# The Variational Multiscale Formulation for the Fully-Implicit Log-Morphology Equation as a Tensor-Based Blood Damage Model

## TL;DR

This paper introduces a variational multiscale finite element formulation for a tensor-based blood damage model that enhances numerical stability and accuracy, especially in complex blood flow simulations involving ventricular assist devices.

## Contribution

The paper develops a novel VMS stabilization method for the log-morph equation, improving numerical behavior over existing GLS and SUPG methods in blood damage modeling.

## Key findings

- VMS stabilization outperforms GLS in 2D stirrer tests.
- VMS method shows clear advantages over SUPG in VAD simulations.
- Logarithmic tensor description prevents unphysical eigenvalues.

## Abstract

We derive a variational multiscale (VMS) finite element formulation for a viscoelastic, tensor-based blood damage model. The tensor equation is numerically stabilized by a logarithmic shape tensor description that prevents unphysical, negative eigenvalues. The resulting VMS stabilization terms for this so-called log-morph equation are presented together with their special numerical treatment. Results for a 2D rotating stirrer test case obtained from log-morph simulations with both SUPG and VMS stabilization show significantly improved numerical behavior if compared with Galerkin/least squares (GLS) stabilized untransformed morphology simulation results. The newly proposed method is also successfully applied to a state-of-the-art centrifugal ventricular assist device (VAD), and clear advantages of the VMS stabilization compared to the SUPG stabilized formulation are presented.