Finite Element Methods & Analysis
Refer to Finite Element Basics
The need to solve complex elasticity and structural analysis problems in Civil and Aeronautical Engineering created a need for Finite Element Analysis (FEA). R. Courant's approach developed in early 1940s divides the domain into finite triangular sub-regions to solve second order elliptic partial differential equations (PDEs) that arise from the problem of torsion of a cylinder. The other pioneers in this field are A. Hrennikoff and Ioannis Argyris including Leonard Oganesyan and K. Feng. The real impetus, however, started with the development of Finite Element Methods (FEM) in 1960s and 1970s along with the availability of funded Open Source FEM Software [1].
The description of the laws of physics (conservation laws, laws of classical mechanics, and laws of electromagnetism) for space and time dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with Analytical Methods. So, FEM discretizes mathematical models into corresponding numerical models. The discretized equations are solved and the results are analyzed, hence the term Finite Element Analysis (FEA) [2].
As an analogy, the linear equation becomes the basis function or trial function if the straight line is considered as the element of constructing a circle. Through the connection (linear combination) of these elements or lines or basis functions, the circle can be approximated. In Hooke's Law and Harmonic Oscillator, basic equations for the system are both sine and the cosine. Hence, FEM encompasses all the methods for connecting many simple element equations over many small sub-domains (finite elements) to approximate a more complex equation over a larger domain. It therefore gives an approximate solution to the mathematical model equations. The difference between the approximate solution to the numerical equations and the exact solution to the mathematical model equations is the error. The variational methods such as Galerkin [3] is used to minimize the error function or residual so that the approximation can reach close to the actual solution. When an estimated error tolerance is reached, convergence occurs.Â
Smaller but more the number of lines i.e. denser the mesh, the closer the approximate solution gets to the actual circle (solution). In practice, computing an approximation for a very much finer mesh than those of interest can be difficult. So, mesh refinement technique is used to reduce the error evaluating different meshes through mesh convergence method.
References:
[1] Wikipedia