iMechanica - Comments for "A general question on dynamic structure problem: spatial resolution of high-frequency modes typically is poor by using the..."
https://www.imechanica.org/node/7670
Comments for "A general question on dynamic structure problem: spatial resolution of high-frequency modes typically is poor by using the..."enExplicit time integration critical time step
https://www.imechanica.org/comment/13672#comment-13672
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<p><em>In reply to <a href="https://www.imechanica.org/node/7670">A general question on dynamic structure problem: spatial resolution of high-frequency modes typically is poor by using the...</a></em></p>
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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.
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delta_t_c = 2/omega_max,
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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.
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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
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delta_t <=L/c
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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.
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Does that help?
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Matt Lewis<br />
Los Alamos, New Mexico
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</ul>Fri, 26 Feb 2010 22:25:41 +0000Matt Lewiscomment 13672 at https://www.imechanica.orgI am afraid that there is a
https://www.imechanica.org/comment/13671#comment-13671
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<p><em>In reply to <a href="https://www.imechanica.org/comment/13654#comment-13654">Laurent Champaney, thank a</a></em></p>
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I am afraid that there is a confusionbetween modal analysis (frequency domaine, first question) and dynamics (time domain, second question).
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For me, the two ctireria are disconnected, and related to dynamics with propagation.
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- 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 :
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^ ^
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/ \ / \
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V
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- 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
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PS: in modal analysis, there is no time, so no time step and no propagation.
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L. Champaney<br />
LMT Cachan/ENS Cachan<br />
France
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</ul>Fri, 26 Feb 2010 20:21:33 +0000Laurent Champaneycomment 13671 at https://www.imechanica.orgShort answer
https://www.imechanica.org/comment/13652#comment-13652
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<p><em>In reply to <a href="https://www.imechanica.org/node/7670">A general question on dynamic structure problem: spatial resolution of high-frequency modes typically is poor by using the...</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
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.
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L. Champaney<br />
LMT Cachan/ENS Cachan<br />
France
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</ul>Fri, 26 Feb 2010 20:08:08 +0000Laurent Champaneycomment 13652 at https://www.imechanica.orgLaurent Champaney, thank a
https://www.imechanica.org/comment/13654#comment-13654
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<p><em>In reply to <a href="https://www.imechanica.org/node/7670">A general question on dynamic structure problem: spatial resolution of high-frequency modes typically is poor by using the...</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Laurent Champaney, thanks a lot for your explanation!
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Now I have another question. We usually say that for explicit time integration, the crital time step should be
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delta_t <= 2/omiga_max, (omiga_max is the maximum mode angular frequency) some times we say
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delta_t <=L/c, L is the effective lengh of all the element, c is the sound of speed in this material.
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Are these two equivalant?
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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?
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Thanks!
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</ul>Thu, 25 Feb 2010 22:29:28 +0000Xiaogai Licomment 13654 at https://www.imechanica.org