Geometrical and computational -0.1cm reduction strategies ...Gianluigi Rozza in collaboration with Al o Quarteroni, Andrea Manzoni, Toni Lassila MATHICSE - CMCS Modelling and Scienti

Reduction Strategies Geometrical reduction Computational Reduction FSI problems

Geometrical and computational

reduction strategies for the approximation of

viscous flows in parametrized domains

Gianluigi Rozzain collaboration with Alfio Quarteroni, Andrea Manzoni, Toni Lassila

MATHICSE - CMCS Modelling and Scientific Computing

Ecole Polytechnique Federale de Lausanne

Workshop on Modern Techniques in theNumerical Solution of Partial Differential Equations

Heraklion, Crete, Greece, September 19-23, 2011

Acknowledgements: A.T. Patera, D.B.P. Huynh (MIT)

Sponsors: Swiss National Science Foundation, European Research Council - Mathcard Project, Progetto Roberto Rocca Politecnico di Milano-MIT


Outline

1. Introduction

Motivation and ingredients

Inverse problems related with shape variation

(e.g. shape optimization, parameter identification and fluid-structure

interaction problems)

2. Geometrical Parametrization

Free-Form Deformations (FFD) and parametric coupling

Radial Basis Functions (RBF)

3. Computational Reduction

Reduced Basis (RB) methodology

Approximation, stability, a posteriori error bounds for Navier-Stokes flows

4. Applications to problems arising in (Newtonian) haemodynamics

A reduced model for the description of FSI effects in a stenosed artery

Shape optimization of bypass grafts

Real-time medical evaluations and geometrical reconstruction


Reduction strategies for simulation/optimization of viscous flows

Goal: to achieve the accuracy and reliability of a high fidelity approximation

but at greatly reduced cost of a low order model

Real-time or many-query problems related with shape variation in haemodynamics

Evaluation of indexes related with geometry that measure arteries occlusion risk

Shape optimization of cardiovascular geometries (e.g. bypass grafts)

Way: coupling suitable shape parametrizations with reduced basis methods

Introduce a low-dimensional shape parametrization (geometrical reduction)

Bring geometry variations back to the equation coefficients

Evaluate PDEs/output using reduced basis methods (computational reduction)



Goal: to achieve the accuracy and reliability of a high fidelity approximation

but at greatly reduced cost of a low order model

Real-time or many-query problems related with shape variation in haemodynamics

Evaluation of indexes related with geometry that measure arteries occlusion risk

Shape optimization of cardiovascular geometries (e.g. bypass grafts)

Way: coupling suitable shape parametrizations with reduced basis methods

Introduce a low-dimensional shape parametrization (geometrical reduction)

Bring geometry variations back to the equation coefficients

Evaluate PDEs/output using reduced basis methods (computational reduction)



Input and Output

Input parameters: µ ∈D ⊂ Rp → geometry, fluid properties, BCs, sources

Output of interest: J(µ) = s(U(µ)) → viscous energy dissipation, vorticity, stresses, ...

Field variables: U(µ) = (u(µ),p(µ)) velocity, pressure → satisfy a µ-parametrized PDE

Essential ingredients of RB methods[early works in 80’s: Noor, Peters, Peterson,.., Ito, Ravindran]

Galerkin projection onto a space spanned by PDE solutions for N selected µ1, . . . ,µN

Offline/Online computational stratagem [Maday, Patera,...]

Rigorous a posteriori error estimation procedures

rapid = minimization of the marginal cost in input/output evaluation

reliable = error bounds of input/output evaluation or field variable



Given an observation operator s : X (Ωo (π))→ Y and a target observation s∗ ∈ Y , find

the π∗ ∈D that solves

minπ∈Dπ

Jo (µ) := ‖s∗− so (Uo (µ))‖2Y + α(Mπ,π)RPπ , ∀ω ∈Dω . (1)

or

minπ∈Dπ

maxω∈Dω

Jo (µ) := ‖s∗− so (Uo (µ))‖2Y + α(Mπ,π)RPπ , (2)

where Uo (µ) is the solution of the state problem

Ao (Uo (µ),W ) = Fo (W ) ∀W ∈ X (Ωo (π)). (3)

µ = (π,ω) ∈D ⊂ RP consists of a control parameter π ∈Dπ ⊂ RPπ and an

uncertainty parameter ω ∈Dω ⊂ RPω . We assume that π characterizes the

geometric configuration, ω is related to physical properties, BCs or sources.

Y = space of observables, X (Ωo (π)) = state space defined on the domain Ωo (π)

target observation s∗ may be polluted by noise and/or measurement error

α > 0, M : RPπ → RPπ is a SPD matrix s.t. (Mπ,π)RPπ is a regularization term

Ao : X (Ωo (π))×X (Ωo (π))→ R is a continuous, inf-sup stable bilinear form,

Fo : X (Ωo (π))→ R is a continuous linear form.



When the parameters π control the domain Ωo (π) of the state problem, traditional

discretization techniques are too expensive in inverse-like problems:

changing the underlying geometry requires an expensive mesh deformation or

remeshing process, followed by the reassembly of the entire linear system

iterative procedures for optimization require multiple evaluations of outputs

depending on field variables and geometry

We consider only fixed domain approaches:

The family of admissible domains Ωo (π) is given as the image of a smooth

parametric map T ( · ;π) : Ω→Ωo (π) (fewer parameters: geometrical reduction)

Both the solution of the state problem (3) and the observations (e.g. (1)) can be

transformed by a change of variables to the fixed reference domain Ω:

minπ∈Dπ

J(µ) := ‖s∗− s(U(µ))‖2Y + α(Mπ,π)RPπ , ∀ω ∈Dω , (4)min

π∈Dπ

JN (µ) := ‖s∗− s(UN (µ))‖2Y + α(Mπ,π)RPπ , ∀ω ∈Dω , (4)

where the state problem on the fixed domain is

A(U(µ),W ; µ) = F (W ; µ) ∀W ∈ X (Ω). (5)A(UN (µ),W ; µ) = F (W ; µ) ∀W ∈ XN (Ω). (5)

The state problem and the related output are approximated by means of

reduced basis methods (computational reduction)














minπ∈Dπ

J(µ) := ‖s∗− s(U(µ))‖2Y + α(Mπ,π)RPπ , ∀ω ∈Dω , (4)min

π∈Dπ



A(U(µ),W ; µ) = F (W ; µ) ∀W ∈ X (Ω). (5)A(UN (µ),W ; µ) = F (W ; µ) ∀W ∈ XN (Ω). (5)
















minπ∈Dπ

J(µ) := ‖s∗− s(U(µ))‖2Y + α(Mπ,π)RPπ , ∀ω ∈Dω , (4)

minπ∈Dπ



A(U(µ),W ; µ) = F (W ; µ) ∀W ∈ X (Ω). (5)

