may be
EQUATIONS
Appendix
element
method.
It
for the
as
theories
of
constitutive
or
stress-strain
an extension of
necessary,
a distinct and
principles is
the use of
five
has
shown
that
to
a
description
of
the
linked
with
the
use
of
ter
6
It is recom-
learning
various
two-dimensional
easy
introduction
that
the
reader
be
advanced
chap-
ters
1-14
are
stated
and
references
are
include
In
the
residual
and
procedures
for
solution
of
algebraic
; I
would
for his assistance
write
at
an
such
final
assemblage
error.
In
perhaps
the planet
we live
on is
provided
the
survey
performed,
a
 closing
error
would
be
moment distribution
were used.
photographs
discretization
such
the
concept
of
finite
Figure
1-6
fore, space
solu-
Archimedes called
(Fig.
1-8),
convergence
implies
 
we
take
from
circle.
The
difference
error in the
can
system
useful to
basic laws,
are
3. (a)
5.
Derive
the
Laplace
equation
for
Fig.
1-12.
1—
x
Figure
1-12
Home
in
a
very
general
sense.
The
effects
as
fluid
head
(p.
by
using
conven-
tional
of
the
stiffness-
load
relationship.
principles
in
must
remain
con-
the interfaces
would
approximate
the
continuous
medium
as
closely
if
problems, it may
second-order
quadratic
function
to
dimensional
Then
a
point
two-dimensional problem [Fig.
the same
computed solution
in the
we must also
it is
defined
to reflect precisely the behavior of the material or the system, the
results
from
equations
terms
the
discretized
body.
A
number
of
alternatives
energy methods and the
knowledge of variational calculus.
element method, we
through the use
consistent states
values
a
of
finding
considered
introduce and
ential calculus.
and
a body under load. If
the body, say, a
used
internal strain
S
2
U
P
=3
2
U-S
2
W
P
>0.
(2-7)
a
symbol
expressed.
For
of unknowns (at the
principle
of
stationary
energy
schemes
are
employed
under
the
MWR,
among
which
are
collocation,
subdomain,
least
governing
a
problem.
As
a
u*
—
forcing function.
elasticity,
in
the
subsequent
chapters.
ELEMENT
EQUATIONS
Use
obtain
equations
body
or
or
residual
procedure
is
the displacements
of two
adjacent or
we
may
need
to
enforce
the
continuity
the case of
the
boundary
conditions.
Only
when
we
introduce
these
conditions
can
we
constraints
can stand
conditions.
 TT
s,
(a)
standard familiar
of these methods is
change
depending
on
Eq.
(2-22).
For
quantity can
It is
often very
ask
REFERENCES
[1]
Plane Stress
Analysis, in
Sept. 1960.
Element
provide
column,
the
global coordinate system.
of
a
local
tions extremely simple
tions
by
us
from
node
point
ever. Point
B is
define
the
those
with
respect
to
coordinate system
which we can define local
coordinate systems
3-2(b)
and
(c).
from
then the global coordinate of any point in the element
is
and
2,
respectively.
Note
that
say, at
point
property that imparts
In this equation,
In simple
coordinate
y
of
a
point
in
the
element.
For
example,
in use,
by
using
called
the
isoparametric
2,
respectively.
REQUIREMENTS
interelement
compatibility
is, the
the first
case
of
the
one-dimensional
should
be
complete;
For instance,
by
the
strain
in the
column, the
bending.
be
a
approximation
body
motion
scope
and
the
reader
interested
element method
completeness
requirement
can
be
further
explained
by
using
the
we
consider
these
two
relations.
to use
(3-12a)
leads
to
e,
tion matrix. Because this is
a
because we have
of the
well-known Hooke's
described
in
Chapter
2
and
in
Appendix
1,
While doing this
as
SW
6Wr
(3-17)
or
perturbation.
For
the
spring as the datum
load in
constant,
this
term
does
minimum
by
examining
a
minimum
value
at
v
to and fro
STl
p
by
using
the
finite
element
ments. Consequently, it is
most economical and direct
cumbersome and often
In
general,
how-
ever, the minimization will involve calculus of variations. In most of the
treatment in
write FL
v
t
joint or
nodal forces
element
instead
Eq. (3-2 Id)
(3-2 lc). The last term denotes
sum-
mation,
P
u
v
l
and (3-23c)
majority of derivations
(3-25)
lead
to
^0>,
coefficients in matrix
and the
joint loads.
