Reduced basis method for the reliable model reduction of Navier-Stokes equations in cardiovascular modelling

Transcription

1 Reduced basis method for the reliable model reduction of Navier-Stokes equations in cardiovascular modelling Toni Lassila, Andrea Manzoni, Gianluigi Rozza CMCS - MATHICSE - École Polytechnique Fédérale de Lausanne In collaboration with Alfio Quarteroni (EPFL & Politecnico di Milano) Supported by the ERC-Mathcard Project (ERC-2008-AdG ), the Swiss National Science Foundation (Project ), and the Emil Aaltonen Foundation Model Reduction for Complex Dynamical Systems, TU Berlin, December 2-4, 2010

2 1 Introduction 2 Navier-Stokes equations 3 Reduced basis approximation 4 Application in cardiovascular modelling

3 Challenges in modelling the human cardiovascular system Human cardiovascular system is a complex flow network of different spatial and temporal scales. When investigating fluid flow processes the flow geometries are changing over time. The geometric variation causes a strong nonlinearity in the equations. Medical professionals are interested in accurate simulation of spatial quantities, such as wall shear stresses at the location of a possible pathology. Computational costs can become unacceptably high, especially if the objective is to model the entire network, and strategies to reduce numerical efforts and model order are being developed.

4 Parametric incompressible Navier-Stokes equations (steady case) We consider the following model problem: For a given parameter vector µ D R P, find U(µ) X s.t. a(u(µ),v ; µ) = f (V ; µ) V X, µ D where U := (u,p) and V := (v,q) consist of the velocity field and the pressure, the product space X = V Q [H 1 (Ω)] 2 L 2 (Ω), and the problem consists of a linear part a 0 and a nonlinear (quadratic in U) part a 1 : a(u,v ; µ) := a 0 (U,V ; µ) + a 1 (U,U,V ; µ) U,V X, µ D

5 Parametric incompressible Navier-Stokes equations (steady case) We consider the following model problem: For a given parameter vector µ D R P, find U(µ) X s.t. a(u(µ),v ; µ) = f (V ; µ) V X, µ D where U := (u,p) and V := (v,q) consist of the velocity field and the pressure, the product space X = V Q [H 1 (Ω)] 2 L 2 (Ω), and the problem consists of a linear part a 0 and a nonlinear (quadratic in U) part a 1 : a(u,v ; µ) := a 0 (U,V ; µ) + a 1 (U,U,V ; µ) U,V X, µ D For example, if the parameter is simply µ = ν (fluid viscosity), we have a 0 (U,V ; µ) = [µ u : v p div(v) q div(u)] dω Ω a 1 (U,W,V ) = v (u )w dω Ω + appropriate boundary conditions.

6 Parametric incompressible Navier-Stokes equations (steady case) We consider the following model problem: For a given parameter vector µ D R P, find U(µ) X s.t. a(u(µ),v ; µ) = f (V ; µ) V X, µ D where U := (u,p) and V := (v,q) consist of the velocity field and the pressure, the product space X = V Q [H 1 (Ω)] 2 L 2 (Ω), and the problem consists of a linear part a 0 and a nonlinear (quadratic in U) part a 1 : a(u,v ; µ) := a 0 (U,V ; µ) + a 1 (U,U,V ; µ) U,V X, µ D For example, if the parameter is simply µ = ν (fluid viscosity), we have a 0 (U,V ; µ) = [µ u : v p div(v) q div(u)] dω Ω a 1 (U,W,V ) = v (u )w dω Ω + appropriate boundary conditions. Typically we are interested in linear functionals of the field solutions (outputs) s(µ) := l(u(µ)), i.e. need to find a reduced model s(µ) that has is within certified tolerance of the actual outputs: s(µ) s(µ) < TOL for all µ D.

7 Finite element approximation to the Navier-Stokes solution Starting from initial guess U 0, solve at each step k of a FP iteration for U k s.t. until convergence. a 0 (U k,v ; µ) + a 1 (U k 1,U k,v ) = f (V ) V V Q Stable discretization with P 2 /P 1 FE spaces for velocity and pressure V h := {v C(Ω,R d ) : v K [P 2 (K)] 2, K T h } V Q h := {q C(Ω,R) : q K P 1 (K), K T h } Q. Galerkin projection in FE space: solve at each step k for U k h s.t. until convergence. a 0 (U k h,v h; µ) + a 1 (U k 1 h,u k h,v h) = f (V h ) V h V h Q h Similar approach for the Newton s method...

