Multi-Objective Optimization of Spatially Truss Structures Based on Node Movement

Round 1

Reviewer 1 Report

The manuscript presents an improved Multi-object Evolutionary Algorithm (MOEA) to realize topology optimization of spatial truss structures. The optimization variable were the coordinates of the nodes and the optimization goals were the weight and the maximum displacement. The authors described the used methodology and its computational implementation. Two examples are presented, one related with a cantilever truss beam and another one related with a tower. From the obtained results, the authors conclude about the feasibility of using the modified MOEA algorithm for the structural design of spatial truss structures.

The topic that is developed in the presented study is interesting and still needs further research. Structural optimization is an important study topic and the proposed methods are of great interest for practice.

This manuscript is a revised version and a resubmission of a previous one. This new version of the manuscript has some minor improvements in comparison with the previous revised one. I consider that the revised manuscript submitted by the authors can be accepted in the present form to be published.

Author Response

Thank you for the reviewer’s comments concerning our manuscript entitled “Multi-objective optimization of spatially truss structures based on node movement” (ID: applsci-725299).

Reviewer 2 Report

Please see the attachd comments.

Comments for author File: Comments.pdf

Author Response

Response to the reviewers

On behalf of my co-authors, we thank you very much for giving us an opportunity to revise our manuscript, we appreciate editor and reviewers very much for their positive and constructive comments and suggestions on our manuscript entitled “Multi-objective optimization of spatially discrete structures based on node movement,” (ID: applsci-725299).

We have studied reviewer’s comments carefully and have made revision by using the track changes mode within the document. We have tried our best to revise our manuscript according to the comments. Attached please find the revised version, which we would like to submit for your kind consideration.

The major revisions were marked in red. We would like to express our great appreciation to you and reviewers for comments on our paper. Looking forward to hearing from you. The point-by-point answers to the comments and suggestions were listed as below. Thank you and best regards.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Thanks authors for their careful replies. The reviewer's comments have been addressed. The manuscript can be accepted in present form.

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

The paper deals with multi-objective optimization of the truss structures. The two components of the vector-valued objective function are defined as the weight and maximum displacement of the structure, respectively. The design variables are the coordinates of the nodes of the truss. The Ansys commercial system together with the improved version of the original Multi-Object Evolutionary Algorithm (MOEA) implemented in C# language were used in search of the set of Pareto optimal solutions. The paper presents the solutions of vector optimization of two three-dimensional cantilever trusses modelling an elongated “shell” structure with an equilateral triangle or a square cross-section. The subject of the work is very important in practice because the search for optimal solutions for more than one objective function requires properly defined compromise. This is due to the fact that the requirement of minimizing all objectives simultaneously is impossible, because in general individual objectives are in contradiction to each other – an improvement with regard to one objective function causes the deterioration of another.

The work should be allowed to be published, however, after making some important corrections.

Remarks.

