Implicit vs Explicit Scheme - Phụ đề song ngữ
Hello everyone, my name is Pushpindar Pandey.
I am from Indian Institute of Technology Delhi.
Today we will talk about the implicit and explicit method.
what are the nuances associated with these two methods, I'm really hopeful that you will be able to understand in next couple of minutes.
So there are first two generic statements applicable for all the schemes, not necessarily for these two implicit and explicit scheme.
So these two are analytical methods to solve your equations.
The second is the unknown is found for the next step using your known or unknown quantity.
I'll explain in next couple of seconds that what do I mean by known or unknown quantity.
Now to understand this let us take an example of any function.
Alright, which is varying over time.
Let's say in this particular fashion.
Now, we have unknown quantity at t equals to n.
The task is to find out the unknown quantity at t n plus 1.
Alright, now we will find out the y at tn plus 1 using your explicit scheme and your implicit scheme.
First for your explicit scheme, the solution at the next step is found using slope at the known time step.
We will find the unknown by using the slope of the known time step.
The question is this y at n plus 1 will be n,
this is your known quantity plus delta t, delta t is your, the time step, which we are supplying in your finite element tool.
then your differentiation of this particular function at t equals to n.
Now, this is known to us delta t is we are supplying.
So, it is also known to us y equals to n, this is known to us, this was only unknown quantity.
So, we can estimate it easily for your explicit scheme the solution at the next step is found using slope at the unknown
time step here the question look like something this y at n plus 1 equals to y
at n plus delta t dot y dash n plus 1 that is a differentiation of of this
function is done at the unknown step so since this is an unknown quantity so
we cannot use our analytical methods to solve your your unknown quantity we have to
relate methods like Newton-Raphson to iterate it and to find out the value of your unknown.
Now, the one thing which is to be noticed here is that for your explicit scheme, the unknown
and the known quantity these two are lying explicitly on the either side of the equation that all the known quantities are on the arches,
all the unknowns are on the LHS side of the your equations so the name is explicit these two are explicitly defined now for implicit you can see
that the known quantity and the unknown they these two are lying on the same side of the equation
and hence the name is your implicit we have to solve it using your some other methods other than analytical,
it's to solve this nature of equation.
So this was just a basic difference between explicit and implicit.
In next slide we will understand about the detail difference or the similarity between these two in connection to your FE process.
The first is governing equation.
Now, the governing equation is going to remain same, no matter which scheme you are going to utilize, whether you are explicit scheme or you are implicit
scheme.
It will use the basic Newton's law.
Your F external is equals to your mass multiplied by your acceleration plus damping multiplied by your velocity plus stiffness multiplied by your displacement.
Same for your implicit scheme.
For your displacement estimation, here the X is known by using your acceleration.
So we will keep everything on the RHS except your acceleration.
For your implicit scheme, it is found using displacement.
So we will keep everything on the RHS except your displacement.
That's how the equation will become.
All right, now for the CPU time estimation, this is very economical for the large models as a mass matrix inversion is quite easy.
Here the mass matrix in the explicit is lumped,
lumped in the sense that it has only diagonal elements,
all the non diagonal elements are zero, so it becomes really easy for the computer to do the inversion.
and found out the value of it.
For your implicit scheme we are doing the inversion of your stiffness matrix since it becomes really cumbersome and really difficult for the computer to do the inversion.
It takes a lot of time since your larger model consists of millions of degrees of freedom and
to estimation of these two quantities becomes really difficult or very time consuming.
The stability factor here it is conditionally stable,
your time step always has to be less than or equal to your characteristics length divided by the speed of sound in that material
that is basically the time taken by your wave to propagate across the edge of an element.
For your implicit method it is an unconditionally stable we can pick up any large time step
as possible but still for the larger models in no matter how big the time step you are picking up it still takes less time.
convergent factor.
The known estimation,
as I have discussed in the previous slide,
the next state is found used from your current state and the quantities to be estimated can be separated out at one side of equation.
The method is known as forward euler or the central difference method.
For example, you can see that this particular differential equation d y by dt equals to minus y square t belongs to 0 and a.
You can see that y k plus 1 is equals to y k minus delta t dot y k square.
You see that at k plus 1 is occurring on the one side of the equation.
This is your unknown and these two quantities are known to us.
Similarly, for your implicit scheme, here the unknown quantity will occur, unknown and
unknown quantity will to the same side of the equation as observed here.
All these points are, I have already discussed, so I have not going to the details of it.
Alright, so this was about the basic differences and the similarities between your explicit and your implicit method.
Please leave questions in the comment section.
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