artie: an arti cial heat transfer student - icerm.brown.edu · s1. form problem data: pd fps;kbg...

Eggplantslice ProblemContinuum Heat Transfer Cognition

Artie: Artificial Heat Transfer StudentArtie: Query Examples

Artie: An Artificial Heat Transfer Student

Anthony T Patera, MIT

ICERM Workshop on Scientific Machine LearningJanuary 30, 2019

Anthony T Patera, MIT Artie: An Artificial Heat Transfer Student



Table of contents

Eggplantslice Problem

Continuum Heat Transfer CognitionParametrized PDEPIMA Analysis ProcedureArchive-cum-Big DataQuery Example: Eggplantslice Problem

Artie: Artificial Heat Transfer StudentParse IMAP Interrogate MAPI Model-Approximate

Artie: Query ExamplesQuery: Spoon ProblemQuery: Eggplantslice ProblemQuery: Brick Wall Problem




Acknowledgments

Collaboration: Discussions and Software

I Abe Gertler, MIT

I Phuong Huynh, Akselos SA

I Kris Kang, Amazon Alexa

I Rohit Karnik, MIT

I Reza Malek-Madani, USNA/ONR

I James Penn, MIT

I Tommaso Taddei, INRIA-Bordeaux

I Masayuki Yano, University of Toronto

Pedagogical 'Data': 2.051, Introduction to Heat Transfer (UG)

Financial Support: MIT Ford Professorship




Relevant and Related Work

Variational Methods for PDEs: [30, 27, 31, 11, 21]

Parametrized PDEs: [26, 13, 5]

Natural Mathematical Language for PDEs: [19, 28]

Heat Transfer Engineering Analysis: [17, 3]

Heat Transfer Expert Systems: [1, 4, 9, 29]

Natural Language Understanding: [12, 10, 8]

Natural Language Understanding forMathematical and Physical Problems: [7, 23, 18]

Artificial Students: [15, 6, 24]




Eggplantslice and Cork Trivet

Figure 1EP0 PIMA




Eggplantslice and Cork Trivet

Dimensions: L = 0.026m; Hmax = 0.096m; Wmax = 0.16m.

Cross-Section Characterization: Area A = 0.012m2;Perimeter P = 0.43m.




Kitchen

Components:

I Refrigerator

I Table

I Roomwalls

Figure 2EP0 PIMA




Experiment

Procedure:

1. Place eggplantslice on cork trivet in refrigerator.

2. Allow eggplantslice on cork trivet to reach equilibrium withrefigerator air.

3. Move eggplantslice on cork trivet from refrigerator to kitchentable at time t ≡ 0.

4. Record IR thermometer reading for eggplantslice top center(red dot) to obtain tmeas

j ,Tmeasj j=1,...,nmeas .




Transducer: ETEKCITY Laser Grip 800

Temperature Accuracy:≈ 0.5C-1.0C .

Temperature Resolution:≈ 0.1C .

Spatial Resolution:≈ 0.02m × 0.02m.

Response Time: ≈ 0.5 seconds.




Data: tmeasj ,Tmeas

j j=1,...,nmeas

2.051 Project: assimilate data; predict surface temperature.




Parametrized PDEPIMA Analysis ProcedureArchive-cum-Big DataQuery Example: Eggplantslice Problem

Mathematical Model: Elliptic Case

Model: PDE

I spatial domain Ω ⊂ Rd , d ∈ 1, 2, 3I function space X = X (Ω) : (H1

0 (Ω))dv ⊂ X (Ω) ⊂ (H1(Ω))dv

I residual g : X × X 7→ RI output functional F : X 7→ Y

Equation: find u ∈ X (Ω) such that

g(u, v) = 0,∀v ∈ X (Ω).

Assumption: PDE admits a unique solution u ∈ X (Ω).

Output: s = F(u).





Parametrized Mathematical Model: Elliptic Case

Parametrized Model [26, 13, 5]: M ≡ PM,PDEMµ

I parameter domain PM ⊂ RPM

I spatial domain µ ∈ PM → Ωµ ⊂ Rd , d ∈ 1, 2, 3I function space Xµ = X (Ωµ)

I residual µ ∈ PM → gµ : X × X 7→ RI output functional µ ∈ PM → Fµ : Xµ 7→ Yµ

Equation: given µ ∈ PM, find uµ ∈ Xµ such that

gµ(uµ, v) = 0, ∀v ∈ Xµ.

Output: sµ = F(uµ).





Mathematical Model: Parabolic Case

Model: PDE

I domain Ω, space X = X (Ω), residual g : X × X 7→ RI temporal interval I ≡ (0, tfinal]

I initial condition u0 ∈ (L2(Ω))dv

I mass bilinear form m : (L2(Ω))dv × (L2(Ω))dv 7→ RI output functional F : L2(I;X )→ Y

Equation: find u(t) ∈ X such that

u(0) = u0, m(u(t), v) + g(u(t), v ;

t

) = 0,∀v ∈ X , ∀t ∈ I.

Output: s = F(u).





