Project report subject numerical analysis

Young’s modulus compression i.e., negative tension, is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied of a materia

VIET NAM NATIONAL UNIVERSITY, HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

™ ™ 

PROJEC T

REPORT Subject: Numerical Analysis

Lecturer teacher Mr Le Thanh Long :

Project A – Plan 4

Group Information :

2 Members:

1 Trần Nguyên Bảo Student ID: 2052880

2 Nguyễn Việt Khoa Student ID: 2052540

3 Nguyễn Trí Hào Student ID: 2153328

4 Nguyễn Ngọc Phước Tấn Student ID: 2012020

Table of Contents

I Introduction

II Theory

III

IV

V

VI

I Introduction

First of all, the group would like to sincerely thank lecturer Le Thanh Long, who created conditions for the group to have the opportunity to train themselves and master their knowledge through projects

With the development of today's world, the trend of working and operating through computers has become one of the main techniques of science and technology Instead of having to solve problems manually, Numerical Method looks at how to solve problems based on given numerical data and gives the same result faster and more accurately That is also the goal of the subject Numerical Analysis, which is currently taught in technical universities

This project is based on the Numerical Analysis curriculum taught at Ho Chi Minh City University of Technology It includes theoretical basis, a number of exercises and solutions that we summarize and present in a concise but complete way of core concepts It helps students practice skills to synthesize learned knowledge, teamwork skills Thereby, each individual can exchange, consolidate knowledge, practice self-control and sense of responsibility at work

II Theory

1 Young’s modulus

compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied

of a material and is determined using the formula:

Ε =

Young's moduli are typically so large that they are expressed not in pascals but in gigapascals (GPa)

1.1 Usage

Young's modulus enables the calculation of the change in the dimension of a bar

made of an isotropic elastic material under tensile or compressive loads For instance,

it predicts how much a material sample extends under tension or shortens under compression The Young's modulus directly applies to cases of uniaxial stress; that is, tensile or compressive stress in one direction and no stress in the other directions Young's modulus is also used in order to predict the deflection that will occur in a statically determinate beam when a load is applied at a point in between the beam's supports

Other elastic calculations usually require the use of one additional elastic property,

these parameters are sufficient to fully describe elasticity in an isotropic material For homogeneous isotropic materials simple relations exist between elastic constants that allow calculating them all as long as two are known:

Ε = 2G( 1 + v) = 3Κ( 1 – 2v)

1.2 Linear versus non-linear

Young's modulus represents the factor of proportionality in Hooke's law, which

relates the stress and the strain However, Hooke's law is only valid under the assumption of an elastic and linear response Any real material will eventually fail and break when stretched over a very large distance or with a very large force; however, all solid materials exhibit nearly Hookean behavior for small enough strains

or stresses If the range over which Hooke's law is valid is large enough compared to the typical stress that one expects to apply to the material, the material is said to be linear Otherwise, (if the typical stress one would apply is outside the linear range) the material is said to be non-linear

Steel, carbon fiber and glass among others are usually considered linear materials, while other materials such as rubber and soils are non-linear However, this is not an absolute classification: if very small stresses or strains are applied to a non-linear material, the response will be linear, but if very high stress or strain is applied to a linear material, the linear theory will not be enough For example, as the linear theory implies reversibility, it would be absurd to use the linear theory to describe the failure

of a steel bridge under a high load; although steel is a linear material for most applications, it is not in such a case of catastrophic failure

In solid mechanics, the slope of the stress–strain curve at any point is called the tangent modulus It can be experimentally determined from the slope of a stress– strain curve created during tensile tests conducted on a sample of the material

1.3 Directional materials

Young's modulus is not always the same in all orientations of a material Most

metals and ceramics, along with many other materials, are isotropic, and their mechanical properties are the same in all orientations However, metals and ceramics can be treated with certain impurities, and metals can be mechanically worked to make their grain structures directional These materials then become anisotropic, and Young's modulus will change depending on the direction of the force vector.Anisotropy can be seen in many composites as well For example, carbon fiber has a much higher Young's modulus (is much stiffer) when force is loaded parallel to the fibers (along the grain) Other such materials include wood and reinforced concrete Engineers can use this directional phenomenon to their advantage in creating structures