Remark. The authors incorrectly use the term “topology optimization” in the context of their work and shown examples. The paper concerns the classic problem of geometry optimization of the truss structures and not truss topology, because the design variables are only the Cartesian coordinates of the truss nodes, and not, for example, the cross-sectional areas of the bars, for which a value of 0 is allowed (changing the topology of the truss node connections). The term “topology optimization” should be either removed from all those places in the text where it refers to the task of searching for optimal positions of truss nodes or should be replaced by the correct term “geometry optimization”. The purpose of the work is not, in particular, to search for optimal connections of truss nodes with bars with different values of cross-sectional areas (including bars with zero cross-sections). A good example of classic vector topology optimization of elastic bodies is e.g. the paper S. Czarnecki, Edgeworth-Pareto optimal trusses of least compliance, in Mechanics and materials, OWPW, ISBN: 978-83-7814-170-9, p.61-75, 2013 in the context of topology optimization of trusses and the paper S. Czarnecki, T. Lewiński, Pareto Optimal Design of Non-Homogeneous Isotropic Material Properties for the Multiple Loading Conditions, Phys. Status Solidi B, 254, 1600821, 2017 in the context of non-homogeneous isotropic elastic bodies. Remark. The numerical results (page 6) obtained by the authors confirm the correctness of Pareto optimal solutions for two test functions (page 5) obtained on the basis of the improved and implemented in C# MOEA algorithm. The reviewer assumes (maybe wrongly) that equally correct results for the same test functions could be obtained for the original version of the MOEA algorithm. If this is true, what are the advantages of the corrections introduced in the original version of the MOEA algorithm. In the reviewer's opinion, the paper should include at least one additional example of numerical calculations, actually confirming the advantages of the changes introduced by the authors. Remark. The entire chapter 4 should be improved in terms of clarity of description of both examples presented in it. In particular, the boundary conditions, i.e. supports of both trusses together with the applied loads, should be clearly marked on the drawings (loading applies to the first example). Remark. The axes of the 0xyz global coordinate system are also not marked (in the context of sentence "… where the variable is the z direction coordinate of the truss node and its locator variable is eight dimensions …" (171). How to understand a sentence “…its locator variable is eight dimensions” ? Remark. How to understand the words “positive triangle” ? Is it about the term "equilateral triangle" ? Remark. Please explain in the paper the term “inflection point positions” in sentence “…As the number of iterations increased, the points on the front edge of Pareto became more evenly distributed, and the inflection point positions became more blurred…”. Is it correct to use the term “on the front edge of Pareto” at the stage of early iterations ? Remark. Figure 7 does not show any truss topology (incorrect use of this term two times) – see the sentence (182-184): “…The truss topology with typical evolution characteristics is shown in Figure 7, and the result indicates that truss topology appeared disordered and had a poor regularity during the early stage of evaluation …”. See also caption under Fig.10 “…Truss topological evolution diagrams: …” - incorrect use of the term “truss topological evolution” here and in many other places. Remark. How to understand the term "…the total structural displacement…" in sentence “Under this load, the structure was optimized and analyzed to minimize the total structure mass and the total structural displacement.” The word “total” is sometimes used e.g. as “total compliance of the body” but what means “total displacement” ? Maybe the integral of the field of all displacements calculated along all truss elements? Remark. The sentence “…The absolute value of the axial stress was between 0 and 235 MPa …” (209-210) suggests (but perhaps this was not the intention of the authors) that the calculations were carried out taking into account the functional constraints on the maximum stress in the truss elements. If this is the case, then the question arises how in the implementation of the MOEA algorithm the functional conditions for design variables have been taken into account ? It is known that taking into account functional constraints (not only trivial box constraints) in all evolutionary algorithms is one of the most difficult problems, most often solved by introducing various types of penalty functions, but not only. So how have stress limits been implemented ? Remark. The work is mainly descriptive. There are no mathematical formulae except simple expressions (1-2). The reviewer does not want to treat this statement as a disqualification. However, the lack of a strict and correctly mathematically defined definition of the problem of vector optimization together with all the concepts appearing in it, such as: design variables, objective functions, constraints, etc., constitutes a serious obstacle in accepting the work for publication. An example of a clearly and strictly mathematically presented basis of the theory of evolutionary optimization algorithms together with numerous numerical examples can be found e.g. in the monograph A. Osyczka, Evolutionary Algorithms for Single and Multicriteria Design Optimization, Physica-Verlag, A Springer - Verlag Company, Heidelberg, New York, 2002. This monograph cannot fail to be found in the references.

Questions.

Question. The second objective function (maximum displacement) requires costly identification of the node in which the displacement is (for a given set of admissible design parameters) the largest among all displacements of the other nodes. Compliance, i.e. the work of all loads on structure displacements, is a much better function both from a mathematical point of view and its physical interpretation, because it is closely related to elastic energy (see e.g. Castigliano theorem) and is one of the best-studied and most often minimized functionals in the topology optimization of the structures. So why was the maximum displacement chosen as the second objective function? Question. In the sentence “The innovation this study is that it removes the inconsistency of the node position in the optimization of traditionally discrete structures or the Ground Structure Approach and uses the coordinate of the node as the optimization variable for the optimization calculation” (14-17 in Abstract) is not understandable to the reviewer term “…it removes the inconsistency of the node position …”. Please, define the term precisely in the text. Question. In the sentence “…For a given issue, the Pareto optimal solution and frontier is definite, and an MOEA is used to solve the multi-objective optimization problem with the view of obtaining the Pareto optimal set and then confirming the final solution according to other information about the problem (such as boundary conditions, etc.)...” (93-96) what has the term “boundary conditions” to do with the “MOEA” ? The boundary conditions are a fundamental part of the static analysis of the structure and must be included in the numerical calculations (e.g. performed by the Ansys system). So why do they appear in this sentence ? Question. The two sentences “…The fitness function is not calculated by the objective function directly, but by indirect connection of the Pareto strength and the objective function value. The Pareto strength increases during the evolutionary process (shown in Figure 2); this is the individual solution with low density, and then the total disaggregation advances to the Pareto optimal frontier through an evolutionary process to finally obtain the representative optimal solution set …” (106-111) are not understandable. In particular what the term “Pareto strength” means ? English is not the native language of a reviewer who is not an expert in using this language. However, in the reviewer's opinion, the work requires a thorough linguistic correction.

