Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper investigates the interpolation cover method (ICM) for enriching standard 4‑node quadrilateral shell finite elements, and studies how neighbouring‑node selection and polynomial enrichment affect numerical accuracy and robustness. Two benchmark examples, a tensile plate with a central hole and a clamped bending beam, are adopted for numerical validation. Numerical results demonstrate that the ICM can remarkably improve stress prediction accuracy and reduce calculation errors even on relatively coarse meshes. This work addresses ambiguous parameter‑selection issues existing in original ICM research and provides meaningful references for engineering simulation. Nevertheless, several aspects including theoretical derivation details, numerical case completeness, convergence proof and comparison scope can be further improved.

（1）The abstract states that key parameters such as the number and position of neighbouring elements/nodes and polynomial degrees of freedom have not been evaluated for the interpolation cover method. However, relevant references have already discussed node‑selection and polynomial‑order effects for ICM. It is recommended to clarify the boundary of previous contributions in the introduction section, and distinguish which content belongs to brand‑new investigation and which content is refined parametric analysis in the present work. This revision can avoid misleading readers into regarding this research topic as completely unexplored and better highlight the novelty of this paper.

（2）The mass matrix of enriched elements keeps identical to the standard finite‑element form without enrichment treatment, yet potential side effects caused by this simplification are not analysed. Although only static problems are solved in this paper, ICM is expected to be extended to dynamic analysis. It remains unclear whether the unmodified mass matrix would induce errors in dynamic responses or modal results. Additional explanations are suggested to illustrate the rationality and application limitations of this treatment.

（3）Section 2.2 describes the assembly procedure for enriched shell elements, where degrees‑of‑freedom from neighbouring nodes are mapped into global system equations. However, the coupling logic is not explained for practical scenarios where multiple enriched elements share the same set of neighbouring nodes. Overlapping enriched patches commonly appear in real‑world simulations. Supplementary descriptions about assembly strategies for multi‑element enrichment are required to explain how the in‑house code prevents abnormal stiffness‑matrix behaviour.

（4）Fine‑mesh G7 results from LS‑DYNA are directly taken as the reference solution, but adequate mesh convergence verification for G7 is absent. Unproven convergence of the benchmark solution brings uncertainty to all calculated relative errors. Mesh‑convergence data among G5, G6 and G7 should be supplemented to prove that the G7 mesh yields converged results and validate its suitability as the reference benchmark.

（5）Parametric studies conclude that choosing first‑nearest neighbouring nodes yields higher accuracy while distant nodes degrade solution performance. Nevertheless, only numerical phenomena are presented without underlying mathematical interpretation. It is not well explained whether accuracy deterioration arises from matrix ill‑conditioning introduced by distant nodes or excessive interpolation‑cover domains. Brief mechanistic discussion should be added to connect observed numerical trends with theoretical reasoning, instead of merely drawing conclusions from case observations.

（6）Poly8, Poly9 and Poly10 polynomial enrichment schemes produce inconsistent optimal performance across two benchmarks: Poly8 works better for the holed plate while Poly10 gives superior accuracy for the clamped beam. General selection guidelines are not provided based on these discrete case results. Further discussion is expected on how geometrical features and stress‑gradient magnitude influence the choice of enrichment schemes, so that practical selection suggestions can be offered for engineering users.

（7）All numerical examples are limited to linear‑elastic problems, while the conclusion anticipates promising potential of ICM for nonlinear and large‑scale industrial applications. Such prospect lacks supporting nonlinear numerical tests. The authors should objectively state current research limitations in discussion: existing conclusions are only valid for linear‑elastic analysis; further research is still required to assess ICM accuracy and ill‑conditioning risks under material plasticity or large geometric deformation, and over‑optimistic statements about present applicability should be avoided.

Author Response

Comment 1:The abstract states that key parameters such as the number and position of neighbouring elements/nodes and polynomial degrees of freedom have not been evaluated for the interpolation cover method. However, relevant references have already discussed node‑selection and polynomial‑order effects for ICM. It is recommended to clarify the boundary of previous contributions in the introduction section, and distinguish which content belongs to brand‑new investigation and which content is refined parametric analysis in the present work. This revision can avoid misleading readers into regarding this research topic as completely unexplored and better highlight the novelty of this paper.

Combined effect of neighboring-node selection and polynomial degree has received comparatively limited systematic investigation.

Response 1: Authors are thankful to respected reviewer for putting the effort in careful reading the manuscript. The novelty of the present work is not in the first introduction of neighbouring-node selection or polynomial enrichment into ICM. Rather, it is in conducting a systematic parametric investigation of their combined influence on the accuracy and robustness of the ICM when enriching standard 4-node quadrilateral shell finite elements. Specifically, the present study shows the influence of position and number of neighboring nodes combined with polynomial degree of enrichment on the solution accuracy.

The existing literature gives the base to use of neighboring nodes and polynomial degree for enrichment of patches. However, the combined relationship between the size and spatial distribution of the neighbouring set and the polynomial degree was not, to our knowledge, systematically examined by a parametric study for the considered shell finite-element formulation. Our work has thus a refined and systematic parametric investigation.

