A nonlinear elimination preconditioned inexact Newton method for heterogeneous hyperelasticity

We propose and study a nonlinear elimination preconditioned inexact Newton method for the numerical simulation of diseased human arteries with a heterogeneous hyperelas- tic model. We assume the artery is made of layers of distinct tissues and also contains plaque. Traditional Newton methods often work well for smooth and homogeneous arteries but suffer from slow or no convergence due to the heterogeneousness of diseased soft tissues when the material is quasi-incompressible. The proposed nonlinear elimination method adaptively finds a small number of equations causing the nonlinear stagnation and then eliminates them from the global nonlinear system. By using the theory of affine invariance of Newton method, we provide insight into why the nonlinear elimination method can improve the convergence of Newton iterations. Our numerical results show that the combination of nonlinear elimination with an initial guess interpolated from a coarse level solution can lead to the uniform convergence of Newton method for this class of very difficult nonlinear problems.