# Finite Elements

In mathematics, the finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for differential equations. This uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution.The Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, named finite elements, FEM encompasses all the methods for connecting many simple element equations over many small subdomains,  to approximate a more complex equation over a larger domain.

General principles

Subdivision of a whole domain into simpler parts has several advantages:

• Accurate representation of complex geometry
• Inclusion of dissimilar material properties
• Easy representation of the total solution
• Capture of local effects.

The typical work out of the method involves (1) dividing the domain of the problem into a collection of sub domains, followed by (2) systematically recombining all sets of element equations into a global system of equations for the final calculation, with each sub domain represented by a set of element equations to the original problem. Global system of equations has known solution techniques, can be calculated from the initial values of the original problem to obtain a numerical answer.

In the first step above, where the original equations are often partial differential equations (PDE), the element equations are simple equations that locally approximate the original complex equations to be studied. For explain the approximation in this process, FEM is commonly introduced as a special case of Galerkin method. Process in mathematics language, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms. Residual is the error caused by the trial functions, the weight functions are polynomial approximation functions that project the residual. Process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with

• a set of algebraic equations for steady state problems,
• A set of ordinary differential equations for transient problems.

Mathematics equation sets are the element equations. They are linear if the underlying PDE is linear, and vice versa. The Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler's method or the Runge-Kutta method.

In step (2) above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains' local nodes to the domain's global nodes. It spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system.  Process is often carried out by FEM software using coordinate data generated from the sub domains.

The FEM is best understood from its practical application, known as finite element analysis (FEA).The FEA as applied in engineering is a computational tool for performing engineering analysis. This includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.

The FEA is a good choice for analyzing problems over complicated domains (like cars and oil pipelines), when the desired precision varies over the entire domain, or when the solution lacks smoothness, when the domain changes (as during a solid state reaction with a moving boundary). To instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation). The Another example would be in numerical weather prediction, or eddies in the ocean) rather than relatively calm areas ,where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as tropical cyclones in the atmosphere. The FEM mesh created by an analyst prior to finding a solution to a magnetic problem using FEM software. The Colors indicate that the analyst has set material properties for each zone, a ferromagnetic component (perhaps iron) in light blue; and air in grey,in this case a conducting wire coil in orange.The Although the geometry may seem simple, using equations alone,it would be very challenging to calculate the magnetic field for this setup without FEM software.

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