The Immersed Structural Potential Method for haemodynamic applications

In this paper, a new fluid–structure interaction immersed computational methodology, based upon the original Immersed Boundary Method (IBM) [1] is outlined with the final aim of modelling cardiovascular phenomena, specifically, heart valve related problems. The principal characteristic of such immersed techniques is the representation of any deformable or rigid body immersed within an incompressible viscous flow field as a momentum forcing source in the Navier–Stokes equations. A number of shortcomings within the immersed formulation still require further investigation and improvement, including the excessive numerical diffusion caused by the interpolation/spreading process, the need to include realistic viscoelastic composite constitutive models describing more accurately the nature of cardiovascular tissues and also the need to capture more effectively stresses developed at the fluid–structure interface. By following the same philosophy as the original IBM, a more sophisticated formulation is derived in this paper, the “Immersed Structural Potential Method (ISPM)”. The method introduced presents an alternative approach to compute the equivalent fluid–structure interaction forces at the fluid mesh, accounts for a sophisticated viscoelastic fibre-reinforced constitutive model to better describe the mechanics of cardiovascular tissues and utilises a novel time-integration methodology for the computation of the deformation gradient tensor which ensures compliance with the incompressibility constraint. A series of numerical examples will be presented in order to demonstrate the robustness and applicability of this new methodology.