Alfio Quarteroni EPFL Lausanne Switzerland MOX Politecnico di

Alfio Quarteroni EPFL - Lausanne, Switzerland MOX - Politecnico di Milano, Italy GEOMETRIC PREPROCSSING MODEL VALIDATION Reduced Numerical Methods for Multiphysics OUTCOME TU Eindhoven, 6 December 2006

REDUCING COMPLEXITY IN THE NUMERICAL APPROXIMATION OF PDEs BY GEOMETRIC PREPROCSSING MODEL REDUCTION MODEL VALIDATION OUTCOME GEOMETRICAL REDUCTION REDUCED BASIS APPROXIMATION INTERFACE REDUCTION IN FSI TU Eindhoven, 6 December 2006

GEOMETRIC PREPROCSSING REDUCING COMPLEXITY BY MODEL REDUCTION (Differential level) MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

Model hierarchy in CFD GEOMETRIC PREPROCSSING Compressible Euler equations Incompressible Navier-Stokes equations Neglecting viscous effects Neglecting compressibility effects MODEL VALIDATION Considering the fluid as irrotational OUTCOME Compressible Navier-Stokes equations Considering the fluid as irrotational Neglecting nonlinear terms and time Full potential equation Laplace equation TU Eindhoven, 6 December 2006

NS-Stokes Coupling: sharp interface GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME Streamfunction TU Eindhoven, 6 December 2006

GEOMETRIC PREPROCSSING 1 MODEL REDUCTION by 1 BOUNDARY CONTROL on OVERLAPPING DOMAINS 2 MODEL VALIDATION 2 OUTCOME TU Eindhoven, 6 December 2006

Boundary control for Advection-Diffusion GEOMETRIC PREPROCSSING The original problem: with The heterogeneous coupling: MODEL VALIDATION Virtual controls OUTCOME TU Eindhoven, 6 December 2006

control: solve GEOMETRIC PREPROCSSING with Lemma. If all data are smooth enough and if then admits a solution. MODEL VALIDATION Theorem. If we set OUTCOME if we let and all other data being fixed, then (Gervasio, J. -L. Lions, Q. , Numerische Mathematik 2001) TU Eindhoven, 6 December 2006

Boundary layer approximation GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

GEOMETRIC PREPROCSSING REDUCING COMPLEXITY BY GEOMETRICAL REDUCTION MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

Geometrical reduction for hydraulics modelling GEOMETRIC PREPROCSSING 2 D/3 D W 1 MODEL VALIDATION W 2 w w 1 1 D 1 D OUTCOME w 3 w 2 TU Eindhoven, 6 December 2006

Different mathematical models GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME 3 D (or 2 D) Free Surface Equations 1 D Free Surface Equations TU Eindhoven, 6 December 2006

Interface conditions: 2 D 1 D GEOMETRIC PREPROCSSING At interfaces between the 2 D and 1 D models we demand the continuity MODEL of VALIDATION 1. i. the cross-section area 2. II. the discharge 3. III. the incoming characteristic OUTCOME Notice that i. + iii. TU Eindhoven, 6 December 2006

1 D-2 D-1 D simulation (a test case) GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

Coupling by Overlapping and Virtual Control GEOMETRIC PREPROCSSING MODEL VALIDATION At interfaces between the 2 D and 1 D models use Lagrange multipliers, then satisfy a minimization principle in the overlapping area OUTCOME TU Eindhoven, 6 December 2006

Coupling by virtual control GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME where domain. is a suitable extension of the 1 D elevation to the 2 D TU Eindhoven, 6 December 2006

Coupling by virtual control: a test case Solitary wave travelling in a channel with an obstacle. Solution at time GEOMETRIC PREPROCSSING t=5 s. Full 2 D 2 D-1 D, overlap=3 MODEL VALIDATION 2 D-1 D, overlap=1 OUTCOME 2 D-1 D, overlap=0. 5 Mesh size: h=0. 05. TU Eindhoven, 6 December 2006