Mathematical Model: Parabolic Case

Model: PDE

I domain Ω, space X = X (Ω), residual g : X × X 7→ RI temporal interval I ≡ (0, tfinal]

I initial condition u0 ∈ (L2(Ω))dv

I mass bilinear form m : (L2(Ω))dv × (L2(Ω))dv 7→ RI output functional F : L2(I;X )→ Y

Equation: find u(t) ∈ X such that

u(0) = u0, m(u(t), v) + g(u(t), v ; t) = 0,∀v ∈ X , ∀t ∈ I.

Output: s = F(u).





Parametrized Mathematical Model: Parabolic Case

Parametrized Model: M ≡ PM,PDEMµ

I parameter µ ∈ PM ⊂ RPM

I domain Ωµ, space Xµ = X (Ωµ), residual gµ : X × X 7→ RI temporal interval Iµ ≡ (0, tfinal]

I initial condition u0µ ∈ (L2(Ωµ))dv

I mass bilinear form mµ : (L2(Ωµ))dv × (L2(Ωµ))dv 7→ RI output functional Fµ : L2(Iµ;Xµ)→ Yµ

Equation: given µ ∈ PM, find uµ(t) ∈ Xµ such that

uµ(0) = u0µ, mµ(uµ(t), v) + gµ(u(t), v ; t) = 0,∀v ∈ Xµ, ∀t ∈ I.

Output: sµ = Fµ(uµ).





Approximation Methods

Approximation method for model M = PM,PDEMµ :

AM : P 7→ Y such that µ ∈ PM → AM(µ; δ) ≡ sδµ (≈ sµ) ,

EM : PM 7→ R such that |sµ − sδµ| / EM(µ; δ) ≡ E δµ ;

here δ is an approximation hyperparameter.

Library of approximation methods, LM ≡ (AM,EM)kk=1,... :

closed-form symbolic approximations — transparent, most fast;

quasi-closed-form numerical approximations for Ω ⊂ Rd≡1

— semi-transparent, fast;

numerical approximations for Ω ⊂ Rd>1 — opaque, least fast.

Model Class: CM ≡ M,LM ≡ PM,PDEMµ ,L

M.





Inputs: Archive

and Query

KnowledgeBase (KB) for Heat Transfer: text, symbolic, numeric

I experiential commons: subject-predicates

I geometry dictionary: subject-predicates, scalars, functions

I thermophysical property database: scalars, functions

I empirical correlation compendium: scalars, functions

Model class dictionary, D ≡ CMii=0,... , for Heat Transfer.

Problem statement, PS:

I problem observables, measured quantities

I problem deliverables — output(s)

expressed in natural language, PStext, and figures, PSimages.





Inputs: Archive and Query







Problem statement, PS:








Inputs: Archive and Query







Problem statement, PS: no prescription








Parse-Interrogate-Model-Approximate (vt) 2.051

S1. Form problem data: PD ≡ PS;KB ≡ PStext, PSimages;KB.S2. Parse PD 7→ PDparse.

S3. Interrogate PDparse.

S4. For any model class CM ≡ PM,PDEMµ ,L

M ∈ Dad(missible)

S4a. Identify parameter value: µquery ∈ PM.

S4b. Choose approximation method: (AM,EM) ∈ LM; δ.

S5. Find approximate output: sδµquery = AM(µquery; δ).

S6. Evaluate error indicator: E δµquery = EM(µquery; δ).

Error Budget : ‖ smeas − sδµquery ‖Y ≤

‖ smeas − s[M, µquery ] ‖Y +

/Eδµquery︷︸︸︷‖ s − sδµquery(= AM(µquery; δ)) ‖Y .





KnowledgeBase (SI). . .

Experiential Commons: table surface normal aligns with −~g ;air is a gas; a spoon is a solid object; . . .

Geometry Dictionary: Volumesphere(a) = (4/3)πa3;AreaC(y(s), z(s)) =

∮C z(s)dy(s); . . .

Thermophysical Properties Database: AHTT 1

air properties: kair, αair, νair, βair(T )

eggplant properties: kep, αep2

cork properties: kcork, αcork

surface emittances (infrared): εep, εpaint

physical constants: gBoston, σSB

1JH Lienhard IV and JH Lienhard V. A Heat Transfer Textbook, 2018. [17]2SD Ali, HS Ramaswamy, and GB Awuah. J Food Process Engineering, 25,

pages 417-433, 2002. [2]Anthony T Patera, MIT Artie: An Artificial Heat Transfer Student

http://web.mit.edu/lienhard/www/ahtt.html




. . . KnowledgeBase (SI)

Empirical Correlation Compendium: Heat Transfer Coefficient

(−ksolid∇T · n)|Γ,solid︸︷︷︸wall heat flux in solid

= h (T |Γ,solid − T∞,fluid)

h = 0: zero wall heat flux ≡ insulated (natural: Robin)

I Natural Convection Plate to Air3: ∆T ≡ Tplate − T∞,fluid

Nu` ≡h `

kair, Ra` ≡

βairg |∆T |`3

αairνair, Prair =

νair

αair

Nu` = Fcn(Ra`,Prair, sgn(∆T ~g ·nplate)),

` = Fcn(plate geometry)

I Radiation from Body to Large Enclosure: AHTT

h = εbodyσSB(273.15 + T )3, T = Fcn(Tbody,Tenclosure)PIMA ArtiePIMAs ArtiePIMAe

3ASHRAE Handbook — Fundamentals, Table 9, page 4.20, 2017. [3]Anthony T Patera, MIT Artie: An Artificial Heat Transfer Student








= h (T |Γ,solid − T∞,fluid)


Nu` ≡h `

kair, Ra` ≡

βairg |∆T |`3

αairνair, Prair =

νair

αair













= h (T |Γ,solid − Tenclosure)


Nu` ≡h `

kair, Ra` ≡

βairg |∆T |`3

αairνair, Prair =

νair

αair










Model M0 ∈ D: Component n

Parameter: µn ≡ Ln,H,W , kn, hnlat,Tn∞, .

