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EME3156 COMPUTATIONAL METHODS FOR MECHANICAL ENGINEERING CHAPTER 1

Trimester 2, 2015/2016 1

1. Introduction

This chapter begins with the review of differential equations, physical applications of

the equations in various fields of study, the boundary and initial conditions, the

numerical methods in solving these equations, and the software packages.

1.1 Differential equations

Differential equations are equations involving the derivatives of functions. They are

broadly classified into two: ordinary and partial. Ordinary differential equations (ODE)

contain derivatives of a dependent variable with respect to one independent variable.

For example, ifyis the dependent variable and xis the independent variable, an ODE

takes the form of Eq. (1.1):

(1.1)

Usually, integration is performed to obtain y as a function of x. The difficulty of the

integration depends on the function yxf , . Specifically, iffis only a function of xin

the form of a polynomial, the basic integration technique is adequate. For example:

(1.2)

In Eq. (1.2),Cis a constant that can be obtained after applying the additional condition

of the problem. This additional condition is named the boundary condition or the initial

condition depending on the physical representation of the problem, and will be

elaborated in Sec. 1.3.

Eq. (1.1) is named the ______________________, because the derivative is of the first

order. There are higher orders ODE, like second order, third order, etc. The higher order

ODE may contain lower order of derivatives in the equation, For example,

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yxyy 2 is the second order ODE containing the first derivative, and also f is

now a function ofxandy.

When 0,

yxf , the ODE is named ________________________ ODE. For example,

0 yy is the second order homogeneous ODE. Conversely, an ODE with

0, yxf is named non-homogeneous ODE.

When the index of the derivatives is not equal to one, the ODE is a ________________

ODE. For example, 122 xy . A non-linear ODE is also obtained when the

derivatives multiply with each other, for example 12 xyy .

On the other hand, a partial differential equation (PDE) contains the partial derivatives

of one dependent variable with respect to several independent variables. For example, in

Eq. (1.3), uis the dependent variable whilexandyare the independent variables.

(1.3)

As in ODE, PDE are also classified into first order, second order, etc. Among the

second order PDE, there is another classification depending on the equation. To explain

this classification, Eq. (1.4) shows the general equation of a second order PDE.

(1.4)

When 042 ACB , it is an elliptic PDE.

When 042 ACB , it is a parabolic PDE.

When 042 ACB , it is a hyperbolic PDE.

1.2 Physical applications of differential equations

In Eq. (1.1), the independent variable is denoted by the symbol x. Ifxrepresents time, it

is usually changed to t, thus the derivative is now written as _______, meaning the

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change of variable y with respect to time. Specifically, if y represents velocity, it is

changed to v, thus the derivative is now written as dtdv , meaning the change of

velocity with respect to time, or simply acceleration. This derivative may appear in the

equation representing the fall of an object with air resistance: kvgdtdv

, wheregis

the acceleration due to gravity, and kis a constant related to air resistance. This equation

is solved to get the velocity of the object as a function of time.

In another example, if y represents the mass of matter, it is changed to m, thus the

derivative is now written as dtdm , meaning the change of mass with respect to time.

This derivative usually appears in the nuclear decay equation: kmdtdm , where kis

a constant related to the rate of decay of a radioactive matter. This equation is solved to

get the mass of the matter as a function of time.

An application of the second order ODE is in a mass-damper-spring oscillation system,

written in the form: ___________________, where mis the mass of the system, cis the

damping coefficient and kis the spring stiffness. Sinceyrepresents the displacement of

the mass, thus y and y represents acceleration and velocity of the mass respectively.

This equation is solved to get the displacement of the mass as a function of time.

The first order PDE takes the form 0yx

vu , which represents the continuity

equation in a fluid flow. There are two dependent variables in this equation: uand v,

representing the x-component and y-component of the velocity respectively. Knowing

that uand vcannot be solved by just having one equation, another equation is required

to complement the problem, named the momentum equation (a second order PDE):

Xuudxdpvuuu yyxxyx . The solution of uand vare usually used to

solve another related equation named the energy equation:

XTTkvTuTC yyxxyx , where Tis the temperature of the domain. These three

equations (continuity, momentum and energy) frequently appear in fluid flow and heat

transfer problems.

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The second order PDE shown in Eq. (1.4) is the most common among all types of PDE,

thus its physical application will be discussed as follows, and the corresponding

numerical methods used to analyse this type of PDE will be explained in later chapters.

A physical application of the elliptic PDE is the two-dimensional steady state heat

conduction, and the PDE usually takes the form fuu yyxx , where u represents the

temperature and f is the heat generation. The same equation can also be used to

represent the torsion of a bar with a non-circular cross-section, where unow represents

the stress andfis the angle of twist. In both examples, xandyare the coordinate system

of the domain.

