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PROFESSOR: Ladies and gentlemen,
welcome to
00:00:23.250 --> 00:00:25.410
lecture number 8.
00:00:25.410 --> 00:00:27.780
In this lecture, I would like
to discuss with you the
00:00:27.780 --> 00:00:32.950
numerical integration and some
modeling considerations.
00:00:32.950 --> 00:00:36.010
You might recall that in the
isoparametric formulation that
00:00:36.010 --> 00:00:39.220
we discussed in the early
lectures, we used numerical
00:00:39.220 --> 00:00:40.910
integration.
00:00:40.910 --> 00:00:44.920
And this is an important fact,
and I want to discuss with you
00:00:44.920 --> 00:00:47.820
in this lecture the Newton-Cotes
formulas that
00:00:47.820 --> 00:00:50.420
we're using, and in particular,
the Gauss
00:00:50.420 --> 00:00:51.440
integration--
00:00:51.440 --> 00:00:54.010
the Gauss numerical integration
that we are using
00:00:54.010 --> 00:00:57.550
in finite element analysis.
00:00:57.550 --> 00:00:59.880
There are various
considerations.
00:00:59.880 --> 00:01:02.030
First of all, I'd like to
present to you a little bit of
00:01:02.030 --> 00:01:05.230
the theory that is being used
and in particular, however, I
00:01:05.230 --> 00:01:08.770
also like to present to you some
practical considerations.
00:01:08.770 --> 00:01:14.380
And finally, together with these
aspects, I would like to
00:01:14.380 --> 00:01:18.380
also present to you some
considerations regarding the
00:01:18.380 --> 00:01:21.190
choice of elements, and, of
course, regarding the choice
00:01:21.190 --> 00:01:23.810
of the order of numerical
integration that has to be
00:01:23.810 --> 00:01:27.120
used in the finite element
formulation.
00:01:27.120 --> 00:01:31.740
Well, just to refresh your
memory, on this view graph, I
00:01:31.740 --> 00:01:34.570
put together once the
various matrices
00:01:34.570 --> 00:01:36.870
that we have to evaluate.
00:01:36.870 --> 00:01:39.280
Here, we have to stiffness
matrix, which of course, is
00:01:39.280 --> 00:01:45.130
obtained by the integral of
B transpose C, B. B is the
00:01:45.130 --> 00:01:48.260
strain displacement
transformation matrix.
00:01:48.260 --> 00:01:49.870
C is the stress strain law.
00:01:49.870 --> 00:01:53.290
Of course, this B-matrix is a
function of x and y in the
00:01:53.290 --> 00:01:57.680
physical space, but if you use
isoparametric interpolation in
00:01:57.680 --> 00:02:00.320
two-dimensional analysis, this
B-matrix would be a function
00:02:00.320 --> 00:02:03.500
of r and s, as we discussed
earlier.
00:02:03.500 --> 00:02:07.970
The m matrix is given here, the
RB load vector is given
00:02:07.970 --> 00:02:12.780
here, the RS load vector is
given here, and the initial
00:02:12.780 --> 00:02:14.990
stress load vector
is given here.
00:02:14.990 --> 00:02:20.040
We derived all of these
quantities in an earlier
00:02:20.040 --> 00:02:24.060
lecture and all I like to now
discuss with you is how do we
00:02:24.060 --> 00:02:28.700
evaluate these quantities using
numerical integration.
00:02:28.700 --> 00:02:32.510
In isoparametric finite element
analysis, I mentioned
00:02:32.510 --> 00:02:36.260
already the displacement
transformation matrix is a
00:02:36.260 --> 00:02:38.210
function of r, s, and t.
00:02:38.210 --> 00:02:40.880
The strain-displacement
interpolation matrix is also a
00:02:40.880 --> 00:02:43.860
functional of r, s, and t in
three-dimensional analysis.
00:02:43.860 --> 00:02:46.440
Of course, in one- or
two-dimensional analysis, we
00:02:46.440 --> 00:02:50.110
only talk about r and
s, respectively.
00:02:50.110 --> 00:02:54.230
Remember please, that r, s,
and t vary from minus 1 to
00:02:54.230 --> 00:02:57.980
plus 1, and that we, therefore,
need the volume
00:02:57.980 --> 00:03:02.720
transformation here, dV being
the determinant of J, where J
00:03:02.720 --> 00:03:05.730
is a Jacobian transformation
between the x- and y-, and r-
00:03:05.730 --> 00:03:09.490
and s-axis, or x-, y-, and z-
and r-, s-, and t-axis.
00:03:09.490 --> 00:03:13.190
And of course, dr, dr, dt are
the respective differentiates
00:03:13.190 --> 00:03:14.940
that have to be used
here, too.
00:03:14.940 --> 00:03:19.170
We discussed that earlier, and
just to refresh your memory
00:03:19.170 --> 00:03:25.490
again, what we came up with is
that the stiffness matrix K
00:03:25.490 --> 00:03:28.470
two-dimensional analysis is
given in terms of of B
00:03:28.470 --> 00:03:32.830
transpose C, B, times
determinant of J, dr, ds, and
00:03:32.830 --> 00:03:35.930
we are integrating from
minus 1 to plus 1.
00:03:35.930 --> 00:03:40.330
The mass matrix is given, as
shown here, where H is, of
00:03:40.330 --> 00:03:42.630
course, once again,
the displacement
00:03:42.630 --> 00:03:44.370
transformation matrix.
00:03:44.370 --> 00:03:47.080
[? Row ?] is the mass density.
00:03:47.080 --> 00:03:52.920
Just a preliminary thought,
please recognize that B
00:03:52.920 --> 00:03:57.050
involves derivatives of
displacements, We have here
00:03:57.050 --> 00:04:00.510
derivatives of displacements,
here we have the actual
00:04:00.510 --> 00:04:01.540
displacements.
00:04:01.540 --> 00:04:06.030
So the product of B transpose
B will, in general, be of
00:04:06.030 --> 00:04:11.530
lower order than the product of
H transpose H. Therefore,
00:04:11.530 --> 00:04:14.690
when we talk about numerical
integration, we will find that
00:04:14.690 --> 00:04:18.209
to evaluate the M matrix
exactly, we will have to use a
00:04:18.209 --> 00:04:23.090
higher order integration than in
the exact evaluation of the
00:04:23.090 --> 00:04:24.540
stiffness matrix.
00:04:24.540 --> 00:04:28.180
I will refer to that
again later.
00:04:28.180 --> 00:04:32.780
We also mentioned already that
in the numerical integration,
00:04:32.780 --> 00:04:38.700
the basic process is that we
are evaluating the function
00:04:38.700 --> 00:04:47.120
Fij, as shown here, at
particular points, i, j,
00:04:47.120 --> 00:04:48.720
denoted by i, j.
00:04:48.720 --> 00:04:52.460
And the alpha i, j, are
weight coefficients.
00:04:52.460 --> 00:04:57.310
Therefore, what we do is, once
again, to evaluate this
00:04:57.310 --> 00:05:02.630
product at a particular point
within the element, multiply
00:05:02.630 --> 00:05:06.870
that product by a weight
coefficient, and then do the
00:05:06.870 --> 00:05:12.180
same for a certain selected
number of points, and sum all
00:05:12.180 --> 00:05:17.080
these evaluations up, as
shown in this equation.
00:05:17.080 --> 00:05:19.960
Now, what this means, of course,
is that we really have
00:05:19.960 --> 00:05:21.890
to use specific points--
00:05:21.890 --> 00:05:26.060
and I have to discuss with you
what points we are using--
00:05:26.060 --> 00:05:29.100
and how many points is another
question that we
00:05:29.100 --> 00:05:30.490
have to also discuss.
00:05:30.490 --> 00:05:36.440
Well, in 2x2 integration, in
isoparametric finite element
00:05:36.440 --> 00:05:42.150
analysis, we would use four
points altogether.
