Finite Element Method
Introduction
The Finite Element Method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for partial differential equations. It is also referred to as finite element analysis (FEA). As such, it is a specific method of approximating the solutions of both boundary-value problems and initial-value problems.
History
The Finite Element Method was developed from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to the work by A. Hrennikoff and R. Courant in the early 1940s. Though the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements.
Basic Concepts
FEM is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other. From a mathematical point of view, FEM is a technique for solving differential equations by replacing them with a system of algebraic equations. The basic concept in FEM is the subdivision of the mathematical model into disjoint (non-overlapping) components of simple geometry called finite elements or elements for short. They are connected at specific points called nodes or nodal points. Elements may have physical properties such as thickness, coefficient of thermal expansion, density, Young's modulus, shear modulus etc.
Mathematical Foundation
The Finite Element Method is based on the premise that an approximate solution to any complex engineering problem can be reached by subdividing the problem into smaller, simpler parts known as finite elements. This method transforms the complicated equations that describe the problem into a set of simple equations that govern the individual elements, which are then solved collectively to depict the behavior of the actual problem.
Applications
FEM has been widely applied in engineering and physical sciences. It has uses in a wide range of applications, such as in structural design, heat transfer, fluid flow, mass transport, and electromagnetic potential. It is also used in the modelling of biological systems, such as blood flow in arteries, the growth of biological tissues, the displacement of the earth's crust, and the motion of stars in galaxies.
Advantages and Limitations
The Finite Element Method has many advantages over other methods of solving PDEs. It can handle irregular shapes and complex boundary conditions. It can also handle any type of material behavior. However, FEM also has its limitations and disadvantages. It requires a lot of computational resources for large problems. It also requires a good understanding of the underlying theory to use effectively.