Parameter Domain: Pn ⊂ RPn ≡ 4 .

Spatial Domain (Solid): Right Cylinder Rectangular Cross-Section

Ωnµ : 0 < x ′ < Ln, (y ′, z ′) ∈ [0,H]× [0,W ] ⊂ Rd≡3.

∂Ωnµ : Γn

bµ ≡ x ′ = 0 ∪ Γntµ ≡ x ′ = Ln ∪ Γn

latµ.

Ln

H

Wx ′

z ′ y ′

Γnlatµ





Model M0 ∈ D: System of Nco Components ArtiePIMA

Parameter: µ ≡ µ1, . . . , µNco, hb, ht,T∞,b,T∞,t,H,W .Parameter Domain: PM0 ≡

∏Ncon=1×Psystem ⊂ RPM0 ≡ 4Nco+6 .

Spatial Domain (Solid): Right Cylinder

Ωµ = ∪Ncon=1Ωn

µ.

∂Ωµ : Γbµ ≡ x = 0 ∪ Γtµ ≡ x = L ∪ Γlatµ.

L =∑Nco

n=1 Ln

H

Wx

z y

Γlatµ

Γbµ Γtµ





Model M0 ∈ D: System of Nco Components ArtiePIMA

Parameter: µ ≡ µ1, . . . , µNco, hb, ht,T∞,b,T∞,t,H,W .Parameter Domain: PM0 ≡

∏Ncon=1×Psystem ⊂ RPM0 ≡ 4Nco+6 .


Ωµ = ∪Ncon=1Ωn

µ.


Function Space: Xµ = H1(Ωµ).

Equation: find Tµ ∈ Xµ such thatNco∑n=1

(∫Ωnµ

kn∇Tµ · ∇v dV + hnlat

∫Γnlat µ

(Tµ − T n∞)v dS

)+∑f=b,t

hf

∫Γf µ

(Tµ − T∞,f)v dS = 0, ∀v ∈ Xµ .

Output: sµ = Tµ ∈ X .





Approximation Method (AM0

1 ,EM0

1 ) ∈ LM0 ArtiePIMA

Parameter: µ ≡ µ1, . . . , µNco, hb, ht,T∞,b,T∞,t,H,W .

Spatial Domain Ω1dµ ≡ (0, L); Function Space X 1d

µ ≡ H1(Ω1dµ ).

Solution Approach: Symbolic.

Discrete Equation: find Tµ ∈ X 1dµ such that

Nco∑n=1

(∫ Ln

0kn

dTµdx

dv

dxdx + hnlat

(A

P

)−1∫ Ln

0(Tµ − T n

∞)v dx

)+∑f=b,t

hf (Tµ|f − T∞,f) = 0,∀v ∈ X 1dµ .

Output: sδµ = Tµ .

Error Indicator (H1(0, L)): E δµ ≡ c1 maxn∈1,...,Ncohnlat(A/P)

kn .





Model M1 ∈ D PIMA ArtiePIMA

Parameter: µ ≡ L,A/P, k , α, hb, ht, hlat,Ti,T∞, Cref.Parameter Domain: PM1 ∈ RPM1 ≡ 9+ℵ0 .


Ωµ : 0 < x < L, (y , z) ∈ AP Cref(Convex) ⊂ Rd≡3.


L

A

PCref

x

zy

Γlatµ

Γbµ Γtµ

Example: Cref ≡ Squareref ≡ [0, 4]× [0, 4].










Example: Cref ≡ Eggplant Cross-Sectionref .










Function Space: Xµ = H1(Ωµ).

Equation: find Tµ(t) ∈ Xµ such that Tµ(t = 0) = Ti (∈ X ) and∫Ω

k

αTµ(t)v + k∇Tµ(t) · ∇v dV + hlat

∫Γlat

(Tµ(t)− T∞)v dS

+∑f=b,t

hf

∫Γf

(Tµ(t)− T∞)v dS = 0,∀v ∈ Xµ,∀t ∈ (0, tfinal] .

Output: for f ∈ b, t, sµ = |Γf |−1∫

ΓfTµ(t) dS , t ∈ (0, tfinal] .





Approximation Method (AM1

1 ,EM1

1 ) ∈ LM1 PIMA ArtiePIMA

Parameter: µ ≡ L,A/P, k , α, hb, ht, hlat,Ti,T∞, Cref.Spatial Domain Ω1d

µ ≡ (0, L); Function Space X 1dµ ≡ H1(Ω1d

µ ).

Solution Approach: Numerical Discretization δ ≡ (∆x ,∆t)

FE in space: XFE∆x ⊂ X 1d;

FD in time: ∆t ≡ tfinal/J, tj = j∆t, 0 ≤ j ≤ J.

