A matrix-form GSM–CFD solver for incompressible fluids and its application to hemodynamics

A GSM–CFD solver for incompressible flows is developed based on the gradient smoothing method (GSM). A matrix-form algorithm and corresponding data structure for GSM are devised to efficiently approximate the spatial gradients of field variables using the gradient smoothing operation. The calculated gradient values on various test fields show that the proposed GSM is capable of exactly reproducing linear field and of second order accuracy on all kinds of meshes. It is found that the GSM is much more robust to mesh deformation and therefore more suitable for problems with complicated geometries. Integrated with the artificial compressibility approach, the GSM is extended to solve the incompressible flows. As an example, the flow simulation of carotid bifurcation is carried out to show the effectiveness of the proposed GSM–CFD solver. The blood is modeled as incompressible Newtonian fluid and the vessel is treated as rigid wall in this paper.

[1]  T. Tezduyar,et al.  Fluid–structure interaction modeling of a patient-specific cerebral aneurysm: influence of structural modeling , 2008 .

[2]  Xiao-Chuan Cai,et al.  A fully implicit domain decomposition based ALE framework for three-dimensional fluid-structure interaction with application in blood flow computation , 2014, J. Comput. Phys..

[3]  Tayfun E. Tezduyar,et al.  Modelling of fluid–structure interactions with the space–time finite elements: Arterial fluid mechanics , 2007 .

[4]  Gui-Rong Liu,et al.  A gradient smoothing method (GSM) for fluid dynamics problems , 2008 .

[5]  Guirong Liu,et al.  Numerical modeling and simulation of pulsatile blood flow in rigid vessel using gradient smoothing method , 2012 .

[6]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[7]  Gui-Rong Liu,et al.  An adaptive gradient smoothing method (GSM) for fluid dynamics problems , 2009 .

[8]  Adélia Sequeira,et al.  A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries , 2010, J. Comput. Appl. Math..

[9]  David A. Steinman,et al.  An image-based modeling framework for patient-specific computational hemodynamics , 2008, Medical & Biological Engineering & Computing.

[10]  K. Y. Dai,et al.  A Smoothed Finite Element Method for Mechanics Problems , 2007 .

[11]  Jiri Blazek,et al.  Computational Fluid Dynamics: Principles and Applications , 2001 .

[12]  Dimitrios I. Fotiadis,et al.  Computer simulation of three-dimensional plaque formation and progression in the carotid artery , 2013, Medical & Biological Engineering & Computing.

[13]  K. Y. Dai,et al.  Theoretical aspects of the smoothed finite element method (SFEM) , 2007 .

[14]  F. N. van de Vosse,et al.  The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model. , 1999, Journal of biomechanics.

[15]  Wayne A. Smith,et al.  Implicit Solution of Preconditioned Navier- Stokes Equations Using Algebraic Multigrid , 1999 .

[16]  Guirong Liu.  A GENERALIZED GRADIENT SMOOTHING TECHNIQUE AND THE SMOOTHED BILINEAR FORM FOR GALERKIN FORMULATION OF A WIDE CLASS OF COMPUTATIONAL METHODS , 2008 .

[17]  S. Paras,et al.  Modeling the effect of blood viscosity on hemodynamic factors in a small bifurcated artery , 2012 .

[18]  G. Liu,et al.  Immersed smoothed finite element method for fluid–structure interaction simulation of aortic valves , 2012 .

[19]  Alejandro F. Frangi,et al.  Efficient pipeline for image-based patient-specific analysis of cerebral aneurysm hemodynamics: technique and sensitivity , 2005, IEEE Transactions on Medical Imaging.

[20]  D. Ku.  BLOOD FLOW IN ARTERIES , 1997 .

[21]  T. Tezduyar,et al.  Arterial fluid mechanics modeling with the stabilized space–time fluid–structure interaction technique , 2008 .

[22]  Harvey Ho,et al.  Blood Flow Simulation in a Giant Intracranial Aneurysm and Its Validation by Digital Subtraction Angiography , 2011 .

[23]  D J Phillips,et al.  Carotid Artery Velocity Patterns in Normal and Stenotic Vessels , 1980, Stroke.

