THE STATIC CONDENSATION REDUCED BASIS ELEMENT
METHOD FOR A MIXED-MEAN CONJUGATE HEAT EXCHANGER
MODEL _
SYLVAIN VALLAGHE _ y AND ANTHONY T. PATERA
z
Abstract.
We propose a new approach for the simulation of conjugate heat exchangers. First, we introduce
a dimensionality-reduced mathematical model for conjugate (uid-solid) heat transfer: in the uid
channels, we consider a mixed-mean temperature de_ned on 1D _laments; in the solid we consider
a detailed PDE conduction representation. We then propose a Petrov-Galerkin _nite element (FE)
numerical approximation which provides suitable stability and accuracy for our mathematical model.
We next apply the static condensation reduced basis element (scRBE) method: a domain decomposition with parametric model order reduction at the intradomain level to populate a Schur complement
at the interdomain level. We start by building a library of \components," each corresponding to a
subdomain with a simple uid channel geometry; for each component, we prepare Petrov-Galerkin
reduced basis bubble approximations (and error bounds). We then form the linear system by static
condensation, and solve for the temperature distribution in the whole domain.
System-level error bounds are derived from matrix perturbation arguments; we also introduce a new output error
bound which is sharper than the original scRBE estimator. We present numerical results for a 2D
car radiator model which demonstrate the exibility, accuracy and computational e_ciency of our
approach.
Key words.
Reduced basis, conjugate heat transfer, domain decomposition
AMS subject classi_cations.
65N12, 65N15, 65N30, 65N55
1. Introduction.
Heat exchangers are designed to improve the heat transfer
between two di_erent media. They are commonly used to prevent car engines from
overheating: a uid circulates in the engine so that it receives some of the heat
produced by the engine; then the uid goes into a radiator (the heat exchanger)
which is cooled by the air owing past it; _nally the uid goes back to the engine,
ready to exchange heat as it is now at lower temperature than the engine.
Being able to predict the performance of a heat exchanger is an important step
of its design. Some analytical methods such as the Log-Mean Temperature Di_erence
(LMTD) and the Number of Transfer Units (NTU) allow to calculate the total heat
transfered in a heat exchanger. Yet they both require estimation of the overall heat
transfer coe_cient and exchange area, which can be complicated for a heat exchanger
with complex media and/or geometry.
A re_ned analysis of a non-trivial heat exchanger hence requires a conjugate
heat transfer model, describing the thermal interactions between the di_erent solids
and uids involved. This requires derivation of the appropriate partial di_erential
equations (PDE) for the temperature, and application of typically rather expensive
numerical solution procedures. For the case of interest here, car radiators, we must
consider the uid owing into the radiator tubes, the radiator solid part, and the
air ow past the radiator. The simplest situation is a uid ow in a straight channel with walls of constant thickness. In this case the conjugate heat transfer can be
readily solved analytically. However, in more realistic con_gurations with more complex channels and also _nned solid exchange surfaces, numerical approaches must be
_
Submitted to SIAM J Sci Comput on August 9, 2012.
Massachusetts Institute of Technology ( svallagh@mit.edu ).
z
Massachusetts Institute of Technology ( patera@mit.edu ).
y
1
2
S. VALLAGHE_AND A.T. PATERA
invoked [1, 19, 4, 15]. The latter typically preclude interactive or conceptul design.
Hence in this paper we consider a compromise between _delity and exibility, as we
now describe.
We propose some model order reduction strategies to eventually be able to simulate a complete 2D heat exhanger, while keeping a detailed description of the local
thermal interactions. We _rst introduce a 2D mathematical model for some simple
channel \component" situations where we consider the heat equation in the solid,
and the uid bulk temperature equation in the channels. The uid bulk temperature
is de_ned as a 1D variable along the channel axis, and we assume that the Nusselt
number at the channel walls is constant. As such, our model for conjugate heat transfer is not as detailed as others considering 2d equations for the uid and/or variable
Nusselt number [1, 19, 4, 15]. It can be viewed as a model order reduction to the
predominant dimension of the problem considered (the direction of the ow in the
channel), and is somehow similar to the methods presented in [5, 3, 13]. We then
derive a Petrov-Galerkin _nite element (FE) approximation of this model.
In a second stage, we consider the entire heat exchanger and we break it down into
subdomains having a simple channel geometry described by our mathematical model.
