Numerical Methods - University of North Carolina at Charlotte · PDF fileWhy Use Numerical...
Transcript of Numerical Methods - University of North Carolina at Charlotte · PDF fileWhy Use Numerical...
Numerical Methods
Why Use Numerical Methods?
Most problems in engineering involve the following steps:
� Development of a mathematical model to represent/model a physicalsystem or manufacturing process
� Derive governing equations by application of physical laws (New-ton’s laws, conservation of mass, energy, etc.)
� Solution of the equations
� Interpretation of the solution
ITCS 4133/5133: Numerical Comp. Methods 1 Introduction
Difficulties in Solution of Equations
� Equations may be linear or non-linear, may be ordinary or partialdifferential equations, or equations involving integrals or derivatives
� Equations may or may not admit closed form or analytic solutions
� Very few practical systems admit analytical solutions
� Analytical solutions require simplifying assumptions
What are Numerical Methods?
� Solutions that cannot be expressed in the form of mathematical ex-pressions (analytical form)
� Athough tedious and computationally intensive, the use of powerfuland inexpensive computers can be effectively used to find solutionsto complex engineering problems.
ITCS 4133/5133: Numerical Comp. Methods 2 Introduction
Example
Consider
I1 =
∫ b
a
xe−x2
dx
can be written as
I1 =
∫ b
a
d(−e−x2
2) =−e−x2
2
resulting in the analytical solution.
ITCS 4133/5133: Numerical Comp. Methods 3 Introduction
Example(contd)
However,
I2 =
∫ b
a
f (x)dx =
∫ b
a
e−x2
dx
does not have a closed form solution and requires a numerical evalua-tion.
Solution:
⇒ As the integral is simply the area under the curve f (x), break thisarea into small rectangular regions and sum up the areas
⇒ Relevant methods : Trapezoidal, Simpson Rules.
ITCS 4133/5133: Numerical Comp. Methods 4 Introduction
Sample Problems and Numerical Methods
Solutions(Roots) of Nonlinear Equations
� Can be algebraic, polynomial equation
� Generally of the form
f (x) = 0
� Goal is to solve the system (determine the roots)
� Applications: turbulent fluid flow, vibration systems
ITCS 4133/5133: Numerical Comp. Methods 5 Introduction
Sample Problem 1: Fixed-Point Iteration
� Equations of the form x2 = c, rewritten as x = 12(x + c
x), forming the
basis for an iterative solution.
ITCS 4133/5133: Numerical Comp. Methods 6 Introduction
Solutions of Simultaneous Linear Equations
� In applications such as heat transfer, fluid mechanics, the governingPDEs are solved using finite difference or finite element techniques
� This converts the problem into a system of linear algebraic equa-tions, which can be solved using techniques such as Gaussian elim-ination
� Example 2 variable system:
a1x1 + a2x2 = b1a3x1 + a4x2 = b2
ITCS 4133/5133: Numerical Comp. Methods 7 Introduction
Numerical Solution: Gaussian Elimination
ITCS 4133/5133: Numerical Comp. Methods 8 Introduction
Solutions of Eigen Value Problems
� In applications such as vibration of structures, we usually have a setof homogeneous linear algebraic equations
� If there are n equations, then there are n + 1 unknowns in thesesystems
� Example 2 equation system:
(a1 − λ)x1 + a2x2 = b1a3x1 + (a4 − λ)x2 = b2
where λ is the eigen value and ~X =
[x1x2
]is the Eigen vector are the
unknowns.
ITCS 4133/5133: Numerical Comp. Methods 9 Introduction
Curve Fitting and Interpolation
� These methods attempt to evaluate a function at unknown pointsusing the function values at known points
� Interpolation: Use function values at neighboring points to estimatefunction value at unknown point, for instance, using a weighted av-erage
� Curve Fitting: Use the known function values to fit a curve(linear,quadratic, cubic, etc); evaluate the curve at the unknown functionlocations.
ITCS 4133/5133: Numerical Comp. Methods 10 Introduction
Numerical Differentiation and Integration
� These methods provide ways to determine the derivative and inte-grals of functions that have known values at only limited numbers ofpoints and have no known expression.
� Can fit polynomial, and then determine the derivative or integral, oruse differencing or area estimation techniques.
ITCS 4133/5133: Numerical Comp. Methods 11 Introduction
Approximating an Integral:Trapezoid Rule
ITCS 4133/5133: Numerical Comp. Methods 12 Introduction
Solution of Ordinary Differential Equations
� These arise in applications such as dynamics, heat and mass trans-fer, usually in the form of PDEs, which can be transformed intoODEs.
dy
dx= f (x, y)
� Can be solved by approximating the derivative as the slope of thefunction y(x) at different values of x.
y1 − y0 = f (x0, y0)(x1 − x0)
or
y1 = y0 + hf (x0, y0)
ITCS 4133/5133: Numerical Comp. Methods 13 Introduction
ITCS 4133/5133: Numerical Comp. Methods 14 Introduction