A(UN (µ),W ; µ) = F (W ; µ) ∀W ∈ XN (Ω). (5)
















minπ∈Dπ

J(µ) := ‖s∗− s(U(µ))‖2Y + α(Mπ,π)RPπ , ∀ω ∈Dω , (4)

minπ∈Dπ



A(U(µ),W ; µ) = F (W ; µ) ∀W ∈ X (Ω). (5)

A(UN (µ),W ; µ) = F (W ; µ) ∀W ∈ XN (Ω). (5)




Parametrized weak formulation of the Navier-Stokes equations

Evaluate/optimize s(U(µ)) where U(µ) ∈ X (Ω) solves

A(U(µ),W ; µ) = F (W ; µ), ∀W ∈ X (Ω)

U(µ) := (u(µ),p(µ)) ∈ X (Ω) = V ×Q ⊂ [H1(Ω)]2×L2(Ω)

A(U,W ; µ) = a(u,w; µ) + b(p,w;π) + b(q,u;π) + c(u,u,w;π)

a(u,w; µ) =∫

Ω

∂u

∂xiνij (x,µ)

∂w

∂xjdΩ, b(p,w; µ) =−

∫Ω

pχij (x,π)∂wj

∂xidΩ

c(u,w,z; µ) =∫

Ωui ηij (x,π)

∂vk

∂xjzk dΩ, F (W ; µ) =

∫Ω

f ·w|JT | dΩ + BC terms(ω)

The parametrized (original) domain Ωo (π) is the image of a fixed (reference) domain

Ω through a parametric map T (·;π) : Ω→Ωo (π) and

ν(x,µ) = ν(ω)J−1T J−T

T |JT | and χ(x,π)≡ η(x,π) = J−1T |JT |

being JT = JT (x,π) = Jacobian of T (x,π)

Output (cost functional): being, e.g. Q = Q(·) a physical quantity of interest,

s(u(µ)) =∫

ΩQ T (U(µ))|JT |dΩ.


Geometrical Parametrization

X RB framework requires a geometrical map T (·;π) : Ω→Ωo (π) in order to

combine discretized solutions for the space construction

X This procedure enables to avoid shape deformation and remeshing (that, e.g.

normally occur at each step of an iterative optimization procedure)

X Reduction in the complexity of parametrization: versatility, low-dimensionality,

automatic generation of maps, capability to represent realistic configurations, ...

Left: Different carotid bifurcation specimens obtained by autopsy (adults aged 30-75);

picture taken from Z. Ding et al., Journal of Biomechanics 34 (2001),1555-1562.

Right: Different carotid bifurcation obtained through radial basis functions techniques.


Shape Parametrization Techniques

Cartesian geometries:

Affine/nonaffine mapping “by hands”

Complex realistic geometries:

Automatic affine transformation (DD) rbMIT

Free-shape nonaffine transformations based on

control points (e.g. Free-Form Deformation

[Sederberg & Parry], Radial Basis Functions

[Bookstein, Buhmann])

Transfinite Mappings [Gordon, Hall]


Free-Form Deformation (FFD) Techniques

Ingredients:

a fixed rectangle D s.t. Ω⊂D, an invertible map x = Ψ(x) s.t. Ψ(D) = (0,1)2

a lattice of control points Pl ,m = [l/L,m/M]T , l = 0, . . . ,L, m = 0, . . . ,M

Construction of the FFD mapping:

Ωo (π) = Ψ−1 T Ψ(Ω,π), T (x,π) =L

∑l=0

M

∑m=0

bL,Ml ,m (x)(Pl ,m + π l ,m)

bL,Ml ,m (x) =

(Ll

)(Mm

)(1− x1)L−l x l

1(1− x2)M−m xm2 (Bernstein pol. tensor products)

Parameters π1, . . . ,πP = displacements of some (selected) control points


Radial Basis Function Techniques

T : R2→ R2, T (x) = Qr (x) +k

∑i=1

wi σ(‖x−Xi‖)

Ingredients:

Xjkj=1,Yjk

j=1 ∈ Rk×2 initial/deformed position of control points

Qr (·) is a low-degree polynomial function (in our case r = 1, Q1(x) = c + Ax)

wjkj=1, wi ∈ R2 set of weights corresponding to the k control points

σ(·) is the basis function; e.g. σ(h) = h3, exp(−Ch2), h2 log(h), ...

Construction of T (x) = c + Ax + W T s(x):

RBF is function of 2k + 6 coefficients: c ∈ R2, A ∈ R2×2, W ∈ Rk×2 given by S Ik X

ITk 0 0

XT 0 0

W (π)

(c(π))T

(A(π))T

=

Y(π)

0

0

where s(x) = (σ(‖x−X1‖), . . . ,σ(‖x−Xk‖))T ∈ Rk and Sij = σ(‖Xj −Xi‖)

Constraints: 2k interpolation (Yj ) = T (Xj ) + 6 “side” ITk W = X T W = 0


Radial Basis Function Techniques

Control positions can be freely chosen

(they can be scattered in the domain and

do not have to reside on a regular lattice

RBF techniques are interpolatory: each

control point of the initial shape is

mapped onto the corresponding control

point of the deformed one

Depending on the choice of control

points, either global or localized

deformations can be describedGlobal and local deformation obtained with RBF techniques

(parameters = horizontal displacements of the • control points


Computational Reduction

Acknowledgement: Anthony T. Patera (MIT) - augustine.mit.edu


Reduced Basis Methods: Construction

Pb(µ;U(µ))

µ-PDE, weak formulation

U(µ) ∈ X : A(U(µ),V ; µ) = F (V ) ∀V ∈ X

J(µ) = s(U(µ))

PbN (µ;UN (µ))

Truth approximation (FEM)

UN (µ) ∈ XN : A(UN (µ),V ; µ) = F (V ) ∀V ∈ XN

JN (µ) = s(UN (µ))

Truth Hypothesis: UN (µ) “indistinguishable” from U(µ).

RB Motivation: µ → UN (µ), JN (µ) too expensive and slow in

many-query and real-time contexts.

