Model Reduction Methods

Transcription

1 Martin A. Grepl 1, Gianluigi Rozza 2 1 IGPM, RWTH Aachen University, Germany 2 MATHICSE - CMCS, Ecole Polytechnique Fédérale de Lausanne, Switzerland Summer School "Optimal Control of PDEs" Cortona (Italy), July 12-17, 21

2 rbmit: - Reduced Basis Software Library in Matlab(C) Environment developed at MIT by DBP Huynh, CN Nguyen, G Rozza and AT Patera based on extensive use of Matlab ToolBoxes like Symbolic, PDEs, Optimization The user must describe the problem. The input can be separated into three parts: The User Input geometry: Ω o(µ) is defined by providing points coordinates, straight/curvy edges describing all regions Ω k o(µ) material properties: coefficients are provided for differential operator in each region Ω k o(µ) and for boundary conditions. parameter control and settings: parameter domain D, reference parameters and other RB information (e.g. N max)

3 rbmit Users Interface Geometry Parameters PDE and BC Output

4 Problem Formulation, Offline and Online Steps The Problem Formulation Step Domain Decomposition/geometric transformations: coefficients Θ q(µ) are generated for each sub-domain (coupled with material µ-properties) FE mesh is generated, discrete FE stiffness matrices/vectors are assembled for each sub-domain to form the µ-independent components The RB Offline Step By a greedy algorithm the RB parameter sample set is obtained FE/RB matrices are saved in order to be used by the Online Step The RB Online Step Given µ D, the RB Online Evaluator returns output and error bound Online_RB (probname, µ, outputname,...): µ s N N (µ), s N (µ) The RB Visualizer renders the field variable(s) and provides error bounds Vis_RB (probname, µ): µ Ω, u N N (x; µ) for all x in Ω o(µ)

5 Problem Formulation, Offline and Online Steps The Problem Formulation Step Domain Decomposition/geometric transformations: coefficients Θ q(µ) are generated for each sub-domain (coupled with material µ-properties) FE mesh is generated, discrete FE stiffness matrices/vectors are assembled for each sub-domain to form the µ-independent components The RB Offline Step By a greedy algorithm the RB parameter sample set is obtained FE/RB matrices are saved in order to be used by the Online Step The RB Online Step Given µ D, the RB Online Evaluator returns output and error bound Online_RB (probname, µ, outputname,...): µ s N N (µ), s N (µ) The RB Visualizer renders the field variable(s) and provides error bounds Vis_RB (probname, µ): µ Ω, u N N (x; µ) for all x in Ω o(µ)

6 Problem Formulation, Offline and Online Steps The Problem Formulation Step Domain Decomposition/geometric transformations: coefficients Θ q(µ) are generated for each sub-domain (coupled with material µ-properties) FE mesh is generated, discrete FE stiffness matrices/vectors are assembled for each sub-domain to form the µ-independent components The RB Offline Step By a greedy algorithm the RB parameter sample set is obtained FE/RB matrices are saved in order to be used by the Online Step The RB Online Step Given µ D, the RB Online Evaluator returns output and error bound Online_RB (probname, µ, outputname,...): µ s N N (µ), s N (µ) The RB Visualizer renders the field variable(s) and provides error bounds Vis_RB (probname, µ): µ Ω, u N N (x; µ) for all x in Ω o(µ)

7 rbmit Example: the Thermal Fin problem Engineering aspects Heat sink designed for cooling of high-density electronic components Shaded domain due to assumed periodicity and symmetry (multi-fin sink) Flowing air is modelled though a simple convection HT coefficient: to compute temperature at the base of the spreader Physical and geometrical parametrization µ 1 = Bi = h c dper/ κ fin Biot number µ 1 [.1,.5] µ 2 = L = L/ d per nondimensional fin height µ 2 [2, 8] µ 3 = κ = κ sp/ κ fin spreader/fin conductivity ratio µ 3 [1, 1]

