# Finite element analysis

History[ edit ] The origin of finite method can be traced to the matrix analysis of structures   where the concept of a displacement or stiffness matrix approach was introduced. Finite element concepts were developed based on engineering methods in s. The finite element method obtained its real impetus in the s and s by John Argyrisand co-workers; at the University of Stuttgartby Ray W. Earlier books such as by Zienkiewicz  and more recent books such as by Yang  give comprehensive summary of developments in finite-element structural analysis. ## Finite Element Analysis - Thermal, Stress, Vibration & Fatigue

The subdivision of a whole domain into simpler parts has several advantages: A typical work out of the method involves 1 dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to Finite element analysis original problem, followed by 2 systematically recombining all sets of element equations into a global system of equations for the final calculation.

The global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer.

In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations PDE.

To explain the approximation in this process, FEM is commonly introduced as a special case of Galerkin method. The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to Finite element analysis.

In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual.

The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with a set of ordinary differential equations for transient problems. These equation sets are the element equations. They are linear if the underlying PDE is linear, and vice versa.

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.

This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system. The process is often carried out by FEM software using coordinate data generated from the subdomains. FEA as applied in engineering is a computational tool for performing engineering analysis.

It 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 equationthe heat equationor the Navier-Stokes equations expressed in either PDE or integral equationswhile the divided small elements of the complex problem represent different areas in the physical system.

FEA is a good choice for analyzing problems over complicated domains like cars and oil pipelineswhen the domain changes as during a solid state reaction with a moving boundarywhen the desired precision varies over the entire domain, or when the solution lacks smoothness.

FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations. Another example would be in numerical weather predictionwhere it is more important to have accurate predictions over developing highly nonlinear phenomena such as tropical cyclones in the atmosphere, or eddies in the ocean rather than relatively calm areas.

Colours indicate that the analyst has set material properties for each zone, in this case a conducting wire coil in orange; a ferromagnetic component perhaps iron in light blue; and air in grey. Although the geometry may seem simple, it would be very challenging to calculate the magnetic 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 the cylinder by diverting the magnetic field created by the coil rectangular area on the right.

The color represents the amplitude of the magnetic flux densityas indicated by the scale in the inset legend, red being high amplitude. The area inside the cylinder is low amplitude dark blue, with widely spaced lines of magnetic fluxwhich suggests that the shield is performing as it was designed to. History[ edit ] While it is difficult to quote a date of the invention of the finite element method, the method originated 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 s. Another pioneer was Ioannis Argyris. In the USSR, the introduction of the practical application of the method is usually connected with name of Leonard Oganesyan.

Feng proposed a systematic numerical method for solving partial differential equations. The method was called the finite difference method based on variation principle, which was another independent invention of the finite element method.

Although the approaches used by these pioneers are different, they share one essential characteristic: Hrennikoff's work discretizes the domain by using a lattice analogy, while Courant's approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations PDEs that arise from the problem of torsion of a cylinder.

Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by RayleighRitzand Galerkin.

The finite element method obtained its real impetus in the s and s by the developments of J. Argyris with co-workers at the University of StuttgartR.Finite element analysis requires a working knowledge of stress analysis and materials principles to get the answer right - the first time.

Our engineers are multi-disciplined in areas of materials, design, metallurgy and manufacturing - each with more than 25 years of experience. The extended finite element method (XFEM) is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM).

It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions.

The finite element method (FEM), or finite element analysis (FEA), is a computational technique used to obtain approximate solutions of boundary value problems in engineering. Nov 09,  · Assembly and solution of finite element equations can be simulated interactively and graphically so that the process of finite element analysis can be .

The finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. Finite element analysis requires a working knowledge of stress analysis and materials principles to get the answer right - the first time.

Our engineers are multi-disciplined in areas of materials, design, metallurgy and manufacturing - each with more than 25 years of experience.

Free Mechanical Engineering: Finite Element Analysis