A stabilized method for high Peclet number Advection-Diffusion-Reaction systems in Cardiac and Vascular applications

  • Balmus, Maximilian (King's College London)
  • Bonini, Mia (University of Michigan)
  • Qureshi, Ahmed (King's College London)
  • de Vecchi, Adelaide (King's College London)
  • Aslanidi, Oleg (King's College London)
  • Nordsletten, David (King's College London)

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Advection-diffusion-reaction systems represents a versatile class of problems with numerous cardiovascular applications, ranging from modelling temperature regulation and thrombogenesis to the mapping of blood residence time. However, most of these applications tend to be convection-dominated and, as a result, their solutions present sharp gradients which generally cannot be captured by practical discretization resolutions. Hence, most finite element based solvers typically incorporate some form of stabilization in order to inhibit spurious osciallations. Common examples include streamline upwind/Petrov-Galerkin (SUPG) combined with shock capturing (i.e. artificial diffusivity), the continuous interior penalty and local projection stabilization methods. In practice, these methods must balance the needs to: capture accurate solutions, maintain tractable solution times, and stabilize deleterious oscillation. In this work, we build on the stabilization scheme proposed by Lynch et al. [1], based on SUPG and artificial diffusivity in the style proposed by Carmo and Galeao [2], and augment it with two additional terms. The first of these schemes takes the form of a secondary shock capturing term scaling as a function of the normalized gradient of the concentration. Its design is meant to target regions of the solutions poorly stabilized by its original counterpart. The second component is a boundary term which penalizes negative value concentrations on the portions of the boundary characterized by zero-advective flux, constraining the values from exceeding physical bounds. The new approach is tested on a series of problems with increasing complexity, ranging from 2D steady flow benchmarks to a 3D moving-domain left heart model. Here, the new scheme is shown to significantly improve the attenuation of spurious oscillations while at the same time retaining the key solution features. This proves the method to be a valuable tool for stabilizing transport problems in cardiovascular applications, allowing for more accurate approximations with less artificial diffusivity and without introducing excessive computational cost.