The following changes has been done in the abstract and introduction sections.

Abstract: The existing studies show that the researchers used the neighboring nodes and polynomial enrichment to construct interpolation covers but the combine effect of selection of number of neighboring nodes and their position in the patches and polynomial enrichment is not explored yet. This study therefore performs a parametric investigation of these factors for enriched 4-node quadrilateral shell finite elements.

Introduction: The previous studies show that the neighboring nodes and polynomial enrichment in interpolation covers enhances the solution accuracy without mesh refinement. But the combined effect of the number and spatial selection of neighbouring nodes with the polynomial degree has received comparatively limited systematic investigation, particularly for the enrichment of standard 4-node quadrilateral shell finite elements. The present study addresses this issue by a systematic parametric investigation of their combined influence of selection of number and position of neighboring nodes with polynomial degree of enrichment by using interpolation covers on the accuracy and robustness of the ICM when enriching standard 4-node quadrilateral shell finite elements.

Comment 2:The mass matrix of enriched elements keeps identical to the standard finite‑element form without enrichment treatment, yet potential side effects caused by this simplification are not analysed. Although only static problems are solved in this paper, ICM is expected to be extended to dynamic analysis. It remains unclear whether the unmodified mass matrix would induce errors in dynamic responses or modal results. Additional explanations are suggested to illustrate the rationality and application limitations of this treatment.

Response 2: Authors are thankful to respected reviewer for raising this important issue. The current study focusses on the static structural problems. Therefore, the governing finite-element equation contains only the internal virtual work associated with the stiffness matrix, and the mass matrix does not contribute to the solution. For this reason, retaining the conventional finite-element mass matrix does not affect any of the static displacement, stress, or strain results reported in this paper. Authors have clarified this limitation of solving elastic structural problems in the revised manuscript.

The following paragraph has been added in the assembly procedure for the enriched shell element section.

In the present paper, interpolation covers method is used to solve exclusively static structural applications so that the mass matrix does not affect the solution accuracy in terms of stress, strain and displacement because mass matrix does not contribute to the equilibrium equation. For simplicity, the conventional finite-element mass matrix is consequently retained. For dynamic or eigenvalue analysis, however, this assumption requires further consideration. Since the interpolation cover enriches the displacement approximation, a consistent dynamic formulation should derive the mass matrix from the complete enriched displacement field. In general, the corresponding mass matrix contains contributions from both the conventional and enriched approximation functions, including their coupling terms. Retaining the standard mass matrix in such an application may therefore introduce discrepancies in the computed natural frequencies, mode shapes, or transient responses. The standard mass matrix adopted in the present work is intended only for the static formulation considered here and should not be regarded as a validated mass formulation for dynamic ICM analysis. A consistent enriched mass matrix and its effects on modal and transient responses require further investigation.

Comment 3: Section 2.2 describes the assembly procedure for enriched shell elements, where degrees‑of‑freedom from neighbouring nodes are mapped into global system equations. However, the coupling logic is not explained for practical scenarios where multiple enriched elements share the same set of neighbouring nodes. Overlapping enriched patches commonly appear in real‑world simulations. Supplementary descriptions about assembly strategies for multi‑element enrichment are required to explain how the in‑house code prevents abnormal stiffness‑matrix behaviour.

Response 3: Authors are thankful to respected reviewer for pointing out this important issue. We agree that the assembly procedure needs clarification in the case where multiple enriched elements share overlapping interpolation covers or neighbour nodes. This situation may arise when interpolation covers are formed around multiple nodes and the global assembly needs to account for degrees of freedom associated with the same neighbour node.

In the revised manuscript, we clarify that enriched degrees of freedom associated with a neighbour node are obtained via a unique global degree-of-freedom numbering procedure. Consequently, enriched elements that use the same neighbour node and enrichment parameters do not generate independent degrees of freedom but refer to the same global degrees of freedom.

The following subsection has been added in the manuscript.

Assembly of multiple enriched elements with overlapping covers:

In practice, interpolation covers associated with different enriched elements can overlap and several enriched elements can share the same neighbour nodes. The situation is illustrated in Figure 1 for two enriched elements with interpolation covers that partially overlap. In the current implementation, the problem of overlapping interpolation covers is addressed by utilising a unique degree-of-freedom numbering procedure for all enriched elements. In particular, enriched degrees of freedom associated with a particular neighbour node and enrichment order are uniquely identified by their indices and consequently, enriched elements that share the same neighbour nodes have some degrees of freedom in common. For each enriched element, the local stiffness matrix is assembled using both standard and interpolation-cover degrees of freedom and a local-to-global mapping array. The element matrix is then assembled into the global stiffness matrix according to the local-to-global mapping. As a result, if multiple enriched elements share the same neighbour node, their contributions are added to the same degree-of-freedom locations in the global stiffness matrix. In this way, overlapping interpolation covers can be used to enrich finite-element approximations while ensuring that neighbour elements are coupled correctly through shared degrees of freedom. The assembly procedure follows the same procedure as in standard finite element analysis with the exception that the local degree-of-freedom arrays also include enrichment degrees of freedom. Using a unique numbering for enrichment degrees of freedom and local-to-global mapping ensures that multiple enriched elements that share the same neighbour node and enrichment parameters do not have independent, equal degrees of freedom but use the same global degree-of-freedom indices. Additionally, the same local-to-global mapping is used for all enriched elements such that overlapping interpolation covers generate identical contributions to the global stiffness matrix.

