How does the finite element method, a theoretical model, solve the problem of impracticality in some physical testing in engineering?
Justin Fu
Although all falling under the name STEM, physics, maths, and engineering are quite different. Physics and maths are disciplines of theory; engineering is a practical endeavour. However, practicality can sometimes be limited in physical testing for engineering. For instance, the testing of the performance of material used in the landing gear of an aircraft when experiencing different levels of vibration can be difficult to conduct in real life. To investigate methods that engineering uses to tackle the impracticality of some physical testing, it is necessary to research theoretical models within the field of engineering that tackle the impracticability of physical testing. Linear algebra is one possible theoretical model. However, linear methods of problem-solving are very idealized. For instance, the use of SUVAT equations to deduce the trajectory of an object is based on the assumption that the object is modified as a particle, air resistance is ignored, and the acceleration of the object is constant. Questioning the utility of these theoretical models, I decided to explore problems that exclude these idealized conditions — ones involving variables (calculus). Different from linear algebra, calculus is a mathematical tool that studies change using differentiation and integration. For instance, ‘ the determination of the total electrical energy transferred to a component until it fails at a current — denoted as Ifail — if the current (I) through the circuit is given in terms of a quadratic equation with respect to time (I = kt2) ’ is a classical model of circuits involving calculus. Due to both variable t (time) and I (current in terms of t), the original linear equation of E = VIt is not applicable. Instead, if we denote t in this case as dt (an infinitely small change in time) and E as dE (an infinitely small change in energy), a differential equation is formed: ∫ 𝑑𝐸 = ∫ 𝑉(𝑘𝑡 2 )𝑑𝑡 And using integration, the continuous summation of a function, we can find the total electrical energy transferred to the component until it fails by setting the upper limit of the integrand as Ifail. Through the basic use of calculus, theoretical models in physics can be a lot more applicable to a wider range of problems. Although using calculus in physics can significantly increase the applicability of a theoretical model to real-world problems, basic theoretical models of physics, even with calculus, only apply in the two dimensions. In secondary level calculus, differentiation and integration are mostly limited to the X-Y plane (two-dimensional); however, real-world problems occur in three dimensions, which in Euclidean vector space includes the Z plane. Thus, this increase in variables will require a further calculus model that includes multiple variables — multivariable calculus. And as I furthered my search for an applicable theoretical method for real-world analysis, this led to the discovery of an intricate numerical method — the finite element method.
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