For each channel geometry type, we introduce some parameters (ow rate, heat transfer coe_cients) and perform parametric model order reduction using reduced basis
(RB) approximations [17] based on our FE formulation. At this point, we are able to
compute very rapidly the solution on a subdomain with a simple channel geometry
for any parameter values. To simulate the entire heat exchanger, we _rst choose parameters for each subdomain and compute the corresponding RB approximations; we
then invoke static consensation to assemble these component-level contributions in a
complete system description. This approach is denoted static condensation reduced
basis element (scRBE) method, and has been introduced in [7]. It allows to deal
e_ciently with very big parametric systems which can be broken down into many
subdomains having a few parameters each.
The key contributions of this paper are: _rst, a dimensionality-reduced mathematical model for heat exchangers; second, a Petrov-Galerkin FE numerical approximation for this mathematical model; third, a scRBE formulation for heat exchangers,
which in particular extends the reach of the scRBE to convection problems; fourth,
an improved a posteriori error bound for scRBE approaches in general.
The paper proceeds as follows. Section 2 gives a complete description of our model
for conjugate heat transfer within a single channel; we also provide the associated
Petrov-Galerkin _nite element approximation. In section 3, we recall the principle
of the RB method and give details about our particular RB spaces based on the
Petrov-Galerkin FE previously introduced. In section 4, we present some extensions
of our model to more complicated channel con_gurations, in which either a channel
splits in two, or two channels merge into one. In section 5, we present the core of our
approach, namely the static condensation reduced basis element (RBE) method [7].
Finally, in section 6, we provide numerical results. We start with a simple 1D problem
having an analytical solution, to compare our approach to a ground truth. We then
consider a 2D system, potentially large, corresponding to a car radiator model. We
show di_erent results on this system, reporting the accuracy and computational cost
of our approach.
2. Mathematical model.
We consider the heat transfer problem shown in _gure 2.1: a uid ows in a plane channel which goes through a solid surrounded by air.
We do not assume a particular shape of the solid domain , but we show some _ns
3
SCRBE FOR CONJUGATE HEAT TRANSFER
h ext
@ ext
solid
k
uid
_;c
x =0
F
y
g
sf
x
z
@ int
h int
x= L
g
@
int
@
ext
solid
Fig. 2.1
in _gure 2.1 as it will be our main geometry for the applications later. The geometry
and material properties are symmetric with respect to the xz plane. The material
properties are constant in the solid and the uid respectively.
The ow is assumed to be independent of the coordinate z. We will consider the
uid bulk temperature Tb, de_ned on a 1D _lament corresponding to the channel axis
(the dotted line in _gure 2.1), where for the bulk temperature we take the mixed-mean
temperature [12]. We denotesf the _lament coordinate and we suppose that we have
a mapping x = g( sf ), which here is simply the identity. Following [12], the uid bulk
temperature Tb( sf ) satis_es the following equation:
_cvA
dTb
= P hint ( T@
dsf
dim
int
( x ) s Tb( g( sf )))
0_ x _ L
S. VALLAGHE_AND A.T. PATERA
4
where A is the cross-sectional area of the channel,P is the perimeter of the channel,
_ is the mass density, c is the speci_c heat of the uid,
v is the average velocity,
and hint is the heat transfer coe_cient between the solid and the uid in the channel.
The superscript dim stands for \dimensional", as we will later non-dimensionalize the
equations. As a boundary condition, we set Tb@) = Tinlet .
The equation for the solid domain is
8 s k _ T = f sdim
>
>
>
>
< k @T = hext ( Ta s T )
@n
>
@T
>
>
k
= hint ( Tb s T )
>
:
@n
dim
in
on @
dim
ext
on @
dim
int
where hext is the heat transfer coe_cient between the solid and the air, and k is the
thermal conductivity of the solid.
Note that we will assume that the heat transfer coe_cient hint is constant along
the channel. Hence this model is most appropriate for turbulent ows which enjoy
relatively short development lengths and for which transport is largely dictated by
uid properties and lateral spatial scales. To support this assumption, we refer to the
Gnielinski correlation [6] for the Nusselt number of turbulent ows: it depends only on
Prandtl and Reynolds number which are both constant for a given channel diameter
and small temperature variations of the uid. In particular it does not depend on the
channel wall conditions as opposed to laminar ows.