Comment 4: Fine‑mesh G7 results from LS‑DYNA are directly taken as the reference solution, but adequate mesh convergence verification for G7 is absent. Unproven convergence of the benchmark solution brings uncertainty to all calculated relative errors. Mesh‑convergence data among G5, G6 and G7 should be supplemented to prove that the G7 mesh yields converged results and validate its suitability as the reference benchmark.

Response 4: Thank you for this important comment. To address this issue, we have supplemented the numerical results with a mesh-convergence study using the G6 and G7 meshes. The relevant response quantities obtained from LS-DYNA are compared across these two mesh levels. In particular, the changes between successive mesh refinements G6→G7 are reported to assess whether further mesh refinement produces only negligible changes in the benchmark response.

The results show that the difference between the G6 and G7 solutions is substantially smaller that indicates the solution is approaching a mesh-independent value. Therefore, the G7 mesh provides a sufficiently converged reference solution for the purpose of evaluating the relative errors of the ICM-enriched elements investigated in this study. We have added the corresponding convergence data and discussion to the revised manuscript.

The following paragraph has been added in the result and discussion section.
Before evaluating the relative errors of the ICM-enriched elements, a mesh-convergence study was performed using the G6 and G7 meshes in LS-DYNA. The selected response quantities were compared between successive mesh refinements to assess the sensitivity of the benchmark solution to mesh density. The results show that the variation from G6 to G7 is considerably smaller in case of plate with a hole 0.115% and in clamped plate bending 1.55%. That indicates the solution is approaching mesh convergence. On this basis, the G7 solution is adopted as the reference solution for the subsequent error evaluation. It should be noted that G7 is considered a numerically converged benchmark within the mesh sequence examined here, rather than an exact analytical solution. Consequently, the relative errors presented subsequently represent deviations from the converged G7 LS-DYNA results.

Comment 5: Parametric studies conclude that choosing first‑nearest neighbouring nodes yields higher accuracy while distant nodes degrade solution performance. Nevertheless, only numerical phenomena are presented without underlying mathematical interpretation. It is not well explained whether accuracy deterioration arises from matrix ill‑conditioning introduced by distant nodes or excessive interpolation‑cover domains. Brief mechanistic discussion should be added to connect observed numerical trends with theoretical reasoning, instead of merely drawing conclusions from case observations.

Response 5: Thank you for this valuable comment. In the revised manuscript, we have added a mechanistic discussion based on the structure of the interpolation-cover enrichment. In the present formulation, the enrichment functions are constructed from the standard shape functions multiplied by the local parametric coordinates. Consequently, the additional interpolation terms are directly influenced by the spatial location of the neighbouring nodes used to construct the interpolation cover.

When the first-nearest neighbouring nodes are selected, the interpolation cover remains relatively local to the enriched element. The additional degrees of freedom therefore represent the displacement variation in the immediate vicinity of the target element, and the enriched approximation can describe the local higher-order variation without introducing a large spatial extrapolation region. This locality is particularly beneficial for representing rapidly varying displacement and stress fields, such as those occurring around the hole boundary or in regions with high bending gradients.

In contrast, selecting more distant neighbouring nodes increases the spatial extent of the interpolation cover. Although these nodes provide additional degrees of freedom, their displacement information is associated with locations farther from the target element. The resulting enrichment therefore becomes less local and may introduce an inappropriate spatial scale into the higher-order interpolation. In addition, as the geometric separation of the nodes increases, the resulting interpolation terms can become less favourably scaled, which may contribute to deterioration of the conditioning of the enriched algebraic system. Thus, the observed reduction in accuracy should not be attributed solely to an increase in the number of neighbouring nodes; rather, it is related to the spatial distribution of the nodes and the resulting locality and conditioning of the enriched interpolation.

The following paragraph has been added in iInfluence of selection of the neighboring nodes section of clamped plate bending.

The influence of neighbouring-node selection can be interpreted from the locality of the interpolation-cover enrichment. In the present formulation, the enrichment functions are constructed in such a way that the additional interpolation terms introduce higher-order spatial variation into the standard bilinear approximation. When first-nearest neighbouring nodes are selected, the interpolation cover remains localized around the target element, and the additional nodal information primarily represents the displacement field in its immediate vicinity. This is advantageous for capturing local displacement and stress gradients without introducing an unnecessarily large interpolation region. In contrast, selecting more distant neighbouring nodes enlarges the spatial extent of the interpolation cover and reduces the locality of the enrichment. The resulting higher-order approximation is then influenced by nodal information from a larger region, which may be less representative of the local field surrounding the target element. Furthermore, the increased spatial separation of the participating nodes can lead to less favourable scaling and conditioning of the enriched interpolation system. These effects provide a mathematical interpretation of the observed reduction in accuracy for distant-node selections. Therefore, the superior performance of first-nearest neighbouring nodes is attributed primarily to the improved locality of the interpolation cover, while the deterioration associated with distant nodes may additionally be accompanied by conditioning effects.

