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PROFESSOR: Ladies and gentlemen, welcome to this
lecture on nonlinear finite element analysis of solids and
structures.
In the previous lectures, we talked about the basic process
of the incremental solution that is used in
finite element analysis.
And we talked about quite some important continuum mechanics
variables that we are employing in nonlinear finite
element analysis.
This was really preparatory work for the discussion of a
general formulation that is very widely used to analyze
nonlinear systems using finite elements.
And this formulation is called the total Lagrangian
formulation.
This is the formulation that I'd like to discuss with you
now in this lecture.
The total Lagrangian formulation is a formulation
that refers the stress and the strain variables at time t
plus delta t to the original configuration at time zero.
Total, in fact, means reference to the original
configuration here.
And notice we, of course, encountered this equation
already earlier in the early lectures.
We have on the left hand side again the total internal
virtual work.
And on the right hand side, the total external virtual
work all at time t plus delta t.
We have here the second Piola-Kirchhoff stress tensor.
Here we have the Green-Lagrange strain tensor.
And on the right hand side, of course, all the body forces,
surface forces et cetera would enter.
We discussed in a previous lecture how the right hand
side is calculated in general and what it all contains.
We also discussed that this equation here is quite
equivalent to this equation.
And in this equation, we have the Cauchy stress tensor
operating on the virtual strain tensor, an
infinitesimally small strain tensor.
And this product is integrated over the volume at time
t plus delta t.
On the right hand side, we have the same
quantity as up there.
Now notice that we really want to solve, of course, this
equation here in finite element analysis.
But we discussed earlier that in order to do so effectively,
we need to introduce a new stress measure, a new strain
measure, and that, of course, brings us then to this
equation up here.
Let us recall briefly some of the important points that we
made earlier in the other lectures.
We said that this equation here, applied at time t plus
delta t, is an expression of the equilibrium,
compatibility, and the stress-strain law at time t
plus delta t.
Let's just look at this equation once more, where
these quantities enter.
Well, if equilibrium is satisfied locally everywhere
in the continuum, then this equation must be holding for
any virtual displacements that satisfy the displacement
boundary conditions, and corresponding virtual strains.
Notice that the virtual displacements enter on the
right hand side, and these here are the corresponding
virtual strains.
The compatibility enters in here because we would, of
course, use compatible virtual displacement, and we would
calculate also the stresses from compatible displacements.
The stress-strain law enters in the calculation of the
stress here from the strains given.
And once again we apply this at time t plus delta t.
We also noted earlier that we will be using
an incremental solution.
Namely, that the solution at time t plus delta t will be
calculated from the solution at time t by incrementing the
displacements by ui.
Notice that we will assume, of course, that
the tui's are known.
That the displacementts at time t are known, and that
these are the unknown quantities that
we're looking for.
Of course, that gives us then the unknown quantities that we
actually want to solve for.
Our goal is, for the finite element solution to linearize
the equation of the principle of virtual work, so as to
obtain a set of equations, finite element equations, that
read as follows.
Here we have a stiffness matrix, a
tangent stiffness matrix.
Here we have an incremental displacement vector.
And on the right hand side, we have a load vector, a vector
of nodal point forces corresponding to the
externally applied loads at time t plus delta t.
And here we have a nodal point force vector that is
equivalent to the current element stresses, in this
particular instance at time t, as indicated by the
superscript t here.
Now this incremental displacement vector, of
course, will be added to the displacement that we know
already, and that correspond to time t.
And that will give us then a first estimate for the
solution, displacement solution, at time
t plus delta t.
Notice this is also summarized down here.
Notice, however, that this vector here is only an
approximation to the actual vector that we're looking for,
because of the linearization process that leads us to this
system of equations.
The equation tku equals r minus f is, of course,
applicable for a single element as well as a total
assemblage of elements.
In other words, this equation here, tk delta u1 equals t
plus delta r minus tf, would be applicable to an element in
which n then would mean just the degrees of freedom, the
number of degrees of freedom of that element.
And also for a total structure in which n then, of course
means the total number of structural degrees of freedom.
The tk matrix, the r vector, and the f vector would be
constructed using the direct stiffness method the way we
are used to it in the linear elastic analysis.
In other words, there is nothing new here to be
discussed really, as far as a construction of these matrices
go, when we talk about a total element assemblage from the
individual element matrices.
