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Parametrized Partial Differential Equations Heat Transfer Back-of-the-Envelope Calculations: Model Simplification, Model Order Reduction Anthony T Patera, MIT Mathematics of Reduced Order Models ICERM Providence, RI, USA February 19, 2020

  1. The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Governing Equations: Dimensional Find [ V ≡ ( V 1 , V 2 , V 3 ) , T ]( x , t ) T s = T | Ω s , T f = T | Ω f ∂ V ∂ t + V · ∇ V = −∇ p + g β ( T f − T ∞ ) e 2 + ν ∇ 2 V in Ω f , t > 0 , ρ ∞ ∇ · V = 0 in Ω f , t > 0 , ∂ T f ∂ t + V · ∇ T f = α f ∇ 2 T f in Ω f , t > 0 , ∂ T s ∂ t = α s ∇ 2 T s in Ω s , t > 0 , n + ε r σ SB ( T 4 s − T 4 T s = T f , − k s ∇ T s · ˆ n = − k ∇ T f · ˆ ∞ ) on Γ sf , t > 0 , T s ( · , t = 0) = T i in Ω s , T f ( · , t = 0) = T ∞ in Ω f . 16/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  2. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Fluid Domain and Wall S Austin 2.51 17/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  3. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation HTC c : Definition Consider solid body B surrounded by fluid; define wall Γ w ≡ ∂ B . Given: wall Γ w approximately iso thermal at temperature T w ; fluid far from wall at temperature T ∞ (and quiescent). The spatial-averaged convection HTC c is defined as � Q w � η iso � ¯ c � [ T w ] ≡ | Γ w | ( T w − T ∞ ) for � Q w ≡ Γ w q w dS ≡ heat transfer rate from wall to fluid, q w ≡ heat flux from wall to fluid, | Γ w | ≡ the surface area of wall Γ w ; � · � ≡ steady-state or long-time-average operator. 18/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  4. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Newton’s Law of Cooling � η iso By construction: � Q w � ≡ Γ w q w dS = � ¯ c � [ T w ] · | Γ w | ( T w − T ∞ ). Heat flux q w : q w ≡ − k f ∇ T f · ˆ n Fourier’s Law (in fluid) ( T ∞ − T w ) δ bl : thermal boundary layer ; ≈ − k f δ bl ( x s ) but for laminar natural convection, δ bl depends weakly on x s , δ bl ( x s ) ∼ α 1 / 2 ( g β | T w − T ∞ | ) − 1 / 4 x 1 / 4 f hence η iso q w ≈ � ¯ c � [ T w ] · ( T w − T ∞ ) on Γ w uniform . 19/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  5. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Boundary Layer Visualization S Austin 2.51 Background-Oriented Schlieren � � δ bl ( x s ) ∼ α f t L-E ( x s ) = α f x s / U buoy ( x s ) � U buoy ( x s ) ∼ g β | T w − T ∞ | x s 20/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  6. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Solid and Fluid Domains 21/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  7. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Dirichlet-Neumann Map ⇒ Robin Condition Now assume T w is not known, but part of solution for T s in B . Boundary condition on solid body B : − k s ∇ T s · ˆ n = − k f ∇ T f · ˆ n (First Law) = q w η iso ≈ � ¯ c � [ T w ] · ( T w − T ∞ ). 22/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  8. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Dirichlet-Neumann Map ⇒ Robin Condition Now assume T w is not known, but part of solution for T s in B . Boundary condition on solid body B : − k s ∇ T s · ˆ n = − k f ∇ T f · ˆ n (First Law) = q w η iso ≈ � ¯ c � [ T f ] · ( T f − T ∞ ) 22/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  9. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Dirichlet-Neumann Map ⇒ Robin Condition Now assume T w is not known, but part of solution for T s in B . Boundary condition on solid body B : − k s ∇ T s · ˆ n = − k f ∇ T f · ˆ n (First Law) = q w η iso ≈ � ¯ c � [ T s ] · ( T s − T ∞ ) 22/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  10. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Dirichlet-Neumann Map ⇒ Robin Condition Now assume T w is not known, but part of solution for T s in B . Boundary condition on solid body B : − k s ∇ T s · ˆ n = − k f ∇ T f · ˆ n (First Law) = q w η iso ≈ � ¯ c � [ T s ] · ( T s − T ∞ ) if isothermal wall condition is approximately satisfied. Condition for approximately isothermal wall: either η iso Bi c [ T w ] (Biot Number) ≡ � ¯ c � [ T w ] L ≪ 1 , k s for L an appropriate length scale in solid body. Argument: k s (∆ T ) in B c � [ T w ] · ( T w − T ∞ ) ⇒ (∆ T ) in B η iso ≈ � ¯ | T w − T ∞ | ≪ 1 . L 22/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  11. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Dirichlet-Neumann Map ⇒ Robin Condition Now assume T w is not known, but part of solution for T s in B . Boundary condition on solid body B : − k s ∇ T s · ˆ n = − k f ∇ T f · ˆ n (First Law) = q w η iso ≈ � ¯ c � [ T s ] · ( T s − T ∞ ) if isothermal wall condition is approximately satisfied. Condition for approximately isothermal wall: or η iso Bi c [ T w ] (Biot Number) ≡ � ¯ c � [ T w ] L ≫ 1 , k s for L an appropriate length scale in solid body. Argument: k s (∆ T ) in B η iso ≈ � ¯ c � [ T w ] · ( T w − T ∞ ) ⇒ T w → T ∞ . L 22/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  12. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation HTC c : Measurement [11, 25] Given heat source Q source in solid body, measure wall temperature at several locations, { T w } , measure farfield fluid temperature, T ∞ , Q source c � [ T avg η iso evaluate � ¯ w ] = − T ∞ ) . | Γ w | ( T avg w Confirm condition for isothermal wall: c � [ T avg η iso w ] (Biot Number) ≡ � ¯ w ] L theory: Bi c [ T avg ≪ 1 ; k s experiment: std dev { T w } ≪ | T w − T ∞ | . 23/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  13. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation HTC c Functions: Experimental Correlations [35, 2] For given HTC c configuration: Introduce length scale associated with Γ w , B : ℓ . Form nondimensional groups: η iso � Nu ℓ � ≡ � ¯ c � [ T w ] ℓ ; k f ℓ ≡ g β | T w − T ∞ | ℓ 3 , Pr ≡ ν Ra w . α f ν α f Define parameter: µ ≡ (Ra w ℓ , Pr) ∈ P ⊂ R 2 + . Fit to data: F HTC c : µ ∈ P �→ � Nu ℓ � ∈ R + ; c � [ T w ] = k f η iso � ¯ ℓ � Nu ℓ � . 24/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  14. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Example: HTC c Correlation — Vertical Plate [35] Extension: orientation relative to gravity, ( θ g , ϕ g ). 25/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  15. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Example: HTC c Correlation — Horizontal Cylinder [35] 26/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  16. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Stefan-Boltzmann Law: Graybodies Wall flux: for convex body in large enclosure η iso q w = � ¯ c � [ T w ]( T w − T ∞ ) + ε r σ SB ( T 4 w − T 4 ∞ ) 27/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  17. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Stefan-Boltzmann Law: Graybodies Wall flux: for convex body in large enclosure η iso q w = � ¯ c � [ T w ]( T w − T ∞ ) + ε r σ SB ( T 2 w + T 2 ∞ )( T 2 w − T 2 ∞ ) 27/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  18. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Stefan-Boltzmann Law: Graybodies Wall flux: for convex body in large enclosure η iso q w = � ¯ c � [ T w ]( T w − T ∞ ) + ε r σ SB ( T 2 w + T 2 ∞ )( T w + T ∞ )( T w − T ∞ ) ; 27/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  19. The Dunk(ing) Problem Heat Transfer to Fluid Conjugate Framework Boundary Condition on Solid Body Convection Heat Transfer Coefficient Experimental Program Classical (HTC) Framework Incorporation of Radiation Stefan-Boltzmann Law: Graybodies Wall flux: for convex body in large enclosure η iso q w = � ¯ c � [ T w ]( T w − T ∞ ) + ε r σ SB ( T 2 w + T 2 ∞ )( T w + T ∞ )( T w − T ∞ ) ; ˜ η � �� � q w = ˜ (˜ η c + ˜ η r )( T w − T ∞ ) . η nlin ( T w ) Nonlinear Case : ˜ '' exact '' η nlin η iso η nlin = ε r σ SB ( T 2 w + T 2 ˜ = � ¯ c � [ T w ]; ˜ ∞ )( T w + T ∞ ). c r η lin ( T lin,c , T lin,r ) Linear(ized) Case : ˜ η lin η iso η lin = ε r σ SB ( T 2 lin,r + T 2 ˜ c = � ¯ c � [ T lin,c ]; ˜ ∞ )( T lin,r + T ∞ ). r where (say) T lin,c = T lin,r = T i . 27/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  20. The Dunk(ing) Problem Conjugate Framework Formulation Convection Heat Transfer Coefficient Small-Biot Regime Classical (HTC) Framework Motivation and Notation P Phan 2.51 28/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  21. The Dunk(ing) Problem Conjugate Framework Formulation Convection Heat Transfer Coefficient Small-Biot Regime Classical (HTC) Framework An Idealized Configuration Let Ω ⊂ R 3 , Ω = Ω s ∪ Ω f : Ω f ≡ fluid (air) domain: effectively infinite; Ω s ≡ solid domain: convex, (single, scale) parameter ℓ ; sf ≡ Ω s ∩ Ω f \ Γ ad sf ∪ Γ ad Γ s ; ∂ Ω s ≡ Γ s uniformly large enclosure: dist(Ω s , ∂ Ω) ≫ ℓ ; coordinate system: x ≡ ( x 1 , x 2 , x 3 ), { e i } i ; gravity g = − g e 2 . Initial conditions: T | Ω s ≡ T s = T i uniform, T | Ω f ≡ T f = T ∞ ; assume T i > T ∞ (wlog). Farfield conditions: quiescent fluid; T f = T ∞ (on ∂ Ω) — implicit. 