00:05:42.150 --> 00:05:45.390
2x2 means two points
this direction, and
00:05:45.390 --> 00:05:47.550
two points that direction.
00:05:47.550 --> 00:05:49.610
In three- dimensional analysis,
we would, of course,
00:05:49.610 --> 00:05:51.970
talk about 2x2x2.
00:05:51.970 --> 00:05:55.030
You would have another two
points in this direction, in
00:05:55.030 --> 00:05:56.660
the t direction.
00:05:56.660 --> 00:06:01.540
These points in Gauss
integration, are given by
00:06:01.540 --> 00:06:04.390
these values here.
00:06:04.390 --> 00:06:07.320
In other words, point 4, for
example, would have the
00:06:07.320 --> 00:06:16.150
values, r equals plus 0.577
and s equal to plus 0.577.
00:06:16.150 --> 00:06:20.800
Point 1 would have the same
values, but both negative.
00:06:20.800 --> 00:06:24.370
These values here are the
Gauss values, and I will
00:06:24.370 --> 00:06:27.340
discuss briefly with you a
little later how these have
00:06:27.340 --> 00:06:28.240
been obtained.
00:06:28.240 --> 00:06:32.910
Of course, Gauss derived them
for us some 100 years ago, and
00:06:32.910 --> 00:06:34.930
we are now simply using them.
00:06:34.930 --> 00:06:39.140
In 3x3 integration, we talk
about nine integration points
00:06:39.140 --> 00:06:40.580
altogether.
00:06:40.580 --> 00:06:44.830
Once again, three integration
points lay, as so to say, in
00:06:44.830 --> 00:06:47.680
this direction, and into
that direction.
00:06:47.680 --> 00:06:49.790
If we use three-point
integration in
00:06:49.790 --> 00:06:53.510
three-dimensional analysis, we
would have altogether 27
00:06:53.510 --> 00:06:56.880
integration points because
each of these layers, of
00:06:56.880 --> 00:07:01.290
course, applies then three
times, and because we also
00:07:01.290 --> 00:07:04.220
have three layers into the t
direction, and therefore
00:07:04.220 --> 00:07:09.270
altogether, 9x3, being 27
integration points.
00:07:09.270 --> 00:07:15.280
The basic process of numerical
integration is that if we want
00:07:15.280 --> 00:07:17.180
to integrate a function.
00:07:17.180 --> 00:07:21.490
And I've shown here an actual
function-- this heavy line is
00:07:21.490 --> 00:07:25.130
the actual function f-- if we
want to integrate that actual
00:07:25.130 --> 00:07:30.640
function, with respect
to x here.
00:07:30.640 --> 00:07:35.420
What we do is we are putting
an interpolating polynomial
00:07:35.420 --> 00:07:38.060
down as an approximation
to that function.
00:07:38.060 --> 00:07:43.140
Now, if we want to integrate
this function from a to b, the
00:07:43.140 --> 00:07:46.480
simplest interpolating
polynomial would be to put a
00:07:46.480 --> 00:07:50.920
straight line from this
point to that point.
00:07:50.920 --> 00:07:56.740
And then the area under the
straight line is assumed to be
00:07:56.740 --> 00:08:03.450
approximately, at least, really
sufficiently close, to
00:08:03.450 --> 00:08:07.790
the area under the actual
function f.
00:08:07.790 --> 00:08:12.410
So what we are doing then is we
are replacing the integral
00:08:12.410 --> 00:08:19.140
of the actual function f, over
this integral, by the integral
00:08:19.140 --> 00:08:26.250
of the interpolating polynomial
over that integral.
00:08:26.250 --> 00:08:30.620
If we do that with just two
points, and the two points are
00:08:30.620 --> 00:08:37.130
these two, we will derive
the trapezoidal rule.
00:08:37.130 --> 00:08:42.169
If we use those three points,
and I have here a view graph
00:08:42.169 --> 00:08:45.370
that shows how we would
use three points.
00:08:45.370 --> 00:08:48.665
One point here, second point
there, and now we have also a
00:08:48.665 --> 00:08:49.930
third point there.
00:08:49.930 --> 00:08:53.290
Then we would obtain
the Simpson's rule.
00:08:53.290 --> 00:08:56.260
Once again, instead of
integrating the actual
00:08:56.260 --> 00:09:04.820
function, f, which is shown by
this line here, we are, in
00:09:04.820 --> 00:09:09.880
fact, integrating the area
under this red line.
00:09:09.880 --> 00:09:11.760
This is a little the weak.
00:09:11.760 --> 00:09:14.371
Well, now there it comes.
00:09:14.371 --> 00:09:18.190
We are integrating the area
under the red line.
00:09:18.190 --> 00:09:23.830
In other words, the assumption
is then that this area is
00:09:23.830 --> 00:09:31.220
close enough to the area under
blue line, and what we need to
00:09:31.220 --> 00:09:38.260
do is to evaluate, of course,
the actual function, f, at
00:09:38.260 --> 00:09:41.260
these three points.
00:09:41.260 --> 00:09:45.950
And we will have to somehow
develop certain constants, the
00:09:45.950 --> 00:09:51.180
alpha i, j's that I talked about
earlier that multiply
00:09:51.180 --> 00:09:58.120
this function, f, in order to
evaluate the red area here
00:09:58.120 --> 00:09:59.990
accurately.
00:09:59.990 --> 00:10:03.590
Well, if we do that--
00:10:03.590 --> 00:10:09.890
in other words, if we integrate
the red area using
00:10:09.890 --> 00:10:15.200
these three function points then
we obtain the Simpson's
00:10:15.200 --> 00:10:20.470
rule, and this is just one
rule of the more general
00:10:20.470 --> 00:10:23.930
Newton-Cotes integration.
00:10:23.930 --> 00:10:27.360
In the Newton-Cotes integration,
we use
00:10:27.360 --> 00:10:29.860
equally-spaced sampling
points.
00:10:29.860 --> 00:10:35.680
In other words, if we want to
integrate from a to b, the
00:10:35.680 --> 00:10:39.790
simplest way of proceeding is to
use a sampling point here,
00:10:39.790 --> 00:10:41.860
and a sampling point there.
00:10:41.860 --> 00:10:44.540
And then, of course, as I
pointed out earlier, we are
00:10:44.540 --> 00:10:46.820
putting, really, a straight line
down and we're assuming
00:10:46.820 --> 00:10:51.610
that the area under this
straight line is equal to the
00:10:51.610 --> 00:10:57.650
area to under the actual
function, f, that might be
00:10:57.650 --> 00:11:00.280
looking like that.
00:11:00.280 --> 00:11:06.310
Well, that is the trapezoidal
rule and Newton-Cotes
00:11:06.310 --> 00:11:13.080
integration with n equal to 1.
00:11:13.080 --> 00:11:17.470
In other words, with two
constants here, is equal to
00:11:17.470 --> 00:11:20.180
this trapezoidal rule.
00:11:20.180 --> 00:11:23.970
What we are saying is that the
area from a to b-- now I used
00:11:23.970 --> 00:11:29.150
here r, this is the r-axis.
say, from a to b--
00:11:29.150 --> 00:11:30.590
this is the actual
integration--
00:11:30.590 --> 00:11:38.370
is replaced by the summation
of the value at this point,
00:11:38.370 --> 00:11:43.640
the F0 value, the F1 value, and
each of these values is
00:11:43.640 --> 00:11:47.030
multiplied by a certain
constant.
00:11:47.030 --> 00:11:51.840
So in this particular case, for
the trapezoidal rules that
00:11:51.840 --> 00:11:55.080
I'm talking about, the right
hand side looks as follows-- b
00:11:55.080 --> 00:12:01.200
minus a, that is this length
here, times two values, the
00:12:01.200 --> 00:12:08.050
C01, n is equal to 1 in this
particular case, times F0.
00:12:08.050 --> 00:12:16.200
Plus C1 1 1 times F1.