However, an important point is that we cannot simply
linearize the principle of virtual work when it is
written in this form here.
And the reason was, of course, that we cannot integrate over
an unknown volume.
The volume t plus delta t is unknown, or the volume at time
t plus delta t is unknown, and we cannot directly increment
the Cauchy stresses the way we discussed it already earlier.
To linearize then, we need to choose a known reference
configuration.
And one known reference configuration that is a very
natural one to use really, is the one
corresponding to time zero.
In this case, if we use that reference configuration, we
talk about the total Lagrangian formulation.
If we use as a reference configuration the
configuration corresponding to time t, then we talk about the
updated Lagrangian formulation.
I'd like to make one point here.
Namely, that of course we have calculated already from all
the configurations from time zero to time t.
So you may ask yourself, why is he not choosing as a
reference configuration, for example, the configuration t
minus delta t?
In other words, one configuration prior to time t
as a possibility.
Well, it would be a possibility, but if you choose
to do so, you would lose the advantages that you have in
the updated Lagrangian formulation, and the
advantages that you would have in the total Lagrangian
formulation.
Of course, there are differences here and we will
talk about the advantages and disadvantages of either of
these formulations.
And you would be left with that choice of t minus delta t
reference configuration, with basically all the
disadvantages only.
And therefore, we generally want to choose either this one
or that one as the reference configuration.
Let us now discusses the total Lagrangian formulation.
In the total Lagrangian formulation, we use the
reference configuration zero, and therefore we talk about
this stress.
We use this stress and that strain, this was a second
Piola-Kirchhoff stress referred to the configuration
at time zero.
This is the Green-Lagrange strain referred to the
configuration at time zero.
The principle of virtual work once again, originally in this
form, is now written in this form.
We have had that already before on viewgraphs, and I
don't think I need to go through those details again.
But the important point is that this is the starting
point of the total Lagrangian formulation.
The formulation proceeds as follows.
We know the solution at time t, therefore this stress, this
displacement derivative, all the static and kinematic
variables in fact, are known corresponding to the
configuration at time t.
And we can now decompose the stress corresponding to time t
plus delta t into one known value and one unknown value.
Similarly we proceed with the strain measures.
The strain measures, the Green-Lagrange strain at time
t plus delta t is decomposed into one value that we know
already, and an increment.
And, of course, remember these two increments are unknown.
They are unknown, whereas these are known because we
have calculated already the configuration
corresponding to time t.
In terms of displacements, we can directly develop that the
Green-Lagrange strain at time t is written as shown here.
We went over these different components already earlier.
Notice once again that we have a product of displacement
derivatives here, that this is, of
course, a nonlinear term.
And once again, as I pointed out in an earlier lecture, we
have nothing neglected here.
You see only quadratic terms.
There are no cubic terms or higher-order terms.
We have nothing neglected.
This is a strain measure that holds for any amount of
deformation, any amount of strain.
Then we can also write this same strain, Green-Lagrange
strain, corresponding to time t plus delta t, and this would
be the result.
Notice whatever we had here as t superscript has now become a
t plus delta t superscript.
That's in fact the only difference.
If you subtract from this strain, that strain, we get
the increment, and that increment is
written out right here.
Now, there are some interesting points regarding
this incremental strain.
This here is an increment in the strain that is linear in
the displacement components.
Similarly this one.
Notice this one here, is, of course a product, product and
may look, at first sight, as a nonlinear term.
When you see products of displacements, you think of a
nonlinearity.
However, if you look closer, you find that
there is a t here.
Therefore, this part is known.
Since this part is known, and this part is unknown, this
total term is really linear in the incremental displacement.
In fact, the incremental displacements don't even go in
here, because we're differentiating here the
displacements at time t with respect to the original
coordinates.
Therefore, this total term here is really
still linear in ui.
Linear in the incremental displacements.
We call this part here, as given here, the initial
displacement effect.
The initial displacement effect because the initial
displacements go into these derivatives here.
We are differentiating the displacement at time t with
respect to the original coordinates.
This term, on the other hand here, is nonlinear in ui.
It is nonlinear because here we're taking one term that
depends on ui, and we multiply it by another term that
depends on ui.
Notice this a differentiation of uk, the incremental
displacement, into the k direction with respect to the
original coordinates.
