FINITE ELEMENT ANALYSIS

INTRODUCTION

The finite-element method originated from the needs for solving complex

elasticity, structural analysis problems in all fields of engineering and. Its

development can be traced back to the work by Alexander Hrennikoff (1941)

and Richard Courant (1942). While the approaches used by these pioneers are

dramatically different, they share one essential characteristic: mesh

discretization of a continuous domain into a set of discrete sub-domains.

Development of the finite element method began in earnest in the middle to late

1950s for airframe and structural analysis and picked up a lot of steam at the

University of Stuttgart through the work of John Argyris and at Berkeley

through the work of Ray W. Clough in the 1960s for use in civil engineering.,

and has since been generalized into a branch of applied mathematics for

numerical modeling of physical systems in a wide variety of engineering

disciplines, e.g., electromagnetism and fluid dynamics.

AIM

Aim of this paper is to femilarise students with FINITE ELEMENT ANALYSIS

FINITE ELEMENT METHOD (FEM)

is used for finding approximate solution of partial differential equations (PDE)

as well as of integral equations such as the heat transport equation. The solution

approach is based either on eliminating the differential equation completely

(steady state problems), or rendering the PDE into an equivalent ordinary

differential equation, which is then solved using standard techniques such as

finite differences, etc.

In solving partial differential equations, the primary challenge is to create an

equation that approximates the equation to be studied, but is numerically stable,

meaning that errors in the input data and intermediate calculations do not

accumulate and cause the resulting output to be meaningless. There are many

ways of doing this, all with advantages and disadvantages. The Finite Element

Method is a good choice for solving partial differential equations over complex

domains (like cars and oil pipelines), when the domain changes (as during a

solid state reaction with a moving boundary), or when the desired precision

varies over the entire domain

The finite difference method (FDM) is an alternative way for solving PDEs.

The differences between FEM and FDM are:

• The finite difference method is an approximation to the differential

equation; the finite element method is an approximation to its solution.

• The most attractive feature of the FEM is its ability to handle complex

geometries (and boundaries) with relative ease. While FDM in its basic

form is restricted to handle rectangular shapes and simple alterations

thereof, the handling of geometries in FEM is theoretically

straightforward.

• The most attractive feature of finite differences is that it can be very easy

to implement.

• There are several ways one could consider the FDM a special case of the

FEM approach. One might choose basis functions as either piecewise

constant functions or Dirac delta functions. In both approaches, the

approximations are defined on the entire domain, but need not be

continuous. Alternatively, one might define the function on a discrete

domain, with the result that the continuous differential operator no longer

makes sense, however this approach is not FEM.

• There are reasons to consider the mathematical foundation of the finite

element approximation more sound, for instance, because the quality of

the approximation between grid points is poor in FDM.

• The quality of a FEM approximation is often higher than in the

corresponding FDM approach, but this is extremely problem dependent

and several examples to the contrary can be provided.

Generally, FEM is the method of choice in all types of analysis in structural

mechanics (i.e. solving for deformation and stresses in solid bodies or dynamics

of structures) while computational fluid dynamics (CFD) tends to use FDM or

other methods (e.g., finite volume method

Galerkin methods

In mathematics, in the area of numerical analysis, Galerkin methods are a

class of methods for converting an operator problems (such as a differential

equation) to a discrete problem. In principle, it is the equivalent of applying the

method of variation to a function space, by converting the equation to a weak

formulation. The approach is credited to the Russian mathematician Boris

Galerkin.

Rayleigh-Ritz method

In applied mathematics and mechanical engineering, the Rayleigh-Ritz method

is a widely used, classical method for the calculation of the natural vibration

frequency of a structure in the second or higher order. It is a direct variational

method in which the minimum of a functional defined on an normed linear

space is approximated by a linear combination elements from that space. This

method will yield solutions when an analytical form for the true solution may

be intractable.It is used for finding the approximate real resonant frequencies of

multi degree of freedom systems, such as spring mass systems or flywheels on a

shaft with varying cross section. It is an extension of Rayleigh's method. It can

also be used for finding buckling loads for columns, as well as more esoteric

uses.

FINITE ELEMENT ANALYSIS (FEA)

Is a computer simulation technique used in engineering analysis. It uses a

numerical technique called the finite element method (FEM). FEA consists of a

computer model of a material or design that is stressed and analyzed for specific

results. It is used in new product design, and existing product refinement. A

manufacturer is able to verify 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 utilized 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.