29/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  22. The Dunk(ing) Problem Conjugate Framework Formulation Convection Heat Transfer Coefficient Small-Biot Regime Classical (HTC) Framework Governing Equations: Dimensional linear(ized) Temperature T s ( x , t ) satisfies ∂ T s ∂ t = α s ∇ 2 T s in Ω s , t > 0 , η lin ( T i , T i ) − k s ∇ T s · ˆ n = ˜ ( T s − T ∞ ) on ∂ Ω s ≡ Γ sf , t > 0 , � �� � � �� � Fourier’s Law HTC T s ( · , t = 0) = T i in Ω s . Dunk pPDE: M [1] [Ω geo ], geo ∈ { P , C , S } s � � µ [1] ≡ η lin , T ∞ , T i , t final ∈ P [1] geo , ℓ, α s , k s , ˜ �→ T s ( x , t ) , x ∈ Ω s , t ∈ (0 , t final ]; o = O [1] ( T s ) . Here O [1] is a linear bounded output functional. Remark Dimensional formulation for expositional convenience. 30/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  23. The Dunk(ing) Problem Conjugate Framework Formulation Convection Heat Transfer Coefficient Small-Biot Regime Classical (HTC) Framework Governing Equation η lin | Ω s | Let Bi dunk ≡ ˜ sf | . k s | Γ For Bi dunk ≪ 1, T s ( x , t ) ≈ ˆ T s ( t ) satisfies | Ω s | d ( ˆ k s T s − T ∞ ) η lin | Γ sf | ( ˆ + ˜ T s − T ∞ ) = 0 , α s dt subject to ( ˆ T s − T ∞ )( t = 0) = ( T i − T ∞ ). Dunk pPDE: M [1] [ − ] , geo = lumped µ [1] ≡ � � η lin , T ∞ , T i , t final ∈ P [1] geo , | Ω s | , | Γ sf | , k s , α s , ˜ �→ ˆ T s ( t ) , t ∈ (0 , t final ]; o = O [1] ( ˆ T s ) . Here O [1] is a linear output functional. Remark pMOR (parametrized Model Order Reduction). 31/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  24. The Fin Problem Classical Formulation Small-Biot Regime (Steady-State) Heat Transfer 101 the Fin Problem 34/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  25. The Fin Problem Classical Formulation Small-Biot Regime (Steady-State) Motivation and Notation skillethandle x 1 skilletpan root tip x 1 = L skillet 35/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  26. The Fin Problem Classical Formulation Small-Biot Regime (Steady-State) An Idealized Configuration Let Ω ⊂ R 3 , Ω = Ω s ∪ Ω f : Ω f ≡ fluid domain: effectively of infinite extent, ∂ Ω f = ∂ Ω; Ω s ≡ solid domain: Ω s ≡ Ω s − ( x 1 ≤ 0) ∪ Ω s+ ( x 1 ≥ 0); Ω s+ ≡ Right Cylinder { 0 < x 1 < L , ( x 2 , x 3 ) ∈ D cs } : D cs ≡ cross section: convex; area A cs , perimeter P cs ; ∂ Ω s+ ≡ Γ sr ∪ Γ sf ∪ Γ st : Γ sf ≡ ]0 , L [ × ∂ D cs , P cs L / A cs ≫ 1; uniformly large enclosure: dist(Ω s , ∂ Ω) ≫ ℓ ; coordinate system: x ≡ ( x 1 , x 2 , x 3 ), { e i } i ; gravity g = − g e 3 ; Farfield conditions: quiescent fluid; T f = T ∞ (on ∂ Ω) — implicit. Insulated Tip: − k s ∂ T s ∂ x 1 = 0 on Γ st , natural — implicit. 36/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  27. The Fin Problem Classical Formulation Small-Biot Regime (Steady-State) Temporal Stages Stage I. Steady-State: T ss s ( x ) estimate or measure steady-state temperature over Γ sr , T root ( > T ∞ , wlog) uniform; predict temperature T ss s ( x ) ≡ T s ( x , t → ∞ ) , x ∈ Ω s+ . Stage II. Cooldown: T cd s ( x , t ) impose zero flux boundary condition on Γ sr ; provide initial condition, T cd s ( x , t = 0) = T ss s ( x ) , x ∈ Ω s+ (reset time); predict temperature T cd s ( x , t ) , x ∈ Ω s+ , t > 0. Notation: denotes spatial average over cross section. 37/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  28. The Fin Problem Classical Formulation Small-Biot Regime (Steady-State) Governing Equations: Dimensional Steady-State Stage Temperature T s ≡ T ss s ( x ) satisfies − k s ∇ 2 T s = 0 in Ω s+ , η lin ( T root , T root ) − k s ∇ T s · ˆ n = ˜ ( T s − T ∞ ) on Γ sf , � �� � � �� � Fourier’s Law HTC T s = T root on Γ sr , − k s ∇ T s · ˆ n = 0 (insulated tip) on Γ st . Cooldown Stage: incorporate ∂ T s and initial condition T ss s . ∂ t 38/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  29. The Fin Problem Formulation Classical Formulation pMOR Interpretation Small-Biot Regime (Steady-State) Governing Equations: Dimensional η lin A cs Let Bi fin ≡ ˜ . k s P cs For Bi fin ≪ 1 , P cs L ≫ 1 , T s ( x ) ≈ ˆ T s ( x 1 ) satisfies A cs d ( ˆ T s − T ∞ ) + η lin P cs ( ˆ − k s A cs T s − T ∞ ) = 0 , 0 < x 1 < L , dx 2 1 d ( ˆ T s − T ∞ ) ˆ T s = T root at x 1 = 0 , − k s = 0 at x 1 = L . dx 1 Fin pPDE: M [2] � � µ [2] ≡ η lin , T ∞ ∈ P [2] k s , A cs , P cs , ˜ �→ ˆ T s ( x 1 ) , 0 ≤ x 1 ≤ L ; o = O [2] ( ˆ T s ) . Here O [2] is a linear output functional. 