Research on Internal Response in Three-Dimensional Elastodynamic Time Domain Boundary Element Method

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1. The manuscriptmentions the time-domain boundary element method (TD-BEM) for handling three-dimensional elastodynamic problems but lacks a detailed explanation of its theoretical framework and specific implementation processes.
2. Multiple references are cited in the manuscript, but there is no detailed explanation of their specific contributions and roles in the development of TD-BEM. It is suggested to provide additional clarification.
3. The Cote quadrature formula is mentioned for directly addressing spatial singularities, but there is no detailed explanation of its specific application steps and effects.
4. The "three-dimensional time-domain fundamental solutions" (ikp* and iku*) are defined, but their physical significance and derivation process are not explained in detail. It is recommended to add this information.
5. The coefficient expressions in equations (2), (3), (5), and (6) are provided, but their physical significance and derivation process are not explained in detail. It is suggested to include these details.
6. The importance of "internal response calculation" is mentioned, but the specific calculation methods and steps are not explained in detail. It is recommended to provide more information.
7. The "element influence coefficients" are given, but their physical significance and calculation methods are not explained in detail. It is suggested to add this information.
8. The "wavefront singularity" is mentioned as affecting calculations, but its specific manifestations and handling methods are not explained in detail. It is recommended to provide more information.
9. The stability issues of the "direct time-domain boundary element method" are mentioned, but the specific analysis methods and results are not explained in detail. It is suggested to provide additional information.
10. The "numerical stability analysis" is mentioned as important in TD-BEM, but the specific analysis methods and results are not explained in detail. It is recommended to provide more information.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Dear authors,

Please see review in the attached file.

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

This study extends the traditional Time-Domain Boundary Element Method (TD-BEM) to calculate internal displacements and stresses, rather than just boundary values. Unlike boundary calculations, internal computations avoid spatial singularities, making them more stable. Key steps include Deriving the internal stress boundary integral equation, Numerical discretization in time and space, Handling wavefront singularities analytically, Validating the method through two numerical examples. The results confirm that the proposed 3D elastodynamic TD-BEM is accurate and reliable, providing a valuable tool for engineering applications.

Here are comments to challenge and critique the paper from a reviewer’s perspective:

1. Lack of Experimental Validation; The study relies solely on numerical examples for validation. Including experimental results or comparisons with real-world data would strengthen confidence in the method’s practical applicability.
2. Handling of Wavefront Singularities; The paper claims to treat wavefront singularities analytically, but it does not clearly justify why this approach is superior to other existing numerical techniques. More discussion on the limitations of this method would improve the analysis.
3. Computational Efficiency and Complexity; The paper does not provide a runtime analysis or a comparison of computational cost with other TD-BEM approaches. Is this method significantly more efficient than existing techniques, or does it introduce new computational challenges?
4. Limited Scope in Engineering Applications; While the method is presented as valuable for engineering design, the paper does not explore practical case studies. Can this approach handle complex geometries, material heterogeneities, or real-world boundary conditions?
5. Unclear Influence of Discretization; The study acknowledges cumulative errors in stress computation but does not systematically analyze how mesh refinement or time-step selection impacts accuracy. A sensitivity study on these parameters would provide better insights.
6. Missing Discussion on Elastoplasticity; The conclusion mentions that the study does not extend to elastoplastic dynamics, but does not explore how the current framework could be adapted for such problems. A discussion on possible extensions or limitations would be beneficial.

Comments on the Quality of English Language

The paper is technically sound, but the quality of English can be improved for better readability and clarity. Specific areas that need attention:

1. Complex sentence structures – Some sentences are overly long and difficult to follow. Breaking them into shorter, more concise statements would enhance readability.
2. Technical explanations – Certain sections, particularly those discussing wavefront singularities and influence coefficients, assume a high level of prior knowledge. Providing clearer transitions and explanations would make the content more accessible.
3. Minor grammatical inconsistencies – Some grammatical errors and awkward phrasing are present, which can be refined through careful proofreading or professional language editing.
4. Equation explanations – Some equations are introduced with minimal explanation, making it difficult for readers to follow the logical flow. Adding more descriptive context would improve clarity.

Recommendation: A thorough proofreading and slight simplification of technical descriptions will enhance clarity without compromising scientific rigor.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

All issues have been addressed. The manuscript can be accepted in the present form.