Sampling (Greedy)

Space Construction

(Hierarchical Lagrange basis)

OFFLINE

SN = µ i , i = 1, . . . ,N

XN = spanUN (µi ), i = 1, . . . ,N

dim(XN ) = N N = dim(XN )

PbN (µ;UN (µ))

Galerkin projection

ONLINE

Reduced Basis (RB) approximation


JN (µ) = s(UN (µ))



Pb(µ;U(µ))


U(µ) ∈ X : A(U(µ),V ; µ) = F (V ) ∀V ∈ X

J(µ) = s(U(µ))

PbN (µ;UN (µ))



JN (µ) = s(UN (µ))




Sampling (Greedy)

Space Construction


OFFLINE

SN = µ i , i = 1, . . . ,N

XN = spanUN (µi ), i = 1, . . . ,N


PbN (µ;UN (µ))

Galerkin projection

ONLINE



JN (µ) = s(UN (µ))



Pb(µ;U(µ))


U(µ) ∈ X : A(U(µ),V ; µ) = F (V ) ∀V ∈ X

J(µ) = s(U(µ))

PbN (µ;UN (µ))



JN (µ) = s(UN (µ))




Sampling (Greedy)

Space Construction


OFFLINE

SN = µ i , i = 1, . . . ,N

XN = spanUN (µi ), i = 1, . . . ,N


PbN (µ;UN (µ))

Galerkin projection

ONLINE



JN (µ) = s(UN (µ))



Pb(µ;U(µ))


U(µ) ∈ X : A(U(µ),V ; µ) = F (V ) ∀V ∈ X

J(µ) = s(U(µ))

PbN (µ;UN (µ))



JN (µ) = s(UN (µ))




Sampling (Greedy)

Space Construction


OFFLINE

SN = µ i , i = 1, . . . ,N

XN = spanUN (µi ), i = 1, . . . ,N


PbN (µ;UN (µ))

Galerkin projection

ONLINE



JN (µ) = s(UN (µ))


Reduced Basis Methods: smooth parametric dependency

M N = UN (µ) ∈ X N ; µ ∈D

XN = spanUN (µi ), i = 1, . . . ,N

How to be rigorous, rapid and reliable?1 depends on the sampling procedure for parameter exploration (greedy algorithm)2 exploits an Offline/Online stratagem based on the affinity assumption:

A(V ,W ; µ) =Qa

∑q=1

Θqa (µ)Aq(V ,W ), F (W ; µ) =

Qf

∑q=1

Θqf (µ)F q(w)

3 relies on a posteriori error analysis1

1Review on [R., Huynh, Patera 08], [Quarteroni, R., Manzoni 11]


Reduced Basis Method: approximation stability

The steady Navier-Stokes problem


For µ ∈D , evaluate JN (µ) = s(uN (µ),pN (µ)), (uN (µ),pN (µ)) ∈ Vµ

N ×QN :a(uN (µ),w; µ) + b(pN (µ),w; µ) + c(uN (µ),uN (µ),w; µ) = F1(w; µ) ∀w ∈ V

µ

N

b(q,uN (µ); µ) = F2(q; µ) ∀q ∈ QN

Reduced basis spaces:

QN (pressure) and Vµ

N (velocity) given by [R., Veroy 07, R. 09]

QN = spanζn := pN (µn), n = 1, . . . ,N

Vµ

N = spanσn := uN (µn), T µ

ζn, n = 1, . . . ,N

T µ : Q→ V is the supremizer operator given by

(T µ q,w)V = b(q,w; µ) ∀ w ∈ V ;

the spaces pair V µ

N ,QN guarantees the fulfillment of an equivalent

parametrized Brezzi inf-sup condition

infq∈QN

supw∈V

µ

N

b(q,w; µ)

‖w‖V ‖q‖Q=: βN (µ)≥ β0 > 0, ∀ µ ∈D

also on the reduced basis spaces.


Reduced Basis Methods: Offline/Online decomposition

Under the assumption of affine parametric dependence

a(u,w; µ) =Qa

∑q=1

Θqa (µ)aq(u,w), b(p,w; µ) =

Qb

∑q=1

Θqb(µ)bq(p,w), . . .

we have to solve the linearized (e.g. fixed point) system: given (u(0)N ,p

(0)N ), ∀k > 0

a(u(k)N (µ),σn; µ) + b(p

(k)N (µ),σn; µ) + c(u

(k−1)N (µ),u

(k)N (µ),σn; µ) = F1(σn; µ) ∀n = 1, . . . ,2N

b(ζn,u(k)N (µ); µ) = F2(ζn; µ) ∀n = 1, . . . ,N

2N

∑j=1

Qa

∑q=1

Θqa Aq

ij u(k)Nj +

N

∑l=1

Qb

∑q=1

ΘqbBq

il p(k)Nl +

2N

∑j=1

2N

∑s=1

Qc

∑q=1

u(k−1)Nj Θq

c C qij (σ s )u

(k)Nj =

QF1

∑q=1

ΘfqF q

1,i , 1≤ i ≤ 2N

2N

∑j=1

Qb

∑q=1

ΘqbBq

jl u(k)Nj =

QF2

∑q=1

Θgq F q

2,l , 1≤ l ≤N

2N

∑j=1

Qa

∑q=1

Θqa Aq

ij u(k)Nj +

N

∑l=1

Qb

∑q=1

ΘqbBq

il p(k)Nl +

2N

∑j=1

2N

∑s=1

Qc

∑q=1

u(k−1)Nj Θq

c C qij (σ s )u

(k)Nj =

QF1

∑q=1

ΘfqF q

1,i , 1≤ i ≤ 2N

2N

∑j=1

Qb

∑q=1

ΘqbBq

jl u(k)Nj =

QF2

∑q=1

Θgq F q

2,l , 1≤ l ≤N

Offline pre-processing: build basis ZN = [ζ1| . . . |ζN ], Z2N = [σ1| . . . |σN |T µ ζ1| . . . |T µ ζN ]],

store matrices [Aq ]m,n = aq(σn,σm), [Bq ]l ,n = bq(ζl ,σn), [C q(σ s )]m,n = cq(σ s ,σn,σm)

Note (e.g.): Aq = ZT2N Aq

N Z2N being [AqN ]i ,j = aq(φ

Ni ,φN

j )

Online evaluations: evaluate coefficients Θq...(µ), assemble RB matrices

AN = ∑Qaq=1 Θq

a (µ)Aq , BN = ∑Qbq=1 Θq

b(µ)Bq , CN (u(k−1)N ) = ∑

2Ns=1 u

(k−1)Nj ∑

Qcq=1 Θq

c (µ)C q(σ s )

and solve, ∀k > 0

AN u

(k)N +BN p

(k)N +CN (u

(k−1)N )u

(k)N = F1,N

BN u(k)N = F2,N

(3N×3N)

