GEOMETRIC PRE- PROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006 Reduced Numerical Methods for Multiphysics Alfio Quarteroni EPFL - Lausanne,

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TU Eindhoven, 6 December 2006

Reduced Numerical Reduced Numerical Methods for Methods for MultiphysicsMultiphysics

Alfio QuarteroniEPFL - Lausanne, Switzerland

and MOX - Politecnico di Milano, Italy

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REDUCING COMPLEXITY IN THE NUMERICAL REDUCING COMPLEXITY IN THE NUMERICAL APPROXIMATION OF PDEs BYAPPROXIMATION OF PDEs BY

MODEL REDUCTIONMODEL REDUCTION

GEOMETRICAL REDUCTIONGEOMETRICAL REDUCTION

REDUCED BASIS APPROXIMATIONREDUCED BASIS APPROXIMATION

INTERFACE REDUCTION IN FSIINTERFACE REDUCTION IN FSI

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REDUCING COMPLEXITY BYREDUCING COMPLEXITY BY

MODEL REDUCTIONMODEL REDUCTION

(Differential level)(Differential level)

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Compressible Navier-Stokes equations

Neglecting nonlinear terms and time

Full potential equation

Considering the fluid as irrotational

Compressible Euler equations

Neglecting viscous effects

Incompressible Navier-Stokes equations

Neglecting compressibility effects

Laplace equation

Considering the fluid as irrotational

Model hierarchy

in CFD

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NS-Stokes Coupling: sharp interface

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

Streamfunction

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MODEL REDUCTION byMODEL REDUCTION by

BOUNDARY CONTROL onBOUNDARY CONTROL on

OVERLAPPING DOMAINSOVERLAPPING DOMAINS

1

2

2

1

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Boundary control for Advection-Diffusion

The original

problem:

with

The heterogeneous coupling:

Virtual controls

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control: solve

with

Lemma. If all data are smooth enough and if

thenadmits a solution.

Theorem. If we set and

all other data being fixed, thenif we let

(Gervasio, J.-L. Lions, Q. , Numerische Mathematik 2001)

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Boundary layer approximation

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REDUCING COMPLEXITY BYREDUCING COMPLEXITY BY

GEOMETRICAL REDUCTIONGEOMETRICAL REDUCTION

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Geometrical reductionfor hydraulics modelling

1

2

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3

2D/3D

1D

1

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1

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3

1

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1

2

3

1

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1

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2D/3D

1D

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Different mathematical models

3D (or 2D) Free Surface Equations 1D Free Surface Equations

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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 characteristici. + iii. ii.Notice that

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

QuickTime™ and a decompressor


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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

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Coupling by virtual control

where is a suitable extension of the 1D elevation to the 2D domain.

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Full 2D

2D-1D, overlap=3

2D-1D, overlap=1

2D-1D, overlap=0.5

Mesh size: h=0.05.

Coupling by virtual control: a test case

Solitary wave travelling in a channel with an obstacle. Solution at time

t=5s.

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From global to local

GEOMETRICAL REDUCTION IN BLOOD FLOW MODELLING

Blood is a suspension of

red cells, leukocytes

and platelets on a liquid

called plasma

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Velocity profiles in the carotid bifurcation (rigid boundaries, Newtonian)

MATHEMATICAL MODEL

(M.Prosi)



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Cardiovascular System: Functionality

Pulmonary circulation

The Heart

Right Ventricle

Right atrium

The Veins

Vena Cava

Veins

Venules

PulmonaryArteries

Pulmonary Veins

The Heart

Left Atrium

Left Ventricle

The Arteries

Aorta

Arteries

ArteriolesThe Capillaries

The lungs

ArteriesArteriolesCapillariesVenules

Veins

Systemic Circulation

MATHEMATICAL MODEL

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QuickTime™ and aMotion JPEG OpenDML decompressor


Local (level1): 3D flow model

Global (level 2): 1D network of

major arteries and

veins

Global (level 3): 0D capillary

network

A local-to-global approach

MATHEMATICAL MODEL

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Stents for abdominal aortic aneurysms (AAA)

endograft

QuickTime™ and a decompressor


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3D-1D for the carotid: pressure pulse

MATHEMATICAL MODEL

(A.Moura)

QuickTime™ and aMotion JPEG OpenDML decompressor


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A 0D model of the whole circulation

Continuity of fluxes and pressure yields the DAE system:

MATHEMATICAL MODEL

Fluid dynamics Electrical circuits

Pressure Voltage

Flow rate Current

Blood viscosity Resistance R

Blood inertia Inductance L

Wall compliance Capacitance C

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LSA

RPA

LPA

AoD

CS

INN

LCA

COR

AoA

Central Shunt (CS)

