Finite element method
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The finite element method is the numerical method of solving systems of differential equations. They are used extensively in many fields of engineering because they require very little knowledge of mathematics beyond basic algebra to use. It belongs to the Methods of Weighted residuals in that the problem is formulated such that some conditions are satisfied exactly others are satisfied only approximately or numerically. I.e. the 'residual' being the difference between the exact solution and the approximate solution is weighted and minimized to get the best approximation.
While the ease of setting up a problem using finite elements and getting a solution is the reason for its popularity, its is also the cause of its frequent misuse and distrust of the answers obtained. Typically, it is recommended or required that experimental verification be performed before a model is acceptable for use. It is the author's observation that the use finite elements in structural analysis is in particular poorly managed. Engineers typically begin working with FE having very little knowledge of numerical methods or knowledge of finite element behavior. There are no standards and guidelines for modeling and few managers who understand well enough to regulate modeling. It has lead to mistrust of the method, its arrested development, and some very bad analysis being performed that is sadly emulated. Much aircraft structural design is analyzed using FE.
