ch03 continuous systems-1d

Introduction to Continuous Systemsand

One-Dimensional Systems

Jayadeep U. B.

M.E.D., NIT Calicut

2

Department of Mechanical Engineering, National Institute of Technology Calicut

Introduction

Continuous distribution of matter as well as loads – Components or Elements are not obvious – Discretization Procedure is necessary.

What are Continuous Systems?

Instead of a finite number of state variables, we have a continuous variation of the primary (& secondary) variables – could be a function of spatial coordinates & time – Infinite Degrees of Freedom (d.o.f.).

Leads to P.D.E.’s, while we get system of algebraic equations in Discrete systems – May not be possible to solve exactly.

For solution, we need to convert them into a system of algebraic equations (numerically!) – Conversion of Infinite d.o.f. system to a finite d.o.f. system.

FEM is one such numerical procedure – Much efficient and versatile in doing it!

Lecture - 01

3


Introduction

Static Stress Analysis of Pressure Vessels.

Examples of Continuous Systems:

Heat Transfer Analysis of I.C. Engine Head Block.

Transient Stress & Heat Transfer Analysis of Rolling Process.

Frequency Analysis of Centrifugal Impeller

Lecture - 01

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1D Continuous Systems

The geometry and variables (both primary and secondary) are assumed to be dependent only on one coordinate.

We will have a mathematical model (P.D.E.), with only one independent variable.

1D Steady-state heat transfer.

qin qout

A vertical column under self weight

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Considering steady state heat conduction equation with heat generation per unit length “Q”, heat balance gives:

xq A x x

dq A q A dx

dx

dx

0

x x x

x

dq A q A q A dx Q dx

dx

dq A Q

dx

Example 1: Steady State 1D Heat Transfer

Lecture - 01

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Using the heat conduction equation:

0

x

dq k

dxd d

k A Qdx dx

Boundary Condition 1: Dirichlet Boundary Condition

0 on

In the above problem, it becomes:

at 0 or (end points of domain)x x L

As a general convention, denotes the boundary,

while denotes the domain of the problem.

Note :

Example 1: Steady State 1D Heat Transfer contd. …

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Example 1: Steady State 1D Heat Transfer contd. …

Boundary Condition 2: Neumann Boundary Condition

0 onn qq q

In the above problem, it becomes:

0 at 0 or (end points of domain)d

k q x x Ldx

In general, find the distribution of Φ = Φ(x), which obey the (Partial) differential equation inside the domain, while satisfying the boundary conditions.

To solve this problem means find temperature profile Φ = Φ(x), which obey the differential equation inside the domain, while satisfying Dirichlet or Neumann boundary conditions as specified in the problem.

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Example 2: 1D Problem in Static Stress Analysis

If “b” gives the body force per unit length, Force Equilibrium in x-direction gives:

x A x x

dA A dx

dx

dx

0

( ) 0

x x x

x

dA A A dx b dx

dx

dA b

dx

b dx

Lecture - 01

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Example 2: 1D Problem in Static Stress Analysis contd. …

Using relation between stress and displacement:

0

x

duE

dxd du

EA bdx dx

Boundary Condition 1: Dirichlet Boundary Condition

0 on (at 0 or )uu u x x L

Boundary Condition 2: Neumann Boundary Condition

0 on (or )

0 at 0 or

n s

duE x x L

dx

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Example 2: 1D Problem in Static Stress Analysis contd. …

To solve this problem means find displacement function u = u(x), which obey the differential equation inside the domain, while satisfying Dirichlet or Neumann boundary conditions as specified in the problem.

Home Work: Formulate the 1D Heat Transfer problem, considering the convective heat transfer from the surface (Hint: Assume the surface area per unit length as a function of “x” and constant coefficient of convection).

If we assume that there is no heat generation in the domain, this becomes similar to the problem of heat transfer through a fin.

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Second Order Differential Equation of a Single Variable

General form:0 for 0

d dua cu q x L

dx dx

Many other problems lead to this type of equation:

Transverse deflection of a cable under tension with distributed transverse load.

Flow through pipe (Hagen-Poiseuille Flow)

Flow through porous media

Electric flux in a dielectric medium

Refer the book, “An Introduction to the Finite Element Method”, by J. N. Reddy for more details (Page no. 71, Table 3.2 in Second Edition)

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A Finite Difference Scheme: Central Difference MethodConsidering the 1D heat transfer equation with the simplification that “k” is constant:

2

20

dk Q

dx

Assuming that the domain (line) is divided into a number (N) of small intervals of equal size Δx, we have the approximation for first derivative at xl:

x0 = 0 x1 x2 xl-1 xl xl+1 xN-1 xN = L

1 1

2l

l l

x x

d

dx x

Similarly, the approximation for second derivative is:2

1 12 2

2

l

l l l

x x

d

dx x

Lecture - 01

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Central Difference Method in 1D Heat Transfer contd. …

A similar equation arises at each of the interior grid points. Considering, the Φl’s as the unknowns & assuming Dirichlet B.C., we get the system of equations:

21

2 1 0

22

3 2 1

2

1 1

21

1 2

2

2

2

2

ll l l

NN N N

x Q

kx Q

k

x Q

k

x Q

k

Substituting in the D.E. gives:

1 12

20l l l

lk Qx

Lecture - 01

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Writing as a Matrix system:

2

1 0

2

21 2 1

2

1

{ } { }2 1 0 0

1 2 1 0

0 0 1 2

{ } ( , , , ) and { }TN

N N

K f

K

xQ

k

xQ

f k

xQ

k

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However, there is a significant difference between FDM and FEM – In FEM we approximate the functions of primary variables and not the definition of derivatives as in FDM. Functions used for approximation should be simple enough, so that we can exactly calculate the derivatives of it.

This Linear Algebraic System can be solved for the unknown primary variables – The Temperatures at the Inner Nodes.

Refer Chapter 1 of the book “Finite Elements and Approximation” by Zienkiewicz and Morgan for more details and about enforcing the Neumann Boundary Conditions.

Lecture - 01

ch03 continuous systems-1d

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