39/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  30. The Fin Problem Formulation Classical Formulation pMOR Interpretation Small-Biot Regime (Steady-State) Weak Form Let X E = { v ∈ H 1 (Ω s+ ) | v | Γ sr = T root } = { v ∈ H 1 (Ω s+ ) | v | Γ sr = 0 } . X Then T s ∈ X E satisfies � � k s ∇ ( T s − T ∞ ) · ∇ v + η lin ( T s − T ∞ ) v = 0 , ∀ v ∈ X . Ω s+ Γ sf X E = { v ∈ X E | v function of x 1 only } ⊂ X E Let ˆ ˆ X = { v ∈ X | v function of x 1 only } ⊂ X . X E such that Find ˆ T s ∈ ˆ optimal in energy norm � � k s ∇ ( ˆ ( ˆ T s − T ∞ ) v = 0 , ∀ v ∈ ˆ η lin T s − T ∞ ) · ∇ v + ˜ X . Ω s+ Γ sf 40/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  31. The Fin Problem Formulation Classical Formulation pMOR Interpretation Small-Biot Regime (Steady-State) Weak Form Let X E = { v ∈ H 1 (Ω s+ ) | v | Γ sr = T root } = { v ∈ H 1 (Ω s+ ) | v | Γ sr = 0 } . X Then T s ∈ X E satisfies � � k s ∇ ( T s − T ∞ ) · ∇ v + η lin ( T s − T ∞ ) v = 0 , ∀ v ∈ X . Ω s+ Γ sf X E = { v ∈ X E | v function of x 1 only } ⊂ X E Let ˆ ˆ X = { v ∈ X | v function of x 1 only } ⊂ X . X E such that Find ˆ T s ∈ ˆ optimal in energy norm � L � L d ( ˆ T s − T ∞ ) dv η lin P cs ( ˆ k s A cs dx 1 + ˜ T s − T ∞ ) v dx 1 dx 1 dx 1 0 0 = 0 , ∀ v ∈ ˆ X . 40/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  32. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation Heat Transfer Back-of-the Envelope (BE ) Framework Formulation 41/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  33. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation General Form Given solid artifact A from set of artifacts (or natural objects); environment ; environment conditions E from set of environment conditions ; process applied to artifact; process conditions P from set of process conditions ; output operator O : X (Ω A s ) → Y ; provide numeric estimate for output, o est ≈ O ( T phy ( A,E,P )) s quantitative justification for proposed answer. show your work Remark Problem Statement is non-prescriptive . 42/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  34. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation General Form Given Teacher solid artifact A from set of artifacts (or natural objects); environment ; environment conditions E from set of environment conditions ; process applied to artifact; process conditions P from set of process conditions ; output operator O : X (Ω A s ) → Y ; provide numeric estimate for output, o est ≈ O ( T phy ( A,E,P )) s quantitative justification for proposed answer. show your work Remark Problem Statement is non-prescriptive . 42/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  35. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation General Form Given Teacher solid artifact A from set of artifacts (or natural objects); environment ; environment conditions E from set of environment conditions ; process applied to artifact; process conditions P from set of process conditions ; output operator O : X (Ω A s ) → Y ; provide Student : BE Single-Screen Script numeric estimate for output, o est ≈ O ( T phy ( A,E,P )) s quantitative justification for proposed answer. show your work Remark Problem Statement is non-prescriptive . 42/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  36. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation Summary 1. Material property function: material �→ k s , α s , k f , α f , ν, β, ε r . 2. Set of convection heat transfer coefficient (HTC c ) functions S HTC c ≡ { Plate( θ g ) , Circular Cylinder , Sphere } for forced and natural convection. 3. Set of radiation heat transfer coefficient (HTC r ) functions S HTC r ≡ { Parallel Plates , Convex Body in Enclosure } for graybody heat exchange. 4. Set of pPDE models S pPDEs ≡ { M [1] , M [2] , M [3] , M [4] } for heat transfer in solid body in communication with environment. 43/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  37. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation Summary 1. Material property function: material �→ k s , α s , k f , α f , ν, β, ε r . 2. Set of convection heat transfer coefficient (HTC c ) functions S HTC c ≡ { Plate( θ g ) , Circular Cylinder , Sphere } for forced and natural convection. Nu(sselt) pPDE models 3. Set of radiation heat transfer coefficient (HTC r ) functions S HTC r ≡ { Parallel Plates , Convex Body in Enclosure } for graybody heat exchange. 