[24]  Guirong Liu,et al.  Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments , 2010 .

[25]  Stefan Turek,et al.  FEM multigrid techniques for fluid-structure interaction with application to hemodynamics , 2012 .

[26]  Tayfun E. Tezduyar,et al.  Sequentially-Coupled Arterial Fluid-Structure Interaction (SCAFSI) technique , 2009 .

[27]  Foad Kabinejadian,et al.  Compliant model of a coupled sequential coronary arterial bypass graft: effects of vessel wall elasticity and non-Newtonian rheology on blood flow regime and hemodynamic parameters distribution. , 2012, Medical engineering & physics.

[28]  Guirong Liu.  Mesh Free Methods: Moving Beyond the Finite Element Method , 2002 .

[29]  L. Antiga,et al.  On the importance of blood rheology for bulk flow in hemodynamic models of the carotid bifurcation. , 2011, Journal of biomechanics.

[30]  Tayfun E. Tezduyar,et al.  Multiscale sequentially-coupled arterial FSI technique , 2010 .

[31]  Tayfun E. Tezduyar,et al.  Patient-specific computational analysis of the influence of a stent on the unsteady flow in cerebral aneurysms , 2013 .

[32]  Rainald Löhner,et al.  Blood flow modeling in carotid arteries with computational fluid dynamics and MR imaging. , 2002, Academic radiology.

[33]  Y. Chatzizisis,et al.  Non-Newtonian models for molecular viscosity and wall shear stress in a 3D reconstructed human left coronary artery. , 2008, Medical engineering & physics.

[34]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[35]  Eric Li,et al.  Simulation of steady and unsteady incompressible flow using gradient smoothing method (GSM) , 2012 .

[36]  Tayfun E. Tezduyar,et al.  Space–time finite element computation of arterial fluid–structure interactions with patient‐specific data , 2010 .

[37]  J-F Gerbeau,et al.  External tissue support and fluid–structure simulation in blood flows , 2012, Biomechanics and modeling in mechanobiology.

[38]  C. Putman,et al.  Characterization of cerebral aneurysms for assessing risk of rupture by using patient-specific computational hemodynamics models. , 2005, AJNR. American journal of neuroradiology.

[39]  Barbara M. Johnston,et al.  Non-Newtonian blood flow in human right coronary arteries: steady state simulations. , 2004, Journal of biomechanics.

[40]  Tayfun E. Tezduyar,et al.  Modelling of fluid–structure interactions with the space–time finite elements: Solution techniques , 2007 .

[41]  Guirong Liu,et al.  A novel linearly-weighted gradient smoothing method (LWGSM) in the simulation of fluid dynamics problem , 2011 .

[42]  G. Y. Li,et al.  A gradient smoothing method (GSM) with directional correction for solid mechanics problems , 2007 .

[43]  Cyrus K. Aidun,et al.  Lattice-Boltzmann Method for Complex Flows , 2010 .

[44]  Guirong Liu,et al.  A gradient smoothing method (GSM) based on strong form governing equation for adaptive analysis of solid mechanics problems , 2008 .

[45]  Kenji Takizawa,et al.  Patient‐specific arterial fluid–structure interaction modeling of cerebral aneurysms , 2011 .

[46]  Guirong Liu,et al.  A moving-mesh gradient smoothing method for compressible CFD problems , 2013 .

[47]  Thomas D. Economon,et al.  Stanford University Unstructured (SU 2 ): An open-source integrated computational environment for multi-physics simulation and design , 2013 .

[48]  Tayfun E. Tezduyar,et al.  Wall shear stress calculations in space–time finite element computation of arterial fluid–structure interactions , 2010 .

[49]  T. Tezduyar,et al.  Fluid–Structure Interaction Modeling of Patient-Specific Cerebral Aneurysms , 2014 .

[50]  J. Monaghan.  Smoothed particle hydrodynamics , 2005 .

[51]  G. R. Liu,et al.  On Smoothed Finite Element Methods , 2010 .

[52]  Charles A. Taylor,et al.  Patient-specific modeling of cardiovascular mechanics. , 2009, Annual review of biomedical engineering.

[53]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .