Minimization and Variational Problems

[Somefantastik]

I just jumped into a finite element methods course, and we are finding minimization problems and variational problems for various PDE's. However, the book never really explains what these guys are and their purpose and what they do, and before I continue, I'd like to understand this. I googled it and came up with more specific information than for what I was looking.

Can someone please explain in a top-level, low-detail manner what these problems are, what they are used for, and how they are implemented in general?

 

[fresh_42]

From Wikipedia:

The FE method can be used to calculate problems from different physical disciplines, since it is basically a numerical method for solving differential equations. First, the calculation area ("component") is divided into a large number of elements - sufficiently fine. These elements are finally finite, i.e. their actual size remains relevant and is therefore not infinitely small. Splitting the area / part into a certain number of finite-size elements, which can be described by a finite number of parameters, gave the method the name "finite element method".

For these elements, there are mapping functions (e.g. local Ritz approaches per element) that describe how it responds to external influences and boundary conditions. Inserting these approach functions into the differential equations to be solved, which describe the laws of physics, together with the initial, boundary, and transition conditions, yields a system of equations. Solving it (at least approximately) is the task of the FE equation solver. The size of the system of equations to be solved depends largely on the number of finite elements. His approximate solution is ultimately the numerical solution of the considered differential equation - solved for all elements, as they behave under loads, so this has also resulted in the reaction of the entire component.

The use of FEM in practice began in the 1950s with a structural calculation of aircraft wings in the aerospace industry (Turner, Clough 1956) and very soon also in vehicle construction. The method is based here on the work at the Daimler AG in Stuttgart, which used the self-developed FEM program ESEM (Elastostatic Element Method) long before the computer-aided design (CAD) in the early 1980s made its entry. The term finite element method was first proposed in 1960 by R. W. Clough and has been widely used since the 1970s.