2.1. Non-dimensional equations.
We de_ne the following dimensionless quanT X Ta
Tb X Ta
x
tities: _ = T inlet
;
_
=
;
_
=
X Ta
W , where W is a characteristic length (typT inlet X T a
ically the width of the base of the solid along the
y coordinate). Similarly we set
_cv A
W
W
L
_ = Wn and _ = W
. We also de_ne F = P k ; Bi int = h intk ; Bi ext = h extk . The
equations then become
8
<
_ _= fs
d_
F
= Bi int ( _@
:
d_
in
int
s _)
0_ __ _
B.1)
with the following boundary conditions
8
>
>
>
>
>
<
>
>
>
>
>
:
@_
= s Bi int ( _ s _ )
@_
@_
= s Bi ext _
@_
_ @) = 1
on @ int
on @ ext
B.2)
As a standard hypothesis we suppose thatF > 0, Bi int > 0 and Bi ext > 0.
2.2. Weak form. We will now derive a variational formulation for B.1) B.2).
To this end, we introduce some functional spaces. For the temperature _ in the
solid, we suppose that it has a homogeneous Dirichlet condition on some part of the
boundary _ _ @, such that _ \ @ int = ? and _ \ @ ext = ? . This subset _ of
the boundary will later correspond to \ports" where \components" can be connected
(section 5). Hence we take
_ 2 Y;
Y = H _1 () ;
H 01 ()
_ Y _ H 1 () :
5
SCRBE FOR CONJUGATE HEAT TRANSFER
For the uid bulk temperature
_ , we take
V = f ' 2 H 1 ([0; _]) ;' @) = 0 g:
_ 2 V;
To simplify the presentation we shall assume that the inhomogeneous boundary condition for the uid is lifted so that we can avoid the a_ne manifold associated with
the nonzero inlet condition, so in practice we consider _ @) = 0. We also de_ne the
following spaces: W = L 2 ([0; _]); X = Y _ V ; Z = Y _ W ; and when needed we
will aggregate the solid and uid temperature variable into one, using the following
notations: w = ( _;_ ) 2 X for the trial function; v = ( #;' ) 2 Z for the test function.
We also de_ne the following norms onX and Z
Z
kwk2X =
jr _j 2 +
2
Z
d_
+ _ 2 (_) ;
dx
[0 ; _]
Z
kvk2Z =
jr # j 2 +
Z
': 2
[0 ; _]
We are now ready to derive the variational formulation. We _rst de_ne the bilinear
form a on X _ Z such that for w 2 X and v 2 Z ,
a( w;v ) =
Z
Z
r _r # + Bi ext
@
_# + Bi int
Z
( _ s _ ) # s Bi int
@
ext
Z
@
int
( _ s _)' + F
int
Z
d_
'
[0 ; _] dx
B.3)
The bilinear form a thus has an obvious a_ne decomposition with respect to the
parameters _ = ( Bi ext ;Bi ;in t F ),
Q
a( w;v ; _ ) = X _q ( _ ) aq ( w;v ) ;
B.4)
q=0
with Q = 3 and _0 ( _ ) = 1, _1 ( _ ) = Bi ext , _2 ( _ ) = Bi int , _3 ( _ ) = F . We will drop
for now the dependency of a on _ and we will reintroduce it later for reduced basis
approximation. We also de_ne the linear form f such that for v 2 Z ,
f ( v) =
Z
fvs s a( z;v ) ;
where z 2 X is the lifted inhomogeneous boundary condition at the uid inlet.
Now taking the scalar product of B.1) with v 2 Z , and taking care of the boundary conditions B.2), we obtain the following weak form: u 2 X satis_es
a( u;v ) = f ( v) ; 8v 2 Z:
B.5)
Lemma 2.1. The bilinear form a is inf-sup stable and continuous on X _ Z .
