Smoothed finite element methods in simulation of active contraction of myocardial tissue samples

In modelling and simulation of cardiac mechanics, tetrahedral meshes are often used due to the easy availability of efficient meshing algorithms. This is beneficial in particular when complex geometries such as cardiac structures are considered. The gold standard in simulating the cardiac cycle is to solve the mechanical balance equations with the finite element method (FEM). However, using linear shape functions in the FEM in combination with nearly-incompressible material models is known to produce overly stiff approximations, whereas higher order elements are computationally more expensive. To overcome these problems, smoothed finite element methods (S-FEMs) have been proposed by Liu and co-workers. So far, S-FEMs in 3D have been utilised only in simulations of passive mechanics. In the present work, different S-FEMs are for the first time used for simulation of an active cardiac contraction on three-dimensional myocardial tissue samples. Further, node-based S-FEM (NS-FEM), face-based S-FEM (FS-FEM) and selective FS/NS-FEM are for the first time implemented as user subroutine in the commercial software Abaqus. Our results confirm that all S-FEMs perform softer than linear FEM and volumetric locking is reduced. The FS/NS-FEM produces solutions with the relative error in maximum displacement and rotation being less than 5% with respect to the reference solution obtained by the quadratic FEM for all considered mesh sizes, although linear shape functions are used. We therefore conclude that in particular FS/NS-FEM is an efficient and accurate numerical method in the simulation of an active cardiac muscle contraction.