4. Set of pPDE models S pPDEs ≡ { M [1] , M [2] , M [3] , M [4] } for heat transfer in solid body in communication with environment. 43/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  38. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation S pPDEs : Set of pPDEs M [1] : Dunk(ing) Bi dunk ≪ 1 M [1] [ − ] geo = lumped ; s ≡ ] − ℓ, ℓ [ ×D ad ; M [1] [Ω P geo = P : Ω P s ] 2 ) < ℓ 2 } × D ad ; M [1] [Ω C geo = C : Ω C s ≡ { ( x 2 1 + x 2 s ] M [1] [Ω S geo = S : Ω S s ≡ { ( x 2 1 + x 2 2 + x 2 3 ) < ℓ 2 } . s ] Bi fin ≪ 1 M [2] : Fin . M [3] : Wall . M [4] : Semi-Infinite Body . Remark PDE complexity: IBVP in time and one spatial coordinate. 44/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  39. Problem Statement (PS) BE Instruction Set pPDE Instantiation BE Procedure Truth Model Simplification BE Motivation Transformation Framework No Composition Given PS, define notional '' truth '' PDE model: M PS : ( A,E,P ) �→ Ω A , o phy = O ( T phy s , T phy ); s s in general, M PS can not (certainly will not) be evaluated. Notation: phy denotes noise-free measurement of physical artifact. 45/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  40. Problem Statement (PS) BE Instruction Set pPDE Instantiation BE Procedure Truth Model Simplification BE Motivation Transformation Framework No Composition Given PS, define notional '' truth '' PDE model: M PS : ( A,E,P ) �→ Ω A , o phy = O ( T phy s , T phy ); s s in general, M PS can not (certainly will not) be evaluated. Notation: phy denotes noise-free measurement of physical artifact. Choose n ] ∈ S pPDEs model selection n ∈ { 1 , . . . , 4 } : a pPDE M [¯ ¯ n ] ∈ P [¯ n ] associated to M [¯ n ] parameter selection µ [¯ ¯ such that o est ≡ o [¯ n ] = O [¯ n ] ( T [¯ n ] µ [¯ n ] )) ≈ o phy ; s (¯ or declare that Problem Statement is “outside envelope.” 45/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  41. Problem Statement (PS) BE Instruction Set pPDE Instantiation BE Procedure Truth Model Simplification BE Motivation Transformation Framework No Composition Given PS, define notional '' truth '' PDE model: M PS : ( A,E,P ) �→ Ω A , o phy = O ( T phy s , T phy ); s s in general, M PS can not (certainly will not) be evaluated. Notation: phy denotes noise-free measurement of physical artifact. Choose n ] ∈ S pPDEs model selection n ∈ { 1 , . . . , 4 } : a pPDE M [¯ ¯ n ] ∈ P [¯ n ] associated to M [¯ n ] parameter selection µ [¯ ¯ such that o est ≡ o [¯ n ] = O [¯ n ] ( T [¯ n ] µ [¯ n ] )) ≈ o phy ; s (¯ or declare that Problem Statement is “outside envelope.” n ] preliminary; n , M [¯ Approach: classification PS ( A , E , P , O ) �→ ¯ simplification M PS �→ M [¯ n ] (¯ µ [¯ n ] ) and confirm ¯ n . 45/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  42. Problem Statement (PS) BE Instruction Set pPDE Instantiation BE Procedure Truth Model Simplification BE Motivation Techniques Replace Conjugate Framework with Classical Framework. Modify Geometry Materials and Thermophysical Properties Initial and Boundary Conditions Heat Transfer Coefficients: HTC c , HTC r . Apply (Parametrized) Model Order Reduction — Dimensional ity Reduction 46/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  43. Problem Statement (PS) BE Instruction Set pPDE Instantiation BE Procedure Truth Model Simplification BE Motivation Justifications Invoke PDE (and domain) knowledge: order-of-magnitude estimates, stability and perturbation results, asymptotic analysis, closed-form solutions, approximation theory, variational methods, computational studies, experimental observations, often with sign information for ( o est − o phy ). 47/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  44. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation Requirements → Objectives and Applications BE Instruction Set functions are shared by large community: continual verification. BE Instruction Set functions are encapsulated: blunder prevention. BE Instruction Set functions are fast: rapid response for design and optimization. BE Code is transparent: assessment of proposed output estimate, o est ; blunder detection. 50/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  45. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation Requirements → Objectives and Applications BE Instruction Set functions are shared by large community: continual verification. BE Instruction Set functions are encapsulated: blunder prevention. BE Instruction Set functions are fast: rapid response for design and optimization. BE Code is transparent: assessment of proposed output estimate, o est ; blunder detection within BE Code. 50/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  46. Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation Requirements → Objectives and Applications BE Instruction Set functions are shared by large community: continual verification. BE Instruction Set functions are encapsulated: blunder prevention. BE Instruction Set functions are fast: rapid response for design and optimization. BE Code is transparent: assessment of proposed output estimate, o est ; blunder detection of large-scale simulation. 50/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  47. Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Heat Transfer Back-of-the-Envelope Framework Examples of Parameter Selection: Truth Model Simplification 51/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  48. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Assessment Artifact and Environment E Miller 2.51 Artifact: Bagelhalf Environment: Kitchen; T ∞ ≈ 20 ◦ C. Remark Proximity of bagelhalf to back wall. 52/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  49. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Assessment Process and Outputs Process: 1. Remove Bagelhalf from toaster. 2. Place Bagelhalf on cooling rack in vertical orientation. 3. Measure Bagelhalf (mid-radius) surface temperature: T Bagelhalf ( t = 0) ≡ T i ≈ 135 ◦ C. surface Output: Temperature T Bagelhalf ( t ) , t > 0. surface Validation Experiment: Measure with IR thermometer T Bagelhalf ( t ) , t > 0. surface 53/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  50. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Assessment Key Simplifications Modifications to Truth PDE: Conjugate → Classical Geometry: Ω s ≡ ] − ℓ, ℓ [ ×D ; D ≡ ]0 , L horiz [ × ]0 , L vert [. Justification: material addition small in relevant metrics. Boundary Conditions: lateral surfaces ] − ℓ, ℓ [ × ∂ D insulated . Justification: large aspect ratio. Regime: Bi dunk ≈ 0 . 5 not small : apply M [1] [Ω geo = P arallelepiped ] — IBVP( x 1 , t ). s Convection HTC : Vertical Plates, L eff = L vert ; T lin,c = T i . Radiation HTC : Convex graybody in enclosure; ε r = 0 . 96; T lin,r = T i (UB); T lin,r = T ∞ (LB);. 54/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  51. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Assessment Simplified Geometry 55/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  52. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Assessment Surface Temperature 56/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  53. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Artifact: Cast-Iron Skillethandle skillethandle x 1 skilletpan root tip x 1 = L skillet 57/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  54. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Artifact: Chamfer Details Remark Sharp corners: (weak) singularities. 58/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  55. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Artifact: Cross Section Area and Perimeter 59/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  56. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Environment: James Penn’s Kitchen Elements: ◮ Gas Range ◮ Cork Trivet on Chair ◮ IR Camera Jig ◮ Roomwalls Temperature of room and roomwalls, T ∞ ≈ 22 . 6 ◦ C. 60/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  57. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Process Sequence of steps: Stage I: Steady-State 1. Boil water in skilletpan until reach steady state. 2. Remove water from skillet pan, and immediately. . . 3. Measure (or estimate) temperature at skillethandle root, T root ≈ 78 . 6 ◦ C. Stage II: Cooldown 4. Place skillet on trivet. 61/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  58. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Outputs Stage I: Steady-Stage Skillethandle temperature at t = 0: ss T s ( x 1 ) , 0 ≤ x 1 ≤ L . Stage II: Cooldown Skillethandle root temperature for t > 0: cd cd T root ( t ) = T s ( x 1 = 0 , t ). Skillethandle tip temperature for t > 0: cd cd T tip ( t ) = T s ( x 1 = L , t ). 