Section 4: Implementation of Finite Element
Analysis Other Elements
1. Quadrilateral Elements
2. Higher Order Triangular Elements
3. Isoparametric Elements
Implementation of FEA:
Other Elements
-1-
Section 4.1: Quadrilateral Elements
Refers in general
to
2
1. There is no closed form solution for the error function
erf() =
2
0
Use the two-point Gauss quadrature approach to estimate erf(1.5). [The exact value is
0.966105.]
2. The depths of a river H are measured at equally spaced distances across a chann
CE208: Numerical Methods for Civil and Infrastructure Engineering
Assignment#3 (Solution of Linear System)
Code all methods stated below in MATLAB to find solution of a well-conditioned linear system of nequations with n-unknowns. The program should take
CE208: Numerical Methods for Civil and Infrastructure Engineering
Assignment#1 (Error Analysis)
1. Problem on truncation error:
Use Taylor series expansion with n = 0 to 2 to approximate f(x) =
tan(x) at xi+1=/4 on the basis of the value of f(x) and its
d
Boot Strapping and Bagging
Need for a Diverse Dataset
If we had access to a classifier with perfect generalization performance, there would be no need to resort
to ensemble techniques. The realities of noise, outliers and overlapping data distributions, h
~
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR
Date:
(FN/AN)
No. of Students: 175
Subject No: EV20001
2nd Year B. Tech.: GG, CY, EE, HS, ME
Full Mark.;: 50
Time: 3 hrs.
Spring End Semester Examination, 2012 - 20 I 3
Subject Name: Environmental Science
Sectio
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CE208: Numerical Methods for Civil and Infrastructure Engineering
Assignment#2 (Finding Roots)
1. Find the root of the following equation correct up to 3 digits after
decimal point: x sin( x) 1 0 in [0, 2.5] using following methods. (a)
Bisection method,
Problem # 1
Given function:
True value of f(x)
f (x) x
x+1
x
f (100) = 4.98756211208899
f (1) = 0.414213562373095
f (10) = 1.5434713018702
f (10000) = 49.9987500624854
f (1000) = 15.8074374289576
f (100000) = 158.113487725586
Calculation of f(1) using 6-
Gerschgorin Circle Theorem
Eigenvalues
In linear algebra Eigenvalues are defined for a square matrix M. An Eigenvalue
for the matrix M is a scalar such that there is a non-zero vector x such that
Mx=x. In linear algebra we saw that the Eigenvalues are als
MA231, Tutorial Sheet: Numerical Analysis
Numerical Methods for Solving Nonlinear Equations
(1) Find a real root of the equation f (x) = x3 x 1 = 0.
(2) Find the real root of the equation x2 + 4 sin x = 0 correct to four
decimal places by Newton-Raphson m
Fredholmes Alternatives
Gauss elimination
Gauss Siedel reduction
Gauss Jordan Method
LU decomposition method
Inverse of a matrix
Jacobis Method
Cholesky Method diagonally dominant matrix
- Sylvester criteria
Eigen values and Eigen Vectors-Band-width Matri
1. Find the root of x * cos[(x)/ (x-2)]=0
The graph of this equation is given in the figure.
Let a = 1 and b = 1.5
Iteration
No.
1
2
3
4
5
6
a
b
c
f(a) * f(c)
1 1.51.133 0.159 (+ve)
1.1331.51.194 0.032 (+ve)
1.1941.51.214 3.192E-3 (+ve)
1.2141.5 1.22 2.58
Module
for
Pivoting Methods
Background
In the Gauss-Jordan module we saw an algorithm for solving a general linear
system of equations
consisting of n equations and n unknowns where it is
assumed that the system has a unique solution. The method is attrib
Indeterminate Analysis
Force Method1
The force (flexibility) method
expresses the relationships
between displacements and
forces that exist in a structure.
Primary objective of the force
method is to determine the
chosen set of excess unknown
forces and
Numerical Methods for Civil Engineers
Lecture 8
Interpolation
Linear Interpolation
Quadratic Interpolation
Polynomial Interpolation
Piecewise Polynomial Interpolation
Mongkol JIRAVACHARADET
SURANAREE
INSTITUTE OF ENGINEERING
UNIVERSITY OF TECHNOLOGY
SCHOO
Numerical Analysis
Chapter 4
4.1
Goal
Interpolation and Approximation
Polynomial Interpolation
Given n + 1 data points
(x0 , y0), (x1, y1 ), (xn , yn ),
to nd the polynomial of degree less than or equal to n that passes through these points.
Remark
There
Solutions to Problems
on the Newton-Raphson Method
These solutions are not as brief as they should be: it takes work to
be brief. There will, almost inevitably, be some numerical errors. Please
inform me of them at adler@math.ubc.ca. We will be excessivel
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Pu
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DateFrom: 01/10/2015
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Pu
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AvgPeriod: 24 Hours
DateFrom: 01/10/2015
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Pu
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Date :Tuesday, October
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CENTRAL POLLUTION CONTROL BOARD
CONTINUOUS AMBIENT AIR QUALITY
Date :Tuesday, October
Time :11:2
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City: Delhi
Station: Punjabi Bagh
Parameter: CO,O3
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DateFrom: 02/12/2015
DateTo: 02/03/2016
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CENTRAL POLLUTION CONTROL BOARD
CONTINUOUS AMBIENT AIR QUALITY
Date :Tuesday, October
Time :11:1
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City: Delhi
Station: Punjabi Bagh
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CENTRAL POLLUTION CONTROL BOARD
CONTINUOUS AMBIENT AIR QUALITY
Date :Tuesday, October
Time :11:2
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City: Delhi
Station: Punjabi Bagh
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CENTRAL POLLUTION CONTROL BOARD
CONTINUOUS AMBIENT AIR QUALITY
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Time :11:2
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CENTRAL POLLUTION CONTROL BOARD
CONTINUOUS AMBIENT AIR QUALITY
Date :Monday, October 24, 2016
Time :11:11:41 AM
State: Delhi
City: Delhi
Station: Punjabi Bagh
Parameter: NO2,O3,PM10
AvgPeriod: 24 Hours
DateFrom: 01/10/2015
DateTo: 01/10/2016
Punjab