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A meshless (LBIE) method for the solution of the Navier - Stokes equations. by Sellountos J Euripides & Adelia Sequeira Instituto Seperior Tecnico CEMAT. Haemodel, Bergamo September 2006. Motivation of Meshless Methods: Easy to model.

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## by Sellountos J Euripides & Adelia Sequeira Instituto Seperior Tecnico CEMAT

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**A meshless (LBIE) method for the solution of the Navier -**Stokes equations by Sellountos J Euripides & Adelia Sequeira Instituto Seperior Tecnico CEMAT Haemodel, Bergamo September 2006**Motivation of Meshless Methods:**Easy to model • Meshing and remeshing of complex geometries relevant to blood flow problems (stenosed, curved or bifurcating vessels) is easy with the addition-movement of nodal points**Motivation of Meshless Methods:**Easy to model • Meshing and remeshing of complex geometries relevant to blood flow problems (stenosed, curved or bifurcating vessels) is easy with the addition-movement of nodal points • Meshless methods: Computational method related to surface reconstruction techniques**Motivation of Meshless Methods:**Easy to model • Meshing and remeshing of complex geometries relevant to blood flow problems (stenosed, curved or bifurcating vessels) is easy with the addition-movement of nodal points • Meshless methods: Computational method related to surface reconstruction techniques • Local solution of the boundary-domain integral equations**Motivation of Meshless Methods:**Easy to model • Meshing and remeshing of complex geometries relevant to blood flow problems (stenosed, curved or bifurcating vessels) is easy with the addition-movement of nodal points • Meshless methods: Computational method related to surface reconstruction techniques • Local solution of the boundary-domain integral equations • Approximation of the unknown field with randomly distributed nodal points only • System of equations are in band form • In small vessels blood behaves as a shear thinning (and viscoelastic fluid)**Nodal Support, Connectivity and Interpolation**Support domain of a nodal point Every nodal point has an associated circular region of influence**Nodal Support, Connectivity and Interpolation**Neighborhood of a nodal point Support domain of a nodal point Every nodal point has an associated circular region of influence**Nodal Support, Connectivity and Interpolation**Support domain of a nodal point Every nodal point has an associated circular region of influence Neighborhood of a nodal point Interpolation of unknown field**Generalized Navier – Stokes equations**• Conservation of mass • Conservation of momentum • Shear stress • Vorticity • Strain rate tensor • Viscosity is assumed to be shear strain rate or shear stress dependant Armin Leuprecht and Karl Perktold**Generalized Navier – Stokes equations**Velocity - vorticity scheme • The fluid motion scheme is partitioned to kinematics • and kinetics • decomposition of velocity and viscosity to a mean and a perturbed value**Generalized Navier – Stokes equations**Velocity - vorticity scheme Kinematics Integral Representation Skerget and Hribersek**Generalized Navier – Stokes equations**Velocity - vorticity scheme Kinetics Integral Representation**Generalized Navier – Stokes equations**Velocity - vorticity scheme Kinematics Local Integral Representation Compation solution • Satisfies linear part of the differential operator • Equals to the fundamental on the local boundary**Generalized Navier – Stokes equations**Velocity - vorticity scheme Kinetics Local Integral Representation**Discretization and Numerical Evaluation of Integrals**Involved Integrals**Discretization and Numerical Evaluation of Integrals**• Arc integrals Involved Integrals**Discretization and Numerical Evaluation of Integrals**• Boundary integrals Involved Integrals**Discretization and Numerical Evaluation of Integrals**• Volume integrals Involved Integrals**Discretization and Solution Procedure**Approximation of Boundary Vorticity ω Computation of shear rate and new nodal viscosity**Discretization and Solution Procedure**Check vorticity’s convergence / Iteration decision**Conclusions – Future work**• Mesh free method, only points are needed for the interpolation • Solution of boundary integral equation • Use of other test functions instead of fundamental solution • Hypersingular integral equation for boundary points in kinematics equation Thanks for your attention

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