Author Response

Dear Reviewer,

The manuscript applsci-652370 has been carefully revised, and the major revisions were marked in red in revised manuscript. We appreciate the detailed and useful comments and suggestions from you and referees. In order to give a clearer one-to-one answer, I divided the reviewer's Remarks into 10 sections and Questions into 4 sections, then answered them one by one. 

Response to Remarks：

Response to Remark1: we accept your suggestion, we use geometry optimization instead of topology optimization in the analysis of typical discrete structure optimization section.

Response to Remark2: The improved MOEA method introduces the idea of Pateto strength order, and several formulas are added to the revised manuscript for explanation. Please check.

Response to Remark3: We accept the referee’s suggestion. We updated Figure 4, and added load and boundary condition.

Response to Remark4: “z direction” refers to color number direction, and we added coordinates to the original picture for illustration, and used positions to replace dimensions. I mean the chord of the truss is evenly divided into eight segments, and the coordinates of the nodes move around the nodes.

Response to Remark5: We accept the referee’s suggestion. We used “equilateral triangle” to replace” positive triangle”, please check.

Response to Remark6: The inflection point in this paper is the data point where the Angle of Pareto front appears. The emergence of inflection point often means that the calculation is still in the early stage of the whole optimization process. With the deepening of the calculation, Pareto frontier curve will become more and more smooth. The front edge of Pareto and Pareto frontier have the same meaning, therefore it is right to use the term “on the front edge of Pareto” at the stage of early iterations.

Response to Remark7: We accept the referee’s suggestion. Figure 7 was replaced with Figure 10. We delete truss topological evolution here and other places in the revised manuscript. We used shape change diagram of truss instead of truss topological evolution.

Response to Remark8: I'm sorry, “total” doesn't use it right, it really should not be applied to the displacement of all nodes, so the correct sentence is “the structure was optimized and analyzed to minimize the total structure mass and the maximum structural displacement.”

Response to Remark9: We accept the referee’s suggestion, that's not really what I meant, so we deleted the sentence “…The absolute value of the axial stress was between 0 and 235 MPa …” in the revised manuscript.

Response to Remark10: We accept the Suggestion of reviewer, and add the Formula (1) ~ Formula (3) and description based on Pareto strength order method in the improvement of MOEA in section 2.2, hoping to make it easier for readers to understand.

 

Response to Questions：

Response to Question1：Since the structural weight and node displacement are a pair of opposite variables under the action of load, the increase of the structural weight inevitably leads to a small node displacement, but the construction cost also increases accordingly. Therefore, the maximum displacement of the node selected by the second objective function is to find the equilibrium point of the two in a single calculation.

Response to Question2：The traditional topology optimization is based on continuous structure, while this paper is based on discrete structure. The traditional optimization algorithm for discrete body structure (Ground Structure Approach) has the stable coordinates, but this paper takes the node coordinates of discrete body structure as the optimization variable, indicating that the node coordinates can be changed.

Response to Question3：The solution on the front of Pareto is many solutions rather than a single solution, and the boundary conditions define the range of this set of solutions, so once the boundary conditions are given, people can choose the appropriate optimal solution according to their own needs

Response to Question4：Each individual in the population is distributed with a strength value, which represents the number of individuals dominated by individual in the evolutionary population. In view of the above problems, we have asked the native speaker to polish the language of this paper. Thank you for your comments and suggestions.