8 Reduced basis approximation of the finite element solution 1 Assumption: parametric manifold of FE solutions M h X h is 1) low dimensional and 2) depends smoothly on µ (valid for small Reynolds number) 2 Choose a representative set of parameter values µ 1,..., µ N 3 Snapshot solutions u h (µ 1 ),...,u h (µ N ) span a subspace V N h for the velocity and p h (µ 1 ),...,p h (µ N ) span a subspace Q N h for the pressure 4 Galerkin reduced basis: given µ D, find U N h (µ) X N h s.t. a 0 (U k,n h,vh N ; µ) + a 1(U k 1,N h,uh k,v h N ) = f (V h N ) V h N X h N 5 Adaptive sampling procedure (greedy algorithm) for the choice of µ 1,..., µ N M = {U(µ) X ; µ D} M h = {U h (µ) X h ; µ D}

9 Reduced basis approximation of the finite element solution 1 Assumption: parametric manifold of FE solutions M h X h is 1) low dimensional and 2) depends smoothly on µ (valid for small Reynolds number) 2 Choose a representative set of parameter values µ 1,..., µ N 3 Snapshot solutions u h (µ 1 ),...,u h (µ N ) span a subspace V N h for the velocity and p h (µ 1 ),...,p h (µ N ) span a subspace Q N h for the pressure 4 Galerkin reduced basis: given µ D, find U N h (µ) X N h s.t. a 0 (U k,n h,vh N ; µ) + a 1(U k 1,N h,uh k,v h N ) = f (V h N ) V h N X h N 5 Adaptive sampling procedure (greedy algorithm) for the choice of µ 1,..., µ N Reliability / accuracy? 1 is based on the quality of the sampling 2 relies on computable and rigorous a posteriori error estimator N (µ): M = {U(µ) X ; µ D} M h = {U h (µ) X h ; µ D} X N h = span{u h(µ i ), i = 1,...,N} U h (µ) U N h (µ) X N (µ), s(µ) s N (µ) s N (µ) = l X h N(µ)

10 A posteriori error estimation of the reduced basis approximation (Veroy-Patera 2005) If τ N (µ) < 1 and β h (µ) > 0 there exists a unique solution U h (µ) s.t. Here: U h (µ) Uh N (µ) X N (µ) =: β [ h(µ) 1 ] 1 τ ρ(µ) N (µ) β h (µ) is the Babuska inf-sup constant that needs to be estimated da(u h (µ); µ)(w,v ) inf sup = β h (µ) > β 0 > 0 W X h V X h W V for the Fréchet derivative of a(u,w,v ) w.r.t first argument at U h ρ(µ) is a Sobolev embedding constant that needs to be estimated τ N (µ) := 2ρ(µ)ε N (µ) β h (µ) 2, where ε N (µ) := f ( ; µ) a(uh N, ; µ) X h is the RB residual

11 A posteriori error estimation of the reduced basis approximation (Veroy-Patera 2005) If τ N (µ) < 1 and β h (µ) > 0 there exists a unique solution U h (µ) s.t. Here: U h (µ) Uh N (µ) X N (µ) =: β [ h(µ) 1 ] 1 τ ρ(µ) N (µ) β h (µ) is the Babuska inf-sup constant that needs to be estimated da(u h (µ); µ)(w,v ) inf sup = β h (µ) > β 0 > 0 W X h V X h W V for the Fréchet derivative of a(u,w,v ) w.r.t first argument at U h ρ(µ) is a Sobolev embedding constant that needs to be estimated τ N (µ) := 2ρ(µ)ε N (µ) β h (µ) 2, where ε N (µ) := f ( ; µ) a(uh N, ; µ) X h is the RB residual Note: for large viscosity we obtain the Stokes equations and the estimator simplifies to U h (µ) Uh N (µ) N(µ) = ε } N(µ) key ingredients β h (µ)

12 Generalization of reduced basis method to parametric geometries Parametrized formulation on fixed reference domain (Rozza 2009) Evaluate output of interest s(µ) = l(u h (µ)) s.t. U h = (u h (µ),p(µ)) V h ( Ω) Q h ( Ω) solves a(u h (µ),v h ; µ) = f (V h ; µ) V h X ( Ω) v a((v,p),(w,q); µ) = ν ij (x, µ) w dω pχ ij (x, µ) w j dω qχ ij (x, µ) v j dω, Ω x i x j Ω x i Ω x i The parametrized (original) domain Ω(µ) is the image of a reference domain Ω through a parametric mapping T ( ; µ) : Ω Ω(µ) One possible parametrization using free-form deformations (L.-Rozza 2009) Transformation tensors (J T = J T (x, µ) = Jacobian of T (x, µ)) ν(x, µ) = J 1 T νo J T T J T and χ(x, µ) = J 1 T χo J T Problem reduced to a parametric PDEs system on Ω (reference domain)

13 Reduced basis offline/online computational framework Offline stage involves precomputation of structures required for the certified error estimate and choice of the reduced basis functions. Online stage has complexity only depending on N and allows evaluation of output s(µ) for any µ D with a certified error bounds.