(equally)
loading; we
by
differentiating
the
expanded
function
for
Tl
p
(minimum)
value.
The
have
measured
For instance,
that the
stiffness or
Simi-
by
adding
by
the
called
into
action
only
the
concept
of
boundary
conditions
ishes, and
gradient or
base is
given
called
natural:
the
column base,
experiences
a
given
These properties, which
that
v
4
0,
and
R
A
requirements,
it
becomes
advisable
and
necessary
linear
or
methods used for
elimination and back
Assembly of
the three
before ;
only
us an idea that for
linear
problems,
where
Kij
do
not
only the back substitution needs
to be
For the displacement formulation based on potential energy, nodal dis-
placements
are the
(3-32).
The
from
the
type
of
2
within
is
consid-
ered
satisfactory.
require addi-
middle.
1,
the entire
While applying Galerkin's method for
finite element analysis, we
Galerkin's method
Ni
to
clarify
a
number
of
aspects
Figure
3-1
v*
defined
the
1,
we have
we have
the
have
nonzero
ments
contribute
left-hand side reduces to
other
words,
the
natural
boundary
condition
final
results
if
that
point
has
a
prescribed
geometric
boundary
condition.
residual
procedure
is
self-adjoint
[8].
In
can be
involve large matrices,
of
the
complementary
matrix
as unknowns
joint
or
nodal
forces.
Hence,
the
approach
specialize
convenient
stiffness
matrix.
Example
3-3
the flexibility
matrix is
is loaded,
[k]
as
in
be used in con-
Eq.
(3-8d)
be
force,
surface
A
since the zeros can appear as  pivots in the denominator
(Appendix
2).
pivoting
[7],
which
involves
exchanges
proce-
(3-69)
lb)
variational
methods,
the
attention
has
solution for displacement.
ments.
This
is
depicted
supports or constraints in the structure and makes it stiffer.
In the case of the
complementary energy
approach, the
approximate value
flexibility value from
distribution
of
displacement.
Physically,
this
may
be
construed
to
and overlaps in the
structure. This makes the structure weaker or less stiff than what it really is.
Figure
3-15
Bounds in
stress approach.
the
makes it so
other classes of
modifications.
For
instance,
one-dimensional
stress
deformation
(Chapter
3),
Figure
3-17
E,
for the
and stresses in
at
(a)
Distribution
of
 
problem
occur
essentially
to
can
porous
media
{LIT)
93
of change
of potential
or temperature
due to
for
<p
are
similar
to
<p
(p.
In
other
subsequently in
the flow behavior through
that we can
use is Darcy's
Galerkin's
method.
(4-1)
can
be
written
as
Q
It
is
only
is
0.0333
cm/sec.
4. They
introducing
Oden, J.
T., Taylor,
pressures
act
is relatively easier
when
temperature
is
unknown
interaction
or
coupling
between
by
e
y
same
as
p
is
differentiated.
residual
load
{Q}.
we may know
for
phenomena.
by
using
The
ing
are found
stresses in
the bar.
TIME-DEPENDENT PROBLEMS
in
permeability
3
domain.
We shall first follow
Fig.
3-2.
Step
A
Use the
tions
[5,
6]
p
n
}dL
the
functional
involves
time
variational
on Eq.
element equations
thermal
q
be written
of
introduction.
In
fact,
a
number
 
Eq.
the
initial
conditions
[Eq.
(5-10)]
(5-26)
known.
The
state.
Method
*<*>
=
3,
we
consider
a
generic
element
tem-
peratures
T
t
a
the
length
a
3
1
and
2,
constitute condi-
5-la)
through
which
three elements
assumed to be of
Here S
at nodes 1 and
(/
+
Under a
direction; then the
of the
effective
stress
can be
the flow
(5-9)
we assume that the top and bottom of the consolidating
mass
are
pervious,
and
or
Dirichlet-type
boundary
condition.
It
is
possible
to
Eq. (5-50b)
is achieved
An important observation can be made at
this
stage,
Example
5-3
Figure
5-5
shows
a
are given:
Initial conditions:
represents
time-
dependent
cooling
of
a
A pictorial distribution
temperatures in
due
ment;
hence,
just
integrated sense.
and
bottom
boundaries
are
pervious,
can
assume
that
p(0,
shows
values of excess pore water pressure along the depth at various
values
of
U
procedure can
yield history
the
a\
since
a,
a'
Eq.
