Dynamic Error Estimation in Higher-Order Finite Elements
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors1)An effort on performance in English must be taken into consideration , like
"...to numerically assess modal parameters.."
Replace with: to evaluate numerically modal parameters
Must be improved
Author Response
Comment:
1)An effort on performance in English must be taken into consideration , like
"...to numerically assess modal parameters.."
Replace with: to evaluate numerically modal parameters
Response: We have, accordingly, revised the whole manuscript for improving the English. The revisions are left in track changes on the submitted document.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe article proposed Dynamic Error Estimation in Higher-order Finite Elements. The article can be considered for publication in " designs " after revising the following questions. The comments are below.
1. Many of the abbreviations in the manuscript were not explained when they first appeared. We recommend adding the full names of these abbreviations when they first appear in the text.
2. Wrong number of conclusion should be corrected.
3. In the methods section, we recommend describing the construction process of the high-level finite element formula, the specific implementation steps of modal analysis and transient dynamic analysis, including the software tools used, meshing strategies, and solver setup so that the reader can reproduce or verify the study results.
4. The unit of error of error in the simulation result should be supplemented or stated if there is no dimension.
Comments on the Quality of English LanguageA few expressions need to be corrected, such as abbreviations, units.
Author Response
Comment 1: Many of the abbreviations in the manuscript were not explained when they first appeared. We recommend adding the full names of these abbreviations when they first appear in the text.
Response 1: Thank you for pointing this out. We checked all abbreviations making sure that they are all explained in full at their first appearance.
Comment 2: Wrong number of conclusion should be corrected.
Response 2: Thank you for pointing this out. The conclusion section has been renumbered accordingly.
Comment 3: In the methods section, we recommend describing the construction process of the high-level finite element formula, the specific implementation steps of modal analysis and transient dynamic analysis, including the software tools used, meshing strategies, and solver setup so that the reader can reproduce or verify the study results.
Response 3: We agree with the reviewer concerns about allowing reproducibility. Regarding the requests of describing the construction process we modified the sentences at the beginning of section 2 (lines 113-116) to make it clear that the details of the implementation are already described in another publication (ref [24]). We also added a sentence regarding the Software Tool used.
The mathematic background of modal analysis and transient dynamic was already detailed in sections 2.2 and 2.4, whereas the specific implementation steps are given in a bullet points list at lines 339-345. Additional details regarding the eigen solution implementation in Matlab are given at lines 350-353, including the solver setup indications.
A comment on the meshing strategy was added at lines 311.
Implementation details regarding the transient dynamic were added at line 431.
Comment 4: The unit of error of error in the simulation result should be supplemented or stated if there is no dimension.
Response 4: Agree. We have added a sentence at the beginning of Section 4, highlighting that all error metrics used in the study are non dimensional.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors have proposed a very detailed consideration of the formulated problem.
However, it should be specified whether ready-to-use procedures for error estimation have been employed or whether the authors have developed/customized the algorithms used in the reported research.
Furthermore, if the conclusions have been derived using available procedures and finite element types (for example in ABAQUS) could they be generalized for other software packages (for example, ANSYS)?
The boundary conditions for the cube (Figure 1) should be explicitly formulated in the manuscript.
I would suggest using different symbols for vectors and matrices. For example, vectors in bold and something else for matrices.
Author Response
Comment 1: The authors have proposed a very detailed consideration of the formulated problem. However, it should be specified whether ready-to-use procedures for error estimation have been employed or whether the authors have developed/customized the algorithms used in the reported research.
Response 1: Thank you for pointing this out. The whole FE formulation was implemented in a ad-hoc Matlab library, for the scope of conducting the research study. Several additions were made in response to request of reviewer 2 for clarifying the implementation details of the whole procedures, which was done in scratch Matlab.
Regarding the error metrics used, the full mathematical background was given in section 3. All the used error metrics are largely known in the fields. The goal of this study is to build a relation among them.
Comment 2: Furthermore, if the conclusions have been derived using available procedures and finite element types (for example in ABAQUS) could they be generalized for other software packages (for example, ANSYS)?
Response 2: Thank you for the interesting question. As already clarified, the FE procedures used in the study were not commercially available but implemented from scratch using MATLAB. We therefore can assume, that our findings can be reproduced also in any commercial software, provided that the formulation is tuned to adhere to a full-integration Gaussian iso-parametric formulation described in the paper.
