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Reduced Numerical Methods for Multiphysics

Alfio Quarteroni EPFL - Lausanne, Switzerland and MOX - Politecnico di Milano, Italy. Reduced Numerical Methods for Multiphysics. REDUCING COMPLEXITY IN THE NUMERICAL APPROXIMATION OF PDEs BY MODEL REDUCTION GEOMETRICAL REDUCTION REDUCED BASIS APPROXIMATION INTERFACE REDUCTION IN FSI.

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Reduced Numerical Methods for Multiphysics

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  1. Alfio Quarteroni EPFL - Lausanne, Switzerland and MOX - Politecnico di Milano, Italy Reduced Numerical Methods for Multiphysics

  2. REDUCING COMPLEXITY IN THE NUMERICAL APPROXIMATION OF PDEs BY MODEL REDUCTION GEOMETRICAL REDUCTION REDUCED BASIS APPROXIMATION INTERFACE REDUCTION IN FSI

  3. REDUCING COMPLEXITY BY MODEL REDUCTION (Differential level)

  4. Compressible Euler equations Incompressible Navier-Stokes equations Neglecting viscous effects Neglecting compressibility effects Considering the fluid as irrotational Considering the fluid as irrotational Neglecting nonlinear terms and time Full potential equation Laplace equation Model hierarchy in CFD Compressible Navier-Stokes equations

  5. NS-Stokes Coupling: sharp interface Streamfunction

  6. 1 1 2 2 • MODEL REDUCTION by • BOUNDARY CONTROL on • OVERLAPPING DOMAINS

  7. Virtual controls Boundary control for Advection-Diffusion The original problem: with The heterogeneous coupling:

  8. control: solve with Lemma.If all data are smooth enough and if then admits a solution. Theorem.If we set and if we let all other data being fixed, then (Gervasio, J.-L. Lions, Q. , Numerische Mathematik 2001)

  9. Boundary layer approximation

  10. REDUCING COMPLEXITY BY GEOMETRICAL REDUCTION

  11. W W 1 1 W W 2 2 w w 1 1 w w 3 3 w w 2 2 Geometrical reductionfor hydraulics modelling 2D/3D 2D/3D W W 1 1 W W 2 2 w 1 1 w w 1D 1D 3 3 w w 2 2

  12. Different mathematical models 3D (or 2D) Free Surface Equations 1D Free Surface Equations

  13. i. + iii. ii. Interface conditions: 2D 1D At interfaces between the 2D and 1D models we demand the continuity of i. the cross-section area II. the discharge III. the incoming characteristic Notice that

  14. 1D-2D-1D simulation (a test case)

  15. Coupling by Overlapping and Virtual Control At interfaces between the 2D and 1D models use Lagrange multipliers, then satisfy a minimization principle in the overlapping area

  16. Coupling by virtual control where is a suitable extension of the 1D elevation to the 2D domain.

  17. Coupling by virtual control: a test case Solitary wave travelling in a channel with an obstacle. Solution at time t=5s. Full 2D 2D-1D, overlap=3 2D-1D, overlap=1 2D-1D, overlap=0.5 Mesh size: h=0.05.

  18. GEOMETRICAL REDUCTION IN BLOOD FLOW MODELLING From global to local Blood is a suspension of red cells, leukocytes and platelets on a liquid called plasma

  19. MATHEMATICAL MODEL Velocity profiles in the carotid bifurcation (rigid boundaries, Newtonian) (M.Prosi)

  20. MATHEMATICAL MODEL Pulmonary circulation The Heart Right Ventricle Right atrium The Heart Left Atrium Left Ventricle Pulmonary Arteries Pulmonary Veins The lungs Arteries Arterioles Capillaries Venules Veins The Arteries Aorta Arteries Arterioles The Veins Vena Cava Veins Venules The Capillaries Systemic Circulation Cardiovascular System: Functionality

  21. MATHEMATICAL MODEL A local-to-global approach Local (level1):3D flow model Global (level 2):1D network of major arteries and veins Global (level 3):0D capillary network

  22. Stents for abdominal aortic aneurysms (AAA) endograft

  23. MATHEMATICAL MODEL 3D-1D for the carotid: pressure pulse (A.Moura)

  24. MATHEMATICAL MODEL A 0D model of the whole circulation Continuity of fluxes and pressure yields the DAE system:

  25. Central Shunt (CS) PULMONARY Flow (%) UPPER BODY CORONARY LCA LSA 80 INN 60 CS 40 AoA RPA 20 COR LPA AoD 0 CS3 CS3.5 CS4 Shunt for restoring heart-pulmonary circulation • Relevant clinical issues: • shunt radius choice • systemic/pulmonary flux balancing • coronary flux

