Designing Parallel Adaptive Laplacian Smoothing for Improving Tetrahedral Mesh Quality on the GPU

Round 1

Reviewer 1 Report

Designing Parallel Adaptive Laplacian Smoothing for Improving Tetrahedral Mesh Quality on the GPU

The authors present a useful numerical study by exploiting the parallelism features of the GPU. They propose a parallel adaptive Laplacian smoothing algorithm for improving the quality of large-scale tetrahedral meshes. The manuscript reads well although there are minor language errors to be corrected and the clarity of some parts to be improved. There also technical deficiencies to be addressed. Improving suggestions are listed below:

1. Abstract (line 13): the numbering of the results outcomes should be removed and be written as normal text.
2. Introduction:
   1. 1^st paragraph: “Generally, the study area is discretized into triangular, tetrahedral or hexahedral meshes [1].”, “The reliability of FEM simulation results is often inextricably impacted by the mesh quality; therefore, it is necessary to further optimize the mesh to improve its quality after the initial generation of FEM meshes [2].” The first paragraph needs improvement. Please refer to the type and dimensions of the solution domains and make these statements in a better way using appropriate references on the influence of the mesh quality to certain types of FEM problems. These two statements are very general are now written.
   2. (line 36) “Laplacian mesh smoothing is computationally expensive.”, When? Why? In which type of meshes? Please be more precise when using such “strong” statements.
   3. (line 56) “However, the proposed ordinary algorithm for tetrahedral mesh cannot avoid the disadvantage of creating potential invalid nodes in the concave area of the mesh, which reduces the accuracy of the algorithm.” Is there a percentage to mention for the reduction? Does it hold for every test case applied? Very general writing.
3. 1.2 (line 94) “The adaptive Laplacian mesh smoothing method is effective dealing with complex curved surface models and successfully avoids invalid nodal iterative calculations in the basic algorithm.” Please mention some computational cases that this is crucial to highlight the importance of this work.
4. (line 96) “At the same time, the adaptive implementation of the algorithm is relatively simple, and the quality calculation is easy to implement on a GPU in parallel. Therefore, the adaptive Laplacian mesh smoothing algorithm is better than the ordinary Laplacian mesh smoothing algorithm in terms of computational accuracy.” These statements should be moved at the end of the manuscript.
5. 1.3. (lines 101-108). Please improve the writing. This paragraph is confusing as written.
6. 2 (line124): “Generally, the SoA data layout can obtain higher global memory performance.”. Please avoid these general comments. In the next sections we see that this statement does not hold for all cases.
7. 1.1 (lines 259-262) This paragraph is very confusing and must be improved. The expressions “slightly better”, “better efficient performance” should be quantified and better described. The results graphs of Fig. 8 should be better explained for the reader to understand the influence of the AoS and SoA layouts.
8. Figures 12 and 13 should be presented in one figure to be easily compared.
9. 3 (lines 313-320). The shortcomings (1)-(3) should be mentioned in the beginning of the proposed study and written in a better way, followed by the advantages of the proposed algorithm.
10. The English language must be improved. Correct minor language errors and improve the narrative and clarity of text.

Author Response

Dear Reviewer,

We would like to submit our revised paper entitled “Designing Parallel Adaptive Laplacian Mesh Smoothing for Improving Tetrahedral Mesh Quality on the GPU” for your consideration for publication in the journal Applied Sciences.

We have made a point-by-point response to the reviewers’ comments and suggestions, including a detailed description of any requested or suggested revisions.

We have also carefully checked and corrected the writing format and errors to make our revised manuscript conform to the journal style.

All the modifications and explanations in this revised version are listed in detail in the following “Responses to Reviewer's Comments”.

We deeply appreciate your consideration and reviewers’ helpful comments and suggestions.

Yours Sincerely,

Ning Xi, Yinjie Sun*, Lei Xiao, Gang Mei*

School of Engineering and Technology, China University of Geosciences (Beijing)

Email: [email protected] (G. Mei)

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper presents extension of authors’ work on Laplacian mesh smoothing, and development and implementation of a new parallel adaptive Laplacian smoothing algorithm on GPUs. Their work is focused on tetrahedral meshes where for large scale applications improvement of mesh quality is computationally expensive. They use the aspect ratio of radii of inscribed and circumscribed spheres of a tetrahedron to judge the smoothing algorithm. The algorithm is tested with five sets of tetrahedron meshes ranging from around 5K to around 100K vertices. They have implemented two forms (named A and B) of smoothing iterations where one uses old nodal locations from the previous iteration and the other uses  new locations of the neighboring nodes from the current iterations; data exchange requirements of both have implications on performance. The paper describes the proposed new search method extending on their earlier work referred in the paper and the new method alleviates the issues of the older method such as being unable to determine the length of the array of neighboring nodes and difficulty in determining all the neighbors of nodes at the same time. They provide details of their search method, including using the aspect ratio as a metric for evaluating the quality of tetrahedral meshes, and implementation on GPUs.
Their GPU implementation results compare various results such as performance against CPU results, between the two forms (A and B), among data layouts (Array of Structures versus Structure of Arrays), single block GPU versus multiple block GPU, etc. They also point out what have not been considered in the paper, such as misaligned AoS data, ignoring the influence of weights of different neighboring nodes etc.

In the results section it is not clear if the CPU results are from a single core CPU run or the CPU results were run on multi-cores of the CPU as the CPU has 48 cores i.e. if the CPU run was for a parallel code also. This should be clearly stated and if it has any impact on analysis of results, it should be added.

Overall the paper is well written and describes well the new method and all the issues and improvements associated with it and compared to the existing method. There is thorough analysis of results. To some extent the novelty is built upon extending the existing work and seems a continuing work (as many future directions are mentioned and which are not considered in this work) and still this is a new incremental work and shows good parallel performance results. And as such it is important information for the community interested in and involved in this research.

Some minor English typos and edits should be done. For example in the second paragraph of section 5.2 it says "It is clearly in Figure 3, there are
approximately 2000 tetrahedrons with aspect ratiso close to 0.0." There are typos (it should be Figure 13?) and English edits that need to be corrected. There are similar typos in other places also that need to be corrected by having a thorough edit.

Author Response

Dear Reviewer,

We would like to submit our revised paper entitled “Designing Parallel Adaptive Laplacian Mesh Smoothing for Improving Tetrahedral Mesh Quality on the GPU” for your consideration for publication in the journal Applied Sciences.

We have made a point-by-point response to the reviewers’ comments and suggestions, including a detailed description of any requested or suggested revisions.

We have also carefully checked and corrected the writing format and errors to make our revised manuscript conform to the journal style.

All the modifications and explanations in this revised version are listed in detail in the following “Responses to Reviewer's Comments”.

We deeply appreciate your consideration and reviewers’ helpful comments and suggestions.

Yours Sincerely,

Ning Xi, Yinjie Sun*, Lei Xiao, Gang Mei*

School of Engineering and Technology, China University of Geosciences (Beijing)

Email: [email protected] (G. Mei)

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The revised paper has a better design, the English language and style have been edited and the quality of presentation has been improved. The details presented, in text and figures, have also been improved. Authors have amended all my suggestions and comments properly.