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Journal Club Theme of May 2009: Configurational forces in finite elements - energy-based mesh adaptation

*This issue of the Journal Club has been prepared as a group by Ankush Aggarwal, Mo Bai, Mainak Sarkar, and Jee E Rim, and with comments and suggestions from Bill Klug .

Most of us are familiar with continuum mechanics of homogeneous solids, and the use of finite elements in these applications. An oft-arising practical issue is that of meshing - the results and sometimes even the tractability of a finite element problem depend on the suitability of the mesh. In particular, for problems with large differences in the deformation gradient, a mesh that adapts to the most optimal configuration is desirable but difficult to obtain. In this issue of the Journal Club, we propose to discuss a method of mesh adaptation - energy-based mesh-adaptivity and it's connection to configurational forces. Though none of us leading this discussion are experts in the field, we think that this is a very interesting field that is useful to several important finite element problems.

The theory of configurational forces can be traced back to the original article by Eshelby (1951), where the concept of forces acting on a singularity was introduced into the classic theory of elasticity by defining the energy-momentum tensor (or the Eshelby stress tensor). In simple terms, the variation of strain energy density in the material due to a defect or inhomogeneity leads to "configurational" forces acting on these inhomogeneities, which are allowed to move through the material. The resulting stress tensor is the superposition of the strain energy density and a transformation of the Cauchy stress. In keeping with the original concept, historically, the configurational or material forces have mainly been used in analyzing physical defects such as interstitial atoms in solids, dislocation lines, interfaces, and cracks. A nice article with various such examples and further references for each is Gross et al.(2003)

A relatively new application of configurational forces is in the context of finite elements, where Braun (1997) recognized that the finite element discretization can be seen as a kind of material inhomogeneity. We're used to thinking of the finite element approximation of the deformation as being a function of the spatial nodal positions. However, the approximation also depends on the global shape functions and their supports, which are determined by the material nodal point positions. Therefore, the discrete potential energy I can be regarded as a function of both x and X, using the standard notation of x for spatial and X for material points of the domain: I = I(x, X). It naturally follows that by minimizing the discrete potential energy with respect to X, or, in other words, enforcing a balance of configurational forces at each material nodal point, an optimal mesh may be found. If this is a little confusing to visualize, we found this article by Braun (recommended article 1) listed below very helpful: it offers a simple one-dimensional example where the configurational forces and their application in mesh optimization can be followed in detail and explicitly in closed-form.

These ideas are quite simple and pleasing, since an optimal mesh defined as that minimizing the total energy using variational principles makes sense physically as well as mathematically. In addition, measures such as error estimates or mesh adaptation indicators are completely unnecessary. However, in practice, the complete variation of energy with respect to discretization is very difficult, as is the numerical implementation of the minimization problem. This is partly due to the fact that unlike a physical defect which may move freely through the material, the movement of material nodes is restricted by the connectivity of the mesh. In particular, the movement of certain material nodes, such as those on the domain boundary, is further restricted by the necessity of maintaining the domain geometry. In addition, the resulting adapted mesh has to be of acceptable quality, i.e., with positive Jacobians and a small enough distortion, meaning that constraints - either implicit or explicit, need to be introduced. The main issues to be resolved in our opinion therefore lie in the efficient and practical application of the concept of configurational force balance for mesh adaptation. Here, we suggest two articles as examples of how these difficulties have been (partially) addressed, by invoking suitable approximations and solution schemes. Both demonstrate significant improvements in the finite element solutions of crack-growth problems with mesh adaptation.

Recommended reading:

  1. M. Braun, "Configurational forces in discrete elastic systems", Archive of Applied Mechanics, 2007, 77:85-93. (http://dx.doi.org/10.1007/s00419-006-0076-y)

    We recommend this paper as a starting point. By dealing with a simple 1-D finite-element problem, this paper provides a clear and easy-to-understand explanation of the concept of configurational forces and investigates its role in mesh optimization. The finite-element solution for nodal displacements are explicitly derived as functions of the material nodal positions, and thereby the direct dependence of the discrete potential energy on the material nodal positions.

  2. P. Thoutireddy and M. Ortiz, "A variational r-adaption and shape-optimization method for finite-deformation elasticity", International Journal for Numerical Methods in Engineering, 2004, 61:1-21 (http://dx.doi.org/10.1002/nme.1052)

    The authors of this paper address the problem of movement of material nodes while maintaining the mesh quality by edge-face or octahedral swapping. This allows limited changes in mesh-topology (the total number of nodes is unchanged), so that while the resulting mesh is not likely to be the minimizer of I, the nodes may move closer towards the minimum without the mesh becoming entangled. While effective, this method is admittedly ad hoc. The flexibility of mesh connectivity is also somewhat undermined by the necessity of maintaining the integrity of the domain boundary - the boundary nodes are constrained to stay within the boundary. These issues were addressed further in a later paper by Mosler and Ortiz (2006) .