Nonaffine cases: empirical interpolation method treatment...[Maday et al. 2004]




a(u,w; µ) =Qa

∑q=1


Qb

∑q=1



(0)N ), ∀k > 0

a(u

(k)N (µ),σn; µ) + b(p

(k)N (µ),σn; µ) + c(u

(k−1)N (µ),u

(k)N (µ),σn; µ) = F1(σn; µ) ∀n = 1, . . . ,2N

b(ζn,u(k)N (µ); µ) = F2(ζn; µ) ∀n = 1, . . . ,N

2N

∑j=1

Qa

∑q=1

Θqa Aq

ij u(k)Nj +

N

∑l=1

Qb

∑q=1

ΘqbBq

il p(k)Nl +

2N

∑j=1

2N

∑s=1

Qc

∑q=1

u(k−1)Nj Θq

c C qij (σ s )u

(k)Nj =

QF1

∑q=1

ΘfqF q

1,i , 1≤ i ≤ 2N

2N

∑j=1

Qb

∑q=1

ΘqbBq

jl u(k)Nj =

QF2

∑q=1

Θgq F q

2,l , 1≤ l ≤N

2N

∑j=1

Qa

∑q=1

Θqa Aq

ij u(k)Nj +

N

∑l=1

Qb

∑q=1

ΘqbBq

il p(k)Nl +

2N

∑j=1

2N

∑s=1

Qc

∑q=1

u(k−1)Nj Θq

c C qij (σ s )u

(k)Nj =

QF1

∑q=1

ΘfqF q

1,i , 1≤ i ≤ 2N

2N

∑j=1

Qb

∑q=1

ΘqbBq

jl u(k)Nj =

QF2

∑q=1

Θgq F q

2,l , 1≤ l ≤N





Ni ,φN

j )


AN = ∑Qaq=1 Θq


b(µ)Bq , CN (u(k−1)N ) = ∑

2Ns=1 u

(k−1)Nj ∑

Qcq=1 Θq

c (µ)C q(σ s )

and solve, ∀k > 0

AN u

(k)N +BN p

(k)N +CN (u

(k−1)N )u

(k)N = F1,N

BN u(k)N = F2,N

(3N×3N)





a(u,w; µ) =Qa

∑q=1


Qb

∑q=1



(0)N ), ∀k > 0

a(u

(k)N (µ),σn; µ) + b(p

(k)N (µ),σn; µ) + c(u

(k−1)N (µ),u

(k)N (µ),σn; µ) = F1(σn; µ) ∀n = 1, . . . ,2N

b(ζn,u(k)N (µ); µ) = F2(ζn; µ) ∀n = 1, . . . ,N

2N

∑j=1

Qa

∑q=1

Θqa Aq

ij u(k)Nj +

N

∑l=1

Qb

∑q=1

ΘqbBq

il p(k)Nl +

2N

∑j=1

2N

∑s=1

Qc

∑q=1

u(k−1)Nj Θq

c C qij (σ s )u

(k)Nj =

QF1

∑q=1

ΘfqF q

1,i , 1≤ i ≤ 2N

2N

∑j=1

Qb

∑q=1

ΘqbBq

jl u(k)Nj =

QF2

∑q=1

Θgq F q

2,l , 1≤ l ≤N

2N

∑j=1

Qa

∑q=1

Θqa Aq

ij u(k)Nj +

N

∑l=1

Qb

∑q=1

ΘqbBq

il p(k)Nl +

2N

∑j=1

2N

∑s=1

Qc

∑q=1

u(k−1)Nj Θq

c C qij (σ s )u

(k)Nj =

QF1

∑q=1

ΘfqF q

1,i , 1≤ i ≤ 2N

2N

∑j=1

Qb

∑q=1

ΘqbBq

jl u(k)Nj =

QF2

∑q=1

Θgq F q

2,l , 1≤ l ≤N





Ni ,φN

j )


AN = ∑Qaq=1 Θq


b(µ)Bq , CN (u(k−1)N ) = ∑

2Ns=1 u

(k−1)Nj ∑

Qcq=1 Θq

c (µ)C q(σ s )

and solve, ∀k > 0

AN u

(k)N +BN p

(k)N +CN (u

(k−1)N )u

(k)N = F1,N

BN u(k)N = F2,N

(3N×3N)





a(u,w; µ) =Qa

∑q=1


Qb

∑q=1



(0)N ), ∀k > 0

a(u

(k)N (µ),σn; µ) + b(p

(k)N (µ),σn; µ) + c(u

(k−1)N (µ),u

(k)N (µ),σn; µ) = F1(σn; µ) ∀n = 1, . . . ,2N

b(ζn,u(k)N (µ); µ) = F2(ζn; µ) ∀n = 1, . . . ,N

2N

∑j=1

Qa

∑q=1

Θqa Aq

ij u(k)Nj +

N

∑l=1

Qb

∑q=1

ΘqbBq

il p(k)Nl +

2N

∑j=1

2N

∑s=1

Qc

∑q=1

u(k−1)Nj Θq

c C qij (σ s )u

(k)Nj =

QF1

∑q=1

ΘfqF q

1,i , 1≤ i ≤ 2N

2N

∑j=1

Qb

∑q=1

ΘqbBq

jl u(k)Nj =

QF2

∑q=1

Θgq F q

2,l , 1≤ l ≤N

2N

∑j=1

Qa

∑q=1

Θqa Aq

ij u(k)Nj +

N

∑l=1

Qb

∑q=1

ΘqbBq

il p(k)Nl +

2N

∑j=1

2N

∑s=1

Qc

∑q=1

u(k−1)Nj Θq

c C qij (σ s )u

(k)Nj =

QF1

∑q=1

ΘfqF q

1,i , 1≤ i ≤ 2N

2N

∑j=1

Qb

∑q=1

ΘqbBq

jl u(k)Nj =

QF2

∑q=1

Θgq F q

2,l , 1≤ l ≤N





Ni ,φN

j )


AN = ∑Qaq=1 Θq


b(µ)Bq , CN (u(k−1)N ) = ∑

2Ns=1 u

(k−1)Nj ∑

Qcq=1 Θq

c (µ)C q(σ s )

and solve, ∀k > 0

AN u

(k)N +BN p

(k)N +CN (u

(k−1)N )u

(k)N = F1,N

BN u(k)N = F2,N

(3N×3N)



Reduced Basis Method: Certification

A posteriori error estimation for the Stokes/Navier-Stokes case2

(‖uN (µ)−uN (µ)‖2V +‖pN (µ)−pN (µ)‖2

Q )1/2 ≤∆N (µ)

Stokes case [Rovas 03, R., Huynh, Manzoni 10]

∆N (µ) =‖rS (·; µ)‖X′

β LBS (µ)

rS (w; µ) = F (W ; µ)−AS (UN (µ),W ; µ)

0 < βLBS (µ)≤ βS (µ) = inf

Y∈X Nsup

W∈X N

AS (Y ,W ; µ)