Discrete Equation: find T jµ ∈ X 1d

µ such that T 0µ = Ti and∫ L

0

k

α

T jµ−T j−1

µ

∆tv + k

dT jµ

dx

dv

dxdx + hlat

(A

P

)−1∫ L

0T jµ v dx

+∑f=b,t

hf(Tjµ|f − T∞)v |f = 0,∀v ∈ X 1d, 1 ≤ j ≤ J .

Output: for f ∈ b, t, sδµ = T jµ|f , 0 ≤ j ≤ J .

Error Indicator (L2(0, tfinal)): E δµ ≡ c1hlat(A/P)

k + c2(∆x)3/2 + c3∆t.





(Reduced) Big Data. . .





. . . (Reduced) Big Data

AHTT






Problem Statement: EP0

An eggplantslice on cork trivet, Figure 1 , is removed from arefrigerator and placed on a table in a kitchen, Figure 2 . Both theair and roomwalls are maintained at temperature T∞. Theeggplantslice is initially at temperature Ti; we consider the timeinterval 0 < t < tfinal.

Find the temperature at the center of the top surface of theeggplant, indicated by the red dot of Figure 1 , as a function of time.

You are given the following data: Hmax = 0.096m, Wmax = 0.16m,L = 0.026m, Ti = 6.7C, T∞ = 24.1C, and tfinal = 3000s.

PIMA





Parse-Interrogate-Model-Approximate

S1. Form problem data: PD ≡ EP0 , [ Figure 1 , Figure 2 ]; KB .S2. Parse PD 7→ PDparse.


S4. For model class CM1 (∈ Dad)

S4a. Identify parameter value µquery ∈ PM1 .

S4b. Choose approximation method: (AM11 ,E

M11 ) ∈ LM1 ; δ.

S5. Find approximate output: sδµquery = AM11 (µquery; δ).

S6. Evaluate error indicator: E δµquery = EM11 (µquery; δ).

Parser and Interrogator : ATP.





S4a. Parameterquery (SI)

µquery ≡ L,A/P, k, α, hb, ht, hlat,Ti,T∞, CrefI L← 0.028I A/P ← 0.0268: image boundary extraction [22]

I k ← kep ≡ 0.28I α← αep ≡ 5× 10−8

I hb ← 0 (≡ insulated): cork-eggplant resistance(t) ratio

I ht ← hnc + 4σSB(Ti + 273.15)3; ∆T = Ti − T∞, ` = A/P,

hnc = kair` 0.27Ra0.25

`

I hlat ← hnc + 4σSBT3i ; ∆T = Ti − T∞, ` = L,

hnc = kair` (0.68 + 0.67Ra0.25

` )(1 + ( 0.492Prair

)9

16 )−49

I Ti ← 6.7I T∞ ← 24.1






µquery ≡ L,A/P, k, α, hb, ht, hlat,Ti,T∞, CrefI L← 0.028I A/P ← 0.0268: image boundary extraction [22]I k ← kep ≡ 0.28I α← αep ≡ 5× 10−8

I hb ← 0 (≡ insulated): cork-eggplant resistance(t) ratio



`


hnc = kair` (0.68 + 0.67Ra0.25

` )(1 + ( 0.492Prair

)9

16 )−49

I Ti ← 6.7I T∞ ← 24.1





S5. Approximate Output sδµquery= AM1(µquery)





S5. Data Assimilation for Ti





S5. Effect of Model

µquery ≡ L,A/P, k, α, hb, ht, hlat,Ti,T∞, Crefµ′query ≡ L,A/P, k , α, hb, ht, 0 ,Ti,T∞, Cref





S5. Penetration Depth (in x)




Parse IMAP Interrogate MAPI Model-Approximate

Syntax Analysis: PStext,KB→ S

Tokens: Name, Part-of-Speech, Tense

Parts-of-Speech:

I Standard: noun, verb, adj, adv, conj, adp, punct, num,. . .

I Custom: MVD ($ $ or % %), MV, EQD (! !), EQ

S(excerpt) =

Google Syntax Analyzer API/Matlab Client (P Huynh) [20, 14, 16]





Entity Analysis: S→ PDparse

PDparse:

I snippet:: subject-predicate

I entity:: compound noun + adpositional phrase

I entity:.attribute:: predicate

snippet1 = 'an eggplantslice is placed on a cork table'entity1 = 'eggplantslice'

entity1.attribute::1 'is placed on a cork table'4 'is initially at temperature % Ti %'9 '( ) is a solid object'

10 '( ) geometry is approximated as a right cylinder'





Image Analysis: PSimage → B(s)

PSimage B(s) [22]





A Sequence of Questions: Unhidden Layers I

1. Which entities are physical (tangible) entities?

2. Which physical entities communicate thermally?

3. Which physical entities are solid? fluid?

4. Which physical entities may be treated as thermal insulators?

5. Which physical entities are related as parent-child?

6. Which physical entities are related as archetype-instantiation?

7. Which physical entities represent degrees of freedom— defined as components?

Assume desired model is defined over Ω ≡ Ωsolid.

Criteria: solid, non-insulator, non-parent, non-archetype,non-negligible thermal resistance(t).