00:12:16.200 --> 00:12:21.800
And of course then, plus an
error, R in this particular
00:12:21.800 --> 00:12:25.870
case, n, once again, is
equal to 1, plus R1.
00:12:25.870 --> 00:12:29.670
This error of course, is assumed
to be very small in
00:12:29.670 --> 00:12:32.850
the analysis, and we would
actually neglect it.
00:12:32.850 --> 00:12:35.880
So once again, what we are
saying is that we can replace
00:12:35.880 --> 00:12:42.050
the integration of F of R dr,
from a to b, via this
00:12:42.050 --> 00:12:43.240
summation here.
00:12:43.240 --> 00:12:47.250
Notice that b minus a times
that constant, that is our
00:12:47.250 --> 00:12:50.710
alpha 1, that I referred
to earlier.
00:12:50.710 --> 00:12:53.960
And b minus a times that
constant would be another
00:12:53.960 --> 00:12:55.830
alpha value.
00:12:55.830 --> 00:12:59.280
And, of course, these alpha
values here multiply the
00:12:59.280 --> 00:13:04.770
actual function values at this
station, and at that station.
00:13:04.770 --> 00:13:10.560
Well in the Simpson rule, n is
equal to 2, b minus a is still
00:13:10.560 --> 00:13:17.707
there, but now we're talking
about C0 2 0 times F0 plus C1
00:13:17.707 --> 00:13:26.640
2 times F1 plus C2 2 times
F2, and of course, an
00:13:26.640 --> 00:13:29.880
error of 2's there.
00:13:29.880 --> 00:13:37.000
Well, the constants, the
C0 1, C1 1 and so on.
00:13:37.000 --> 00:13:43.440
The constant Cin have been
evaluated for us, and they are
00:13:43.440 --> 00:13:47.190
tabulated in this table here.
00:13:47.190 --> 00:13:52.340
Notice, that for the trapezoidal
rule, we have the
00:13:52.340 --> 00:13:55.740
simple constants, 1/2, 1/2.
00:13:55.740 --> 00:14:01.740
The error here is shown here,
10 to the minus 1 times the
00:14:01.740 --> 00:14:05.370
second derivative of F. So if
the second derivative of the
00:14:05.370 --> 00:14:11.220
actual function is 0, well, then
our integration is exact.
00:14:11.220 --> 00:14:15.670
Of course, what that means is
that our function is indeed a
00:14:15.670 --> 00:14:16.950
straight line.
00:14:16.950 --> 00:14:20.640
Because only then the second
derivative is 0.
00:14:20.640 --> 00:14:23.590
So our integration
is exact when we
00:14:23.590 --> 00:14:25.040
integrate a straight line.
00:14:25.040 --> 00:14:29.910
Well, if you look at our earlier
graph here, surely, if
00:14:29.910 --> 00:14:32.630
our actual red line is a
straight line between this
00:14:32.630 --> 00:14:36.190
point and that point, then our
integration, using the
00:14:36.190 --> 00:14:38.350
trapezoidal rule,
must be exact.
00:14:38.350 --> 00:14:42.570
And that is verified by looking
at the error bound,
00:14:42.570 --> 00:14:44.460
given in this table.
00:14:44.460 --> 00:14:50.700
The Simpson's rule has
constants, 1/6, 4/6, 1/6.
00:14:50.700 --> 00:14:53.340
n is equal to 2 in this
particular case, as I
00:14:53.340 --> 00:14:55.040
mentioned earlier.
00:14:55.040 --> 00:14:59.720
The error here is now given
to the force derivative.
00:14:59.720 --> 00:15:01.280
And so on.
00:15:01.280 --> 00:15:03.510
Here, we have integrals
1 to 6.
00:15:03.510 --> 00:15:08.220
In practice, however, we would
only use this the trapezoidal
00:15:08.220 --> 00:15:11.840
rule, the Simpson rule, and
we might then still
00:15:11.840 --> 00:15:15.940
use this rule here.
00:15:15.940 --> 00:15:20.450
Notice that the error of this
rule here is of the same order
00:15:20.450 --> 00:15:23.850
as the error of this rule,
involving five points.
00:15:23.850 --> 00:15:27.920
So the four-point integration
rule, of course, is preferable
00:15:27.920 --> 00:15:30.880
when you compare to the
five-point integration rule.
00:15:30.880 --> 00:15:35.230
The other important point
is that if we use the
00:15:35.230 --> 00:15:41.010
Newton-Cotes formally, we are
really always involving the
00:15:41.010 --> 00:15:42.750
endpoints of the integration.
00:15:42.750 --> 00:15:50.090
In other words, if we want to
integrate from a to b then we
00:15:50.090 --> 00:15:53.625
will involve these two points
in the actual integration.
00:15:56.220 --> 00:15:59.020
So if we talk about the
integration through the
00:15:59.020 --> 00:16:02.760
thickness off the shell, or
through a beam element
00:16:02.760 --> 00:16:07.250
thickness, then we would have
here the top and bottom
00:16:07.250 --> 00:16:08.500
surfaces involved.
00:16:15.380 --> 00:16:16.200
Bad point--
00:16:16.200 --> 00:16:19.570
well, I shouldn't say bad, but
certainly it's not a very
00:16:19.570 --> 00:16:24.070
advantageous point, of the
Newton-Cotes integration is
00:16:24.070 --> 00:16:29.910
that we need to involve a
fairly large number of
00:16:29.910 --> 00:16:33.130
integration points, or
integration point stations to
00:16:33.130 --> 00:16:37.240
obtain an acceptable accuracy
when we talk about finite
00:16:37.240 --> 00:16:38.790
element analysis.
00:16:38.790 --> 00:16:43.250
And therefore, the Gauss
numerical integration is a
00:16:43.250 --> 00:16:45.350
more attractive scheme.
00:16:45.350 --> 00:16:48.230
And, in fact, we will find
that in finite element
00:16:48.230 --> 00:16:53.730
analysis, we almost always use
Gauss numerical integration.
00:16:53.730 --> 00:16:58.010
They're only certain special
cases in analysis of beams,
00:16:58.010 --> 00:17:01.250
plates, and shells where we, in
fact, use the Newton-Cotes
00:17:01.250 --> 00:17:02.030
integration.
00:17:02.030 --> 00:17:04.290
And the reason then, of course,
is that we want to
00:17:04.290 --> 00:17:08.560
obtain also surface stresses
and strains.
00:17:08.560 --> 00:17:10.310
That would be, for example,
one reason.
00:17:10.310 --> 00:17:13.710
Now, in Gauss numerical
integration, the basic concept
00:17:13.710 --> 00:17:17.069
is quite similar to the concept
00:17:17.069 --> 00:17:18.130
that we just discussed.
00:17:18.130 --> 00:17:22.170
That we again use an
interpolating polynomial to
00:17:22.170 --> 00:17:27.630
lay through the function over
the interval, a to b.
00:17:27.630 --> 00:17:32.520
The important point, however,
now is that we use--
00:17:32.520 --> 00:17:35.200
if we look at our
function here.
00:17:35.200 --> 00:17:38.280
Here is the interval, a to b.
00:17:38.280 --> 00:17:39.980
Our rx is here.
00:17:39.980 --> 00:17:43.470
If you look at that function,
it's that we, in Gauss
00:17:43.470 --> 00:17:49.220
numerical integration, do not
use equal-spaced intervals.
00:17:49.220 --> 00:17:52.770
In Newton-Cotes integration, we
use equal-spaced intervals,
00:17:52.770 --> 00:17:55.520
and involves the end points.
00:17:55.520 --> 00:17:59.900
In Gauss numerical integration,
we are using
00:17:59.900 --> 00:18:07.960
points, stations, 1 to 2, and
our stations that can be
00:18:07.960 --> 00:18:10.400
anywhere in the interval.
00:18:10.400 --> 00:18:16.550
And Gauss has developed the
optimum stations for us.