Comment 6: Poly8, Poly9 and Poly10 polynomial enrichment schemes produce inconsistent optimal performance across two benchmarks: Poly8 works better for the holed plate while Poly10 gives superior accuracy for the clamped beam. General selection guidelines are not provided based on these discrete case results. Further discussion is expected on how geometrical features and stress‑gradient magnitude influence the choice of enrichment schemes, so that practical selection suggestions can be offered for engineering users.

Response 6: Thank you for this constructive comment. The revised manuscript now discusses the selection of the polynomial enrichment scheme in relation to the spatial characteristics of the problem. The effectiveness of a higher-order enrichment is not determined solely by the polynomial degree; it also depends on the complexity of the local displacement field, the magnitude and spatial variation of the stress/strain gradients, and the geometric features responsible for these gradients.

For the plate with a central hole, the dominant response is strongly influenced by the local stress concentration around the hole boundary. In this case, the relevant field variation is highly localized around a relatively well-defined geometric feature. The results indicate that Poly8 provides a better balance between enrichment capability and numerical performance for this problem. Increasing the polynomial order beyond this level does not necessarily provide additional accuracy because the additional higher-order terms may not correspond to a significant improvement in representing the dominant local field.

For the clamped bending beam, the deformation and stress fields are governed by bending and the associated variation of curvature and stress through the beam. The resulting field variation is different from that of the stress-concentration problem, and the numerical results show that the higher-order Poly10 enrichment provides improved accuracy for this case. This indicates that a higher polynomial order can be advantageous when the displacement/stress field exhibits more pronounced spatial variation that cannot be adequately represented by a lower-order enrichment.

Based on these observations, we have clarified that there is no universal rule that a higher polynomial degree always produces a more accurate solution. Instead, the enrichment order should be selected according to the complexity of the local field that needs to be represented. As a practical guideline, lower-order enrichment such as Poly8 may be appropriate for problems dominated by relatively localized and well-defined field variations, whereas higher-order enrichment such as Poly10 may be beneficial when stronger or more distributed gradients and higher spatial variation are present. Poly9 can be regarded as an intermediate choice when the required field complexity lies between these two cases.

The following paragraph has been added in influence of selection of enrichment terms in clamped plate bending section.

The results indicate that the optimal polynomial enrichment order depends on the characteristics of the local solution field rather than increasing monotonically with polynomial degree. For the plate with a hole, the dominant feature is the localized stress concentration around the hole boundary. In this case, Poly8 provides a favourable balance between the ability to represent the local field variation and the additional complexity introduced by higher-order enrichment. For the clamped bending beam, the solution is governed primarily by bending deformation and the associated spatial variation of curvature and stress, for which the higher-order Poly10 enrichment provides improved accuracy. These results suggest that the polynomial order should be selected according to the complexity and spatial extent of the displacement and stress gradients generated by the geometry and loading conditions. As a practical guideline, a relatively lower-order enrichment can be considered for problems in which the dominant field variation is localized and can be adequately represented by the lower-order basis, whereas a higher-order enrichment may be advantageous when pronounced or more distributed field gradients are present. Therefore, Poly8, Poly9, and Poly10 should be regarded as alternative enrichment levels whose suitability depends on the characteristics of the specific engineering problem, rather than as universally ranked schemes.

Comment 7: All numerical examples are limited to linear‑elastic problems, while the conclusion anticipates promising potential of ICM for nonlinear and large‑scale industrial applications. Such prospect lacks supporting nonlinear numerical tests. The authors should objectively state current research limitations in discussion: existing conclusions are only valid for linear‑elastic analysis; further research is still required to assess ICM accuracy and ill‑conditioning risks under material plasticity or large geometric deformation, and over‑optimistic statements about present applicability should be avoided.

Response 7: Thank you for this important comment. We acknowledge that the original conclusion regarding the potential application of ICM to nonlinear and large-scale industrial problems was too broad in relation to the numerical evidence presented in this work. We have therefore revised the Conclusions section to explicitly state the scope and limitations of the present study. The conclusions drawn from the numerical investigations are now limited to the linear-elastic shell problems considered in this paper.

In particular, the revised manuscript clarifies that the present results demonstrate the accuracy and parameter sensitivity of the ICM-enriched 4-node quadrilateral shell element for the investigated linear-elastic problems, but they do not establish its performance for nonlinear analysis. The behaviour of the enriched formulation under material plasticity requires further investigation because the interaction between the interpolation-cover enrichment and the evolving constitutive response may affect both accuracy and numerical robustness.

