Convergence Trend with Unsymmetric Newton Raphson

Why do for some of the hyperelastic problems show a better convergence trend with the unsymmetric Newton Raphson (NROPT,UNSYM) solver? Does mixed u-P formulation result in unsymmetric stiffness matrix?

Mixed u-P element formulation itself doesn’t introduce an unsymmetric stiffness matrix. The stress stiffening matrix is unsymmetric, and pressure load stiffness matrices are also unsymmetric. Usually a procedure of symmetrizing the tangent stiffness matrix is invoked and then solved with symmetric Newton Raphson (NROPT,FULL) solver.

When the contribution of stress stiffening and pressure load stiffness matrices are significant in obtaining a consistent tangent stiffness matrix for a problem, then NROPT,UNSYM will lead to a better convergence trend. But there is trade-off: The inversion of an unsymmetric matrix takes longer and uses more memory to compute than a symmetric matrix. This is because we need to store the full matrix and not only the upper triangular half.

Below are two Ansys help theory manual links that discuss this concept.

3.3. Large Rotation (ansys.com)

3.4. Stress Stiffening (ansys.com)