1.4 Temperature dependence

The Young's modulus of metals varies with the temperature and can be realized

through the change in the interatomic bonding of the atoms, and hence its change is found to be dependent on the change in the work function of the metal Although classically, this change is predicted through fitting and without a clear underlying mechanism (for example, the Watchman's formula), the Rahemi-Li model[4] demonstrates how the change in the electron work function leads to change in the Young's modulus of metals and predicts this variation with calculable parameters, using the generalization of the Lennard-Jones potential to solids In general, as the temperature increases, the Young's modulus decreases via Ε(Τ) = β where the electron work function varies with the temperature as φ(Τ) = and γ is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC) is the electron work function at T=0 and β is constant throughout the change

1.4 Calculation

� engineering extensional strain, ε, in the elastic (initial, linear) portion of the physical stress–strain curve:

Ε = = =

Where

Ε is the Young's modulus (modulus of elasticity)

F is the force exerted on an object under tension

A is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force

∆L is the amount by which the length of the object changes (∆L is positive if the material is stretched, and negative when the material is compressed)

is the original length of the object

1.4.1 Force exerted by stretched or contracted material

The Young's modulus of a material can be used to calculate the force it exerts

under specific strain

F =

where F is the force exerted by the material when contracted or stretched by ∆L Hooke's law for a stretched wire can be derived from this formula:

F = () ∆L = kx

where it comes in saturation

k ≡ and x ≡ ∆L

But note that the elasticity of coiled springs comes from shear modulus, not Young's modulus

1.4.2 Elastic potential energy

The elastic potential energy stored in a linear elastic material is given by the

integral of the Hooke's law:

= =

now by explicating the intensive variables:

= = =

This means that the elastic potential energy density (that is, per unit volume) is given by:

=

or, in simple notation, for a linear elastic material: = = , since the strain is defined ε

≡

In a nonlinear elastic material the Young's modulus is a function of the strain, so the second equivalence no longer holds, and the elastic energy is not a quadratic function

of the strain:

= ≠ E

2 Cross section

2.1 Definition

If a plane intersects a solid (a 3-dimensional object), then the region common to the plane and the solid is called a cross-section of the solid A plane containing a

cross-section of the solid may be referred to as a cutting plane

The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid For instance, while all the cross-sections of a ball are disks, the cross-sections of a cube depend on how the cutting plane is related to the cube If the cutting plane is perpendicular to a line joining the centers of two opposite faces of the cube, the cross-section will be a square, however, if the cutting plane is perpendicular to a diagonal of the cube joining opposite vertices, the cross-section can

be either a point, a triangle or a hexagon

2.2 Usage

The cross-sectional area (A’) of an object when viewed from a particular angle is

the total area of the orthographic projection of the object from that angle For example, a cylinder of height h and radius r has A’ = π when viewed along its central axis, and A’ = 2rh when viewed from an orthogonal direction A sphere of radius r has A’ = π when viewed from any angle More generically, A’ can be calculated by evaluating the following surface integral:

A’ = ,

Where is the unit vector pointing along the viewing direction toward the viewer, dA

is a surface element with an outward-pointing normal, and the integral is taken only over the top-most surface, that part of the surface that is "visible" from the perspective of the viewer For a convex body, each ray through the object from the viewer's perspective crosses just two surfaces For such objects, the integral may be taken over the entire surface (A) by taking the absolute value of the integrand (so that the "top" and "bottom" of the object do not subtract away, as would be required by the Divergence Theorem applied to the constant vector field ) and dividing by two:

A’ =

2.3 Examples in science

In geology, the structure of the interior of a planet is often illustrated using a

diagram of a cross-section of the planet that passes through the planet's center, as in the cross-section of Earth at right

