Why Mathematical Modeling Is as Crucial as Experiments in Modern Engineering

In engineering, mathematical models are regarded as equally important as experiments because they provide quantitative predictions for design decisions, material selection, and system sizing; they allow analysis of complex or high‑performance systems that cannot be reproduced experimentally; they are essential for understanding phenomena that occur rarely, such as reliability of highly dependable components; and modern computers enable simulations of chemical reactions, stress calculations, and fluid flows.

The goal of applying mathematics in engineering is to comprehend phenomena, create new value, construct mathematical models, solve them analytically or numerically, and compare the solutions with experimental or survey data. When the model explains the data well, it is considered correct. Both analytical methods and numerical/computer simulations are used, and statistical techniques may be employed to derive models from data. Optimization methods arise from this modeling process.

For example, during the 1991 Gulf War the U.S. Department of Defense commissioned a study on the potential climate impact of Iraqi oil‑well fires. Using the fundamental equations of fluid dynamics and heat transfer, a mathematical model was built and solved via computer simulation. The results indicated severe regional effects but no global climate change, influencing military planning and illustrating the concept of a "mathematical war."

Another example is aircraft design. Historically, each change in parameters such as wingspan or airfoil required building a new scale model and conducting wind‑tunnel tests, which was costly and time‑consuming. Advances in mathematical modeling, particularly the Navier‑Stokes equations, now allow accurate simulation of fluid flow around aircraft, reducing the need for physical prototypes and accelerating design cycles. Similar model‑based simulation approaches are used for missile development and other high‑technology engineering systems, dramatically cutting time, cost, and risk.

Shen Wenxuan, Yang Qingtiao, "Mathematical Modeling Attempts"