d_
Proof. For any w = ( _;_ ) 2 X , we de_ne w_ = ( _;_ + _ dx ) 2 Z with _ being a
_
small constant. Replacing v with w in the bilinear form gives:
_
a( w;w ) =
Z
Z
jr _j 2 + Bi ext
@
s _Bi int
Z
@
int
_2 + Bi int
ext
Z
@
( _ s _)2
int
Z
d_
d_ 2 1
_
+ _F
+ ( F + _Bi int ) _ 2 (_) :
dx
2
[0 ; _] dx
S. VALLAGHE_AND A.T. PATERA
6
Using the identity (for c 2 R, d 2 R, _ 2 R+)
1 2
2j cjj dj_
c + _d;2
_
we obtain
Z
s 2_
@
int
d_ Z
_
dx
@
B.6)
2
1 2
d_
_ s _
:
_
dx
s
int
B.7)
From the trace theorem [16], and recalling that _ is zero on some part of @ int , we
also know that there is a constant _int > 0 such that
Z
@
F
Bi int
We choose_ =
jr _j 2 :
int
in B.7) so that
Z
s _Bi int
Z
_2 _ _int
_
@
int
2
int
d_
1 _inBi
t
_s
_
F
dx
2
Z
Z
1
_F
2
@
jr _js2
int
2
int
d_
:
dx
So _nally
2
int
1 _inBi
t
_
F
2
_
a( w;w ) _ A s
)
Z
jr _j 2 +
Z
1
_F
2
@
d_ 2 1
+ ( F + _Bi int ) _ 2 (_)
dx
2
int
_ K kwk2X ;
B.8)
for _ small enough. Also using integrating we can show that
Z
_2 _ _
2
Z
[0 ; _]
[0 ; _]
d_
;
dx
8_ 2 V:
and as a consequence
Z
(_ + _
[0 ; _]
2
2
Z
d_ 2 Z
d_
d_
) _
2(_ 2 + _2
) _ 2(_ + _2 )
:
dx
dx
[0 ; _]
[0 ; _] dx
De_ning C = max _ 1; p 2(_ + _2 ) _ , it follows that
kw_ kZ_
C kwkX
8w 2 X:
B.9)
This proves the inf-sup stability of a:
inf sup
w 2 X v2 Z
a( w;v )
a( w;w _ )
K kwkX
K
_ inf
_ inf
_
> 0;
_
w 2 X kw kk
w 2 X kw _ kZ
kwkk
C
X v kZ
X w kZ
and hence _0 _ KC is a lower bound for the inf-sup constant.
The continuity of a is obvious from B.3) using the trace theorem and the CauchySchwarz inequality. We give here a few details about how to bound the terms in
a( w;v ):
j
j
Z
Z
@
j
r _r # j_ (
_#j_ (
@
1
_2 ) 2 (
@
_' j_ (
int
1
jr _j 2 ) 2 (
Z
int
Z
Z
@
_2 ) (
int
1
# 2 ) 2 _ _2int (
@
1
2
1
jr # j 2 ) 2 _k wkk
X v kZ
Z
int
Z
Z
1
jr _j 2 ) 2 (
Z
1
jr # j 2 ) 2 _ _2int kwkk
X v kZ
int
Z
@
Z
1
' 2) 2 _
int
p
_(
Z
[0 ; _]
p
1
d_ 2 12 Z
) (
' 2 ) 2 _ _ kwkk
X vkZ
dx
[0 ; _]
7
SCRBE FOR CONJUGATE HEAT TRANSFER
Applying the same ideas to all the terms in a( w;v ), we get
j a( w;v ) j_
where
0
0
0 kwkk
X
vkZ
8w 2 X ; v 2 Z;
is an upper bound for the continuity constant de_ned as
p
p
= 1 + Bi ext _2ext + Bi in
_ t 2int + _ Bi in
_ t int + Bi in
_ t int + _ Bi int + F :
Note that _ext is the constant in the trace inequality on @
Z
Z
_2 _ _ext
@
ext ,
such that
jr _j 2 :
ext
Note that our assumption of Dirichlet conditions on _ can be relaxed such that
we consider what we shall denote the \all natural" problem in which Y = H 1 (). In
this case we invoke the completeH 1 norm on to retain well-posedness. Throughout
this paper (for convenience of exposition and simplicity) the rigorous analysis will be
applied to Y as de_ned above (i.e., with homogeneous Dirichlet conditions over part
of @), however on occasion for purposes of interpretation we shall consider the \all
natural" problem.
2.3. Finite element discretization.
Let Th be a simplicial mesh of and Sh a
mesh of [0; _] (Figure 2.1). We assume that the restriction of Th to the top or bottom
part of @ int is equal to Sh so that the mapping between [0; _] and @ int is trivial. We
introduce the following discrete spaces:Yh _ Y , the P1 _nite element space associated
with Th ; Vh _ V , the P1 _nite element space associated with Sh ; Wh _ W , the P0
_nite element space associated withSh ; X h = Yh _ Vh ; Z h = Yh _ W:h Note that the
dimensions of Vh and Wh are the same thanks to the condition _ @) = 0. Also, for
_ h 2 Vh , we denote _ h 2 Wh its average over each element inSh .
