46
FEM in Heat TransferPart 2Part 2
Marcela B. Goldschmit
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CONTENTS
Part 1Introduction to heat transferHeat transfer equationsNon-dimensional numbersThe finite element method in heat transferB d diti t l ti f d tiBoundary conditions: natural convection, forced convectionBoundary conditions: radiation, boilingExamples on conduction-convection heat transfer problems
Part 2Boundary conditions: reviewBoundary conditions: condensationBoundary conditions: condensationTime integrationNon-linear equations: Picard method, Newton-Raphson method, BFGSExamples on transitory thermal problems
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Examples on transitory thermal problems
CONTENTS
Part 3Non-linear heat transferNon-linear heat transfer: thermal conductivity, forces term, volumetric termNon-linear heat transfer: radiation BCNon linear heat transfer: phase changeNon-linear heat transfer: phase changeModeling of heat transfer: weldingExamples on non-linear heat transfer problems
Part 4Inverse thermal problemsInverse thermal problemsExamples on phase change problems
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Boundary conditions: review
�
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Boundary conditions: review
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Boundary conditions: reviewNatural and Forced Convection
( )[ ]4
32
1
Pr/12
1
n
nn
n
A
RaAANu⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++=
( )[ ]Pr/1 3A ⎪⎭⎪⎩ +
Lh C/fkLhNu =
kCp
Cpkμ
ρρμ
==/
/PrPrGrRa =
( ) 3β LTT( )( )2
3
/ ρμβρ LTTGr w ∞−
=
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Boundary conditions: reviewB ili gBoiling
Nucleate
Boiling
Transition
Boiling
Film
Boiling
Free
Conv.
qcrit)
Boiling Boiling BoilingConv.
BCHF
log
(qs)
qmin C
A
MHF
log (ΔTe)
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Boundary conditions: condensation
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Time integration
( ) xQTktTCp Ω∈=∇⋅∇−∂∂ρ
xqq
xTTt
Tnn
Timp
impΤ∈=
Τ∈=∂
( ) conditioninitialTxT 00, =
( ) xHxtTxHtxT ˆ)(ˆ)(,~ ⋅=⋅=
( )( )
=Ω∇⋅∇−Ω∂∂
∫∫ ΩΩdTkHd
tTCpH TT ~~
ρ Part integration
( ) Τ−+Ω ∫∫ ΤΩdqqHdQH
qimpnn
TT
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•
Time integration
FTKTM =⋅+⋅•
ˆˆ
dBkBK
dhChM
G
jpie
Gij
e
Ω
Ω=
∫∑
∫∑Ω
ρ
kk
k ⎥⎥⎤
⎢⎢⎡
= 0000
hhhT
dBkBK
n
pjmpime
Gij
e
~21
⎥⎤
⎢⎡ ∂∂∂
⎥⎤
⎢⎡ ∂
Ω= ∫∑Ω
L niiG
i qhdQhF
k
∑∫∑∫ −Ω=
⎥⎥⎦⎢
⎢⎣ 00
TBT
hhhyh
yh
yh
xxx
yTx
n ˆˆ
~
~21 ⋅=⋅
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
∂∂∂∂∂
∂∂
∂∂
∂∂∂
=
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
∂∂∂
L
imp
q
ne
ie
ii qhdQh ∑∫∑∫ΤΩ
zh
zh
zh
zT n21 ⎥
⎥
⎦⎢⎢
⎣ ∂∂
∂∂
∂∂
⎥⎥⎥
⎦⎢⎢⎢
⎣ ∂∂ L
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Time integrationTime integration with convection therms
( ) QTkTvCTC pp =∇⋅⋅∇−∇⋅+∂ ρρ
( ) FTKNTM =⋅++⋅•
ˆˆ
( ) Qt pp ∂
ρρ
( ) FTKNTM =⋅++⋅
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Time integration
Time integration with convection therms
dBvhNdBkBK pjpi
eGijpjmpim
eGij ; Ω=Ω= ∫∫
ee∫∫ΩΩ
impnivi
eGi qhdqhF ∫∫ −Ω=
imp
eqe
nivii ∫∫ΤΩ
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Time integrationTime integration with convection therms
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Time integrationAlpha Method
( ) FTKNTM =++•
ˆˆ ( ) FTKNTM =⋅++⋅
The objective is to obtain an approximation for given the value of
and
Ttt Δ+ Tt
FF ttt Δ+
Alpha Method seeks to satisfy the differential equation in Alpha Method seeks to satisfy the differential equation in
10; ≤≤Δ+ αα tt
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Time integration: Alpha Method
( ) FTKNTM =⋅++⋅•
ˆˆ
tTTT
ttttt
Δ+•Δ+
Δ−
=αˆˆˆ
( ) ( ) TTttTTTT tttttt
ttt Δ+Δ+
Δ+ +−=Δ+Δ−
+= ααϑαα ˆˆ1ˆˆˆˆ 2( ) ( )
( ) FFF
ttt
ttttt Δ+Δ+ +
Δ
αα
ααϑα
α 1( ) FFF +−= αα1
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Time integration: Alpha Method
αααα
α
1=α Implicit Euler backward Method, unconditionally stable ( )tΔϑ
0=α Explicit Euler forward Method, conditionally stable
( )
( )tΔϑ
21
=α Implicit trapezoidal rule, unconditionally stableCranck Nicolson method
( )2tΔϑ
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2 Cranck Nicolson method
Time integration: Alpha Method
From Zienkiewicz & Taylor, The FiniteElement Method
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Element Method
Time integration: Alpha Method
Approximation error
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Time integrationPenetration depth measures the distance or thickness of thermal energy propagating into Penetration depth measures the distance or thickness of thermal energy propagating into the surface through conduction.