Coordinate axis xj here particular.
So this is here, clearly a nonlinear term in ui.
A quadratic term, as a matter of fact.
We will look at this strain, of course, a little
bit more just now.
But keep in mind that there is one linear term that contains
an initial displacement effect and a nonlinear term.
I might add here that, of course, this initial
displacement effect is zero if the initial
displacements are zero.
In other words, in our incremental solution, for
example, if we are just starting the solution process,
then the initial displacements are still zero.
This term drops out.
We note now regarding the formulation that the variation
on the total Green-Lagrange strain tensor corresponding to
time t plus delta t, is really equal to the variation in the
increment of the Green-Lagrange strain tensor,
from time t to time t plus delta t.
In this picture we show what we mean by that.
Here on the left hand side in black, you have the body in
its original configuration.
It moves to a configuration at time t.
And then to a configuration, here shown in green, that
corresponds to time t plus delta t.
Notice here we have the displacement corresponding to
time t, and then the displacement corresponding
from time t to time t plus delta t.
The total displacement, of course, here is the
displacement from time zero to time t plus delta t.
What we are doing, of course in the principle of virtual
work, is to impose a variation in displacements about the
configuration at time t plus delta t.
And that variation is here indicated by the blue line.
And if we impose this variation here, since the
displacements at time t are constant, surely whatever
these displacements were should not matter.
And that is, in fact, expressed by this
relationship here.
If we vary here, since the displacements to ui are
constant, they do not affect the variation on the strains.
And therefore, we can simply vary the incremental strain,
and we find that variation is indeed equal to the variation
of the total strain.
Of course, you can also prove that to yourself
mathematically by just going through the arithmetic.
In other words, applying the variation to this quantity,
and then applying the variation to that quantity,
which I've given you on the previous viewgraph, and you
would find that indeed you get identically the same
expressions.
We can define a linear strain term, linear strain increment,
as 0 eij, and the nonlinear strain increment as 0 eta ij.
Notice that once again, here we have the quadratic terms.
And, of course, when we add these two terms together, we
get back our total strain increment from time t to time
t plus delta t.
If we take a variation on that increment, that is equivalent
to taking a variation on the linear term and
the nonlinear term.
Notice that so far we have only talked
about continuum mechanics.
Of course we want to use the continuum mechanics principles
that we discussed so far in the finite element
discretization.
But notice that these strain terms here that we talked
about, are really continuum mechanics variables.
There is an interesting observation.
Namely, if we talk about this strain measure and that strain
measure, rather the linear incremental strains and the
nonlinear incremental strains, linear and nonlinear in ui,
then if we apply these in finite element analysis, we
have to remember that the displacements are interpolated
in terms of nodal point variables.
And what we're really looking for in finite element analysis
are the nodal points variables.
In isoparametric finite element analysis, which is the
final element procedure that we will be using extensively,
we interpolate the internal element displacements, tui, in
terms of the nodal point displacements.
Let me define a little bit more what this is here.
This is, of course, the displacement from time zero to
time t into the i's direction.
We are summing here over all the finite element nodal
points, the N nodal points.
We have interpolation functions, hk, that we will
talk more about later on.
And we have here the nodal point displacement at time t
into the i direction of the nodal point k.
This k goes with that k.
And we're summing over all the nodal points.
Now notice that this is the linear relationship in
isoparametric finite element analysis of
solids, I should say.
This is a linear relationship.
And therefore, our linear strain terms that we talked
earlier about, being linear in the incremental continuum
mechanics displacements, are also linear in the incremental
nodal point displacements.
Similarly, this term here, which was nonlinear in the
increments of displacements within the domain, from a
continuum mechanics point of view, will also be nonlinear
in the nodal point displacement increments.
However, in the formulation of structural elements, with the
incremental displacements, in fact also the total
displacements, I interpolated using nodal point
displacements and nodal point rotations.
In other words, we also deal with nodal point rotations in
order to calculate the total displacements of the element.
And in that case, we have to recognize that the exact
linear strain increment, the way we have defined it earlier
linear in the incremental displacements, is still
properly given as a linear strain increment in the nodal
point variables.
However, the term that we defined earlier as a nonlinear
strain increment, this one, is not the full story of all the
nonlinear strain increments.
Because the rotations will put additional nonlinear strain
increments into this term here.