We now invoke these spaces to provide a Petrov-Galerkin approximation of B.3).
We de_ne the bilinear form ah on X h _ Z h such that for wh 2 X h and vh 2 Z h ,
ah ( u;v
h
h)
=
Z
r _h r # h + Bi ext
Z
_#
h
@
+ Bi int
Z
@
h
ext
( _h s _ h ) # h s Bi int
Z
@
int
( _h s _ h ) '
h
+ F
Z
[0 ; _]
int
d_h
': h
dx
B.10)
Note that ah ( w;v
h
h ) di_ers from a( w;v
h
h ) due to the average of _ h .
The problem for the discrete solution uh can then be stated as: uh 2 X h satis_es
ah ( u;v
h
h)
= f ( vh ) ; 8vh 2 Z:h
B.11)
Lemma2.2. The bilinear form ah is inf-sup stable and continuous on X h _ Z h .
Proof. The proof is very similar to the continuous case, except that the de_nition
_
of wh_ is slightly di_erent. For any wh = ( _;_
h
h ) 2 X h , we de_ne wh = ( _;_
h
h +
d_ h
_ dx ) 2 Z h with _ being a small constant. We must take the average of_ h to stay in
the required discrete test space. Then exactly as before, we arrive at
ah ( w;w
h
_
h)
_ A s
_ K
1 _inBi
t
_
F
2
kuh k2X
:
2
int
)
Z
jr _h j 2 +
Z
1
_F
2
@
int
d_h 2 1
+ ( F + _Bi int ) _ 2h (_)
dx
2
B.12)
S. VALLAGHE_AND A.T. PATERA
8
Using Simpson's rule for integration which is exact for _ 2h on each element ofSh and
the inequality B.6), we get
Z
Z
2
_h _
[0 ; _]
_;2h
8_ h 2 V:h
[0 ; _]
As a consequence, although there is a slight change by introducing_ h , the following
inequality still holds
kwh_ kZ_
C kwh kX
8wh 2 X:h
B.13)
This proves the inf-sup stability of ah :
inf
sup
w h 2 X h vh 2 Z h
ah ( w;v
h
h)
kwh kX kvh kZ
_
inf
wh 2 X h
_
ah ( w;w
K
h
h)
_
> 0:
_
kwh kX kwh kZ
C
B.14)
The continuity proof for ah is exactly the same than for a.
We thus note that _0 _ KC as introduced earlier is also a lower bound for the
inf-sup constant of ah . We now provide details for the computation of this quantity,
as we will need it later for RB a posteriori error estimation. First of all, we choose _
such that
1s
1 _inBi
t
_
F
2
2
int
=
1
_F()
2
2
_=
F +
_ int Bi
F
2
int
:
Then for K , we can take
K = min
_1
_
1
_F ; ( F + _Bi int ) :
2
2
C has been de_ned earlier as
C = max _ 1; p 2(_ + _2 ) _ :
In practice, we will have _ _ 1 and _ _ 1, so the actual lower bound that we compute
is
_hLB _
_F
:
p
2 2(_ + _2 )
Let u and uh be the solutions to the continuous and discrete
Proposition 2.3.
problems respectively, andh the mesh step size. Assuming that u 2 X \ H 2 () _
H 2 ([0; _]) , we then have the following a priori error estimate:
ku s uh kX
_O ( h)( kukX + j uj 2 ) ;
where j_j 2 is the Sobolev semi-norm onH 2 () _ H 2 ([0; _]) .
Proof. We use the * superscript as de_ned in the proof of lemma 2.2. For any
wh 2 X h , we have from B.5),B.11),B.14)
_
_
_0 kuh s wh kX kuh s wh kZ_
_
_
_
_
_
_
ah ( uh s w;u
h
h
h
h s wh ) + a( u s w;u
h s wh ) s a( u s w;u
h s wh )
_
_
_
_
_
_
_ a( u s w;u
h
h
h
h s wh ) + ( a( w;u
h s wh ) s ah ( w;u
h s wh ))
_
_
_
_
_
_
_ 0 ku s wh kX kuh s wh kZ + j a( w;u
h
h
h s wh ) s ah ( w;u
h s wh ) j ;
9
SCRBE FOR CONJUGATE HEAT TRANSFER
using the fact that u_h s wh_ belongs both to Z and Z h . Dividing by ku_h s wh_ kZ , we
get
s wh_ ) s ah ( w;u
h
ku_h s wh_ kZ
j a( w;v
h
h ) s ah ( w;v
h
0 ku s wh k
X + sup
kvh kZ
vh 2 Z h
_0 kuh s wh kX _
0 ku
_
s wh kX +
_
h
j a( w;u
h
_
h
s wh_ ) j
h )j
Let _ h be the component in Vh of wh . Then from B.3) and B.10), it follows that
a( w;v
h
h)
s ah ( w;v
h
h)
= Bi int
Z
@
( _ h s _ h )( '
h
s #h ) ;
int
and it is straightforward to show that
sup
j a( w;v
h
vh 2 Z h
s ah ( w;v
h
kvh kZ
h)
h )j
_ M1
Z
( _ h s _h )
2
!