00:18:16.550 --> 00:18:22.110
in other words, what he has
said, at that time is let us
00:18:22.110 --> 00:18:26.630
optimize the particular
locations of the integration
00:18:26.630 --> 00:18:31.990
points stations, and in
addition, let us optimize the
00:18:31.990 --> 00:18:35.620
weights, alpha 1 to alpha n.
00:18:35.620 --> 00:18:40.000
In the Newton-Cotes integration,
we only optimize,
00:18:40.000 --> 00:18:42.540
so to say, the alpha values.
00:18:42.540 --> 00:18:46.210
The stations were fixed because
we involved, for
00:18:46.210 --> 00:18:49.340
example, in the trapezoidal
rule, we involved this end
00:18:49.340 --> 00:18:50.880
point, and that point.
00:18:50.880 --> 00:18:54.070
In the Simpson's rule,
we used the midpoint.
00:18:54.070 --> 00:18:55.070
And so on.
00:18:55.070 --> 00:18:57.820
In Gauss integration, we say,
well, let us does not
00:18:57.820 --> 00:19:00.680
necessarily fix those stations
but rather let us just
00:19:00.680 --> 00:19:03.370
optimize the location
of these stations.
00:19:03.370 --> 00:19:08.460
Well, since we then can optimize
the values of the
00:19:08.460 --> 00:19:14.410
alpha 1, to alpha n, this of
course are n, components, and
00:19:14.410 --> 00:19:18.530
the stations R1 to Rn, which
are also n components.
00:19:18.530 --> 00:19:22.530
We really can optimize
two n components.
00:19:22.530 --> 00:19:26.350
Now, if we look at the fact that
we're dealing with two n
00:19:26.350 --> 00:19:30.990
components that we optimize
now, and the fact that the
00:19:30.990 --> 00:19:34.400
interpolating polynomial of
course, also involves one
00:19:34.400 --> 00:19:39.670
constant, then we find that the
interpolating polynomial
00:19:39.670 --> 00:19:43.230
can now be of order,
2n minus 1.
00:19:43.230 --> 00:19:45.650
2n minus 1.
00:19:45.650 --> 00:19:51.340
Whereas in the Simpson's rule,
the interpolating polynomial
00:19:51.340 --> 00:19:55.940
with n stations could only
be of order, n minus 1.
00:19:55.940 --> 00:20:00.200
You see, in the Simpsons rule,
for example, with three
00:20:00.200 --> 00:20:06.220
stations, the n was, in the
Simpson's rule equal to 3, but
00:20:06.220 --> 00:20:11.280
the order of the interpolating
polynomial was 2.
00:20:11.280 --> 00:20:17.630
In other words, a parabola could
be integrated exactly.
00:20:17.630 --> 00:20:21.860
Now in the Gauss integration,
we are talking about the
00:20:21.860 --> 00:20:26.640
interpolating polynomial that
can be of order, 2n minus 1, a
00:20:26.640 --> 00:20:31.880
much higher order
for n stations.
00:20:31.880 --> 00:20:35.660
Here, with the Newton-Cotes rule
for n stations, we only
00:20:35.660 --> 00:20:39.100
get an interpreting polynomial
of n minus 1.
00:20:39.100 --> 00:20:48.180
Well, Gauss has developed the
specific stations, R1 to Rn,
00:20:48.180 --> 00:20:50.270
to be used [UNINTELLIGIBLE]
00:20:50.270 --> 00:20:54.630
in order to fit the
interpolating polynomial
00:20:54.630 --> 00:21:01.030
through the actual function
that we want to integrate.
00:21:01.030 --> 00:21:05.680
And of course, we also know
now the error that is
00:21:05.680 --> 00:21:07.860
involved, Rn.
00:21:07.860 --> 00:21:11.190
The error of course, being
much smaller than in the
00:21:11.190 --> 00:21:14.110
Newton-Cotes formula because
we are dealing with a
00:21:14.110 --> 00:21:17.340
higher-order interpolating
polynomial.
00:21:17.340 --> 00:21:21.100
The Gauss values
are given here.
00:21:21.100 --> 00:21:25.655
When n is equal to 1, well, we
only use the midpoint and the
00:21:25.655 --> 00:21:27.820
alpha value is 2.
00:21:27.820 --> 00:21:30.600
When n is equal to 2, in other
words, we are talking about
00:21:30.600 --> 00:21:35.670
two-point integration, then
ri is equal to plus
00:21:35.670 --> 00:21:38.750
minus 0.577, and 1.
00:21:38.750 --> 00:21:44.330
In this particular case, we
are using the fact that we
00:21:44.330 --> 00:21:48.660
want to integrate 4 minus
1, to plus 1.
00:21:48.660 --> 00:21:52.230
In other words, our
r here, is this.
00:21:52.230 --> 00:21:54.880
We coordinate here, and
we're integrating from
00:21:54.880 --> 00:21:57.450
minus 1 to plus 1.
00:21:57.450 --> 00:22:03.090
So in two-point integration, we
using a point here, which
00:22:03.090 --> 00:22:07.290
is minus 0.577, and another
point here,
00:22:07.290 --> 00:22:11.090
which is plus 0.577.
00:22:11.090 --> 00:22:14.740
Notice that we are feeding in
these points to a higher-order
00:22:14.740 --> 00:22:19.760
accuracy in a computer program,
typically, in a
00:22:19.760 --> 00:22:21.990
computer program that runs
on a CDC machine
00:22:21.990 --> 00:22:23.800
with 14-digit precision.
00:22:23.800 --> 00:22:25.350
We would actually
put these points
00:22:25.350 --> 00:22:28.440
in to 14-digit precision.
00:22:28.440 --> 00:22:33.660
Of course, they are actually put
into this [? outputing. ?]
00:22:33.660 --> 00:22:35.880
They are there, and the computer
program just picks up
00:22:35.880 --> 00:22:38.850
this value and evaluates then,
of course, the function that
00:22:38.850 --> 00:22:41.740
we want to integrate
at that station.
00:22:41.740 --> 00:22:45.420
The factor that the function
has to be multiplied by is
00:22:45.420 --> 00:22:47.690
given here.
00:22:47.690 --> 00:22:51.960
In three-point integration,
we use three points.
00:22:51.960 --> 00:22:53.720
This one is the easiest
to look at--
00:22:53.720 --> 00:22:56.590
it's right to center point, and
then there are two more
00:22:56.590 --> 00:23:03.050
points, one at minus 0.77
and one at plus 0.77.
00:23:03.050 --> 00:23:07.945
In finite element analysis,
we know, of course, in
00:23:07.945 --> 00:23:10.410
isoparametric finite element
analysis, I should really say,
00:23:10.410 --> 00:23:12.530
we know, of course, that our
integration runs from
00:23:12.530 --> 00:23:13.910
minus 1 to plus 1.
00:23:13.910 --> 00:23:17.510
So these are the actual
values to be used.
00:23:17.510 --> 00:23:19.390
And these are the actual values
that you would find,
00:23:19.390 --> 00:23:23.220
for example, in the computer
program such as ADINA.
00:23:23.220 --> 00:23:32.720
If we want to integrate an
interval from a to b--
00:23:32.720 --> 00:23:36.130
I'd like to simply mention
that, of course, the same
00:23:36.130 --> 00:23:39.530
scheme can directly be used.
00:23:39.530 --> 00:23:43.540
If we have the ri and alpha i
for the interval from minus 1
00:23:43.540 --> 00:23:49.370
to plus 1, then these would be
the value to be used for the
00:23:49.370 --> 00:23:51.330
integration point stations.
00:23:51.330 --> 00:23:54.690
If our interval is not from
minus 1 to plus 1. but from a
00:23:54.690 --> 00:23:58.680
to b, and this would be the
weight that would have to be
00:23:58.680 --> 00:24:03.030
used for this particular case.
00:24:03.030 --> 00:24:07.440
The i in alpha i, once again,
are the same ri and alpha 1
00:24:07.440 --> 00:24:11.640
that I listed here
in this table.
