Talk:Arc-length method

Responses to follow-up comments after the first revision

Hi, Marco Chemello (WMIT)@Marco Chemello (WMIT): Thank you for the review and for the suggestive comments. The draft has been revised to address the points raised.

• The response to the AFC reviewer comments is placed at the talk page, and the old maintenance templates are removed.
• A second paragraph is added to the lead section, covering the mechanical application and practical issues.
• The predictor-corrector section is shortened, and some detailed mathematical equations are deleted.
• Two websites from official finite element manuals are added to support the text.
• Now some used terms are removed and replaced by plain words, some are linked to the existing articles, and the others are explained in the subordinate clauses immediately after the terms.
• The sentence with "Today" in the lead is rewritten.
• The copyedit is completed.

Thank you for your valuable time! Arelativelygoodname (talk) 14:10, 8 July 2026 (UTC)

Responses to AFC reviewer comment

Hi, Ldm1954@Ldm1954:

Thank you for the review and for the constructive suggestions. I have revised the draft substantially to address the points raised.

Lead section: The lead has been rewritten to provide an overview of the arc-length method with as little technical language as possible. Only simple and necessary mathematical and mechanical words are included.

Definitions: Explanations of the used terms have been added immediately after their first appearance in the article. Some mathematical-style expressions are rewritten or deleted.

Theoretical content: The mathematical and theoretical sections have been greatly simplified to avoid assuming that readers are already familiar with nonlinear structural analysis. The article now presents only the fundamental concepts and formulations needed to explain the method.

Applications: The applications section has been expanded substantially. The section has been written in detailed with the support of many references.

Written style: I have removed promotional and subjective wording. The wording has been changed to a more neutral encyclopedic tone.

I would appreciate your valuable time and efforts in helping revise this article. Arelativelygoodname (talk) 14:10, 8 July 2026 (UTC)

First suggestions

@Arelativelygoodname Thak you for your edits. In the first phrase, please put arc-length method in bold, as explained in Wikipedia:Manual of Style/Lead section. Marco Chemello (WMIT) (talk) 14:19, 1 July 2026 (UTC)

Review

I have reviewed this article. It is a good article indeed, but there are some problems that need to be addressed.

Mathematical notation: The load term is written with $\Phi ^{2}$. Therefore, the coefficients $a$, $b$, and $c$ in the derived quadratic equation should use $\Phi ^{2}$ consistently. Alternatively, both the notations should be written as $\Phi$.

Incremental variables: In the quadratic derivation, the vector $\mathbf {v} ^{i}$ should probably be defined using the accumulated displacement increment within the current step, i.e.

$$\mathbf {v} ^{i}=\Delta \mathbf {u} _{p}^{i}+\delta \mathbf {u} _{R}^{i},$$

rather than the absolute displacement vector $\mathbf {u} _{p}^{i}$.

Xiaotong Lian (talk) 12:48, 3 July 2026 (UTC)

Reviwer 2

Contrary to reviewer 1 I cannot speak for the content and the correctness of the formula.

What I've added:

• A wikilink on numerical continuation
• "(i+1)th" changed to "$(i+1)$th"

What I suggest:

1. Inline math notation: I would suggest using "$(i+1)$-th", "$i$-th", and "$p$-th" instead of "$(i+1)$th", "$i$th", and "$p$th",
2. Grammar "around" math formulas: the formulas should follow the grammar of the sentence right before and right after (e.g., if after a formula you start a new sentence, the formula should end with a period and the new sentence should start with a capital letter. In general pay attention to that.
3. Add citations on the "Limitations" bullet points. If those are the ones listed right after "Several issues remain important in practical applications", I would suggest to spread them over the bullet list items

I have not further comments since I can't directly contribute to such a topic. Nice job :) Cerlonghish (talk) 13:49, 3 July 2026 (UTC)

Reviwer 3

Great job on publishing this comprehensive academic article! It is a good article indeed, but there are some problems that need to be addressed.

I think the arc-length method is a path-following or continuation method rather than a scalar root-finding algorithm, the category Root-finding algorithms may not be appropriate. The categories Computational mechanics, Finite element method, or Structural analysis may be recommended.