The reason being, that for large rotations, of course, we
have cosine sine terms in these rotations and when we
evaluate the total nonlinear strain term, we find that
there are additional terms coming up here.
Well, what is the effect of these two statements?
It means that the right hand side force vector, or rather
the vector t 0 f, the vector of nodal point forces
corresponding to the internal elements stresses, is always
correctly calculated.
However, the stiffness matrix, t 0 k, for the formulation is
only approximated because some of the nonlinear terms here
have been dropped in the case of structural elements.
Of course, you can also include the nonlinear terms.
Then you would have to add these terms to this expression
here, and then you would get a true tangent stiffness matrix.
This means, of course, for an actual analysis, that when you
form the equations, we have on the left hand side-- let me
just put all of it in here--
for the first iteration, we talked about the fact that we
have a stiffness matrix times an increment in displacements.
I already put a 1 there because it will be only an
approximation to the total incremental
displacement vector.
Well, we notice then that this matrix here is approximated if
you were to drop these additional terms.
But the right hand side vector f here is properly calculated.
Therefore if you iterate, you always get the exact solution,
the correct solution.
But the convergence in the iteration might be a bit
slower than what you could have obtained if you had
included these additional terms here that I referred to.
Let us now continue with the continuum mechanics
formulation.
The equation of the principle of virtual work becomes, if we
substitute the quantities that I defined earlier, directly
this equation.
Notice here we have now the increment in stress times the
variation of the incremental Green-Lagrange strain, from
time t to t plus delta t.
Of course, it's integrated over the original volume plus
this integral here.
Here we have the total stress, second Piola-Kirchhoff stress,
corresponding to time t operating on the nonlinear
strain increment of the total
Green-Lagrange strain increment.
And once again, we are integrating over the total
volume corresponding to time zero.
And on the right hand side, we have the external virtual work
minus the stress at time t, operating on the linear strain
increment corresponding to the total
Green-Lagrange strain increment.
We have so far made no approximations, and this is a
relationship that has to hold for any arbitrary variation in
displacements which go in here, or any arbitrary virtual
displacement that go in here.
And of course, corresponding strains that appear here.
In essence, what we have been doing is that we have
established a variation on the configuration at time
t plus delta t.
We have looked at the principle of virtual work
corresponding to time t plus delta t, but we have
introduced this increment in displacement.
Notice that this increment in displacement is measured from
the displacement at time t.
All we have done therefore, is to rewrite the principle of
virtual work in terms of tu and u instead of dealing with
t plus delta tu.
No approximation being done yet so far.
Of course, we really haven't been talking about finite
elements either yet, when we just look at this basic
equation of principle of virtual work.
It is this principle or the equation that we are dealing
with, is in general, a very complicated function of the
unknown displacement increment.
We obtain an approximate relationship in finite element
analysis, by linearizing the governing equation, and, of
course, then in the finite element sense cast it into
this set of linear equations.
The process of linearizing is a very important process,
interesting process for us to study, and that's what
I want to do now.
We begin to look, in this linearization at all the
individual terms that appear in the integral.
And let us look first at this term here.
This term is actually already linear in ui.
So there is nothing much to linearize.
t 0 sij does not contain ui.
It's a known quantity.
It's a known stress value.
And notice that when we take the variation on this
nonlinear strain increment, we directly get this expression,
where this part here is a variation on the
displacement increment.
Similarly here.
And of course, these are going to be constants.
Therefore, this term here is linear in the incremental
displacements.
Linear in ui.
So nothing to linearize there.
The term on this viewgraph here, contains an incremental
stress and an incremental strain.
Notice this incremental strain contains the linear and the
nonlinear term.
And this, of course, this product is a linear functional
off ui, sorry it contains linear terms,
nonlinear terms as well.
This stress here is, in general, a nonlinear function
of 0 epsilon ij.
The variation that we are taking here means we are
taking a variation on the linear part and
the nonlinear part.
We will see actually that this one here is constant.
But this one carries also terms of ui, so this total
expression is linear in ui.
And if we multiply these terms here, we want to end up in
just one expression that has neglected all higher-order
terms in ui, but just contains ui.
Well, let us do so.
The objective is expressed once more here, and we
recognize that, of course, the variation on 0 epsilon ij
contains only constant and linear terms in ui, the way I
just expressed it.
