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.

翻译：心肺模拟在诊断和手术规划中的应用日益广泛，这要求开发出显著快于现有技术的计算方法。为实现这一目标，我们利用这类流动的时间周期性，将方程离散化从时域转换到频域。该方法显著减小了离散问题的规模，从而降低了模拟成本。基于此动机，我们提出了一种用于模拟物理稳定时间周期流动的有限元方法。该时-谱方法通过在基线伽辽金法中引入最小二乘罚项来构建。该罚项由正定稳定化张量加权，该张量通过求解涉及速度卷积矩阵与单元度量张量缩并的特征值问题计算得出。最终得到一种形式上稳定的残差基方法，在模拟稳态流动时能复现标准时域方法的行为。因此，它保留了标准方法的优良特性，包括在强对流中的稳定性以及速度-压力同阶插值函数的使用便利性等。该方法在雷诺数和沃默斯利数分别设为500和10的名义患者特异性Fontan模型上进行了测试，证明其仅使用7个模态、以11%的计算成本即可复现传统时域模拟结果。由于具有更高的局部-处理器计算密度，所提方法还展现了更好的并行可扩展性，从而能高效利用计算资源快速模拟时间关键型应用。