A Simultaneous Augmented Lagrange Approach for the Simulation of Soft Biological Tissue

Abstract

In this paper, we consider the elastic deformation of arterial walls as occurring, e.g., in the process of a balloon angioplasty, a common treatment in the case of atherosclerosis. Soft biological tissue is an almost incompressible material. To account for this property in finite element simulations commonly used free energy functions contain terms penalizing volumetric changes. The incorporation of such penalty terms can, unfortunately, spoil the convergence of the nonlinear iteration scheme, i.e., of Newton's method, as well as of iterative solvers applied for the solution of the linearized systems of equations. We show that the augmented Lagrange method can improve the convergence of the linear and nonlinear iteration schemes while, at the same time, implementing a guaranteed bound for the volumetric change. Our finite element model of an atherosclerotic arterial segment, see Fig. 1, is constructed from intravascular ultrasound images; for details see [4]. © Springer-Verlag Berlin Heidelberg 2013.