0 sij can be written as a Taylor series expansion in 0
epsilon ij.
This is the general expression.
Here we take the partial of the stress at time t, with
respect to the Green-Lagrange strain at time t, here we are
having this bar with the t that denotes that we are
actually doing this evaluation at time t.
And here we have an increment in this strain, and of course,
there are higher-order terms.
These higher-order terms we will neglect, and therefore we
get directly for this stress here, this expression.
Notice we already have substituted for 0 epsilon rs,
these two quantities.
This one here is quadratic in ui, this one is linear in ui.
And if we linearize this term, we get directly this one here.
Notice that this is the appropriate stress-strain law
that has to be used corresponding
to this stress measure.
And we will later on talk more about how we evaluate this
stress-strain law.
As an example, we may look here at this schematic
solution, computed solution, over a number of time steps.
We're going time 1, 2, 3, 4, 5, up to t minus delta
t and then to t.
The stress-strain law relates basically the stress increment
here to the strain increment.
And notice it represents a tangent to this this curve.
This green line here is a tangent to the red or the blue
curve, the blue one now overlapping the red curve
right there.
So we have here the tangent, or the slope of this tangent
is giving us the material tensor 0c.
Of course, here we're talking just about one element.
In general we have many elements corresponding to the
strain and stress measures that we are dealing with.
If we now substitute the result that we just looked at
into the general equation--
sorry to take this one down--
into the general equation here, we see directly that
this part here, of course, is the stress.
And this part here gives us that part.
We simply multiply out and look at the individual terms.
We notice that this one does not contain ui.
We notice that this one is linear in ui.
So since this one is already linear in ui, this term will
be quadratic in ui.
We will have to drop it.
This term here is linear in ui, but this one is constant,
so this total term is linear in ui.
And this is the one we keep, and which is then the
linearized result.
In other words, in summary once more, this total term
here linearized is obtained via this expression here.
The final linearized equation that we are then dealing with,
now that we're substituting all these results into the
original equation that we have developed, is as follows.
Here we have one term that comes from the incremental
stress times the variation on the total incremental
Green-Lagrange strain.
That all has reduced to this term.
Here we did not have to linearize at all.
We kept what we already had.
And on the right hand side, did not linearize either,
because this was a term that we had already there, and of
course this is the external virtual work.
Now it's interesting to note that these two terms here
result into this expression, where this is the tangent
stiffness matrix.
Notice that tangent stiffness matrix contains the material
tensor, as well as the current stresses.
This incremental displacement here, vector, comes from this
strain part and that strain part.
Notice that this virtual displacement vector here comes
from that strain part and also from that one here.
Below here we have, of course, that the external virtual work
results into a vector of nodal point forces times these
virtual displacements.
And this one here results into the force vector corresponding
to the internal element stresses.
And this is, of course, a very important quantity that we
have to calculate accurately, as I pointed out earlier.
Because we want ultimately in the iteration that this vector
is equilibriating that vector, and if we do make a mistake in
calculating this vector, then we might have converged, but
we have converged to the wrong solution.
So it's very important to recognize that this vector
must be accurately calculated, by all means.
However, the k matrix here, if we go back once more to the
discussion of this matrix, the k matrix here is a tangent
stiffness matrix.
And this tangent stiffness matrix is selected such as to
obtain, of course, in the incremental solution, an
appropriate incremental displacement.
We will see later on that this matrix is
updated in the iteration.
It depends on what kind of intuitive scheme we are using,
depending on that scheme we are updating this matrix
differently.
In any case, it's a matrix that we will in quotes, "play
around with", in order to accelerate the convergence.
Therefore there is no unique matrix that can be used here,
that should be used here.
There are different possibilities that we will
discuss in later lectures.
But on the right hand side once again, this F vector is
the F vector obtained from the current stresses, and that one
has to be calculated uniquely in the correct way.
Well, an important point is that this relationship here on
the left hand side, is equal to the relationship given here
on the right hand side.
And this holds because this term here is
equal to that term.
We interpret, in fact, this right hand side as an out of
balance virtual work term.
Let's look once why this holds, because this might be a
bit of a surprise.
The mathematical explanation is given on this viewgraph.
We had earlier already that the variation on the total
Green-Lagrange strain is nothing else than the
variation on the incremental total Green-Lagrange strain.