00:24:11.640 --> 00:24:17.290
Well, so far we really talked
about one-dimensional
00:24:17.290 --> 00:24:19.180
integration.
00:24:19.180 --> 00:24:22.130
However, the same concept can
directly be used for multiple
00:24:22.130 --> 00:24:23.790
dimension integration.
00:24:23.790 --> 00:24:29.590
In other words, if we talk
about the integration,
00:24:29.590 --> 00:24:32.460
minus 1 to plus 1.
00:24:32.460 --> 00:24:35.150
For example, as we would
integrate, of course, in the
00:24:35.150 --> 00:24:37.810
formation of a tress element.
00:24:37.810 --> 00:24:40.250
If we now want to go through
the integration of a
00:24:40.250 --> 00:24:43.750
two-dimensional element, then
we would have to integrate a
00:24:43.750 --> 00:24:47.630
two-dimensional function in
r and s, and that means
00:24:47.630 --> 00:24:50.495
integrating twice, once with
respect to r, once with
00:24:50.495 --> 00:24:54.030
respect to s, and we can use the
same concept that we used
00:24:54.030 --> 00:24:54.620
in [? analytical ?]
00:24:54.620 --> 00:24:58.880
integrations, and we first
integrate with respect to r.
00:24:58.880 --> 00:25:01.110
That gives us one alpha
i here, and then
00:25:01.110 --> 00:25:03.040
with respect to s.
00:25:03.040 --> 00:25:07.440
And the final result is shown
here, that we have to bring in
00:25:07.440 --> 00:25:11.020
a weight factor, alpha i, and
a weight factor, alpha j.
00:25:11.020 --> 00:25:13.312
This one for the i integration,
that one for the
00:25:13.312 --> 00:25:14.340
a integration.
00:25:14.340 --> 00:25:18.750
And of course, stations ri and
sj are the stations at which
00:25:18.750 --> 00:25:20.840
we would evaluate the
actual function.
00:25:20.840 --> 00:25:25.440
Notice one important point that
we could use a different
00:25:25.440 --> 00:25:27.900
order of integration into
the two directions.
00:25:27.900 --> 00:25:31.320
For example, if we have an
element such as this one,
00:25:31.320 --> 00:25:33.250
two-dimension element,
two nodes
00:25:33.250 --> 00:25:36.020
here, in the s direction.
00:25:36.020 --> 00:25:38.440
But say three nodes in
the r direction.
00:25:38.440 --> 00:25:45.665
So our r-axis is this one,
s-axis is that one.
00:25:45.665 --> 00:25:48.420
Then in this particular case,
since we're using a higher
00:25:48.420 --> 00:25:52.290
order interpolation into the r
direction, in this particular
00:25:52.290 --> 00:25:55.340
case, we might say, well, let us
use three-point integration
00:25:55.340 --> 00:26:00.610
this way, but two-point
integration into the s-axis.
00:26:00.610 --> 00:26:03.350
So here, two-point integration
into the s-axis, but
00:26:03.350 --> 00:26:06.150
three-point integration
into the r-axis.
00:26:06.150 --> 00:26:08.990
And of course, the same concept
applies also in
00:26:08.990 --> 00:26:10.430
three-dimensional analysis.
00:26:10.430 --> 00:26:14.060
In three-dimensional analysis
as an example here, we would
00:26:14.060 --> 00:26:19.370
have the F now being a function
of r, s, and t.
00:26:19.370 --> 00:26:22.470
Three times integration from
minus 1 to plus 1.
00:26:22.470 --> 00:26:26.440
And the final result just
derived the same way as in
00:26:26.440 --> 00:26:28.270
two-dimensional analysis
as given here.
00:26:28.270 --> 00:26:31.980
Notice once again, we can
use different orders of
00:26:31.980 --> 00:26:35.330
integration into the
three directions.
00:26:35.330 --> 00:26:38.830
I should also mention at this
point that when we talk about
00:26:38.830 --> 00:26:42.210
an actual stiffness matrix, of
course, we talk about an
00:26:42.210 --> 00:26:48.920
array, n by n, for
three-dimensional element, say
00:26:48.920 --> 00:26:50.460
a 20-node brick.
00:26:50.460 --> 00:26:55.250
This would be a 60 by 60
stiffness matrix, and every
00:26:55.250 --> 00:26:59.700
one of these elements here, in
that stiffness matrix would be
00:26:59.700 --> 00:27:03.490
an F, such an F. Every one of
these elements would be
00:27:03.490 --> 00:27:05.310
integrated as shown here.
00:27:08.140 --> 00:27:11.150
Let us now look at the practical
use of numerical
00:27:11.150 --> 00:27:13.140
integration.
00:27:13.140 --> 00:27:20.070
Well, the first important point
is that the order of
00:27:20.070 --> 00:27:24.060
integration that is required to
evaluate a specific element
00:27:24.060 --> 00:27:28.950
matrix can, of course, be
evaluated by studying the
00:27:28.950 --> 00:27:31.300
function, f, to be integrated.
00:27:31.300 --> 00:27:36.990
In other words, if I want to
evaluate a certain stiffness
00:27:36.990 --> 00:27:40.880
matrix, K. Here is the stiffness
matrix given.
00:27:40.880 --> 00:27:45.140
If I want to evaluate that
stiffness matrix exactly, I
00:27:45.140 --> 00:27:47.990
could look at each all of these
elements, and there are
00:27:47.990 --> 00:27:51.550
now many of these F functions
in there.
00:27:51.550 --> 00:27:53.970
Let us look at that one.
00:27:53.970 --> 00:27:59.880
I would look at that F value and
identify its dependency on
00:27:59.880 --> 00:28:02.270
i, s, and t.
00:28:02.270 --> 00:28:05.280
Linear, cubic, parabolic
and so on.
00:28:05.280 --> 00:28:09.500
And then I could, by
entering into--
00:28:09.500 --> 00:28:12.980
by the use of the Gauss numeric
integration, or the
00:28:12.980 --> 00:28:17.780
Newton-Cotes integration, and
by looking up the errors
00:28:17.780 --> 00:28:22.000
involved, I could identify what
integration order I need
00:28:22.000 --> 00:28:27.500
in order to evaluate this
integral exactly.
00:28:27.500 --> 00:28:34.080
That can be done and for simple
elements, relatively
00:28:34.080 --> 00:28:34.920
simple elements--
00:28:34.920 --> 00:28:39.770
in other words, 4-noded
elements, that are square, or
00:28:39.770 --> 00:28:42.020
8-noded elements that
are square.
00:28:42.020 --> 00:28:46.050
We can fairly simply evaluate
the order of integration that
00:28:46.050 --> 00:28:47.090
is required.
00:28:47.090 --> 00:28:49.500
However, when the element
is curved.
00:28:49.500 --> 00:28:52.040
In other words, when we
talk about a general
00:28:52.040 --> 00:28:54.030
elements like that--
00:28:54.030 --> 00:28:58.930
the required integration order
is not that easily assessed,
00:28:58.930 --> 00:29:02.290
in order to evaluate the
stiffness matrix exactly.
00:29:02.290 --> 00:29:04.840
And the reason for it is that
we have the Jacobian
00:29:04.840 --> 00:29:07.300
transformation in there.
00:29:07.300 --> 00:29:11.400
Therefore, in practice, the
integration is frequently not
00:29:11.400 --> 00:29:15.530
performed exactly, but we have
to remember that the
00:29:15.530 --> 00:29:19.060
integration order must
be high enough.
00:29:19.060 --> 00:29:23.020
Well, considering then the
evaluation of the element
00:29:23.020 --> 00:29:28.560
matrices, we should notice the
following requirements.
00:29:28.560 --> 00:29:31.190
For the stiffness matrix
evaluation, we note that the
00:29:31.190 --> 00:29:35.650
element matrix should not
contain any spurious zero
00:29:35.650 --> 00:29:37.400
energy modes.
00:29:37.400 --> 00:29:38.570
What do I mean by that?