A Sequence of Questions: Unhidden Layers II

8. What characterizes a component:

I geometry class: type, surfaces, ports, parameters?

I boundary conditions on (non-port) surfaces?

I material, or thermophysical properties?

9. How are components connected to form system:set of (component-port, component-port) pairs?

10. What are the boundary conditions on ∂Ωµ:

I temperature, flux, heat transfer coefficient?

I convection, natural convection, radiation?

I 'sink'(fluid, enclosure) temperature?





A Sequence of Questions: Unhidden Layers III

11. What are the the initial conditions (if time-dependent)?

12. What are the quantities-of-interest ≡ output functional?





Application of Model Classes

Identify mapping from PS symbols to model class symbols.

Deduce set of admissible model classes, Dad ⊂ D.

For each CM ∈ Dad:

a. instantiate parameter value: µquery ∈ PM ⊂ RPM;

b. for each approximation (AM,EM) ∈ LM:I form component mass and stiffness matrices, source vectors;I assemble system matrix, system load and output vectors;I evaluate output sδµquery

;I evaluate error indicator for ‖sµquery

− sδµquery‖Y , E δ

µquery.

‖smeas − sδµquery‖Y ≤ ‖smeas − sµquery‖Y + ‖sµquery − sδµquery‖Y︸︷︷︸

/Eδµquery

.





Comparison Opportunities

Triangle Inequality: given f ∈ Y , g [1] ∈ Y , g [2] ∈ Y ,

max(‖f − g [1]‖Y , ‖f − g [2]‖Y ) ≥ 12‖g [1]− g [2]‖Y .

Application I: Blunder Detection

I f = sµ for given model

I g [1] ≡ sδµ[1] and g [2] ≡ sδµ[2] different approximations to s

Conclusion: 12‖s

δµ[1]− sδµ[2]‖Y > max(E δµ[1],max(E δµ[2])⇒ blunder.

Application II: Model Selection (approximation error negligible)

I f = smeas

I g [1] ≡ sµ[1] and g [2] ≡ sµ[2] different model predictions

Conclusion: 12‖sµ[1]− sµ[2]‖Y > εtol ⇒ at least one model deficient.




Query: Spoon ProblemQuery: Eggplantslice ProblemQuery: Brick Wall Problem

Problem Statement: Spoon. . . ArtiePIMA

Modified 2.051 Midterm Exam Question

A cup contains tea. Air is in contact with the tea; also airsurrounds the cup. The tea is maintained at temperature Tliq; theair is maintained at temperature T∞. The tea-air interface is atx = 0. A spoon comprises two parts: a head connected to ahandle. The head rests in the cup. The head is immersed in thetea; the handle is exposed to the air.

The spoon geometry is approximated as a right cylinder withrectangular cross-section for dimensions a and b; the axialcoordinate is x . The head is of length L1; the handle is of lengthL2. For our coordinate system, the head extends from x = −L1 tox = 0, and the handle extends from x = 0 to x = L2.





. . . Problem Statement: Spoon

Let h1,bot denote the heat transfer coefficient from head to teaprescribed over the axial face at x = −L1; let h2,top denote theheat transfer coefficient from handle to air prescribed over theaxial face x = L2; let h1,lat denote the heat transfer coefficientfrom head to tea prescribed over the head lateral face; let h2,lat

denote the heat transfer coefficient from handle to air prescribedover the handle lateral face.

The head has thermal conductivity k1; the handle has thermalconductivity k2; the cup is an insulator.

Plot the temperature as a function of the axial coordinate. Youmay use the following numerical values: a = 0.002, b = 0.01,L1 = 0.05, L2 = 0.12, h1,bot = 10.0, h2,top = 5, h1,lat = 10.0,h2,lat = 5.0, k1 = 50, k2 = 50, Tliq = 90, and T∞ = 23.


A cup contains tea. Air is in contact with the tea; also air surrounds the cup. The tea is maintained at temperature $Tliq$; the air is maintained at temperature $Tinf$. The tea - air interface is at ! x = 0 !. A spoon comprises two parts: a head connected to a handle. The head rests in the cup. The head is immersed in the tea; the handle is exposed to the air.

The spoon geometry is approximated as a right cylinder with rectangular cross-section for dimensions $a$ and $b$; the axial coordinate is $x$. The head is of length $L1$; the handle is of length $L2$. For our coordinate system, the head extends from ! x = 0 - L1 ! to ! x = 0 !, and the handle extends from ! x = 0 ! to ! x = L2 !.

Let $h1bot$ denote the heat transfer coefficient from head to tea prescribed over the axial face at ! x = 0 - L1 !; let $h2top$ denote the heat transfer coefficient from handle to air prescribed over the axial face ! x = L2 !; let $h1lat$ denote the heat transfer coefficient from head to tea prescribed over the head lateral face; let $h2lat$ denote the heat transfer coefficient from handle to air prescribed over the handle lateral face.

The head has thermal conductivity $k1$; the handle has thermal conductivity $k2$; the cup is an insulator.

