Numerical Design of an Optimal Bypass for a Partially Blocked Artery

A parallel domain decomposition method is introduced for numerical design of an optimal bypass for a partially blocked artery. The optimal bypass is described as the solution of a shape optimization problem governed by the steady-state incompressible Navier-Stokes equations that are used to model the blood flow. The problem is discretized with a finite element method on unstructured moving meshes and then solved by a parallel one-shot Lagrange-Newton-Krylov-Schwarz algorithm. In order to accelerate the convergence of the inexact Newton method, we introduce a two-level inexact Newton method which solves a coarse grid problem to generate a good initial guess for the fine grid inexact Newton method. Numerical experiments show that our algorithms perform well on a supercomputer with hundreds of processors.

[1]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[2]  A. Quarteroni,et al.  OPTIMAL CONTROL AND SHAPE OPTIMIZATION OF AORTO-CORONARIC BYPASS ANASTOMOSES , 2003 .

[3]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[4]  Robert T. Biedron,et al.  Recent Enhancements To The FUN3D Flow Solver For Moving-Mesh Applications , 2009 .

[5]  H. M. Bücker,et al.  Sensitivity of optimal shapes of artificial grafts with respect to flow parameters , 2010 .

[6]  M. Heinkenschloss,et al.  Shape Optimization in Stationary Blood Flow : A Numerical Study of Non-Newtonian Effects , 2004 .

[7]  Max Gunzburger,et al.  Perspectives in flow control and optimization , 1987 .

[8]  P. Hoskins,et al.  Numerical investigation of physiologically realistic pulsatile flow through arterial stenosis. , 2001, Journal of biomechanics.

[9]  Maatoug Hassine,et al.  Optimal shape design for fluid flow using topological perturbation technique , 2009 .

[10]  Olof B. Widlund,et al.  Domain Decomposition Algorithms with Small Overlap , 1992, SIAM J. Sci. Comput..

[11]  D. Zingg,et al.  Mesh Movement for a Discrete-Adjoint Newton-Krylov Algorithm for Aerodynamic Optimization , 2007 .

[12]  Natalia Alexandrov,et al.  Multidisciplinary design optimization : state of the art , 1997 .

[13]  J. Zolésio,et al.  Introduction to shape optimization : shape sensitivity analysis , 1992 .

[14]  Subhendu Bikash Hazra.  Multigrid One-shot Method for Aerodynamic Shape Optimization , 2008, SIAM J. Sci. Comput..

[15]  Xiao-Chuan Cai,et al.  Parallel Multilevel Restricted Schwarz Preconditioners with Pollution Removing for PDE-Constrained Optimization , 2007, SIAM J. Sci. Comput..

[16]  P. Hoskins,et al.  Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses. , 2007, Journal of biomechanics.

[17]  Q. Long,et al.  Study of the collateral capacity of the circle of Willis of patients with severe carotid artery stenosis by 3D computational modeling. , 2008, Journal of biomechanics.

[18]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[19]  James F. Antaki,et al.  Computational Strategies For Shape Optimization Of Time-Dependent Navier-Stokes Flows , 1997 .

[20]  Xiao-Chuan Cai,et al.  A parallel two-level method for simulating blood flows in branching arteries with the resistive boundary condition , 2011 .

[21]  O. Pironneau,et al.  SHAPE OPTIMIZATION IN FLUID MECHANICS , 2004 .

[22]  S. Schmidt,et al.  Shape derivatives for general objective functions and the incompressible Navier-Stokes equations , 2010 .

[23]  Xiao-Chuan Cai,et al.  Two-Level Newton and Hybrid Schwarz Preconditioners for Fluid-Structure Interaction , 2010, SIAM J. Sci. Comput..

[24]  A. Marsden,et al.  The impact of uncertainty on shape optimization of idealized bypass graft models in unsteady flow , 2010 .

[25]  Eyal Arian,et al.  Shape Optimization in One Shot , 1995 .

[26]  M. Heinkenschloss,et al.  Shape optimization in steady blood flow: A numerical study of non-Newtonian effects , 2005, Computer methods in biomechanics and biomedical engineering.