00:29:38.570 --> 00:29:44.550
Well, I mean that if I have
an element such as
00:29:44.550 --> 00:29:46.420
that element here.
00:29:49.750 --> 00:29:54.540
We know that if you're talking
about a plane stress element,
00:29:54.540 --> 00:29:57.370
there only three rigid
body modes.
00:29:57.370 --> 00:30:01.390
One rigid translation this way,
one rigid translation
00:30:01.390 --> 00:30:03.810
that way, and one rigid
rotation of the
00:30:03.810 --> 00:30:04.950
elements that way.
00:30:04.950 --> 00:30:09.250
These are the only three rigid
body modes that the element
00:30:09.250 --> 00:30:12.000
stiffness matrix
should contain.
00:30:12.000 --> 00:30:17.870
It should, of course, contain
these in order to satisfy the
00:30:17.870 --> 00:30:20.360
convergence requirements that
we discussed earlier.
00:30:20.360 --> 00:30:24.230
What I'm saying now is that my
integration order should be
00:30:24.230 --> 00:30:29.800
high enough so that there is
no additional 0 igon value.
00:30:32.300 --> 00:30:36.160
If there is an additional 0 igon
value, I might run into
00:30:36.160 --> 00:30:40.230
serious numerical difficulties
in the solution of the overall
00:30:40.230 --> 00:30:42.030
governing equilibrium
equations.
00:30:42.030 --> 00:30:45.500
And I will give you just now
a very simple example.
00:30:45.500 --> 00:30:50.010
The second requirement is that
the element stiffness matrix
00:30:50.010 --> 00:30:53.600
should contain the required
constant strain states.
00:30:53.600 --> 00:31:01.200
So basically, what I'm saying
here is that the numerically
00:31:01.200 --> 00:31:05.100
integrated stiffness matrix
should still contain the
00:31:05.100 --> 00:31:09.720
requirements that we discussed,
which the element
00:31:09.720 --> 00:31:14.170
must satisfy for convergence.
00:31:14.170 --> 00:31:18.170
For the mass matrix evaluation,
here we only need
00:31:18.170 --> 00:31:21.150
to satisfy that the total
mass must be included.
00:31:21.150 --> 00:31:24.140
In other words, the numerical
integration should pick up the
00:31:24.140 --> 00:31:26.150
actual mass of the element.
00:31:26.150 --> 00:31:30.850
If that were not the case, then
considering the total
00:31:30.850 --> 00:31:33.750
analysis, we would have
lost some mass.
00:31:33.750 --> 00:31:36.570
The force vector evaluation
valuation here, also is a
00:31:36.570 --> 00:31:37.350
total loads.
00:31:37.350 --> 00:31:40.550
Must be included, must be picked
up, so to say, by the
00:31:40.550 --> 00:31:43.270
numerical integration.
00:31:43.270 --> 00:31:47.890
Let us look at this requirement
here, and then I
00:31:47.890 --> 00:31:50.436
have a very simple example.
00:31:50.436 --> 00:31:55.820
Here we have the 8-node element,
plane stress element,
00:31:55.820 --> 00:31:58.360
which is supported here
on a hinge, and
00:31:58.360 --> 00:31:59.990
here on a stiff string.
00:31:59.990 --> 00:32:02.870
I apply a load, p, there.
00:32:02.870 --> 00:32:06.680
If I use 2x2 Gauss integration,
00:32:06.680 --> 00:32:09.200
I get absurd results.
00:32:09.200 --> 00:32:12.190
If I use 3x3 Gauss
integration, I
00:32:12.190 --> 00:32:13.540
get the correct results.
00:32:13.540 --> 00:32:16.510
The correct results, of
course, being that the
00:32:16.510 --> 00:32:19.780
elements simply rotates
about this point, a.
00:32:19.780 --> 00:32:23.340
It simply rotates about the
point, a, as a rigid body.
00:32:26.570 --> 00:32:29.090
And, of course, extending
in that rotation's a
00:32:29.090 --> 00:32:31.610
spring at point b.
00:32:31.610 --> 00:32:34.940
The absurd results
here come about--
00:32:34.940 --> 00:32:37.290
using 2x2 integration--
00:32:37.290 --> 00:32:43.710
because the stiffness matrix k
does not contain just three
00:32:43.710 --> 00:32:50.150
rigid body modes, but four rigid
body modes, with 2x2
00:32:50.150 --> 00:32:51.560
Gauss integration.
00:32:51.560 --> 00:32:56.270
Now, three rigid body modes are
suppressed by the hinge
00:32:56.270 --> 00:32:59.880
here and the spring there,
they are suppressed.
00:32:59.880 --> 00:33:03.350
But if I look at the fact that
we are now having four in the
00:33:03.350 --> 00:33:07.250
stiffness matrix, if I suppress
three, I still am
00:33:07.250 --> 00:33:11.990
left with one rigid body
mode, and that
00:33:11.990 --> 00:33:13.540
means one 0 igon value.
00:33:13.540 --> 00:33:19.920
So this is still an unstable
structural stiffness matrix,
00:33:19.920 --> 00:33:22.840
and when I subject that
stiffness matrix two a load p,
00:33:22.840 --> 00:33:25.020
I get absurd results.
00:33:25.020 --> 00:33:27.990
Therefore, this is a very
simple example that
00:33:27.990 --> 00:33:32.000
demonstrates that the Gauss
integration has to be high
00:33:32.000 --> 00:33:36.520
enough in order to prevent
instabilities occurring in the
00:33:36.520 --> 00:33:38.990
solution of the equations.
00:33:38.990 --> 00:33:41.970
In practice, of course, what
happens frequently is that we
00:33:41.970 --> 00:33:46.790
are using these elements in a
mesh, so we have an 8-node
00:33:46.790 --> 00:33:51.290
element here, and another
8-node element here.
00:33:51.290 --> 00:33:54.900
And if we use 2x2 integration
for this element, 2x2
00:33:54.900 --> 00:33:57.720
integration for that element, we
might very well be able to
00:33:57.720 --> 00:34:01.380
solve the equations very nicely,
and we might, in fact
00:34:01.380 --> 00:34:05.830
get good results in a analysis
of a more complicated problem.
00:34:05.830 --> 00:34:10.219
Because the stiffness of this
element and that element,
00:34:10.219 --> 00:34:17.280
they, each other, compensate
for the loss for the rigid
00:34:17.280 --> 00:34:22.320
body mode that is still
available in
00:34:22.320 --> 00:34:24.530
this one element here.
00:34:24.530 --> 00:34:29.350
In other words, although this
one element is unstable, when
00:34:29.350 --> 00:34:31.989
you look at it numerically, when
we solve the equations
00:34:31.989 --> 00:34:34.199
for a single element,
as shown here.
00:34:34.199 --> 00:34:38.449
When the element is surrounded
by other elements, they
00:34:38.449 --> 00:34:41.980
provide enough stiffness into
this element so that the
00:34:41.980 --> 00:34:47.030
overall solution of the mesh can
still be obtained, and we
00:34:47.030 --> 00:34:49.320
obtain, in fact, good results.
00:34:49.320 --> 00:34:52.610
However, if you do use reduced
integration, we call this
00:34:52.610 --> 00:34:56.340
reduced integration because
we are not evaluating the
00:34:56.340 --> 00:34:58.670
stiffness matrix exactly.
00:34:58.670 --> 00:35:01.320
If you do use reduced
integration, you should be
00:35:01.320 --> 00:35:06.190
aware of this fact that for
a single element, or for a
00:35:06.190 --> 00:35:10.600
certain element mesh layouts,
we can run into numerical
00:35:10.600 --> 00:35:12.720
difficulties.
00:35:12.720 --> 00:35:17.370
Let's look briefly at the
stress calculations.
00:35:17.370 --> 00:35:19.890
Surely, stresses can be
calculated at any
00:35:19.890 --> 00:35:22.550
point of the element.
