Civil Engineering Reference
In-Depth Information
g coord
nodal coordinates for all elements
jac
Jacobian matrix
kay
permeability matrix
kc
element conductivity matrix
kv
global conductivity matrix
loads
excess pore pressure values
mass
element lumped mass vector
mm
element “mass” matrix
newlo
new excess pore pressure values
p
“descent” vector used in equation (3.22)
points
integrating point local coordinates
press
excess pore pressure values after
ntime
time steps
prop
element properties
r
holds fixed rhs terms in pcg solver
storpb
stores augmented diagonal terms
store
stores global augmented diagonal terms
storka
stores lhs element matrix
storkb
stores rhs element matrix
store
kc
stores element
kc
matrices
store mm
stores lhs element matrices
u
vector used in equation (3.22)
value
fixed boundary values of excess pore pressure
weights
weighting coefficients
work
working space array
“old” solution vector
x
“new” solution vector
xnew
x(r)
-coordinates of mesh layout
x coords
y(z)
-coordinates of mesh layout
y coords
8.4 Exercises
1. A layer of clay of thickness 2
, free draining at its top and bottom surfaces is
subjected to a suddenly applied distributed load of one unit. Working in terms of a
dimensionless Time Factor given by
D
2
, and using a single finite element,
T
=
c
v
t/D
use the Crank-Nicolson approach (
θ
=
0
.
5) with a time step of
T
=
0
.
1toestimate
the mid-depth pore pressure when
T
=
0
.
3. Compare this result with the analytical
solution to your equation.
(Ans: Numerical 0.40; Analytical 0.41)
2. A rod of length 1 unit and thermal diffusivity 1 unit is initially at a temperature of
zero degrees along its entire length. One end of the rod is then suddenly subjected
to a temperature of 100
◦
and is maintained at that value. You may assume that the
other end of the rod is perfectly insulated (i.e. there is no temperature gradient at
that point). Using two 1D 'rod' elements, and assuming time-stepping parameters
t
