Finite element analysis was first developed for use in the aerospace and nuclear industries where the safety of structures is critical. Today, the growth in usage of the method is directly attributable to the rapid advances in computer technology in recent years. As a result, commercial finite element packages exist that are capable of solving the most sophisticated problems, not just in structural analysis, but for a wide range of phenomena such as steady state and dynamic temperature distributions, fluid flow and manufacturing processes such as injection molding and metal forming.

 

FEA consists of a computer model of a material or design that is loaded and analyzed for specific results. It is used in new product design, and existing product refinement. A company is able to verify that a proposed design will be able to perform to the client's specifications prior to manufacturing or construction. Modifying an existing product or structure is utilised to qualify the product or structure for a new service condition. In case of structural failure, FEA may be used to help determine the design modifications to meet the new condition.

Mathematically, the structure to be analyzed is subdivided into a mesh of finite sized elements of simple shape. Within each element, the variation of displacement is assumed to be determined by simple polynomial shape functions and nodal displacements. Equations for the strains and stresses are developed in terms of the unknown nodal displacements. From this, the equations of equilibrium are assembled in a matrix form which can be easily be programmed and solved on a computer. After applying the appropriate boundary conditions, the nodal displacements are found by solving the matrix stiffness equation. Once the nodal displacements are known, element stresses and strains can be calculated

Within each of these modeling schemes, the programmer can insert numerous algorithms (functions) which may make the system behave linearly or non-linearly. Linear systems are far less complex and generally ignore many subtleties of model loading & behaviour. Non-linear systems can account for more realistic behaviour such as plastic deformation, changing loads etc. and is capable of testing a component all the way to failure.

Despite the proliferation and power of commercial software packages available, it is essential to have an understanding of the technique & physical processes involved in the analysis. Only then can an appropriate & accurate analysis model be selected, correctly defined and subsequently interpreted.

History of FEM & FEA

Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration systems. Shortly thereafter, a paper published in 1956 by Turner, Clough, Martin, & Topp established a broader definition of numerical analysis. This paper centered on the "stiffness and deflection of complex structures".

By the early 70's, FEA was limited to expensive mainframe computers generally owned by the aeronautics, automotive, defense, and nuclear industries, and the scope of analyses were considerably limited. Finite Element technology was further enhanced during the 70's by such people as Zeinkiewicz & Cheung, when they applied the technology to general problems described by Laplace & Poisson's equations. Mathematicians were developing better solution algorithms, the Galerkin, Ritz & Rayleigh-Ritz methods emerged as the optimum solutions for certain categories of general type problems. Later, considerable research was carried out into the modelling & solution of non-linear problems, Hinton & Crisfield being major contributors.

 

While considerable strides were made in the development of the finite element method, other areas did not remain static. Very powerful mesh generation algorithms have been developed. Commercial generators have the capability of meshing all but the most difficult geometry. Superior CAE concepts have also emerged, it is not unusual to have a single CAD model for producing engineering drawings, carrying out kinematic & assembly analysis, as well as being used for finite element modelling.

Due to the rapid decline in the cost of computers and the phenomenal increase in computing power, present day desktop computers are capable of producing accurate results for all kinds of parameters (standard PC's are over 10 times more powerful than the best supercomputers of the early 90's).

The finite element method now has it's roots in many disciplines, the end result is a technology that is so advanced that it is almost indinguishable from magic. The vast catalog of capability that comprises FEA, will no doubt grow considerably larger in the future. CAE is here to stay, but in order to harness it's true power, the user must be familiar with many concepts, including the mechanics of the problem being modelled. All analyses require time, experience & most importantly, careful planning.

What's the difference between FEM & FEA ??

This is a very contentious issue, one that academics love to debate over a cool longneck of a Friday evening. I am going to stick my head on the block here & try to explain the difference, happy chopping my academic friends.

The terms 'finite element method' & 'finite element analysis' seem to be used interchanably in most documentation, so the question arises is there a difference between FEM & FEA ??

The answer is yes, there is a difference, albeit a subtle one that is not really important enough to loose sleep over.

The finite element method is a mathematical method for solving ordinary & elliptic partial differential equations via a piecewise polynomial interpolation scheme. Put simply, FEM evaluates a differential equation curve by using a number of polynomial curves to follow the shape of the underlying & more complex differential equation curve. Each polynomial in the solution can be represented by a number of points and so FEM evaluates the solution at the points only. A linear polynomial requires 2 points, while a quadratic requires 3. The points are known as node points or nodes. There are essentially three mathematical ways that FEM can evaluate the values at the nodes, there is the non-variational method (Ritz), the residual mehod (Galerkin) & the variational method (Rayleigh-Ritz).

FEA is an implementation of FEM to solve a certain type of problem. For example if we were intending to solve a 2D stress problem. For the FEM mathematical solution, we would probably use the minimum potential energy principle, which is a variational solution. As part of this, we need to generate a suitable element for our analysis. We may choose a plane stress, plane strain or an axisymmetric type formulation, with linear or higher order polynomials. Using a piecewise polynomial solution to solve the underlying differential equation is FEM, while applying the specifics of element formulation is FEA, e.g. a plane strain triangular quadratic element.

