@article {Simula.simula.1594,
	title = {Improved Discretisation and Linearisation of Active Tension in Strongly Coupled Cardiac Electro-Mechanics Simulations},
	journal = {Computer Methods in Biomechanics and Biomedical Engineering},
	volume = {17},
	number = {6},
	year = {2014},
	pages = {604-15},
	publisher = { },
	abstract = {Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush-Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.},
	author = {Sundnes, Joakim and Wall, Samuel and Osnes, Harald and Thorvaldsen, Tom and Andrew D. {McCulloch}}
}
@phdthesis {Thorvaldsen.2006.2,
	title = {An Electro-Mechanics Solver for the Heart},
	year = {2006},
	month = {April},
	school = {Department of Informatics, University of Oslo},
	type = {phd},
	author = {Thorvaldsen, Tom}
}
@misc {Thorvaldsen.2005.3,
	title = {A Mixed FEM for Modeling the Passive Mechanical Properties of the Myocardium},
	howpublished = {Talk at the SIAM Conference on Computational Science \& Engineering},
	year = {2005},
	month = {February},
	author = {Thorvaldsen, Tom and Osnes, Harald and Sundnes, Joakim}
}
@article {Thorvaldsen.2005.1,
	title = {A Mixed Finite Element Formulation for a Nonlinear, Transversely Isotropic Material Model for the Cardiac Tissue},
	journal = {Computer Methods in Biomechanics and Biomedical Engineering},
	volume = {8},
	number = {6},
	year = {2005},
	month = {December},
	pages = {369-379},
	abstract = {In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.},
	author = {Thorvaldsen, Tom and Osnes, Harald and Sundnes, Joakim}
}
@inproceedings {Thorvaldsen.2005.2,
	title = {A Mixed Formulation for Modeling the Mechanical Bahavior of the Heart},
	journal = {MekIt05 - Third national conference om Computational Mechanics, Trondheim May 11-12},
	year = {2005},
	month = {May},
	pages = {327-341},
	publisher = {Tapir Academic Press},
	type = {1},
	abstract = {In this paper we present a mixed finite element formulation for modeling the mechanical behavior of the cardiac tissue. The modeling may be divided into a passive and an active part. The passive properties are in this work described by a transversely isotropic, hyperelastic material model. Moreover, the myocardium is assumed to be almost incompressible. In advanced models the active contraction is strongly connected to electro-physiological processes in the muscle cells. In this paper we simplify the active stress contribution by modeling it as a linear function in time. Due to the fact that the heart muscle undergoes large deformations during contraction, both material and geometrical non-linearities must be taken into account in the analysis. Numerical results are presented for different activation patterns applied to a box-shaped test specimen, which represents a piece of the myocardial wall. The proposed mixed method offers reasonable results in the modeling of the mechanical behavior of the test specimen.},
	author = {Thorvaldsen, Tom and Sundnes, Joakim and Osnes, Harald},
	editor = {Bj{\o}rn H. {Skallerud} and Helge I. {Andersson}}
}
@techreport {Thorvaldsen.2005.4,
	title = {Non-Linear Elasticity With Applications to Heart Mechanics},
	number = {2005-16},
	year = {2005},
	publisher = {Simula Research Laboratory},
	abstract = {This report gives an introduction to mathematical and numerical modeling of large-deformation elasticity problems. Basic non-linear continuum mechanics is presented along with a set of hyperelastic material models. Two alternative finite element formulations are described for solving such problems, for compressible and almost incompressible materials. Simulation results are presented for both methods. The elasticity solver is employed in simulating the mechanical behavior of the heart. Mathematical models are presented for describing the passive behavior of cardiac tissue and the active force development. Moreover, we show mathematical models and numerical algorithms for simulating the large deformations of the left ventricle during a heart beat.},
	author = {Thorvaldsen, Tom and Sundnes, Joakim and Osnes, Harald}
}
@techreport {SC.7.Thorvaldsen.2004,
	title = {Numerical Modeling of Isotropic, Hyperelastic Materials Undergoing Large Deformations},
	number = {2004-06},
	year = {2004},
	publisher = {Simula Research Laboratory},
	author = {Thorvaldsen, Tom and Sundnes, Joakim and Osnes, Harald}
}
@inbook {SC.5.Thorvaldsen.2003.a,
	title = {Finite Element Modeling of Elastic Structures},
	booktitle = {Advanced Topics in Computational Partial Differential Equations - Numerical Methods and Diffpack Programming},
	series = {Lecture Notes in Computational Science and Engineering},
	year = {2003},
	pages = {507-576},
	publisher = {Springer},
	organization = {Springer},
	author = {Thorvaldsen, Tom and Hans Petter {Langtangen} and Osnes, Harald},
	editor = {Hans Petter {Langtangen} and Tveito, Aslak}
}
