I would like to know examples of mathematical research which had its origin in engineering problems (I am mainly interested in mechanical engineering and civil engineering) and which was actually helpful to solve those (or related) problems. There is a similar question here (whose only answer is relevant to my question), but it's about mathematical definitions rather than research.
Here are a few examples.
Wavelets. There was early work by Haar, but the subject became intensively studied only decades later, in part due to work by the engineer Jean Morlet, who in fact introduced the term wavelet (well, the French term ondelette). The history of the subject before it really took off in the 1980s involved a lot of independent rediscoveries. See the paper "The Story of Wavelets" by R. Polikar.
Information theory. This was born in Shannon's book The Mathematical Theory of Communication in the 1940s. He worked at Bell Labs and was motivated by practical problems of telephone communication.
Coding theory (error-correcting codes). This is due to Hamming, another employee of Bell Labs. A famous quote of his, related to having to restart computer programs after they stopped two weekends in a row due to mistakes: "Damn it, if the machine can detect an error, why can't it locate the position of the error and correct it?"
O. Heaviside used things akin to the Dirac delta prior to 1900, helping to design transatlantic telegraph cables. For that matter, G. Green did, c. 1838, implicitly, in treatment of fundamental solutions of PDE's, in the context of electromagnetic phenomena. Yes, P. Dirac used such ideas again, in the late 1920's, in physics. And S. Sobolev rigorized a certainly class of distributions/generalized functions in the 1930's (and subsequently). But it was only in the late 1940's that L. Schwartz rigorized a general theory of distributions/generalized functions in a robust fashion. In the early 1950's, A. Grothendieck finished up several important mathematical aspects of this.