1
2
[0 ; _]
_ Mh2
Z
[0 ; _]
2
d_h !
dx
1
2
_ Mh2 kwh kX
_ Mh2 ( ku s wh kX + kukX )
And then we get
kuh s wh kX
_
0
+ Mh2
M2
ku s wh kX +
h ku kX
_0
_0
By choosing for wh the interpolant _u
of u in X h , we obtain from standard results
h
in approximation theory [16]
kuh s _u
h kX_O
( h)( kukX + j uj 2 ) :
And _nally, using the triangular inequality, we have the a priori error estimate
ku s uh kX
_O ( h)( kukX + j uj 2 ) :
3. Reduced Basis Approximation.
3.1. RB Spaces.
parameters
In the following, we now consider the dependency on the
_ = ( Bi ext ;Bi ;in t F )
which belong in a parameter spaceD _ R3 . The FE weak form reads:
uh ( _ ) 2 X h
satis_es ah ( uh ( _ ) ;v h ; _ ) = f ( vh )
8vh 2 Z:h
We then introduce the set of parameters
SN = f _ 1 ;:::; _ N g ;
C.1)
S. VALLAGHE_AND A.T. PATERA
10
as provided by a greedy algorithm [14] and then de_ne
X N = spanf uh ( _ n ) ; 1 _ n _ N g :
C.2)
The uh ( _ n ), 1 _ n _ N max , are often referred to as \snapshots" of the parametric manifold M h = f uh ( _ ) j _ 2 D g. It is clear that, if indeed the manifold is
low-dimensional and smooth, then we would expect to well approximate any member
of the manifold | any solution uh ( _ ) for some _ in D | in terms of relatively few
snapshots.
In order to understand the RB test space we return to the inf-sup discussion of
section 2.3 . In particular, we recall that we de_ned the quantity _ as a function of
the parameters _
2
_=
F +
_ int Bi
F
2
int
:
Now that we consider RB approximations, the parameters are allowed to vary and so
we introduce
2
_min =
;
_ int Bi 2int ; max
F max +
F min
where F2 [F min ; F max ] and Bi int ; max is the maximum of Bi int in D . Hence _min is
independent of _ and setting _ = _min ensures that B.12) remains valid for all _ 2 D .
We also de_ne the * superscript in what follows as
8uh = ( _;_
h
h)
2 X;h
_
uh = ( _;_
h
h
+ _min
d_h
) 2 Z:h
dx
We are now ready to properly de_ne the RB test space
_
Z N = f uN j uN 2 X N g ;
Clearly Z N _ Z h , so we can directly de_ne our RB approximation : uN ( _ ) 2 X N
satis_es
8 vN 2 Z N :
ah ( uN ( _ ) ;v N ; _ ) = f ( vN ) ;
C.3)
The well-posedness of this discrete problem follows from the inf-sup discussion provided above.
We now consider the discrete equations associated with the Petrov-Galerkin approximation C.3). We must _rst choose an appropriate basis for our spaces. Towards
this end, we apply the Gram-Schmidt process in the ( _; _) X inner product to our
snapshots uh ( _ n ), 1 _ n _ N max , to obtain mutually orthonormal functions
_n ,
1 _ n _ N max : ( _;_
n mh) = _nm , 1 _ n;m _ N max , where _nm is the Kronecker-delta
symbol. We then choose the setsf _n g
n =1 ;:::;N as our bases forX N , 1 _ N _ N max .