Comment 3: The boundary conditions for the cube (Figure 1) should be explicitly formulated in the manuscript.
Response 3: Thank for pointing this out. The essential boundary conditions are explicitly assigned, corresponding to the "one-face-clamped" cube description available right after the Figure 1. A sentence describing the formulation of such boundary condition was added in the text (lines 312-314).
Comment 4: I would suggest using different symbols for vectors and matrices. For example, vectors in bold and something else for matrices.
Response 4: We agree with the need of using different styles for vector and matrices. In facts in our submission, we decided to maintain Matrices and Vectors in bold, whereas scalar in normal font. Moreover, Matrices are in upright bold, whereas vector are in bold.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors have addressed the issues raised in the last review of this manuscript. I recommend accepting this article.
Author Response
Comment: The authors have addressed the issues raised in the last review of this manuscript. I recommend accepting this article.
Reply: Thanks
Reviewer 3 Report
Comments and Suggestions for AuthorsFor works using creatively well-established approaches, the discussion of the novel aspects deserves more attention. Please define the original aspects of the employed procedures.
The authors mention that the FE analysis is developed “from scratch.” Therefore, it might not be hard to explicitly define the boundary conditions for the considered problem, clearly in function of the degrees of freedom of the finite element formulations employed. For example (clearly, after defining the coordinate system):
At x=0, u_x=0…
As stated in the conclusion, one of the potential outcomes of the study is “to achieve high-fidelity predictions, which are essential for contemporary industrial applications such as virtual sensing and digital twins.” To demonstrate the FE approach's applicability in algorithms such as digital twins, the computational time for an FE analysis should be compared with that of the alternative methods. Moreover, it is difficult to define the FE analyses outlined in the study as extremely computationally demanding. I suggest not discussing this at this point: the authors could go back to it at subsequent stages of the research.
A very minor remark:
“tessellation” instead of “tesselation” (line 84)
Author Response
Comment 1: For works using creatively well-established approaches, the discussion of the novel aspects deserves more attention. Please define the original aspects of the employed procedures.
Response 1: we understand and appreciate the suggestion of the reviewer. A paragraph highlighting the original points of this study with respect of the literature was already present in the introduction (lines 79-92). We complemented the paragraph by adding another sentence (lines 93-96) making more explicit another original point of this study which is related to the comparison between the NEE metric and the TRAC and FRAC metrics.
Comment 2: The authors mention that the FE analysis is developed “from scratch.” Therefore, it might not be hard to explicitly define the boundary conditions for the considered problem, clearly in function of the degrees of freedom of the finite element formulations employed. For example (clearly, after defining the coordinate system): At x=0, u_x=0…
Response 2: Section 2.1 has been expanded (lines 174-188), to details the derivation of the constrained FE dynamic formulations.
Comment 3: As stated in the conclusion, one of the potential outcomes of the study is “to achieve high-fidelity predictions, which are essential for contemporary industrial applications such as virtual sensing and digital twins.” To demonstrate the FE approach's applicability in algorithms such as digital twins, the computational time for an FE analysis should be compared with that of the alternative methods.
Response 3: In the conclusions, the extract reported from the reviewer belong to the sentence: "This relationship indicates that the NEE metric could be a reliable tool for fine-tuning FE models to achieve high-fidelity predictions, which are essential for contemporary industrial applications such as virtual sensing and digital twins." We kindly remark to the reviewer that the suggested applicability of the NEE metric to steer the virtualization process of FE in the context of Virtual Sensing and Digital Twins it is not related to the FE methods implementation, but to the nature of the metric itself.
Comment 4: Moreover, it is difficult to define the FE analyses outlined in the study as extremely computationally demanding. I suggest not discussing this at this point: the authors could go back to it at subsequent stages of the research.
Response 4: We appreciate the remarks of the reviewer, although it is hard to find such an explicit sentence in the conclusions. We could only speculate that the comment is related to the computational efficiency discussed at the end of the results section. If that was the case, we kindly remark to the reviewer that the reported results are meant to document the comparison between the different orders of the FE formulations. The only claims we made regarding the computational efficiency regard the fact that for a given accuracy target, the cubic order FE formulations showed to be the more effective alternative.
Comment 5: A very minor remark: “tessellation” instead of “tesselation” (line 84)
Response 5: We thank once more the reviewer for the attention paid to minor details such this. The spelling was corrected.
