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Parametrized Partial Differential Equations Heat Transfer - - PowerPoint PPT Presentation

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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

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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

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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pPDEs 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

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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Acknowledgments

Collaborators: ◮ J Penn MIT ◮ K Kaneko, P Fischer, P-H Tsai U Illinois ◮ T Taddei INRIA Bordeaux ◮ P Huynh, D Knezevic, L Nguyen Akselos SA ◮ Students in 2.51 Intermediate Heat and Mass Transfer MIT Financial Support: AFOSR, ONR, ARO

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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3/92 Motivation Disciplinary Focus Outline

Perspective

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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4/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Back-of-the-Envelope Calculation (Figurative)

Definition (Wikipedia) A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic of back-of-the-envelope calculations is the use of simplified assumptions.

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5/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Single-Screen Script (Literal)

Definition (Paterapedia) A single-screen script is a rough prediction, implemented with a limited instruction set in a code which can be viewed in its entirety

  • n a single screen. It is more than a guess but less than an

accurate calculation or mathematical proof. A defining characteristic of single-screen script predictions is the use of model simplification.

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6/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Relevance to Workshop

Definition (Paterapedia) A single-screen script is a rough prediction, implemented with a limited instruction set in a code which can be viewed in its entirety

  • n a single screen. It is more than a guess but less than an

accurate calculation or mathematical proof. A defining characteristic of single-screen script predictions is the use of model simplification. The limited instruction set is (for heat transfer). . .

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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6/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Relevance to Workshop

Definition (Paterapedia) A single-screen script is a rough prediction, implemented with a limited instruction set in a code which can be viewed in its entirety

  • n a single screen. It is more than a guess but less than an

accurate calculation or mathematical proof. A defining characteristic of single-screen script predictions is the use of model simplification. The limited instruction set is (for heat transfer). . . a set of pPDEs.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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7/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Questions to Ponder: 2020

Why do we teach students Back-of-the-Envelope — succinct, transparent, fast — methods of engineering analysis still in 2020?

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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7/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Questions to Ponder: 2020

Why do we teach students Back-of-the-Envelope — succinct, transparent, fast — methods of engineering analysis still in 2020? Why do engineers practice Back-of-the-Envelope calculations — in tandem with large-scale simulation — still in 2020?

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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7/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Questions to Ponder: 2020

Why do we teach students Back-of-the-Envelope — succinct, transparent, fast — methods of engineering analysis still in 2020? Why do engineers practice Back-of-the-Envelope calculations — in tandem with large-scale simulation — still in 2020? How can we study Back-of-the-Envelope engineering analysis through the lens of undergraduate education (2.51)?

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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7/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Questions to Ponder: 2020

Why do we teach students Back-of-the-Envelope — succinct, transparent, fast — methods of engineering analysis still in 2020? Why do engineers practice Back-of-the-Envelope calculations — in tandem with large-scale simulation — still in 2020? How can we study Back-of-the-Envelope engineering analysis through the lens of undergraduate education (2.51)? How can the Back-of-the-Envelope benefit — without losing essential advantages — from computational advances 1960-2020?

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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7/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Questions to Ponder: 2020

Why do we teach students Back-of-the-Envelope — succinct, transparent, fast — methods of engineering analysis still in 2020? Why do engineers practice Back-of-the-Envelope calculations — in tandem with large-scale simulation — still in 2020? How can we study Back-of-the-Envelope engineering analysis through the lens of undergraduate education (2.51)? How can the Back-of-the-Envelope benefit — without losing essential advantages — from pMOR advances 1960-2020?

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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7/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Questions to Ponder: 2020

Why do we teach students Back-of-the-Envelope — succinct, transparent, fast — methods of engineering analysis still in 2020? Why do engineers practice Back-of-the-Envelope calculations — in tandem with large-scale simulation — still in 2020? How can we study Back-of-the-Envelope engineering analysis through the lens of undergraduate education (2.51)? How can the Back-of-the-Envelope benefit — without losing essential advantages — from pMOR advances 1960-2020? Is the Back-of-the Envelope fundamentally a human activity, or can it be viewed more formally as an algorithm or framework?

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8/92 Motivation Disciplinary Focus Outline Definitions Research Agenda

Future Prospects: 2030

Headline:

Artificial Student Earns A+ in MIT Subject 2.51

Implications: in engineering education How should we change what we teach, and how we teach? How should we change our assessment of (human) students? and downstream, in professional engineering practice, How can we enhance prediction procedures? General theme: integrated methodology for mathematical modeling and computation. First (very brittle) steps: Artie [44].

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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9/92 Motivation Disciplinary Focus Outline

Macroscale Heat Transfer. . .

S Austin 2.51

External Flows Conduction, Forced and Natural Convection (Gravity-Induced), Radiation

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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10/92 Motivation Disciplinary Focus Outline

. . . in Everyday Life (and Beyond)

2.51 Project Case Studies

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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11/92 Motivation Disciplinary Focus Outline

Key Topics

Review of Heat Transfer Heat Transfer (2.51) Back-of-the-Envelope Framework Examples from 2.51 Project Case Studies Opportunities for Parametrized Model Order Reduction Thread : Parametrized Partial Differential Equations

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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12/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework

Heat Transfer 101 via the Dunk Problem

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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13/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework

Macroscale Heat Transfer. . .

