Parallel Finite Element Algorithm for Large Elastic Deformations: Program Development and Validation
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe manuscript entitled "Parallel Finite Element Algorithm for Large Elastic Deformations: Program Development and Validation" addresses a significant topic: the development and validation of a parallel finite element program for large elastic deformations. The topic is relevant to multiple domains, including Tectonics, geophysics, structural engineering, and computational mechanics.
The manuscript demonstrates clear effort in presenting a novel computational approach validated against analytical models and applied to a geophysical case study.
However, there are areas for improvement in structure, clarity, methodology, and scientific rigor.
1- The abstract lacks numerical details (e.g., efficiency metrics or computational results) that could highlight the significance of the results.
2- The abstract does not explicitly emphasize the novelty of the parallel computing approach compared to existing methodologies.
Therefore, I would recommend including specific quantitative results, such as accuracy, speedup ratios, and efficiency metrics, to strengthen the abstract. And to emphasize how the work advances the state-of-the-art in finite element analysis or parallel computing.
3- Key terms such as "PFELAC software" and "parallel computing" are introduced without sufficient context or references.
4- Some equations and symbols (e.g., Equation 7 and Equation 23) are introduced abruptly without clear derivations or context.
Here, I suggest including more technical details about the numerical implementation (e.g., convergence criteria, solver type, and computational resources).
5- Captions of Figures 3 and 5 should include more detailed explaining their relevance to the topic.
6- The comparison with analytical results is briefly presented in tabular form but lacks sufficient discussion on potential sources of error. I think the paper would benefit from a more detailed discussion of discrepancies between numerical and analytical results, especially for stress fields.
7- The case study is oversimplified, with limited discussion on geological implications. Please, extend the discussion on the geological significance of the results and how this approach could inform tectonic studies.
8- Figures comparing small and large deformation models lack clarity, especially in terms of units and color scales.
9- The limitations of the study (e.g., assumptions, computational constraints) are not discussed. There is room for that in the Discussion section.
Overall, the study has significant potential and demonstrates substantial computational advancements. However, improvements in clarity, technical rigor, and presentation are necessary to meet the standards of a high-impact journal.
Author Response
Comments and Suggestions for Authors
The manuscript entitled "Parallel Finite Element Algorithm for Large Elastic Deformations: Program Development and Validation" addresses a significant topic: the development and validation of a parallel finite element program for large elastic deformations. The topic is relevant to multiple domains, including Tectonics, geophysics, structural engineering, and computational mechanics.
The manuscript demonstrates clear effort in presenting a novel computational approach validated against analytical models and applied to a geophysical case study.
However, there are areas for improvement in structure, clarity, methodology, and scientific rigor.
- The abstract lacks numerical details (e.g., efficiency metrics or computational results) that could highlight the significance of the results.
Answer: We conducted a quantitative assessment of the benefits of parallel computing frameworks in finite element analysis, incorporating numerical comparisons of efficiency and acceleration ratios into the modified abstract: “As the number of cores increases, the parallel speedup ratio rises, but parallel efficiency decreases. For 16 cores, the speedup ranges from 11.36 to 12.24, with parallel efficiency between 0.71 and 0.77.”
The new Abstract is below:
A comprehensive understanding of large elastic deformation, characterized by its nonlinear strain and stress properties, is vital for examining tectonic deformation across geological timescales. We employ the PFELAC software platform to automate the generation of parallel elastic large deformation finite element codes. By writing only a minimal amount of fundamental finite element language rooted in the principle of virtual work, we significantly enhance program development efficiency. The accuracy of the finite element method is rigorously validated through comparisons with analytical solutions from two idealized models. Furthermore, we investigate the influence of mesh density and CPU core count on computational performance. As the number of cores increases, the parallel speedup ratio rises, but parallel efficiency decreases. For 16 cores, the speedup ranges from 11.36 to 12.24, with parallel efficiency between 0.71 and 0.77. In contrast, for 64 cores, the speedup is between 24.70 and 34.78, while parallel efficiency drops to between 0.39 and 0.43. The program's application to simulate crustal fold deformation reveals marked distinctions between large and small deformation theories, emphasizing the critical importance of large deformation theory in tectonic studies.
2- The abstract does not explicitly emphasize the novelty of the parallel computing approach compared to existing methodologies.
Therefore, I would recommend including specific quantitative results, such as accuracy, speedup ratios, and efficiency metrics, to strengthen the abstract. And to emphasize how the work advances the state-of-the-art in finite element analysis or parallel computing.
Answer: We have incorporated a concise paragraph in the abstract to illustrate the features and advantages of the parallel finite element program developed in this study, along with specific quantitative results pertaining to accuracy, acceleration ratio, and efficiency: “We employ the PFELAC software platform to automate the generation of parallel elastic large deformation finite element codes. By writing only a minimal amount of fundamental finite element language rooted in the principle of virtual work, we significantly enhance program development efficiency.” “As the number of cores increases, the parallel speedup ratio rises, but parallel efficiency decreases. For 16 cores, the speedup ranges from 11.36 to 12.24, with parallel efficiency between 0.71 and 0.77.”
