Fast and accurate nonlinear hyper-elastic deformation with a posteriori numerical verification of the convergence of solution: Application to the simulation of liver deformation.

Abstract

In this paper, we propose a new method to reduce the computational complexity of calculating the tangential stiffness matrix in a nonlinear finite element formulation. Our approach consists in partially updating the tangential stiffness matrix during a classic Newton-Raphson iterative process. The complexity of such an update process has the order of the number of mesh vertices to the power of two. With our approach, this complexity is reduced to the power of two of only the number of updated vertices. We numerically study the convergence of the solution with our modified algorithm. We describe the deformation through a strain energy density function which is defined with respect to the Lagrangian strain. We derive the conditions of convergence for a given tangential stiffness matrix and a given set of updated vertices. We use nonlinear geometric deformation and the nonlinear Mooney-Rivilin model with both tetrahedron and hexahedron element meshing. We provide extensive results using a cube with small and large number of elements. We provide results on nonlinearly deformed liver with multiple deformation ranges of updated vertices. We compare the proposed method to state-of-the-art work and we prove its efficiency at three levels: accuracy, speed of convergence and small radius of convergence.