Discontinuous Galerkin finite element methods for incompressible non-linear elasticity

A discontinuous Galerkin finite element method (DGFEM) for incompressible, non-linear elasticity is derived. One of the limitations of the continuous Galerkin finite element method (CGFEM) when applied to incompressible elasticity is that the accuracy of the numerical solution may be adversely affected by the phenomenon known as locking—this may prevent the use of a low order polynomial approximation to the displacement on each element. We demonstrate using simulations that the DGFEM presented here does not suffer from this drawback. Two further advantages in this setting of this DGFEM over CGFEMs are that: (i) highly anisotropic meshes—meshes containing elements with a very high aspect ratio—may be used without significantly degrading the accuracy of the solution; and (ii) discontinuities in the elasticity parameters are handled more effectively.