We now insert
N
uN ( _ ) = X
uNm ( _ ) _m ;
C.4)
m =1
and vN = _n_ , 1 _ n _ N , into C.3) to obtain the RB \sti_ness" equations
N
X
m =1
_
_
ah ( _m;_ n ; _ ) uNm ( _ ) = f ( _n ) ;
1 _ n _ N;
C.5)
11
SCRBE FOR CONJUGATE HEAT TRANSFER
for the RB coe_cients uNm ( _ ), 1 _ m _ N .
We have the following a priori error result for the RB apProposition 3.1.
proximation
kuh ( _ ) s uN ( _ ) kX_
A+
0(_ )
) inf kuh ( _ ) s wN Xk :
_0 ( _ ) w N 2 X N
Proof. Thanks to our choice for Z N , we can write
inf
sup
u N 2 X N vN 2 Z N
ah ( uN;v N ; _ )
_
kuN Xk kvN Zk
inf
uN 2 X N
ah ( uN;u _N ; _ )
kuN Xk ku_N kZ
C.6)
And because of setting _ = _min , B.12) holds meaning that
ah ( uN;u
_
N
; _ ) _ K ( _ ) kuN k2X ;
where K ( _ ) = min ~12 _min F ; 12 ( F + _min Bi int ) _ : Then we get
ah ( uN;u _N ; _ )
_ _0 ( _ ) > 0;
kuN Xk ku_N kZ
inf
uN 2 X N
C.7)
K (_ )
2
with _0 ( _ ) = C and C = max _ 1; p 2(_ + _min
) _ . So for any wN 2 X N , invoking
Galerkin orthogonality,
_
_
_0 ( _ ) kuN ( _ ) s wN Xk kuN ( _ ) s wN kZ_
_
_
ah ( uN ( _ ) s wN;u N ( _ ) s wN ; _ )
_ ah ( uh ( _ ) s wN;u _N ( _ ) s wN_ ; _ )
_ 0 ( _ ) kuh ( _ ) s wN Xk ku_N ( _ ) s wN_ kZ ;
and using the triangular inequality leads the result.
This shows that the quality of our RB solution depends entirely on the approximation capability of our RB spaces. Under certain assumptions, theoretical results
predict an exponential convergence of the greedy RB spaces [2], meaning
9_> 0;
inf
wN 2 X N
kuh ( _ ) s wN Xk
_ CeX _N
8_ 2 D:
3.2. A Posteriori
Error Estimation.
The central equation in a posteriori
theory is the error residual relationship.
The error e( _ ) _ uh ( _ ) s uN ( _ ) 2 X h
satis_es
ah ( e( _ ) ;v ; _ ) = r ( v; _ ) ;
8 v 2 Z:h
C.8)
Here r ( v ; _ ) 2 Z h0 (the dual space of Z h ) is the residual,
r ( v; _ ) _ f ( v; _ ) s ah ( uN ( _ ) ;v ; _ ) ;
8 v 2 Z:h
C.9)
It is clear that r is bounded since f and ah are bounded. We de_ne the following
error estimator
_ N (_ ) _
kr ( _; _ ) kZ h0
_0 ( _ )
:
S. VALLAGHE_AND A.T. PATERA
12
Lemma 3.2. _ N ( _ ) is a rigorous error bound, meaning
kuh ( _ ) s uN ( _ ) kX_
_ N (_ ):
Proof. From the inf-sup condition and continuity condition, we know thatah ( e( _ ) ;: ; _ )
is bounded and so we can de_ne its Riesz representatione(^_ ) 2 Z h such that
(^e( _ ) ;v ) Z = ah ( e( _ ) ;v; _ ) ;
8v 2 Z:h
It can then be shown [18] that
e
^( _ ) = arg sup
v2 Z h
ah ( e( _ ) ;v; _ )
;
kvkZ
C.10)
8 v 2 Z;h
C.11)
and e
^( _ ) satis_es
(^e( _ ) ;v ) Z = r ( v; _ ) ;
as well as
ke
^( _ ) kZ = kr ( :; _ ) kZ h0 :
C.12)
We can thus also write the error residual equation C.8) as
ah ( e( _ ) ;v ; _ ) = (^e( _ ) ;v ) Z ;
8 v 2 Z:h
C.13)
It follows directly from C.13) for v = ^e( _ ) and the Cauchy-Schwarz inequality that
ah ( e( _ ) ; e
^( _ )) _k e
^( _ ) kk
^( _ ) kZ :
Z e
But from C.6),C.7) and C.10), _0 ( _ ) ke( _ ) kk
^( _ ) kZ_
X e
obtain
jj e( _ ) jj X_
C.14)
ah ( e( _ ) ; e
^( _ ); _ ), hence we
ke
^( _ ) kZ
:
_0 ( _ )
The result directly follows from C.12).
