Pegolotti Luca, Pfaller Martin R, Marsden Alison L, Deparis Simone
SCI-SB-SD, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, EPFL, CH-1015 Lausanne, Switzerland.
Department of Pediatrics (Cardiology), Bioengineering, Stanford University, Clark Center E1.3, 318 Campus Drive, Stanford, CA 94305, USA.
Comput Methods Appl Mech Eng. 2021 Jul 1;380. doi: 10.1016/j.cma.2021.113762. Epub 2021 Mar 27.
We are interested in a reduced order method for the efficient simulation of blood flow in arteries. The blood dynamics is modeled by means of the incompressible Navier-Stokes equations. Our algorithm is based on an approximated domain-decomposition of the target geometry into a number of subdomains obtained from the parametrized deformation of geometrical building blocks (e.g., straight tubes and model bifurcations). On each of these building blocks, we build a set of spectral functions by Proper Orthogonal Decomposition of a large number of snapshots of finite element solutions (offline phase). The global solution of the Navier-Stokes equations on a target geometry is then found by coupling linear combinations of these local basis functions by means of spectral Lagrange multipliers (online phase). Being that the number of reduced degrees of freedom is considerably smaller than their finite element counterpart, this approach allows us to significantly decrease the size of the linear system to be solved in each iteration of the Newton-Raphson algorithm. We achieve large speedups with respect to the full order simulation (in our numerical experiments, the gain is at least of one order of magnitude and grows inversely with respect to the reduced basis size), whilst still retaining satisfactory accuracy for most cardiovascular simulations.
我们对一种用于高效模拟动脉血流的降阶方法感兴趣。血液动力学通过不可压缩的纳维 - 斯托克斯方程进行建模。我们的算法基于将目标几何形状近似域分解为多个子域，这些子域是通过几何构建块（例如直管和模型分支）的参数化变形获得的。在每个这样的构建块上，我们通过对大量有限元解的快照进行本征正交分解来构建一组谱函数（离线阶段）。然后，通过谱拉格朗日乘数耦合这些局部基函数的线性组合来找到目标几何形状上纳维 - 斯托克斯方程的全局解（在线阶段）。由于降阶自由度的数量比其有限元对应物小得多，这种方法使我们能够在牛顿 - 拉夫逊算法的每次迭代中显著减小要求解的线性系统的规模。相对于全阶模拟，我们实现了大幅加速（在我们的数值实验中，增益至少为一个数量级，并且与降基大小成反比增长），同时对于大多数心血管模拟仍保持令人满意的精度。
