Mesh Free Radial Point Interpolation Based Displacement Recovery Techniques for Elastic Finite Element Analysis
Round 1
Reviewer 1 Report
Dear authors and editor!
I have previously reviewed this article many times.
This time the article has been supplemented and I believe that my previous remarks have been taken into account in a sense. Now it is at least clear that the article has developed a new displacement error recovery method. Also added point 8: "Present Study Limitations and future research work", which discusses the possibilities of the proposed method.
Author Response
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Author Response File: 
Author Response.doc
Reviewer 2 Report
Тhe proposed paper is devoted to the development of reliable and cost efficient displacement recovery based error estimation technique using radial point interpolation method in mesh free and mesh dependent environment. The authors use three radial basis functions and circular and rectangular support domain shape to perform error recovery analysis. Finite element analysis on test examples is carried out. The finite element solution errors are calculated directly from the recovered displacement. The characteristics of the radial point interpolation based error recovery with different radial basis function are compared in terms of error convergence properties, effectivity and adaptively refined meshes.
The presentation of the main results is clear and comprehensive. The results are valuable and worthy of being published taking into account the efficiency of the proposed mesh free recovery technique and its suitability for error recovery and adaptive analysis of problems dealing with large domain changes and domain discontinuities.
Minor revisions are suggested to improve the quality of the exposition:
p. 1, line 26: I suggest to write “may prove to be most suitable” instead of “may prove to most suitable”.
p. 1, line 41: The meaning of the acronym FEM should be given before.
p. 2, line 56: “L2” is written differently in different places.
p. 2, line 80: The meaning of the acronym RPIM should be given a line before.
p. 2, line 99: The period is missing after “derivatives”.
p. 3, line 101: The expression “The present study aimed to investigates” should be reformulated.
p. 3, line 117: I suggest the content of the following sections to be described at the end of the Introduction.
p. 3, line 119 and later: The formatting of the equations is different throughout the text.
p. 8, line 259: It is not clear whether “x” in eqs. (36) – (38) means a variable or the multiplication sign.
p. 14, line 353: The captions of figures are inconsistent: sometimes they include the number, sometimes don’t include.
p. 16, line 405: I suggest to write “to the possibility” instead of “to possibility”.
p. 16, line 425: I suggest to write “of the kind of the radial basis function” instead of “of kind of radial basis function”.
p. 17, line 466: I suggest to write “Therefore, the performance” instead of “Therefore, performance”.
p. 17, line 484: It should be suggest to write “It is found from the results” instead of “It is found form the results”.
Author Response
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Author Response File: Author Response.doc
Reviewer 3 Report
Attached
Comments for author File: 
Comments.pdf
Author Response
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Author Response File: Author Response.doc
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
I'm sorry, but due to the confusing notation used by the authors I can't get past page 3 in reading.Reviewer 2 Report
I have reviewed the manuscript "Radial Point Interpolation Based Displacement Recovery Techniques for Elastic Finite Element Analysis", Manuscript ID: mathematics-1240635 that has been submitted for publication in the Mathematics MDPI Journal and I have identified a series of aspects that in my opinion must be addressed in order to bring a benefit to the manuscript.
In this paper, the authors aim to evaluate the displacement error characteristics in the finite element elastic analysis employing the mesh free radial point interpolation (RPI) based recovery approach. I consider that the authors must emphasize more clearly for the reader the meaning, the purpose and the usefulness of the research developed within the paper. Moreover, the paper will benefit if the authors address within the manuscript the following aspects:
Remark 1 – Lines 10-24, the "Abstract" of the manuscript. I consider that the authors should structure the "Abstract" as to cover the most important points of interest: the authors should have positioned the manuscript’s topic in a broad context, therefore covering appropriately the topic’s background; the authors should have presented succinctly the methods they have employed in order to attain the purpose of their study; the authors should have summarized the most important outcomes of their study and the conclusion that one could draw. I consider that the abstract of the manuscript under review will be improved if the authors state and explain concisely at the end of the abstract the clear contribution that their study has brought to the current state of knowledge. In the abstract, the authors must state more clearly for the reader the meaning, the purpose, the usefulness of the research developed within the paper, their methods, their original results and conclusions as well as the novelty of their study. In the actual form of the manuscript, the abstract offers information related only to some of these aspects and even so, their delimitation is unclear.
Remark 2 - the "Introduction" section, the gap in the current state of knowledge. After having performed a critical survey of what has been done up to this point in the scientific literature, the authors must identify and state more clearly in the paper a gap in the current state of knowledge that needs to be filled, a gap that is being addressed by their manuscript. In order to pursue innovative research is mandatory to identify research gaps, consisting in missing pieces or insufficient information in the scientific literature, areas subjected to further research because they are unexplored, under-explored, or outdated. Therefore, in the "Introduction" section, the authors must pinpoint an exact deficiency, an unsolved problem, a gap that still exists in the current body of knowledge that needs to be filled, therefore justifying the need and novelty of their study. Otherwise, without identifying and stating clearly this gap, the study from the Manuscript ID: mathematics-1240635 does not justify its need, importance and novelty. This gap must also be used afterwards by the authors, when discussing the obtained results, when the authors should justify why their approach fills the identified gap in rapport with previous studies from the scientific literature.
