Enforcing a Force–Displacement Curve of a Nonlinear Structure Using Topology Optimization with Slope Constraints

Round 1

Reviewer 1 Report

A topology optimization method to find distributions of Young’s modulus to achieve a target force-displacement curve using a small number of specified target points is introduced. The authors report a good convergence in optimizing structures of both hardening and softening types, achieved by including constraints on the “slopes” at the target points. The study is well-motivated, and the examples effectively demonstrate the benefit of the proposed approach. The article is recommended for publication after some clarifications.

- The authors should go through the entire article to correct any typos, skewed sizes and shapes of symbols embedded in texts and to improve figure quality (use legible and consistent font size in all figures; enlarge Fig. 1 for clarity and shrink Figs. 2, 9 for reasonable use of space).

- Specific design strategies or analysis methods mentioned in the introduction should be explained concisely where they appear, instead of simply using the terminology. For example, line 45: “one-step and multi-step approaches”; line 71: “energy interpolation scheme”.

- The index j in summation in Eq. (8) should belong to a *set of indices* referring to neighboring elements, not the *number* of neighboring elements, Nei.

- Please define r in line 180.

- The use of the term “slope” in reference to the theta’s is confusing, since they do not represent the slopes of the target F-D curve. As the authors point out, these angles can be determined the moment the target points along the curve are decided, as these values are merely another representation of the same information (position in the F-D space). Yet, the optimization process seems to benefit from having constraints based on theta’s. The authors should offer some insight into the different aspects of the u*-based metric and theta-based metric, in relation to how they appear in the optimization problem (objective vs constraints).

- The description in text for the first example stats that the density is set 1 everywhere (line 285), yet the iteration history in Fig. 4 seems to show the volume fraction starting at 0.7. Please clarify.

- In discussing the comparison between the proposed method and a method without the “slope” constraints but using a larger number of target points, in page 14, please include more specific information to make the comparison clear. For example, what caused the different computational time, number of optimization iterations to reach the same level of convergence (if so what are the stopping criteria), or analysis time?

Author Response

- The index j in summation in Eq. (8) should belong to a *set of indices* referring to neighboring elements, not the *number* of neighboring elements, Nei.

=> Thank you for your comment, but the equation seems to correct.

 

- Please define r in line 180.

=> Please see that “ is the number of the set of the neighborhood element i within a certain filter radius r.”

 

- The description in text for the first example stats that the density is set 1 everywhere (line 285), yet the iteration history in Fig. 4 seems to show the volume fraction starting at 0.7. Please clarify.

=> Thank you very much, it is our mistake and we have modified it accordingly.

 

- In discussing the comparison between the proposed method and a method without the “slope” constraints but using a larger number of target points, in page 14, please include more specific information to make the comparison clear. For example, what caused the different computational time, number of optimization iterations to reach the same level of convergence (if so what are the stopping criteria), or analysis time?

=>Thank you for your valuable comment, based on your comment we added below paragraph.

"In summary, in the absence of a slope constraint, a large number of target points are required to obtain a layout that follows the prescribed target points. Of course, however, the computational analysis time takes long in this case. In order to overcome this, this study proposed a slope constraint, which makes it possible to obtain the layout of a structure with a nonlinear F-D curve of the desired with just three target points."

Reviewer 2 Report

This paper describes a study to use topology optimization to fit a non-linear thin-plate structure to a force displacement curve. It differs from its most closely related paper by requiring less points on the curve which improves computation time. As well they guarantee convergence by using the slope for the force-displacement curve represented by a secant matrix. They utilize a density-based approach where 0 is lack of material and 1 is presence of material. SIMP method is used to interpolate the young’s modulus of each element. The paper seems to have appropriate references, is well written and free of grammatical mistakes.
The test cases for the simulation show the performance increase for the proposed method compared to the most related current research. The results seem appropriate and are well explained.

The paper seems a little lengthy and spends a lot of time talking about others works that are being implemented, instead of highlighting the importance of this work. Although implementation of various techniques to make the work more efficient and stable is important it is not a contribution. Instead this work should spend more time discussing the use cases or importance of non-linear force-displacement curves which would give better context.

Figure 7 shows meters as the unit, but then there is a note that it is x10^(-3) which means the units on the graph are mm, it is quite misleading. Better to state in axis title that the units are mm. Additionally graphs should have similar text size. Figure 7 has very small text that is hard to read, while figure 8 has legible text. Similarly check text size for all other figures.

