friendly to include author names.  

4)      Table 1 can be much more useful. It should include: references, computational costs metrics(cheap/ expensive in what terms), and other key parameters like the type of grid.

5)      On page 2, line 50, can you provide insight into how much this improvement will bring? What is expected?

6)      The discussion of the Volume of Fluid (VOF) method (line 74) would be more precise with references to primary and secondary phases.

7)      On page 3, line 90 the authors present Mules and isoAdvector but do not detail their limitations or the criteria for selecting one over the other.

8)      Small typo on line 100 on page 3 (function)

9)      After section 3, images and their discussions are often too far apart. Please review the position of the images.

10)  The way that the NNs interact is not clear. The paper mentions simultaneous operation but also describes information flow between networks. Clarifying this with an updated version of Figure 4 would be quite helpful.

11)  Is the dataset representative of any practical case?

12)  The legend in Figure 7 is too small and difficult to read

13)  The criteria for using exact labels or predicted values during training should be clearly explained.

14)  In page 8, line 242, large errors are mentioned but not justified. Is there any justification?

15)  The choice of 50 epochs in the Optuna-based optimization process should be explained. Was this a compromise due to computational constraints, or were additional epochs found unnecessary

16)  Figures 12a and 12b might be more effective as tables, given their structure and information content.

17)  The paper lacks details when discussing the application to the simulation like

a.       the dimensionless numbers that characterize the system.

b.       You run both stationary or transient simulations. What is expected when running each?

c.      Does it lead to higher computational times to solve the problem? Does this require more in computational terms  for example in terms of memory?

18)  In page 15, line 377, “the results look very similar”. Can you quantify? What means being similar? Did the authors consider showing some contours or visual results?

19)  How sensitive is the method to changes in mesh quality and/or boundary conditions?

20)  Will the implementation be made available as open-source?

 

 

Author Response

Dear Reviewer 1,

Thank you for your thorough and constructive feedback on our manuscript titled "Machine learning model for gas-liquid interface reconstruction". Your insights have been very useful in enhancing the clarity, depth, and accuracy of our research. We have carefully considered each of your suggestions and have implemented several key modifications to address the issues raised.

Here are the detailed modifications made to the new version of the article:

 

1)      A key work pointing to unstructured meshes would be highly recommended.

References and Context: We have incorporated additional references to unstructured meshes to provide better context and support for our claims, especially in the introduction page 2 line 48-57.

 

2)      Several statements throughout the document should be supported by appropriate references (for example the first one in the introduction as well as section 3).

Citations and Claims: We have revised the manuscript to include more precise citations throughout, ensuring that all claims are adequately supported by relevant literature or by some results of our work.

 

3)      Referring to works solely by numbers (e.g., “[5] proposed”) decreases the readability. It would be more reader-friendly to include author names.  

Readability Enhancements: Author names are now included alongside numerical references to improve readability and traceability of the sources.

 

4)      Table 1 can be much more useful. It should include: references, computational costs metrics(cheap/ expensive in what terms), and other key parameters like the type of grid.

Expansion of Table 1: page 2, lines 48-57, we have better introduced Table 1 by defining the accuray, stability terms and the computational cost metrics in order to make our discussion and comparaison clearer. 

 

5)      On page 2, line 50, can you provide insight into how much this improvement will bring? What is expected?

Clarifications and Insights: page 2, lines 58-63, we have provided detailed insights on the expected improvements of the method: the objective of this paper is to improve the accuracy of the interface reconstruction step of the VoF method on unstructured meshes. Computational gains can also be expected when using machine learning methods, however this will be the focus of future work. We have added a comment regarding this topic in the future work section of the conclusion.

 

6)      The discussion of the Volume of Fluid (VOF) method (line 74) would be more precise with references to primary and secondary phases.

Clarifications: page 3, lines 85-97, we have clarified the discussion around the Volume of Fluid (VOF) method, particularly concerning the handling of primary and secondary phases.

 

7)      On page 3, line 90 the authors present Mules and isoAdvector but do not detail their limitations or the criteria for selecting one over the other.

page 3 lines 104-111, we give some hints on the differences of the Mules ands isoAvector and their application domains of preference.

 

8)      Small typo on line 100 on page 3 (function)

Typographical Corrections: All identified typographical errors have been corrected.