(5-44).
Av(t)
[10].
dimen-
used
to
be
timewise meshes
known,
line
(or consolidation)
A
=i(l
-L)A
X
+^(1
Fig.
5-3
with
the
of
Example
5-3.
Use
2,
=
numer-
error
consolidation problem. Hint:
depth
of
of
Finite
Ele-
C. S. Desai (ed.),
Van
is called Deformation
chapter. In the
; a
number
materials
(NMAT),
number
(NSLC), option
for whether
=
temperature,
or
pore
We also compute the length /
(ALL) of the element
corresponding
to
1)
and
IE(M,
2),
auto-
matically
the
areas can be assigned.
=
IAREA
midsections)
are
input.
Input
traction loading
TY is
subset
tions, and
variation
of these values is equal to NTIME.
The third
=
first and
required for stress-deformation
and
can
be
and
steady
flow
problems,
A(I,J)
and
AK(I,J)
are
(3-28)
element
load
vector
{Q}.
vector
to
the
load
vector
that
is
degree of consolidation are also printed
out.
A
only once
=
only the
first and
last elements.
is, input
printed out
which output is desired.
flow.
=
media,
see
Table
6-1.
QK
TIME
TUINIT
If
KODE
and for
potential,
the nodes that are generated
by
the
computer.
are required only for the first and the last element;
data for
intermediate elements
are generated
for 1 1 elements, if there
is
a
0)
that is, 1 1 nodes,
and
adopt
A/
V
57Q
WRITE
(6,1050)
MAIN
583
MAIN1660
IN
17
70
MAIN1310
1/2.0
MAIN2160
IF (N0PT.LT.3)
A(K,M)=0.0
II
MAIN3020
630
CONTINUE
MAIN3030
THAN
0.0001
WRITE
(6,1250)
I,R(I)
GO
TO
530
3 3 I
NODE
PORE
PRESSURE
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
P.
Under
finite element
replaced
by
a
line
[Fig.
7-
1(b)],
necessary
to
w
u
X
at
to x
term of
The
approximation
function
[Eq.
(7-2)]
function
in
Eq.
(7-
14a)
below.
the finite element
given in Appendix 3.
du
proper
energy
to
-
/
s
1
[Wpds
explanation
method
gives
leads
to
Substitution
of
w
two nodes. It may be noted
that
if
when joint load
discretized body,
which denotes
in
approximation
For
element
7,
0.0000
M(at$
grams.
~j^
833.4
closed
yield
moments
at
support in
finite element computations
however,
the
the
force dia-
derivation
or
curvature.
Evaluation
As
-1
leads
7-8.
Problem
7-9
problem
with
a
constant
axial
load
P
and
bending
energies
axial load;
the term
Pu denotes
p
be expressed as
(3-28).
tions
are
essentially
undertake
solution
the
P.
Under
these
circumstances,
Chapters
3
and
15.
conditions of
along
x
can
be
LJ
Eqs.
(7-46)
and
l_
IF
J_
6F
_J_
of Fig. 7-4, we
have, for uniform loading,
zeros
on
the
diagonal.
case
we
if EI
varies
linearly
axial
load
P
boundary conditions
element with the interpola-
p(s)
example
7-12.
Assume
the
linearly
as
variation
in
moment
Variational Methods,
New York,
and
be stated
as
the
first
For derivation
(8-5)
the
normal
derivative
make this
arising from the
second part of
Combination
can be
equa-
[K] in Eq.
time
required
for
more
susceptible
to
the numerical solution
by
Example
8-2
A
solution
Eq.
(8-1)
was
obtained
by
Guymon
[3]
predictions with
a closed
solution. The former
 
of the
Symp.
on
Finite
Element
K.
Front,
Water
Resour.
Res.,
Vol.
11,
No.
2,
April
1975,
pp.
343-347.
Smith,
Res., Vol.
Blacksburg,
but it can often
case.
of the
varying flow
[6],
according
to
S
rainfall
excess
causes
overland
flow
Q
f0
which
discharges
channel flow
solution
for
In the first stage, the
overland flow
r
c
can
be
the
channel
is
replaced
[Q
fn
(9-1)
time
t
discharge is zero
at
(9-13b)
The values of excess rainfall for overland flow and resulting inflow due
to the
of flow
of
areas
at t
reached.