62/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  59. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Key Simplifications Modifications to Truth PDE: Conjugate → Classical Geometry: Ω s+ ≡ right cylinder of circular cross section: � L � L A cs ≡ 1 0 Area( x 1 ) dx 1 , P cs ≡ 1 0 Peri( x 1 ) dx 1 . L L Justification: material modification small in relevant metrics. Regime: Bi fin ≪ 1, P cs L / A cs ≫ 1: apply M [2] . Convection HTC : Horizontal Cylinder 2-D; D = D eff ≡ P cs /π . Justification: D eff preserves boundary-layer length; δ bl ≈ ℓ/ � Nu D � ≪ fin axial length scale. Radiation HTC : Convex graybody in enclosure; ε r = 0 . 95 . Justification: blackbody convex-hull equivalence result. 63/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  60. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Validation Temperature Measurements t = 0 (Stage I) 65/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  61. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Accuracy: Steady State ε r = 0 . 95 Numerical error: [53] � T s − T h s � L ∞ (Ω s ) ≤ 0 . 0001 ( a posteriori indicator). 66/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  62. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Sensitivity to Emissivity ε r = 0 . 50 67/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  63. Problem Statement Hot Bagelhalf Cooling: pPDE Dunk Back-of-the-Envelope Skillethandle: pPDE Fin Validation Experiment Assessment of BE Predictions Accuracy: Cooldown 68/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  64. Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Parametrized Model Order Reduction: Reduced Basis Method [27, 47] Nusselt Number: Slot Flow P-H Tsai, Fischer Group, UIUC 69/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  65. Convection HTC: Slot Flow Formulation RB Mathematical Enabler Temperature Fields Reduced Basis (RB) Method for Slot Flow Computational Cost Motivation: Trombe Wall M Kessler 2.51 pPDE Wall : Parallel Thermal Resistances in Series 70/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  66. Convection HTC: Slot Flow Formulation RB Mathematical Enabler Temperature Fields Reduced Basis (RB) Method for Slot Flow Computational Cost Nusselt Configuration: Air Gap — Idealized [46] Spatial domain: Ω f ≡ ] − ℓ/ 2 , ℓ/ 2[ × ] − 10 ℓ, 10 ℓ [ ⊂ R 2 ; Ω ∗ f ≡ ] − 1 / 2 , 1 / 2[ × ]10 , 10[. Boundary conditions (nondimensional): Θ f = − 1 at x ∗ 1 = − 1 / 2 and Θ f = 1 at x ∗ 1 = 1 / 2; insulated on x ∗ 2 = − 10 and x ∗ 2 = 10. Variable angle of gravity, θ g ∈ P θ g ≡ [0 , 180 ◦ ]: buoyancy force Θ f ( − e 1 cos θ g + e 2 sin θ g ). � 10 1 ∂ Θ f � 2 dx ∗ Nusselt number: � Nu ℓ � ≡ � 2 � . � 1 = − 1 ∂ x ∗ x ∗ 2 · 20 − 10 1 Parameter variation: � Nu ℓ � = � Nu ℓ � ( θ g ; Ra ℓ , Pr); Ra ℓ = 10 3 , Pr = 0 . 71. 71/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  67. Convection HTC: Slot Flow Formulation RB Mathematical Enabler Temperature Fields Reduced Basis (RB) Method for Slot Flow Computational Cost Governing Equations: Nondimensional Nusselt pPDE Find [ V ∗ ≡ ( V ∗ Θ f ( · , t ∗ = 0) = 0 in Ω ∗ 1 , V ∗ 2 , V ∗ 3 ) , Θ f ]( x ∗ , t ∗ ) f ∂ V ∗ ∂ t ∗ + V ∗ · ∇ V ∗ = −∇ p ∗ + Pr 1 ℓ ) − 1 2 (Ra w 2 ∇ 2 V ∗ f , t ∗ > 0 , + Θ f ( − e 1 cos θ g + e 2 sin θ g ) in Ω ∗ ∇ · V ∗ = 0 in Ω ∗ f , t ∗ > 0 , ∂ Θ f ∂ t ∗ + V ∗ · ∇ Θ f = Pr − 1 f , t ∗ > 0 , ℓ ) − 1 2 (Ra w 2 ∇ 2 Θ f in Ω ∗ 1 = ± 1 / 2 and ∂ Θ f 2 = ± 10 , t ∗ > 0 . Θ f = ± 1 at x ∗ ∂ n = 0 on x ∗ � 20 1 ∂ Θ f � 1 =0 dx ∗ Evaluate � Nu ℓ � ≡ � 2 � . � ∂ x ∗ x ∗ 2 · 20 0 1 72/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  68. Convection HTC: Slot Flow Formulation RB Mathematical Enabler Temperature Fields Reduced Basis (RB) Method for Slot Flow Computational Cost Ra ℓ = 10 3 : Steady States — Current Work 73/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  69. Convection HTC: Slot Flow Formulation RB Mathematical Enabler Temperature Fields Reduced Basis (RB) Method for Slot Flow Computational Cost Ra ℓ = 10 4 : Statistically Stationary States — Future Work 74/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  70. Convection HTC: Slot Flow Formulation RB Mathematical Enabler Temperature Fields Reduced Basis (RB) Method for Slot Flow Computational Cost Direct Simulation Hardware (2-D) 8 processors: Intel(R) Xeon(R) CPU E5-2620 v3 a ○ 2.40GHz. Software Nek5000 parallel spectral element code [43, 16]. Computation Time (Wall-Clock) 2-D Spatial Domain, Ω ∗ f ≡ ] − 1 / 2 , 1 / 2[ × [ − 10 , 10[: ≈ 1.7s per C(onvective)T(ime)U(nit)s; ≈ 1000 CTU to reach (statistically) stationary state. 