S Austin 2.51

External Flows Conduction, Forced and Natural Convection (Gravity-Induced), Radiation

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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14/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework

Motivation and Notation

P Phan 2.51

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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15/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework

An Idealized Configuration

Let Ω ⊂ R3, Ω = Ωs ∪ Ωf: Ωf ≡ fluid (air) domain: effectively infinite; Ωs ≡ solid domain: convex, (single, scale) parameter ℓ; Γ

sf ≡ Ωs ∩ Ωf \Γad s ;

∂Ωs ≡ Γ

sf ∪ Γad s

uniformly large enclosure: dist(Ωs, ∂Ω) ≫ ℓ; coordinate system: x ≡ (x1, x2, x3), {ei}i; gravity g = −ge2. Initial conditions: T|Ωs ≡ Ts = Ti uniform, T|Ωf ≡ Tf = T∞; assume Ti > T∞ (wlog). Farfield conditions: quiescent fluid; Tf = T∞ (on ∂Ω) — implicit.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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16/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework

Governing Equations: Dimensional

Find [V ≡ (V1, V2, V3), T](x, t) Ts = T|Ωs, Tf = T|Ωf ∂V ∂t + V · ∇V = −∇ p ρ∞ + gβ(Tf − T∞)e2 + ν∇2V in Ωf, t > 0 , ∇ · V = 0 in Ωf, t > 0 , ∂Tf ∂t + V · ∇Tf = αf∇2Tf in Ωf, t > 0 , ∂Ts ∂t = αs∇2Ts in Ωs, t > 0 , Ts = Tf, −ks∇Ts · ˆ n = −k∇Tf · ˆ n + εrσSB(T 4

s − T 4 ∞) on Γ sf, t > 0 ,

Ts(·, t = 0) =Ti in Ωs, Tf(·, t = 0) = T∞ in Ωf .

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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17/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Fluid Domain and Wall

S Austin 2.51

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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18/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

HTCc: Definition

Consider solid body B surrounded by fluid; define wall Γw ≡ ∂B. Given: wall Γw approximately isothermal at temperature Tw; fluid far from wall at temperature T∞ (and quiescent). The spatial-averaged convection HTCc is defined as ¯ ηiso

c [Tw] ≡

Qw |Γw|(Tw − T∞) for Qw ≡

  • Γw qw dS ≡ heat transfer rate from wall to fluid,

qw ≡ heat flux from wall to fluid, |Γw| ≡ the surface area of wall Γw; · ≡ steady-state or long-time-average operator.

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19/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Newton’s Law of Cooling

By construction: Qw ≡

  • Γw qw dS = ¯

ηiso

c [Tw] · |Γw|(Tw − T∞).

Heat flux qw: qw ≡ −kf∇Tf · ˆ n Fourier’s Law (in fluid) ≈ −kf (T∞ − Tw) δbl(xs) δbl: thermal boundary layer ; but for laminar natural convection, δbl depends weakly on xs, δbl(xs) ∼ α1/2

f

(gβ|Tw − T∞|)−1/4 x1/4 hence qw ≈ ¯ ηiso

c [Tw] · (Tw − T∞) on Γw uniform.

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20/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Boundary Layer Visualization

S Austin 2.51

Background-Oriented Schlieren δbl(xs) ∼

  • αftL-E(xs) =
  • αfxs/Ubuoy(xs)

Ubuoy(xs) ∼

  • gβ|Tw − T∞|xs

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21/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Solid and Fluid Domains

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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22/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Dirichlet-Neumann Map ⇒ Robin Condition

Now assume Tw is not known, but part of solution for Ts in B. Boundary condition on solid body B: −ks∇Ts · ˆ n = −kf∇Tf · ˆ n (First Law) = qw ≈ ¯ ηiso

c [Tw] · (Tw − T∞).

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22/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Dirichlet-Neumann Map ⇒ Robin Condition

Now assume Tw is not known, but part of solution for Ts in B. Boundary condition on solid body B: −ks∇Ts · ˆ n = −kf∇Tf · ˆ n (First Law) = qw ≈ ¯ ηiso

c [Tf] · (Tf − T∞)

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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22/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Dirichlet-Neumann Map ⇒ Robin Condition

Now assume Tw is not known, but part of solution for Ts in B. Boundary condition on solid body B: −ks∇Ts · ˆ n = −kf∇Tf · ˆ n (First Law) = qw ≈ ¯ ηiso

c [Ts] · (Ts − T∞)

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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22/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Dirichlet-Neumann Map ⇒ Robin Condition

Now assume Tw is not known, but part of solution for Ts in B. Boundary condition on solid body B: −ks∇Ts · ˆ n = −kf∇Tf · ˆ n (First Law) = qw ≈ ¯ ηiso

c [Ts] · (Ts − T∞)

if isothermal wall condition is approximately satisfied. Condition for approximately isothermal wall: either Bic[Tw] (Biot Number) ≡ ¯ ηiso

c [Tw] L

ks ≪ 1 , for L an appropriate length scale in solid body. Argument: ks(∆T)in B

L

≈ ¯ ηiso

c [Tw] · (Tw − T∞) ⇒ (∆T)in B |Tw−T∞| ≪ 1 .

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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22/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Dirichlet-Neumann Map ⇒ Robin Condition

Now assume Tw is not known, but part of solution for Ts in B. Boundary condition on solid body B: −ks∇Ts · ˆ n = −kf∇Tf · ˆ n (First Law) = qw ≈ ¯ ηiso

c [Ts] · (Ts − T∞)

if isothermal wall condition is approximately satisfied. Condition for approximately isothermal wall: or Bic[Tw] (Biot Number) ≡ ¯ ηiso

c [Tw] L

ks ≫ 1 , for L an appropriate length scale in solid body. Argument: ks(∆T)in B

L

≈ ¯ ηiso

c [Tw] · (Tw − T∞) ⇒ Tw → T∞.