.The finite element analysis was first developed in 1943 by Richard Courant,

who used 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[1]

established a broader definition of

numerical analysis. Development of the finite element method in structural

mechanics is usually based on an energy principle such as the virtual work

principle or the minimum total potential energy principle. By the early 70's,

FEA was limited to expensive mainframe computers generally owned by the

aeronautics, automotive, defense, and nuclear industries. Since the rapid decline

in the cost of computers and the phenomenal increase in computing power, FEA

has been developed to an incredible precision. Present day supercomputers are

now able to produce accurate results for all kinds of parameters.

HOW DOES FINITE ELEMENT ANALYSIS WORK?

FEA uses a complex system of points called nodes which make a grid called a

mesh (Fig 1).This mesh is programmed to contain the material and structural

properties which define how the structure will react to certain loading

conditions. Nodes are assigned at a certain density throughout the material

depending on the anticipated stress levels of a particular area. Regions which

will receive large amounts of stress usually have a higher node density than

those which experience little or no stress. Points of interest may consist of:

fracture point of previously tested material, fillets, corners, complex detail, and

high stress areas. The mesh acts like a spider web in that from each node, there

extends a mesh element to each of the adjacent nodes. This web of vectors is

what carries the material properties to the

object, creating many elements.

Fig 1: Mesh created on a structure.

Finite element analysis

In general, there are three phases in any computer-aided engineering task:

• Pre-processing – defining the finite element model and environmental

factors to be applied to it

• Analysis solver – solution of finite element model

• Post-processing of results using visualization tools

There are multiple loading conditions which may be applied to a system.

• Point, pressure, thermal, gravity, and centrifugal static loads

• Thermal loads from solution of heat transfer analysis

• Enforced displacements

• Heat flux and convection

• Point, pressure and gravity dynamic loads

An element has to face all the above loading condition which can be

successfully solve by FEA

Each FEA program may come with an element library, which is constructed

over a period of time. Some sample elements are:

• Rod elements

• Beam elements

• Plate/Shell/Composite elements

• Shear panel

• Solid elements

• Spring elements

• Mass elements

• Rigid elements

• Viscous damping elements

Many FEA programs also are equipped with the capability to use multiple

materials within the structure such as:

• Isotropic, identical throughout

• Orthotropic, identical at 90 degrees

• General anisotropic, different throughout

In its applications, the object or system is represented by a geometrically similar

model consisting of multiple, linked, simplified representations of discrete

regions—i.e., finite elements on an unstructured grid. Equations of equilibrium,

in conjunction with applicable physical considerations such as compatibility

and constitutive relations, are applied to each element, and a system of

simultaneous equations is constructed. The system of equations is solved for

unknown values using the techniques of linear algebra or non-linear numerical

schemes, as appropriate. While being an approximate method, the accuracy of

the FEA method can be improved by refining the mesh in the model using more

elements and nodes.

A common use of FEA is for the determination of stresses and displacements in

mechanical objects and systems. However, it is also routinely used in the

analysis of many other types of problems, including those in heat transfer, solid

state diffusion and reactions with moving boundaries, fluid dynamics, and

electromagnetism. FEA is able to handle complex systems that defy closed-

form analytical solutions.

Pre-processing

The first step in using FEA, pre-processing, is constructing a finite element model of

the structure to be analyzed. The input of a topological description of the structure's

geometric features is required in most FEA packages This can be in either 1D, 2D,

or 3D form, modeled by line, shape, or surface representation, respectively, although

nowadays 3D models are predominantly used. The primary objective of the model is

to realistically replicate the important parameters and features of the real model. The

simplest mechanism to achieve modeling similarity in structural analysis is to utilize

pre-existing digital blueprints, design files, CAD models, and/or data by importing

that into an FEA environment. Once the finite element geometric model has been

created, a meshing procedure is used to define and break up the model into small

elements. In general, a finite element model is defined by a mesh network, which is

made up of the geometric arrangement of elements and nodes. Nodes represent

points at which features such as displacements are calculated. FEA packages use

node numbers to serve as an identification tool in viewing solutions in structures

such as deflections. Elements are bounded by sets of nodes, and define localized

mass and stiffness properties of the model. Elements are also defined by mesh

numbers, which allow references to be made to corresponding deflections or stresses

at specific model locations.

Analysis (computation of solution)

The next stage of the FEA process is analysis. The FEM conducts a series of

computational procedures involving applied forces, and the properties of the

elements which produce a model solution. Such a structural analysis allows the

determination of effects such as deformations, strains, and stresses which are

caused by applied structural loads such as force, pressure and gravity.

Post-processing (visualization)

These results can then be studied using visualization tools within the FEA

environment to view and to fully identify implications of the analysis.

Numerical and graphical tools allow the precise location of data such as stresses

and deflections to be identified.