‖Y ‖X ‖W ‖X

Navier-Stokes case: Brezzi-Rappaz-Raviart theory [Patera, Veroy 05, R. Deparis 09]

∆N (µ) =β LB

NS (µ)

ρ2(µ)

(1−

√1− τN (µ)

) τN (µ) =2ρ2(µ)‖rNS (·; µ)‖X ′

(β LBNS (µ))2

, ρ(µ) =√

2 supv∈V N

‖v‖L4(Ω)

‖v‖V N

rNS (w; µ) = F (W ; µ)−ANS (UN (µ),W ; µ)

0 < βLBNS (µ)≤ βNS (µ) = inf

V∈X Nsup

W∈X N

dANS (V ,W ;UN (µ); µ)

‖V ‖X ‖W ‖X

being

AS (Y ,W ; µ) = a(u,w; µ)+b(p,w; µ)+b(q,u; µ)

ANS (Y ,W ; µ) = AS (Y ,W ; µ)+c(u,u,w; µ)F (W ; µ) = F1(w; µ)+F2(q,µ)

2LB by SCM [Huynh, R., Sen, Patera, 2007] Successive Constraint Method (and variants).


Reduced Basis Method: Certification

A posteriori error estimation for the Stokes/Navier-Stokes case2

(‖uN (µ)−uN (µ)‖2V +‖pN (µ)−pN (µ)‖2

Q )1/2 ≤∆N (µ)

Stokes case [Rovas 03, R., Huynh, Manzoni 10]

∆N (µ) =‖rS (·; µ)‖X′

β LBS (µ)

rS (w; µ) = F (W ; µ)−AS (UN (µ),W ; µ)

0 < βLBS (µ)≤ βS (µ) = inf

Y∈X Nsup

W∈X N

AS (Y ,W ; µ)

‖Y ‖X ‖W ‖X

Navier-Stokes case: Brezzi-Rappaz-Raviart theory [Patera, Veroy 05, R. Deparis 09]

∆N (µ) =β LB

NS (µ)

ρ2(µ)

(1−

√1− τN (µ)

) τN (µ) =2ρ2(µ)‖rNS (·; µ)‖X ′

(β LBNS (µ))2

, ρ(µ) =√

2 supv∈V N

‖v‖L4(Ω)

‖v‖V N

rNS (w; µ) = F (W ; µ)−ANS (UN (µ),W ; µ)

0 < βLBNS (µ)≤ βNS (µ) = inf

V∈X Nsup

W∈X N

dANS (V ,W ;UN (µ); µ)

‖V ‖X ‖W ‖X

being

AS (Y ,W ; µ) = a(u,w; µ)+b(p,w; µ)+b(q,u; µ)

ANS (Y ,W ; µ) = AS (Y ,W ; µ)+c(u,u,w; µ)F (W ; µ) = F1(w; µ)+F2(q,µ)

2LB by SCM [Huynh, R., Sen, Patera, 2007] Successive Constraint Method (and variants).


Reduced Basis Method: the “complete game”

Offline stage involves precomputation of FE structures required for the RB space

construction and the certified error estimates.

Online stage has complexity only depending on N and allows evaluation of

solution/output for any µ ∈D with a certified error bound.


Application in Haemodynamics - IFluid-Structure Interaction and Inverse Problems

A normal, healthy artery and a stenosed vessel. The wall thickens as a result of the accumulation of cholesterol (atherosclerosis).

At first, as the plaques grow, only wall thickening occurs without any narrowing.

Stenosis is a late event, which is often the result of repeated plaque rupture and healing responses.


Fluid-Structure Interaction (FSI) problems

With variables (U,ηf ,ηs ) for the fluid solution and displacements of the fluid and

structure domain respectively, the fluid-structure interaction problem is

F (U,ηs ,ηf ) = 0 Fluid

S(U,ηs ) = 0 Structure

G(ηs ,ηf ) = 0 Geometry

The coupling conditions for FSI areηs −ηf = 0 on Γ, geometric continuity

σf ·nf + σs ·ns = 0 on Γ, balance of normal forces.

Both F and S are typically nonlinear operators

Geometric variables ηf ,ηs introduce another strong nonlinearity

Reassembly of matrices corresponding to F and S is required at each subiteration

due to mesh motion when using fully implicit approach

Reduction strategy for FSI problems

The fluid equation is defined in a parametrized domain Ωo (π) and solved through

the RB method

The fluid displacement ηf (π) is described through a FFD parametrization and

recovered from the structural displacement η by solving a parameter

identification problem



With variables (U,ηf ,ηs ) for the fluid solution and displacements of the fluid and

structure domain respectively, the fluid-structure interaction problem is

F (U,ηs ,ηf ) = 0 Fluid

S(U,ηs ) = 0 Structure

G(ηs ,ηf ) = 0 Geometry

The coupling conditions for FSI areηs −ηf = 0 on Γ, geometric continuity

σf ·nf + σs ·ns = 0 on Γ, balance of normal forces.

Both F and S are typically nonlinear operators

Geometric variables ηf ,ηs introduce another strong nonlinearity

Reassembly of matrices corresponding to F and S is required at each subiteration

due to mesh motion when using fully implicit approach

Reduction strategy for FSI problems

The fluid equation is defined in a parametrized domain Ωo (π) and solved through

the RB method

The fluid displacement ηf (π) is described through a FFD parametrization and

recovered from the structural displacement η by solving a parameter

identification problem



Structure equation3 + coupling conditionsε

∂ 4η

∂x41

−kGh∂ 2η

∂x21

+Eh

1−ν2P

η

R0(x1)2= τΓw (u,p), x1 ∈ (0,L)

η(0) = η(L) = 0

η′(0) = η

′(L) = 0

ηf = η , τΓw (u,p) = (σ(u,p) ·n) ·n, on Γw

being

h = wall thickness, k = Timoshenko shear correction factor

G = shear modulus, E = Young modulus,

νP = Poisson ratio, R0 = radius of the reference configuration

τΓw (u,p) = traction applied to the wall by the fluid inside the domain Ωo (π)

FSI-coupled problem

Find (u,p,η) ∈ X (Ωo (π))×Q(Ωo (π))×H(0,L) s.t.a(u,w) + b(p,w) + c(u,u,w) = F1(w) ∀w ∈ X (Ωo (π))

b(q,u) = F2(q) ∀q ∈Q(Ωo (π))

S(η ,φ) = 〈τΓw (u,p),φ〉 ∀φ ∈D(Γw )

where

S(η ,φ) = ε

∫ L

0

∂ 2η

∂x21

∂ 2φ

∂x21

dx1 + kGh∫ L

0

∂η

∂x1

∂φ

∂x1dx1 +

Eh

1−ν2P

∫ L

0

ηφ

R0(x1)2dx1

Weak coupling algorithm (sequential parameter identification)