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23/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

HTCc: Measurement

[11, 25]

Given heat source Qsource in solid body, measure wall temperature at several locations, {Tw}, measure farfield fluid temperature, T∞, evaluate ¯ ηiso

c [T avg w ] =

Qsource |Γw|(T avg

w

− T∞) . Confirm condition for isothermal wall: theory: Bic[T avg

w ] (Biot Number) ≡ ¯

ηiso

c [T avg w ] L

ks ≪ 1 ; experiment: std dev{Tw} ≪ |Tw − T∞|.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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24/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

HTCc Functions: Experimental Correlations

[35, 2]

For given HTCc configuration: Introduce length scale associated with Γw, B: ℓ. Form nondimensional groups: Nuℓ ≡ ¯ ηiso

c [Tw] ℓ

kf ; Raw

ℓ ≡ gβ|Tw − T∞|ℓ3

αfν , Pr ≡ ν αf . Define parameter: µ ≡ (Raw

ℓ , Pr) ∈ P ⊂ R2 +.

Fit to data: FHTCc : µ ∈ P → Nuℓ ∈ R+; ¯ ηiso

c [Tw] = kf

ℓ Nuℓ.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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25/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Example: HTCc Correlation — Vertical Plate

[35]

Extension: orientation relative to gravity, (θg, ϕg).

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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26/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Example: HTCc Correlation — Horizontal Cylinder

[35]

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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27/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Stefan-Boltzmann Law: Graybodies

Wall flux: for convex body in large enclosure qw = ¯ ηiso

c [Tw](Tw − T∞) +

εrσSB(T 4

w − T 4 ∞)

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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27/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Stefan-Boltzmann Law: Graybodies

Wall flux: for convex body in large enclosure qw = ¯ ηiso

c [Tw](Tw − T∞) +

εrσSB(T 2

w + T 2 ∞)(T 2 w − T 2 ∞)

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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27/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Stefan-Boltzmann Law: Graybodies

Wall flux: for convex body in large enclosure qw = ¯ ηiso

c [Tw](Tw − T∞) +

εrσSB(T 2

w + T 2 ∞)(Tw + T∞)(Tw − T∞) ;

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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27/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Heat Transfer to Fluid Boundary Condition on Solid Body Experimental Program Incorporation of Radiation

Stefan-Boltzmann Law: Graybodies

Wall flux: for convex body in large enclosure qw = ¯ ηiso

c [Tw](Tw − T∞) +

εrσSB(T 2

w + T 2 ∞)(Tw + T∞)(Tw − T∞) ;

˜ qw = ˜ η

  • (˜

ηc + ˜ ηr)(Tw − T∞) . Nonlinear Case: ˜ ηnlin(Tw) ''exact'' ˜ ηnlin

c

= ¯ ηiso

c [Tw]; ˜

ηnlin

r

= εrσSB(T 2

w + T 2 ∞)(Tw + T∞).

Linear(ized) Case: ˜ ηlin(Tlin,c, Tlin,r) ˜ ηlin

c = ¯

ηiso

c [Tlin,c]; ˜

ηlin

r

= εrσSB(T 2

lin,r + T 2 ∞)(Tlin,r + T∞).

where (say) Tlin,c = Tlin,r = Ti.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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28/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Formulation Small-Biot Regime

Motivation and Notation

P Phan 2.51

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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29/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Formulation Small-Biot Regime

An Idealized Configuration

Let Ω ⊂ R3, Ω = Ωs ∪ Ωf: Ωf ≡ fluid (air) domain: effectively infinite; Ωs ≡ solid domain: convex, (single, scale) parameter ℓ; Γ

sf ≡ Ωs ∩ Ωf \Γad s ;

∂Ωs ≡ Γ

sf ∪ Γad s

uniformly large enclosure: dist(Ωs, ∂Ω) ≫ ℓ; coordinate system: x ≡ (x1, x2, x3), {ei}i; gravity g = −ge2. Initial conditions: T|Ωs ≡ Ts = Ti uniform, T|Ωf ≡ Tf = T∞; assume Ti > T∞ (wlog). Farfield conditions: quiescent fluid; Tf = T∞ (on ∂Ω) — implicit.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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30/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Formulation Small-Biot Regime

Governing Equations: Dimensional linear(ized)

Temperature Ts(x, t) satisfies ∂Ts ∂t = αs∇2Ts in Ωs, t > 0 , −ks∇Ts · ˆ n

  • Fourier’s Law

= ˜ ηlin(Ti, Ti)

  • HTC

(Ts − T∞)

  • n ∂Ωs ≡ Γ

sf, t > 0 ,

Ts(·, t = 0) = Ti in Ωs . Dunk pPDE: M[1][Ωgeo

s

], geo ∈ {P, C, S} µ[1] ≡

  • geo, ℓ, αs, ks, ˜

ηlin, T∞, Ti, tfinal

  • ∈ P[1]

→ Ts(x, t), x ∈ Ωs, t ∈ (0, tfinal]; o = O[1](Ts) . Here O[1] is a linear bounded output functional. Remark Dimensional formulation for expositional convenience.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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31/92 The Dunk(ing) Problem Conjugate Framework Convection Heat Transfer Coefficient Classical (HTC) Framework Formulation Small-Biot Regime

Governing Equation

Let Bidunk ≡ ˜ ηlin |Ωs| ks |Γ

sf| .