(9-15)
procedure
represent
a
field
situation.
matrix differential
value problem. The resulting matrix [K] is not singular, in
contrast
to
the
stiffness
matrices
(Chapter
3),
at time
Fig.
9-3(a).
global node
nodes
of
two
adjoining
.0
square
miles
around
a
were available from
three
subregions
applied independently
to three
in., were used
=
A 2655 1931
be treated
simply as
Prediction
Methods
and
State
University,
Water
Equations
Land-Use
considered. Although these problems
solutions
waves propagate
the bar.
(10-
la)
is
a
statement
linearly elastic, the stress-strain law is
o
x
fo
x
dx
axial strain.
dh4.
dx
2
dt
2
sound and vibrations
=
+
r
consistent
because
it
is
derived
from
the
it as lumped,
where the total
the
two
nodes:
 1
0
1
m
Apl
2
(10-8(1)
can offer computational
be more
accurate for
the
[K]
{r}
terms
here we
need to
second derivatives instead of the first derivative in the other
three equations.
acceleration
time level
/ to t
discretization of
=
{R}a..
=
—
as {r}^.
t and t
be
time step.
displacement at the fixed
its
derivatives.
be
applied
4,
as an
with closed
various publications
Solution:
[m]
Apir\
4
J.,
Apl
120
UL
u:
u
10-1.
10-4.
Consider
a
single-element
p
of
Univ. of
/.
of
Finite
S.
Differ-
ence
Methods,
Dynamics,
/. Eng.
Mech.
Div.
ASCE,
Vol.
99,
No.
EM2,
April
1973.
only
are based
bar
[Fig.
11
-1(b)].
of the
rotations of
latter
sections
[1,
2].
As
a
consequence,
planes
of
cross
section of
bar. (c)
(11-2)
represents
are nodes
a
(ll-4a)
where [N] is the matrix of interpolation functions and yields linear variation
of u
(11-2)
and
(11-4)
(11-7)
indicates
that
is
[2]
[«J
'dw
there are
only two
for
linear
variation
of
y/,
of the warp-
completeness.
warping
functions.
gradients of
X
function
does not contribute to
results. Equation (11 -17a) can be written in matrix form
as
fcjfar}
[k
r
isotropic
a discretized body. For
x
2
Figure
11
-5(b)
numbering
chosen
the element matrices are the same. In fact, even if
a
different
The
assembly
warping
functions, and [R] is the assemblage nodal forcing parameter vector.
Since
0.
Then
the modified equations are obtained by deleting the first row and column
in
to
express
solution
for
dy//dy
Element 1:
torsional
(see next
section), the
(11
-28a)
are measured from the center point of the bar, and
a
and
b
denote
of an element.
presenting comparisons between
closed form solutions for some of the problems in this
chapter. In
the computed
results yield
realistic solu-
tions of
(p,
according
cp
are
(ll-28a)
W
e
is
the
complementary
poten-
tial
=
function is distributed equally
bar
4
cm
x
discretiza-
can consider and
to the
are
responding to
function
[Eq.
the
twisting
moment
M,
is
given
by
[2]
M
t
represents
twice
the
volume under the stress function distribution over the cross section of
the
bar
bar is
of computed
Figure
11-8
(11-25),
Hence,
from the warping
with
the
Stress
Function
Approach
to the specific
the
four
elements
function
approach,
finer
mesh,
-<Pi
-<Pi
Figure
11-11
shows
plots
refined
mesh.
COMPARISONS
Torsional
Constant.
(11-38)
is
M
t
i
r/7
in
Fig.
11-9
and
(p
to
21.00%
in
r
xz
and
T
yz
shear
stresses
is
(a) Finite
functions
stress
functions
COMPARISONS
Torsional
torsional
For better
accuracy one
for
displacement or
they
of the element. Conversely, it is possible to assume stresses
inside
error
in
the
tion.
will
be
as
y/
a
modified
complemen-
,
function.
The
components
of
{T}
r
similarly.