Plot the temperature as a function of the axial coordinate. You may use the following numerical values: ! a = 0.002!, ! b = 0.01 !, ! L1 = 0.05 !, ! L2 = 0.12!, !h1bot = 10.0!, !h2top = 5!, !h1lat = 10.0!, !h2lat = 5.0!, ! k1 = 50 !, ! k2 = 50!, !Tliq = 90!, and !Tinf = 23!.





artie('spoon')

S1. Form problem data: PD ≡ Spoon ; KB .S2. Parse PD 7→ PDparse.





M01 ) ∈ LM2 ; δ.







S3.Topology: G (Physical Entities,Thermal Connections)





S3,S4a. Geometry: Ωµquery, ∂Ωµquery






µquery ≡ L1, L2, k1, k2, h1lat, h

2lat, ,T

1∞,T

2∞, hb, ht,T∞,b,T∞,t

I L1 ← L1 ≡ 0.05I L2 ← L2 ≡ 0.12I k1 ← k1 ≡ 50.0I k2 ← k2 ≡ 50.0I h1

lat ← h1,lat ≡ 10.0I h2

lat ← h2,lat ≡ 5.0I T 1

∞ ← Tliq ≡ 90.0I T 2

∞ ← T∞ ≡ 23.0I hb ← h1,bot = 10.0I ht ← h2,top = 5.0I T∞,b ← Tliq ≡ 90.0I T∞,t ← T∞ ≡ 23.0





S5,S6. Approximate Output sδµquery; Error Estimate E δ

µquery

Bitransverse ≡ maxcomponents

hlatAP

k: E δµquery = c1Bitransverse.





Problem Statement: EPtext

An eggplantslice is placed on a cork table. The eggplantslice andcork table are surrounded by air. The eggplantslice and cork tableand air are enclosed within roomwalls. Both the air and roomwallsare maintained at temperature T∞ .

The eggplantslice is initially at temperature Ti . We consider thetime interval 0 < t < tfinal.

The eggplantslice geometry is approximated as a right cylinder.The eggplantslice cross-section boundary is highlighted in red inthe figure. The eggplantslice cross-section is of maximum heightHmax and maximum width Wmax . The eggplantslice is of thicknessL. The axial coordinate is x ; in our coordinate system, theeggplantslice extends from x = 0 to x = L.





. . . Problem Statement: EPtext ArtiePIMA

The acceleration of gravity is of magnitude g and direction −x .

The eggplantslice axial face at x = L and the eggplantslice lateralface are exposed to the air and view the roomwalls. Theeggplantslice axial face at x = 0 is insulated.

The eggplantslice has thermal conductivity k and thermaldiffusivity α; the eggplantslice and roomwalls are blackbodies; thecork table is an insulator.

Plot the temperature on the axial face at x = L as a function oftime. You may use the following numerical values: Hmax = 0.096,Wmax = 0.16, L = 0.026, k = 0.28, α = 5× 10−8, Ti = 6.7,T∞ = 24.1 g = 9.8, and tfinal = 3000.


An eggplantslice is placed on a cork table. The eggplantslice and cork table are surrounded by air. The eggplantslice and cork table and air are enclosed within roomwalls. Both the air and roomwalls are maintained at temperature $Tinf$ .

The eggplantslice is initially at temperature $Ti$ . We consider the time interval ! 0 < t < tfinal !.

The eggplantslice geometry is approximated as a right cylinder. The eggplantslice cross-section boundary is highlighted in red in the figure. The eggplantslice cross-section is of maximum height $Hmax$ and maximum width $Wmax$. The eggplantslice is of thickness $L$. The axial coordinate is $x$; in our coordinate system, the eggplantslice extends from ! x = 0 ! to ! x = L ! .

The acceleration of gravity is of magnitude $g$ and direction $- x$.

The eggplantslice axial face at ! x = L ! and the eggplantslice lateral face are exposed to the air and view the roomwalls. The eggplantslice axial face at ! x = 0 ! is insulated.

The eggplantslice has thermal conductivity $k$ and thermal diffusivity $alpha$; the eggplantslice and roomwalls are blackbodies; the cork table is an insulator.

Plot the temperature on the axial face at ! x = L ! as a function of time. You may use the following numerical values: ! Hmax = 0.096 !, ! Wmax = 0.16 !, ! L = 0.026 !, ! k = 0.28 !, !alpha = 0.00000005 !, !Ti = 6.7 !, !Tinf = 24.1 !, !g = 9.8 !, and !tfinal = 3000! .




Problem Statement: EPimageArtiePIMA

Red Boundary Contour: RGB = [255 0 0]






artie('eggplantslice','eggplantslice topview')

S1. Form problem data: PD ≡ EP , figure ; KB .S2. Parse PD 7→ PDparse.





M11 ) ∈ LM1 ; δ.







S3.Topology: G (Physical Entities,Thermal Connections)






µquery ≡ L,A/P, k, α, hb, ht, hlat,Ti,T∞, CrefI L← 0.028I A/P ← 0.0268: image boundary extraction [22]I k ← kep ≡ 0.28I α← αep ≡ 5× 10−8

I hb ← 0: insulated



`


hnc = kair` (0.68 + 0.67Ra0.25

` )(1 + ( 0.492Prair

)9

16 )−49

I Ti ← 6.7: data assimilation ATPI T∞ ← 24.1





S5. Approximate Output sδµquery= AM1

1 (µquery)





S6. Error Estimation

Bitransverse ≡hlat

AP

k: E δµquery = c1Bitransverse + c2(∆x)3/2 + c3∆t︸︷︷︸

negligible

.