For Bidunk ≪ 1, Ts(x, t) ≈ ˆ Ts(t) satisfies ks αs |Ωs|d( ˆ Ts − T∞) dt + ˜ ηlin |Γ

sf|( ˆ

Ts − T∞) = 0 , subject to ( ˆ Ts − T∞)(t = 0) = (Ti − T∞). Dunk pPDE: M[1][−], geo = lumped µ[1] ≡

  • geo, |Ωs|, |Γ

sf|, ks, αs, ˜

ηlin, T∞, Ti, tfinal

  • ∈ P[1]

→ ˆ Ts(t), t ∈ (0, tfinal]; o = O[1]( ˆ Ts) . Here O[1] is a linear output functional. Remark pMOR (parametrized Model Order Reduction).

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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34/92 The Fin Problem Classical Formulation Small-Biot Regime (Steady-State)

Heat Transfer 101 the Fin Problem

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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35/92 The Fin Problem Classical Formulation Small-Biot Regime (Steady-State)

Motivation and Notation

skillethandle skillet x

1

skilletpan root x

1 = L

tip

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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36/92 The Fin Problem Classical Formulation Small-Biot Regime (Steady-State)

An Idealized Configuration

Let Ω ⊂ R3, Ω = Ωs ∪ Ωf: Ωf ≡ fluid domain: effectively of infinite extent, ∂Ωf = ∂Ω; Ωs ≡ solid domain: Ωs ≡ Ωs− (x1 ≤ 0) ∪ Ωs+ (x1 ≥ 0); Ωs+ ≡ Right Cylinder {0 < x1 < L, (x2, x3) ∈ Dcs}: Dcs ≡ cross section: convex; area Acs, perimeter Pcs; ∂Ωs+ ≡ Γsr ∪ Γ

sf ∪ Γst : Γ sf ≡]0, L[ ×∂Dcs, PcsL/Acs ≫ 1;

uniformly large enclosure: dist(Ωs, ∂Ω) ≫ ℓ ; coordinate system: x ≡ (x1, x2, x3), {ei}i; gravity g = −ge3; Farfield conditions: quiescent fluid; Tf = T∞ (on ∂Ω) — implicit. Insulated Tip: −ks ∂Ts

∂x1 = 0 on Γst, natural — implicit.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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37/92 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 Γsr, T root (> T∞, wlog) uniform; predict temperature T ss

s (x) ≡ Ts(x, t → ∞), x ∈ Ωs+.

Stage II. Cooldown: T cd

s (x, t)

impose zero flux boundary condition on Γsr; provide initial condition, T cd

s (x, t = 0) = T ss s (x), x ∈ Ωs+ (reset time);

predict temperature T cd

s (x, t), x ∈ Ωs+, t > 0.

Notation: denotes spatial average over cross section.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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38/92 The Fin Problem Classical Formulation Small-Biot Regime (Steady-State)

Governing Equations: Dimensional

Steady-State Stage Temperature Ts ≡ T ss

s (x) satisfies

−ks∇2Ts = 0 in Ωs+ , −ks∇Ts · ˆ n

  • Fourier’s Law

= ˜ ηlin(T root, T root)

  • HTC

(Ts − T∞) on Γ

sf ,

Ts = T root on Γsr , −ks∇Ts · ˆ n = 0 (insulated tip) on Γst . Cooldown Stage: incorporate ∂Ts ∂t and initial condition T ss

s .

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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39/92 The Fin Problem Classical Formulation Small-Biot Regime (Steady-State) Formulation pMOR Interpretation

Governing Equations: Dimensional

Let Bifin ≡ ˜ ηlin Acs ksPcs . For Bifin ≪ 1, PcsL Acs ≫ 1, Ts(x) ≈ ˆ Ts(x1) satisfies −ksAcs d( ˆ Ts − T∞) dx2

1

+ ηlinPcs( ˆ Ts − T∞) = 0 , 0 < x1 < L , ˆ Ts = T root at x1 = 0 , −ks d( ˆ Ts − T∞) dx1 = 0 at x1 = L . Fin pPDE: M[2] µ[2] ≡

  • ks, Acs, Pcs, ˜

ηlin, T∞

  • ∈ P[2]

→ ˆ Ts(x1), 0 ≤ x1 ≤ L; o = O[2]( ˆ Ts) . Here O[2] is a linear output functional.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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40/92 The Fin Problem Classical Formulation Small-Biot Regime (Steady-State) Formulation pMOR Interpretation

Weak Form

Let X E = {v ∈ H1(Ωs+) | v|Γsr = T root} X = {v ∈ H1(Ωs+) | v|Γsr = 0}. Then Ts ∈ X E satisfies

  • Ωs+

ks∇(Ts − T∞) · ∇v + ηlin

  • Γ

sf

(Ts − T∞)v = 0 , ∀v ∈ X . Let ˆ X E = {v ∈ X E | v function of x1 only} ⊂ X E ˆ X = {v ∈ X | v function of x1 only} ⊂ X . Find ˆ Ts ∈ ˆ X E such that

  • ptimal in energy norm
  • Ωs+

ks∇( ˆ Ts − T∞) · ∇v + ˜ ηlin

  • Γ

sf

( ˆ Ts − T∞)v = 0 , ∀v ∈ ˆ X.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 53

40/92 The Fin Problem Classical Formulation Small-Biot Regime (Steady-State) Formulation pMOR Interpretation