00:35:22.550 --> 00:35:25.910
And they would be evaluated
as shown here.
00:35:25.910 --> 00:35:27.270
The stress strain law is here.
00:35:27.270 --> 00:35:29.260
Here we have to the
strain-displacement
00:35:29.260 --> 00:35:30.330
interpolation matrix.
00:35:30.330 --> 00:35:33.140
Here, we have seen the nodal
point displacement vector.
00:35:33.140 --> 00:35:36.340
Of course, these nodal points
displacements are now known,
00:35:36.340 --> 00:35:37.380
they are now given.
00:35:37.380 --> 00:35:41.400
This is an initial stress vector
which is also given.
00:35:41.400 --> 00:35:43.450
You should notice that the
stresses are, in general,
00:35:43.450 --> 00:35:45.625
discontinuous across
element boundaries.
00:35:48.900 --> 00:35:51.090
If we look at a [? practical ?]
analysis, we
00:35:51.090 --> 00:35:54.040
might actually not evaluate the
stresses at the boundary
00:35:54.040 --> 00:35:57.190
of an element, but we might
evaluate the stresses at the
00:35:57.190 --> 00:35:58.140
integration points.
00:35:58.140 --> 00:36:02.260
So if you look at a mesh of
elements, as shown here.
00:36:02.260 --> 00:36:06.860
We might evaluate the stresses
at the 2x2 integration points,
00:36:06.860 --> 00:36:09.880
shown here for that element.
00:36:09.880 --> 00:36:13.390
If we were to evaluate the
stresses of this element at
00:36:13.390 --> 00:36:16.410
this boundary, and of this
element at that boundary, we
00:36:16.410 --> 00:36:18.900
would see a stress jump.
00:36:18.900 --> 00:36:22.360
I should mention here, just as a
side note, that in fact, one
00:36:22.360 --> 00:36:26.470
can show that the stresses at
the Gauss integration points
00:36:26.470 --> 00:36:29.150
are somewhat better predicted
in a finite element analysis
00:36:29.150 --> 00:36:32.630
than along the boundary
of the element.
00:36:32.630 --> 00:36:36.850
So that is another reason why we
use, rather, why we predict
00:36:36.850 --> 00:36:40.660
stresses rather, at the Gauss
integration points, and not on
00:36:40.660 --> 00:36:45.460
the boundaries of the element,
using directly this approach.
00:36:45.460 --> 00:36:50.150
As an example here, let us look
at a very simple problem.
00:36:50.150 --> 00:36:52.980
Here we have a cantilever
modeled
00:36:52.980 --> 00:36:54.780
with two 8-node elements.
00:36:54.780 --> 00:36:59.100
Now remember, that these two
8-node elements each contain
00:36:59.100 --> 00:37:01.560
the parabolic displacement
variation.
00:37:01.560 --> 00:37:05.950
So since this is a bending
moment that we apply to that
00:37:05.950 --> 00:37:09.550
cantilever, and since this
bending moment, of course,
00:37:09.550 --> 00:37:14.780
gives parabolic displacement
variationn The finite element
00:37:14.780 --> 00:37:17.790
solution would be correct,
will be exact in this
00:37:17.790 --> 00:37:18.810
particular case.
00:37:18.810 --> 00:37:22.880
And therefore, there would
be no stress jump here.
00:37:22.880 --> 00:37:26.590
Since the elements contain the
displacements that shall be
00:37:26.590 --> 00:37:31.920
predicted, that are the
analytically-correct results.
00:37:31.920 --> 00:37:35.770
The finite element mesh will
predict these displacements,
00:37:35.770 --> 00:37:38.800
and, of course, we have
stress continuity.
00:37:38.800 --> 00:37:41.416
This is shown here by--
00:37:41.416 --> 00:37:43.090
we have dissected these
elements here.
00:37:43.090 --> 00:37:47.160
This is element 2, this
is here, element 1.
00:37:47.160 --> 00:37:50.790
And we have plotted the stress
distribution along
00:37:50.790 --> 00:37:52.400
the line a, b, c.
00:37:52.400 --> 00:37:53.590
Here's the line, a, b, c.
00:37:53.590 --> 00:37:56.200
And as you can see,
there's 900 here,
00:37:56.200 --> 00:37:58.150
and there's 900 there.
00:37:58.150 --> 00:38:00.510
And of course, a linear
variation stresses, so no
00:38:00.510 --> 00:38:02.152
stress jumps there.
00:38:02.152 --> 00:38:07.580
However, if you look at the
same problem, the same
00:38:07.580 --> 00:38:11.950
cantilever, I should say, with
now an edge share load, as
00:38:11.950 --> 00:38:13.340
shown here.
00:38:13.340 --> 00:38:18.310
Then these elements cannot
represent the exact
00:38:18.310 --> 00:38:22.190
displacement distribution in
the cantilever because they
00:38:22.190 --> 00:38:25.830
only contain a parabolic
variation in displacement.
00:38:25.830 --> 00:38:30.050
And in this case, we have the
stress discontinuity.
00:38:30.050 --> 00:38:33.250
tau x x, in other words, the
stress into this direction
00:38:33.250 --> 00:38:35.240
here, so normal stress.
00:38:35.240 --> 00:38:39.140
For this element along, for
element 2, along a,b,c, you
00:38:39.140 --> 00:38:42.080
can see 1744 up there.
00:38:42.080 --> 00:38:42.850
And then [UNINTELLIGIBLE]
00:38:42.850 --> 00:38:44.440
a neutral axis because
of symmetry
00:38:44.440 --> 00:38:46.010
conditions, of course.
00:38:46.010 --> 00:38:53.000
And for element 1, along a, b,
c, we have 1502 right there.
00:38:53.000 --> 00:38:56.470
So there is, in fact, this
stressed discontinuity that I
00:38:56.470 --> 00:38:58.740
was talking about earlier.
00:38:58.740 --> 00:39:02.320
The share stresses here have
also a slight discontinuity,
00:39:02.320 --> 00:39:09.000
as you can see, 291.38,
296.48.
00:39:09.000 --> 00:39:14.220
The reason that we have this
stress discontinuity here is,
00:39:14.220 --> 00:39:16.930
of course, that the
elements cannot
00:39:16.930 --> 00:39:18.505
represent the exact solution.
00:39:21.550 --> 00:39:27.710
And therefore, they each try to,
in the integral sense of
00:39:27.710 --> 00:39:31.850
the finite element formulation,
try to predict
00:39:31.850 --> 00:39:37.060
the solution accurately within
its domain, and the result is
00:39:37.060 --> 00:39:38.140
that we have a stress
00:39:38.140 --> 00:39:41.210
discontinuity between elements.
00:39:41.210 --> 00:39:46.430
Finally, let us look then at
some modeling considerations.
00:39:46.430 --> 00:39:51.300
In order to solve a problem,
we should really have a
00:39:51.300 --> 00:39:55.330
qualitative knowledge of the
response to be predicted.
00:39:55.330 --> 00:39:58.000
We should also have a thorough
knowledge of the principles of
00:39:58.000 --> 00:40:02.470
mechanics and the finite element
procedures available.
00:40:02.470 --> 00:40:05.820
And this, of coure, is
the subject of my
00:40:05.820 --> 00:40:08.880
lectures here to you.
00:40:08.880 --> 00:40:12.530
If we have these two knowledges,
then we should
00:40:12.530 --> 00:40:16.320
still know a little bit more
about what kind of elements we
00:40:16.320 --> 00:40:23.410
should use, how one element
performs versus another.
00:40:23.410 --> 00:40:27.200
And here, the first remark that
I like to make is that
00:40:27.200 --> 00:40:31.270
parabolic undistorted elements
are usually most effective.
00:40:31.270 --> 00:40:35.910
Here, please see that I'm saying
usually most effective
00:40:35.910 --> 00:40:39.190
because in some earlier lecture,
I talked about the
00:40:39.190 --> 00:40:41.510
analysis of fracture mechanics
problems, where we, in fact,
00:40:41.510 --> 00:40:44.550
distort elements to pick up
stress singularities.