Archive: Model and Approximations

Model Class M2:

I Ωµ: conforming union of bricks

I ∂Ωµ: lateral surfaces — normal ey or ezaxial surfaces — x = xleft, x = xright

I steady conduction

I insulated on lateral surfaces

I prescribed heat transfer coefficient on axial surfaces

I output: nondimensional heat transfer rate through wall

Approximations:

I (AM21 ,EM2

1 ): insulator-superconductor variational bounds

I (AM22 ,EM2

2 ): finite element (FE) approximation





Problem Statement: Brick Wall I

A wall separates inside air and outside air. The wall consists offour bricks: Brick 1, Brick 2, Brick 3, and Brick 4. Brick 1connects to Brick 2; Brick 2 also connects to Brick 3 and Brick 4.Brick 1 is in communication with inside air at temperature Tin;Brick 2, Brick 3, and Brick 4 are in communication with outside airat temperature Tout.

The coordinates are denoted x1, x2, and x3; x1 corresponds todistance through the wall. Each brick is a parallelepiped ofrectangular cross-section of dimensions a (in x2), b (in x2), L (inx1). The spatial domain of Brick 1 is 0 < x1 < L, 0 < x2 < a,0 < x3 < b; the spatial domain of Brick 2 is L < x1 < 2L,0 < x2 < a, 0 < x3 < b; the spatial domain of Brick 3 is





Problem Statement: Brick Wall II

L < x1 < 2L, 0 < x2 < a, b < x3 < 2b; the spatial domain of Brick4 is L < x1 < 2L, 0 < x2 < a, −b < x3 < 0.

Each brick has thermal conductivity kb.

Brick 1 is exposed to inside air over the face at x1 = 0 through heattransfer coefficient hin. Brick 2, Brick 3, and Brick 4 are exposedto outside air over the faces at x1 = 2L through heat transfercoefficient hout. The remainder of the boundary is insulated.

We introduce a nondimensional heat transfer rate H given byQ/(kb(Tin − Tout)a); here Q denotes the heat transfer rate intoBrick 1 over the face at x1 = 0. Develop a lower bound and alsoan upper bound for H.





Problem Statement: Brick Wall III

You may use the following parameter values: Tin = 23, Tout = 0,a = 0.1, b = 0.1, L = 0.05, hin = 10, hout = 100, and kb = 0.5.


A wall separates inside air and outside air. The wall consists of four bricks: brick 1, brick 2, brick 3, and brick 4. Brick 1 connects to brick 2; brick 2 also connects to brick 3 and brick 4. Brick 1 is in communication with inside air at temperature $Tin$; Brick 2, Brick 3, and Brick 4 are in communication with outside air at temperature $Tout$ .

The coordinates are denoted $x1$,$x2$,and $x3$; $x1$ corresponds to distance through the wall. Each brick is a parallelepiped of rectangular cross-section of dimensions $a$ (in $x2$), $b$ (in $x3$), $L$ (in $x1$). The spatial domain of Brick 1 is !0 < x1 < L, 0 < x2 < a, 0 < x3 < b!; the spatial domain of Brick 2 is !L < x1 < 2*L, 0 < x2 < a, 0 < x3 < b!; the spatial domain of Brick 3 is !L < x1 < 2*L, 0 < x2 < a, b < x3 < 2*b!; the spatial domain of Brick 4 is !L < x1 < 2*L, 0 < x2 < a, 0 - b < x3 < 0!.

Each brick has thermal conductivity $kb$ .

Brick 1 is exposed to inside air over the face at !x1 = 0! through heat transfer coefficient $hin$ . Brick 2, Brick 3, and Brick 4 are exposed to outside air over the faces at !x1 = 2*L! through heat transfer coefficient $hout$. The remainder of the boundary is insulated.

We introduce a nondimensional heat transfer rate $H$ given by !Q/(kb*(Tin-Tout)*a)!; here $Q$ denotes the heat transfer rate into Brick 1 over the face at !x1 = 0!. Develop a lower bound and also an upper bound for $H$. You may use the following parameter values:!Tin = 23!, !Tout = 0!, !a = 0.1!, !b = 0.1!, !L = 0.05!,!hin = 10!, !hout = 100!, and !kb = 0.5!.




S3. Topology: G (Physical Entities,Thermal Connections)





S5,S6. Approximate Output sδµquery; Error Estimate E δ

µquery

Approximation: (AM2

1 ,EM2

1 ) — variational bounds

sδµquery lower bound = 0.64516, sδµquery upper bound = 0.84507

⇓sδµquery = 0.7451,E δµquery = 0.100.

Approximation: (AM2

2 ,EM2

2 ) — FE approximation (2D)4

sδµquery = 0.72198, E δµquery = 7.71× 10−6.

Comparison test: no blunders detected.

4FE Templates (M Yano) [31, 25]: adaptive refinement; extrapolation errorestimators.





S5. Field Visualization




References




[1] NH Afgan and M Da Graca Carvalho.

A confluence-based expert system for the detection of heat exchangerfouling.

Heat Transfer Engineering, 19(2):28–35, 1998.