Remark 3 - the final part of the "Introduction" section. It will benefit the paper if in the final part of the "Introduction" section, the authors present the main contributions of their paper, eventually synthetized within a bulleted list. At the end of the "Introduction" section, the authors should preview the structure of their paper under the form: "The rest of the paper is structured as follows: Section 2 contains…".
Remark 4 - the equations within the manuscript. The equations within the manuscript should be explained, demonstrated or cited, as there are some equations that have not been introduced in the literature for the first time by the authors and that are not cited.
Remark 5 - the software and the detailed hardware configuration. It will benefit the paper to specify details regarding the version numbers for the software and the detailed hardware configuration used to obtain the results.
Remark 6 - the chosen numerical examples, from Section 6. In section 6, at Lines 226-227, the authors state: "The quality of recovery method is obtained by adaptive finite element analysis of two elastic plate problems...". First of all, I consider that the authors should highlight more the generalization capability of their approach in order to be able to justify a wider contribution that has been brought to the current state of art. Secondly, the authors should provide more details regarding the chosen numerical cases by explaining the source of data contained by Tables 1-14 (it is unclear if these values have been retrieved from somewhere or computed someway).
Remark 7 – Pages 9-14 of the manuscript. In the actual form of the manuscript, there are a lot of insufficiently interpreted/explained tables and figures, spreading along the pages 9-14. The authors must explain and analyze in detail all the tables and figures that have been inserted within the manuscript, it is not suitable to put the reader in the situation of interpreting, analyzing, continuing or refining the study from the manuscript under review.
Remark 8 – the "Discussion" section. In order to validate the usefulness of their research, when discussing their obtained results, the authors should make a comparison between their study from the manuscript and other ones that have been developed and used in the literature for the same or similar purposes. In its actual form, the manuscript does not mention anything about the way in which the obtained approach can be perceived in perspective of previous studies that have tackled similar problems. There are many valuable studies in the scientific literature related to the subject of the manuscript to which the authors can compare to and this comparison will highlight even more the insights that their paper has brought in contrast to the existing studies. This comparison is mandatory in order to highlight the clear contribution to the current state of knowledge that the authors have brought and the advantages of the developed approach in contrast to the existing ones from the literature.
Remark 9 – limitations and future research work. When discussing their obtained results, the authors should also highlight in the manuscript current limitations of their study, and briefly mention some precise directions that they intend to follow in their future research work.
Remark 10 - insights. The paper will benefit if the authors make a step further, beyond their approach and provide an insight when discussing their obtained results regarding what they consider to be, based on the obtained results, the most important benefits of the research conducted within the manuscript, taking also into account its practical applicability. The authors should highlight whether, or to what extent they have managed to address the necessity that should have been previously identified within the "Introduction" section. In this section the authors must present in a clearer manner the learnings and insights of their research.
Remark 11 – the sections' and subsections' numbering and order. From a certain point, the sections and subsections of the manuscript are numbered in a random manner. Therefore, in the actual form of the manuscript, section "5. Measurement of Errors" does not contain any subsection and is followed by section "6. Elastic Plate Problems" that has the following subsections: "6.1 Square Plate Problem", "5.1.1 Meshfree and Mesh Dependent Displacement Recovery Techniques", "5.1.2 RPI Recovery Technique and Radial Basis Function Type", "5.1.3 RPI Recovery Technique and Patch Configuration", "6.1 Square Plate with Opening Problem", and afterwards, while the section 7 is missing, it comes section "8. Discussion". Please address these issues.
Remark 12 – the Figures' citations within the paper. The first Figure cited within the paper is Figure 4 (at Page 3), followed by Figures 1,…,8, in the correct order. Please address this issue by renumbering the Figures.
Remark 13 – the Tables' citations within the paper. In the actual form of the manuscript, one can often find tables' citation under the following manner: "Table 1-6" (Line 249), "Table 1-3" (Line 261), "Table 4-6" (Line 278), "Table 7-9" (Line 293), "Table 11-13" (Line 322), "Table 1-10" (Line 373), "Table 11-14" (Line 375 and Lines 415-416). This kind of citation is inappropriate, due to the fact that is used the singular noun "Table" along with more than one table. Please address this issue.
Reviewer 3 Report
The manuscript is a resubmission of the previous one which was titled «Comparison of Meshfree Displacement and Stress Error Recovery of Finite Element Solutions using Moving Least Squares Interpolation».
In contrast to the previous manuscript, the radial point interpolation method has been added, which has been used by many researchers over the past two decades. The introduction has also been significantly revised, it has become more motivating.
Computational experiments are performed in computational domains of simple geometry and are illustrative. It is not clear how the methods used in this manuscript will behave in real problems. It is worth mentioning this in the discussion section.
The novelty of the work is not clear. I highly recommend explicitly emphasize what is new was proposed by the authors.