Now if we look down here, we recognize that this variation
here at ui equal to 0, is nothing else than the
variation on the Green-Lagrange
strain at time t.
If ui is zero, then this term here is nothing else than the
Green-Lagrange strain at time t.
So we're looking here at the variation of the
Green-Lagrange at time, t if I put ui equal to 0.
Now let's look at these expressions here.
If I evaluate this term here, at ui equal to 0, we find that
this term here, being a constant, is recovered.
But notice that this term here, with ui equal to 0,
simply turns out to be 0.
Therefore, this term is equal to that term.
And this means, therefore, that the variation on t zero
epsilon ij is equal to the variation on 0 eij, which we
wanted to prove, of course.
So this is the reason why this term here is equal to that
term, as we have used it on the previous viewgraph.
This result also makes physical sense, because if we
look at the governing equation, we find that this
term here, replaced, is nothing else than tr.
Should be tr.
And of course, the script r the external virtual
work at time t.
And this, of course, is here the out of balance load term.
Now if, for example, the material is elastic, and the
external virtual work has not changed, then clearly the
displacements would not change.
In other words, t plus delta tui would be equal to tui, and
the incremental displacements would be 0.
Therefore, equilibrium would be satisfied.
Hence, this term, which we have here, must actually be
equal to t 0 sij times variation t 0 epsilon ij, the
way we discussed it just now.
We may rewrite the linearized governing equation as given on
this viewgraph.
And I am rewriting it in this form because we anticipate
that we need an iteration.
We simply introduce here, an iteration counter 1, with the
delta in front of the incremental strain value.
And similarly we introduce a delta here, and an iteration
counter 1 there.
And on the right hand side, we introduce also an iteration
counter, but one lower.
In other words, 0 here, whereas we have a 1 there.
Notice 0 here, a 1 here.
And notice furthermore, that this term is nothing else than
that term, and this term here is nothing else than the
variation on t 0 epsilon ij.
This would be the equation that we have developed
already, just having introduced now a different
notation which leads us towards what we want to deal
with in the iteration.
The governing equations from a finite element point of view,
then would be as shown here.
Notice tangent stiffness matrix, del u1, iteration
counter, the external loads, which we still assume to be
constant, by the way.
We still assume deformation independent loading if we have
deformation dependent loading, these would also
change in the iteration.
And here we have the vector corresponding to the internal
element stresses at time t.
We write that vector as t plus delta t 0, 0 there.
And that means that we are really talking, at this point,
about this vector.
We calculate this incremental displacement vector and add it
to the previous displacements, which, of course, in this
particular instance, in the first iteration, are nothing
else than tu.
We add those, and obtain our first estimates of
displacements, namely the estimate at the end of
iteration 1.
Having obtained now this estimate on displacements, we
can repeat the process.
And that is shown here on the next viewgraph.
Notice that now I am talking about the right hand side,
about the Green-Lagrange strains corresponding to time
t plus delta t, and the first iteration.
In other words, these are the Green-Lagrange strains
corresponding to the end of the first iteration.
They include now the incremental displacements that
we just calculated.
Similarly, these are the stresses corresponding to the
end of the first iteration.
And this is an out of balance virtual work term.
If that is non-zero, we will have further increment in
displacements.
And that increment results, of course, into an
increment of strains.
Namely, the second increment.
Here is the 2 that denotes the increment.
The finite element equations then would look as shown here.
Same equation as before, but now with a 1
here and a 2 there.
We calculate the incremental displacements, we add them to
what we had already and get a better approximation to the
displacements corresponding to time t plus delta t.
This process, of course, can now be repeated, and on this
viewgraph we show how it's being done for
every iteration k.
In a particular iteration, we have now on the right hand
side the external virtual work corresponding to time t plus
delta t, the stresses corresponding to time t plus
delta t and iteration k minus 1.
At the end of iteration k minus 1, the strains
corresponding to time t plus delta t, and at the end of
iteration k minus 1.
This is the out of balance virtual work term, and on the
left hand side we have the same quantities as before, but
now with the iteration counter k.
When discretized, we obtain these equations here.
Delta uk, the increment in the iteration k of displacements,
r minus f, computed, of course, from the current
displacements.
From the current displacements we calculate that vector here.
Notice this vector is obtained from this integral here.
