Introducing a Harmonic Balance Navier-Stokes Finite Element Solver to Accelerate Cardiovascular Simulations

TL;DR

This work tackles the high cost of patient-specific transient CFD in cardiovascular applications by presenting a harmonic balance approach in the frequency domain combined with a stabilized finite element discretization. By representing time-dependence with a small set of Fourier modes and employing FFT-based transformations, the method achieves cost scaling of $O(N\log N)$ and substantial speedups (approximately 10–100×) over traditional time-stepping solvers while preserving accuracy when enough modes are used. The approach is validated on three patient-specific geometries (Glenn pulmonary flow, cerebral arteries, and left main coronary arteries), showing good agreement with conventional solvers and rapid convergence, with reasonable memory requirements due to careful matrix-splitting and FFT strategies. The results indicate strong potential for rapid, high-fidelity 3D cardiovascular simulations to support diagnosis and surgical planning, while highlighting avenues for extension to more complex boundary conditions, resistive outlets, and fluid-structure interaction.

Abstract

The adoption of cardiovascular simulations for diagnosis and surgical planning on a patient-specific basis requires the development of faster methods than the existing state-of-the-art techniques. To address this need, we leverage the periodic nature of these flows to accurately capture their time-dependence using spectral discretization. Owing to the reduced size of the discrete problem, the resulting approach, known as the harmonic balance method, significantly lowers the solution cost when compared against the conventional time marching methods. This study describes a stabilized finite element implementation of the harmonic balanced method that targets the simulation of physically-stable time-periodic flows. That stabilized method is based on the Galerkin/least-squares formulation that permits stable solution in convection-dominant flows and convenient use of the same interpolation functions for velocity and pressure. We test this solver against its equivalent time marching method using three common physiological cases where blood flow is modeled in a Glenn operation, a cerebral artery, and a left main coronary artery. Using the conventional time marching solver, simulating these cases takes more than ten hours. That cost is reduced by up to two orders of magnitude when the proposed harmonic balance solver is utilized, where a solution is produced in approximately 30 minutes. We show that that solution is in excellent agreement with the conventional solvers when the number of modes is sufficiently large to accurately represent the imposed boundary conditions.

Figures (20)

• Figure 1: Simulation convergence speed for four pseudo time step sizes, $\Delta \Tilde{t}$. The y-axis is the relative residual compared to the residual at the initialization step, $\|\hbox{\boldmath{$r$}}\|/\|\hbox{\boldmath{$r$}}^{(0)}\|$.
• Figure 2: Clinically obtained Glenn pulmonary geometry with a sectional view of the mesh. SVC, superior vena cava; LPA, left pulmonary artery; RPA, right pulmonary artery.
• Figure 3: Discretized harmonic balance boundary conditions at each time point (black circle) on the SVC inlet. The dotted line represents the reconstructed profile from the time points. For reference, the solid line is the original flow profile in time. (a) $N = 7$, (b) $N = 13$, (c) $N = 19$.
• Figure 4: The cost and speed-up of the harmonic balance solver for the pulmonary flow case. The left axis shows the wall-clock time of the harmonic balance simulations. The right axis shows the speedup of the harmonic balance simulations compared to the conventional time-stepping simulation. A linear scaling line is included for reference.
• Figure 5: Harmonic balance results using $N = 7$, 13, and 19 compared to the conventional time results for the pulmonary flow case. The (a) pressure and (b) velocity magnitude contours are taken at $t = 0.2$ seconds of the cardiac cycle.