Application Areas

In essence, the finite element is a mathematical method for solving ordinary & partial differential equations. Because it is a numerical method, it has the ability to solve complex problems that can be represented in differential equation form. As these types of equations occur naturally in virtually all fields of the physical sciences, the applications of the finite element method are limitless as regards the solution of practical design problems.

Due to the high cost of computing power of years gone by, FEA has a history of being used to solve complex & cost critical problems. Classical methods alone usually cannot provide adequate information to determine the safe working limits of a major civil engineering construction. If a tall building, a large suspension bridge or a neuclear reactor failed catastrophically, the economic & social costs would be unacceptably high.

In recent years, FEA has been used almost universally to solve structural engineering problems. One discipline that has relied heavily on the technology is the aerospace industry. Due to the extreme demands for faster, stronger, lighter & more efficient aircrafts, manufacturers have to rely on the technique to stay competitive. But more importantly, due to safety, high manufacturing costs of components & the high media coverage that the industry is exposed to, aircraft companies need to ensure that none of their components fail, that is to cease providing the service that the design intended.

FEA has been used routinely in high volume production & manufacturing industries for many years, as to get a product design wrong would be detrimental. For example, if a large manufacturer had to recall one model alone due to a piston design fault, they would end up having to replace up to 10 million pistons. Similarly, if an oil platform had to shut down due to one of the major components failing (platform frame, turrets, etc..), the cost of lost revenue is far greater than the cost of fixing or replacing the components, not to mention the huge environmental & safety costs that such an incident could incurr.

The finite element method is a very important tool for those involved in engineering design, it is now used routinely to solve problems in the following areas:

• Structural strength design

• Structural interation with fluid flows

• Analysis of Shock (underwater & in materials)

• Acoustics

• Thermal analysis

• Vibrations

• Crash simulations

• Fluid flows

• Electrical analyses

• Mass diffusion

• Buckling problems

• Dynamic analyses

• Electromagnetic evaluations

• Metal forming

• Coupled analyses

Nowadays, even the most simple of products rely on the finite element method for design evaluation. This is because contemporary design problems usually cannot be solved as accurately & cheaply using any other method that is currently available. Physical testing was the norm in years gone by, but now it is simply too expensive.

What can go wrong with FEA?

As finite-element analysis spreads to designers who may lack formal training in numerical procedures, practitioners must ask whether the most appropriate techniques are being used—and whether they are producing accurate details.

Finite-element methods are in abundant use in today's engineering practice through various general-purpose commercial computer programs and many special-purpose programs written for specific applications. These techniques are, to an increasing extent, being used to help identify good new designs and improve designs with respect to performance and cost.

Considering the important role that finite-element methods now play in various areas of engineering, practitioners need to ask themselves whether their procedures are the most appropriate techniques available and whether the methods will lead to accurate results. These questions are particularly important because more and more design engineers who have not necessarily been trained in numerical procedures are applying finite-element techniques in their work.

A schematic plot of strain energies Eh of finite-element solutions is given when the element size h is decreased. The plot shows convergence as the mesh is refined.

As the use of these methods expands to a larger and more diverse group, users must address the important question of what can go wrong in finite-element analysis. This article is not intended to resolve the question in the broadest sense; rather, we shall focus on some aspects of the reliability of finite-element methods and their accurate use. To illustrate, we will consider linear elastic solutions and assume that the algebraic finite-element equations are solved exactly. For a more complex analysis, the same considerations hold, but additional requirements need to be addressed as well.

First of all, engineers should recall that the finite-element method is used to solve a mathematical model, which is the result of an idealization of the actual physical problem considered. The mathematical model is based on assumptions made regarding the geometry, material conditions, loading, and displacement boundary conditions. The governing equations of the mathematical model are in general partial differential equations subjected to boundary conditions. These equations cannot be solved in closed analytical form. Therefore, engineers resort to the finite-element method to obtain a numerical solution.

Consider, for example, an analysis of a valve housing of axisymmetric geometry and axisymmetric loading. In such a case, it is reasonable to assume axisymmetric conditions for analysis. The complete mathematical model and thus the analysis problem is obtained by specifying the geometry and dimensions, support conditions, material constants, and loading.

While engineers cannot, in general, obtain analytically the exact solution for the posed mathematical model in closed form, the exact solution of the mathematical model does exist, the solution is unique, and an approximation of this exact solution can be obtained with very high accuracy using finite-element methods.

To quantify these observations, the notion of convergence must be introduced. Let E denote the (unknown) exact strain energy of the mathematical model. Also, let Eh denote the strain energy corresponding to the finite-element solution with a mesh of element size h. Then convergence means:

The schematic plot on above shows how convergence is reached. As the mesh is refined (that is, the element size h is decreased), the strain energy Eh approaches a value E. The rate at which the error between E and Eh is reduced depends on the problem solved as well as on the element and meshes used. Clearly, higher-order elements will reduce the error at a faster rate with mesh refinement than lower-order elements.