The parabolic PDE takes the formtxx

uu , and typically appears in one-dimensional

transient heat conduction. Thus, x and t represent the coordinate system and time

respectively.

A hyperbolic PDE takes the formxxtt

uu , usually appears in one-dimensional wave

equation. As in the parabolic PDE, x and t represent the coordinate system and time

respectively. However, unow represents the displacement of the wave.

1.3 Boundary and initial conditions

As shown in Eq. (1.2), the solution of a differential equation is not complete without

stating the additional conditions. Mathematically speaking, the value of C must be

obtained to complete the solution process.

Generally, the number of conditions required to complete the solution process depends

on the order of the differential equation. For an ODE, the number of conditions is equal

to the order of the equation, meaning that first order ODE requires one condition,

second order two conditions, etc.

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For first order ODE, when the independent variable is time, this additional condition is

usually specified at the beginning of the time (t = 0), hence it is also named the

_____________________ (IC). Since almost all first order ODE is having time as the

independent variable, thus a first order ODE is also named the initial-value problem.

Using Eq. (1.2) as an example, the complete formulation can now be written as follows:

(1.5) 12 tdt

dy; IC: 2,0 yt or in short ___________

A second order ODE needs two conditions to completely solve the problem. Usually,

these conditions are specified at the boundary of the problem (the beginning and the end

of the independent variable). Hence this type of problem is also named the boundary

value problem, and the conditions are named the ______________________. An

example of complete statement of the boundary value problem is given in Eq. (1.6),

where the boundary is located at 0x and 1x .

(1.6) yxdx

dy

dx

yd

2

2

2

BC: 2,0 yx and 10,1 yx or in short ______________________

1.4 The meaning of domain and boundary

Before specifying the boundary conditions, it is important to understand the physical

location and meaning of the boundary of the problem, so that the BC is completely and

correctly stated to represent the physical problem. The boundary relates closely with the

physical shape of the domain. The meaning of boundary will be explained in terms of

the shape of the domain.

From the shape point of view, since this is a three-dimensional world, all objects

generally have a three-dimensional shape. For example, a cube has width, height and

depth. The boundary of the cube (three-dimensional shape) is the exterior surfaces of

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the cube. When one of the dimensions are negligible compared to the other two (like a

piece of paper), the shape is two-dimensional. In this case, the boundary of the paper is

the edge of the paper. If another dimension is reduced, it becomes a one-dimensional

shape, like a long string. The boundary of the string is the end points of the string.

Finally, there is the zero-dimension where the shape is simply one solitary point.

1.5 Non-dimensional form of differential equations

Consider a one-dimensional heat transfer problem in Eq. (1.7), where u is the

temperature with the unit K or C; x is the distance with unit metres; Q is a constant

related to heat source with unit K/m2.

(1.7) 02

2

Qdx

ud, 00 u , 1002 u

The solution of this problem is only valid for the specifications mentioned above. If the

domain is changed to five metres long, theoretically the solution procedure needs to be

repeated. However, there is a generalised method to solve this governing equation that

can represent a general solution for the same type of problem.

Firstly, these variables can be non-dimenionalised by multiplying with constants. For

example, the non-dimensional form ofxcan be achieved by dividingxwith the lengthL

of the domain, and given a new symbolX. Similarly, the temperature ucan also be non-

dimensionalised by dividing with a constant Tref representing a reference temperature,

thus a given a new symbol . Eq. (1.8) is the non-dimensional form of (1.7):

(1.8)

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After solving for , the actual temperature ucan be obtained using the relation in Eq.

(1.8). The non-dimensionalisation of governing equations can also be done for higher

order ODE and PDE.

1.6 Engineering software

There are numerous commercial software that solve engineering problems. As in any

computer program, the user interface and data entry must be user friendly and easy to

use, and concentrate on the presentation of the physics of the problem more than the

mathematical operations in solving the problem. Therefore, these software do not

usually require the user to specify the type of equations to be solved; rather the physical

representation of the problem (for example fluid flow, heat transfer, structural,

vibration, etc). After specifying the type of problem, the software enters the input mode

where the geometry, material properties, meshing, boundary or initial conditions are

inserted. In each of the steps above, the software runs background operations like

transforming the parameters into mathematical equations. Finally, the results are

presented in graphical form including the use of colours to represent the values of the

dependent variables.

The role of an engineer does not stop at running the software and printing the results.

An engineer should also discuss the results obtained in terms of the accuracy and the

reliability of the results. Ultimately, the engineer should judge if the problem has been

accurately solved and the results can be used for further improvements or taken as the

final values for design and fabrication purposes. These additional steps require technical

knowledge on the mathematical (or numerical) analysis of the software, or the

knowledge on the physics of the problem.

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