 

9)      After section 3, images and their discussions are often too far apart. Please review the position of the images.

Image and Discussion Placement: The positioning of images and their corresponding discussions has been adjusted to ensure they are closely aligned, enhancing the flow of information. On acceptation, the final manuscript will be reviewed together with the editor to ensure correct figure placement.

 

10)  The way that the NNs interact is not clear. The paper mentions simultaneous operation but also describes information flow between networks. Clarifying this with an updated version of Figure 4 would be quite helpful.

We thank the reviewer for his remark. Indeed, the trained model is composed of several elementary components that can be trained in two distinct phases and subsequently combined during inference.

The complete model is presented in Figure 4d. It consists of three elementary sub-models. The inputs to these sub-models are represented by the respective arrows pointing towards each sub-model. From an implementation perspective, it would have undoubtedly been more efficient to design a single model with the spatial distribution of concentrations represented as a graph at the input and the labels of interest (i.e., normal, curvature, center, area) as the output.

However, not all labels depend on the spatial distribution of the concentrations. Some variables can be retrieved using information local to the element. For instance, the plane defining the concentration within an element is uniquely determined by the center and the normal. Similarly, the area can be uniquely calculated given the element and the intersecting plane. For this reason, a model that divides the aforementioned tasks into three components proves to be more effective.

To this end, the model is divided into three sub-models:
Model NN-1 computes the interface normal and curvature.
Model NN-2 calculates the plane's center in the VOF approximation.
Model NN-3 computes the area of the plane intersecting the element of interest.

All these explanations have been introduced page 5, line 168-176.

 

11)  Is the dataset representative of any practical case?

Dataset Representativeness: We have assessed and discussed the representativeness of the dataset used, ensuring it aligns with practical applications. Thus as long as, the application simulations lead  to surfaces with curvature corresponding to the range of curvatures selected to generate the dataset, this one is representative of the simulation surfaces.

 

12)  The legend in Figure 7 is too small and difficult to read.

 We have enlarged the corresponding legend.

 

13)  The criteria for using exact labels or predicted values during training should be clearly explained.

We thank the reviewer for the remark. As previously explained, the model is divided into three sub-models. We have added the following precisions in the full manuscript: 

The prediction of sub-model NN-2 depends on the output of NN-1, and the prediction of NN-3 depends on the output of NN-2. In this configuration, the backpropagation of gradients becomes less effective due to the depth of the entire model. At the same time, the errors from intermediate predictions are propagated to the subsequent models.

To mitigate the challenges of ineffective training caused by the model's depth and to avoid propagating errors from intermediate predictions, auxiliary loss functions were introduced. These losses account for the true labels (wl) as inputs to the models used for computing the center and the area. These auxiliary loss functions help to address the vanishing gradient problem and prevent the compounding of errors from intermediate predictions.

 

14)  In page 8, line 242, large errors are mentioned but not justified. Is there any justification?

Page 9, line 270-271 : we have detailed the meaning of large errors.

 

15)  The choice of 50 epochs in the Optuna-based optimization process should be explained. Was this a compromise due to computational constraints, or were additional epochs found unnecessary

Page 10, lines 286-293. The hyperparameter optimisation process description has been clarified. We have detailled the practical reasons for the choice of number of trials and number of maximum epochs.

 

16)  Figures 12a and 12b might be more effective as tables, given their structure and information content.

Figure and Table Adjustments: Page 11 line 321-323. We have better introduced the color legend that improves the readability of the table structured information content presented in Figure 12. That helps to analyze the trade-off between computational time and accuracy and to determine the choice of the dataset size.

 

17)  The paper lacks details when discussing the application to the simulation like
a.       the dimensionless numbers that characterize the system.
b.       You run both stationary or transient simulations. What is expected when running each?
c.      Does it lead to higher computational times to solve the problem? Does this require more in computational terms  for example in terms of memory?

a. Simulation Details: Page 15, lines 377-383. More comprehensive details on the application to simulations, including discussions on dimensionless numbers and expectations for different types of simulations, have been added.

b. The stationary test case aims to evaluate the accuracy of the method measuring the energy of residual streams that may appear due to induced geometrical property errors of the methods. The transient test case aims to highlight the effect of the new GNN method in the advection scheme computations. 

c. Our GNN methods lead to a lower computation time at the cost of higher memory storage (but reasonable on our current computers compared with tabulation methods, for example). However, as indicated in the answers above and highlighted in the article revision, the computational efficiency aspects will be the subject of further work.

