Accelerating cardiac and vessel mechanics simulations: An energy-transform variational formulation for soft-tissue hyperelasticity

Abstract

Computational modeling constitutes a powerful tool to understand the biomechanical function of the heart and the aorta. However, the high dimensionality and non-linear nature of current models can be challenging in terms of computational demands. In this work, we present a novel energy-transform variational formulation (ETVF) for accelerating the numerical simulation of hyperelastic biosolids. To this end, we propose a mixed variational framework, where we introduce auxiliary fields that render the strain energy density into a quadratic form, at the expense of adding unknown fields to the problem. We further reduce the non-linearity of the problem by transforming the constraints that arise due to auxiliary fields in a Lagrange multiplier formulation. The resulting continuous problem is solved by using multi-field non-linear finite-element schemes. We assess the numerical performance of the ETVF by solving two benchmark problems in cardiac and vessel mechanics and one anatomically-detailed model of a human heart under passive filling that assumes an orthotropic heterogeneous constitutive relation. Our results show that the ETVF can deliver speed-ups up to 2.28 in realistic cardiovascular simulations only by considering the proposed reformulation of the hyperelastic problem. Further, we show that the ETVF can decrease the wall-clock time of simulations solved in parallel architectures (8-cores) by 55%. We argue that the decrease in computational cost is explained by the ability of the ETVF to reduce the condition number of tangent operators. We believe that the ETVF offers an effective framework to accelerate the numerical solution of general hyperelastic problems, enabling the solution of large-scale problems in attractive computing times. Codes are available for download at https://github.com/dehurtado/ETVF.