finite elasticity

Solving incompressible elasticity has been quite challenging numerically. The conventional approach for handling incompressibility is the so-called penalty method. A volumetric energy term enters into the strain energy and penalizes the volumetric deformation. One straightforward issue is that the penalty parameter goes directly into the tangent matrix. The bigger the penalty parameter, the worse the condition number of the matrix. This is really a manifestation of the ill-posedness of theories based on the Helmholtz free energy, in my opinion [3].