  3. M. Scherer, R. Denzer, and P. Steinmann, "Energy-based r-adaptivity: a solution strategy and applications to fracture mechanics", International Journal of Fracture, 2007, 147:117-132. (http://dx.doi.org/10.1007/s10704-007-9143-9)

    Here, in contrast to the above article, the element connectivities are held fixed during the mesh adaptation. Instead, excessive distortion of the mesh is prevented by introducing inequality constraints that dictate an admissible maximum distortion for each element. The measure for the element distortional deformation can be chosen to be twice continuously differentiable, so that the explicit derivatives can easily be implemented in a finite element method. An advantage of this approach is that the element constraints can be made independent of the original element by defining the distortion measure relative to an arbitrary reference element - thus the original quality of the mesh is not important.

In general, the simultaneous minimization of the discrete potential energy I(x, X)with respect to both x and X is problematic due to the non-convexity of I. Thus, both articles 2 and 3 employ a staggered scheme (conjugate gradient and Newton, respectively), for the minimization: first, the configurational forces are computed from equilibrated displacement fields, then the material nodes are shifted to satisfy the configurational force balance, and lastly a new mechanical equilibrium is found for the new material nodal positions. Unfortunately, not enough information is presented for a comparison of computational cost between the two strategies. We envision ongoing and future development in strategies for increased robustness and efficiency - as well as investigations into broader classes of problems.

Further References:

  1. J. D. Eshelby,  "The force on an elastic singularity", Philos. Trans. Roy. Soc. London Ser. A, 1951, 244:87-112.
  2. D. Gross, S. Kolling, R. Mueller, I. Schmidt,  "Configurational forces and their application in solid mechanics", Eur. J. Mech. A, 2003, 22:669-692.
  3. M. Braun, "Configurational forces induced by finite-element discretization", Proc. Estonian Acad. Sci. Phys. Math, 1997, 46:24-31.
  4. J. Mosler, M. Ortiz, "On the numerical implementation of variational arbitrary Lagrangian-Eulerian (VALE) formulations", Int. J. Numer. Meth. Engng, 2006, 67:1272-1289.

Comments

Arash_Yavari's picture

Dear Jee and other friends:

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.

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.

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.

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.
    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.

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].

[1] J. D. Eshelby. The force on a disclination in a liquid crystal. Philosophical Magazine A,
42(3):359–367, 1980.

[2] F. R. N. Nabarro. Material forces and configurational forces in the interaction of elastic singularities.
Proceedings of the Royal Society of London, A398:209–222, 1975.

[3] J. L. Ericksen. Remarks concerning forces on line defects. ZAMP, 46:247–271, 1995.

[4] J. L. Ericksen. On nonlinear elasticity theory for crystal defects. International Journal of
Plasticity, 14(1-3):9–24, 1998.

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"?

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.

Regards,
Arash

 

Dear  Arash,

   Thank you for your very helpful comments about configuration force!  The following are

  my unstanding.

   From my point of view, it seems better to understand the configuration force from

 optimization point of view. For some mechanical systems, their equilibirum status can be

obtained by extremizing some functionals. The values of these  functionals are often

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

    \partial U\tilde (b)/\partial b= \partial U/\partial b + (\partial U/\partial u).(\partial u/\partial b)=0.

  Since for fixed b, the equilibrium state should satisfy  \partialU/\partialu=0,  then the final optmal value of b should staisfy

       \partial U/\partial b=0 at b=b^opt and u=u(b^opt) .

   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". 

  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.

    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.  

   best regards

   Xu Guo

 

 

  

 

 

  

Arash_Yavari's picture

Dear Xu:

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.

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).

Regards,
Arash

Dear Arash,

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 Askes (2005) et al.? The configurational forces in such a system are the forces that are conjugate to the intial truss joint positions.

  1. H. Askes, S. Bargmann, E. Kuhl, P. Steinmann, "Structural optimization by simultaneous equilibration of spatial and material forces", Commun. Numer. Meth. Engng, 2005, 21:433-442.

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?

Best Regards,
Jee.

 

Arash_Yavari's picture

Dear Jee:

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.

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.

Regards,
Arash

Konstantin Volokh's picture

I agree with remarks of Arash. Let us regard the Theory of Configurational
Forces as a mystical chapter of Solid Mechanics Laughing

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