Similarly, large geometric deformation has not been considered in the present study. Under geometrically nonlinear conditions, the changing configuration may alter the spatial relationship between the target element and its neighbouring nodes and may therefore affect the effectiveness and conditioning of the interpolation-cover enrichment. Consequently, the possibility of ill-conditioning or other numerical difficulties under large deformation cannot be excluded based on the current results.

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript presents an investigation into the Interpolation Cover Method (ICM) applied to the 4-node quadrilateral shell element (Q4) for structural stress analysis. By enriching standard bilinear FE shape functions with localized polynomial cover functions across neighboring node patches, the formulation seeks to improve stress gradient resolution and mitigate displacement locking without global mesh refinement. The authors evaluate the performance of linear (p=1, Poly8) and quadratic (p=2, Poly9 and Poly10) enrichment covers, as well as the spatial arrangement of neighboring nodes (1^st, 2^nd, and 3^rd nearest neighbors), across two benchmark problems: a plate with a central hole under tensile loading and a clamped plate under bending. The results demonstrate that incorporating immediate (1^st nearest) neighboring nodes yields optimal stress recovery, reducing peak von Mises stress errors significantly compared to standard Q4 elements in LS-DYNA. However, the manuscript exhibits critical discrepancies in physical loading parameters, missing figure cross-references, unaddressed computational latency and system matrix bandwidth trade-offs, and a lack of mesh distortion evaluations required to substantiate the claimed robustness. A major revision is required to resolve these theoretical, numerical, and editorial deficiencies.

1. Resolved cross-references and citation artifact cleanup: Several figure cross-references in the text are unrendered and appear as broken placeholders (`Figure ??`). Specifically, line 270 ("highlighted by the black circle in Figure ??)"), line 273 ("on the exact same mesh (Figure ??)."), and line 344 ("indicated by black circles in Figure ??).") lack correct figure numbers. Furthermore, the captions for Figure 26 on page 20 contain raw citation syntax artifacts (e.g., `(b) Poly8 %[cite: 1]`, `(b) Poly9 %[cite: 1]`, and `(c) Poly10 %[cite: 2]`). The authors must thoroughly audit and fix all figure references and citation strings throughout the manuscript.
2. Reconciliation of loading condition discrepancies in clamped plate benchmark: In Section 3.2 (line 329), the text specifies an applied transverse bending load of F = 100 N. However, Figure 15 explicitly labels the applied tip load as F = 500 N. This contradiction creates ambiguity regarding the boundary conditions and load magnitude utilized in the numerical simulations and baseline comparisons. The authors must clarify the exact force magnitude applied in the finite element models and ensure complete consistency between textual descriptions, figure diagrams, and reported stress values.
3. Quantitative evaluation of computational complexity and matrix bandwidth: The paper emphasizes that ICM avoids mesh refinement and additional global sub-element meshing. However, enriching quadrilateral elements with cover functions introduces extra degrees of freedom (e.g., increasing local element DOFs from 24 to 48 or 72, as described in Section 2.2) and expands the stencil sparsity pattern of the global stiffness matrix. The manuscript lacks any quantitative reporting of solver execution times, CPU memory overhead, or condition numbers of the global stiffness matrix relative to standard h-refined FE meshes. The authors should include a comparative table or table-free quantitative narrative detailing CPU runtime and matrix bandwidth for standard vs. enriched formulations to objectively justify the computational efficiency claims.
4. Empirical verification of robustness under mesh distortion: The manuscript title explicitly highlights the "Robustness of the Interpolation Cover Method". However, both benchmark problems (plate with hole and clamped beam) are discretized using regular or mildly graded orthogonal quad meshes (Figure 4 and Figure 16). In practical structural analysis, elements often suffer from severe angular skewness, aspect ratio distortion, or non-planar warping. To validate the claimed robustness of ICM against element distortion and locking phenomena, the authors should evaluate the method on standard benchmark tests with distorted or skewed meshes (such as the MacNeal skewed arm or irregular quad patches).
5. Justification and analysis of zero Poisson's ratio assumption: In both numerical applications (Section 3.1, line 260; Section 3.2, line 329), the material properties are set with a Poisson's ratio of zero. While setting value to zero simplifies plane stress decoupling, engineering structural materials (e.g., steel, aluminum, composites) possess Poisson's ratios typically ranging between 0.25 and 0.35. The authors should discuss the physical rationale for selecting zero value and present sensitivity evaluations showing how realistic Poisson's ratios affect volumetric locking behavior, lateral strain coupling, and the accuracy of stress recovery under enriched cover stencils.
6. Discussion of higher-order cover stability and linear dependence handling: In Section 3.1.2 and Section 3.2.2, higher-order polynomial enrichments (Poly9 and Poly10) are tested. As noted in the introductory literature review (lines 63–65, 87), cover-based enrichment schemes can encounter rank deficiency or linear dependence in the global stiffness matrix when higher-degree polynomials are used over overlapping patches. The authors should explain how linear dependence and stiffness matrix ill-conditioning are avoided or conditioned when implementing Poly9 and Poly10 formulations, particularly when selecting larger surrounding node clusters.