Weak Form

Let X E = {v ∈ H1(Ωs+) | v|Γsr = T root} X = {v ∈ H1(Ωs+) | v|Γsr = 0}. Then Ts ∈ X E satisfies

  • Ωs+

ks∇(Ts − T∞) · ∇v + ηlin

  • Γ

sf

(Ts − T∞)v = 0 , ∀v ∈ X . Let ˆ X E = {v ∈ X E | v function of x1 only} ⊂ X E ˆ X = {v ∈ X | v function of x1 only} ⊂ X . Find ˆ Ts ∈ ˆ X E such that

  • ptimal in energy norm

ksAcs L d( ˆ Ts − T∞) dx1 dv dx1 dx1 + ˜ ηlin Pcs L ( ˆ Ts − T∞)v dx1 = 0, ∀v ∈ ˆ X.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 54

41/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation

Heat Transfer Back-of-the Envelope (BE ) Framework Formulation

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 55

42/92 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;

  • utput operator O: X(ΩA

s) → Y ;

provide numeric estimate for output, oest ≈ O(T phy

s

(A,E,P)) quantitative justification for proposed answer.

show your work

Remark Problem Statement is non-prescriptive.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 56

42/92 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;

  • utput operator O: X(ΩA

s) → Y ;

provide numeric estimate for output, oest ≈ O(T phy

s

(A,E,P)) quantitative justification for proposed answer.

show your work

Remark Problem Statement is non-prescriptive.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 57

42/92 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;

  • utput operator O: X(ΩA

s) → Y ;

provide Student: BE Single-Screen Script numeric estimate for output, oest ≈ O(T phy

s

(A,E,P)) quantitative justification for proposed answer.

show your work

Remark Problem Statement is non-prescriptive.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 58

43/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation

Summary

  • 1. Material property function: material → ks, αs, kf, αf, ν, β, εr .
  • 2. Set of convection heat transfer coefficient (HTCc) functions

SHTCc ≡ {Plate(θg), Circular Cylinder, Sphere} for forced and natural convection.

  • 3. Set of radiation heat transfer coefficient (HTCr) functions

SHTCr ≡ {Parallel Plates, Convex Body in Enclosure} for graybody heat exchange.

  • 4. Set of pPDE models

SpPDEs ≡ {M[1], M[2], M[3], M[4]} for heat transfer in solid body in communication with environment.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 59

43/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation

Summary

  • 1. Material property function: material → ks, αs, kf, αf, ν, β, εr .
  • 2. Set of convection heat transfer coefficient (HTCc) functions

SHTCc ≡ {Plate(θg), Circular Cylinder, Sphere} for forced and natural convection. Nu(sselt) pPDE models

  • 3. Set of radiation heat transfer coefficient (HTCr) functions

SHTCr ≡ {Parallel Plates, Convex Body in Enclosure} for graybody heat exchange.

  • 4. Set of pPDE models

SpPDEs ≡ {M[1], M[2], M[3], M[4]} for heat transfer in solid body in communication with environment.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 60

44/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation

SpPDEs: Set of pPDEs

M[1]: Dunk(ing) M[1][−] geo = lumped ; Bidunk ≪ 1 M[1][ΩP

s ]

geo = P : ΩP

s ≡ ] − ℓ, ℓ[×Dad ;

M[1][ΩC

s ]

geo = C : ΩC

s ≡ {(x2 1 + x2 2) < ℓ2} × Dad ;

M[1][ΩS

s ]

geo = S : ΩS

s ≡ {(x2 1 + x2 2 + x2 3) < ℓ2} .

M[2]: Fin . Bifin ≪ 1 M[3]: Wall . M[4]: Semi-Infinite Body . Remark PDE complexity: IBVP in time and one spatial coordinate.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 61

45/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation pPDE Instantiation Truth Model Simplification

Transformation Framework No Composition

Given PS, define notional ''truth'' PDE model: MPS : (A,E,P) → ΩA

s, T phy s

, ophy = O(T phy

s

); in general, MPS can not (certainly will not) be evaluated. Notation: phy denotes noise-free measurement of physical artifact.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 62

45/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation pPDE Instantiation Truth Model Simplification

Transformation Framework No Composition

Given PS, define notional ''truth'' PDE model: MPS : (A,E,P) → ΩA

s, T phy s

, ophy = O(T phy

s

); in general, MPS can not (certainly will not) be evaluated. Notation: phy denotes noise-free measurement of physical artifact. Choose ¯ n ∈ {1, . . . , 4}: a pPDE M[¯

n] ∈ SpPDEs model selection

¯ µ[¯

n] ∈ P[¯ n] associated to M[¯ n] parameter selection

such that

  • est ≡ o[¯

n] = O[¯ n](T [¯ n] s (¯

µ[¯

n])) ≈ ophy;

  • r declare that Problem Statement is “outside envelope.”

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 63

45/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation pPDE Instantiation Truth Model Simplification

Transformation Framework No Composition

Given PS, define notional ''truth'' PDE model: MPS : (A,E,P) → ΩA

s, T phy s

, ophy = O(T phy

s

); in general, MPS can not (certainly will not) be evaluated. Notation: phy denotes noise-free measurement of physical artifact. Choose ¯ n ∈ {1, . . . , 4}: a pPDE M[¯

n] ∈ SpPDEs model selection

¯ µ[¯

n] ∈ P[¯ n] associated to M[¯ n] parameter selection

such that

  • est ≡ o[¯

n] = O[¯ n](T [¯ n] s (¯

µ[¯

n])) ≈ ophy;

  • r declare that Problem Statement is “outside envelope.”