At each iteration k (until convergence) we must solve the following “inverse problem”: find

πk+1 giving the best fit (in least square sense) to the structural displacement η∗ = η∗(·,πk )

πk+1 := argmin

π

∫Γw

|ηf (·, π)−η∗(·,πk )|2 dΓ

to find the configuration of the fluid domain Ωo (µk+1) at the next iteration, where

ηf (·; π) = T (·; π)−T (·;0) is given by FFD (no further equations to solve)

η∗ = η∗(·,πk ) is the structural displacement (related to load computed in Ωo (πk ))

3Generalized 1D string model for arterial wall [Q., Tuveri, Veneziani 00]Existence by fixed point argument + regularity assumption [Grandmont 98]




∂ 4η

∂x41

−kGh∂ 2η

∂x21

+Eh

1−ν2P

η

R0(x1)2= τΓw (u,p), x1 ∈ (0,L)

η(0) = η(L) = 0

η′(0) = η

′(L) = 0


being





FSI-coupled problem


b(q,u) = F2(q) ∀q ∈Q(Ωo (π))

S(η ,φ) = 〈τΓw (u,p),φ〉 ∀φ ∈D(Γw )

where

S(η ,φ) = ε

∫ L

0

∂ 2η

∂x21

∂ 2φ

∂x21

dx1 + kGh∫ L

0

∂η

∂x1

∂φ

∂x1dx1 +

Eh

1−ν2P

∫ L

0

ηφ

R0(x1)2dx1




πk+1 := argmin

π

∫Γw

|ηf (·, π)−η∗(·,πk )|2 dΓ








∂ 4η

∂x41

−kGh∂ 2η

∂x21

+Eh

1−ν2P

η

R0(x1)2= τΓw (u,p), x1 ∈ (0,L)

η(0) = η(L) = 0

η′(0) = η

′(L) = 0


being





FSI-coupled problem


b(q,u) = F2(q) ∀q ∈Q(Ωo (π))

S(η ,φ) = 〈τΓw (u,p),φ〉 ∀φ ∈D(Γw )

where

S(η ,φ) = ε

∫ L

0

∂ 2η

∂x21

∂ 2φ

∂x21

dx1 + kGh∫ L

0

∂η

∂x1

∂φ

∂x1dx1 +

Eh

1−ν2P

∫ L

0

ηφ

R0(x1)2dx1




πk+1 := argmin

π

∫Γw

|ηf (·, π)−η∗(·,πk )|2 dΓ






Example: a reduced model for FSI in a stenosed artery

Parametrization and RB space construction

Fluid domain displacement is performed with FFD by using a 12×2 grid of

control points; only the 8 central points on the upper row can move freely in the

x2-direction, with πi ∈ [−0.25,0.25] for i = 1,2, . . . ,8 (P=8 parameters)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Number of FE dof Nf ≈ 35,000

Number of RB functions N 8

Error tolerance RB εRBtol 5×10−2

Affine components Q 106

Error tolerance EIM εEIMtol 1×10−4

Nonlinear system reduction 1,500

FE fluid simulation tFE ≈ 20′

RB fluid simulation tonlineRB ≈ 3.2′′

1 2 3 4 5 6 7 810−2

10−1

100

N

∆ N(µ

)

Greedy RB construction



Parameter identification for FSI coupling: each inner minimization problem is

solved by means of sequential quadratic programming

The solution of a coupled FSI problem takes about 15 RB fluid solution/output

evaluations (≈ 50s)

Reynolds number: Re ≈ 80

Vorticity and velocity streamlines of the flow in a rigid stenosed artery (RB simulation)

In the case that the artery is rigid does not deform; the stenosis induces a strong

double vortex downstream, resulting in an area of low wall shear stress

immediately after the stenosed part

The shape of the upper wall has a very strong effect on the type of vortices

created and ultimately the potential growth of the stenosis



To explore the uncertainty related to the arterial wall properties we define the

uncertainty parameters ω = (G ,E) as the shear modulus and Young modulus, where

G ∈ [0.2,1.7] ·106dyn/cm2, E ∈ [0.35,1.85] ·106dyn/cm2.

To measure the effect of the uncertainty in the wall properties we look at four

different output functionals:

the total viscous energy dissipation

J1(u) =ν

2

∫Ωo

|∇u|2 dΩo ,

the minimum downstream shear rate

J2(u) = ν minx∈[γs ,L]

∂u(x ,y)

∂y

∣∣∣y=0

,

the mean downstream shear rate

J3(u) =ν

|L− γs |

∫ L

γs

∂u(x ,y)

∂y

∣∣∣y=0

dx ,

the mean pressure drop in the stenosed section

J4(u) =∫ H

0p(0,y) dy −

∫ 1

0p(L,y) dy .



Vorticity and velocity streamlines

of the steady incompressible

Navier-Stokes flow in compliant

stenosed artery (RB simulation)

for four different elastic moduli

values

1. The more compliant the arterial wall is, the larger is the recirculation region

created behind the stenosis

2. The effect of Young modulus E is considerably larger than the effect of shear

modulus G on the outputs (at least for a simpified structural model)



0 0.5 1 1.5 20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Young modulus E

She

ar m

odul

us G

Viscous energy dissipation J1

0.445

0.45

0.455

0.46

0.465

0.47

0.475

0.48

0.485

0 0.5 1 1.5 20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Young modulus E

She

ar m

odul

us G

Minimum downstream shear rate J2

−0.66

−0.64

−0.62

−0.6

−0.58

−0.56

−0.54

0.5 1 1.5 20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Young modulus E

She

ar m

odul

us G

Mean downstream shear rate J3

0.4

0.41

0.42

0.43

0.44

0.45

0 0.5 1 1.5 20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Young modulus E

She

ar m

odul

us G

Mean pressure drop J4

14.8

15

15.2

15.4

15.6

15.8

16

16.2

16.4

16.6

16.8

3. The stiffer the arterial wall, the more dissipation is observed: atherosclerotic stiff

arteries are in fact at greater risk of stenosis occurrence.

4. The minimum shear rate depends primarily on E and behaves in a nonlinear way.

RB comp. times 70 mins vs. FEM comp. times 40 hours


Applications in Haemodynamics - IIShape design and optimization of cardiovascular geometries


Shape optimization of femoro-popliteal bypass grafts

Shape optimization of cardiovascular geometries helps to avoid

post-surgical complications

Thickening caused by atherosclerosis is the most important

cause of disease in bypass grafting

Blood vorticity and/or viscous energy dissipation are related

to artery occlusion risk and highly depends on bypass shape

[Previous study case: aorto-coronaric bypass grafts, Stokes flows]

Shape Optimization problem:

minπ∈Dπ

Jo (π) =∫

Ωo (π)|∇u|2dΩo s.t.

