In this work, we analyse space-time reduced basis methods for the efficient numerical simulation of haemodynamics in arteries. The classical formulation of the reduced basis (RB) method features dimensionality reduction in space, while finite differences schemes are employed for the time integration of the resulting ordinary differential equation (ODE). Space-time reduced basis (ST-RB) methods extend the dimensionality reduction paradigm to the temporal dimension, projecting the full-order problem onto a low-dimensional spatio-temporal subspace. Our goal is to investigate the application of ST-RB methods to the unsteady incompressible Stokes equations, with a particular focus on stability. High-fidelity simulations are performed using the Finite Element (FE) method and BDF2 as time marching scheme. We consider two different ST-RB methods. In the first one - called ST--GRB - space-time model order reduction is achieved by means of a Galerkin projection; a spatio-temporal supremizers enrichment procedure is introduced to guarantee stability. The second method - called ST-PGRB - is characterized by a Petrov-Galerkin projection, stemming from a suitable minimization of the FOM residual, that allows to automatically attain stability. The classical RB method - denoted as SRB-TFO - serves as a baseline for theoretical development. Numerical tests have been conducted on an idealized symmetric bifurcation geometry and on the patient-specific one of a femoropopliteal bypass. The results show that both ST-RB methods provide accurate approximations of the high-fidelity solutions, while considerably reducing the computational cost. In particular, the ST-PGRB method exhibits the best performance, as it features a better computational efficiency while retaining accuracies in accordance with theoretical expectations.
翻译:在这项工作中,我们分析对动脉中血液动力学高效数字模拟的降低时空基础方法。对降低基数(RB)方法的典型提法以空间的维度减少为特点,而对于由此产生的普通差异方程式(ODE)的时间整合则采用有限的差异办法。空间降低基数(ST-RB)方法将维度减少模式扩大到时间层面,将全序问题投射到低维空间中。我们的目标是调查ST-RB方法对不稳定的不固定的Stokes 方程式的应用,特别侧重于稳定性。在进行高密度模拟时使用Fite Element(FE)方法和BDFDF2作为时间进化计划的时间整合时,我们考虑两种不同的ST-RB方法。在第一个方法中,称为ST-GRB-时空模型的减少,通过Galeralkin预测实现全位;在Statiernoral Stal-stopredition 浓缩程序上引入了ST-PROD-S-deal-dealalalalalal-deal-dealal lavelopal laftal laft laft laft laft laft laft lax lax lax laisal lady lax laft laft laft lax labal laft lax lax lady lady lautal laft lax lax lax lax lax lautal 方法,通过Sl lax lax lax lax 一种最佳成本效率,通过S-mod lax lax lax a 一种叫制S-sal 方法使S-modald 一种最佳S-mod-mod-mod 方法使S-modal demod labal devaldald ladaldaldaldal-mocalalalal lad lad 方法使S-mod-mod-mod-mod-modal-modal-mod-modaldaldaldaldal-mod lax lax lax 一种