Author Response

Comment 1: Resolved cross-references and citation artifact cleanup: Several figure cross-references in the text are unrendered and appear as broken placeholders (`Figure ??`). Specifically, line 270 ("highlighted by the black circle in Figure ??)"), line 273 ("on the exact same mesh (Figure ??)."), and line 344 ("indicated by black circles in Figure ??).") lack correct figure numbers. Furthermore, the captions for Figure 26 on page 20 contain raw citation syntax artifacts (e.g., `(b) Poly8 %[cite: 1]`, `(b) Poly9 %[cite: 1]`, and `(c) Poly10 %[cite: 2]`). The authors must thoroughly audit and fix all figure references and citation strings throughout the manuscript.

Response 1: We sincerely thank the Reviewer for carefully identifying these formatting and cross-referencing issues. The authors provided latex files along with pdf file during submission of the manuscript. These errors may come from compiling the latex files to get pdf file. However, we have thoroughly audited the entire manuscript to identify and correct all broken figure cross-references, unrendered placeholders, and citation syntax artifacts. Specifically, the three occurrences of Figure ?? identified by the Reviewer have been corrected to the appropriate figure numbers at the relevant locations (lines 270, 273, and 344).

We have also corrected the citation artifacts appearing in the caption of Figure 26. The raw strings such as %[cite: 1] and %[cite: 2] have been removed and replaced with the appropriate citation format. Furthermore, all figure captions and in-text citations were checked for similar rendering or formatting artifacts.

Comment 2: Reconciliation of loading condition discrepancies in clamped plate benchmark: In Section 3.2 (line 329), the text specifies an applied transverse bending load of F = 100 N. However, Figure 15 explicitly labels the applied tip load as F = 500 N. This contradiction creates ambiguity regarding the boundary conditions and load magnitude utilized in the numerical simulations and baseline comparisons. The authors must clarify the exact force magnitude applied in the finite element models and ensure complete consistency between textual descriptions, figure diagrams, and reported stress values.

Response 2: We sincerely thank the Reviewer for identifying this inconsistency. We agree that the discrepancy between the force value stated in Section 3.2 and that indicated in Figure 15 could cause ambiguity regarding the loading condition used in the numerical simulations.

After carefully reviewing the finite element model, simulation setup, and corresponding results, we confirm that the actual transverse tip load applied in the finite element simulations was F = 500 N. The value of F = 100 N reported in Section 3.2 was an inadvertent typographical error.

To eliminate this inconsistency, we have revised Section 3.2 and replaced F = 100 N with F = 500 N. Figure 15 has also been carefully checked to ensure that the loading condition is correctly represented.

Comment 3: Quantitative evaluation of computational complexity and matrix bandwidth: The paper emphasizes that ICM avoids mesh refinement and additional global sub-element meshing. However, enriching quadrilateral elements with cover functions introduces extra degrees of freedom (e.g., increasing local element DOFs from 24 to 48 or 72, as described in Section 2.2) and expands the stencil sparsity pattern of the global stiffness matrix. The manuscript lacks any quantitative reporting of solver execution times, CPU memory overhead, or condition numbers of the global stiffness matrix relative to standard h-refined FE meshes. The authors should include a comparative table or table-free quantitative narrative detailing CPU runtime and matrix bandwidth for standard vs. enriched formulations to objectively justify the computational efficiency claims.

 

Response 3: We sincerely thank the Reviewer for this important observation. We acknowledge that the introduction of cover functions increases the number of local degrees of freedom and consequently modifies the sparsity pattern of the global stiffness matrix. Therefore, in the revised manuscript, we have added a quantitative comparison between the conventional FE formulation and the enriched ICM formulation in terms of CPU time.

The comparison was performed using the same benchmark problem, material properties, boundary conditions, loading conditions, convergence criteria, and computational platform to ensure a meaningful comparison. The corresponding results are summarized in the revised manuscript in Table 3.

The quantitative results demonstrate that, although the enriched formulation increases the number of degrees of freedom at the element level and produces a somewhat wider matrix stencil, it avoids the generation and assembly of a globally refined sub-element mesh. Consequently, the proposed ICM formulation provides a favorable balance between local enrichment and global discretization complexity. In particular, the computational cost associated with the additional enrichment degrees of freedom remains lower than that associated with the equivalent globally h-refined FE model for the benchmark considered.

The revised comparison also makes clear that the computational advantage of ICM should not be interpreted as implying that enrichment introduces no additional computational cost. Rather, the advantage arises from introducing additional approximation capability locally through the cover functions without requiring a corresponding global increase in the number of physical finite elements and associated mesh-generation/assembly operations.