−ν∆u + (u ·∇)u + ∇p = f in Ωo (π)

∇ ·u = 0 in Ωo (π)

u = ug on Γow

−p ·n + ν∂u

∂n= 0 on Γo

out

Steady NS flow: moderate velocity, mid-size arteries

Minimization of the viscous energy dissipation

Acknowledgement: P. Crosetto, CMCS-EPFL (pictures and mesh)




FFD: 6×4 grid of control points, only 6 chosen points can move freely in the


Optimization requires 260 seconds and 42 RB Online field/output evaluations

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5

−0.4

−0.2

0

0.2

0.4

0.6

Number of FE dof N ≈ 16,000





Nonlinear system reduction 250



−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

0.1

0.2

0.3

0.4

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

−0.15

−0.1

−0.05

0

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

10

20

30

40

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

10

20

30

40

Optimized bypass anastomosis and Navier-Stokes flow (velocity magnitude and pressure)

Velocity gradient (squared magnitude) for the unperturbed (left) and optimal (right) configuration

Reduction of viscous energy dissipation = 16.3%




FFD: 6×4 grid of control points, only 6 chosen points can move freely in the


Optimization requires 260 seconds and 42 RB Online field/output evaluations

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5

−0.4

−0.2

0

0.2

0.4

0.6

Number of FE dof N ≈ 16,000





Nonlinear system reduction 250



−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

0.1

0.2

0.3

0.4

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

−0.15

−0.1

−0.05

0

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

10

20

30

40

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

10

20

30

40

Optimized bypass anastomosis and Navier-Stokes flow (velocity magnitude and pressure)

Velocity gradient (squared magnitude) for the unperturbed (left) and optimal (right) configuration

Reduction of viscous energy dissipation = 16.3%



How to choose the (moderate number of) shape parameters?

Build a trial parametrization which can effectively explore all the parametric

variability of the PDE system (e.g. FFD with all the control points activated).

Build (iteratively) a reduced parameter set D ′ which enables to well explore the

parametric variability of the functional J(U(µ)).

Experimental Design based on parametric sensitivities

1 Define parametric variations around a reference value µ ∀p = 1, . . . ,P

µp,min∗ = [ µ1, . . . ,µ

minp , . . . , µP ], µ

p,max∗ = [ µ1, . . . ,µ

maxp , . . . , µP ]

2 Evaluate the µ-component snapshots by computing the FE approximations

UN (µp,min∗ ) and UN (µ

p,max∗ ) ∀p = 1, . . . ,P

3 Alternative 1: Select the parameter µp with the largest parametric sensitivity

D ′→D ′ ∪µp := argmax|J(UN (µ

p,max∗ ))−J(UN (µ

p,min∗ ))|

|µp,max∗ −µ

p,min∗ |

Alternative 2: Perform a greedy procedure on a train sample of FE solutions

and select the parameter µp which maximizes the correlation w.r.t. the set of

µ-component snapshots



Comparison of parameters chosen by the different strategies

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5

−0.4

−0.2

0

0.2

0.4

0.6

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5

−0.4

−0.2

0

0.2

0.4

0.6

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5

−0.4

−0.2

0

0.2

0.4

0.6

Empirical Choice

Parametric Sensitivities

Greedy Procedure



Comparison of parameters chosen by the different strategies

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

0.1

0.2

0.3

0.4

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

−1 −0.5 0 0.5 1 1.5 2 2.5 3

−0.4

−0.2

0

0.2

0.4

0.6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Empirical Choice

initial value J(µ(0)) = 0.429

optimal value J(µ) = 0.359

reduction 16.3%

Parametric Sensitivities



reduction 41.4%

Greedy Procedure



reduction 46.3%



Is the optimal bypass graft robust w.r.t. residual flows in the occlusion?

Robust Shape Optimization problem:

minπ∈Dπ

maxω∈Dω

Jo (µ) =∫

Ωobs (π)|∇×u|2dΩo

s.t.

−ν∆u + (u ·∇)u + ∇p = f in Ωo (π)

∇ ·u = 0 in Ωo (π)

u = uin on Γin

u = uc (ω) on Γc

u = 0 on Γw

−p ·n + ν∂u

∂n= 0 on Γout

If the occlusion in the artery is total, some

vortices may be generated around the

bypass-graft anastomosis, yielding to a

possible re-occlusion after surgery

The residual flow is modelled as a

parametrized Dirichlet BC, where ω ∈ [0,8]

is the uncertain amplitude of the flow

Shape parametrization: FFD + selection of control points (greedy procedure)

RB construction: as in the previous case ...



Optimal shape (velocity stream lines and vorticity) for ω = ωmin (left) and ω = ωmax (right)

Initial shape (velocity stream lines and vorticity) for ω = ωmin (left) and ω = ωmax (right)

Robust optimization requires 1750 s and 400 RB Online field/output evaluations

Optimal shape is robust wrt the presence/magnitude of the residual flow: no

vortices are created around the anastomosis even in absence of residual flow



Optimal shape (velocity stream lines and vorticity) for ω = ωmin (left) and ω = ωmax (right)

Initial shape (velocity stream lines and vorticity) for ω = ωmin (left) and ω = ωmax (right)

Robust optimization requires 1750 s and 400 RB Online field/output evaluations

Optimal shape is robust wrt the presence/magnitude of the residual flow: no

vortices are created around the anastomosis even in absence of residual flow



ω = ωmin = 0 ω = ωmax = 8

0 2 4 6 8237.8

238

238.2

238.4

238.6

238.8

239

239.2

239.4

239.6

239.8

ω

J 4

Local vorticity (robust optimal shape)

output evaluations cpu time output reduction

Shape Optimization 42 240s 46.3%

Robust Shape Optimization 400 1750s 49.2%

Optimal shape is not sensible (∆J < 1%) if varying the amplitude of residual flow

A reduced order model is necessary when performing robust shape optimization


Applications in Haemodynamics - IIITowards geometrical reconstruction and real-time blood flow simulation

Different carotid bifurcation specimens obtained by autopsy (adults aged 30-75).

Picture taken from Z. Ding et al., Journal of Biomechanics 34 (2001), 1555-1562.


Real-time blood flow simulation

Vessels geometry strongly influences haemodynamics behaviour

Study the influence of the vessel shape on blood flow

Real-time evaluation of flow indexes related with geometry variation

that assess/measure arteries occlusion risk (e.g. vorticity, viscous

energy dissipation) [MQR11]

Output evaluation problem:

evaluate Jo (π) =∫

Ωo (π)|∇u|2dΩo s.t.

