iMechanica - education - Comments

hi contact me by

mehdi_eft — Sat, 11 Feb 2012 15:34:54 -0500

contact me by e-mail

send your model and Iwill help you

mehdi_eft@yahoo.com

Thank you so much

mksahu — Wed, 08 Feb 2012 06:35:10 -0500

Thanks sir for sharing this great book

stiffness matrix

Lihua Jin — Mon, 06 Feb 2012 14:57:47 -0500

The physical meaning of the component of a stiffness matrix c(i,j) is the force adding on node i, if a unit displacement is added on node j.

Ajit R. Jadhav — Thu, 02 Feb 2012 02:04:14 -0500

Ajit R. Jadhav — Thu, 02 Feb 2012 02:03:53 -0500

Ajit R. Jadhav — Thu, 02 Feb 2012 02:03:12 -0500

Ajit R. Jadhav — Thu, 02 Feb 2012 02:02:49 -0500

Re: randomly varying coefficients

Biswajit Banerjee — Wed, 01 Feb 2012 15:44:17 -0500

Amit,

You've "dealt with" the question quite well :) Weak convergence has been used to justify why PDEs with smooth coefficients are so good at predicting the behavior of structures even though there is considerable (and non-smooth) variation in microscopic properties.

Tartar writes : " ... One may have discontinuities of the coefficients, so that the partial dierential equations should be
understood in the sense of distributions, ... in the case of piece-wise smooth coefficients showing a discontinuity along a smooth interface, it is equivalent to writing the partial differential equations in a classical way on each side, and adding adapted transmissions conditions at the interface."

If you look at the literature on stochastic finite elements, people assume that one can differentiate a random variable in the standard way (and I'm not talking about discrete calculus). That has been bothering me and I wonder if there is a way of showing that such an assumption does not make any practical difference.

-- Biswajit

spatially varying coefficents

Amit Acharya — Wed, 01 Feb 2012 10:17:30 -0500

Biswajit,

I presume you mean what one could say about the limiting forms of energy in the case the material properties oscillate a lot.

The one result I know of is the following (by Tartar) - for the scalar, linear, second order wave equation with L^\inf coefficients - so only essentially bounded but otherwise could vary wildly (would accommodate what you describe) - one can take a sequence of approximating weak solutions that converge to a limit (say zero) in the sense of weak convergence - thus in the limit you have a solution that is wildly oscillating with mean zero. Then the action - the kinetic energy *minus* the potential energy evaluated along this sequence tends to the limit 0. This proves a type of equipartition where the total energy is exactly equally divided into the potential energy and the kinetic energy in the limit (note this meaning of equipartition is different from that used in statistical mechanics for discrete systems).The main issue in the above question is that the kinetic energy and the potential energy are both nonlinear functions of the state.

To evaluate the common limit of the potential energy and the kinetic energy in this case, one needs to assume more about the smoothness of the coefficients - I believe it is C^1. Then Tartar characterizes (i.e. gives a formula for) the limit by the H-measure of the sequence.

So, for the explicit formula one does not have a rigorous result for the type of material property variation you mention - but note that even a C^1 function can be made to oscillate very, very wildly.

The interesting thing about the above result (for me) is that this is probably the only rigorous result I know of where you assume almost nothing and have internal energy emerge, i.e. the difference of (the limit of the kinetic energy + potential energy along the sequence) And (the kinetic energy + potential energy evaluated on the weak limit of the sequence) is non-zero in general and we have an explicit formula for it.

I don't think there are comparable results in the system (even linear) case.

May be the above helps. I had to interpret what you meant by 'dealt with.'

- Amit

Re: Tartar's light

Biswajit Banerjee — Tue, 31 Jan 2012 20:36:12 -0500

Tartar mentions that discontinuities in the governing equations can be dealt with by invoking the integrals -> distribution concept. Is that true when each material point has a different property?

-- Biswajit

Plasma

Amit Pandey — Tue, 31 Jan 2012 09:52:15 -0500

Thanks Dr Acharya

I enjoyed it although I did not get much at the end of it. I remember learning as a kid that there are 3 states of matter and later on realized everyone ignored plasma. And now eqn's representing such state of matter seems like a living object unrevealing some hidden stories. ..fascinating!!

Regards

Amit

Looking for a PhD student

pschiavone — Mon, 30 Jan 2012 18:06:55 -0500

Linear Elasticity, Applied Mathematics

Do you know if there is any

random — Mon, 30 Jan 2012 09:33:59 -0500

Do you know if there is any other way of doing it? I need to mainly convert the geometry i have in the ls dyna file to abaqus

I know a way in

sspadhee — Sun, 29 Jan 2012 03:29:24 -0500

I know a way in HyperMesh.

Step 1. Open HM in Abaqus mode

Step 2. Import the .k file

Step 3. Re-define the element type and material card.

Step 4. Export it as a .inp file

How to import Ls-Dyna files(*.k) to Abaqus CAE 6.7 Edu version

random — Sat, 28 Jan 2012 18:37:30 -0500

Could you please let me know if you figured out how to import LS dyna .k file to abaqus