GEOMETRICAL REDUCTION IN BLOOD FLOW MODELLING From global to local GEOMETRIC PREPROCSSING Blood is a suspension of red cells, leukocytes and platelets on a liquid MODEL VALIDATION called plasma OUTCOME TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL Velocity profiles in the carotid bifurcation (rigid boundaries, Newtonian) GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME (M. Prosi) TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL Cardiovascular System: Functionality GEOMETRIC PREPROCSSING Pulmonary circulation The Heart Pulmonary Arteries Pulmonary Veins Right Ventricle Right atrium MODEL VALIDATION The Veins Vena Cava Left Atrium The lungs Left Ventricle Arteries Arterioles Capillaries Venules Veins The Arteries Veins OUTCOME Venules The Heart Aorta Arteries The Capillaries Arterioles Systemic Circulation TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL A local-to-global approach GEOMETRIC PREPROCSSING Local (level 1): 3 D flow model Global (level 2): 1 D network of major arteries and MODEL veins VALIDATION Global (level 3): 0 D capillary network OUTCOME TU Eindhoven, 6 December 2006

Stents for abdominal aortic aneurysms (AAA) GEOMETRIC PREPROCSSING MODEL VALIDATION endograft OUTCOME TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL 3 D-1 D for the carotid: pressure pulse GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME (A. Moura) TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL A 0 D model of the whole circulation GEOMETRIC PREPROCSSING MODEL VALIDATION Continuity of fluxes and pressure yields the DAE system: OUTCOME Fluid dynamics Electrical circuits Pressure Voltage Flow rate Current Blood viscosity Resistance R Blood inertia Inductance L Wall compliance Capacitance C TU Eindhoven, 6 December 2006

Shunt for restoring heart-pulmonary circulation GEOMETRIC PREPROCSSING Central Shunt (CS) PULMONARY UPPER BODY CORONARY Flow (%) LCA LSA 80 INN 60 CS 40 MODEL VALIDATION Ao. A RPA 20 COR LPA Relevant clinical issues: OUTCOME • shunt radius choice Ao. D • systemic/pulmonary flux balancing 0 CS 3. 5 CS 4 • coronary flux TU Eindhoven, 6 December 2006

A multiscale 3 D-0 D model GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME Shunt TU Eindhoven, 6 December 2006

Flow reversal in the pulmonary artery GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME (F. Migliavacca) TU Eindhoven, 6 December 2006

GEOMETRIC PREPROCSSING REDUCING COMPLEXITY BY REDUCED BASIS APPROXIMATION MODEL VALIDATION OUTCOME Acknowledgments: G. Rozza, A. T. Patera (MIT) TU Eindhoven, 6 December 2006

Application in Haemodynamics: coronary bypass GEOMETRIC PREPROCSSING *Study of a preliminary configuration based on geometrical parameters and their ratio (sensitivity). Inputs: MODEL VALIDATION tk bypass diameter, Dk arterial diameter, Sk stenosis distance, Lk outflow distance, k graft angle, Hk bridge height OUTCOME Output: TU Eindhoven, 6 December 2006

Reduced Basis Methods (for Design and Optimization) Computational methods that allow accurate and efficient real-time evaluation of input-output (geometry/design quantities/indexes) GEOMETRIC PRErelationship governed by parametrized PDEs PROCSSING The approximation is based on global FEM solutions (at different parameter value) and on Galerkin projection properties. Basic Idea MODEL VALIDATION OUTCOME [Prud’homme, Patera, Maday et al. “Reliable Real Time Solution of parameterized PDEs: Reduced basis output bound methods”. J. Fluids Eng. (172), 2002. ] TU Eindhoven, 6 December 2006

Real Time outputs’ sensitivity Bypass diam. /arterial diam. GEOMETRIC PREPROCSSING Stenosis/Diameter Graft Angle MODEL VALIDATION Synthesis t/D Parameters Hierarchy: in order of importance (sensitivity): OUTCOME • Ratio tk/Dk • Ratio Sk/Dk • Graft angle k TU Eindhoven, 6 December 2006