8 rbmit Example: the Thermal Fin problem Modeling: temperature u o(µ) over Ω o(µ) satisfies the steady heat equation Output: average temperature over the base of the spreader (component to be cooled, being the hottest location in the system)» «µ3 u x o i µ 3 x o j o(µ) = in Ω 1 o {z } κ 1 o ij» «1 u x o i 1 x o j o(µ) = in Ω 2 o (µ 2) {z }, µ2, µ , µ2 3 2, µ2 2 κ 2 o ij n o i κ 1 u o o ij (µ) = 1 on Γ x o 1 o j n o i κ 2 u o o ij (µ) + (µ x 1 )u o = on Γ R = Γ o 5 Γ 6 o j n o i κ o ij u x o j o(µ) = on Γ \ (Γ o 1 Γ R ) Z output: T o av(µ) = 2 u o(µ) Γ o 1, 3 5 (, ) 3 2, , ,

9 rbmit Example: the Thermal Fin problem Example of geometry and field visualizations provided by rbmit package Example of initial geometry, domain decompostion, FE mesh and RB solution visualization for a thermal fin problem

10 rbmit Example: the Thermal Fin problem Approximation property # of mesh nodes N 4198 # of RB functions N 1 Reduced Basis vs Finite Elements RB online.13s (N = 7) evaluation time.15s (N = 13) FEM sol. µ s N (µ) 1.96s Reduction of 4:1 in linear system dimension Online evaluation 5 6% of the FEM computational cost μ1 RB output/error bars -[s N (µ) s N (µ), s N (µ) + s N (µ)] as a function of µ 1 for µ 2 = 2, µ 3 = 1 and N = 6. RB temperature field for different choices of parameters: µ = (.5, 2, 1), µ = (.5, 2, 5), µ = (.1, 2, 1).

11 Geometrical Parametrization: the goal The parametrized (original) domain Ω o (µ) is the image of a fixed (reference) domain Ω through a map T ( ; µ) : Ω Ω o (µ) In order to recover the affine parameter dependence, the parametric map T ( ; µ) has to be an affine map. The rbmit software allows to deal with more complex configurations by means of automatically built affine mappings

12 Geometrical Parametrization: Domain Decomposition Domain decomposition: definition Original Domain Ω o (µ), u e o Xe o (Ω o(µ)) Ω o (µ) = K dom k=1 Ωk o (µ) ; Reference domain Ω, u e X e (Ω) Ω = K dom k=1 Ωk, common configuration where Ω = Ω o (µ ref ) for µ ref D. For Ω k, Ω k o (µ) we choose in R2 triangles, elliptical triangles and curvy triangles. In R 3 we choose parallelepipeds (and in theory tetrahedra). Connectivity requirement: subdomain intersections must be an entire edge, a vertex, or null.

13 Geometrical Parametrization: Domain Decomposition Domain decomposition: definition Original Domain Ω o (µ), u e o Xe o (Ω o(µ)) Ω o (µ) = K dom k=1 Ωk o (µ) ; Reference domain Ω, u e X e (Ω) Ω = K dom k=1 Ωk, common configuration where Ω = Ω o (µ ref ) for µ ref D. For Ω k, Ω k o (µ) we choose in R2 triangles, elliptical triangles and curvy triangles. In R 3 we choose parallelepipeds (and in theory tetrahedra). Connectivity requirement: subdomain intersections must be an entire edge, a vertex, or null.

14 Geometrical Parametrization: Affine Mappings Require where Ω k o (µ) = T aff,k (Ω k ; µ), 1 k K dom, T aff,k (x; µ) = C aff,k (µ) + G aff,k (µ)x, µ D is an invertible affine mapping from Ω k onto Ω k o (µ). Further require T aff,k (x; µ) = T aff,k (x; µ), x Ω k Ω k, 1 k, k K dom, µ D to ensure a continuous piecewise-affine global mapping T aff ( ; µ) from Ω onto Ω o (µ). It follows that for w o H 1 (Ω o(µ)), w o T aff = H 1 (Ω).

15 Geometrical Parametrization: Affine Mappings Require where Ω k o (µ) = T aff,k (Ω k ; µ), 1 k K dom, T aff,k (x; µ) = C aff,k (µ) + G aff,k (µ)x, µ D is an invertible affine mapping from Ω k onto Ω k o (µ). Further require T aff,k (x; µ) = T aff,k (x; µ), x Ω k Ω k, 1 k, k K dom, µ D to ensure a continuous piecewise-affine global mapping T aff ( ; µ) from Ω onto Ω o (µ). It follows that for w o H 1 (Ω o(µ)), w o T aff = H 1 (Ω).

