A new stabilized time-spectral finite element solver for fast simulation of blood flow

TL;DR

A finite element method for simulating time-periodic flows that are physically stable by discretizing equations in the frequency domain instead of the time domain is introduced and exhibits improved parallel scalability for fast simulation of time-critical applications.

Abstract

The increasing application of cardiorespiratory simulations for diagnosis and surgical planning necessitates the development of computational methods significantly faster than the current technology. To achieve this objective, we leverage the time-periodic nature of these flows by discretizing equations in the frequency domain instead of the time domain. This approach markedly reduces the size of the discrete problem and, consequently, the simulation cost. With this motivation, we introduce a finite element method for simulating time-periodic flows that are physically stable. The proposed time-spectral method is formulated by augmenting the baseline Galerkin's method with a least-squares penalty term that is weighed by a positive-definite stabilization matrix. An error estimate is established for the convective-diffusive system, showing that the proposed method emulates the behavior of existing standard time methods including optimal convergence rate in diffusive regimes and stability in strong convection. This method is tested on a patient-specific Fontan model at nominal Reynolds and Womersley numbers of 500 and 10, respectively, demonstrating its ability to replicate conventional time simulation results using as few as 7 modes at 11% of the computational cost. Owing to its higher local-to-processor computation density, the proposed method also exhibits improved parallel scalability for fast simulation of time-critical applications.

Figures (13)

• Figure 1: The Fontan geometry considered in this study with boundary faces marked and a sectional view of the mesh.
• Figure 2: Inlet flow rate for both the SVC and the IVC, $Q_{\mathrm{in}}(t)$. The reference profile used for the time simulation (filled circles) and its best approximation with $N=$ 4, 7, and 10 modes (dashed, solid, and dash-dot lines, respectively) used for time-spectral simulations is shown. The inset shows the flow energy spectrum ($\left| \Tilde{Q}_n\right|^2 = \Tilde{Q}_n \Tilde{Q}_n^*$).
• Figure 3: The norm of the residual as a function of simulation wall-clock time in performing time-spectral simulation with different pseudo-time step sizes.
• Figure 4: Normalized residual (solid line) and $L_2$-norm error in the predicted flow rate through the LPA (dashed line) versus pseudo-time step.
• Figure 5: The solution obtained from the GLS method for the convection-diffusion problem subjected to a steady boundary condition at the IVC. The steady, the first, the second, and the third mode are shown in order from top to bottom rows. The left and right columns show the solution that is in-phase and out-of-phase, respectively, with the prescribed inflow boundary condition.

Theorems & Definitions (16)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof