Solution of engineering problems

ME6603 – Finite Element Analysis

INTRODUCTION

ME 6603 FEA Erode Sengunthar Engineering College

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Structural Problems◦ Stress analysis of beam, Fatigue analysis of

mechanical component, Modal analysis of beams, etc.

Thermal problems◦ Heat transfer analysis of plate, temperature

distribution in fin, Heat flux distribution in a object Computational Fluid Dynamics Problems

- High resolution predictions that hold across a large range of flow conditions, motion for the dispersed phase, shock capturing, simulation of turbulent flows

Engineering Problems


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AnalyticalMethod

Classical approach

100 % accurate

Closed form solution

Complete in itself

Methods for solving Engineering problems

NumericalMethod

•Mathematical representation•Approximate assumptions•Results must be verified experimentally

Experimental method

• Correct procedure

• Time consuming

• Expensive• Physical

prototype required (3-5 protype reqd.)


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Methods for solving Engineering problems (contd.)

AnalyticalMethod

• Provides Closed form solution

• Example: Theory of Bending

NumericalMethod

• Examples - FAM - FEM - FDM - FVM - BEM

Experimental method

• Examples- Strain gauge

measurements- Photoelasticity

analysis- Vibration

measurement- Sensors for

temperature measurement

- Fatigue test


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An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally accepted set.

A closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions but e.g. usually no limit.

Closed form solution


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Continuum Modeling Discrete Modelling

TYPES OF MODELLING

Continuum Modeling Continuum mechanics is a branch of mechanics that

deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles.

Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the particular material studied is added through constitutive relations.

7ME 6603 FEA Erode Sengunthar

Engineering College


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Model obtained using finite number of well defined components

Number of elements is very large Overcomes the intractability of realistic

types of continuum problems

Discrete Modeling


Types- Functional Approximation method

(FAM) - Finite Element Method (FEM)

- Finite Difference Method (FDM) - Boundary Element Method (BEM) - Finite Volume Method (FVM)

Numerical methods

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It can be applied for linear, non linear and continuum problems

Problems mentioned in terms of differential equations or mathematical expressions are solved

The whole system called the domain is expressed by differential equations

Examples : Rayleigh Ritz Methods, Weighted Residual Method

Functional Approximation Method (FAM)


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Nature of problems- Equilibrium problem or boundary value problems- Eigen Value Problem - Propagation problems or initial value problems

In Equilibrium Problems, the domain is closed and the boundary conditions are prescribed around the entire boundary. They are expressed as elliptic equations

Functional Approximation methods


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Finding a function to describe the temperature of this idealised 2D rod is a boundary value problem

Equilibrium problem or Boundary value problems


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Eigen Value Problem are the special problems where solution exists only for a special values of a parameter of the problem.- Mechanical vibration problems are eigen value problems

EIGEN VALUE PROBLEM


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The Eigen values provide the natural frequencies of the system. The eigenvectors represent the mode shapes of the system.

The solution of an Eigenvalue problem can be quite cumbersome (especially for problems with many degrees of freedom), but fortunately most math analysis programs have Eigen value routines

EIGEN VALUE PROBLEM

https://en.wikipedia.org/wiki/Mode_shape


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Propagation problems are intial value problems in open domain in which the solution in the domain of interest.

Propagation problems are governed by parabolic or hyperbolic PDE’s

Propagation problems are to predict the subsequent stresses or deformation states of a system under the time-varying loading and deformation states.

It is called initialvalue problems in mathematics or disturbance transmissions in wave propagation.

Propagation problems


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Comparison of Engg. Analysis


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Numerical method with unknown functions of the problem domain is approximated by piecewise defined functions

Complex regions defining the domain is divided into smaller elements called finite elements

Physical properties like shape, dimensions and other boundary conditions are imposed

The elements are assembled in a proper way and the solution for the entire system can be revealed.

Finite Element Method (FEM)


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Approximated the derivatives in the governing differential equation using difference equations

FDM replaces derivative terms in the differential equations by the difference equivalents

Used for solving heat transfer and fluid mechanics problems

Method cannot be used effectively for regions having curved and irregular boundaries

Finite Difference Method (FDM)


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Unit Volume is considered as finite volume Variable properties such as Pressure,

Velocity, Area, Mass, etc. can be assessed. Based on Navier Stokes Equation (Mass,

Momentum and energy conservation equilibrium equations)

Computational fluid Dynamics (CFD) problems are based on FVM

Finite Volume Method (FVM)


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The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations

The boundary element method is often more efficient than other methods, including finite elements, in terms of computational resources for problems where there is a small surface/volume ratio

Solves Acoustics or Noise vibration Harshness problems

Solving the problem faster Reduces the dimensionality of the problem

BOUNDARY ELEMENT METHOD (BEM)


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1906 – Civil engineering problems for structure analysed for 1 D problems

1909 – Ritz Variational method (FAM)1915 – Galerikin Weighted Residual methods (FAM)1940 – Courant , Pragger and Synge Mathematical

foundation for present form of FEA1941 – Hreinkoff solution for elasticity problems “Frame

work method”1943 – Piecewise polynomial interpolation over triangular

elements (FEA)

History of Numerical methods


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1950’s – Argyris, Kelsery, Turner: Direct Continuum elements, Aerospace industry engineers formulated Stiffness problems

1956 – Turner derived stiffness matrix for beam, truss and other elements

1960 – FAM used for stress analysis, fluid flow, heat transfer problems and other areas

1967 – First FEA book published by Zienkiewicz and Chung

1972 – Oden’s book on non-linear problems (Development of main frames appeared)1980 – Graphical and computational development1990’s – Emergence of low cost powerful PC

workstations and FEA adopted by mid and small scale industries

History of Numerical methods (Contd.)

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Engineering College

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Solution of engineering problems

Engineering

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