Non-Monotone Projected Gradient Method in Linear Elasticity Contact Problems with Given Friction

Round 1

Reviewer 1 Report

The contribution at hand focuses on new algorithms for solving large-scale convex optimization problems which are typical for linear elasticity contact problems discretized by finite element techniques. After a short review of the
theory and the authors examine the behavior and the efficiency of a modified spectral projected gradient method for the abovementioned Quadratic Programming problems. The developed algorithm is applied on the solution of a benchmark example in Matlab software.

In the opinion of the reviewer the submitted article is of high scientific quality and should be considered for publication. However, it would be very interesting to see how the calculated tangential contact forces are distributed in the contact zone. In this way the presented results in Fig. 3 could become clearer.

Author Response

Dear reviewer,

we would like to thank you for your interest in our work and we appreciate your comments. Based on your suggestions, we decided to separate the presentation of the final results into two larger figures - the displacement and the computed friction forces in the contact zone. We hope that this way of presentation will make results much more clear, please see Fig. 4 in the new version of manuscript.

With best regards,

authors

Reviewer 2 Report

The manuscript entitled ‘Non-monotone Projected Gradient Method in Linear Elasticity Contact Problems with Given Friction’ The article deals with a solver for contact finite element problem.

The interest is apply Spectral Projected Gradient for solving the quadratic programming problem. The study is interesting, the paper is well written.

I think it is of interest for Sustainability and propose it for publication.

Some minor remarks :
- There is only one example : are the result similar if we choose a different geometry ?
- is the method sensitive to level error (1E-6) to stop iteration regarding the results ?

Author Response

Dear reviewer,

we would like to thank you for your interest in our work and we appreciate your comments.
You are completely right - there is not a motivation to develop a new algorithm without the motivation from the practical problems and the paper should include the results of the problems on more complex geometries. However, it is a common way in optimization papers (which present a new algorithm) to examine the behavior of the algorithm on the benchmark, which can be easily reproduced. We chose this simple geometry, because the stiffness matrix has a trivial form and it is quite well conditioned. Additionally, we are able to simply "scale" the dimension of the problem and present the dependency of the number of performed iterations (and the computation time) on the size of the problem. In the case of more complex geometries, the situation would be, of course, different. However, we believe that the algorithm based on "simple" gradient descend (such as SPG-QP, which we used in the paper) should be able to deal even with these problems, because of the robustness. To support this statement, we updated figures in the paper - now we solve the dual problem to much more larger precision, please, see Fig. 5.
Additionally, you hit the question, which also bothers us - how the precision of the solution of dual problem influences the error of the primal solution. This question is not trivial. In the case of our paper, we suppose that 1e-6 should be sufficient to obtain reasonable results.
Nowadays we start to cooperate with colleagues from the Department of Structures. They are able to perform the real-world tests of the materials as well as perform the simulations on commercial software (such as Ansys). We are preparing a paper, where we examine and compare the efficiency different types of methods (e.g., interior-point, active-set, and gradient descent methods), commecial software, and results from real-world simulations.
If you are interested, please, follow our future research.

With best regards,

authors

Reviewer 3 Report

I recommend it for publication in present form.

Author Response

Dear reviewer,

we would like to thank you for your interest in our work and we appreciate your comments.
However, based on the comments from other reviewers, we decided to present new Fig. 4 (the final friction forces in the contact zone) and Fig. 5 (now we are solving the problem to higher precision). We hope that you will still find our paper to be ready for publishing.

With best regards,

authors