Relevant clinical issues:

• shunt radius choice

• systemic/pulmonary flux balancing

• coronary flux

PULMONARY UPPER BODYCORONARY

0

20

40

60

80

CS3 CS3.5 CS4

Flow (%)

Shunt for restoring heart-pulmonary circulation

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A multiscale 3D-0D model

PULMONARY ARTERIES

CORONARY SYSTEM

HEART

LOWER BODY

UPPER BODY

AORTA

SHUNT

Shunt

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TU Eindhoven, 6 December 2006 (F.Migliavacca)

QuickTime™ and aCinepak decompressor


Flow reversal in the pulmonary artery

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REDUCING COMPLEXITYREDUCING COMPLEXITY

BY REDUCED BASIS APPROXIMATIONBY REDUCED BASIS APPROXIMATION

Acknowledgments:

G.Rozza,A.T.Patera (MIT)

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Application in Haemodynamics: coronary bypass

*Study of a preliminary configuration based on geometrical parameters and their ratio (sensitivity).

Inputs:

Output:

tk bypass diameter, Dk arterial diameter, Sk stenosis distance,

Lk outflow distance, k graft angle, Hk bridge height

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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 .

[Prud’homme, Patera, Maday et al. “Reliable Real Time Solution of parameterized PDEs: Reduced basis output bound methods”. J.Fluids Eng. (172), 2002.]

Basic Idea

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Real Time outputs’ sensitivity

Graft Angle

Stenosis/Diameter

Bypass diam./arterial diam.

t/DSynthesis

Parameters Hierarchy: in order of

importance (sensitivity):

• Ratio tk/Dk

• Ratio Sk/Dk

• Graft angle k

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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]

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REDUCING COMPLEXITY BY REDUCING COMPLEXITY BY

INTERFACE REDUCTIONINTERFACE REDUCTION

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Model of the arterial vessel

INTIMA

MEDIA

ADVENTITIA

MATHEMATICAL MODEL

Mechanical interaction(Fluid-wall coupling)

Biochemical interactions(Mass-transfer processes:macromolecules, drug delivery, Oxygen,…)

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Fluid-vessel mechanical interaction

Blood-flow equations:

Vessel equation:

Coupling equations:

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Dimensional reduction: working at interface

Fluid

Interface StructureFluid

Role of Interface

MATHEMATICAL MODEL

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Construction of the Steklov-Poincare’ (Dirichlet-to-Neumann) maps SPf and SPs:

Steklov-Poincare’ equation

Interface Problem: Domain Decomposition Formulation, I

MATHEMATICAL MODEL

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1. Compute the residual stress from a given displacement

3. Update displacement

2. Apply the inverse of the preconditioner to the stress

recover displacement

DD Formulation, II: Preconditioned Iterations

MATHEMATICAL MODEL

(S. Deparis, M. Discacciati, G.Fourestey and A.Q. 2004)

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Flowfield and Vessel Wall Deformation

INTERFACE SOLUTION

QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.

(G.Fourestey)

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• 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)

FSI in America’s Cup sailing boats

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Yacht’s components of AC Class

•Main

•Genoa

•Riggings

•Mast

•Boom

•Hull

•Keel

•Bulb

•Winglets

•Rudder

•Hula ?

Hull

Keel

BulbWinglets

Rudder

Main GenoaSpinnaker

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Water Phase

Air Phase

Interface Conditions

Two-phase flow equations

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From design to simulation



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Pressure on hull and appendages



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Pressure forces on two boats during downwind leg



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Forces on sails

Sail analysis

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• 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)

streamlines velocity and flow separation

Flow around spinnaker and mainsail

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Which cost to achieve the desired accuracy ?

CPU Time

RAM Memory

Number of Processors

• 24 hours

• 30 gigabytes of RAM

• 32 processors

15.000.000 elements, 135.000.000 unknowns

Computational Cost

Mizar Cluster @ epfl (450 AMD Opteron processors, 900 Gb distributed RAM, Myrinet Network)

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Really want to Really want to

REDUCE?REDUCE?

Conclusions

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QuickTime™ and aTIFF (LZW) decompressor


Before reduction ….(the three Lavaredo peaks in the Dolomites)

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…. After reduction

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Documents

GEOMETRIC PRE- PROCSSING MODEL VALIDATION OUTCOME TU Eindhoven, 6 December 2006 Reduced Numerical Methods for Multiphysics Alfio Quarteroni EPFL - Lausanne,