4 k
( ) 010
4
<−
=
initial
p
TT
tC
γ
ργ
is the minimum time at which temperature resultsare desired.
mint
( ) 01.0<− initialBC
initial
TTγ
N is the elements number to discriteze thepenetration depth
4 k
θTTipically N = 6 to 10
min4 t
Ck
Nx
pρ=Δ
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From Bathe, Finite Element Proceduresθ=TTipically N = 6 to 10
Non-linear equations
( ) RTF =ˆ ( ) RTTK =⋅ ˆSteady State
⎟⎠⎞⎜
⎝⎛=⎟
⎠⎞⎜
⎝⎛ Δ+Δ+Δ+ TRTTF
tttttttt ˆˆ,ˆTransient State ⎠⎝⎠⎝
( ) ( )TRTTTK tttttttt Δ+Δ+Δ+ =⋅ ˆ,
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Non-linear equations: Picard Method
It is called successive substitutions method.
Starting with an initial guess
Evaluate kk += 1( ) ( ) ( )TRTTTK
kktttktttktt Δ+Δ+−Δ+
=⋅⎟⎠⎞⎜
⎝⎛
+=
ˆ,ˆ
11
⎠⎝
Until the result no longer changes to within a specified toleranceUntil the result no longer changes to within a specified tolerance
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Non-linear equations: Picard Method
Picard's method is the easiest method to program and usually has large areas of convergence .g
Converges linearly and for many problems its convergence rate is very smooth
The most important application of Picard's method is to use it as the first iterations of the Newton-Raphson method .
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Non-linear equations: Newton-Raphson Method
Historical Note.
Newton's work was done in 1669 but published muchNewton s work was done in 1669 but published muchlater. Numerical methods related to the Newton Method were used by al-Kash, Viete, Briggs, and Oughtred, all many years before Newton.
Raphson, some 20 years after Newton, got close to Newton Equation, but only forpolynomials of degree 3, 4, 5, . . . , 10.
Raphson like Newton seems unaware of the connection between hisRaphson, like Newton, seems unaware of the connection between hismethod and the derivative. The connection was made about 50 years later(Simpson, Euler), and the Newton Method finally moved beyond polynomialequations. The familiar geometric interpretation of the Newton Method mayhave been first used by Mourraille (1768). Analysis of the convergence ofthe Newton Method had to wait until Fourier and Cauchy in the 1820s.
