A Co-Rotational Meshfree Method for the Geometrically Nonlinear Analysis of Structures
Round 1
Reviewer 1 Report
Review of
A Co-rotational Meshfree Method for Geometrically Nonlinear Analysis of Structures
by Wen Cheng Yeh
The author presents a corotational beam formulation discretized with the differential reproducing kernel (DRK) method for large deformation problems. The meshfree method provides advantages over the finite element method for large deformations and geometrically nonlinear problems. Additionally, the DRK method improves upon the standard RK method by approximating the derivatives in the stong form of the governing equations directly. Several numerical examples are provided that demonstrate the effectiveness of the method.
The structure of the manuscript is good and the included references are sufficient. The numerical examples and comparison with published data are also sufficient. However, the manuscript would benefit from considerable editing for English grammar. The English can be difficult to read in places. The manuscript could also be improved by addressing the following comments and questions.
1. Page 4. Equation 3 - should this be e_{XYZ} = dr(s)/ds? Also, in line 164, the reference to Equation (3) is confusing - it seems to reference a previous equation rather than the Equation 3 given below.
2. Page 11, lines 360-362 state, "In the following examples, the quadratic basis functions and five neighboring nodes, Ne=5, were adapted in the DRK approximations." However, in the first example different basis functions and number of neighbors are used.
3. Page 13, Figure 6, are the R values in the legend the approximate convergence rates? Is the optimal number of neighbors used for each basis order this plot? Can more description be provided here?
4. Page 13, Figure 7, this figure does not add much information. Would recommend removing it or providing a more detailed description in the caption and text.
5. Figures 12, 13, 20 and 21 are difficult to read.
Author Response
Please see the attachment
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Line 233, the constitutive law links internal forces and generalized strains.
For test 4.2 it is worth adding the results obtained in a recent paper [1].
Furthermore, in that paper [1] it is also reported the number of iterations required to obtain the deformed configuration by a novel scheme with indepedent stress [2]. Are stress variables used as independent variables in your iterative solution process? The number of iterations required by your method should be reported for the sake of completeness.
[1] "A large rotation finite element analysis of 3D beams by incremental rotation vector and exact strain measure with all the desirable features"
[2] "How to improve efficiency and robustness of the Newton method in geometrically non-linear structural problem discretized via displacement-based finite elements"
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 3 Report
The present paper shows an interesting development of a FEM method for beam element that account for 3D elements and large deformation. Though it shows many potentials and interest to the scientific community, the reviewer thinks it needs to be improved before any publication. First, the English quality really needs to be improved. Many sentences begin with “And” / Many sentences has no verb, many verbs do not have the correct tense. And many typos (space, missing words). The reviewer did not pointed out all of them, though he began to do so. Please read carefully or ask to a collaborators to check and correct English mistakes
Second, too many figures have poor quality. Graphs (Figures 5, 6, 8, 10, 12, 13, 15, 16, 18, 19, 20 and 21) must not be blurry, have readable and explained (or straightforward) axis legend, readable legend. In general, please increase the font size. Figures that described the case study could be more helpful to understand the subject investigated. Sometimes axis, forces, or dimensions are missing. Please include all relevant dimensions directly on the Figures
Third, The presentation of the case studies is often a bit fuzzy and could be improved for the sake of understandability.
Additionally, some specific (non-exhaustive) comments follow:
Line 17: “the concept meshfree method of having no mesh”: not very elegant. It also seems that the sentence is not grammatically correct.
Line 60-61: The sentence is neither clear not correct. Please clarify.
Line 64-65: the sentence is not grammatically correct
Line 73: missing spaces before the references.
Line 76: have taken or took instead of taken.
Line 81-84: the sentence does not seem correct. It is also a bit hard to follow. Consider rewriting.
Line 84: undergone does not seem correct.
Line 85: a “that” seems to miss between “known” and “the”.
Pay attention to the English style used along the document. Either choose American English (characterize, modeling, modeled, behavior, color, discretized, analyze) or British English (characterise, modelling, modelled, behaviour, colour, discretised, analyse) and keep it for the rest.
Line 170: ex, ey and ez are already defined just above.
Please increase the quality of Figure 5 and 6. Axis are neither describe, not readable. Please clarify. Also, the paragraph Line 372-382 is not very clear. The author should better state what he wants to show with these figures. The analysis of the best compromise (Figure 4 and 5) misses the computational time required (at least mentioned). Also, the number of points is very limited and does not help the reader to understand the point of the author.
The reviewer understands that the author developed an extended methodology for 3D problems. The author first show some application in 2D to compare the results with existing literature. Please state this more clearly in the beginning of Section 4.
In general, the case studies are not well described. Please improve.
The reviewers does not understand the interest of Figure 7. By the way, the figure states 41 nodes while the text 31 (Line 385). If useless, please remove the figure.
In the reviewer’s opinion, it’s difficult to state the accuracy of the method using only three nodes (Figure 8). What happens if we increase the number of nodes? Do all the nodes align with the analytical solution? It should be checked for a case with 10 or even more nodes.
The third case study misses the axis (X, Y, Z) which prevents the reader from understanding Figure 13.
Table 2: last line, last column. 897. Seems to miss some digits.
The description of Case study 5 is not complete. We do not know if the length refers to the curved length or the distance between supports. Also, based on Figure 17, the arc is not perpendicular to the cone of 60º. If this is on purpose, please also provide the height of the arch. The dimensions of the cross-section, though probably evident, are also missing. Some forces and moments (Fzb, Mza) are not defined. It makes the understanding of the example very complicated.
Figure 18 and 19 show Fyb while in the text, Fzb is written. The reader is lost.
Author Response
Please see the attachment
Author Response File: Author Response.pdf
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.