Some sentences still read like academic papers or textbooks rather than encyclopedia entries, such as "as can be seen" and "with the aid of".

Elvis Cristale (talk) 14:15, 3 July 2026 (UTC)

Follow-up comments after the recent revision

Hi Arelativelygoodname, thank you for revising the draft. I compared the earlier version with the current one, and the article is now noticeably closer to an encyclopedic draft.

The lead is much clearer than before. The earlier version began with a dense technical explanation of a path-following algorithm, nonlinear systems, limit points, load control, and an augmented system of equations. The current lead gives a simpler definition first, explains the role of the load parameter more gradually, and places the method in nonlinear structural analysis.

The basic theory section has also improved. Several terms are now explained immediately after the equations, rather than leaving the reader to infer the meaning of the symbols. This is a useful response to the earlier comment that readers should not be expected to know what terms such as load, residual, or nonlinear equation mean.

The applications section is now much stronger. It has changed from a short list into prose paragraphs covering post-buckling, snap-through, material nonlinearity, damage, fracture, and finite element software. That directly addresses the earlier request to expand the applications rather than focusing almost entirely on the mathematical path-following formulation.

The tone is also more neutral. Much of the previous essay-like language has been reduced, and the current version is less like a technical note or course handout.

I think a few issues still remain before resubmission:

• Please remove the personal response to the reviewer from the draft itself. The text beginning with "Hi, Ldm1954" and the list of changes belongs on the talk page, not in the article draft. The draft page should contain only the article text and the AfC templates/comments. You can move here messages, using {{ping}} template to notify to the user.
• The old maintenance templates are still present. If you believe the problems have been addressed, review each tag individually and remove only the ones that are no longer accurate, with an edit summary explaining the cleanup. Do not remove the AfC reviewer comment itself.
• The lead could still be slightly more complete. It now defines the method, but it does not yet summarize the main applications and limitations discussed later in the article. Consider adding one short second paragraph mentioning that the method is used for post-buckling, snap-through, softening, damage/fracture problems, and that practical issues include step-length selection, root selection, and branch switching.
• The predictor-corrector section is still rather mathematical for a general encyclopedia article. Per MOS:MATH, consider shortening the derivation. The article probably does not need every intermediate equation for the Taylor expansion, Crisfield reformulation, coefficients of the quadratic equation, and quadratic formula. A concise explanation of the predictor, corrector, constraint equation, and root-selection issue may be enough, with the detailed derivation left to the cited literature.
• Some terms still need plain-language explanation when first used. In particular, "monotonic function", "limit point", "load-displacement response", "snap-through", "snap-back", "post-buckling", "strain softening", and "branch switching" should either be briefly explained in context or linked where a suitable article exists.
• Please check that broad statements are supported by broad sources. Statements about where the method is used, what most variants have in common, or how finite element software implements arc-length procedures are better supported by textbooks, review articles, manuals, or overview papers. Recent primary research papers are fine for specific examples, but they should not be used to imply broad field-wide conclusions unless the source itself says so.
• The final sentence of the lead says "Today, it is implemented...". Since "today" becomes dated, consider replacing it with wording such as "The method has been implemented..." or "Arc-length-type procedures are implemented...".
• Please do one final copyedit for grammar and style. For example, "the residual equation" should not be preceded by "solve" without "the", and "behaviour" should be consistent with the chosen variety of English throughout the article.

After making these edits, you can resubmit through AfC. If the AfC box on the draft shows a "Submit for review" or "Resubmit" button, click that button. If the button is not visible, open the draft in source editing mode and add {{subst:submit}} at the top of the draft. When the draft shows a yellow box saying that the review is waiting, the resubmission has been completed. Please do not move the page manually to mainspace or copy-paste it into a new mainspace article; let the AfC reviewer handle acceptance. Please be patient: if you follow suggestions I'm confident the article may be republished (moved) soon. Thank you. Marco Chemello (WMIT) (talk) 09:22, 8 July 2026 (UTC)