{qjqGF[H]^[G]{q
for
U
and
U
c
(ll-54a)
[Gf
AGA
Hz
Al
2
23
[G]
A
2
(y
m
en
expressed
in
terms
the
displace-
hybrid approach
We
consider
x
2
as
boundary
elements
degree
of
and
that from
equilateral triangle.
obtained
exact
solution,
the displacement
to side AB (Fig.
Ixz
Exact
-7.136
method
Displacement
-7.145
The strain displacement
and
-
V
V
285
(11-61)
5
2
Application
dU
M
<5ru
We now
because
detailed
scope.
left to
Before
we
proceed
which
a
it
is
Figure
11-18
(b) Quadri-
at the inner node(s)
to be solved
The
do
ment compatibility
at the
assembly
procedure.
The
(1
respectively.
Notice
and
the element
twisting
moment
by
by
using
and torsional
of
twisting
moment
M
t
vs.
up
to
n
of
Elasticity.
/.
1924.
[6]
Murphy.
G.,
Conf.
Matrix
12,
Academic
Press,
New
Mech.
field
problems.
They
are
k
x
%g
(12-lb).
POTENTIAL
FLOW
by a
special form
of Eq.
is
irrotational
can
be
assumed
are
Chapter 12
a
triangular
element
will
be
essentially
identical
to the one described for torsion in Chapter 11. Hence, we
present
a
formula-
tion
by
using
a
possible to
yields
a
bilinear
tion of
s
and
easier differentiations and
problem
governed
problem
gov-
erned
Flow 305
y
in
as
(12-16b)
Equation
the
three terms obtained
A
where
[I]
look
like
k
9ll
[3]
formula
is
m
is
the
number
of
function
coordinate
directions
[Fig.
W
[3].
Example
12-1.
Numerical
Integration
It
this
element
are
also
given
in
Chapter
13.
the
square,
which
square. Now,
[3],
of integration
12-4(c),
and
the
weighting
points
only
:1
in
a
(12-32)
is
modified
a
number
on
side
1-2
applied
fluxes
V
n
is
the
velocity
9
is
the
8
x
8
(0,
0)
etc.,
denote
coordinates.
i
1
potentials
12-1
gives
at the
the
negative
x
direction
6
is
v
x
in
element(s)
adjoin-
Figure
12-10
If neces-
are different. The
unknown can now
and
q
is
the
medium, T, and the
the quadrilateral element
into 9 elements
x
323
(12-59)
elevation
head
[Fig.
12-12(a)].
surfaces:
q>
12
U
(a)
Free
surface
12-12
parentheses.
The
computed
values
of
velocities
are
shown
in
Table
12-4.
These
values
com-
computed
from
Dairy's
law:
TABLE
12-4
quantity
can permit hand calculations.
in Fig.
coefficients of permeability. The soil is assumed to be isotropic,
that
be
included in the finite element procedure. The structure itself is assumed to
be
Diriehlet)
boundary
downstream
nodes
are
p.
change
of
g
van:
sr.es
Figure
12-13
governing
differ-
ential
equation,
method
{Q}
integration
shown
in
Example
12-1.
3-6-9
medium
such
section.
12-8.
Consider
Prob.
12-7
but
include
q
wall
Engineering,
of
[6]
Desai,
Under Drawdown, /. Soil
ments
T.,
Taylor,
C,
and
Zien-
kiewicz,
Struct.,
Vol.
4,
No.
1,
1968.
Zienkiewicz,
assumptions,
which
is possible
to approximate
bending
stresses
with
assume that
sider
in the
w
in
the
z
nonzero
stress
components
is
applied.
The
initial
strain
a
quadrilateral
elements
have
we shall discuss
in the energy function,
along the
compatible since
approximation function
{q}
energy; the
potential energy
is given
components
more
follows
M
and so
-3b) and
the b
quadrilateral
order
matrix
[B].
Referring
to
Eq.
(12-20c),
which is one-fourth of the area of the square. Therefore,
use
of
^
^*
u
«
—
—
•*
—
oq cqoq cqoq
oq
u+
as
Buisi,
t
t
u
i
stresses
as
follows:
and
a
as
element
1
degrees
and local
are
(1,
2),
(3,
4),
(5,
6),
and
(7,
8)
(1,
4),
1, 2,
2,
13
by
potential energy
by
diagonal
elements.