The new Abstract is below:
A comprehensive understanding of large elastic deformation, characterized by its nonlinear strain and stress properties, is vital for examining tectonic deformation across geological timescales. We employ the PFELAC software platform to automate the generation of parallel elastic large deformation finite element codes. By writing only a minimal amount of fundamental finite element language rooted in the principle of virtual work, we significantly enhance program development efficiency. The accuracy of the finite element method is rigorously validated through comparisons with analytical solutions from two idealized models. Furthermore, we investigate the influence of mesh density and CPU core count on computational performance. As the number of cores increases, the parallel speedup ratio rises, but parallel efficiency decreases. For 16 cores, the speedup ranges from 11.36 to 12.24, with parallel efficiency between 0.71 and 0.77. In contrast, for 64 cores, the speedup is between 24.70 and 34.78, while parallel efficiency drops to between 0.39 and 0.43. The program's application to simulate crustal fold deformation reveals marked distinctions between large and small deformation theories, emphasizing the critical importance of large deformation theory in tectonic studies.
3- Key terms such as "PFELAC software" and "parallel computing" are introduced without sufficient context or references.
Answer: We have added a paragraph to introduce “PFELAC software” and “parallel computing”: We developed a parallel finite element program for elastic large deformation analysis based on the principle of virtual work, using the PFELAC 2.2 platform (Element Computation Technology Co., 2018a, 2018b; Xu et al., 2022; Shi et al., 2023). The program utilizes a domain decomposition approach for parallelization, with a master process responsible for assembling the global stiffness matrix and load vector, while multiple sub-processes handle the computation of element-level stiffness matrices and load vectors (new lines 172-177).
4- Some equations and symbols (e.g., Equation 7 and Equation 23) are introduced abruptly without clear derivations or context.
Answer: Equation (7) is directly derived from Equations (1) and (2) and is presented in tensor form for mathematical clarity. The derivation of Equation (23) is detailed in Refs. 40-41, which provide comprehensive definitions of the various symbols, particularly the inner product symbol, along with the fundamental syntax of the finite element language.
Here, I suggest including more technical details about the numerical implementation (e.g., convergence criteria, solver type, and computational resources).
Answer: We employ the bi-conjugate gradient stabilization (BiCGSTAB) method, a variant of the bi-conjugate gradient (BiCG) method, for solving nonsymmetric matrices. The iterative algorithm terminates when the relative residuals are below . A comparison of CPU core usage for the computations is presented in Table 3. The simulations were run on Huawei's TaiShan 200 (2280) server, equipped with two 48-core Kunpeng 920 processors, four 32 GB DDR4 RDIMM modules, and four 1200 GB general-purpose hard drives. (new lines 205-211)
5- Captions of Figures 3 and 5 should include more detailed explaining their relevance to the topic.
Answer: We modified the Captions of Figures 3 and 5.
Figure 3. Schematic of the ideal model for tensile and rotational elasticity in parallel program testing. (new lines 234-235)
Figure 5 Schematic of a simple shear ideal model for parallel program testing. (new line 284)
6- The comparison with analytical results is briefly presented in tabular form but lacks sufficient discussion on potential sources of error. I think the paper would benefit from a more detailed discussion of discrepancies between numerical and analytical results, especially for stress fields.
Answer: The sources of numerical error in the finite element calculations primarily include the finite nature of the mesh (up to 8 million cells), interpolation errors arising from the choice of shape functions, and the truncation errors associated with the nonlinear iterative algorithm. The numerical error in the displacement field is generally smaller, whereas the error in the stress field is larger due to the least squares method used for stress computation, which does not yield absolute accuracy in the simulated stress values at the nodes. (new lines 416-421)
7- The case study is oversimplified, with limited discussion on geological implications. Please, extend the discussion on the geological significance of the results and how this approach could inform tectonic studies.
Answer: The application case of crustal tectonic shortening is chosen because it is a common geological phenomenon that effectively illustrates the differences between large and small deformation theories. This case serves as a valuable research tool for the numerical simulation of tectonic deformation in more complex scenarios, such as globally distributed folded thrust belts. (new lines 421-426)
8- Figures comparing small and large deformation models lack clarity, especially in terms of units and color scales.
Answer: We have revised the relevant figures. Although there is a noticeable difference in some stress results between the small and large deformation models, the actual difference is nearly zero.
9- The limitations of the study (e.g., assumptions, computational constraints) are not discussed. There is room for that in the Discussion section.
Answer: The sources of numerical error in the finite element calculations primarily include the finite nature of the mesh (up to 8 million cells), interpolation errors arising from the choice of shape functions, and the truncation errors associated with the nonlinear iterative algorithm. The numerical error in the displacement field is generally smaller, whereas the error in the stress field is larger due to the least squares method used for stress computation, which does not yield absolute accuracy in the simulated stress values at the nodes. (new lines 416-421)
Overall, the study has significant potential and demonstrates substantial computational advancements. However, improvements in clarity, technical rigor, and presentation are necessary to meet the standards of a high-impact journal.
