Compatibility of constrained displacement field in the case of simply connected body

In nonlinear elasticity we regard the compatibility as a restriction of the motion (of a simple body) in terms of the strain measures: deformation tensor F or right Cauchy-Green deformation tensor C (also left Cauchy-Green deformation tensor b).

Commonly when we talk about the compatibility we consider an inverse problem. Given F (or C) we need to derive the deformation map. To do this certain mathematical restrictions must be placed upon the strain measures in order that the inverse problem have a solution. We call these mathematical restrictions the compatibility conditions or more precisely the integrability conditions, since in general the compatibility is equivalent to path independence of the solution dealing with integrations of the strains. So, the main question to answer is,  given F (or C or b), is there a deformation mapping?


Now we consider the forward problem. Given a deformation mapping (smooth immersion), it is easy to compute the strain measure. And following Truesdell C.A, Toupin R.A., Ericksen J.L, "The Classical Field Theories",

In a problem formulated entirely in terms of the deformation, the conditions of compatibility need not be regarded, since they are satisfied automatically in virtue of the definitions of b and C. 

It's look simple.

Now the question is raised when the deformation mapping itself is constrained. For instance, instead of assuming that here is the deformation mapping from the abstract space R^3 to the Euclidean space E^3, we consider the mapping from R^3 to  R^3 times  R^N times R^M times SO(3). What's going on the compatibility conditions in this case?

2013: Still with FASTCAP. Towards ExaFMM?

Electrostatics. Looking for the parallel fast field solver.

I successfully used the FASTCAP fast field solver long years ago for capacitance calculation and electrostatic charge density and forces derivation. Incorporating this solver into a finite element code, solution of multiphysics coupled problems can be obtained quickly. With staggered scheme of course.
 

Recently coming back  to the electrostatic problem calculation, I tried to find a new generation of the fast field codes. My feeling that here is still no ready open source code except the premium FASTCAP.


The FASTCAP itself is acceptable from the accuracy standpoint, particularly for my problems in the hand. I have compared the BEM results with FEM reference solutions  and found it quite acceptable. Also for the science community it is a common practice to compare new and more efficient solvers versus FASTCAP. The problem is the speed of runs. The code was born in 90's, in the era of single-core computations. Nowadays impressed by "Biomolecular electrostatics using a fast multipole BEM on up to 512 gpus and a billion unknowns" I think to write a custom electrostatic code with ExaFMM library. Obviously it would take a long time and  perhaps I'll go flop. As a result the simple, open source,  parallel performance, electrostatic solver is still looking for.