RB extension: shape design • The use of Reduced Basis in more complex models provides computational savings not only in sensitivity analysis, but also GEOMETRIC PREin solving shape optimization (and optimal control) problems using PROCSSING 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 MODEL (Agoshkov, Q. , Rozza, SIAM J. Num. An. 06). VALIDATION OUTCOME Role of Curvature in bypass optimization problem and vorticity (Dean numbe [Sherwin, Doorly] TU Eindhoven, 6 December 2006

GEOMETRIC PREPROCSSING REDUCING COMPLEXITY BY INTERFACE REDUCTION MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL Model of the arterial vessel Mechanical interaction (Fluid-wall coupling) GEOMETRIC PREPROCSSING Biochemical interactions (Mass-transfer processes: macromolecules, drug delivery, Oxygen, …) INTIMA MEDIA MODEL VALIDATION ADVENTITIA OUTCOME TU Eindhoven, 6 December 2006

Fluid-vessel mechanical interaction Blood-flow equations: GEOMETRIC PREPROCSSING Vessel equation: MODEL VALIDATION Coupling equations: OUTCOME TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL Dimensional reduction: working at interface GEOMETRIC PREPROCSSING Role of Interface Fluid Structure MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL Interface Problem: Domain Decomposition Formulation, I GEOMETRIC PREPROCSSING Steklov-Poincare’ equation Construction of the Steklov-Poincare’ (Dirichlet-to-Neumann) maps SPf and SPs: MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

MATHEMATICAL MODEL DD Formulation, II: Preconditioned Iterations GEOMETRIC PREPROCSSING 1. Compute the residual stress from a given displacement 2. Apply the inverse of the preconditioner to the stress recover displacement MODEL VALIDATION 3. Update displacement OUTCOME (S. Deparis, M. Discacciati, G. Fourestey and A. Q. 2004) TU Eindhoven, 6 December 2006

INTERFACE SOLUTION Flowfield and Vessel Wall Deformation GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME (G. Fourestey) TU Eindhoven, 6 December 2006

FSI in America’s Cup sailing boats • Premier international yacht race GEOMETRIC PREPROCSSING • First race in 1851 around Isle of Wight • “America” was the name of the winner yacht of the first edition MODEL VALIDATION OUTCOME • 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) TU Eindhoven, 6 December 2006

Yacht’s components of AC Class Main Genoa Spinnaker Keel Rudder • Main GEOMETRIC PREPROCSSING • Genoa • Riggings • Mast • Boom Winglets Hull Bulb • Hull MODEL VALIDATION • Keel • Bulb • Winglets OUTCOME • Rudder • Hula ? TU Eindhoven, 6 December 2006

Two-phase flow equations GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME Air Phase Interface Conditions Water Phase TU Eindhoven, 6 December 2006

From design to simulation GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

Pressure on hull and appendages GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

Pressure forces on two boats during downwind leg GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

Sail analysis GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME Forces on sails TU Eindhoven, 6 December 2006

GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

Flow around spinnaker and mainsail • Boat speed: 5. 540 m/s (~ 15. 45 kts) • True Wind Angle: 148 Deg • True Wind Speed at 10 m: 5. 660 m/s (~ 17. 54 kts) GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME streamlines velocity and flow separation TU Eindhoven, 6 December 2006

Computational Cost 15. 000 elements, 135. 000 unknowns GEOMETRIC PREPROCSSING Which cost to achieve the desired accuracy ? CPU Time MODEL VALIDATION RAM Memory • 24 hours • 30 gigabytes of RAM • 32 processors Number of Processors OUTCOME Mizar Cluster @ epfl (450 AMD Opteron processors, 900 Gb distributed RAM, Myrinet Network) TU Eindhoven, 6 December 2006

Conclusions GEOMETRIC PREPROCSSING Really want to REDUCE? MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

Before reduction …. (the three Lavaredo peaks in the Dolomites) GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

…. After reduction GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006

GEOMETRIC PREPROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006