TAM 470 HW Set 1; due 9/11/15
Note: The exercises below refer to pages 8-10 in the Moin text. When a problem asks you
to write a program, do so do not work the solution by hand as a means to avoid writing
code. Please program your own interpolation algori

TAM 470 HW set 6; due 12/4/15
All students:
1. Let E3 be an open domain, and let u, v H 2 (). Write expanded (long-form) expressions
for u H 2 () . (5 pts.)
u2 + (u,x )2 + (u,y )2 + (u,z )2 + (u,xx )2 + (u,xy )2 + (u,xz )2 + (u,yy )2 + (u,yz )2 + (u,zz )2

Iterative (Indirect) Solvers for Linear Systems
R. B. Haber
1
Overview
This document summarizes mathematical structures and algorithmic features that are common to a variety iterative solution techniques systems of
linear (i.e., matrix) equations. Suppose

Pad Approximation
e
R. B. Haber
For a centered, three-point stencil, cfw_xj1 , xj , xj+1 , we write fj := f (xj ) in terms of
the values of f at the three stencil points and the values of f at the neighbor points, xj1
and xj+1 .
fj =
ai fj+i
bi fj+i + e

Modied Wave Number Analysis
R. B. Haber
1
Overview
Modifed wave number analysis provides an alternative to order-of-accuracy
estimates for assessing the accuracy of nite dierence approximations. With
Fourier series approximations in mind, consider for exa

Solution HW3
October 13, 2015
1
Moin text, pp 44-45, Exercise 8 parts (a) and (b) (20 points)
Given integral
1
I=
0
100
1
+
dx.
x + 0.01 (x 0.3)2 + 0.001
We evaluate above integral numerically using three methods- the trapezoidal, Simpsons, and trapezoid

Introduction to Vector Spaces
R. B. Haber
1
Introduction
Most students entering this class are familiar with vector spaces in which the vectors are
directed line segments in n-dimensional Euclidean space, denoted by En . Each vector is
characterized by it

Exercise 1
A computer program for Lagrange interpolation is implemented in Matlab. The
program implementation is summarized below.
I. A function named inter_lagrange is implemented. It takes three input arrays
xp, yp, and x and returns an output array y.

TAM 470 Computational Mechanics:
Finite Element Methods
R. B. Haber
Fall Semester 2015
2
Abstract
These notes provide a brief introduction to nite element methods
for elliptic boundary value problems. We begin with some preliminary
mathematical material,

TAM 470 HW5 Solutions
November 16, 2015
1
1
Moin text p 156; Exercise 6 (25 points)
1.a
Stable range of CFL number
Given discrete equation to solve
1
t
(n+1)
uj
u
t
= c u is
x
1 (n)
(n)
u
+ uj1
2 j+1
=
c
(n)
(n)
u
uj1 ,
2x j+1
(1)
which can be arranged a

Numerical Integration Rules for Discrete Data
R. B. Haber
1
Overview
This document describes several common integration schemes for integrating functions where the data is only available and discrete locations in
the domain of the function, as is typical

TAM 470 HW set 6; due 12/4/15
All students:
1. Let E3 be an open domain, and let u, v H 2 (). Write expanded (lnog-form) expressions
for u H 2 () . (5 pts.)
2. Given: f = sin x, = cfw_x ]0, [ (open set), and the model elliptic problem,
Find u C2 ()
u + f

TAM 470 / CSE 450 Computational Mechanics Exam 2
18 November 2015
Prof. Haber
SOLU T ION KEY
Name
Instructions:
There are four problems (length and credit for each problem varies). This exam will last 2
hours.
Show all of your work; credit will not be giv

HW2 Solution
October 5, 2015
1
Moin text p 26 Exercise 2 (points 20)
The objective is to determine the most accurate formula for fi , given fi1 , fi , fi+1 , and fi+2 , where
xi1 , xi , xi+1 , and xi+2 are equally spaced points with spacing h.
The error,

TAM 470 / CSE 450 Computational Mechanics Exam I
21 October 2015
Prof. Haber
SOLU T ION KEY
Name
Instructions: There are four problems. Show all of your work in this booklet; credit will
not be given without supporting work. Continue your work in the spac

TAM 470 HW Set 4; due MONDAY 10/26/15 (No automatic extensions to 10/28
for 90% credit will be granted for this assignment).
All students:
1. Moin text, p 96; Exercise 23 parts (a) and (b). Be sure to include the required
discussion of the results for bot

TAM 470 HW Set 3; due 10/9/15
All students:
1. Moin text, pp 4445; Exercise 8 parts (a) and (b). Be sure to include a discussion of
the results for both parts. Implement a program to carry out the parameter studies
in the number of panels; dont do this ma

TAM 470 HW Set 5; due 11/6/15.
All students:
1. Moin text, p 156; Exercise 6. (25 points)
2. Moin text, pp 156158; Exercise 9, Part 1 (a). (25 points)
3. Moin text: pp 164165, Exercise 22, all parts. (25 points)
In each of parts (b) and (c), show isotherm

TAM 470 HW4 Solutions
November 10, 2015
1
1
1.a
Moin text; Exercise 23 (a) and (b) (25 points)
ode23s
The solution from Matlabs ode23s, a sti ODE solver, is shown in Fig 1.
We implement a function in Matlab such that it takes y = [y1 , y2 , y3 , y4 , y5 ]

Finite Element Interpolation, Dierentiation, and Integration
R. B. Haber
1
1.1
Interpolation
Lagrange interpolation
Interpolation is one of the key technologies used to develop nite element approximations
to solutions of boundary-value problems in mechani