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

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. This penalty term is weighted by a positive-definite stabilization tensor, computed by solving an eigenvalue problem that involves the contraction of the velocity convolution matrix with the element metric tensor. The outcome is a formally stable residual-based method that emulates the standard time method when simulating steady flows. Consequently, it preserves the appealing properties of the standard method, including stability in strong convection and the convenient use of equal-order interpolation functions for velocity and pressure, among other benefits. 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, thereby enabling efficient utilization of computational resources for the rapid simulation of time-critical applications.