Approach: classification PS (A,E,P,O) → ¯ n, M[¯

n] preliminary;

simplification MPS → M[¯

n](¯

µ[¯

n]) and confirm ¯

n.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 64

46/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation pPDE Instantiation Truth Model Simplification

Techniques

Replace Conjugate Framework with Classical Framework. Modify Geometry Materials and Thermophysical Properties Initial and Boundary Conditions Heat Transfer Coefficients: HTCc, HTCr . Apply (Parametrized) Model Order Reduction — Dimensionality Reduction

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 65

47/92 Problem Statement (PS) BE Instruction Set BE Procedure BE Motivation pPDE Instantiation Truth Model Simplification

Justifications

Invoke PDE (and domain) knowledge:

  • rder-of-magnitude estimates,

stability and perturbation results, asymptotic analysis, closed-form solutions, approximation theory, variational methods, computational studies, experimental observations,

  • ften with sign information for (oest − ophy).

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 66

50/92 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, oest; blunder detection.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 67

50/92 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, oest; blunder detection within BE Code.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 68

50/92 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, oest; blunder detection of large-scale simulation.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 69

51/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin

Heat Transfer Back-of-the-Envelope Framework Examples of Parameter Selection: Truth Model Simplification

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 70

52/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Assessment

Artifact and Environment

E Miller 2.51

Artifact: Bagelhalf Environment: Kitchen; T∞ ≈ 20◦C. Remark Proximity of bagelhalf to back wall.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 71

53/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope 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

surface

(t = 0) ≡ Ti ≈ 135◦C. Output: Temperature T Bagelhalf

surface

(t), t > 0. Validation Experiment: Measure with IR thermometer T Bagelhalf

surface

(t), t > 0.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 72

54/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Assessment

Key Simplifications

Modifications to Truth PDE: Conjugate → Classical Geometry: Ωs ≡ ] − ℓ, ℓ[×D; D ≡]0, Lhoriz[×]0, Lvert[. Justification: material addition small in relevant metrics. Boundary Conditions: lateral surfaces ] − ℓ, ℓ[×∂D insulated. Justification: large aspect ratio. Regime: Bidunk ≈ 0.5 not small: apply M[1][Ωgeo=Parallelepiped

s

] — IBVP(x1, t). Convection HTC: Vertical Plates, Leff = Lvert; Tlin,c = Ti. Radiation HTC: Convex graybody in enclosure; εr = 0.96; Tlin,r = Ti (UB); Tlin,r = T∞ (LB);.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 73

55/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Assessment

Simplified Geometry

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 74

56/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Assessment

Surface Temperature

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 75

57/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Artifact: Cast-Iron Skillethandle

skillethandle skillet x

1

skilletpan root x

1 = L

tip

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 76

58/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Artifact: Chamfer Details

Remark Sharp corners: (weak) singularities.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 77

59/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Artifact: Cross Section Area and Perimeter

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 78

60/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope 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.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 79

61/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope 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.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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62/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Outputs

Stage I: Steady-Stage Skillethandle temperature at t = 0: T

ss s (x1), 0 ≤ x1 ≤ L.

Stage II: Cooldown Skillethandle root temperature for t > 0: T

cd root(t) = T cd s (x1 = 0, t).

Skillethandle tip temperature for t > 0: T

cd tip(t) = T cd s (x1 = L, t).

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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63/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Key Simplifications

Modifications to Truth PDE: Conjugate → Classical Geometry: Ωs+ ≡ right cylinder of circular cross section: Acs ≡ 1

L

L

0 Area(x1)dx1 , Pcs ≡ 1 L

L

0 Peri(x1)dx1 .

Justification: material modification small in relevant metrics. Regime: Bifin ≪ 1, PcsL/Acs ≫ 1: apply M[2]. Convection HTC: Horizontal Cylinder 2-D; D = Deff ≡ Pcs/π. Justification: Deff preserves boundary-layer length; δbl ≈ ℓ/NuD ≪ fin axial length scale. Radiation HTC: Convex graybody in enclosure; εr = 0.95 . Justification: blackbody convex-hull equivalence result.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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65/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Validation Temperature Measurements t = 0 (Stage I)

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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66/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Accuracy: Steady State εr = 0.95

Numerical error:

[53]

Ts − T h

s L∞(Ωs) ≤ 0.0001 (a posteriori indicator).

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 84

67/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Sensitivity to Emissivity εr = 0.50

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 85

68/92 Hot Bagelhalf Cooling: pPDE Dunk Skillethandle: pPDE Fin Problem Statement Back-of-the-Envelope Validation Experiment Assessment of BE Predictions

Accuracy: Cooldown

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 86

69/92 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

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 87

70/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Formulation Temperature Fields Computational Cost

Motivation: Trombe Wall

M Kessler 2.51

pPDE Wall: Parallel Thermal Resistances in Series

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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SLIDE 88

71/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Formulation Temperature Fields Computational Cost

Nusselt Configuration: Air Gap — Idealized

[46]

Spatial domain: Ωf ≡] − ℓ/2, ℓ/2[×] − 10ℓ, 10ℓ[⊂ R2; Ω∗

f ≡] − 1/2, 1/2[×]10, 10[.

Boundary conditions (nondimensional): Θf = −1 at x∗

1 = −1/2 and Θf = 1 at x∗ 1 = 1/2;

insulated on x∗

2 = −10 and x∗ 2 = 10.