  26. A multiscale 3D-0D model Shunt

  27. Flowreversal in the pulmonary artery (F.Migliavacca)

  28. REDUCING COMPLEXITY BY REDUCED BASIS APPROXIMATION Acknowledgments: G.Rozza,A.T.Patera (MIT)

  29. Application in Haemodynamics: coronary bypass *Study of a preliminary configuration based on geometrical parameters and their ratio (sensitivity). Inputs: tk bypass diameter, Dk arterial diameter, Sk stenosis distance, Lk outflow distance, k graft angle, Hk bridge height Output:

  30. Reduced Basis Methods (for Design and Optimization) Computational methods that allow accurate and efficient real-time evaluation of input-output (geometry/design quantities/indexes) relationship governed by parametrized PDEs The approximation is based on global FEM solutions (at different parameter value) and on Galerkin projection properties . Basic Idea [Prud’homme, Patera, Maday et al. “Reliable Real Time Solution of parameterized PDEs: Reduced basis output bound methods”. J.Fluids Eng. (172), 2002.]

  31. Real Time outputs’ sensitivity Bypass diam./arterial diam. Stenosis/Diameter Graft Angle Synthesis t/D • Parameters Hierarchy: in order of • importance (sensitivity): • Ratio tk/Dk • Ratio Sk/Dk • Graft angle k

  32. RB extension: shape design • The use of Reduced Basis in more complex models provides • computational savings not only in sensitivity analysis, but also • in solving shape optimization (and optimal control) problems using • e.g. curved shape functions. • [non affine mapping, “empirical interpolation” (Maday, Patera, 2004)]. • With this approach we can use Reduced Basis for Shape Design • Problems, for example, in small flow perturbation case • (Agoshkov,Q.,Rozza, SIAM J. Num. An. 06). Role of Curvature in bypass optimization problem and vorticity (Dean number) [Sherwin, Doorly]

  33. REDUCING COMPLEXITY BY INTERFACE REDUCTION

  34. MATHEMATICAL MODEL INTIMA MEDIA ADVENTITIA Model of the arterial vessel Mechanical interaction(Fluid-wall coupling) Biochemical interactions(Mass-transfer processes:macromolecules, drug delivery, Oxygen,…)

  35. Fluid-vessel mechanical interaction • Blood-flow equations: • Vessel equation: • Coupling equations:

  36. MATHEMATICAL MODEL Structure Fluid Fluid Interface Dimensional reduction: working at interface Role of Interface

  37. MATHEMATICAL MODEL Interface Problem: Domain Decomposition Formulation, I Steklov-Poincare’ equation Construction of the Steklov-Poincare’ (Dirichlet-to-Neumann) maps SPf and SPs:

  38. MATHEMATICAL MODEL DD Formulation, II: Preconditioned Iterations 1. Compute the residual stress from a given displacement 2. Apply the inverse of the preconditioner to the stress recover displacement 3. Update displacement (S. Deparis, M. Discacciati, G.Fourestey and A.Q. 2004)

  39. INTERFACE SOLUTION Flowfield and Vessel Wall Deformation (G.Fourestey)

  40. FSI in America’s Cup sailing boats • Premier international yacht race • First race in 1851 around • Isle of Wight • “America” was the name of • the winner yacht of the first edition • Held by U.S.A. for 132 years, the cup was won in 1983 by the winged-keel Australia II • New Zealand’s Black Magic has dominated the two editions of 1995 and 2000 • Alinghi has won the last edition (ended March 2, 2003)

  41. Main Genoa Spinnaker Keel Rudder Hull Bulb Winglets Yacht’s components of AC Class • Main • Genoa • Riggings • Mast • Boom • Hull • Keel • Bulb • Winglets • Rudder • Hula ?

  42. Two-phase flow equations Air Phase Interface Conditions Water Phase

  43. From design to simulation

  44. Pressure on hull and appendages

  45. Pressure forces on two boats during downwind leg

  46. Sail analysis Forces on sails

  47. Flow around spinnaker and mainsail • Boat speed: 5.540 m/s (~ 15.45 kts) • True Wind Angle: 148 Deg • True Wind Speed at 10m: 5.660 m/s (~ 17.54 kts) velocity and flow separation streamlines

  48. 15.000.000 elements, 135.000.000 unknowns Which cost to achieve the desired accuracy ? • 24 hours • 30 gigabytes of RAM • 32 processors CPU Time RAM Memory Number of Processors Computational Cost Mizar Cluster @ epfl (450 AMD Opteron processors, 900 Gb distributed RAM, Myrinet Network)

  49. Conclusions Really want to REDUCE?

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