Comment 4: Empirical verification of robustness under mesh distortion: The manuscript title explicitly highlights the "Robustness of the Interpolation Cover Method". However, both benchmark problems (plate with hole and clamped beam) are discretized using regular or mildly graded orthogonal quad meshes (Figure 4 and Figure 16). In practical structural analysis, elements often suffer from severe angular skewness, aspect ratio distortion, or non-planar warping. To validate the claimed robustness of ICM against element distortion and locking phenomena, the authors should evaluate the method on standard benchmark tests with distorted or skewed meshes (such as the MacNeal skewed arm or irregular quad patches).

Response 4: We sincerely thank the Reviewer for this important observation. We would like to clarify, however, that the robustness of the Interpolation Cover Method (ICM) with respect to mesh distortion has already been investigated in previous studies of the interpolation-cover formulation [1-3].

Kim and Bathe [1] developed the finite element method enriched by interpolation covers and demonstrated that the use of cover functions can significantly improve the convergence and accuracy of low-order triangular and tetrahedral elements, including cases involving distorted elements. Furthermore, Jeon et al. [2] investigated an MITC3 shell finite element enriched with interpolation covers and demonstrated good convergence behavior for distorted meshes through several numerical examples, including plate and shell problems. More recently, Lee et al. [3] explicitly investigated mesh distortion in an adaptive interpolation-cover formulation and reported robust performance for distorted meshes.

Therefore, the present manuscript does not intend to establish mesh-distortion robustness for the first time. Rather, the objective of the present work is to investigate the performance of the ICM formulation for the structural benchmark problems considered here and to demonstrate its accuracy and convergence characteristics relative to the conventional finite element formulation. The use of regular and controlled meshes in these benchmark examples was intentional so that the effects of the interpolation-cover enrichment could be assessed without introducing an additional variable associated with mesh quality.

References:

[1] Kim J. and Bathe K.J., 2013. The finite element method enriched by interpolation covers. Computers & Structures, 116, pp.35-49.

[2] Jeon H.M., Lee P.S. and Bathe K.J., 2014. The MITC3 shell finite element enriched by interpolation covers. Computers & Structures, 134, pp.128-142.

 [3] Lee J., Yun G. and Kim D.N., 2025. An adaptive procedure for the analysis of transient wave propagation using interpolation covers. Computer Methods in Applied Mechanics and Engineering, 445, p.118167.

Comment 5: Justification and analysis of zero Poisson's ratio assumption: In both numerical applications (Section 3.1, line 260; Section 3.2, line 329), the material properties are set with a Poisson's ratio of zero. While setting value to zero simplifies plane stress decoupling, engineering structural materials (e.g., steel, aluminum, composites) possess Poisson's ratios typically ranging between 0.25 and 0.35. The authors should discuss the physical rationale for selecting zero value and present sensitivity evaluations showing how realistic Poisson's ratios affect volumetric locking behavior, lateral strain coupling, and the accuracy of stress recovery under enriched cover stencils.

Response 5: We sincerely thank the Reviewer for raising this important point. We agree that a Poisson's ratio of zero should not be interpreted as a representative material property for common engineering materials such as steel or aluminum. The value of ν = 0 was selected in the present benchmark problems primarily as a simplifying reference assumption, rather than to represent a particular engineering material.

The main objective of the numerical examples is to assess the performance of the proposed Interpolation Cover Method (ICM), including its stress approximation characteristics, while minimizing the influence of additional material-coupling effects. With ν = 0, the normal strain components are decoupled in the constitutive relation, which provides a convenient baseline for evaluating the interpolation-cover enrichment and stress-recovery procedure. Thus, the selected value should be understood as a benchmark parameter used to isolate the numerical behavior of the proposed formulation and not as a claim that ν = 0 is typical of structural materials.

We agree with the Reviewer that the response of the formulation can change when a realistic non-zero Poisson's ratio is introduced because the constitutive matrix then contains coupling between the normal strain components. We have therefore clarified this point in the revised manuscript and explicitly stated that the ν = 0 assumption limits the direct assessment of lateral-strain coupling and Poisson-ratio-dependent locking effects.

It is also important to distinguish the present benchmark objective from the nearly incompressible limit. Volumetric locking becomes particularly relevant as ν approaches 0.5, whereas ν = 0 represents the opposite limiting case in terms of volumetric constraint. Consequently, the present examples were not intended to constitute a dedicated investigation of volumetric locking.

The following paragraph has been added in plate with a hole, result and discussion section in the revised manuscript.

In the present numerical examples, a Poisson's ratio of ν = 0 is adopted as a controlled benchmark assumption. This choice is made to eliminate lateral constitutive coupling and thereby isolate the influence of the interpolation-cover enrichment on the convergence and stress-recovery characteristics. The selected value should not be interpreted as representative of a particular engineering material; rather, it provides a simplified reference configuration for assessing the numerical performance of the proposed formulation. Since volumetric locking is primarily associated with nearly incompressible materials as ν approaches 0.5, the present examples are not intended to investigate locking in the nearly incompressible regime. The interpolation-cover formulation itself is not restricted to ν = 0 and can be employed with nonzero Poisson's ratios for applications involving specific engineering materials.