Answer: We have carefully addressed the reviewers' comments, incorporating additional text and making improvements to the abstract, methods, figures, and discussion sections. We thank the reviewers for their valuable feedback and suggestions.
Submission Date
27 December 2024
Date of this review
17 Jan 2025 13:05:00
Author Response File: 
Author Response.docx
Reviewer 2 Report
Comments and Suggestions for AuthorsComments are provided in the attached file.
Comments for author File: Comments.docx
Author Response
In the introduction chapter the part relative to what the aims are and what has been achieved in this paper is too short. Improve it by adding something about your motivation and some hints on the main conclusions of the research.
Answer: The elastic large deformation theory uses the Green-Lagrange strain tensor, which accurately captures the coupling of finite rotation and deformation. Unlike small deformation theory, which assumes linear strain and limits strain to 5%, this theory is applicable in scenarios with large deformations. It is widely used in biomechanical engineering, such as simulating heart valve motion (with strains up to 40%), designing foldable screens (with curvature radii < 3 mm), developing flexible electronics, and advancing new materials like hydrogel smart materials (with strains ranging from 200% to 800%).
The study lacks a detailed discussion of potential limitations, such as the assumptions made in the models, numerical stability concerns, or constraints in handling highly complex geometries. Addressing these would provide a more balanced evaluation of the method.
Answer: The sources of numerical error in the finite element calculations primarily include the finite nature of the mesh (up to 8 million cells), interpolation errors arising from the choice of shape functions, and the truncation errors associated with the nonlinear iterative algorithm. The numerical error in the displacement field is generally smaller, whereas the error in the stress field is larger due to the least squares method used for stress computation, which does not yield absolute accuracy in the simulated stress values at the nodes.
The conclusion effectively summarizes findings but could be strengthened by emphasizing specific practical implications of the research.
Answer: The application case of crustal tectonic shortening is chosen because it is a common geological phenomenon that effectively illustrates the differences between large and small deformation theories. This case serves as a valuable research tool for the numerical simulation of tectonic deformation in more complex scenarios, such as globally distributed folded thrust belts.
Some figures, especially those comparing small and large deformation results, could benefit from clearer labelling and annotations to highlight key differences. Additional explanation of colour scales and stress distribution patterns would improve readability.
Figure 4: reduce the decimals from the secondary y-axis
Figures 3 and 5 are way too large
Figure 6: same as for fig. 4
Answer: We have redrawn Figures 4-6 in accordance with the reviewer's comments.
Figure 7: this figure must be improved. What are the symbols indicating? The arrows?
Answer: Fig. 7: Geometrical model of fold deformation during the geological period. The model has a thickness of 1 km, a length of 6 km, and a depth of 2.5 km, featuring two layers of folds above and below. The yellow bands represent crustal folds at varying depths, while the green area denotes the surrounding stratigraphic medium. The left boundary experiences uniform horizontal compression with free sliding in tangent direction, as indicated by a transverse arrow. The upper boundary is unconstrained, while the other boundaries prevent normal displacement and allow tangential sliding, represented by planar rollers.
Author Response File: Author Response.docx
Reviewer 3 Report
Comments and Suggestions for AuthorsThe problem for tectonic deformation across geological timescales is considered.
In this study, a parallel finite element program for elastic large deformation using the PFELAC software platform is developed. It is validated this program by comparing its results with analytical solutions from two conceptual models. Also it is compared the computation time of 1, 4, 16, and 64 cores from different models and quantitatively evaluated speedup and parallel efficiency of parallel finite element program for elastic large deformation.
The authors demonstrate their programming skills and their good knowledge of a complex problem.
The presentation is clear.
The results are well explained.
I recommend correcting (reducing) the sizes of some graphs, especially figure 3 and figure 5.
I recommend publishing this article.
Author Response
The problem for tectonic deformation across geological timescales is considered.
In this study, a parallel finite element program for elastic large deformation using the PFELAC software platform is developed. It is validated this program by comparing its results with analytical solutions from two conceptual models. Also it is compared the computation time of 1, 4, 16, and 64 cores from different models and quantitatively evaluated speedup and parallel efficiency of parallel finite element program for elastic large deformation.
The authors demonstrate their programming skills and their good knowledge of a complex problem.
The presentation is clear.
The results are well explained.
Answer: We thank the reviewer for the careful review and comments. We have revised the abstract, methodology, and discussion to further emphasize the features and advantages of our parallel elastic large deformation finite element program by introducing parallel efficiency and acceleration ratios.
I recommend correcting (reducing) the sizes of some graphs, especially figure 3 and figure 5.
Answer: We reduced the size of Figures 3 and 5.
I recommend publishing this article.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsGood job, thank you!
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors responded and improved all the critical issues I highlighted in the first review.
