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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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)

  6. Nodal Support, Connectivity and Interpolation Support domain of a nodal point Every nodal point has an associated circular region of influence

  7. 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

  8. 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

  9. 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

  10. 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

  11. Generalized Navier – Stokes equations Velocity - vorticity scheme Kinematics Integral Representation Skerget and Hribersek

  12. Generalized Navier – Stokes equations Velocity - vorticity scheme Kinetics Integral Representation

  13. 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

  14. Generalized Navier – Stokes equations Velocity - vorticity scheme Kinetics Local Integral Representation

  15. Meshless LBIE – Integral representations

  16. Meshless LBIE – Integral representations

  17. Meshless LBIE – Integral representations

  18. Discretization and Numerical Evaluation of Integrals Involved Integrals

  19. Discretization and Numerical Evaluation of Integrals • Arc integrals Involved Integrals

  20. Discretization and Numerical Evaluation of Integrals • Boundary integrals Involved Integrals

  21. Discretization and Numerical Evaluation of Integrals • Volume integrals Involved Integrals

  22. Discretization and Solution Procedure

  23. Discretization and Solution Procedure

  24. Discretization and Solution Procedure Approximation of Boundary Vorticity ω Computation of shear rate and new nodal viscosity

  25. Discretization and Solution Procedure

  26. Discretization and Solution Procedure

  27. Discretization and Solution Procedure Check vorticity’s convergence / Iteration decision

  28. Newtonian Flow

  29. Newtonian Flow

  30. Newtonian Flow

  31. Newtonian Flow

  32. Newtonian Flow

  33. Stenosis problem - qualitative example

  34. Stenosis problem- qualitative example

  35. Stenosis problem- qualitative example

  36. Stenosis problem- qualitative example

  37. Stenosis problem- qualitative example

  38. Stenosis problem- qualitative example

  39. Stenosis problem- qualitative example

  40. Stenosis problem- qualitative example

  41. Stenosis problem- qualitative example

  42. Stenosis problem- qualitative example

  43. Stenosis problem- qualitative example

  44. Stenosis problem- qualitative example

  45. Stenosis problem- qualitative example

  46. Stenosis problem- qualitative example

  47. 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