Robustness in dual shape function calculation

[dewenyushu]

Reason

During mortar-based mechanical contact simulations, issue comes up sometimes with dual shape function calculation where the dual coefficient calculation may error out while inverting a matrix that is not numerically positive definite. Efforts should be focused on: 1) the origination of a non-positive mass matrix, and 2) how to make the calculation more robust regardless of the positive definiteness of the matrix.

Design

• Investigate into other factorization approaches that applies to matrices that are not strictly positive definite
• Look into the origin of the non-positive definite mass matrices and fix issues there if any

Impact

Make the mortar-based mechanical contact simulations more robust

[lindsayad]

Can't you just do an lu_decomposition?

[dewenyushu]

Can't you just do an lu_decomposition?

IIRC, I started with LU factorization and was suggested to switch to Cholesky since it is computationally cheaper. Switching back to LU will be my first thing to try, but I wonder if there may be other issues besides the factorization.

(original discussions are here https://github.com/libMesh/libmesh/pull/2552)