Variable angle of gravity, θg ∈ Pθg ≡ [0, 180◦]: buoyancy force Θf (−e1 cos θg + e2 sin θg). Nusselt number: Nuℓ ≡ 1 2 · 20 10

−10

∂Θf ∂x∗

1

  • x∗

1 =− 1 2 dx∗

2 .

Parameter variation: Nuℓ = Nuℓ(θg; Raℓ, Pr); Raℓ = 103, Pr = 0.71.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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72/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Formulation Temperature Fields Computational Cost

Governing Equations: Nondimensional Nusselt pPDE

Find [V ∗ ≡ (V ∗

1 , V ∗ 2 , V ∗ 3 ), Θf](x∗, t∗)

Θf(·, t∗ = 0) = 0 in Ω∗

f

∂V ∗ ∂t∗ + V ∗ · ∇V ∗ = −∇p∗ + Pr

1 2 (Raw

ℓ )− 1

2 ∇2V ∗

+ Θf(−e1 cos θg + e2 sin θg) in Ω∗

f , t∗ > 0 ,

∇ · V ∗ = 0 in Ω∗

f , t∗ > 0 ,

∂Θf ∂t∗ + V ∗ · ∇Θf = Pr− 1

2 (Raw

ℓ )− 1

2 ∇2Θf in Ω∗

f , t∗ > 0 ,

Θf = ±1 at x∗

1 = ±1/2 and ∂Θf

∂n = 0 on x∗

2 = ±10, t∗ > 0.

Evaluate Nuℓ ≡ 1 2 · 20 20 ∂Θf ∂x∗

1

  • x∗

1 =0 dx∗

2 .

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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73/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Formulation Temperature Fields Computational Cost

Raℓ = 103: Steady States — Current Work

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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74/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Formulation Temperature Fields Computational Cost

Raℓ = 104: Statistically Stationary States — Future Work

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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75/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Formulation Temperature Fields 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.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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75/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Formulation Temperature Fields 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.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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76/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow

Parametric Manifold Steady-State

[V ∗, Θf]h ∈ X h high-dimensional ⊂ X(Ω∗

f )

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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76/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow

Parametric Manifold Steady-State

[V ∗, Θf]h ∈ X h high-dimensional ⊂ X(Ω∗

f )

[V ∗, Θf]h ∈ Mh ≡ {[V ∗, Θf]h(θg) | θg ∈ Pθg }

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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77/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow

Manifold Snapshots Steady-State

Snapshots: ξm ≡ [V ∗, Θf]h(ˆ θm

g ∈ Pθg ), m = 1, . . . , M.

Raℓ = 103: Nek5000, t∗ → ∞; stable steady states.

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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78/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Ingredients: Steady Flow Numerical Results: Raℓ = 103 Next Steps

Bare Necessities

RB Spaces (hierarchical): X N

RB ⊂ span{ξm, m = 1, . . . , M}, 1 ≤ N ≤ Nmax.

Weak-Greedy [54] or Proper Orthogonal Decomposition (POD) Galerkin Projection: θg ∈ Pθg → [V ∗, Θf]N

RB(θg) ∈ X N RB.

A Posteriori Error Indicator: [54, 14] [V ∗, Θf]h − [V ∗, Θf]N

RBX 1 βh est

inf sup residualh X ′ h .

Affine Expansion in Functions of Parameter: A0[V ∗, Θf] + cos(θg)A1[V ∗, Θf] + sin(θg)A2[V ∗, Θf] = F ∈ X ′ . Offline-Online Decomposition: Online complexity independent of dim(X h) .

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78/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Ingredients: Steady Flow Numerical Results: Raℓ = 103 Next Steps

Bare Necessities

RB Spaces (hierarchical): X N

RB ⊂ span{ξm, m = 1, . . . , M}, 1 ≤ N ≤ Nmax.

Weak-Greedy [54] or Proper Orthogonal Decomposition (POD) Galerkin Projection: θg ∈ Pθg → [V ∗, Θf]N

RB(θg) ∈ X N RB.

A Posteriori Error Indicator: [54, 14] [V ∗, Θf]h − [V ∗, Θf]N

RBX 1 βh est

inf sup residualh X ′ h .

Affine Expansion in Functions of Parameter: A0[V ∗, Θf] + cos(θg)A1[V ∗, Θf] + sin(θg)A2[V ∗, Θf] = F ∈ X ′ . Offline-Online Decomposition: real-time, many-query contexts Online complexity independent of dim(X h) .

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Bare Necessities

RB Spaces (hierarchical): X N

RB ⊂ span{ξm, m = 1, . . . , M}, 1 ≤ N ≤ Nmax.

Weak-Greedy [54] or Proper Orthogonal Decomposition (POD) Galerkin Projection: θg ∈ Pθg → [V ∗, Θf]N

RB(θg) ∈ X N RB.

A Posteriori Error Indicator: [54, 14] [V ∗, Θf]h − [V ∗, Θf]N

RBX 1 βh est

inf sup residualh X ′ h .

Affine Expansion in Functions of Parameter: A0[V ∗, Θf] + cos(θg)A1[V ∗, Θf] + sin(θg)A2[V ∗, Θf] = F ∈ X ′ . Offline-Online Decomposition: BE HTCc Functions Online complexity independent of dim(X h) .

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79/92 Convection HTC: Slot Flow RB Mathematical Enabler Reduced Basis (RB) Method for Slot Flow Ingredients: Steady Flow Numerical Results: Raℓ = 103 Next Steps

Accuracy: POD Bifurcation [26]

RB: N = 14, N = 16 (← POD spectrum); Newton continuation.