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Non-linear equations: Newton-Raphson Method
Steady state problem ( ) ( ) ( ) TTKTFTFR ˆˆˆ;0ˆ ⋅==−y p ( ) ( ) ( );0
( )1−∂
kFLinearized ( ) ( )
( ) ( ) ( )
,11 ;ˆ
ˆ =− Δ
∂∂
+= NEQjk
jj
k TTFFF
( ) ( ) ( )1ˆˆˆ −−=Δ kj
kj
kj TTT
iFK ∂Call tangent matrix
j
iT T
Kij ˆ∂
∂=
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Non-linear equations: Newton-Raphson Method
( ) ( ) ( )0ˆ11 =Δ⋅−− −− kkk TKFR 0=Δ⋅−−
TTKFR
( ) ( ) ( )11 ˆ −− Δ kkk FRTK ( ) ( )
( ) ( ) ( )1ˆˆˆ −−=Δ
−=Δ⋅kkk
T
TTT
FRTK
Start conditions( )
TT ˆˆ 0=Start conditions dataTT =
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Non-linear equations: Newton-Raphson Method
Transient state problem
TTKFFRtttttttttttt ˆˆ;0
Δ+Δ+Δ+Δ+Δ+Δ+ ⋅⎟⎠⎞⎜
⎝⎛==−
Linearized ( )( )
( ),1
1
1 ;ˆˆ =
Δ+
−
Δ+
Δ+−Δ+Δ+ Δ
∂∂
+= NEQjk
jtt
k
jtt
ttktttt T
TFFF
( ) ( ) ( )1ˆˆˆ −Δ+Δ+Δ+ −=Δ kj
ttkj
ttkj
tt
j
TTT
Call tangent matrixj
tti
tt
Ttt
TFK
ij ˆΔ+
Δ+Δ+
∂∂
=
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Non-linear equations: Newton-Raphson Method
( ) ( ) ( )0ˆ11 =Δ⋅−− Δ+−Δ+−Δ+Δ+ kttkttktttt TKFR 0Δ
TTKFR
( ) ( ) ( )11 ˆ −Δ+Δ+Δ+−Δ+ Δ kttttkttktt FRTK ( ) ( ) ( )
( ) ( ) ( )1
11
ˆˆˆ −Δ+Δ+Δ+
Δ+Δ+Δ+Δ+
−=Δ
−=Δ⋅kttkttktt
kttttttkT
tt
TTT
FRTK
Start conditions
( ) ( ) ( ) FFKKTT tttT
tT
ttttt=== Δ+Δ+Δ+ 000
;;ˆˆ
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Non-linear equations: Newton-Raphson Method
For one degree of freedom
( )1−kx( )kx
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Non-linear equations: Newton-Raphson Method
E l W th N t R h M th d t fi d l ti Example: We use the Newton-Raphson Method to find a non-zero solution of
x = 2 sinx
(a) Start x(0)= 1.1
(b) Start x(0)= 1.5
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Non-linear equations: Newton-Raphson Method
If the initial estimate is not close enough to the root, the Newton-RaphsonM h d h Method may not converge, or may converge to the wrong root.
The successive estimates of the Newton-Raphson Method may converge to the root too slowly, or may not converge at all.
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Non-linear equations: Newton-Raphson Method
T
T
( )1T̂Δ ( )2T̂Δ
Ttt ˆΔ+Tt ˆ Temperature
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Non-linear equations: Newton-Raphson Method
T
( )0T
tt
T
KTf
t
Δ+−=∂∂
T( )
( )1
1T
tt
T
KTf
tt
Δ+−=∂∂
Δ+
( )1T̂Δ ( )2T̂Δ Ttt ˆΔ+Tt ˆTemperature
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Ttt Δ+
Non-linear equations: Newton-Raphson Method
Convergence
Quadratic convergence when converges
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Non-linear equations: Newton-Raphson Method
Convergence
(1) First property
• If the tangent matrix is nonsingular( )1−Δ+ ktt K• If the tangent matrix is nonsingular
• If and its first derivatives with respect to are
TK
( )1−Δ+ ktt F( )1ˆ −Δ+ ktt
T
continuous in a neighborhood of the solution
• If will be closer to ( )1ˆ −Δ+ ktt
T *T̂
tt Δ+
*T̂
tt Δ+
• If will be closer to
than and the sequence of iterative solutions converges to
T T
( )kttT̂
Δ+ *T̂
tt Δ+
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Non-linear equations: Newton-Raphson Method
Convergence
(2) Second property – Lipschitz continuity
• If the tangent matrix satisfies( ) ( ) ( ) ( )11 ˆˆ −Δ+Δ+−Δ+Δ+ −≤−
kttkttkT
ttkT
tt TTLKK
for all in the neighborhood of
d L 0
( ) ( )1ˆˆ −Δ+Δ+ kttkttTandT
*T̂
tt Δ+
and L>0
then convergence is quadratic.
This means that if the error after iteration (k) is the order e, then the error after iteration (k+1) will be of the order e2
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Modified Newton-Raphson Method
N-R iteration is recognized as an expensive computational cost per iteration due
to the calculation and factorization of the tangent matrix. Then, the use of a
modification of the full N-R algorithm can be effective.
Maintains the o tangent matrix constant during the iterations or
it is modified each n iterations
( )1−Δ+ kT
tt K( )xf '
Advantage: saving computational effort
Disadvantage: loss of quadratic convergence g q g
The choice of time step depend on the degree of non-linearities
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Quasi-Newton Methods: BFGS
As an alternative to forms of N-R iteration, a class of methods known as matrix update methods or quasi-Newton methods has been developed .
