g TU Munich, 16-20 September 2013, RB Summer School An introduction to geometrical parametrizations for the applications of reduced order modelling: learning by examples FUNDAMENTALS [RHP , 2008, ARCME, Vol.15, 229-275] Gianluigi Rozza Collaboration Network MOX (A. Quarteroni, F. Ballarin, P . Pacciarini) EPFL (T. Lassila, F. Negri, P . Chen, D. Forti) MIT ( A.T. Patera, D.B.P. Huynh, C.N. Nguyen) SISSA (A. Manzoni, D. Devaud), U. Konstanz (L. Iapichino) 22.02.13 SISSA MathLab Tuesday, September 10, 2013
Outline Simple Elliptic µ PDEs: Setting Problem Scope: Geometry Problem Scope: Bilinear Forms Working Examples: TBlock AMass EBlock3D Rozza G. Certified Reduced-Basis Methods 1
Statement Simple Elliptic µ PDEs R P , Given µ ∈ D ⊂ I evaluate s e ( µ ) = ℓ ( u e ( µ )) † where u e ( µ ) ∈ X e (Ω) satisfies ∀ v ∈ X e . a ( u e ( µ ) , v ; µ ) = f ( v ) , † Here e refers to “exact.” Rozza G. Certified Reduced-Basis Methods 3
Statement Simple Elliptic µ PDEs Definitions and . . . µ : input parameter; P -tuple D : parameter domain; s e : output; ℓ : linear bounded output functional; u e : field variable; 0 (Ω)) ν ⊂ X e ⊂ ( H 1 (Ω)) ν ; X e : function space ( H 1 Rozza G. Certified Reduced-Basis Methods 4
Statement Simple Elliptic µ PDEs . . . Hypotheses a ( · , · ; µ ): bilinear, continuous, symmetric, µ PDE coercive form, ∀ µ ∈ D ; f : linear bounded functional. COMPLIANT case: ℓ = f (and a symmetric). Rozza G. Certified Reduced-Basis Methods 5
Statement Simple Elliptic µ PDEs Affine Parameter Dependence † Definition: � Q Θ q ( µ ) a q ( w, v ) a ( w, v ; µ ) = q =1 where for q = 1 , . . . , Q Θ q : D → I µ - dependent functions ; R , a q : X e × X e → I R , µ - independent forms . † In fact, broadly applicable to many instances of property and geometry parametric variation. Rozza G. Certified Reduced-Basis Methods 6
FE Approximation Simple Elliptic µ PDEs Galerkin Projection R P , Given µ ∈ D ⊂ I evaluate s N ( µ ) = f ( u N ( µ )) † where u N ( µ ) ∈ X N ⊂ X e satisfies a N ∀ v ∈ X N . a ( u N ( µ ) , v ; µ ) = f ( v ) , † Here X N is a sequence of FE approximation spaces indexed by dim( X N ) = N . Rozza G. Certified Reduced-Basis Methods 7
FE Approximation Simple Elliptic µ PDEs Typical Triangulation 4 3 2 1 0 -1 -2 -3 -2 -1 0 1 2 3 Rozza G. Certified Reduced-Basis Methods 8
Goal Simple Elliptic µ PDEs For any ε des > 0 , evaluate ACCURACY µ ∈ D → s N N ( µ ) ( ≈ s N ( µ ) ) that provably achieves desired accuracy RELIABILITY | s N ( µ ) − s N N ( µ ) | ≤ ε des but at (very low) marginal cost ∂t comp † EFFICIENCY independent of N as N → ∞ . † ∂t comp : time to perform one additional certified evaluation µ → s N N ( µ ) . Rozza G. Certified Reduced-Basis Methods 9
Goal Simple Elliptic µ PDEs Relevance Real-Time Context (parameter estimation, . . . ): t 0 + ∂t comp : s N → t 0 : µ N ( µ ) . “need” “response” Many-Query Context (dynamic simulation, . . . ): t comp ( µ j → s N N ( µ j ) , j = 1 , . . . , J ) = ∂t comp J as J → ∞ . Rozza G. Certified Reduced-Basis Methods 10
Domain Decomposition Problem “Scope”: Geometry Definition u e o ∈ X e Original Domain Ω o ( µ ) , o (Ω o ( µ )) Ω o ( µ ) = � K dom k k =1 Ω o ( µ ) ; u e ∈ X e (Ω) reference domain Ω , Ω = � K dom k , k =1 Ω common configuration where Ω = Ω o ( µ ref ) for µ ref ⊂ D † . † Connectivity requirement: subdomain intersections must be an entire edge, a vertex, or null. Rozza G. Certified Reduced-Basis Methods 11
Domain Decomposition Problem “Scope”: Geometry Building Blocks o ( µ ) we choose in R 2 † , For Ω k , Ω k (Parallelograms — by hand); EBlock3D Triangles; Elliptical Triangles*; and Curvy Triangles*. † In R 3 , we choose Parallelepipeds (and in theory Tetrahedra). Rozza G. Certified Reduced-Basis Methods 12
Affine Mappings Problem “Scope”: Geometry Local ∀ µ ∈ D Require k k ; µ ) , 1 ≤ k ≤ K dom , o ( µ ) = T aff ,k (Ω Ω where T aff ,k ( x ; µ ) = C aff ,k ( µ ) + G aff ,k ( µ ) x , k onto Ω k is an invertible affine mapping from Ω o ( µ ) . Rozza G. Certified Reduced-Basis Methods 13