Comment 6: Discussion of higher-order cover stability and linear dependence handling: In Section 3.1.2 and Section 3.2.2, higher-order polynomial enrichments (Poly9 and Poly10) are tested. As noted in the introductory literature review (lines 63–65, 87), cover-based enrichment schemes can encounter rank deficiency or linear dependence in the global stiffness matrix when higher-degree polynomials are used over overlapping patches. The authors should explain how linear dependence and stiffness matrix ill-conditioning are avoided or conditioned when implementing Poly9 and Poly10 formulations, particularly when selecting larger surrounding node clusters.

Response 6: We sincerely thank the Reviewer for raising this important issue concerning the numerical stability of higher-order polynomial enrichments. We agree that increasing the polynomial order and the number of surrounding nodes can potentially introduce redundant enrichment contributions, which may affect the rank and conditioning of the global stiffness matrix.

We would like to clarify that the Poly8, Poly9, and Poly10 formulations used in the present work are constructed using a local enrichment strategy, in which the additional polynomial terms are associated with selected surrounding nodes of the target element. As described in Eqs. (22)–(24), Poly8 employs four additional terms, Poly9 employs five additional terms, and Poly10 employs six additional terms. Accordingly, the number of surrounding nodes is increased consistently with the number of additional enrichment terms rather than introducing an arbitrarily large number of enrichment nodes.

The local nature of this construction is important for maintaining the independence of the enrichment contributions. As described in Section 2.2, the additional degrees of freedom associated with the surrounding nodes are assembled into their corresponding global nodal locations. Thus, the higher-order enrichment does not introduce a separate global polynomial field; instead, the additional approximation capability is incorporated locally through the selected surrounding-node cluster.

Furthermore, the influence of the enrichment order and surrounding-node selection has already been examined numerically in the present manuscript. Figure 14 compares Poly8, Poly9, and Poly10 for the plate-with-a-hole benchmark. The maximum von-Mises stress varies only slightly among the three formulations, with the difference remaining approximately within 1%. This limited variation indicates that the higher-order enrichment does not introduce an unstable or highly sensitive response for the configurations investigated.

We also investigated the influence of the surrounding-node configuration. As shown in Figures 12 and 13 in the manuscript, the first, second, and third nearest surrounding node configurations produce closely comparable stress distributions, although a small variation in the maximum von-Mises stress is observed. In addition, the comparison presented in Table 3 demonstrates that the selection of the enrichment polynomial and surrounding node configuration can slightly influence the numerical result, but the resulting stresses remain close to the reference solution.

Therefore, in the present implementation, the polynomial order and surrounding-node number are selected together according to the enrichment basis. For Poly9 and Poly10, the additional terms are supported by five and six surrounding nodes, respectively, as defined in Eqs. (23) and (24). The numerical results demonstrate that these higher-order configurations remain stable for the benchmark problems considered.

The following paragraph has been added in the section 3.1.2.

The influence of the enrichment polynomial basis on the numerical response of the ICM formulation is investigated using the plate-with-a-hole benchmark. In the Poly8 formulation, four additional polynomial terms are introduced into the standard Q4 shell approximation where additional nodes can be selected by the user according to the desired enrichment order. The number of surrounding nodes is selected consistently with the number of additional enrichment terms. Figure 14 compares the von-Mises stress distributions obtained using Poly8, Poly9, and Poly10. The maximum von-Mises stress shows only a small variation among the three enrichment bases, with the difference remaining approximately within 1% for the investigated case. The results indicate that increasing the polynomial order does not produce a significant deterioration of the numerical response for the considered enrichment configurations. The influence of the surrounding-node selection was also investigated. As shown in Figures 12 and 13, the 1st , 2nd , and 3rd nearest neighboring-node configurations produce similar stress distributions, although a small variation in the maximum von-Mises stress is observed. These results indicate that the enrichment response is relatively insensitive to the investigated variations in the surrounding-node configuration. However, the results also demonstrate that the location and number of surrounding nodes can have a small influence on the recovered stress field. It should be noted that higher-order polynomial enrichment can generally lead to linear dependence or ill-conditioning if redundant enrichment functions or inappropriate surrounding-node configurations are introduced. In the present ICM implementation, the enrichment is applied locally through the selected surrounding-node clusters, and the number of surrounding nodes is selected consistently with the enrichment basis. The Poly8, Poly9, and Poly10 configurations considered here produced stable numerical solutions and no singular system was encountered in the investigated benchmark cases.

 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have comprehensively addressed my comments by resolving broken figure cross-references and citation artifacts, reconciling the transverse tip load consistency (F = 500 N), incorporating quantitative CPU execution time benchmarking across standard FE and enriched ICM formulations, contextualizing mesh-distortion robustness within existing literature, justifying the zero Poisson's ratio benchmark decoupling assumption, and elucidating the numerical stability and linear independence preservation of higher-order cover enrichments (Poly8–Poly10). The revised manuscript is technically sound, methodologically rigorous, and significantly strengthened; I recommend it for publication in Materials.