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I know comparatively little about general engineering methodologies, but it strikes me that the (now outdated) "waterfall" method is possibly very apt for mechanical engineering. Perhaps advances in rapidly changing software engineering could also be applied.

Are software methodologies influencing other engineering disciplines? Or vice versa?

Wikipedia's page on engineering says:

Koen argues that the definition of what makes one an engineer should not be based on what he produces, but rather how he goes about it.

Which vaguely implies that development processes are to some degree ad-hoc, given that neither the methodology section of that page, nor the problem solving subsection of the engineering page make any references to development stages in the precise ways that is widely encouraged in software engineering.

So why the apparent lack of processes?

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It is my observation and practical experience that most engineering product development processors are industry and institution based. Waterfall though old are still used in some industries. But thanks to the software industry and globalization agile methodology is fast gaining acceptance is engineering discipline. The automotive industry is one that has had to change.

The US automotive industry in the past used to take for ever produce new products. But automotive manufactures from east as force the US manufactures to fast adopt, or be left behind.

But there are legitimate barriers, primarily cost. Unlike software some tools are very expensive. Interrupting engineering manufacturing process to make modifications has significant impact in productivity or in other words long downtime. On such situations the good old waferfall methodology is the only viable option.

Thanks to advances in technology such as 3D printing, quick turn PCB, advanced CAD tools agile methodology is fast been adopted in engineering product development processes. The follow graphic from eetimes better explains the shift.

enter image description here

Here is another example from stage-gate.

enter image description here

Below are some references where you might finds specific public examples.

References:

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Waterfall

Waterfall is the basic method that a lot of engineering is done with already. I would say that "waterfall" was the original method for software engineering because software engineering came about much later than the other areas of engineering, e.g. software already borrowed from other areas of engineering.

Agile (and others)

Agile is used in software engineering and not in other engineering areas because there are big differences in risk. Agile sounds terrible (impossible) in construction. It also will not work in areas where there are actual consequences for problems (and a change is harder than an update away).

Middle ground

There are lots of middle areas between the two extremes, so not everything will fall into these areas, e.g. rapid prototyping could be considered a form of agile.

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  • $\begingroup$ To add a bit to this, a semi-agile strategy is what Boeing used to use/uses. Qualitative part design while simultaneously estimating likely cost and manufacturing issues. Part designers working with process designers. This method can be expensive as process designers can be expensive to use in early design as they likely have a lot of later process design work to do. So as with everything there are tradeoffs to "agile-like" methods vs waterfall-like methods. $\endgroup$
    – do-the-thing-please
    May 10, 2016 at 19:35
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Software engineering is a relatively newer form of engineering so borrows more than leads - the agile methodologies borrowed heavily from lean manufacturing, Toyota and Kanban. The Wiki summary of Toyota principles is very close to the agile manifesto.

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