Reliability in finite-element methods means that in the solution of a well-posed mathematical model, the finite-element procedures will have two attributes: The finite-element solutions will converge to the exact solution of the mathematical model as h approaches 0 for any (physically realistic) material data, displacement boundary conditions, and loading applied; and for a reasonable finite- element mesh, a reasonable finite-element solution will be obtained. Furthermore, the quality of the finite-element solution does not change drastically when the material data (or thickness of a shell) are changed.

These conditions are of crucial importance. If the first condition (convergence) is violated, then with mesh refinement a solution is approached that is not the exact solution of the mathematical model. Such erroneous solutions could lead to wrong design decisions and disastrous consequences. Of course, finite-element methods that violate the first condition should not be used.

To consider the second condition—obtaining a reasonable solution using a reasonable mesh—assume, for example, that the valve housing is made of steel (Young's modulus is 200,000 megapascals; Poisson's ratio, n, is 0.30). The analysis using a reasonable mesh has given acceptable results (that is, the error |E—Eh| is acceptably small). Suppose that we now change the material to a plastic, for which Poisson's ratio is 0.49—close to the incompressible condition of 0.50. This change in material condition might result in a relatively small change in the exact solution, and it should then result only in the corresponding small change in the finite-element solution. Unfortunately, finite-element formulations are still used that violate the second condition and yield a solution grossly in error when Poisson's ratio is changed to 0.49.

This solution phenomenon is observed with displacement-based finite elements. The large errors are present because the elements are much too stiff when v is close to 0.50, and cannot be used when V=0.50. The underlying mechanical reason is that, considering the stresses, p=Kev, where p is pressure, K is bulk modulus, and ev is volumetric strain. As n approaches 0.50, K becomes very large and is infinite when V=0.50. Also, as n approaches 0.50, in the exact solution, ev becomes very small and is zero when V= 0.50. Therefore, in an almost incompressible analysis, the pressure is given by a very large number (the bulk modulus) multiplied by a very small number (the volumetric strain), and must be accurately calculated to balance the applied surface tractions. While the bulk modulus becomes very large, the pressure is always a finite number and usually does not change much as n approaches 0.50.

As a remedy in displacement-based finite-element methods, "reduced integration" is employed. This means that in the numerical integration of the element stiffness matrices, the exact matrices are not evaluated. The method is simple to program and requires less computation time to establish the matrices, and with experience acceptable results are frequently obtained. However, the technique can also lead to very large errors.

Consider the frequency analysis of the bracket. In this case, using nine-node elements, 323 Gauss integration corresponds to full numerical integration and 222 Gauss integration is "reduced integration." Since no analytical closed-form solution exists for the frequencies of the bracket, a very fine mesh (the 16264 element model) was used to obtain an accurate solution. Of interest are the solutions obtained with the coarse 16-element mesh. As predicted by theory, using full integration, the frequencies of the coarse 16-element mesh are larger than the accurate solution frequencies. When reduced integration is used, some frequencies are better approximated than when using full integration, but among those few listed is a phantom frequency. Phantom frequencies do not physically exist but are only introduced by the reduced-integration scheme. If a dynamic step-by-step solution is performed, such phantom frequencies are not noticed and absorb energy, introducing large errors into the solution. Error measures would detect that the errors are large, but they are frequently not available in dynamic analysis. Thus, reduced integration is unreliable and should be avoided.

Instead, only reliable mixed finite-element formulations, which do not require reduced integration, should be used. Such formulations satisfy the first and second conditions, are currently available, and can be employed confidently for the solution of mathematical models with any material properties, loading, and boundary conditions. A displacement/pressure-based formulation is particularly attractive when incompressible materials are considered and was used to analyze a plastic bracket with Poisson's ratio equal to 0.499 (see Analysis Clinic, March). The formulation is also used effectively in nonlinear analysis, where almost incompressible conditions are widely encountered. Inelastic conditions of plasticity and creep, and rubberlike conditions, give rise to (almost) incompressible behavior.

Similar difficulties, as described for almost-incompressible analysis, are also encountered in the analysis of plate and shell structures. Here, too, some finite-element technology still uses reduced integration, and the results can be very much in error. Again, reliable formulations, which are now available, should be used instead of such reduced-integration schemes.

To sum up, finite-element methods can now be employed with great confidence, but only the methods considered reliable should be used. Earlier technology based on reduced integration should not be used, or should at any rate be employed with great care. By proceeding in this way, practitioners can have confidence that a finite-element analysis will be effective and will not go wrong

ENGINEERS CADD CENTRE PRIVATE LIMITED

F10, 11, 12 – Eureka Court, Beside BATA

Near Chermas, Ameerpet

Hyderabad, AP 500 073
INDIA

D. ANAND K. REDDY, Managing Director, ECC

Register Now