−ν∆u + (u ·∇)u + ∇p = f in Ωo (π)

∇ ·u = 0 in Ωo (π)

u = ug on Γow := ∂Ωo \Γout ,

−p ·n + ν∂u

∂n= 0 on Γout

A case of interest: carotid artery bifurcation (e.g. in presence of stenosis)

Shape reconstruction through parameter identification

Shape sensitivity analysis



Shape reconstruction through parameter identificationGiven a target shape, find the closest configuration in a set of parametrized shapes

Parameter identification problem

µ = arg minπ∈Dad

E(π)

being

E(µ) =nr

∑j=1

‖T (xrj ;π)−yr

j ‖2 + β

nc

∑i=1

‖T (xci ;π)−yc

i ‖2

and

xci

nci=1 control points; xr

j nrj=1 registration points

yci

nci=1 target control points; yr

j nri=1 target registration points

Shape Parametrization: RBF, polynomial kernel (φ(r) = r3), Pπ = 7 input parameters(displacements of • control points)

Assumption: scaled/translated shapes, surrogate target data

(random FFD perturbation of a reference shape)



2 4 6 8 10 12 14 1610

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

N

∆

N(µ)

error (min)

error (max)

error (average)

Number of FE dof Nv +Np 24046


Number of design variables Pπ 7

Nonlinear system dimension reduction 500:1

FE evaluation tFE (s) 217.76

RB evaluation tonlineRB (s) 2.31

• Error estimation and • true error RB vs. FE approximation

Shape reconstruction

t = 5.35s

RB flow simulation

t = 2.31s

Output evaluation

t = 1.54s

−6 −4 −2 0 2 4 6

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Computational times are obtained as an average over 50 shape reconstructions/RB Online evaluations



−6 −4 −2 0 2 4 6

−2

−1

0

1

2

−6 −4 −2 0 2 4 6

−2

−1

0

1

2

−6 −4 −2 0 2 4 6

−2

−1

0

1

2

Reconstructed and target shapes, velocity and pressure fields, vorticity and viscous stresses



Shape sensitivity analysis and output evaluation

−6

−4

−2

02

46

−1.

5

−1

−0.

50

0.51

1.5

00.05

0.1

0.15

0.2

0.25

0.3

−6

−4

−2

02

46

−1.

5

−1

−0.

50

0.51

1.5

−

0.68

39

−0.

342

0

−6

−4

−2

02

46

−1.

5

−1

−0.

50

0.51

1.5

00.1

0.2

0.3

0.4

0.5

−6

−4

−2

02

46

−1.

5

−1

−0.

50

0.51

1.5

−

1.2

−1

−0.

8

−0.

6

−0.

4

−0.

2

0

Blood flows in different stenosed parametrized geometries

Shape Parametrization: RBF, Gaussian kernel (φ(r) = exp(−Ch2)), Pπ = 4 inputparameters (horizontal displ. of • control points)

Each RB online evaluation take about 2.5 seconds



Shape sensitivity analysis and output evaluation

!!

!"

!#

$#

"!

!#

!%&'!%

!$&'$

$&'%

%&'#

dc

db

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Viscous energy dissipation sN(d

c,d

b)

dc

d b

0.065

0.07

0.075

0.08

0.085

0.09

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

dc

RB error bound ∆N(d

c,d

b)

d b

1

2

3

4

5

6

7

8

x 10−4

(Left) Viscous energy dissipation in 1000 configurations w.r.t. diameters dc = dc (π1 ,π2) and db = db (π3 ,π4)

(Right) A posteriori estimation ∆N (µ) of the error between the RB field solution and the corresponding FE solutions

Flow disturbances caused by stenoses lead to higher values of the dissipated

energy, the maximum occurring for the smallest diameters on both sections

[MQR11]

Simple response surface or nonlinear regression models may be introduced to

explain the parametric dependence


Conclusions

Although very simple (no flow pulsatility, steady flow, ...) the reduced model

presented allows to characterize (almost) in real-time blood flows in complex

geometries, by capturing several features related e.g. to vessel shape, fluid

behavior and structural parameters.

Coupling geometrical reduction and computational reduction proves to be

necessary for optimal shape design and more in general for inverse problems

related with shape variation.

The robust shape optimization, as well as the exploration of the parameter space

for the FSI problem, took around 1 hour of computational time by using the RB

method for the fluid simulation.

The same problem with the full FEM simulation would have taken around 30-40

hours of computational time – and considerably more if shape deformations had

been handled by standard methods and not taking advantage of the geometrical

reduction afforded by the shape parametrization.


References

LM+11 T. Lassila, A. Manzoni, A. Quarteroni, G. Rozza. A reduced computational and geometrical framework forinverse problems in haemodynamics, in preparation, 2011.

LQR10 T. Lassila, A. Quarteroni, G. Rozza. Reduced Formulation of Steady Fluid-structure Interaction withParametric Coupling, submitted, 2010.

LR10 T. Lassila, G. Rozza. Parametric free-form shape design with PDE models and reduced basis method,Comput. Methods Appl. Mech. Engrg., 199: 1583–1592, 2010.

MQR10 A. Manzoni, A. Quarteroni, G. Rozza. Shape optimization of cardiovascular geometries by reduced basismethods and free-form deformation techniques, submitted, 2011.

MQR11 A. Manzoni, A. Quarteroni, G. Rozza. Model reduction techniques for fast blood flow simulation inparametrized geometries, accepted for publication in Int. J. Num. Meth. Biomed. Engrg., 2011.

QMR11 A. Quarteroni, G. Rozza, A. Manzoni. Certified reduced basis approximation for parametrized PDEsin industrial applications, J. Math. Ind., 3(1), 2011.

RHM10 G. Rozza, D.B.P. Huynh, A. Manzoni. Reduced basis approximation and a posteriori error estimation forStokes flows in parametrized geometries: roles of the inf-sup stability constants, submitted, 2010.

RHP08 G. Rozza, D.B.P. Huynh, A.T. Patera. Reduced basis approximation and a posteriori error estimation foraffinely parametrized elliptic coercive PDEs. Arch. Comput. Methods Engrg.,15: 229–275, 2008.

For more information see

http://sma.epfl.ch/∼rozza

http://cmcs.epfl.ch/

http://augustine.mit.edu

Documents

Geometrical and computational -0.1cm reduction strategies ...Gianluigi Rozza in collaboration with Al o Quarteroni, Andrea Manzoni, Toni Lassila MATHICSE - CMCS Modelling and Scienti