Test velocities in FEM
santa claws
point on a proof, but I can't realize what's wrong with my assumption.
First, let's say this (from the book):
- d is the displacement
- v is the velocity (time differentiation of d: dd/dt)
- a is the acceleration (dv/dt)
- x_n means the quantity x at time n (for example v_n, v_(n+1), ...)
- T is the time-gap between two successive discretized quantities as
in x_(n+1) and x_n
The book says that a property holds for any test(v), and derives a
condition using one particular test velocity:
test(v) = ( d_(n+1) - d_n ) / T
That's fine, and I can find the same conditions he gets. What I have
tried is this: why not using another test velocity? For example, why
not use:
test(v) = v_n
If I expand all the steps, I end up finding *another* condition, not
the same as before. What's wrong with this? Is it ok, and the proof
holds for the intersection between the two, or am I missing a
theoretical point here? And what if I use another test velocity, like
something derived from the acceleration... would it be ok? I am
studying this, and I am trying to make all the steps in the book, but
this is beyond my (poor) knowledge.
Thanks!
jbriggs444
> Hi, I am studying the finite element method, and I came across a weird
> point on a proof, but I can't realize what's wrong with my assumption.
>
> First, let's say this (from the book):
> - d is the displacement
> - v is the velocity (time differentiation of d: dd/dt)
> - a is the acceleration (dv/dt)
> - x_n means the quantity x at time n (for example v_n, v_(n+1), ...)
> - T is the time-gap between two successive discretized quantities as
> in x_(n+1) and x_n
>
> The book says that a property holds for any test(v), and derives a
> condition using one particular test velocity:
> test(v) = ( d_(n+1) - d_n ) / T
You've not indicated what is meant by the phrase "test velocity" or,
in general by "test displacement", "test acceleration" or "test
<anything>". I've not read your book and you've not referenced it.
And I'm not familiar with finite element methods. So this post may be
worth every penny you paid for it.
I'm going to assume that "test <x>" is a formula for calculating an
approximation for <x> based on the entities that you are explicitly
tracking in your model.
> That's fine, and I can find the same conditions he gets. What I have
> tried is this: why not using another test velocity? For example, why
> not use:
> test(v) = v_n
At some point you're going to be stepping your model into the future,
using model values from step n (and prior) to obtain approximate
values at step n+1.
Setting v_(n+1) based on v(n+1) is not going to work.
Given bounds on actual acceleration one can guarantee that the
instantaneous velocity v_n is approximately equal to the leading
average velocity ( d_(n+1) - d_n ) / T or the trailing average
velocity ( d_n - d(n-1) ) / T or the average of the two, ( d_(n+1) - d
(n-1) ) / 2T.
Anything that is true of any of those three measures is going to be
approximately true of the actual instantaneous velocity.
> If I expand all the steps, I end up finding *another* condition, not
> the same as before. What's wrong with this? Is it ok, and the proof
> holds for the intersection between the two, or am I missing a
> theoretical point here? And what if I use another test velocity, like
> something derived from the acceleration... would it be ok? I am
> studying this, and I am trying to make all the steps in the book, but
> this is beyond my (poor) knowledge.
You're doing a pretty good job of failing to provide details here.
I expect that any approximation will do. But obviously, some
approximations are better than others. Less obviously, some will have
systematic biases.
An approximation to velocity that factors in acceleration is going to
predict future velocity more accurately than a formula that only
factors in historical displacement.
An approximation to velocity that factors in predicted future
acceleration is going to predict future velocity more accurately than
a formula that only factors in historical acceleration.
Google Runge Kutta or Euler's Method for additional perspective on
techniques for solving differential equations with numerical methods.
At a guess, this is what the text book is leading toward, albeit
possibly not with quite the same fancy words.