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Journal Club Theme of May 2009: Configurational forces in finite elements - energy-based mesh adaptation
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.
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.
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) .
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.
- J. D. Eshelby, "The force on an elastic singularity", Philos. Trans. Roy. Soc. London Ser. A, 1951, 244:87-112.
- D. Gross, S. Kolling, R. Mueller, I. Schmidt, "Configurational forces and their application in solid mechanics", Eur. J. Mech. A, 2003, 22:669-692.
- M. Braun, "Configurational forces induced by finite-element discretization", Proc. Estonian Acad. Sci. Phys. Math, 1997, 46:24-31.
- J. Mosler, M. Ortiz, "On the numerical implementation of variational arbitrary Lagrangian-Eulerian (VALE) formulations", Int. J. Numer. Meth. Engng, 2006, 67:1272-1289.