 

18)  In page 15, line 377, “the results look very similar”. Can you quantify? What means being similar? Did the authors consider showing some contours or visual results?

Quantitative Statements: We have quantified the statements regarding the similarity of results and included additional visual results to support these claims.

 

19)  How sensitive is the method to changes in mesh quality and/or boundary conditions?

It is an interesting question. This study is an on going project and will be the subject of future.

 

20)  Will the implementation be made available as open-source?

Open-source Availability: The project is right know available in the github project : https://github.com/gratienj/openfoam.git branch dev-ml4cfd. On going works aims to improve the computational efficiency of the method with a better integration of pytorch algorithms in a C++ code. We plan to make our implementation available for the OpenFOAM community, to facilitate wider use and verification of our results.

 

We hope that these modifications adequately address your concerns and enhance the manuscript's contribution to the field of computational fluid dynamics using machine learning. We appreciate the opportunity to improve our work with your expert guidance and look forward to your further suggestions.

Warm regards,

Tamon NAKANO and Jean-Marc GRATIEN

 

 

 

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript proposes a VOF method based on graph neural network (GNNs) for machine learning enhancement to accelerate the interface reconstruction on general unstructured grids. A method is developed to generate synthetic data sets based on discrete paraboloids on unstructured grids, so as to obtain data sets with configurations similar to those encountered in industrial environments. Then, an optimized GNN architecture is trained on this data set. The subject is relevant, and interesting. However, before being recommended for publication, the author should address the following issues, concerns in the revised version:

 1. In line 352 on page 15, there is a typographical error in the formula.

2. Compared with the traditional VOF method, can the VOF method based on Graph Neural Networks (GNNs) effectively improve the computational efficiency in unstructured grid computing? Some quantitative analysis results are suggested to be provided.

3. The author gives a very detailed error analysis in the bubble rise simulation in Section 6.2. However, the author uses Graph Neural Networks (GNNs) method to reconstruct the interface, so it is suggested that the author compare the bubble morphology of VOF method based on Graph Neural Networks (GNNs), traditional VOF method and other numerical methods, so as to observe the difference of interface morphology more intuitively.

4. This paper selects 500,000 graph data sets and divides them into 7:2:1 ratio for training, verification and testing. Why did the author choose 500,000 data sets? Is there a basis for selecting the sample size?

5. How does the number of graph data sets affect the final training results?

6. During the training process, the author divides the data set into 7:2:1 ratio. What is the effect of dividing different ratios on the training results?

Author Response

Dear Reviewer 2,

Thank you for your thorough and constructive feedback on our manuscript titled "Machine learning model for gas-liquid interface reconstruction". Your insights have been very useful in enhancing the clarity, depth, and accuracy of our research. We have carefully considered each of your suggestions and have implemented several key modifications to address the issues raised.

Here are the detailed modifications made to the new version of the article:

 

1. In line 352 on page 15, there is a typographical error in the formula.

Page 15, line 389. The link to the referenced equation has been corrected.

 

2. Compared with the traditional VOF method, can the VOF method based on Graph Neural Networks (GNNs) effectively improve the computational efficiency in unstructured grid computing? Some quantitative analysis results are suggested to be provided.

In our work, we focus on the accuracy improvement of the GNN method on the computation of interface geometrical properties on unstructured mesh.
We also performed experiments regarding the computational efficiency in unstructured grid, however the results highly depend on the amount of di-phasic cells in the simulation and on the hardware configuration (use of GP-GPU, of many-core CPUs, etc.). We plan to focus on computational efficiency in future work. We have modified the text to reflect this, both in the introduction as well as in the conclusion.

 

3. The author gives a very detailed error analysis in the bubble rise simulation in Section 6.2. However, the author uses Graph Neural Networks (GNNs) method to reconstruct the interface, so it is suggested that the author compare the bubble morphology of VOF method based on Graph Neural Networks (GNNs), traditional VOF method and other numerical methods, so as to observe the difference of interface morphology more intuitively.

