Finite element method.
Among the up-to-date methods of stress state analysis, the finite element method (abbreviatedas FEM below, oroften as FEA foranalyses as well) dominates clearly nowadays; it is used also in other fields of engineeringanalyses (heat transfer, streaming of liquids, electricity and magnetismfields, etc.).
In mechanics, the FEM enables us to solve the following types of problems:
The fundamentals of FEM are quite different from the analytical methods of stress-strainanalysis. While the analytical methods of stress-strain analysis are based on the differentialand integral calculus, FEM is based on the variation calculus which is generallynot so well known; it seeks for a minimum of some functional.
Explanation of the concept – analogy with functions:
Among all the allowable displacements (i.e. those which meet the boundary conditions of the problem, its geometric and physical equations), only those displacements can come into existence between two close loading states (displacement change by a variation δu) which minimize the quadratic functional ΠL. This functional (called Lagrange potential) represents the total potential energy of the body, and the corresponding displacements, stresses and strains minimizing its value represent the seeked elasticity functions.
This principle is called Lagrange variation principle.
Lagrange potential can be described as follows:
ΠL = W – P
where W - total strain energy of the body
P – total potential energy of external loads
They can be distinguished from the viewpoint of assumptions the element is based on (bar assumptions, axisymmetry, Kirchhoff plates, membrane shells, etc.), or for what a family of problems the element is proposed.