16 Geometrical Parametrization: Affine Mappings Elliptical Triangles: definition Outwards: Inwards: { }} { O(µ) = (x cen o1, xcen o2 ( )T ) cos φ(µ) sin φ(µ) Q rot (µ) = sin φ(µ) cos φ(µ) S(µ) = diag(ρ 1 (µ), ρ 2 (µ))

17 Geometrical Parametrization: Affine Mappings Elliptical Triangles: constraints Given x 2 o (µ), x3 o (µ), find x1 o (µ), x4 o (µ) ( T aff,1&2 ) } (i) produce desired elliptical arc µ D; (ii) satisfy internal angle criterion these conditions ensure continuous invertible mappings. Explicit recipes for admissible x 1 o(µ) (Inwards case) and x 4 o(µ) (Outwards case) are readily obtained.

18 Geometrical Parametrization: Affine Mappings Elliptical Triangles: example (CinS triangulation) Ω o (µ) : µ = (µ 1, µ 2,...) D [.8, 1.2] 2...

19 Affine Mappings Elliptical Triangles: example (CinS triangulation) Ω = Ωo (µref = (1, 1)) Ωo (µ = (.8, 1.2)) 1 2

20 Geometrical Parametrization: Affine Mappings Curvy Triangles: definition Outwards: Inwards: { }} { O(µ) = (x cen o1, xcen o2 ( )T ) cos φ(µ) sin φ(µ) Q rot (µ) = sin φ(µ) cos φ(µ) S(µ) = diag(ρ 1 (µ), ρ 2 (µ))

21 Geometrical Parametrization: Affine Mappings Curvy Triangles: constraints Given x 2 o (µ), x3 o (µ), find x1 o (µ), x4 o (µ) ( T aff,1&2 ) } (i) produce desired curvy arc µ D; (ii) satisfy internal angle criterion these conditions ensure continuous invertible mappings. Quasi-explicit recipes for admissible x 1 o (µ) and x4 o (µ) can (sometimes) be obtained in the convex/concave case.

22 Geometrical Parametrization: Affine Mappings Elliptical Triangles: example (Cosine triangulation) (say) Ω o (µ) : µ = (µ 1,...) D [ 1 6, 1 2 ]...

23 Geometrical Parametrization: Affine Mappings Elliptical Triangles: example (Cosine triangulation) Ω = Ω o (µ ref = 1 3 ) Ω o(µ = 1 2 )

24 Geometrical Parametrization: Tensor Transformations Transformation: Formulation on original domain (IR 2 ) For w, v H 1 (Ω o (µ)) a o (w, v; µ) = Kdom k=1 Ω ko (µ) [ w x o1 w x o2 w u e o (µ) H1 (Ω o(µ)) ] K k oij (µ) v x o1 v x o2 v where K k o : D R3 3, SPD for 1 k K dom (note K k o affine in x o is also permissible). We consider the scalar case; the vector case (linear elasticity) admits an analogous treatment.

25 Geometrical Parametrization: Tensor Transformations Transformation: Formulation on reference domain For w, v H 1 (Ω) u e (µ) H 1 (Ω) a(w, v; µ) = Kdom k=1 Ω k [ w x 1 w x 2 w ] K k ij (µ) v x 1 v x 2 v K k (µ) = det G aff,k (µ) D(µ)K k o (µ)dt (µ), and D(µ) = (Gaff,k ) 1. 1

26 Geometrical Parametrization: Tensor Transformations Transformation: Affine form Expand a(w, v; µ) = K 1 11 }{{ (µ) w v +... } Ω 1 x 1 x }{{ 1 } Θ 1 (µ) a 1 (w,v) We can often greatly reduce the requisite Q. with as many as Q = 4K terms. Achtung! Many interesting problems are not affine (or require Q very large). For example, K k o (x; µ) for general x dependence; and nonzero Neumann conditions on curvy Ω yield non-affine a(, ; µ).