Cross-sections are often used in anatomy to illustrate the inner structure of an organ,

as shown at the left

A cross-section of a tree trunk, as shown at left, reveals growth rings that can be used

to find the age of the tree and the temporal properties of its environment

3 Finite difference method

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences Both the spatial domain and time interval (if applicable) are discretized,

or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points

Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be nonlinear, into a system of linear equations that can be solved by matrix algebra techniques Modern computers can perform these linear algebra computations efficiently which, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis.[1] Today, FDM are one of the most common approaches to the numerical solution of PDE, along with finite element methods

3.1 Derivation from Taylor's polynomial

First, assuming the function whose derivatives are to be approximated is properly

behaved, by Taylor's theorem, we can create a Taylor series expansion

ƒ( = ƒ() + + + … + + ,

where n! denotes the factorial of n, and is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function We will derive

an approximation for the first derivative of the function "f" by first truncating the Taylor polynomial:

ƒ(+h) = ƒ() + ƒ’()h + ,

Setting, = a, we have:

ƒ(a + h) = ƒ(a) + ƒ’(a) h +

Dividing across by h gives:

= + ƒ’(a) +

Solving for f'(a):

ƒ’(a) =

Assuming that is sufficiently small, the approximation of the first derivative of "f" is: ƒ’(a) ≈

This is, not coincidentally, similar to the definition of derivative, which is given as: ƒ’(a) =

except for the limit towards zero (the method is named after this)

3.2 Accuracy and order

The error in a method's solution is defined as the difference between the

approximation and the exact analytical solution The two sources of error in finite difference methods are round-off error, the loss of precision due to computer rounding

of decimal quantities, and truncation error or discretization error, the difference between the exact solution of the original differential equation and the exact quantity assuming perfect arithmetic (that is, assuming no round-off)

To use a finite difference method to approximate the solution to a problem, one must first discretize the problem's domain This is usually done by dividing the domain into

a uniform grid (see image to the right) This means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner

An expression of general interest is the local truncation error of a method Typically expressed using Big-O notation, local truncation error refers to the error from a single application of a method That is, it is the quantity ƒ’() - if ƒ’() refers to the exact value and to the numerical approximation The remainder term of a Taylor polynomial is convenient for analyzing the local truncation error Using the Lagrange form of the remainder from the Taylor polynomial for ƒ(), which is ) = , where

the dominant term of the local truncation error can be discovered For example, again using the forward-difference formula for the first derivative, knowing that ƒ() = ƒ(), ƒ( = ƒ() + ƒ’( +

and with some algebraic manipulation, this leads to

= ƒ’( +

and further noting that the quantity on the left is the approximation from the finite difference method and that the quantity on the right is the exact quantity of interest plus a remainder, clearly that remainder is the local truncation error A final expression of this example and its order is:

= ƒ’() + O(h)

This means that, in this case, the local truncation error is proportional to the step sizes The quality and duration of simulated FDM solution depends on the discretization equation selection and the step sizes (time and space steps) The data quality and simulation duration increase significantly with smaller step size Therefore, a reasonable balance between data quality and simulation duration is necessary for practical usage Large time steps are useful for increasing simulation speed in practice However, time steps which are too large may create instabilities and affect the data quality.[3][4]

The von Neumann and Courant-Friedrichs-Lewy criteria are often evaluated to determine the numerical model stability

4 ANSYS APDL

4.1 Definition

ANSYS APDL or ANSYS Parametric Design Language is the primary language

used to commute with the Mechanical APDL solver APDL is commonly used to automate the task or even make a complete parametric model It comprises of a wide array of different features such as matrix and vector operations, do-loop and if-then-else functions ANSYS APDL is a time-tested old-fashioned command-driven program that allows the user to input line by line codes one at a time and execute them on demand Using APDL instead of the modern user-interface is said to give the users more power and control over each step of their simulation

4.2 Five common ANSYS APDL commands users should know

4.2.1 /SOLU – /POST1 – /PREP7 /FINISH – Processors

/SOLU – this command will take you into the solution processor This processor

comprises of all the design modification and changes that need to be done to your geometry This is the most visited processor in product design and testing

/POST1 – this command will take you to the post-processor The post-processor allows the users to work with their results, makes graphical outputs, and do in-depth post-processing