These methods involve updating the coefficient matrix to provide a secantapproximation to the matrix from iteration (k-1) to (k).
BFGS: Broyden, Fletcher, Goldfarb and Shanno method
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Quasi-Newton Methods: BFGSInicialize las
k = k+1
Inicialize lasmatrices y vectores,
k=0
k = k+1
t+∆t∆T(k) = (t+∆tKT‐1)(k‐1) . (t+∆tR ‐ t+∆tF(k‐1))
Calcular t+∆tF(k)( t+∆tT(k) )
(t+∆tKT‐1)(k) = A(k)T . (t+∆tKT‐1)(k‐1) . A(k)t+∆tT(k) = t+∆tT(k‐1) + t+∆t∆T(k)
F ( T )
Converge en alguna nonorma
siSiga con el paso
de tiempo sig iente
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de tiempo siguiente
Quasi-Newton Methods: BFGS
(k)T( )TkA( )kA
TT
TTT T
TTT
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Quasi-Newton Methods: BFGS
( ) ( ) ( )
( ) ( ) ( )1
1ˆˆˆ−Δ+Δ+Δ+
−Δ+Δ+Δ+
Δ
−=Δkttkttktt
kttkttktt
FFF
TTT( ) ( ) ( )1Δ+Δ+Δ+ −=Δ kttkttktt FFF
Since the product( )kA( )TkA
is positive definite and symmetric, to avoid numerically problems, the condition number is calculated.
The update is performed if:( ) ( )510=< nexampleasnc k
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BFGS with linear searches ( ) ( ) ( )1ˆˆˆ −Δ+Δ+Δ+ Δ
kttkttktt TTT β( ) ( ) ( )Δ+ −=Δtt TTT β
b is a scalar multiplier
It is varied until the component of the out-of-balance loads in the
direction is small.( )ktt T̂ΔΔ+ TΔ
( ) ( )( ) ( ) ( )( )1ˆˆ −Δ+Δ+Δ+Δ+Δ+Δ+ Δ−ΔΔ≤Δ−ΔΔ kttttTkttktttt
Tktt FRTTOLFRT ( ) ( )O
Linear searches are made with simple algorithms such as bisection
Linear searches are computationally expensive because they must calculate multiple times in each iteration ( )ktt FΔ+
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BFGS with linear searches Inicialize las
k = k+1
matrices y vectores, k=0
t+∆t∆T(k) = (t+∆tKT‐1)(k‐1) . (t+∆tR ‐ t+∆tF(k‐1))
Calcular t+∆tF(k)( t+∆tT(k) )
(t+∆tKT‐1)(k) = A(k)T . (t+∆tKT‐1)(k‐1) . A(k)t+∆tT(k) = t+∆tT(k‐1) +β t+∆t∆T(k)
Converge en alguna norma
no
siSiga con el paso
de tiempo siguiente
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Convergence criteria
1) Convergence in temperatures ( )DTOLT
ktt <ΔΔ+ ˆ
2) Convergence in porcentual values
( )
( ) ETOLT
ktt
ktt
<Δ
−Δ
Δ+
1ˆ
ˆ
( )T
ktt ΔΔ+ 1
( ) in
i
in
i
i aaaaaa max;;1
11
2
2===
∞==∑∑
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Examples on transitory heat transfer problemsExercise 1: Obtain the FEA formulation for the Linear Transitory heat transfer problem
considering convection. Analyze the stability of the different timeintegration
Exercise 2: Consider the transitory heat transfer problem in a 1D beam discretizedy pwith 10 regular elements. Solve the finite element model with time integration for different alpha values (0; 0.5 and 1) for the following cases:
Heat transfer equation
Border Condition
Initial Condition
Use this non-dimensional numbers for the analysis:
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Examples on transitory heat transfer problems
Exercise 3: Consider a 90º semi-infinite cylinder. Sides AB and BC are subjected to prescribed temperature of 50º. The initial temperature profile is 0º. The heat capacity of the material is constant. Perform a transient analysis to calculate the temperature distribution within the semi infinite domain at different values of time Use the Euler distribution within the semi-infinite domain at different values of time. Use the Euler Backward, Cranck Nicholson and Euler Forward Method.
The domain is discretized using a 10 × 10 mesh of 4 node 2 D conduction elementsof 4-node 2-D conduction elements.The conduction matrix is evaluated using a consistent heat capacity matrix. The timestep is Δt = 0.016.
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