# Piecewise Constant Enrichment of the continuous galerkin method

I am interested to study crack propagation in a hyperelastic material in a variational setting. The crack surface exhibits a jump discontinuity. The function space for displacement field should consist of discontinuous functions.

The discontinuous galerkin finite element (DGFEM) approximation space consist of piecewise discontinuous functions which seems to a valid choice to solve crack problem. However, the required degree of freedom is extremely large for a high resolution discretization.

The recently proposed, enriched galerkin finite element method (EGFEM) enriched the continuous galerkin approximation space with a piecewise constant function. This allows discontinuities along the element face. For example, a one dimensional function $f(x)$ can be approximated in a unit interval as,

$f(x) = f_1 \phi_1(x) + f_2 \phi_2(x) + f_3$

where $\phi_1(x)$ and $\phi_2(x)$ are basis functions. EGFEM seems to be a better choice to solve crack problem as required degree of freedom is quite less than for DGFEM.

EGFEM does introduce the discontinuity along the edges/faces, but in the deformed configuration the disjoint edges/faces would always be parallel. This is one of the limitation of EGFEM.

Before going to implement these methods, I would like to understand advantage and disadvantage of DGFEM and EGFEM approach to solve the hyperelastic energy minimization problem.

What all points I should consider to evaluate these methods ?

• Your calculation is not correct. If you take into account that each of the two functions $\phi_1,\phi_2$ are shared between two cells, then EGFEM has $\frac{1}{2}(1+1)+1=2$ shape functions per cell. The DGFEM also has 2 shape functions per cell. You are correct in 2d and 3d, though. Oct 30 '17 at 3:20
• @WolfgangBangerth Sorry, I could not understand your calculation of shape function for EGFEM. Can you please explain it again assuming $\phi_1$ and $\phi_2$ are shared between two cells ?
– hari
Oct 30 '17 at 5:33
• @WolfgangBangerth Yes, you are right. The number of shape functions on average would be same for both EGFEM and DGFEM for 1d.
– hari
Oct 30 '17 at 14:19
• Why not choose the POU concept in XFEM or GFEM to enrich the standard approximation basis space? Oct 31 '17 at 0:01
• @WenjinXing I dont' know the spatial location of crack interface. I am estimating it by formulating a variational problem based on principal of minimum energy. POU methods probably requires knowledge of cracks precisely in a grid, which is unknown in our case.
– hari
Oct 31 '17 at 7:18