The
trace
relatively finer
entire shear
geometry
stress
o
y
at
stiffness
ing
stress concentrations,
by a rock
of the gallery. By using PLANE-2DFE and
assuming plane
y
be
the
that the horizontal displacement
a rigid
providing
loads.
and
of
the
results, it
loading caused
hydro-
static
dures that
problems, let
meshes
as
shown
in
Fig.
13-16.
Note
mesh
TABLE
13-1
results in a faster
given
Thus there exists
a
bending
stress
improves
with corresponding
the
element.
This
be
six
nodes,
and
of
equations
Which of the two approaches, refinement of mesh with a
lower-order
approxima-
wide in
13-9.
13-2.
N
t
in
€
Example
13-2
by
rearranging
the
water level
load due to the
or
is
required
to
(or any
other suitable
loading. Here
beneficial
we need to
one-
structure resting on a bed of (individual) springs representing the
founda-
consist
of
axial
displacement
w,
lateral
displacements
their nodal
given
by
S
=-r
(14-3d)
(14-3e)
(14-4)
where
f
14
{QF
or slabs
components. It
is not
hence, we have
only orthogonal (horizontal
Fig.
14-3(b)
and v [Fig.
have
14-3(b)], s
The interpolation functions
polation
functions.
For
appropriate partial
integration.
Their
(13-24).
(14-14b).
The
soil
solved
by
x
computed from
0.01568 cm.
Figure 14-7(a) shows a plate
fixed
at
two
[3,
8].
This
plate
14-8
the
plate
two
the redistribution of moments
Element
building
levels.
Two
analyses
were
performed
; one
without
floor
significantly reduced
if floor
y
Example
14-5.
Effect
of
Springs
at
Supports
total restraint
analyses for frames.
Chapter
14
our
or common
involved transformations that
slabs
in the building frame, and curved structures (e.g., shells), it is often
necessary to
systems. Then it is
achieved
by
using
a
between
local
and
global
[t]
in Eq.
(14-8a) as
the beam-column
z
axis.
Solution:
1
[k,]
Building, M.E.
covered
relatively
simple
problems
for
which
conventional
solutions
ranging
initial conditions
improved
element
mathematical
properties
these topics.
normed spaces.
and Convection
16. Evaluation and
—
the
available
schemes
Crandall
[1].
a
selected
number
as
n
*-g.
(Al-4)
and
this
(b)
(a)
Collocation,
displacement,
p(x)
terms
of
conditions in
Eq. (A
Chapter
7,
when
conditions [Eq. (Al-8a)]
expression in Eq.
7.57
x
lCPai
the
following
approximation
w
(p
t
Hence,
in
this
to
11,
are beyond the scope
complete the
various
textbooks
[1].
The
finite
difference
continuous
derivative
in
the
5 in. 6
in. 8 in.
shear
forces
by
using
the
presented
herein,
the
are
merits
and Abel, J. F., Introduction to the Finite Element Method,
Van
Jacobi, Gauss-Seidel,
successive over-
[1-3].
We
these techniques.
Gaussian Elimination
and
a
common
method
for
the
idea
of
creating
of
but
than
consider
(A2-3)
by
superscript
(1)
as
first equation
equation is
multiplied by
to the
the
unknown
and
Gauss-
Doolittle
can
equa-
tions
[1-3].
adjacent diagonals,
symmetric;
has
lOO^*
estimate
is
by
selecting
a
small
acceptable
we
accept
x}
l)
as
the
approximate
\xi
m)
solution
by
setting
the
finite
schemes
a
to
the
numerical
characteristics
accuracy and
reliability of
diagonal
elements
a
ti
which
become
number
details of
MODELS FOR CONVERGENCE
plastic. As shown in Fig. 1-8, they can illustrate the
idea
of
of
sides
are
circle.
7)
models. Figure
A3-2(b)
of
d
2
The effect can
can
understand
relevant
teacher and
for solving advanced
sequences such
such as
and
situations such as
Function channels quantity
of
flow
retaining
walls.
can
plot
zones
of
equal
also available.
1 A code for two-dimensional nonlinear analysis of problems idealized
as plane
The
stress-strain
behavior
handling
soil-structure
between the structure
slopes, and
bearing
capacity,
be used to simu-
nodal
direct stiffness
395
concept
weighted residuals,
Transformation
matrix,
finite element
topics
will
also
be
useful
to
the