iMechanica - Comments for "Journal Club Theme of May 2009: Configurational forces in finite elements - energy-based mesh adaptation "
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Comments for "Journal Club Theme of May 2009: Configurational forces in finite elements - energy-based mesh adaptation "enMysticism
https://www.imechanica.org/comment/10801#comment-10801
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<p><em>In reply to <a href="https://www.imechanica.org/node/5377">Journal Club Theme of May 2009: Configurational forces in finite elements - energy-based mesh adaptation </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>I agree with remarks of Arash. Let us regard the Theory of Configurational<br />
Forces as a mystical chapter of Solid Mechanics <img src="/modules/tinymce/includes/jscripts/tiny_mce/plugins/emotions/images/smiley-laughing.gif" border="0" alt="Laughing" title="Laughing" /></p>
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</ul>Tue, 12 May 2009 06:22:16 +0000Konstantin Volokhcomment 10801 at https://www.imechanica.orgoptimization
https://www.imechanica.org/comment/10795#comment-10795
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<p><em>In reply to <a href="https://www.imechanica.org/comment/10793#comment-10793">understanding configuration force from optimization point of vie</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>Dear Xu:</p>
<p>Thank you for your comments. I agree with you on your comment regarding optimization. "Energy" (or some other functional) can be optimized with respect to many different variables. It's perfectly fine to adaptively change a mesh using energy optimization but this doesn't mean that the conjugate forces should behave or be governed by balance laws similar to those of standard (Newtonian) forces. It's perfectly fine to think of J-integral as a "configurational force". But why not "define" a configurational "mass", etc.? Perhaps one can do all that but do those mathematical constructs mean anything? My point is that why don't we stick to what is useful and not worry about abstractions that don't solve any real problem and/or do not lead to any real insights.</p>
<p>Regarding your other comment, I agree that one can define many things in a discrete system like a carbon nanotube or a crystal with a defect but do we really need "configurational forces" there? I don't think so. In a crystal, one can, in principle, keep track of the motion of defects without any use of "cofigurational forces" (using a knowledge of atomic bonds and F=ma, of course above the Debye temperature). </p>
<p>Regards,<br />
Arash</p>
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</ul>Mon, 11 May 2009 22:16:48 +0000arash_yavaricomment 10795 at https://www.imechanica.orgRe: Re: Configurational Forces
https://www.imechanica.org/comment/10794#comment-10794
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<p><em>In reply to <a href="https://www.imechanica.org/comment/10789#comment-10789">Re: Configurational Forces</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>Dear Jee:</p>
<p>The paper you mentioned is another interesting work and in some sense a 1D version of what you discussed earlier. By an intrinsically discrete system I mean something like a collection of atoms. Is there an atomistic version of "material forces"? Of course, configurational forces are useful in shape optimization (and many other things) but still some fundamental issues remain unanswered.</p>
<p>Regarding your question, I guess the answer should be yes. If you have a variational principle, you can extend it to a larger configuration space and the new variations will give you some new Euler-Lagrange equations that may be useful.</p>
<p>Regards,<br />
Arash</p>
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</ul>Mon, 11 May 2009 21:37:54 +0000arash_yavaricomment 10794 at https://www.imechanica.orgunderstanding configuration force from optimization point of vie
https://www.imechanica.org/comment/10793#comment-10793
<a id="comment-10793"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/10748#comment-10748">configurational forces</a></em></p>
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Dear Arash,
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Thank you for your very helpful comments about configuration force! The following are
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my unstanding.
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From my point of view, it seems better to understand the configuration force from
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optimization point of view. For some mechanical systems, their equilibirum status can be
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obtained by extremizing some functionals. The values of these functionals are often
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dependent both on some state variables (displacement field, stress field etc) and some parameters (the positions of point dislocations, length of the crack, etc) of the system. Generally, the state variables are also the implicit functionals/functions of the parameters, i.e. U=U(u(b),b)=U\tilde (b), where u is the state variable and b is the parameter. For fixed value of b, we have \partialU/\partialu=0, here \partial U/\partial u represents the "physical" unbalance force. If the value of b is allowed to vary (the system is evolved) WITHOUT constraint (for simplicity), then the final state of the structure should satisfy
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\partial U\tilde (b)/\partial b= \partial U/\partial b + (\partial U/\partial u).(\partial u/\partial b)=0.
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Since for fixed b, the equilibrium state should satisfy \partialU/\partialu=0, then the final optmal value of b should staisfy
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\partial U/\partial b=0 at b=b^opt and u=u(b^opt) .
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This is just the optimality criteria of the considered optimization problem. Therefore, in some sense, \partial U/\partial b can be viewed as the "configuration " or "material" unbalance force and the optimallity criteria \partial U/\partial b=0 can be viewd as the "balance of configurational linear momentum" or "configurational balance law".
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For the mesh adapation problem, as we know, every conformal FEM discretization of the structure will give a upper bound of the true total potential energy of the system. So if the measure of the mesh quality is the total potential energy of the system, then the best discretization (representing by the node position) in a suitable admissible space should attains the lowest total potential energy. I think this is the theory behind using configuration force to optimize the mesh. Of course, for other kind of mesh quality measures, using \partial U/\partial X_i as configuration force is questionable, at least theoretically.
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For "intrinsically discrete system" as you metioned, I think that we can also define the corresponding configuration force. For example, under the applied force, the S-W defect in a carbon nanotube can be evolved. If you take the position vector X of the defect as a configuration parameter, then \paritial U/ \partial X can be defiend as the corresponding configuration force.
