Analysis using FEM-(Fea)

Analysis using FEM-(Fea)BY:-ANKIT KUMAR -2K10/ME/029LALIT-2K10/ME/062MAHESH KR. MEENA-2K10/ME/064MIITESH SOLANKI-2K10/ME/070DEFINITION:-FEM stands for finite element method FEM is amethod for dividing up a very complicated problem into small elements that can be solved in relation to each other.The finite element method (FEM) (its practical application oftenknown as finite element analysis (FEA)) is a numerical technique for finding approximate solutions to partialdifferential equations (PDE) and their systems, as well as (less often) integral equations.FEM is a special case of the more general Galerkin method with polynomial approximation functions. The solutionapproach is based on eliminating the spatial derivatives from the PDE. This approximates the PDE with a system of algebraic equations for steady state problems, a system of ordinary differential equations for transient problems.These equation systems are linear if the underlying PDE is linear, and vice versa. Algebraic equation systems aresolved using numerical linear algebra methods. Ordinary differential equations that arise in transient problems arethen numerically integrated using standard techniques such as Euler's method or the Runge-Kutta method.Now a days this method (FEM-FEA)is very popular and used in analysing a lot of cases.As stated above this method particularly deals with making partial differential eqns. Then analysing them and then solving them and representing the results.

HISTORY OF FEMoriginated from the need for solving complex elasticity and structural analysis problems.Development in the middle to late 1950s for airframe and structural analysis.development of the finite element software NASTRAN in 1965Apart from thisRichard Courant, Alexander Hrennikoff and Olgierd Zienkiewicz from Imperial College done a lot in the field of developing better possibilities for the method.Courant's contribution was evolutionaryVarious types of finite element methods

AEM The Applied Element Method, or AEM combines features of both FEM and Discrete element method, or (DEM).

Generalized finite element method The Generalized Finite Element Method (GFEM) uses local spaces consisting of functions, not necessarily Polynomials.

hp-FEM The hp-FEM combines adaptively, elements with variable size h and polynomial degree p in order to achieveexceptionally fast, exponential convergence rates.hpk-FEMThe hpk-FEM combines adaptively, elements with variable size h, polynomial degree of the local approximations pand global differentiability of the local approximations (k-1) in order to achieve best convergence rates.

The finite difference method (FDM) is an alternative way of approximating solutions of PDEs.FDM in its basic form is restricted to handle rectangular shapes and simple alterationsThere are several ways one could consider the FDM a special case of the FEM approach. E.g., first order FEM isidentical to FDM for Poisson's equation, if the problem is discretized by a regular rectangular mesh with eachrectangle divided into two trianglesGenerally, FEM is the method of choice in all types of analysis in structural mechanicsFEA as discussed in first slide is application of FEM is used in most analysing cases. It consist of following steps

Preprocessing: formulating the problem.

Analysis

Post processing: A typical postprocessor display overlays colored contoursUsing of softwares for computation in earlier times FEM used to solve tough equation for geometry of surface and other applications and numerical values were obatined which were then used in the results.But now a days there are already made programs by the programmers so that one need not to write or think of the difficult algorithm of the problems solution instead the program can be used to calculate the results which not only save times but also increases the speed of computation.

FEM mesh created by an analyst priorto finding a solution to a magnetic problem using FEM software. Colours indicate that the analyst has set materialproperties for each zone, in this case a conducting wire coil in orange; a ferromagnetic component (perhaps iron) inlight blue; and air in grey. Although the geometry may seem simple, it would be very challenging to calculate themagnetic field for this setup without FEM software, using equations alone.

FEM solution to the problem at left,involving a cylindrically shaped magnetic shield. The ferromagnetic cylindrical part is shielding the area inside thecylinder by diverting the magnetic field created by the coil (rectangular area on the right). The color represents theamplitude of the magnetic flux density, as indicated by the scale in the inset legend, red being high amplitude. Thearea inside the cylinder is low amplitude (dark blue, with widely spaced lines of magnetic flux), which suggests thatthe shield is performing as it was designed to.