stable elements for mixed elasticity

Dear Alessio,

I think that you are referring to the PEERS element introduced by Arnold D.N, Brzzi F., Douglas J. in Jap J. Appl. Math (see web page http://www.ima.umn.edu/~arnold/publications.html for the paper) that uses unsymmetric approximations for teh stress and imposes a weakened symmetry condition through the use of Lagrange multiplier.

The idea however originally was purpoted by Fraijs de Veubeke. 

 Hope this helps.

School of Engineering, Swansea University

Hello Alessio,

You are right, in general one cannot interpolate the six components without additional constraints imposing symmetry of the wretched symmetric 2nd order stress tensor!

The equations of elasticity denote conservation of Linear momentum, the angular momentum conservation however, is tacitly buried in the symmetry of the stress tensor. When one tries to do mixed interpolation for such a system, with stresses being dual variables the underlying finite element space must be made cognizant of this. It is to the point very abstruse to explain, but think on the following lines ... when you try to evaluate the integral \int(\sigma_{ij}(u):\sigma_{ij}(v) dx you implicitly assume a symmetry on \sigma_{_{ij}}(v) without noticing that \sigma_{ij}(v) is a test function and comes from a finite element space (polynomials) and knows nothing about the fact that (\sigma_{ij}(u) = \sigma_{ji}(u)) hence the assumption and everything that follows (computation, matrices) etc. is wrong!. Indeed what happens is we observe a locking like behavior (no convergence at all!) with either mesh refinement or increasing order of interpolation.

Another way to understand this is to think that integrals essentially represent projection operator (dot product, thats why inner product is an integral), in this case one is trying to project a symmetric quantity \sigma_{ij}(u) onto a a test space \sigma_{ij}(v) which does not contain only symmetric tensors (by design) and hence picks up spurious modes from this space which can be detrimental to the approximation. Note that quite often you might get a solution that "looks" just fine but will refuse to converge, other times you might not get a solution at all.

Mixed finite elements have been traditionally designed for scalar-vector systems (temp-heatflux, velocity-pressure) but elasticity is a vector-tensor based where the tensor is symmetric. Efforts to surmount this issue of symmetry dates back upto four decades (Arnold et. al) and there is no clear solution. Some low order polynomial spaces have been suggested) which (by design) interpolate symmetric tensors. However question on how to scale these design strategies to higher orders or higher dimension or hierarchical interpolations remains unanswered.

I stumbled upon this issue during my research and was spun off for a while before I figured the easiest way to address this issue is to simply interpolate the Gradients!, which are not symmetric and hence do not require a crafty interpolations. In literature this issue is addressed primarily by giving up on the symmetry of stress and then weakly imposing it using Lagrange multipliers (LM) (see PEERS element and in thesis you have posted). This is not cheap as it burdens an already inflated(mixed) system with another set of variable namely LM to be interpolated. I find the solution I proposed above to be cheap and effective way to obtain convergent finite elements for elasticity. Since stresses are just linear multiples of gradients (modify your constitutive law to map gradients directly to stresses) this is not bad at all and you get optimally convergent mixed finite elements for the first time. I used this approach in a discontinuous Galerkin formulation but I believe the remedy is truly independent of that fact. I have presented these findings in a soon to be published paper in CMAME.

Amidst all this, remember inf-sup (LBB) condition has nothing to do with symmetry issue and the two are different animals. The inf-sup condition ensures that the bilinear form is coercive on the given FE spaces. Inf-sup is essentially the "coercivity" condition as applied to a mixed bilinear form. A coercive bilinear form leads to stability, crudely amounts to a full rank system, one that can then be solved. Also remember that inf-sup condition is "sufficient" but not "necessary" all the time :) ...

To keep the discussion short, the inf-sup condition for elliptic system is well studied just follow this article by Bochev and Gunzburger (SIAM J.Numer. Anal, 43:340–362, 2005.) which sums it up very well. The so called Dirichlet (Minimize potential energy w.r.t strain-displacement as constraint) form is easier to construct. Here the crux of the matter is whether your formulation contains a gradient (Dirichlet) or a divergence (Kelvin) operator. Very Quickly let me just say, with gradient operator it is easy to construct a stable pair of spaces (why??) :) ...

Hope this helps to any extent.

-saurabh