A BRIEF OVERVIEW OF FINITE ELEMENT METHOD

WHAT IS FEM?

The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics. Also referred to as Finite element analysis which is the studying and analysing of a phenomenon using FEM. FEM is a numerical technique, requiring good knowledge of Engineering Mathematics which is usually covered in undergraduate engineering curriculum. Also, conceptual knowledge of Mechanics, Heat Transfer and Fluid Mechanics is essential as FEM is extensively employed in solving practical problems of these areas, which are also covered at undergraduate level.

ABOUT HISTORY OF FEM

The development of the finite element method, cannot be attributed to a single person, there have been many pioneers who have contributed to the field at different stages in time. Originally, the method was devised following the need to solve complex problems in elasticity and structural analysis in civil and aeronautical engineering. However the development of the method can be attributed to the works by A. Hrennikoff and R. Courant in the early 1940s. Another contribution to the field was done Loannis Argyris. Leonard Organysean of the USSR, is responsible for the introduction of its practical application. Later, other countries also started working on the method and it gained more prominence with time.

FEM ANALYSIS AND TECHNOLOGY

Finite Element Method being diverse and flexible in its use is finding its way in the analysis and solution for a wide variety of engineering problems. The rapid advancement in computer science and technology has fostered this method tremendously making its application easier, faster and more efficient, since the computer is the basic need for the application of this method, due to the nature of the complexities of the problems it deals with. A variety of popular brand of finite element analysis packages are available commercially and industrially. Some of the popular packages are STAAD-PRO, GT-STRUDEL, NASTRAN, NISA and ANSYS. These softwares analyse several complex structures, with their technical variables and specifications and provide understandable and relevant solutions in matter of minutes, which is impossible if done manually.

 

FEM TECHNICAL OVERVIEW

In engineering, problems are solved with various unknowns. Once found, the behaviour of the entire structure can reliably be predicted. The basic unknowns known as the Field variables which are generally encountered in the engineering problems are displacements in solid mechanics, velocities in fluid mechanics, electric potentials and magnetic potentials in electrical engineering and temperatures in heat flow problems. Talking about a continuum, where these unknowns are infinite, the finite element procedure works on reducing these unknowns to a finite number by dividing the solution region into small parts/sections called elements and by expressing the unknown Field variables in terms of assumed approximating functions (also known as Interpolating functions/Shape functions) within each element. The approximating or interpolating functions are defined in terms of field or unknown variables of specified points called nodes or nodal points. Thus, in the finite element analysis the unknowns are the field variables of the nodal points. Once these are found the field variables at any point can be found by using interpolation functions. After selecting elements and nodal unknowns next step in finite element analysis is to assemble element properties for each element.

Typically the method involves the following steps:

  1. First dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem.

 

  1. Then systematically recombining all sets of element equations into a global system of equations for the final calculation.

In the first step above, the element equations are simple equations that locally approximate the original complex equations which has to be analysed, where the original equations are often partial differential equations (PDE). Galerkin method is used to approximate the solution. 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 nothing but polynomial approximation functions that depicts the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with different set of algebraic and ordinary differential equations for steady state and transient problems encountered in Heat Transfer applications.

These are the element equations, which are linear if the underlying PDE is linear, and non-linear if not. 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.

This explains the strong knowledge requirement of engineering mathematics covered in undergraduate curriculum.

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.

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 or the Navier-Stokes equation expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system. Thus, depicting the need for knowledge in Heat Transfer and Fluid Subjects.

 

The software package has all the logic, equations and mathematics predefined. It just analyses, the given set of data, variables and solves accordingly.

 

WHAT IS THE NEED TO STUDY FEM?

The finite element knowledge makes a good engineer better while just user without the knowledge of FEA may produce more dangerous results. To use the FEA software packages properly, the user must know the following points clearly:

  • Which elements are to be used for solving the problem in hand. 
  • How to discretize to get near to accurate results.
  • How to introduce boundary conditions properly.
  • How the element properties are developed and what are their limitations.
  • How the displays are developed in pre and post processor to understand their limitations.

To understand the difficulties involved in the development of FEA programs and hence the need for checking the commercially available packages with the results of standard cases.

Unless user has the background knowledge of Finite Element Methods, he may produce egregious results and proceed with errors. Hence it is quintessential that the users of FEA package should have sound and detailed knowledge of FEA. Calculating manually, the designer always gets the feel of the structure and get rough idea about the expected results. This important aspect cannot be ignored by any designer, whatever be the reliability of the program. Firstly, a complex problem is solved by making drastic assumptions, then checked whether expected trend of the result is obtained. Then slowly more plausible assumptions are made and more refined results with FEA package are obtained. User must remember that structural behaviour is not directed by the computer programs. Hence the designer should develop feel of the structure and make use of the programs to get numerical results which are close to structural behaviour.

WHAT ARE THE APPLICATIONS & BENEFITS?

The finite element analysis originated as a method of stress analysis in the design of aircrafts. It started as an extension of matrix method of structural analysis. Today this method is used not only for the analysis in solid mechanics, but even in the analysis of fluid flow, heat transfer, electric and magnetic fields and many others. Civil engineers use this method extensively for the analysis of beams, space frames, plates, shells, folded plates, foundations, rock mechanics problems and seepage analysis of fluid through porous media. Both static and dynamic problems can be handled by finite element analysis. This method is used extensively for the analysis and design of ships, aircrafts, space crafts, electric motors, heat engines and numerous other areas.

FEA now is always the choice for analysing problems over complicated domains (like cars and oil pipelines), whenever there is a change in the domain (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations eliminate numerous instances of creation and testing of hard prototypes for various high fidelity and critical situations. For example, in a frontal crash simulation FEA make it possible to increase prediction accuracy in "critical" areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation). Another example would be in numerical weather prediction, where it is essential to have accurate predictions over developing highly nonlinear phenomena (tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas FEA also finds it application in military and weapon technology.