finite element chapter 1

8/13/2019 Finite Element Chapter 1

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INTRODUCTION TO FINITE ELEMNT METHOD

1. INTRODUCTION:

The finite element method is a numerical method for solving problems of engineering and

mathematical physics.Typical problem areas of interest:

Structural analysis, heat transfer, fluid flow, mass transport, and Electromagnetic

potential

Useful for problems with complicated geometries, loadings, and material properties where

analytical solutions can not be obtained.

The finite element formulation of the problem results in a system of simultaneous algebraic

equations for solution, rather than requiring the solution of differential equations.

Process of modeling a body by dividing it into an equivalent system of smaller bodies or

units (finite elements) interconnected at points common to two or more elements (nodal

points or nodes) and/or boundary lines and/or surfaces is called discretization.

The solution for structural problems typically refers to determining the displacements ateach nodeand the stresses within each element making up the structure that is subjected

to applied loads.

In nonstructural problems, the nodal unknowns may, for instance, be temperatures or

fluid pressuresdue to thermal or fluid fluxes.


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Brief History:

Grew out of aerospace industry

Post-WW II jets, missiles, space flight Need for light weight structures

Required accurate stress analysis

Paralleled growth of computers

Common FEA Applications

Mechanical/Aerospace/Civil/Automotive

Engineering

Structural/Stress Analysis

Static/Dynamic

Linear/Nonlinear

Fluid Flow

Heat Transfer

Electromagnetic Fields

Soil Mechanics

Acoustics

Biomechanics


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General Steps In Finite Element method Analysis:

There are two general direct approaches traditionally associated with the finite element

method as applied to structural mechanics problems.

One approach, called the force, or flexibility, method, uses internal forces as the

unknowns of the problem. The second approach, called the displacement, or stiffness, method, assumes the

displacements of the nodes as the unknowns of the problem.

For computational purposes, the displacement (or stiffness) method is more desirable

because its formulation is simpler for most structural analysis problems.

1. Discretization:The first step involves dividing the body into an equivalent system of finite elements with

associated nodes and choosing the most appropriate element type to model most closely the

actual physical behavior.

Model body by dividing it into an equivalent system of many smaller bodies or units (finite

elements) interconnected at points common to two or more elements (nodes or nodalpoints) and/or boundary lines and/or surfaces.


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Feature

Obtain a set of algebraic equations to solve for unknown (first) nodal quantity

(displacement).

Secondary quantities (stresses and strains) are expressed in terms of nodal values of

primary quantity.


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Types of Elements:

The primary line elements consist of bar (or truss) and beam elements (1D). They have a

cross-sectional area but are usually represented by line segments. In general, the cross-

sectional area within the element can vary, but it will be considered to be constant for the

sake of this section. These elements are often used to model trusses and frame structures. The simplest line

element (called a linear element) has two nodes, one at each end.

The basic two-dimensional (or plane) elements are loaded by forces in their own plane

(plane stress or plane strain conditions). They are triangular or quadrilateral elements.

The simplest two-dimensional elements have corner nodes only (linear elements) withstraight sides or boundaries The elements can have variable thicknesses throughout or be

constant.

The most common three-dimensional elements are tetrahedral and hexahedral (or

brick) elements; they are used when it becomes necessary to perform a three-

dimensional stress analysis. The basic three-dimensional elements have corner nodes

only and straight sides.


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FEM FORMULATION PROCEDURE:


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Principles of FEA:

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.

Boundary value problems are also called f ield problems. The fieldis the domain of interest and most often represents a physicalstructure.

The f ield var iables are the dependent variables of interest governed

by the differential equation.

The boundary conditions are the specified values of the fieldvariables (or related variables such as derivatives) on the

boundaries of the field.

For simplicity, at this point, we assume a two-dimensional casewith a single field variable (x, y) to be determined at every point

P(x, y) such that a known governing equation (or equations) issatisfied exactly at every such point.


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A finite element is not a differential element of size dx dy.

A node is a specific point in the finite element at which the value of the field variable is

to be explicitly calculated.


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h h i l d h i l


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The Mathematical and Physical FEMMathematical FEM:

The centerpiece in the process steps of the Mathematical FEM is the

mathematical model which is often an ordinary or partial differential equation in

space and time.

A discrete finite element model is generated from of the mathematical model.

The resulting FEM equations are processed by an equation solver, which

provides a discrete solution.

FEMdiscretization may be constructed without any reference to physics. The concept of error arises when the discrete solution is substituted in the

mathematical and discrete models.

This replacement is generically called verification.

The solution error is the amount by which the discrete solution fails to satisfy

the discrete equations. This error is relatively unimportant when usingcomputers. More relevant is the discretization error, which is the amount by

which the discrete solution fails to satisfy the mathematical model.


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PHYSICAL FEM:

The processes of idealization and discretization are carried out concurrently to

produce the discrete model.

Indeed FEMdiscretization may be constructed and adjusted without reference to

mathematical models, simply from experimental measurements.

The concept of error arises in the physical FEM in two ways, known as

verification and validation.

Verification is the same as in the Mathematical FEM: the discrete solution isreplaced into the discrete model to get the solution error. As noted above, this

error is not generally important.

Validation tries to compare the discrete solution against observation by

computing the simulation error, which combines modeling and solution errors.

Since the latter is typically insignificant, the simulation error in practice can be

identified with the modeling error.

Comparing the discrete solution with the ideal physical system would in

principle quantify the modeling errors.


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finite element chapter 1

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