TYPES OF ENGINEERING ANALYSIS

Structural analysis consists of linear and non-linear models. Linear models use

simple parameters and assume that the material is not plastically deformed.

Non-linear models consist of stressing the material past its elastic capabilities.

The stresses in the material then vary with the amount of deformation.

Vibrational analysis is used to test a material against random vibrations, shock,

and impact. Each of these incidences may act on the natural vibrational

frequency of the material which, in turn, may cause resonance and subsequent

failure.

Fatigue analysis helps designers to predict the life of a material or structure by

showing the effects of cyclic loading on the specimen. Such analysis can show

the areas where crack propagation is most likely to occur. Failure due to fatigue

may also show the damage tolerance of the material .

Heat Transfer analysis models the conductivity or thermal fluid dynamics of

the material or structure .This may consist of a steady-state or transient transfer.

Steady-state transfer refers to constant thermo-properties in the material that

yield linear heat diffusion.

Results of Finite Element Analysis

FEA has become a solution to the task of predicting failure due to unknown

stresses by showing problem areas in a material and allowing designers to see

all of the theoretical stresses within. This method of product design and testing

is far superior to the manufacturing costs which would accrue if each sample

was actually built and tested.

APPLICATIONS OF FEA TO THE MECHANICAL ENGINEERING

INDUSTRY

A variety of specializations under the umbrella of the mechanical engineering

discipline such as aeronautical, biomechanical, and automotive industries all

commonly use integrated FEA in design and development of their products.

Several modern FEA packages include specific components such as thermal,

electromagnetic, fluid, and structural working environments. In a structural

simulation FEA helps tremendously in producing stiffness and strength

visualizations and also in minimizing weight, materials, and costs. FEA allows

detailed visualization of where structures bend or twist, and indicates the

distribution of stresses and displacements. FEA software provides a wide range

of simulation options for controlling the complexity of both the modeling and

the analysis of a system. Similarly, the desired level of accuracy required and

the associated computational time requirements can be managed simultaneously

to address most engineering applications. FEA allows entire designs to be

constructed, refined, and optimized before the design is manufactured. This

powerful design tool has significantly improved both the standard of

engineering designs and the methodology of the design process in many

industrial applications. The introduction of FEA has substantially decreased the

time taken to take products from concept to the production line.] It is primarily

through improved initial prototype designs using FEA that testing and

development have been accelerated. In summary, the benefits of FEA include

increased accuracy, enhanced design and better insight into critical design

parameters, virtual prototyping, fewer hardware prototypes, a faster and less

expensive design cycle, increased productivity, and increased revenue.

COMPUTER-AIDED DESIGN AND FINITE ELEMENT ANALYSIS IN

INDUSTRY

The ability to model a structural system in 3D can provide a powerful and

accurate analysis of almost any structure. 3D models, in general, can be

produced using a range of common computer-aided design packages. Models

have the tendency to range largely in both complexity and in file format,

depending on 3D model creation software and the complexity of the model's

geometry. FEA is a growing industry in product design, analysis, and

development in engineering. The trend of utilizing FEA as an engineering tool

is growing rapidly. The advancement in computer processing power, FEA, and

modeling software has allowed the continued integration of FEA in the

engineering fields of product design and development. In the past, there have

been many issues restricting the performance and ultimately the acceptance and

utilization of FEA in conjunction with CAD in the product design and

development stages. The current trend in FEA software & industry in

engineering has been the increasing demand for integration between solid

modeling and FEA analysis.. Designers are now beginning to introduce

computer simulations capable of using pre-existing CAD files, without the need

to modify and re-create models to suit FEA environments.

Dynamic modeling

There is increasing demand for dynamic FEA modeling in the heavy vehicle

industry. Many heavy vehicle companies are moving away from traditional

static analysis and are employing dynamic simulation software. Dynamic

simulation involves applying FEA in a more realistic sense to take into account

the complicated effects of analyzing multiple components and assemblies with

real properties.

Modeling assemblies

Dynamic simulation, used in conjunction with assembly modeling, introduces

the need to fasten together components of different materials and geometries.

Therefore, CAE tools should have comprehensive capabilities to easily and

reliably model connectors, including joints that allow relative motion between

components, rivets, and welds. Typical MSS models are composed of rigid

bodies (wheels, axles, frame, engine, cab, and trailer) connected by idealized

joints and force elements. Joints and links may be modeled as either rigid links,

springs, or dampers in order to simulate the dynamic characteristics of real

truck components. Force transfer across assembly components through

connectors makes them susceptible to high stresses. It is simpler and easier to

idealize connectors as rigid links in these systems. This idealization provides a

basic study of assembly behavior in terms of understanding system

characteristics; engineers must model joining parameters like fasteners

accurately when performing stress analysis to determine how failures might

take place. "Representing connectors as rigid links assumes that connectors

transfer loads across components without deforming and undergoing stress

themselves.

