International Journal for Numerical Methods in Engineering ( IF 2.9 ) Pub Date : 2021-04-29 , DOI: 10.1002/nme.6695 Benjamin E. Grossman‐Ponemon 1 , Adrian J. Lew 1
- DCM is a label to denote a family of methods based on a common idea, but methods in that family present differences among them. In our work, we implemented a particular version of DCM introduced by C. A. Duarte and collaborators in Reference 2. A novel aspect of this version is an averaging scheme introduced and described in § 3.5.1 of their work which does not appear in any other earlier proposed DCM method. As such, it is an original contribution by C. A. Duarte and collaborators. In labeling our implementation “GDD-DCM” we are specifically referring to the method introduced in such reference. For concreteness, the computation of stress intensity factors with Eq. (32) in Reference 2 corresponds to Eq. (3.10) in Reference 1, and it is a unique feature of the DCM introduced by C. A. Duarte and collaborators in Reference 2.
- In the letter authored by C. A. Duarte, the following statement is incorrect: “The DCM formulation adopted by Gupta et al.2 is the same one used much earlier by several authors, as reviewed in 2.” An exhaustive review of all the references cited in Reference 2 (and beyond) shows that the version of DCM introduced in Reference 2, namely the averaging scheme, is novel to that manuscript. C. A. Duarte and collaborators implicitly acknowledge this in the introduction of Reference 2, in which they state “Strategies to handle non-planar crack surfaces with DCM and to improve the accuracy of the method are also presented,” which they do in § 3.5, whose title is “Strategies to improve the Displacement Correlation Method,” and which contains the aforementioned averaging scheme.
- At no point in Reference 1 do we make any statement about the convergence or lack thereof of the stress intensity factors computed with the DCM in combination with displacement fields computed with G/XFEM. In other words, we never state that the method that results from the combination of GDD-DCM and G/XFEM by C. A. Duarte and collaborators in Reference 2 does not converge.** This fact contradicts the assertion in C. A. Duarte's letter that says: “Grossman-Ponemon et al. state (at the end of page 14) the following about GDD-DCM results for the semi-infinite straight crack: ‘Referring to Figure 8 and in contrast to GM-II, we did not observe convergence of the GDD-DCM in any of error metrics and for any of the mesh families.’ This is an incorrect statement about the work by Gupta et al.” Also, we disagree with the statement in the letter by C. A. Duarte that states that it is incorrect to separate the DCM and G/XFEM in Reference 2: “It is incorrect to ignore this recommendation while calling the DCM as the ‘Gupta, Duarte, and Dhankhar's displacement correlation method (GDD-DCM)’ and then state that the method is not convergent.”
In view of these facts, it is evident that referring to the DCM we implemented solely as “DCM” would have been imprecise, and that there was no misrepresentation of the results in Reference 2.
Arguing, as the letter by C. A. Duarte does, that the label “GDD-DCM” leads the reader to incorrect conclusions about Reference 2 fails to acknowledge the clear distinction we make of the DCM and G/XFEM methods in Reference 2 in our manuscript. In the introduction of our manuscript, we state, “More recently, Gupta et al. applied the displacement correlation method to numerical solutions found using the generalized/extended finite element method (XFEM). Here, the enrichment of the FEM function spaces with the analytical asymptotic behavior of the displacement field yields more accurate displacements in the vicinity of the crack front.,” acknowledging that DCM and G/XFEM are separate methods. Later in § 3.1, we refer to GDD-DCM as “the displacement correlation method of Gupta, Duarte, and Dhankhar,” specifically acknowledging the contribution to the DCM family in Reference 2. We believe that a reader of our manuscript is able to distinguish between the DCM and the G/XFEM method in Reference 2.
Finally, and for the record, prior to the submission of our manuscript, on January 9, 2019 we shared with C. A. Duarte the numerical results that comprise Fig. 6 in Reference 1; our correspondence was concluded on January 13, 2019, well before the manuscript was submitted for review on August 14, 2019. In our communication, we clearly explained that we were using the DCM in Reference 2 with FEM. Further, we explicitly asked whether the results we obtained were reasonable in view of his experience with GDD-DCM as a way to certify that no mistake was made on our side with respect to our DCM calculations.