 

Reviewer 2 Report

 

Section 1:

Please define Pareto. I suggest moving the definition of Pareto optimal frontier when it’s first introduced in Section 1. Please justify why “the weight and maximum displacement” of the discrete structure were used as the optimization goal”. Why are these two variables are set to be the goals when improved MOEA method is used?

Section 2:

Line 117: what is Pareto strength value order? Line 121: “… by way of transferring the value to the optimization module through the connector”, what is the optimization module, is it built inside ANSYS or author wrote this module as an external module? What is the “connector”?

Section 4:

If I understand the loading conditions correctly, this truss is working as a cantilever beam (one end fixed, one end free) with a concentrated load (or point load) on the free end. Can the authors provide a schematic of the problem setup? What are the material type and material parameters used for the truss material? how is convergence guaranteed in this algorithm? Figure 7 shows that by step 50, the solution is still evolving. How is it sensitive to the number of points used at the initial step?

Overall, I suggest the authors address the following issues:

How are the solutions comparing to a traditional single-objective optimization problem? By adding multiple constraints, what are the numerical challenges numerically? What is the significance and/or advantage in using nodal degree of freedom and design variables? How do they compare to or outperform other methods? The authors didn’t address clearly the challenges, advantages, and performance that the proposed method has compared to other methods. I suggest the authors do a major revision to address those before this paper is published.

Author Response

Dear Reviewer,

The manuscript applsci-652370 has been carefully revised, and the major revisions were marked in red in revised manuscript. We appreciate the detailed and useful comments and suggestions from you and referees. The point-by-point answers to the comments and suggestions were listed as below.

Response to Section 1:

Thanks, we accept the reviewer’s suggestion. MOEA method (Multi-Objective Evolutionary Algorithm) is to seek one or a group of compromise solutions in the case of multiple conflicting targets. When there are multiple targets, there are usually solutions that cannot be easily compared. This solution is often referred to Pareto optimal solution. We move the definition of Pareto optimal frontier when it’s first introduced in Section 1. Because under the same load, increasing the weight of the structure will inevitably reduce the maximum displacement of the structure. Therefore, taking the two opposite indexes (structure weight and maximum displacement) as optimization variables is to find the best combination of them in the same calculation.

Response to Section 2:

Pareto strength value order: Each individual in the population is distributed with a strength value, representing the number of individuals dominated by individual in the evolutionary population. While the fitness value of an individual represents the sum of the strength values of the dominant individual in the evolutionary population. The smaller the fitness value, the more likely the individual is to be selected for the next evolution. Optimization module means the optimization program that author wrote in C# language, it is outside ANSYS through “Call command program”, and we regard this “Call command program” as connector.

Response to Section 4:

I really appreciate your judgment. Yes, your understanding is right, “The boundary condition is fixed at one end and free at the other end, and the free end is subjected to a downward concentrated force load” is in original text. We accept the reviewer’s suggestion, and update Figure 4. (Truss schematic diagram). Specific optimization algorithm has nothing to do with material, so we do not list material parameter. Pareto frontier continues to advance, proving that the calculation convergence, in the previous calculation method verification, after 50 steps of calculation, it can meet the requirements, and the distribution of each data point is uniform, we can infer calculation reasonable.

Response to following issues:

Thanks for the reviewer’s comments. Compared with the traditional single-target problem, the MOEA multi-target solution adopted in this paper considering the two targets: structure weight and maximum displacement under load, which is more comprehensive. Setting corresponding constraints and boundary conditions in the optimization variables can make the calculation itself more reasonable and more practical in engineering, which will increase the difficulty in the convergence of numerical calculation. Different from the traditional “Ground Structure Approach” this paper takes the node coordinate as the optimization variable, which can reflect more samples and correspondingly overcome the shortcomings of the traditional algorithm in the initial stage of optimization, such as heavy computation and slow speed. In view of these problems, we have made major modifications according to the specific requirements of reviewers. I hope to get your approval.

Reviewer 3 Report

Ms. Ref. No.:  applsci-652370-652370
Title: Multi-objective topology optimization of spatially discrete structures based on node movement

Journal: Applied Sciences

-Reviewer

This manuscript takes practice to develop the solutions for topology optimization of spatially discrete structure. The manuscript has been written in good structure and is acceptable for publication.