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best regards
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Xu Guo
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</ul>Mon, 11 May 2009 16:13:17 +0000Xu Guocomment 10793 at https://www.imechanica.orgRe: Configurational Forces
https://www.imechanica.org/comment/10789#comment-10789
<a id="comment-10789"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/5377">Journal Club Theme of May 2009: Configurational forces in finite elements - energy-based mesh adaptation </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>
Dear Arash,
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Thanks for your thoughtful comments and for providing further perspective. I understand that the nature of the configurational balance laws are still subject to ongoing discussion. But whether the configurational forces are real in the physical sense or a mathematical construct, it seems that they provide a quite general and effective way to analyze material defects and inhomogeneities. Also, for an intrinsically discrete system, may we not consider the optimization of a structural truss as in <cite> Askes (2005) et al.</cite>? The configurational forces in such a system are the forces that are conjugate to the intial truss joint positions.
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H. Askes, S. Bargmann, E. Kuhl, P. Steinmann, "Structural optimization by simultaneous equilibration of spatial and material forces", <em>Commun. Numer. Meth. Engng</em>, 2005, <strong>21</strong>:433-442.
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The last point that you raise, about systems that do not have a well-defined reference configuration, is something I also find very intriguing. For a problem such as the mechanics of a fluid membrane system, where the potential energy does not depend on a previous configuration, the results of a finite element analysis will still be influenced by the initial nodal positions due to the path the solver takes during minimization. Would it be possible to optimize the initial nodal positions systematically using similar ideas?
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<p>Best Regards,<br />
Jee. </p>
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</ul>Mon, 11 May 2009 04:17:31 +0000Jee E Rimcomment 10789 at https://www.imechanica.orgconfigurational forces
https://www.imechanica.org/comment/10748#comment-10748
<a id="comment-10748"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/5377">Journal Club Theme of May 2009: Configurational forces in finite elements - energy-based mesh adaptation </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>Dear Jee and other friends:</p>
<p>Thank you for choosing this nice topic and also thank you for the short description of what one would wish to do in FE mesh adaptation inspired by Eshelby's work. I have a few comments that may be relevant. I should emphasize that I don't mean to criticize any of the papers you cited and/or similar papers by any means.</p>
<p>Eshelby's work was motivated by what he had seen in field theories. In a field theory with a background metric g, if one varies the metric, action would change. Energy-momentum tensor is the variational derivative of action. In the paper you mention, Eshelby shows that if a defect moves in the reference configuration there will be a change in energy and the thermodynamic force conjugate to defect motion is, according to Eshelby, a "defect force" (or a configurational or material force). This is fine and beautiful. Many researchers have been able to find similar expressions for different examples of "defects" like ferromagnetic domain walls, shock waves, etc.</p>
<p>In the case of finite elements, there have been works (including the ones you mention) in which one changes the mesh adaptively, e.g. a more refined mesh around a moving crack tip, etc. These works are all very interesting and the idea of thinking of finite element nodes as "defects" seems to be useful. But what I would like to comment on is that there is no real theory behind any of these works. This doesn't make them not useful but it's something to know. There is much excitement in the literature regarding the so-called configurational mechanics but, in my opinion, there is also so much confusion. Let me mention a few things that I never completely understood and would very much like to understand some day.</p>
<p>In the literature, you would see things like "balance of configurational linear momentum". But what is a configurational force? What is the meaning of a configurational balance law? Is it something convenient that one can use and solve problems with? Or there is something deeper behind it? I don't know and I don't think the existing arguments are convincing. One line of argument is to "pull-back" balance of linear momentum to the reference configuration and then call it "balance of configurational linear momentum". But pulling something back wouldn't give one anything new. This is like defining the first or second Piola-Kirchhoff stress tensors. These are "equivalent" to Cauchy stress and don't give one anything new.<br />
It is also known that if the standard Euler-Lagrange equations are satisfied then action is trivially extremized with respect to variations in the reference configuration (of course these are all valid when everything is smooth). It turns out that this is not the case when one deals with a discrete system (like FE). "Postulating" that action should be extremized with respect to referential variations would give "better" meshes, at least in the examples solved so far. But yet it's not clear why this happens.</p>
<p>Let me mention another controversy in the literature. Eshelby [1] was puzzled by what he found in liquid crystals. He asked if a configurational force on a disclination can be a "real" force. Nabarro [2] had a similar problem with dislocations. There are a couple of other interesting papers that discuss this [3,4]. </p>
<p>[1] J. D. Eshelby. The force on a disclination in a liquid crystal. Philosophical Magazine A,<br />
42(3):359–367, 1980.</p>
<p>[2] F. R. N. Nabarro. Material forces and configurational forces in the interaction of elastic singularities.<br />
Proceedings of the Royal Society of London, A398:209–222, 1975.</p>
<p>[3] J. L. Ericksen. Remarks concerning forces on line defects. ZAMP, 46:247–271, 1995.</p>
<p>[4] J. L. Ericksen. On nonlinear elasticity theory for crystal defects. International Journal of<br />
Plasticity, 14(1-3):9–24, 1998. </p>
<p>Last but not least, let me comment on discrete systems. By discrete I mean an intrinsically discrete system and not discretization of a continuum (like FE). We all know that Cauchy stress can be, at least qualitatively, understood as some "average" of interatomic forces in the underlying particle system. What about "configurational stress"? What is a discrete version of Eshelby's stress? In a particle system there is no well-defined reference configuration. If you look at any known interatomic potential, all you need is the current position of particles to calculate the energy. So, do "configurational forces" have any discrete analogues or they are just artifacts of the "continuum"?</p>
<p>In summary, I believe the ideas you have discussed are interesting and useful but one should not get too excited when there is no real theory behind these techniques, at least not to this date.</p>
<p>Regards,<br />
Arash</p>
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</ul>Thu, 07 May 2009 01:30:11 +0000arash_yavaricomment 10748 at https://www.imechanica.org