 Simulation images of the evolution of the bubble morphology are very similar to the naked eye. Therefore, we chose to provide statistics that better illustrate the differences and similarities between the methods.

 

4. This paper selects 500,000 graph data sets and divides them into 7:2:1 ratio for training, verification and testing. Why did the author choose 500,000 data sets? Is there a basis for selecting the sample size?

5. How does the number of graph data sets affect the final training results?

In section 5.3, page 11 line 317, we discuss the choice of the dataset size, based on the analyze of the impact of the dataset size on the accuracy of the trained model and on the cost of the train process. At the end we have chosen a 500,000 data set considering the trade-off between computational time and accuracy.

 

6. During the training process, the author divides the data set into 7:2:1 ratio. What is the effect of dividing different ratios on the training results?

We have chosen the 7:2:1 ratio as a standard and commonly used ratio. We did not evaluate the impact of these ratio in the analysis presented in section 5.3. As suggested by the reviewer we will realize such analysis in future works.

 

We hope that these modifications adequately address your concerns and enhance the manuscript's contribution to the field of computational fluid dynamics using machine learning. We appreciate the opportunity to improve our work with your expert guidance and look forward to your further suggestions.

Warm regards,

Tamon NAKANO and Jean-Marc GRATIEN

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Authors

Thank you very much for your detailed responses. I appreciate your effort in addressing most of my concerns; however, a few points still require further clarification or improvement.

While I suggested adding a new keyword regarding "unstructured mesh," I do acknowledge that the additions made to the introduction are a nice touch. That said, I would like to point out a small typo on line 50, page 2, which should be corrected.

The meaning of the variables considered in Table 1 is now clearer. However, I am still uncertain about the thresholds used to classify something as "expensive/cheap" or "accurate/inaccurate." I believe this could still benefit from further clarification.

In relation to point 14, while you have added a definition of what constitutes "large errors," my question was more directed towards whether you have identified the underlying causes for these large errors.

I believe there is an issue with the formula provided for the Eötvös number. As this number represents the ratio between gravitational forces and surface tension forces, it should not include the bubble velocity. Kindly verify and correct this.

As for point 19, in line 368, page 15, you mention the use of three meshes. Could you please clarify the specific purpose of each mesh and how they were utilized in your simulations?

I appreciate your continued efforts to refine the manuscript.

Kind regards

Author Response

Thank you for your thorough and constructive feedback on our manuscript titled "Machine learning model for gas-liquid interface reconstruction". Your insights have been very useful and we have carefully considered each of your suggestions to go on improving the manuscript.

Here are the detailed modifications made to the new version of the article:

While I suggested adding a new keyword regarding "unstructured mesh," I do acknowledge that the additions made to the introduction are a nice touch. That said, I would like to point out a small typo on line 50, page 2, which should be corrected.

The typo on line 50, page 2 has been corrected. We have had "unstructured mesh" to the list of key words of the article.

The meaning of the variables considered in Table 1 is now clearer. However, I am still uncertain about the thresholds used to classify something as "expensive/cheap" or "accurate/inaccurate." I believe this could still benefit from further clarification.

Page 1, line 48-65. We have revised the presentation of the table, eliminating references to 'inaccuracy' and the term 'cheap.' We have also introduced a brief discussion on the limitations of the Volume of Fluid (VoF) method due to its computational cost, which confines its use to small physical domains with a low rate of di-phasic grid cells. For complex simulations, such as those needed to model chemical processes with a large number of bubbles, the VoF method is not practical for the entire simulation. However, it is sometimes utilized for its precision, in combination with other methods that are more suitable for large-scale simulations. In multi-scale coupled strategies, the VoF method is instrumental in accurately determining local parameters that model the impact of bubbles, treated as particles, on the overall flow dynamics.

In relation to point 14, while you have added a definition of what constitutes "large errors," my question was more directed towards whether you have identified the underlying causes for these large errors.

Page 10, line 274 to 279. We have improved the paragraph with details of the potential causes of these large errors.

I believe there is an issue with the formula provided for the Eötvös number. As this number represents the ratio between gravitational forces and surface tension forces, it should not include the bubble velocity. Kindly verify and correct this.

