FORMULATION OF 2 AND 3 DIMENSIONAL BVPs
Laplacian Equations, e.g. Heat Conduction, Incompressible Potential Flow
Preliminary
NSD # of space dimensions
open set, domain (not including boundary)
boundary of
,
such that
n
Prescribed traction
Prescribed

Ugly
1 of 2
file:/C:/z%20Northwestern%20Data/C90/c90%20website%20content/Readings/BGugly.html
Ugly Method
First rule of Ugly Method: don't use Ugly Method. It is much easier and quicker and more reliable to use the "grouping" method to construct bond
grap

Junctions
M.A. Peshkin
Common effort / common flow
A system consisting of two elements shares both an effort, and a flow. (Shares magnitude, but not necessarily sign)
A system consisting of three elements is either common effort (0-junction) or common fl

1. Sketch, reduce and annotate bond graph models of the following systems. Gravity
acts downward in all cases and there is no friction.
2. Sketch, reduce and annotate a bond graph model of the following system. There is
friction between m1 and ground. Con

1. Consider the circuit below.
C
Vs
L
R
a.
b.
c.
d.
Develop a bond graph for the circuit. Reduce and annotate.
Identify state variables.
Write junction relations and constitutive relations in proper causal form.
Find state equations.
2. The purpose

Reduction rules
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Reduction rules
You can (and should) reduce a bond graph as far as possible using the reduction rules below. This makes your subsequent work easi

ME 390 INTRODUCTION TO DYNAMIC SYSTEMS
Homework #1
1. Write a differential equation for x(t) using this block diagram:
x(t)
-1
A dt
C
B
+
d
dt
+
1
+
u(t)
You can assume that u(t) is a known input function.
2. Consider a tower crane of the

Dynamic variables and modeling
When you make a physical diagram (picture, circuit diagram,
etc) be sure to show each element distinct from all the others.
This can be easily overlooked for an effect like gravity (you
tend to think of it as part of the mas

Arbitrary Causality and State Equation Loops
When assigning causality to your bond graph, if you find that you can arbitrarily assign
the causality of your resistive elements, you will hit a loop when deriving your state
equations.
Example:
SE
C
R
1
0
I
S

MANE 4240 & CIVL 4240
Introduction to Finite Elements
Prof. Suvranu De
Higher order elements
Reading assignment:
Lecture notes
Summary:
Properties of shape functions
Higher order elements in 1D
Higher order triangular elements (using area coordinates)

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

[Ref: Jacob Fish and Ted Belytschko, The first Course in Finite Element, John Wiley & Son, 2007]
HEAT CONDUCTION:
1. FOURIER LAW (LIKE HOOKES LAW FOR STRESS ANAYSIS)
dT
d TEMPERATURE
g=
HEAT FLUX =
)
k
CONDUCTIVITY (
dx
dx
2. CONSERVATION OF ENEGY
3. T C

1D Elasticity (Boundary Value Problem (BVP) or Strong Form)
Given a bar of cross-sectional area A(x), with thermal expansion coefficient (x), temperature
distribution t(x), Youngs modulus E, acted on by a body force per unit volume b(x), find: u(x)
in the

ME-CEE 327 Finite Element Methods in Mechanics Fall 2016
Instructors:
Professor Wing Kam Liu with Dr. Mark Fleming of Caulfield Engineering as Guest Lecturer
Days and Times:
Tu, Th 12:30pm-1:50pm, Tech LR2
Office hour:
Professor Wing Kam Liu: Tu Th 11:15a

% CEE-ME_327 computer homework #1 %
clear;
%= Preprocessing =%
nel=5; % number of elements
totlength=10; % total length of the domain
b = 10; % body force [force/length]
youngs=1; % elastic modulus
area=1; % element area
nne=2; % number of nodes in an ele

Computer Homework #1
Description:
Starting with the attached Matlab code, develop a program for onedimensional elasticity. The parts that need to be added are the
computational of strain and stress. The program should be able to treat
an arbitrary numb

Chapter 1
Tension and Compression in Bars
1
1 Tension and Compression in
Bars
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Stress.
Strain.
Constitutive Law .
Single Bar under Tension or Compression.
Statically Determinate Systems of Bars .
Statically Indeterminate Sys

About Bond graphs - The System Modeling World
Page 1 of 25
About Bond Graphs
1. Introduction
2. Power variables
3. Standard elements
4. Power directions
5. Bond numbers
6. Causality
7. System equations
8. Activation
9. Example models
10. Art of creating m