Finite element method
The finite element method (FEM) is a numerical method of solving systems of differential equations.[1][2] They are used extensively in many fields of engineering because they require very little knowledge of mathematics beyond basic algebra to use.[1][2] It belongs to the Methods of Weighted residuals[3] in that the problem is formulated such that some conditions are satisfied exactly, while 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, it is also the cause of its frequent misuse and distrust of the answers obtained.[4] Usually you have to check your results with an experiment before your model is acceptable for use. It is the author's observation that the use of finite elements in structural analysis is in particular poorly managed. Engineers typically begin working with FEM 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 led to mistrust of the method, its arrested development, and some very bad analysis being performed that is copied by others. Much aircraft structural design is analyzed using FEM.
Finite Element Method Media
Visualization of how a car deforms in an asymmetrical crash using finite element analysis
A function in H_0^1, with zero values at the endpoints (blue) and a piecewise linear approximation (red)
A piecewise linear function in two dimensions
Solving the two-dimensional problem u_{xx}+u_{yy}=-4 in the disk centered at the origin and radius 1, with zero boundary conditions.(a) The triangulation.
(b) The sparse matrix L of the discretized linear system
(c) The computed solution, u(x, y)=1-x^2-y^2.
3D pollution transport model - concentration field on ground level
3D pollution transport model - concentration field on perpendicular surface
Finite Element Model of a human knee joint.
References
- ↑ 1.0 1.1 Brenner, S., & Scott, R. (2007). The mathematical theory of finite element methods (Vol. 15). Springer Science & Business Media.
- ↑ 2.0 2.1 Oden, J. T., & Reddy, J. N. (2012). An introduction to the mathematical theory of finite elements. Courier Corporation.
- ↑ Finlayson, B. A., & Scriven, L. E. (1966). The method of weighted residuals—a review. Appl. Mech. Rev, 19(9), 735-748.
- ↑ Babuska, I., Strouboulis, T., & Babuška, I. (2001). The finite element method and its reliability. Oxford University Press.