27 Geometrical Parametrization: Tensor Transformations Transformation: Affine form Expand a(w, v; µ) = K 1 11 }{{ (µ) w v +... } Ω 1 x 1 x }{{ 1 } Θ 1 (µ) a 1 (w,v) We can often greatly reduce the requisite Q. with as many as Q = 4K terms. Achtung! Many interesting problems are not affine (or require Q very large). For example, K k o (x; µ) for general x dependence; and nonzero Neumann conditions on curvy Ω yield non-affine a(, ; µ).

28 RB Simulation / Output Evaluation Geometries, Software: Geometrical Problems Flow Control and Optimal Design with RB Methods Basic parametrized geometries Automatic Geometrical Parametrization: Basic setup: AirfoilAirfoil of the NACAExample 4-digits family (symmetric c I in parallel flow Airfoil of the NACA 4-digits family (symmetric case) Ω L u µ1 x2 Thickness distribution 4-digits profile x2 = µ1 ( Ω x x1.126x1.352x x1.121x Airfoil geometry description for the rbmit software.1.2 x1 x2 «x1 x2 «= = 1 «« ±µ1 /2 «±µ1 /2 « t2.2969t.126t2.352t t6.121t8! 1 t2.2969t.126t2.352t t6.121t8!.8 1, t [,, t [.3, 1] Complexity (potential flow): # of subdomains K = 42/84 # of bilinear forms Q = 16/2.3]

29 RB Simulation / Output Evaluation Geometries, Software: Geometrical Problems Flow Control and Optimal Design with RB Methods Basic parametrized geometries Automatic Geometrical Parametrization: Basic setup: AirfoilAirfoil of the NACAExample 4-digits family (symmetric c I in parallel flow Airfoil of the NACA 4-digits family (symmetric case) Ω L u µ1 x2 Thickness distribution 4-digits profile x2 = µ1 ( Ω x x1.126x1.352x x1.121x Airfoil geometry description for the rbmit software.1.2 x1 x2 «x1 x2 «= = 1 «« ±µ1 /2 «±µ1 /2 « t2.2969t.126t2.352t t6.121t8! 1 t2.2969t.126t2.352t t6.121t8!.8 1, t [,, t [.3, 1] Complexity (potential flow): # of subdomains K = 42/84 # of bilinear forms Q = 16/2.3]

30 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Potential Flows: Output Evaluation Parameters: thickness µ 1 [4, 24], angle of attack µ 2 [.1, π/16] and profiles distance (µ 3, µ 4) [.85, 1] [1.2, 1.6] Output: pressure coefficient c p = p p in 1 2 ρ u in 2 around the profiles

31 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Thermal Flows: Output Evaluation Thermal Flows around an airfoil with ground effect included Physical and geometrical parametrization: thickness µ 1 [4, 24], ground distance µ 2 [1.5, 3], Peclet number µ 3 [1, 1] The thermal boundary layer on the ground and on profile becomes thinner and more separated for higher Peclet numbers

32 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Potential/Thermal Flows: Computational Costs Potential Flow mesh nodes N 3693 subdomains 163 affine operator components Q a 12 affine rhs components Q f 3 reduced basis functions N 2 t online F E 17.15s t online RB.8s speedup 195 RB online time as funcion of N (potential flow) Thermal Flow mesh nodes N 6727 subdomains 88 affine operator components Q a 42 reduced basis functions N pr 7 reduced basis functions N du 61 t online F E 16.97s t online RB.26s speedup 65 Average outflow temperature w.r. to ground distance µ 2 (thermal flow)

33 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Potential Flow: Shape Optimization Problem Airfoil inverse design problem Z 1 1/2 min p(s, µ) p target(s) ds«2 + λ ˆα(µ) 5 2, µ D Z Z s.t. u v dω o = fv dω o v H 1 (Ω o(µ)) Ω o(µ) u = on Γ out, Ω o(µ) u u = 1 on Γin, n Choose target airfoil (ex: NACA4412) and compute pressure distribution p target on its surface using the Bernoulli equation (p = p 1 2 u 2 ) Objective: find small perturbation of reference airfoil NACA12 s.t. pressure distribution on the airfoil surface is close to p target Add penalty term to enforce the constraint on the angle of attack (AOA = 5 ) n = elsewhere