I think this paper should be considered for journal publication with minor modifications.

Author Response

Thank you for your valuable comments in advance. 

 

The paper seems a little lengthy and spends a lot of time talking about others works that are being implemented, instead of highlighting the importance of this work. Although implementation of various techniques to make the work more efficient and stable is important it is not a contribution. Instead this work should spend more time discussing the use cases or importance of non-linear force-displacement curves which would give better context.

=>Thank you for your valuable comment, I agree with your opinion that it is a bit lengthy to talk about others works. But, based on other reviewers’ comments that it is good to emphasized very well on the other’s previous works. Hence I would like to kindly ask you for your understanding.

 

Figure 7 shows meters as the unit, but then there is a note that it is x10^(-3) which means the units on the graph are mm, it is quite misleading. Better to state in axis title that the units are mm.

=>Thank you for your critical comment and I also agree with your opinion that the scale should be changed to mm. but in previous relevant works, they also used meter scale, therefore we also used the same scale to remain the consistency with previous results in the references.

 

Additionally graphs should have similar text size. Figure 7 has very small text that is hard to read, while figure 8 has legible text. Similarly check text size for all other figures.

=>Thank you for your valuable comment. To reflect your comment, we have changed the figure accordingly based on your comment. Please see Figure 7 in the updated version and Figure 4 as well.

Reviewer 3 Report

The paper deals with a topology optimization method able to find the shape of a structure that follows a user-prescribed F-D curve. The desired nonlinearity is defined using a small number of target points on the F-D plane and a typical softening or hardening behaviour can be simulated. The control on the target point positions and on the curve slope in correspondence of these targets are able to improve the convergence of the numerical analysis.
The overal quality of the paper is very good, the procedure is explained step-by-step and each point is clearly analysed. The mathematical framework is described with strictness. State of the art of the topic is often recalled to give support to the proposed method. Results are in accordance with others obtained in literature and look as improvements if compared with similar procedures used by commercial softwares.
Some minor issues should be addressed by the authors:
1- in page 9, first results of the optimization on a clamped-clamped test are reported where two procedure are compared: with slope constraints and without. The authors should report the final shape of the optimized layout.
2- in figure 4, the convergence history of the overmentiond analyses are reported. The convergence accuracy imposed in the two cases is the same? Can you quantify it?
3- in the same figure 4, final volume fractions are quite different (0.62 without slope constraints vs 0.52 with slope constraints), this deserves some comments by the authors.
4- for the clamped-free test, can you report the final volume fractions obtained for the optimized layouts in cases of softening and hardening behaviour?

Author Response

Thank you for your overall good evaluation on this manuscript.

 

1- in page 9, first results of the optimization on a clamped-clamped test are reported where two procedure are compared: with slope constraints and without. The authors should report the final shape of the optimized layout.

=> We think that you are a bit confused, the final optimal layout is reported in Figure 6.

2- in figure 4, the convergence history of the overmentiond analyses are reported. The convergence accuracy imposed in the two cases is the same? Can you quantify it?

=> Thank you for your valuable comment. The convergence accuracy is achieved and quantified by stop criteria in the optimization process.

3- in the same figure 4, final volume fractions are quite different (0.62 without slope constraints vs 0.52 with slope constraints), this deserves some comments by the authors.

=> Thank you for your critical comment, to be honest we still haven’t clearly been assured to analyzed the difference of volume fractions.

Reviewer 4 Report

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

Thank you for your valuable comments in advance. 

 

1. One major contribution of this paper is to employ slope constraints to enforce the force-displacement relation. This information should be somehow reflected in the title of the paper.

=>thank you very much for your valuable comment. Based on your comment we have modified title as “Enforcing a force-displacement curve of a nonlinear structure using topology optimization with slope constraints”

 

2. In page 2, line 73-75 “in which softening, hardening, or both nonlinearities”, no example is presented for both nonlinearities. Additional example is needed for this statement.

=> Although there are no cases where both nonlinearities, softening and hardening, have been dealt with at the same time, the applicability has been verified in cases where individual nonlinearities of hardening and softening are dealt with, so it is natural that the proposed method will work even when both nonlinearities are incorporated.

 

3. There is a big jump on the objective function value in Fig 4 b, can the authors comment on that?

=> Thank you very much for your critical comment. To be honest we are still working on the big jump of the objective function for without constraint case.