We repeat this process for iteration k 1, 2, 3 et cetera
until convergence.
We will have to talk about how we measure
convergence and so on.
But in this iteration we're adding up the total
incremental displacements to obtain always a better
estimate for the displacements corresponding to time t plus
delta t, and here, of course, iteration k.
Notice that the counter here goes from j equals 1 to k,
that is this j and this k, of course, is nothing else than
the k that we have on the left hand side here.
Notice once again that the first iteration, when k is
equal to 1, it amounts to nothing else than what we
discussed already at the very beginning a little earlier in
the lecture.
We simply generalized that discussion now to an equation
in which we iterate and always try to calculate another
increment in displacements until the right
hand side is 0.
Notice when the right hand side is 0, the external
virtual work is equilibrated by the internal virtual work,
and of course, that is what we want to reach.
That is what our criterion is for the analysis.
The external virtual work must be equilibrated by the
internal virtual work.
In the finite element discretization, the whole
process is as summarized on this viewgraph.
Initially, we are given tu, the displacements
corresponding to time t, and the externally applied loads
corresponding to time t plus delta t.
We compute the stiffness matrix, the tangent stiffness
matrix corresponding to time t.
We calculate this vector, the nodal point forces
corresponding to the stresses at time t.
Notice this 0 here refers always to the total Lagrangian
formulation.
Notice that this vector here is really the initial
condition for the iteration.
Similar, the displacements at time t are the initial
conditions for the iteration for the displacements.
Zero here.
Zero there, meaning the initial
conditions for our iteration.
We set the iteration counter, k equal to 1, and we now go
into the following loop.
Notice on the right hand side, an out of balance load vector
is calculated.
And that out of balance load vector gives us an incremental
displacement vector.
Of course, the tangent stiffness matrix is involved
in that calculation.
Notice that this incremental displacement vector is added
to the displacements that we had previously calculated, to
obtain a new estimate on the displacements.
And like that, we have now obtained a displacement vector
that, of course, hopefully is closer to satisfying certain
convergence criteria.
We will have to talk about these convergence
criteria later on.
Basically, we check whether equilibrium is satisfied
within a certain tolerance.
If the equilibrium is not satisfied, we go here.
We use these displacements that we just have calculated
to compute a new vector of nodal point forces
corresponding to the element stresses.
Notice this vector now contains the information that
we have just calculated in solving this equation here.
This was information that we obtained.
And that information is contained in this vector, and
that vector is used right here to calculate this nodal point
force vector.
Having calculated that, we increase our iteration
counter, come back here and like that we continue the
iteration, looping around through here
as I have just described.
Notice at convergence once again, we want, of course,
that r is equal to f, meaning that the external loads are
equilibrated by the nodal point forces corresponding to
the internal element stresses.
Well, this brings us to the end of this lecture.
You might recognize that this last viewgraph really contains
a lot of information that, of course, we discussed in this
lecture, but that also I referred to already in the
first lecture.
At that time, I tried to introduce you to this
iteration process using rather physical concepts, not going
through a lengthy mathematical derivation.
We did not talk about continuum mechanics variables.
We did not talk about total Lagrangian formulation, second
Piola-Kirchhoff stresses, Green-Lagrange
strains and so on.
I hoped at that time to give you just a physical feel of
how we are iterating.
The same kind of equations, the same equations we looked
at at that time, now I hope with this lecture, I have
given you the mathematical basis of these equations.
I hope you have learned how to derive these equations, how to
work with these equations a little bit.
And in some of the next lectures, we will look at how
actually we do construct this k matrix
for different elements.
That is, of course, a very important consideration.
How do we calculate this k matrix for 1D, 2D, 3D
elements, shell elements, beam elements?
How do we calculate this force vector for
these different elements?
We will have to discuss that as well.
Notice that in this k matrix goes the material law.
Whether we are dealing with an elasto-plastic material,
rubber type material, all that will affect the actual
ingredients, the actual elements
here in that k matrix.
We we also have to discuss how we can possibly make this
convergence that I'm talking about fast.
In other words, how can we accelerate the convergence of
these iterations?
So in essence, I like to just convey to you that what you
see here you will see a number of times again.
We will talk about the different parts that you have
been looking at here already briefly once again in the
upcoming lectures.
Thank you very much for your attention.