34 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Potential Flow: Shape Optimization Problem Geometrical parametrization: Free-Form Deformation Techniques Geometrical parameters µ 1,..., µ P are chosen as the perturbations of a (small) lattice of FFD control points

35 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Potential Flow: Shape Optimization Problem Pressure distributions and computational cost (online solution of the parametric PDE) Finite element method Reduced basis method CPU time (seconds) Real time barrier (response time < 1 s) 1 1 (a) Inverse design (b) Target airfoil Number of mesh nodes x 1 4 (c) Computational costs Number of mesh nodes N 843 Lattice of FFD control points 6 4 Number of shape parameters 8 Number of reduced basis functions N 52 Error tolerance for RB greedy ε RB tol 1 4 Number of affine expansion terms Q a 8 Error tolerance for EIM greedy ε EIM tol Results from T. Lassila, G. Rozza, Comput. Methods Appl. Mech. Engrg. 199 (21) Reduction of 5:1 in parametric complexity compared to explicit nodal deformation Reduction of 2:1 in linear system dimension

36 Thermal Flows: Optimal Design of Airfoils RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Optimal heat exchange problem» u target 1 min µ D s.t. Z Ω o(µ) Z 2 u dγ + λ [α(µ) α ] 2, Γ out Γ out ε u v + v Z b u dω o = fv dω o u = on Γout, u = T on Γin Γfree, n u = T 1 on Γ surf, u = T 2 on airfoil Objective: find airfoil shape and vertical position s.t. average temperature over outflow equals u target and angle of attack equals α Heat exchange of an airfoil in exterior flow with b = [1; ] and ε =.2 is considered Penalty term enforces the constraint on the angle of attack (AOA = α ) Ω o(µ)

37 Thermal Flows: Optimal Design of Airfoils RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods α = 7, u target = 4.1 α = 5, u target = 4.5 Number of mesh nodes N Lattice of FFD control points 6 6 Number of shape parameters 8 Number of reduced basis functions N 36 Error tolerance for RB greedy ε RB tol 1 5 Number of affine expansion terms Q a 18 Error tolerance for EIM greedy ε EIM tol 1 4 Reduction of 1:1 in parametric complexity compared to explicit nodal deformation Reduction of 436:1 in linear system dimension Results from G. Rozza, T. Lassila, A. Manzoni, Proc. of Icosahom Conference, 29, in press

38 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Environmental Flows: Control of Air Pollution Control of Air pollution min J(µ) = y(u(µ u)) z d µ D ZD 2 dω, o(µ g) Z s.t. (ν(µ p) y v + V(µ p) y) dω o = Ω o(µ g) y = on ΓN, n y = on ΓD Goal: to regulate the pollutant emission by industrial plants in order to keep the pollutant level below a fixed threshold over an observation area Parameters: control input µ u, which define the control function u = u(µ u); physical input µ p (e.g. viscosity, advection velocity); geometrical input µ g (domain configuration) Z Ω o(µ g) u(µ p)v dω o

39 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Environmental Flows: Control of Air Pollution Control input µ u : variable emission rate Use of reduced basis method for the efficient solution of the state and the adjoint parametrized problems Results from A. Quarteroni, G. Rozza, A, Quaini, Advances in Numerical Mathematics, 27, p

40 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Environmental Flows: Control of Air Pollution Physical Input µ p : wind direction Use of reduced basis method for the efficient solution of the state and the adjoint parametrized problems Results from A. Quarteroni, G. Rozza, A, Quaini, Advances in Numerical Mathematics, 27, p

41 RB Simulation / Output Evaluation Flow Control and Optimal Design with RB Methods Environmental Flows: Control of Air Pollution Geometrical Input µ g : parametrized domain Use of reduced basis method for the efficient solution of the state and the adjoint parametrized problems Results from A. Quarteroni, G. Rozza, A, Quaini, Advances in Numerical Mathematics, 27, p