75/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  71. Convection HTC: Slot Flow Formulation RB Mathematical Enabler Temperature Fields Reduced Basis (RB) Method for Slot Flow Computational Cost Direct Simulation Hardware (3-D) 64 processors: Intel(R) Xeon Phi(TM) CPU 7210 a ○ 1.30GHz. Software Nek5000 parallel spectral element code [43, 16]. Computation Time (Wall-Clock) 2-D Spatial Domain, Ω ∗ f ≡ ] − 1 / 2 , 1 / 2[ × [ − 10 , 10[: ≈ 1.7s per C(onvective)T(ime)U(nit)s; ≈ 1000 CTU to reach (statistically) stationary state. 3-D Spatial Domain, Ω ∗ f ≡ ] − 1 / 2 , 1 / 2[ × ] − 10 , 10[ × ] − 10 , 10[: ≈ 5000s per CTU; ≈ 1000 CTU to reach (statistically) stationary state. 75/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  72. Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Parametric Manifold Steady-State [ V ∗ , Θ f ] h ∈ X h high-dimensional ⊂ X (Ω ∗ f ) 76/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  73. Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Parametric Manifold Steady-State [ V ∗ , Θ f ] h ∈ X h high-dimensional ⊂ X (Ω ∗ f ) [ V ∗ , Θ f ] h ∈ M h ≡ { [ V ∗ , Θ f ] h ( θ g ) | θ g ∈ P θ g } 76/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  74. Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Manifold Snapshots Steady-State Snapshots: ξ m ≡ [ V ∗ , Θ f ] h (ˆ θ m g ∈ P θ g ) , m = 1 , . . . , M . Ra ℓ = 10 3 : Nek5000, t ∗ → ∞ ; stable steady states. 77/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  75. Convection HTC: Slot Flow Ingredients: Steady Flow Numerical Results: Ra ℓ = 10 3 RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Next Steps Bare Necessities RB Spaces (hierarchical): X N RB ⊂ span { ξ m , m = 1 , . . . , M } , 1 ≤ N ≤ N max . Weak-Greedy [54] or Proper Orthogonal Decomposition (POD) Galerkin Projection: θ g ∈ P θ g → [ V ∗ , Θ f ] N RB ( θ g ) ∈ X N RB . A Posteriori Error Indicator: [54, 14] inf sup � residual h � X ′ � [ V ∗ , Θ f ] h − [ V ∗ , Θ f ] N 1 RB � X � h . β h est Affine Expansion in Functions of Parameter: A 0 [ V ∗ , Θ f ] + cos( θ g ) A 1 [ V ∗ , Θ f ] + sin( θ g ) A 2 [ V ∗ , Θ f ] = F ∈ X ′ . Offline-Online Decomposition: Online complexity independent of dim( X h ) . 78/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  76. Convection HTC: Slot Flow Ingredients: Steady Flow Numerical Results: Ra ℓ = 10 3 RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Next Steps Bare Necessities RB Spaces (hierarchical): X N RB ⊂ span { ξ m , m = 1 , . . . , M } , 1 ≤ N ≤ N max . Weak-Greedy [54] or Proper Orthogonal Decomposition (POD) Galerkin Projection: θ g ∈ P θ g → [ V ∗ , Θ f ] N RB ( θ g ) ∈ X N RB . A Posteriori Error Indicator: [54, 14] inf sup � residual h � X ′ � [ V ∗ , Θ f ] h − [ V ∗ , Θ f ] N 1 RB � X � h . β h est Affine Expansion in Functions of Parameter: A 0 [ V ∗ , Θ f ] + cos( θ g ) A 1 [ V ∗ , Θ f ] + sin( θ g ) A 2 [ V ∗ , Θ f ] = F ∈ X ′ . Offline-Online Decomposition: real-time, many-query contexts Online complexity independent of dim( X h ) . 78/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  77. Convection HTC: Slot Flow Ingredients: Steady Flow Numerical Results: Ra ℓ = 10 3 RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Next Steps Bare Necessities RB Spaces (hierarchical): X N RB ⊂ span { ξ m , m = 1 , . . . , M } , 1 ≤ N ≤ N max . Weak-Greedy [54] or Proper Orthogonal Decomposition (POD) Galerkin Projection: θ g ∈ P θ g → [ V ∗ , Θ f ] N RB ( θ g ) ∈ X N RB . A Posteriori Error Indicator: [54, 14] inf sup � residual h � X ′ � [ V ∗ , Θ f ] h − [ V ∗ , Θ f ] N 1 RB � X � h . β h est Affine Expansion in Functions of Parameter: A 0 [ V ∗ , Θ f ] + cos( θ g ) A 1 [ V ∗ , Θ f ] + sin( θ g ) A 2 [ V ∗ , Θ f ] = F ∈ X ′ . Offline-Online Decomposition: BE HTC c Functions Online complexity independent of dim( X h ) . 78/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

  78. Convection HTC: Slot Flow Ingredients: Steady Flow Numerical Results: Ra ℓ = 10 3 RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Next Steps Accuracy: POD Bifurcation [26] RB: N = 14, N = 16 ( ← POD spectrum); Newton continuation. 79/92 Anthony T Patera, MIT Model Simplification, Model Order Reduction

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