Affine Mappings Problem “Scope”: Geometry Global ∀ µ ∈ D Further require k ∩ Ω k ′ T aff ,k ( x ; µ ) = T aff ,k ′ ( x ; µ ) , ∀ x ∈ Ω , 1 ≤ k, k ′ ≤ K dom , 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) . Rozza G. Certified Reduced-Basis Methods 14
Elliptical Triangles Problem “Scope”: Geometry Definition Outwards: Inwards: � �� � O ( µ ) = ( x cen o1 , x cen 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 )) Rozza G. Certified Reduced-Basis Methods 15
Elliptical Triangles Problem “Scope”: Geometry Constraints Given x 2 o ( µ ) , x 3 o ( µ ) , find x 1 o ( µ ) , x 4 ( ⇒ T aff , 1&2 ) o ( µ ) � ( i ) produce desired elliptical arc ∀ µ ∈ D ; ( ii ) satisfy internal angle criterion 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. Rozza G. Certified Reduced-Basis Methods 16
Elliptical Triangles Problem “Scope”: Geometry Triangulation: ‘CinS’... Ω o ( µ ): µ = ( µ 1 , µ 2 , . . . ) ⊂ D ≡ [0 . 8 , 1 . 2] 2 × . . . Rozza G. Certified Reduced-Basis Methods 17
Elliptical Triangles Problem “Scope”: Geometry ...Triangulation: ‘CinS’ 2 2 4 4 1.5 1.5 14 11 11 14 24 29 10 13 1 10 13 1 24 29 23 30 25 31 25 31 23 30 28 0.5 0.5 28 26 32 26 32 33 33 2 2 3 3 0 0 12 27 12 27 34 34 17 19 17 19 −0.5 −0.5 9 9 15 22 18 20 15 22 18 20 16 21 −1 −1 6 8 6 8 16 21 5 7 −1.5 −1.5 5 7 1 1 −2 −2 −2 −1 0 1 2 −2 −1 0 1 2 Ω = Ω o ( µ ref = (1 , 1)) Ω o ( µ = (0 . 8 , 1 . 2)) Rozza G. Certified Reduced-Basis Methods 18
Curvy Triangles Problem “Scope”: Geometry Definition Outwards: Inwards: � �� � O ( µ ) = ( x cen o1 , x cen 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 )) Rozza G. Certified Reduced-Basis Methods 19
Curvy Triangles Problem “Scope”: Geometry Constraints Given x 2 o ( µ ) , x 3 o ( µ ) , find x 1 o ( µ ) , x 4 ( ⇒ T aff , 1&2 ) o ( µ ) � ( i ) produce desired curvy arc ∀ µ ∈ D ; ( ii ) satisfy internal angle criterion conditions ensure continuous invertible mappings. † Quasi-explicit recipes for admissible x 1 o ( µ ) and x 4 o ( µ ) can (sometimes) be obtained in the convex/concave case. Rozza G. Certified Reduced-Basis Methods 20
Curvy Triangles Problem “Scope”: Geometry Triangulation: ‘Cosine’... (say) Ω o ( µ ): µ = ( µ 1 , . . . ) ⊂ D ≡ [ 1 6 , 1 2 ] × . . . Rozza G. Certified Reduced-Basis Methods 21
Curvy Triangles Problem “Scope”: Geometry ...Triangulation: ‘Cosine’ 0.5 0.2 6 7 6 8 0 7 8 0 3 −0.2 5 3 −0.4 2 5 4 −0.5 −0.6 2 4 −0.8 1 1 −1 −1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Ω = Ω o ( µ ref = 1 Ω o ( µ = 1 3 ) 2 ) Rozza G. Certified Reduced-Basis Methods 22
Transformation Problem Scope: Bilinear Form R 2 ) Original Domain ( I For w, v ∈ H 1 (Ω o ( µ )) † u e o ( µ ) ∈ H 1 0 (Ω o ( µ )) ∂v ∂x o1 � � � K dom � ∂w ∂w ∂v K k a o ( w, v ; µ ) = ∂x o2 w o ij ( µ ) o ( µ ) Ω k ∂x o1 ∂x o2 k =1 v o : D → R 3 × 3 , SPD for 1 ≤ k ≤ K dom where K k (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. Rozza G. Certified Reduced-Basis Methods 23
Transformation Problem Scope: Bilinear Form Reference Domain For w, v ∈ H 1 (Ω) u e ( µ ) ∈ H 1 0 (Ω) ∂v � � � K dom � ∂x 1 ∂w ∂w ∂v K k a ( w, v ; µ ) = ∂x 2 w ij ( µ ) Ω k ∂x 1 ∂x 2 k =1 v K k ( µ ) = | det G aff ,k ( µ ) | D ( µ ) K k o ( µ ) D T ( µ ) , and 0 ( G aff ,k ) − 1 D ( µ ) = 0 . 0 0 1 Rozza G. Certified Reduced-Basis Methods 24
Transformation Problem Scope: Bilinear Form Affine Form Expand � ∂w ∂v a ( w, v ; µ ) = K 1 11 ( µ ) + . . . � �� � ∂x 1 ∂x 1 Ω 1 � �� � Θ 1 ( µ ) a 1 ( w,v ) with as many as Q = 4 K terms. We (Maple) can often greatly reduce the requisite Q . Rozza G. Certified Reduced-Basis Methods 25
Transformation Problem Scope: Bilinear Form 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 ) . Rozza G. Certified Reduced-Basis Methods 26
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