Page 16, line 392-398. There were, indeed, real issues in this paragraph. We have corrected them, updated the definition the Eötvös number and clarified the role of the bubble velocity U0 when they are no gravity effect in Test1 and Test2 and the gravity velocity for only test3. The different values of the table have been checked and corrected.

As for point 19, in line 368, page 15, you mention the use of three meshes. Could you please clarify the specific purpose of each mesh and how they were utilized in your simulations?

Page 15, line 380-387. We have clarified why we utilized three different meshes. Initially, we refined the original mesh, which had a size of 0.1, twice to obtain two refined meshes with sizes of 0.5=0.1/2 and 0.025=0.1/4. The objective was to conduct a convergence study to observe how our method performs with varying mesh sizes. As the grid cell size decreases, an effect emerges that causes the relative curvature of interfaces to approach zero. This occurs due to the normalization of the GNN graphs to a reference graph of unity size. The purpose of this study is to highlight these effects.

 

We hope that these modifications adequately address your concerns and enhance the manuscript's contribution to the field of computational fluid dynamics using machine learning. We appreciate the opportunity to improve our work with your expert guidance and look forward to your further suggestions.

Warm regards,

Tamon NAKANO and Jean-Marc GRATIEN

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript had been revised according to the review comments and may be considered for acceptance.

Author Response

Thank you for your thorough and constructive feedback on our manuscript titled "Machine learning model for gas-liquid interface reconstruction". Your insights have been very useful and we have carefully considered each of your suggestions to go on improving the manuscript.

Here are the detailed modifications made to the new version of the article:

While I suggested adding a new keyword regarding "unstructured mesh," I do acknowledge that the additions made to the introduction are a nice touch. That said, I would like to point out a small typo on line 50, page 2, which should be corrected.

The typo on line 50, page 2 has been corrected. We have had "unstructured mesh" to the list of key words of the article.

The meaning of the variables considered in Table 1 is now clearer. However, I am still uncertain about the thresholds used to classify something as "expensive/cheap" or "accurate/inaccurate." I believe this could still benefit from further clarification.

Page 1, line 48-65. We have revised the presentation of the table, eliminating references to 'inaccuracy' and the term 'cheap.' We have also introduced a brief discussion on the limitations of the Volume of Fluid (VoF) method due to its computational cost, which confines its use to small physical domains with a low rate of di-phasic grid cells. For complex simulations, such as those needed to model chemical processes with a large number of bubbles, the VoF method is not practical for the entire simulation. However, it is sometimes utilized for its precision, in combination with other methods that are more suitable for large-scale simulations. In multi-scale coupled strategies, the VoF method is instrumental in accurately determining local parameters that model the impact of bubbles, treated as particles, on the overall flow dynamics.

In relation to point 14, while you have added a definition of what constitutes "large errors," my question was more directed towards whether you have identified the underlying causes for these large errors.

Page 10, line 274 to 279. We have improved the paragraph with details of the potential causes of these large errors.

I believe there is an issue with the formula provided for the Eötvös number. As this number represents the ratio between gravitational forces and surface tension forces, it should not include the bubble velocity. Kindly verify and correct this.

Page 16, line 392-398. There were, indeed, real issues in this paragraph. We have corrected them, updated the definition the Eötvös number and clarified the role of the bubble velocity U0 when they are no gravity effect in Test1 and Test2 and the gravity velocity for only test3. The different values of the table have been checked and corrected.

As for point 19, in line 368, page 15, you mention the use of three meshes. Could you please clarify the specific purpose of each mesh and how they were utilized in your simulations?

Page 15, line 380-387. We have clarified why we utilized three different meshes. Initially, we refined the original mesh, which had a size of 0.1, twice to obtain two refined meshes with sizes of 0.5=0.1/2 and 0.025=0.1/4. The objective was to conduct a convergence study to observe how our method performs with varying mesh sizes. As the grid cell size decreases, an effect emerges that causes the relative curvature of interfaces to approach zero. This occurs due to the normalization of the GNN graphs to a reference graph of unity size. The purpose of this study is to highlight these effects.

 

We hope that these modifications adequately address your concerns and enhance the manuscript's contribution to the field of computational fluid dynamics using machine learning. We appreciate the opportunity to improve our work with your expert guidance and look forward to your further suggestions.

Warm regards,

Tamon NAKANO and Jean-Marc GRATIEN