中文翻译：
对“BE Grossman-Ponemon、LM Keer 和 AJ Lew 对三维非平面裂纹的混合模式应力强度因子计算方法”论文的评论（Int. J. Numer. Methods Eng.，2020 年） ”
- DCM 是一个标签，用于表示基于共同思想的一系列方法，但该系列中的方法在它们之间存在差异。在我们的工作中，我们实施了由 CA Duarte 和合作者在参考文献2 中引入的特定版本的 DCM 。此版本的一个新颖方面是在他们工作的第 3.5.1 节中介绍和描述的平均方案，该方案没有出现在任何其他早期提出的 DCM 方法中。因此，它是 CA Duarte 及其合作者的原创贡献。在标记我们的实现“GDD-DCM”时，我们特指此类参考中介绍的方法。对于具体性，应力强度因子的计算公式为。参考文献2 中的(32)对应于方程。参考文献1 中的(3.10)，这是 CA Duarte 及其合作者在参考文献2 中引入的 DCM 的独特功能。
- 在 CA Duarte 撰写的信中，以下陈述是不正确的：“Gupta 等人采用的 DCM 公式。2与多位作者更早使用的相同，如2 中所述。” 对参考文献2（及以后）中引用的所有参考文献的详尽审查表明，参考文献2 中介绍的 DCM 版本，即平均方案，对该手稿来说是新颖的。CA Duarte 和合作者在参考文献2的介绍中含蓄地承认了这一点，其中他们指出“还介绍了使用 DCM 处理非平面裂纹表面并提高方法准确性的策略”，他们在第 3.5 节中进行了说明，其标题是“改进位移相关方法的策略”，以及其中包含上述平均方案。
- 在参考文献1 中，我们从未对使用 DCM 计算的应力强度因子与使用 G/XFEM 计算的位移场相结合的收敛性或缺乏收敛性做出任何陈述。换句话说，我们从不声明参考文献2中CA Duarte 和合作者结合 GDD-DCM 和 G/XFEM 得到的方法不收敛。**这一事实与 CA Duarte 的信中的断言相矛盾：“Grossman-Ponemon 等人。陈述（在第 14 页的末尾）关于半无限直裂纹的 GDD-DCM 结果的以下内容： '参考图 8，与 GM-II 相比，我们没有观察到 GDD-DCM 在任何一个中的收敛错误度量和任何网格系列。这是对 Gupta 等人工作的错误陈述。” 此外，我们不同意 CA Duarte 在信中的声明，即在参考文献2中将 DCM 和 G/XFEM 分开是不正确的：“在将 DCM 称为‘Gupta、Duarte、和 Dhankhar 的位移相关方法 (GDD-DCM)'，然后声明该方法不收敛。”
鉴于这些事实，很明显，将我们实施的 DCM 单独称为“DCM”是不准确的，并且参考文献2中的结果没有误传。
正如 CA Duarte 在信中所做的那样，认为标签“GDD-DCM”导致读者对参考文献2得出错误的结论未能承认我们在参考文献2中对 DCM 和 G/XFEM 方法所做的明确区分在我们的手稿中。在我们手稿的介绍中，我们说：“最近，Gupta 等人。将位移相关方法应用于使用广义/扩展有限元方法 (XFEM) 找到的数值解。在这里，利用位移场的解析渐近行为丰富 FEM 函数空间可以在裂纹前沿附近产生更准确的位移。”承认 DCM 和 G/XFEM 是不同的方法。稍后在第 3.1 节中，我们将 GDD-DCM 称为“Gupta、Duarte 和 Dhankhar 的位移相关方法”，特别承认参考文献2 中对 DCM 系列的贡献。我们相信我们手稿的读者能够区分参考文献2 中的 DCM 和 G/XFEM 方法。
最后，作为记录，在提交我们的手稿之前，我们于 2019 年 1 月 9 日与 CA Duarte 分享了参考文献1 中图 6 的数值结果；我们的通信于 2019 年 1 月 13 日结束，早在 2019 年 8 月 14 日手稿提交审查之前。在我们的通信中，我们清楚地解释了我们将参考文献2 中的 DCM与 FEM 一起使用。此外，鉴于他使用 GDD-DCM 的经验，我们明确询问我们获得的结果是否合理，以此证明我们在 DCM 计算方面没有犯错误。
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