00:40:44.550 --> 00:40:47.380
But if we don't talk about
stress singularities or
00:40:47.380 --> 00:40:51.410
special situations, usually
the parabolic undistorted
00:40:51.410 --> 00:40:54.820
elements are most effective.
00:40:54.820 --> 00:41:00.630
Here, I have put together in a
table the kinds of elements
00:41:00.630 --> 00:41:03.680
that I would recommend
for usage.
00:41:03.680 --> 00:41:06.550
These are typically also the
elements that I used in
00:41:06.550 --> 00:41:09.020
general purpose code
such ADINA.
00:41:09.020 --> 00:41:11.480
Tress or cable element,
the 2-node
00:41:11.480 --> 00:41:13.690
element is quite effective.
00:41:13.690 --> 00:41:16.430
In some cases, we want to use
a three-node element or
00:41:16.430 --> 00:41:19.100
four-node element, but the
2-node element is cheap and
00:41:19.100 --> 00:41:20.970
very effective in modeling.
00:41:20.970 --> 00:41:22.840
For two-dimensional plane
stress, plane strain,
00:41:22.840 --> 00:41:26.300
axisymetric analysis, the 8-node
or 8-node element.
00:41:26.300 --> 00:41:29.070
Those are the elements most
effective, and as I just
00:41:29.070 --> 00:41:32.520
mentioned, we should use them
undistorted, in other words,
00:41:32.520 --> 00:41:35.080
rectangular.
00:41:35.080 --> 00:41:38.600
The three-dimensional
analysis then.
00:41:42.530 --> 00:41:45.360
In three-dimensional analysis,
we would effectively use the
00:41:45.360 --> 00:41:48.550
20-node brick element, which is
really the counterpart of
00:41:48.550 --> 00:41:51.060
the 8- or 9-node element.
00:41:51.060 --> 00:41:57.560
For the 3D that is curved, we
would use three-moded elements
00:41:57.560 --> 00:41:58.920
or four-noded elements-- better
00:41:58.920 --> 00:42:00.450
even four-noded elements.
00:42:00.450 --> 00:42:03.230
Of course, if we have a straight
beam, then we would
00:42:03.230 --> 00:42:06.420
use simply the engineering
[? animation, ?]
00:42:06.420 --> 00:42:08.190
two-noded beam.
00:42:08.190 --> 00:42:11.420
But if you talk about a curved
beam, then the isoparametric
00:42:11.420 --> 00:42:13.630
beam that I presented to you in
00:42:13.630 --> 00:42:17.060
lecture 7 is very effective.
00:42:17.060 --> 00:42:21.880
The plate and shell elements
that I would recommend are
00:42:21.880 --> 00:42:24.520
this element here, for
plate analysis--
00:42:24.520 --> 00:42:26.600
I mentioned that already
earlier.
00:42:26.600 --> 00:42:31.370
And for the shell analysis, the
nine-noded element and the
00:42:31.370 --> 00:42:33.790
16-node element.
00:42:33.790 --> 00:42:37.530
Even for plates, it might pay
sometimes to use the 16-node
00:42:37.530 --> 00:42:40.110
element, which is very
effective, and particularly
00:42:40.110 --> 00:42:43.720
works for very, very thin
plates and shells.
00:42:43.720 --> 00:42:48.000
It does not have the
deteriorating behavior of the
00:42:48.000 --> 00:42:53.790
locking phenomenon that I
referred to in lecture 7.
00:42:53.790 --> 00:43:00.390
Let us look also at some
mesh considerations.
00:43:00.390 --> 00:43:06.350
If we perform an analysis, we
might find that in some area
00:43:06.350 --> 00:43:08.430
of the element idealization.
00:43:08.430 --> 00:43:12.440
We want to use 4-node elements
lower order elements, and in
00:43:12.440 --> 00:43:15.320
another area, we want to use
higher order elements.
00:43:15.320 --> 00:43:19.460
Well, here, I've show a
transition region going from a
00:43:19.460 --> 00:43:22.530
4- to a 5-node to take
an 8-node element.
00:43:22.530 --> 00:43:28.130
Notice that we have full
compatibility along this side.
00:43:28.130 --> 00:43:32.250
So this is a comparable element
transition from 4- to
00:43:32.250 --> 00:43:32.940
8-node elements.
00:43:32.940 --> 00:43:35.850
And similarly, of course, in
three-dimensional analysis,
00:43:35.850 --> 00:43:39.720
you could go from 8-node bricks
to 20-node bricks by
00:43:39.720 --> 00:43:43.960
using such transition
elements.
00:43:43.960 --> 00:43:49.960
An alternative approach would
be to use a 4-node element
00:43:49.960 --> 00:43:53.730
here, and say, two 4-node
elements there, but notice
00:43:53.730 --> 00:43:59.370
that if you do so, the
displacements at node a, have
00:43:59.370 --> 00:44:02.980
to be constraint, if you
want to preserve
00:44:02.980 --> 00:44:04.580
compatibility here.
00:44:04.580 --> 00:44:10.310
Notice that along BC, for this
4-node element, we have a
00:44:10.310 --> 00:44:12.150
linear variation and
displacements because we have
00:44:12.150 --> 00:44:15.340
only UB, VB, UC, VC there.
00:44:15.340 --> 00:44:18.620
And of course, here, for these
two 4-noded elements, we have
00:44:18.620 --> 00:44:21.370
a linear varition from here to
there, and another linear
00:44:21.370 --> 00:44:23.940
variatoin from here to there
in displacements.
00:44:23.940 --> 00:44:27.500
To preserve full compatibility
along this line, these two
00:44:27.500 --> 00:44:30.240
constraint equations
have to be used.
00:44:30.240 --> 00:44:34.810
And that means, really, that VA
and UA are set to the mean
00:44:34.810 --> 00:44:39.720
of UB, UC, and VB, BC,
as shown here.
00:44:39.720 --> 00:44:45.020
Another transition approach is
shown here on the last view
00:44:45.020 --> 00:44:47.290
graphs that I wanted
to present to you.
00:44:47.290 --> 00:44:52.020
And here we're going from 8-node
elements, or 7-node
00:44:52.020 --> 00:44:55.880
elements, if you wanted to use
8-noded elements here, then of
00:44:55.880 --> 00:45:00.050
course, we would have to put
another node in there each.
00:45:00.050 --> 00:45:05.290
So here we're going from two
layers of 8-node elements,
00:45:05.290 --> 00:45:08.100
over into one layer of
8-node not elements.
00:45:08.100 --> 00:45:12.570
Notice that in this particular
transition region here, we are
00:45:12.570 --> 00:45:16.190
using a distorted element.
00:45:16.190 --> 00:45:21.400
Of course, that means the order
of accuracy that we can
00:45:21.400 --> 00:45:25.000
expect here in the stress
predictions and so on, is not
00:45:25.000 --> 00:45:27.260
quite as good as we would
like to see it.
00:45:27.260 --> 00:45:32.110
So this transition region should
be away from the area
00:45:32.110 --> 00:45:34.540
of interest. the area of
interest being somewhere over
00:45:34.540 --> 00:45:37.690
here, far way from the
transition reason.
00:45:37.690 --> 00:45:41.460
However, we do have a compatible
element layout, and
00:45:41.460 --> 00:45:45.390
that, of course, is an important
point, which I
00:45:45.390 --> 00:45:48.570
mentioned to you earlier, that
we would like to preserve
00:45:48.570 --> 00:45:50.930
compatibility as much
as possible.
00:45:50.930 --> 00:45:53.330
And for that reason, of course,
we have developed
00:45:53.330 --> 00:45:56.820
these variable number
node elements.
00:45:56.820 --> 00:45:59.650
This concludes what I wanted
to say in this lecture.
00:45:59.650 --> 00:46:00.900
Thank you for your attention.