MM3CMT:
Thermo-fluids (Finite Difference Method)
Learning Outcome
Know the classification of physical behaviours
and PDEs
Review the governing equations for thermo-fluids
Understand discretisation schemes (spatial &
temporal) in FD
Understand boundary and
MM3 CMT Computer Modelling Techniques
1-D Finite Element Analysis: Supplementary Notes 1
J. F. Webb
28th April, 2014
(last updated 11th April, 2014)
A Governing Equation
For a string with a loading force per unit length f (x) and deection u(x),
with a ten
MM3 CMT Computer Modelling Techniques
2-D Finite Element Analysis: Problems
J. F. Webb
12th May, 2014
Problem 1
With the help of the del operator
+
=
x
y
and the divergence theorem, show that the weak form of the 2-D governing
equation,
2u 2u
+
+ f (x, y)
MM3 CMT Computer Modelling Techniques
1-D Finite Element Analysis: Problems
J. F. Webb
12th May, 2014
Problem 1
Show how to derive the governing equation for a loaded string:
T
d2 u
+ f (x) = 0,
dx 2
xL < x < xR
Problem 2
For boundary conditions u(0) = 0
Part II
One-Dimensional Finite
Element Analysis for a
Continuum System
Governing equation for a tensioned string:
d 2u
T 2 f x 0
dx
1-D form of the Poisson equation
Boundary conditions:
du
u 0 or
0
dx
at x xL and xR
u Deflection of the string
x Longitudin
Part III
Two-Dimensional Finite Element
Analysis for a Continuum System
Physical Problems:
Tensioned membranes
Torsion of prismatic bars
Steady state heat transfer
etc.
Governing equation for a tensioned membrane:
2u 2 u
T 2 + 2 + f ( x, y ) = 0
y
x
i
Finite Element Method
(Part I)
Pin-Jointed Frame Structures
Pin-Jointed
Frame
Structures
1
Review (1st year Mechanics of Solids)
A = Cross section area
L = Length
f
A
L
E = Youngs modulus
f
f
E
f
Design criteria:
Permissible
Known problem:
f
Reinvented
Interpolation over ith element:
u ( x) =
Shape functions
xi +1 x
x xi
ui +
ui +1 = N i ( x ) ui + N i +1 ( x ) ui +1 ,
xi +1 xi
xi +1 xi
xi x xi +1
Correction w u
= Ni ( x )
ui
(i )
( i ) ui
N i +1 ( x ) = [ S ]cfw_
cfw_ =
u
ui +1
i +1
Correction
MM3CMT:
Thermo-fluids (Finite Volume Method)
Learning Outcome
Get to know the FVM formulation of thermo-fluid
governing equations
Simple thermo-fluid examples (conduction,
convection etc.)
1
Finite volume method
The underlying principle of this technique
MM3 CMT Computer Modelling Techniques
1-D Finite Element Analysis: Supplementary Notes 2
J. F. Webb
28th April, 2014
(last updated 11th April, 2014)
The Weak Form of the Weighted Integral
Remember that the weighted integral over the ith element is
xi+1
w(