14 Shape Optimization of Aorto-Coronaric Bypass Grafts Shape optimization of cardiovascular geometries helps to avoid post-surgical complications Local fluid patterns (vorticity) and wall shear stress are strictly related to the development of cardiovascular diseases

15 Shape Optimization of Aorto-Coronaric Bypass Grafts Shape optimization of cardiovascular geometries helps to avoid post-surgical complications Local fluid patterns (vorticity) and wall shear stress are strictly related to the development of cardiovascular diseases Shape optimization problem [Agoshkov-Quarteroni-Rozza 2006] minj(ω; v) s.t. ν v + p = f in Ω v = 0 in Ω v = v g on Γ D := Ω \ Γ out, p n + ν v n = 0 on Γ out J o (Ω;v) = v 2 dω, Ω df J o (Ω;v) = ν v Ω n tdγ o Pictures taken from: Lei et al., J Vasc. Surg. 25(1997), ; Loth et al., Annu. Rev. Fluid Mech. 40(2008),

16 Shape optimization of aorto-coronaric bypass grafts A possible free-form deformation approach [Manzoni-Quarteroni-Rozza 2010] Several analyses show a deep impact of the graft-artery diameter ratio Φ and anastomotic angle α on shear stress and vorticity distributions Oscillatory shear stress with different graft-artery diameter ratios Φ and anastomotic angles α. Picture taken from F.L. Xiong, C.K. Chong, Med. Eng. & Phys. 30(2008),

17 Shape optimization of aorto-coronaric bypass grafts A possible free-form deformation approach [Manzoni-Quarteroni-Rozza 2010] Several analyses show a deep impact of the graft-artery diameter ratio Φ and anastomotic angle α on shear stress and vorticity distributions Oscillatory shear stress with different graft-artery diameter ratios Φ and anastomotic angles α. Picture taken from F.L. Xiong, C.K. Chong, Med. Eng. & Phys. 30(2008), In order to get a low-dimensional FFD parametrization we need to maximize the influence of the control points by placing them close to the sensitive regions 8 parameters (7 vertical and 1 horizontal displacements) to control the anastomotic angle, the graft-artery diameter ratio, the upper side, the lower wall

18 Shape optimization of aorto-coronaric bypass grafts! N (µ) RB approximation space construction RB space dimension N Number of FE dof N v + N p Lattice FFD control points P i,j 5 6 Number of design variables P 8 Number of RB functions N 22 Error tolerance RB greedy εtol RB Affine operator components Q 222 Error estimation (energy norm) for RB space construction (greedy procedure) and selected snapshots µ µ 1 µ 2 µ 3 µ 4 µ 6 µ 7 µ Reduction in linear system dimension 500:1 Computational speedup (single flow simulation) 107 Reduction in parametric complexity w.r.t. explicit nodal deformation 102:1

19 Shape optimization of aorto-coronaric bypass grafts Vorticity Minimization (downfield region) Automatic iterative minimization procedure (sequential quadratic programming) Vorticity evaluation by using the reduced basis velocity at each step [Manzoni-Quarteroni-Rozza 2010] Optimized bypass anastomosis and Stokes flow (velocity magnitude and pressure) Optimal (black) and unperturbed (grey) configurations of FFD parametrization Vorticity magnitude for the unperturbed (left) and optimal (right) configuration

20 Reduction of time-dependent Navier-Stokes (ongoing work) One-dimensional prototype problem (viscous Burgers equation): find u(µ) L 2 (0,T ;V ) C 0 ([0,T ];L 2 (Ω)) s.t. d dt Ω (u(µ),v) + µ u x (µ)v x dx 1 2 u 2 (µ)v x dx = f (v) Ω where Ω = (0,1) and V H 1 (Ω). v V Time discretization with implicit Euler = time-discrete equations Spatial discretization with FEM + reduced basis reduction as before Stability constant ρ N := inf v V da(u N,k h )(v,v) not necessarily positive! v X A posteriori estimator (Nguyen-Rozza-Patera 2009) for k = 1,..., t T ( ) uh k (µ) un,k h (µ) k t µ N (µ) := k m=1 εn 2 (tm ; µ) m 1 j=1 (1 + tρ N(t j ; µ)) k m=1 (1 + tρ N(t m ; µ)) BUT the error bound grows exponentially in time

21 Conclusions Reduced basis methods a reliable MOR method for parametric PDEs Parameters can also describe the (variable) flow geometry Certified error bounds for spatial outputs of reduced field variables Extensions to noncoercive (Stokes) and nonlinear (Navier-Stokes) cases Future work Time-dependent Navier-Stokes, improved error estimates Reduction of coupled multiphysics problems Parameter identification and inverse problems

22 References V.I. Agoshkov, A. Quarteroni and G. Rozza. Shape design in aorto-coronaric bypass using perturbation theory. SIAM J. Num. Anal. 44(1), , D.B.P. Huynh, D.J. Knezevic, Y. Chen, J.S. Hesthaven, and A.T. Patera. A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Engrg. 199: , T. L. and G. Rozza. Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Engrg. 199(23-24): , A. Manzoni, A. Quarteroni, G. Rozza. Shape optimization for viscous flows by reduced basis methods and free-form deformation, J. Comp. Phys., submitted, A. Quarteroni and G. Rozza. Numerical solution of parametrized Navier-Stokes equations by reduced basis methods. Numer. Methods Partial Differential Equations, 23(4): , N.-C. Nguyen, G. Rozza and A.T. Patera. Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers equation. Calcolo 46(3): , G. Rozza, D.B.P. Huynh, A.T. Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive PDEs. Arch. Comput. Methods Engrg.,15: , G. Rozza and K. Veroy. On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Engrg., 196(7): , K. Veroy and A.T. Patera. Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations; rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47(89), , 2005.