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A general question on dynamic structure problem: spatial resolution of high-frequency modes typically is poor by using the...

Hi Everyone,

I have a general question about dynamic strcture problem,

I read from some books that "spatial resolution of high-frequency modes typically is poor by using the conventional finite element spatial domain discretization"?

Could someone explain more on this? Does it because the element usually is not fine enough to capture the high frequency response or...?

Thanks a lot!

BR,

Li

Comments

Laurent Champaney's picture

 High-frequency modes means large number of spacial oscillations per unit of length. So you need a very large number of elements per unit of length to catch these oscillations.That is why, for a given structure and a given mesh, the FE answer is better for the low-frequency modes at for that high-frequency ones.

L. Champaney
LMT Cachan/ENS Cachan
France

Laurent Champaney, thanks a lot for your explanation!

Now I have another question. We usually say that for explicit time integration, the crital time step should be 

delta_t <= 2/omiga_max, (omiga_max is the maximum mode angular frequency) some times we say 

delta_t <=L/c, L is the effective lengh of all the element, c is the sound of speed in this material. 

Are these two equivalant?

 

To me, the first one delta_t <= 2/omiga_max seems it's something related to the modal analysis of the dynamic system. If we have a system, MU''+KU=F, the modes of the system is something related with omiga_max. However, what about delta_t<=L/c?

 Thanks!

 

Laurent Champaney's picture

I am afraid that there is a confusionbetween modal analysis (frequency domaine, first question) and dynamics (time domain, second question).

For me, the two ctireria are disconnected, and related to dynamics with propagation.

 - The first one (delta_t <= 2/omiga_max) is related to a correct time representation of oscillations in time. I think you need at least 4 time steps to represent one oscillation :

  ^       ^

/  \     /   \ 

       V      

 - The second one is related to stability of the integration scheme. The time discretization must be such that one element cannot miss an information that propagate (at velocity c) in the structure. This means that between to time steps, the information must must move on a distance greater than the size L of an element:  delta_t<=L/c

 PS: in modal analysis, there is no time, so no time step and no propagation.

L. Champaney
LMT Cachan/ENS Cachan
France

Matt Lewis's picture

If one does a stability analysis on explicit time integration and the finite element degrees of freedom and stiffness and mass matrices, the first expression Li puts forward for the critical time step is right.  The critical time step is the one at which the largest magnitude eigenvalue of the amplification matrix has a value of unity.

delta_t_c = 2/omega_max,

where delta_t_c is the critical time step and omega_max is the highest natural frequency of the discretized system (i.e. coming from the system mass and stiffness matrices).  The important point is that time steps larger than this limit will cause instability (typically seen as a lot of NANs or floating-point overflows).  Kind of like the speed of light, this is not just a good idea, it's the law.

In explicit codes, we never actually form a stiffness matrix, so we need a good approximation (and it better be a lower bound) for this expression.  We typically use Li's second expression, namely

delta_t <=L/c

where L/c is the smallest ratio of element length to sound speed in the mesh.  And by sound speed, we mean the highest wave speed based on an acoustic analysis.

 

Does that help?

 

Matt Lewis
Los Alamos, New Mexico

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