Augmented Lagrangian Approach for Quasi-Incompressible Cardiac Mechanics

Abstract

Computational models of heart mechanics have a large potential for application in medical research, and may give improved understanding of heart physiology and of important clinical problems such as heart failure. From the mathematical point of view, the passive mechanical behaviour of the heart can be described using the finite deformation theory from the field of solid mechanics. The tissue is typically modeled as an anisotropic, non-linear, hyperelastic and either incompressible or nearly incompressible material. Incompressibility is often enforced using a penalty function, which is known to dramatically change the conditioning of the stiffness matrix. For high penalty parameters, the condition number increases and the performance of iterative solvers will decrease. In this work we apply the Augmented Lagrangian approach with a mixed three field finite element formulation to incorporate quasi-incompressibility. This approach allows smaller penalty parameters, which is expected to result in linear systems with better numerical properties. Another advantage of the approach is that it offers complete control of the volumetric change during a simulation, which is not possible using the standard penalty formulation. The performance of the different approaches are examined using a set of problems from a cardiac mechanics benchmark.