Author Response

Response：Thank you for the reviewer’s comments concerning our manuscript entitled “Multi-objective topology optimization of spatially discrete structures based on node movement” (ID: applsci-652370).

Reviewer 4 Report

The manuscript presents an improved Multi-object Evolutionary Algorithm (MOEA) to realize topology optimization of spatial truss structures. The optimization variable were the coordinates of the nodes and the optimization goals were the weight and the maximum displacement. The authors briefly described the used methodology and its computational implementation. Two examples are presented, one related with a cantilever truss beam and another one related with a tower. From the obtained results, the authors conclude about the feasibility of using the modified MOEA algorithm for the structural design of spatial truss structures.

The topic that is developed in the presented study is interesting and still needs further research. Structural optimization is an important study topic and the proposed methods are of great interest for practice.

The manuscript needs to be thoroughly revised before it can be accepted for publication.

 

Comment 1

The manuscript needs to be largely revised to improve the reading and understanding. Some parts of the article are hard to read and understand. It is recommended that the authors seek for professional help to improve the manuscript on this point.

Several typos need also to be corrected throughout the text.

 

Comment 2

In addition to the previous comment, please revise technical terms. Also, all acronyms must be defined.

 

Comment 3

Section 1, line 30

The reference “1904, Michell [1]” does not match with the reference list. Please check. Also, at line 32 what is the reference of the “subsequent study”?.

 

Comment 4

Section 2

The explanations about the proposed modified MOEA algorithm and its computational implementation must be improved. It is not easy for a reader non-specialist on the topic to understand many parts of Section 2.

Also, Section 2.2 cannot start with Figure 2.

 

Comment 5

Equations (1) and (2)

Explain better equations (1) and (2). In particular, what represents variable “x” and “y”, as well as functions f_1 and f_2?

It seems to me that two optimization variables exist and not just one, as referred at line 136. Please check.

Also, Eq. (2) has formatting issues (subscripts) which need to be solved.

 

Comment 6

Section 4.1,

In the 1^st paragraph, please explain in the manuscript what is “positive triangle” (maybe it is better to call it an “equilateral triangle”) and also what is “left oblique and a right oblique form”. Explain also the limits of “30º to 50º”.

 

Comment 7

Section 4.1

I didn´t understand the 2^nd paragraph (after Figures 4 and 5). It should be entirely rewritten for better understanding.

 

Comment 8

Section 4.1,

At Line 171, please refer what is “z direction”. Also, at line 173, what “elitism” stands for?

 

Comment 9

Conclusions

At Line 232, please correct the sentence. It seems to me that the “maximum displacement” appear twice as the optimization variable.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors of the paper did not refer to the reviewer's opinion contained in the sentence:

“However, the lack of a strict and correctly mathematically defined definition of the problem of vector optimization together with all the concepts appearing in it, such as: design variables, objective functions, constraints, etc., constitutes a serious obstacle in accepting the work for publication.” (response to Remark 10 is not the proper answer in reviewer opinion). 

The reviewer treats this as a serious obstacle to the publication of the article.

In the reviewer's opinion, the definition of the problem could be e.g. written as follows (however, the authors and not the reviewer should correctly define the problem mathematically):

Let X∈Rⁿ be an admissible set of design parameters (of course it is necessary to define properly the variables and constraints, even box type + equilibrium conditions they must meet). The reviewer obviously guesses that these are e.g. z-Cartesian coordinates of the truss nodal points shown e.g. in Figure 6 and static FEM relation of the type K q = Q but it must be formally written and explained in the text for any engineer or researcher interested in this topic. The information about the dimension n (or how to calculate n for arbitrary truss or for special variants of trusses numerically tested in the paper) is of course also very important.

Let f1: X-> R be the function calculating the maximum displacement (rather its maximal absolute value ?) of the truss, while f2: X-> R be the function calculating the weight (just volume) of the truss. Then, a set of all efficient points y* belonging to the image f(X) of the mapping f = (f1, f2): X-> R² is sought together with Pareto optimal solutions x* pertaining to them – please read the basic definitions of terms used here (efficient points, Pareto optimal solutions etc.) in any book (or monograph or article) on vector optimization - for example Claus Hillermeier, Nonlinear Multiobjective Optimization. A Generalized Homotopy Approach. ISNM Vol.135, Birkhauser, 2001.