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Accuracy: Weak Greedy

RB: Newton iteration; initialization Π

X N

RB

H1(Ω) of nearest-θg snapshot;

wall-clock time 4.5ms per θg value → SHTCc.

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Raℓ = 104: Statistically Stationary States

[23, 24][55, 21]

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82/92 Mathematical Enabler PR-RBC Method: Library Thermal Heatsink

Parametrized Model Order Reduction: Port-Reduced Reduced-Basis Component Library Thermal Heatsink L Nguyen, Akselos SA

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Parametrized Model Order Reduction: PR-RBC Library Thermal Heatsink L Nguyen, Akselos SA

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Acoustics Waveguide

Consider a waveguide D⊥ × (0, ∞), and find p(x1, x2, x3) such that −∇2p − κ2p = 0 in D⊥ × (0, ∞) , subject to boundary conditions p = q on (x1, x2) ∈ D⊥, x3 = 0, ∂p ∂n = 0 on (x1, x2) ∈ ∂D⊥ × (0, ∞), p (say) outgoing bounded wave as x3 → ∞.

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Separation of Variables

Restrict attention to the transverse domain D⊥, and find (Υ

i(x1, x2), λi)i=1,... solution of eigenproblem

−∇2

x1,x2Υ = λΥ in D⊥ ,

∂Υ ∂n = 0 on ∂D⊥ ;

  • rder (real) eigenvalues λ1 = 0 < λ2 ≤ λ3 ≤ . . .

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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Evanescence

Define n such that κ ∈ [√λn,

  • λn+1[: then

p(x; κ) =

n

  • j=1

propagating modes

  • cj Υ

j(x1, x2) e−i√ κ2−λj x3

+

∞

  • j=n+1

cj Υ

j(x1, x2) e

real negative

  • −
  • λj − k2 x3

for coefficients cj chosen to realize p(·, ·, x3 = 0) = q. Acoustics : κ > 0 ⇒ n ≥ 1; one or more propagating modes. Heat Conduction : κ = 0 ⇒ n = 1; single “propagating” mode, Υ

1 ≡ Constant .

Equilibrium Elasticity : κ = 0 ⇒ n = 6; rigid-body modes.

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Thermal Heatsink: Thermal Fins in situ

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Library of Parametrized Archetype Components

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pPDE Model: System of Instantiated Components

Encapsulated pPDE Model Simple Heatsink: µSystem ≡ (4Bifin, H, Lfin) ∈ P ≡ [0.01, 0.5] × [1, 2] × [3, ∞[.

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pPDE Model: System of Instantiated Components

Encapsulated pPDE Model Simple Heatsink: BE estimation µSystem ≡ (4Bifin, H, Lfin) ∈ P ≡ [0.01, 0.5] × [1, 2] × [3, ∞[.

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pPDE Model: System of Instantiated Components

Encapsulated pPDE Model Simple Heatsink: BE incorporation µSystem ≡ (4Bifin, H, Lfin) ∈ P ≡ [0.01, 0.5] × [1, 2] × [3, ∞[.

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Offline Stage: Library

[13, 30, 39, 34, 33]

Port Reduction: Evanescence

[18, 10]

Train over all port-compatible archetype component pairs: impose random Dirichlet conditions on unshared ports; consider random parameter values within each component; accumulate restriction of solution to shared port. Perform POD on port restrictions for each port “color.” Bubble Reduction: Component Parametric Manifold Train over all (single) archetype components: for each port mode-cum-Dirichlet data: consider random parameter values within component; identify RB space for solution in interior of component.

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Online Stage

Instantiation and Assembly: map µSystem to archetype component (local) parameters; connect (compatible) ports to form System. Static Condensation: eliminate RB — not FE — bubble degrees of freedom within each instantiated component of System. Direct Stiffness: construct Schur complement for System reduced port degrees of freedom — small and block-sparse. Solution: apply sparse Gaussian elimination to Schur complement to obtain reduced port degrees of freedom. Postprocessing: reconstruct RB bubble approximations in interiors

  • f components from reduced port degrees of freedom.

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Future Prospects: 2030

Headline:

Artificial Student Earns A+ in MIT Subject 2.51

Implications: in engineering education How should we change what we teach, and how we teach? How should we change our assessment of (human) students? and downstream, in professional engineering practice, How can we enhance prediction procedures? General theme: integrated methodology for mathematical modeling and computation. First (very brittle) steps: Artie [44].

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Future Prospects: 2030

Headline:

and Accepts Employment as a ParaEngineer

Implications: in engineering education How should we change what we teach, and how we teach? How should we change our assessment of (human) students? and downstream, in professional engineering practice, How can we enhance prediction procedures? General theme: integrated methodology for mathematical modeling and computation. First (very brittle) steps: Artie [44].

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References

Anthony T Patera, MIT Model Simplification, Model Order Reduction

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A posteriori error for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN, 39(1):157–181, 2005. [24] B Haasdonk and M Ohlberger. Reduced basis method for finite volume approximation of parametrized linear evolution equations. M2AN, 42(2):277–302, 2008. [25] AV Hassani and KGT Hollands. On natural convection heat transfer from three-dimensional bodies of arbitrary shape. ASME Journal of Heat Transfer, 111:363–371, 1989. [26] M Hess, A Alla, A Quaini, G Rozza, and Max G. A localized reduced-order modeling approach for PDEs with bifurcating solutions. Preprint arXiv:1807.08851, July 2018. [27] JS Hesthaven, G Rozza, and B Stamm.

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