[2] SD Ali, HS Ramaswamy, and GB Awuah.

Thermophysical properties of selected vegetables as influenced bytemperature and moisture content.

J Food Process Engr, 25:417–433, 2002.

[3] ASHRAE.

2017 ASHRAE Handbook - Fundamentals.

ASHRAE, Atlanta, GA, 2017.

[4] DG Bagby and RA Cormier.

A heat exchanger expert system.

AHSRAE Transactions, 95(2), 1989.

[5] J Ballani, P Huynh, D Knezevic, L Nguyen, and AT Patera.

PDE Apps for acoustic ducts: A parametrized component-to-systemmodel-order-reduction approach.




In Proceedings ENUMATH 2017, Voss, Norway. Springer, 2018.

[6] JP Birk.

The computer as student: an application of artifical intelligence.

J Chem Educ, 69(4):294–295, April 1992.

[7] DG Bobrow.

Natural language input for a computer problem solving system.

Technical Report Report MAC-TR-1, Project MAC, MIT, June 1964.

[8] D Braun, AH Mendez, F Matthes, and M Langen.

Evaluating natural language understanding services for conversationalquestion answering systems.

In Proceedings of the SIGDIAL 2017 Conference, pages 174–185.Association for Computational Linguistics, 2017.

[9] WJ Cochran, D Hainley, and L Khartabil.

Knowledge based system for the design of heat exchangers.

In SPIE Applications of Artificial Intelligence 1993: Knowledge-BasedSystems in Aerospace and Industry, volume 1963, 1993.




[10] D Das, D Chen, AFT Martins, and NA Smith.

Frame-semantic parsing.

Computational Linguistics, 40(1), 2014.

[11] HG Elrod.

Two simple theorems for establishing bounds on the total heat fow insteady-state heat-conduction problems with convective boundaryconditions.

ASME J Heat Transfer, 96(1):65–70, 1974.

[12] JR Finkel, T Grenager, and C Manning.

Incorporating non-local information into information extraction systems byGibbs sampling.

In Proceedings of the 43nd Annual Meeting of the Association forComputational Linguistics (ACL 2005), pages 363–370, 2005.

[13] JS Hesthaven, G Rozza, and B Stamm.

Certified reduced basis methods for parameterized partial differentialequations.

Springer, 2016.Anthony T Patera, MIT Artie: An Artificial Heat Transfer Student



[14] J Hokanson.

Software urlread2 for http requests and response processing inMATLAB, 2012.

[15] A Hormann.

Gaku: An artifical student.

Behavioral Science, 10:88–107, 1965.

[16] DBP Huynh.

Software GNLRequest: Matlab client for Google Natural LanguageProcessor, 2018.

[17] JH Lienhard IV and JH Lienhard V.

A Heat Transfer Textbook.

Phlogiston Press, Fourth Edition, version 2.12 July 15, 2018, available athttp://ahtt.mit.edu edition, 2018.

[18] N Kushman, Y Artzi, L Zettlemoyer, and R Barzilay.

Learning to automatically solve algebra word problems.

In Proceedings 52nd Annual Meeting of the Association forComputational Linguistics, 2014.


http://ahtt.mit.edu



[19] HP Langtangen and A Logg.

Solving PDEs in Python - The FEniCS Tutorial Volume 1.

Springer, 2017.

[20] Google LLC.

Google Natural Language Processor.

https://cloud.google.com/natural-language/docs/, 2018.

[21] M Magen, BB Mikic, and AT Patera.

Bounds for conduction and forced convection heat transfer.

International Journal of Heat and Mass Transfer, 31(9):1747–1757, 1988.

[22] MathWorks.

MATLAB.

https://www.mathworks.com/.

[23] A Mukherjee and U Garain.

A review of methods for automatic understanding of natural languagemathematical problems.

Artificial Intelligence Review, 29(2):93–122, 2008.Anthony T Patera, MIT Artie: An Artificial Heat Transfer Student

https://cloud.google.com/natural-language/docs/



[24] AT Patera.

Project Artie: An artificial student for disciplines informed by partialdifferential equations.

Technical report, arXiv (math), https://arxiv.org/abs/1809.06637,September 2018.

[25] P-O Persson.

Software DistMesh (MATLAB) for generation and manipulation ofunstructured simplex meshes, 2004-2012.

[26] A Quarteroni, A Manzoni, and F Negri.

Reduced basis methods for partial differential equations.

Springer, 2016.

[27] A Quarteroni and A Valli.

Numerical Approximation of Partial Differential Equations.

Springer, 2008.

[28] L Ridgway Scott.

Introduction to Automated Modeling with FEniCS.Anthony T Patera, MIT Artie: An Artificial Heat Transfer Student



Computational Modeling Initiative, 2018.

[29] Universal Technical Systems.

UTS Software, Heat Transfer 5.0, 2-18.

[30] B Szabo and I Babuska.

Introduction to Finite Element Analysis: Formulation, Verification andValidation.

Wiley, first edition, 2011.

[31] M Yano.

Software fem2d (MATLAB) for finite element solution of PDEs in twospace dimensions, 2018.


artie: an arti cial heat transfer student - icerm.brown.edu · s1. form problem data: pd fps;kbg...

Documents