This remark is all the more justified by the fact that all figures (e.g. 1,3,9) show images f(x)∈R² of the admissible points x∈X⊂Rⁿ , in particular the images of Pareto optimal solutions x∈X⊂Rⁿ lying on the so-called Pareto front – not wrongly named in paper as set of “optimal solutions” , because optimal Pareto solutions are e.g. z coordinates – not its images (efficient points) f(z) in R² lying on Pareto front. Of course, each admissible point x∈X⊂Rⁿ  is represented by an appropriately coded individual i in the population P.

The reviewer will not insist on requesting a change of notation, which is unfortunately quite often used incorrectly by many engineers.

This or similar precise definition of the problem must be explicitly written, because it is necessary even for the correct verification of the results presented in the article by other researchers.

Furthermore, the above text was placed by the reviewer in relation to the unclear notation “>” in the formula (1), i.e. how should the formula (1) be understood.

Question. If P is a population of individuals (vectors or lists properly coded, but there is no precise information in this matter), what does the relation i>j in (1) mean if i and j are the individuals belonging to the population of P?

This relationship probably has a close relationship with order relation in R^k but it is not explained.

The reviewer guesses that the sign | {...} | in (1) means the power of the set {...}, i.e. the number of elements meeting the condition defined in (1).

Of course, a similar question applies to the notation in formula (2) next to the Σ sign.

There are obvious errors in formula (5). The variable y does not appear after f1 or after f2.

The remaining answers to the questions, comments and remarks contained in the first review can be considered satisfactory.

Remark. The sentence “In the single loading case … this theory is only appropriate for simple working conditions but not for practical engineering” (see 1. Introduction) is not true, because for only one objective function and for only one load variant it is possible to model and optimize large engineering structures such as bridges with a span of several kilometers - please read for example an article: Fairclough HE, Gilbert M, Pichugin AV, Tyas A, Firth I. 2018 Theoretically optimal forms for very long-span bridges under gravity loading. Proc. R. Soc. A 474: 20170726. http://dx.doi.org/10.1098/rspa.2017.0726

Author Response

Dear Reviewer, 

Thank you very much for giving us an opportunity to revise our manuscript again, we appreciate editor and reviewers very much for their positive and constructive comments and suggestions on our manuscript

(1) Sorry for the omission of the Remark10 in the first reply. I would like to add here and hope to get your forgiveness.

Design variable: truss node coordinates

Objective functions: truss weight; truss stiffness (maximum displacement)

Constraints: one end is free; the other end is fully constrained, and the free end is subjected to a downward concentrated force load.

(2) We accept your suggestion, We added three equations to give clear explanation

“Multi-objective optimization is different from single-objective optimization, which usually has multiple optimal solutions that meet the conditions. In order to explain how to deal with multiple optimal solutions, the mathematical model of multi-objective optimization problem is given: x = (x1, x2, … , xn)T it satisfies the constraints in Equations (1) and (2) …” please check.

Besides, the total length n of the truss is based on a practical project and has little to do with the algorithm in essence, which is explained here.

(3) i and j are both individuals of the population P, and we add comments to equations (1) and (2).Y is just a variable, and we gave it a range.

(4) Thank you for your approval of the other questions in my first review.

(5) I have checked the article you mentioned, and I did have some problems in introduction, so delete the sentence this theory is only appropriate for simple working conditions but not for practical engineering.

Reviewer 4 Report

I received and read the revised version of the manuscript “Multi-objective topology optimization of spatially discrete structures based on node movement”. I´m generally satisfied with the author´s replies to my earlier comments and I also consider that all my suggestions and concerns have been properly explained and considered by the author to improve the manuscript.

I consider that the revised manuscript submitted by the author can be accepted in the present form to be published.

Author Response

Thank you for giving us so much pertinent advice on our manuscript entitled “Multi-objective topology optimization of spatially discrete structures based on node movement,” (ID: applsci-652370).