Geometrical and computational -0.1cm reduction strategies ...Gianluigi Rozza in collaboration with...
Transcript of Geometrical and computational -0.1cm reduction strategies ...Gianluigi Rozza in collaboration with...
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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 Geometrical reduction Computational Reduction FSI problems
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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Reduction strategies for simulation/optimization of viscous flows
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Inverse problems related with shape variation
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.
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Inverse problems related with shape variation
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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Inverse problems related with shape variation
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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Inverse problems related with shape variation
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)
minπ∈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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Inverse problems related with shape variation
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)
minπ∈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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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.
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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.
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Computational Reduction
Acknowledgement: Anthony T. Patera (MIT) - augustine.mit.edu
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
UN (µ) ∈ XN : A(UN (µ),V ; µ) = F (V ) ∀V ∈ XN
JN (µ) = s(UN (µ))
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
UN (µ) ∈ XN : A(UN (µ),V ; µ) = F (V ) ∀V ∈ XN
JN (µ) = s(UN (µ))
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
UN (µ) ∈ XN : A(UN (µ),V ; µ) = F (V ) ∀V ∈ XN
JN (µ) = s(UN (µ))
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
UN (µ) ∈ XN : A(UN (µ),V ; µ) = F (V ) ∀V ∈ XN
JN (µ) = s(UN (µ))
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Reduced Basis Method: approximation stability
The steady Navier-Stokes problem
Reduced Basis (RB) approximation
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.
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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).
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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).
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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.
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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.
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Fluid-Structure Interaction (FSI) problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Fluid-Structure Interaction (FSI) problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Fluid-Structure Interaction (FSI) problems
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]
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Example: a reduced model for FSI in a stenosed artery
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Example: a reduced model for FSI in a stenosed artery
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 .
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Example: a reduced model for FSI in a stenosed artery
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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Example: a reduced model for FSI in a stenosed artery
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Applications in Haemodynamics - IIShape design and optimization of cardiovascular geometries
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Example: a reduced model for FSI in a stenosed artery
Parametrization and RB space construction
FFD: 6×4 grid of control points, only 6 chosen points can move freely in the
x2-direction, with πi ∈ [−0.2,0.2] for i = 1,2, . . . ,6 (P=6 parameters)
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
Number of RB functions N 20
Error tolerance RB εRBtol 5×10−2
Affine components Q 104
Error tolerance EIM εEIMtol 1×10−4
Nonlinear system reduction 250
FE fluid simulation tFE ≈ 6′
RB fluid simulation tonlineRB ≈ 2.5′′
−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%
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Example: a reduced model for FSI in a stenosed artery
Parametrization and RB space construction
FFD: 6×4 grid of control points, only 6 chosen points can move freely in the
x2-direction, with πi ∈ [−0.2,0.2] for i = 1,2, . . . ,6 (P=6 parameters)
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
Number of RB functions N 20
Error tolerance RB εRBtol 5×10−2
Affine components Q 104
Error tolerance EIM εEIMtol 1×10−4
Nonlinear system reduction 250
FE fluid simulation tFE ≈ 6′
RB fluid simulation tonlineRB ≈ 2.5′′
−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%
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Shape optimization of femoro-popliteal bypass grafts
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Shape optimization of femoro-popliteal bypass grafts
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Shape optimization of femoro-popliteal bypass grafts
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
initial value J(µ(0)) = 0.434
optimal value J(µ) = 0.254
reduction 41.4%
Greedy Procedure
initial value J(µ(0)) = 0.428
optimal value J(µ) = 0.229
reduction 46.3%
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Shape optimization of femoro-popliteal bypass grafts
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 ...
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Shape optimization of femoro-popliteal bypass grafts
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Shape optimization of femoro-popliteal bypass grafts
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Shape optimization of femoro-popliteal bypass grafts
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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.
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Real-time blood flow simulation
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)
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Real-time blood flow simulation
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 RB functions N 16
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Real-time blood flow simulation
−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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Real-time blood flow simulation
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
Real-time blood flow simulation
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
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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.
Reduction Strategies Geometrical reduction Computational Reduction FSI problems
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