Complex Variable Approach for Thermoelastic Boundary Value Problem Using Rational Mapping Techniques
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper proposes a semi-analytical complex variable method to solve steady-state thermoelastic boundary value problems in elastic plates with curvilinear holes. The novelty of the work lies in the use of rational conformal mappings combined with Gaursat functions and collocation techniques, in order to reduce reliance on heavy meshing while capturing stress localization effectively. The manuscript seems to be mathematically rigorous, well-organized, and contains analytical derivations validated with different illustrative cases. However, certain aspects need refinement to improve clarity, generalizability, and contextualization. More specifically:
- Some sentences in the Introduction are repetitive and should be varied.
- Stress analysis is presented mainly via qualitative plots (Figures 6–9), while it is suggested to add some quantitative error metrics, such as stress maxima compared with FEM or known analytical benchmarks.
- Only linear isotropic material is considered herein, where a possible extension to orthotropic or functionally graded materials could be commented.
- The paper is well-referenced, but recent works (2023–2025) on computational conformal mapping and thermoelastic stress analysis could be cited for completeness.
Author Response
We sincerely appreciate the time and effort you invested in reviewing our manuscript, and we are grateful for your insightful comments and valuable suggestions for improving our paper. We have incorporated most of your recommendations. Kindly refer to the attached Pdf document, which provides a detailed point-by-point response to your feedback.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsReview of the article
Mai Taha, Mohamed A. Abdou, Amnah E. Shammaky, Abeer A. Al-Dohiman, Eslam M. Youssef "Complex Variable Approach for Thermoelastic Boundary Value Problem Using Rational Mapping Techniques"
The authors present a novel approach to analyzing stationary thermoelastic boundary value problems in isotropic elastic plates with curvilinear holes using the method of complex variables and rational conformal mappings. The authors consider a physical domain with a non-circular hole, which is conformally mapped to a unit disk. The thermoelastic potential combines the temperature distribution, determined by Laplace's equation, with Neumann boundary conditions. Based on computational experiments, the authors demonstrate that boundary modifications influence stress distribution. According to numerical simulations, stress distribution is more uniform with smoother boundaries, but stress concentration increases with the size of geometric perturbations. The proposed approach reflects the interaction of geometry and thermal effects in two-dimensional thermoelasticity. This approach can become a reliable tool for investigating complex heated elastic domains. The research direction is relevant, and the obtained results possess scientific novelty.
Conclusion. The article can be published in a scientific journal in its current form.
Comments for author File: Comments.pdf
Author Response
We sincerely appreciate the time and effort you invested in reviewing our manuscript, and we are grateful for your insightful comments and valuable suggestions for improving our paper. We have incorporated most of your recommendations. Kindly refer to the attached Pdf document, which provides a detailed point-by-point response to your feedback.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsPlease see the comments in the attachment.
Comments for author File: Comments.pdf
Author Response
We sincerely appreciate the time and effort you invested in reviewing our manuscript, and we are grateful for your insightful comments and valuable suggestions for improving our paper. We have incorporated most of your recommendations. Kindly refer to the attached Pdf document, which provides a detailed point-by-point response to your feedback.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe work has been adequately improved, and it is now suitable for publication in the present form.
Author Response
We sincerely thank the reviewer for the positive feedback and for recognizing the improvements in our manuscript.
Reviewer 3 Report
Comments and Suggestions for Authors- Series Truncation Justification
The Gaursat functions are approximated by truncated power series. A more detailed discussion on truncation error, convergence behavior, and optimal order selection would strengthen the methodology. - Collocation Method Robustness
Although sensitivity analysis for collocation points is included, the study could expand by testing adaptive or non-uniform collocation strategies to better capture stress near singularities or sharp boundary perturbations. - Residual and Stability Analysis
Reporting condition numbers of the resulting least-squares system, alongside residual norms, would provide insight into numerical stability and potential ill-conditioning. - Mapping Function Parameters
The roles of parameters (A, B, a, l) in the rational conformal mapping are described qualitatively. A systematic parametric study quantifying their effect on stress localization would improve clarity and reproducibility. - Benchmarking Against Alternatives
The method is validated against an analytical case of a circular hole. Additional benchmarking with finite element simulations or other semi-analytical methods would further establish reliability. - Material Sensitivity Extension
The paper varies (E, \nu, \alpha) for isotropic materials. Extending the framework to anisotropic, orthotropic, or functionally graded materials would highlight the broader applicability of the method. - Error Quantification in Applications
In the annulus application, including relative error plots (computed vs. analytical/benchmark) across the domain—not just at boundary points—would make the method’s accuracy more transparent. - Reproducibility Measures
Providing implementation details (e.g., algorithmic pseudocode, collocation point generation strategy) or supplementary code would help others replicate and apply the approach.
Author Response
We sincerely appreciate the time and effort you invested in reviewing our manuscript, and we are grateful for your insightful comments and valuable suggestions for improving our paper. We have incorporated most of your recommendations. Kindly refer to the attached Pdf document, which provides a detailed point-by-point response to your feedback.
Author Response File: Author Response.pdf