Soft ware Features

• Captures design intent through 3D parametric definition and

technological attributes

• Directly defines the design's general shape with constrained features such

as cuboids, revolutions, extrusions and cylinders

• Prevents duplication of effort by making standardized parts available

• Locks in designated feature definitions to manage change

• Computes part modifications with associative features

• Stores features in a library using a defined catalog access path for easier

referencing

• Comes with a large catalog starter set of features

• Allows companies and their subcontractors to share the same standards

through the use of the catalog features

CURRENT MODELING TECHNIQUES IN INDUSTRY

Engineers at automobile design model using specialist dynamic FEA software.

Each model contains a flexible body and chassis, springs, roll bars, axles, cab

and engine suspension, the steering mechanism, and any frequency-dependent

components such as rubber mounts..Fig 2.

Fig 2: Engineering Modeling

Dynamic FEA simulation enables a variety of maneuvers to be accurately

tested. Tests such as steady-state cornering, roll-over testing, lane changing, J-

turns, vibration analysis, collisions, and straight-line braking can all be

conducted accurately using dynamic FEA. Non-linear and time-varying loads

allow engineers to perform advanced realistic FEA, enabling them to locate

critical operating conditions and determine performance characteristics. As a

result of the improved dynamic testing capabilities, engineers are able to

determine the ultimate performance characteristics of the vehicle's design

without having to take physical risks. As a result of dynamic FEA, the need for

expensive destructive testing has been lessened substantially.

Fig 3: Visualization of how a car deforms in an asymmetrical crash using finite

element analysis

VIRTUAL TESTING

Virtual Testing is Pira’s generic term for the application of finite element

analysis (FEA) or computational modelling technology to packaging design and

evaluation. FEA is commonplace within automotive, aerospace, defence and

other engineering sectors and selected applications are already in place or being

developed in the packaging sector:

� Metal cans design and formation

� Injection moulding

� Blow moulding and structural assessment of blown containers.

Mathematical modelling allows the rapid assessment of solutions based on a

range of design modifications, without the need for repetative soft tooling.

Common objectives include achieving weight savings, shape optimisation and

performance improvement.

Fig 4: Virtual testing

BENEFITS OF MODELLING

� Reducing time to market

� Avoiding loss of sales through slow development i.e. the process of

refining protoypes

� Scope to reduce/eliminate iterative sampling/tooling during development

� Significant ongoing savings on light weighting and re-engineering

COMMON FEA APPLICATIONS

• Simulation for Structural Failure

Conduct structural / thermal FEA studies on a component / assembly to

study reasons for failure in field. Redesign parts to avoid failure

Fig 5: Simulation for structural failure

• Simulation to Duplicate Testing Conditions

Create testing conditions in the FEA environment and check FEA results

with respect to previous test results. This creates a framework to carry

out future testing using FEA, thus saving the cost of testing.

• Simulation to Validate New Part Design

Simulate the field conditions for a new part / assembly design to validate

strength, durability in field.

• Analyzing joints and modeling sealing solutions

• Optimization

Optimize the design for weight / cost reduction, removal of redundant

components, etc. using static / dynamic FEA.

Fig 6: Optimization models

Biomedical/Bioengineering Visualization

CEI’s visualization products are used widely to analyze and present the results

of CFD and FEA simulations for a variety of biomedical and bioengineering

applications. Since many of these involve transient phenomena, the strong

animation creation and display features of our products are especially popular.

Recently, support for 2D textures has been added, allowing greater realism to

your results, important in a field of science unfamiliar with simulations.

Fig 7: Application in biomedical engineering

Use of FEA for coating

Electroplating is an essential part in the finishing process of manufactured products, but

more often than not, costly trial-and-error, grinding and polishing methods are

employed to achieve uniformity. FEA simulation of this process is a way to solve this

problem. This is achieved by mapping the current density field using finite element

methods to determine the amount deposited on the discrete parts of the component

Fig 8: Application in coating

CONCLUSION

The FEA uses computer simulation tech in Engg Analyis which has reduced the

economy of product, time to market and various effert to making sample and

practically testing on testbench and field. It short it has improved the

development speed of an product by reducing interactive sampling, tooling and

re-engineering work. Hence use of FEA is cost of the product, increase the

speed as development of product and improve the quality as product.