Based on this error estimator, we will use the standard Greedy algorithm [14] to
construct the RB spaces of section 3.1. It should be mention that rather than _0 ( _ )
a sharper lower bound may be obtained by the successive constraint method [9].
3.3. O_ine-Online strategy.
The O_ine-Online strategy is standard and it
thus su_ces to excerpt a brief description from [17]. The system C.5) is nominally
of small size: a set of N linear algebraic equations in N unknowns. However, the
formation of the sti_ness matrix, and indeed the load vector, involves entities
_n ,
1 _ n _ N; associated with our FE approximation space, which is of large dimension
denoted N . If we must invoke FE _elds in order to form the RB sti_ness matrix for
each new value of _ the marginal cost per input-output evaluation _ ! uN ( _ ) will
remain unacceptably large.
13
SCRBE FOR CONJUGATE HEAT TRANSFER
Fortunately, we can appeal to a_ne parameter dependence to construct very
e_cient O_ine-Online procedures, as we now discuss. In particular, we note that our
system C.5) can be expressed, thanks to B.4), as
N
P
_
m =1
Q
_
P _ q( _ ) aq ( _m;_ n_ ) uNm ( _ ) = f ( _n ) ;
C.15)
q=1
1 _ n _ N:
We observe that the_n are now isolated in terms that are independent of_ and hence
that can be pre-computedin an O_ine-Online procedure.
In the O_ine stage, we _rst compute theuh ( _ n ), 1 _ n _ N max , and subsequently
the _n , 1 _ n _ N max ; we then form and store the
f ( _n ) ;
1 _ n _ N max ;
C.16)
and
_
aq ( _m;_ n ) ;
1 _ n;m _ N max ; 1 _ q _ Q:
C.17)
The O_line operation count depends on N max , Q, and N .
In the Online (or \deployed") stage, we retrieve C.17) to form
Q
X _ q( _ ) aq ( _m;_ n_ ) ;
1 _ n;m _ N ;
C.18)
q=1
and we solve the resulting N _ N sti_ness system C.15) to obtain the uNm ( _ ),
1 _ m _ N . The Online operation count is O( QN 2 ) to perform the sum C.18) and
O( N 3 ) to invert C.15) | note that the RB sti_ness matrix is full. The Online cost |
and hence marginal cost and also asymptotic average cost | to evaluate _ ! uN ( _ )
is thus independent of N . The implications are two-fold: _rst, if N is indeed small,
we will achieve very fast response in the real-time and many-query contexts; second,
we may chooseN very conservatively | to e_ectively eliminate the error between the
exact and FE predictions | without adversely a_ecting the Online (marginal) cost.
A similar but more involved O_ine-Online strategy may be developed for the error
bound; we refer the reader to [17].
4. More advanced models.
4.1. Channel splitting.
We consider here the case where the uid channel
splits in two, as shown in _gure 4.1.
We describe the main additions to the theory presented in section 2.2. First, we
de_ne two di_erent variables for the uid bulk temperature, corresponding to the two
di_erent channels: _ 1 is de_ned on [0; _ 1], where _ 1 is the non-dimesionnal length of
the horizontal channel, and _ 2 is de_ned on [0; _ 2], where _ 2 is the non-dimesionnal
length of the vertical channel. The mapping of [0; _ 1] to the channel walls is less trivial
in this case; the di_erent types of mapping are represented by colors in _gure 4.1.
We change the de_nition of the following spaces :
V = f _ 1 2 H 1 ([0; _ 1]) ;_ 1 @) = 0 ;_ 2 2 H 1 ([0; _ 2]) ;_ 2 @) = _ 1 (
W = L 2 ([0; _ 1]) _ L 2 ([0; _ 2]) :
_1
) g;
2
S. VALLAGHE_AND A.T. PATERA
14
@
1
int
@
1
int
@
1
int
g
x
@
y
F
1
int
A G _ )F
_F
@
2
int
@
2
int
Fig. 4.1: A channel splitting component. Note the colors indicate the identity mappings from @to the uid mixed-mean temperature _lament coordinate.
We also use the notations
2 X = Y _ V;
2 ) 2 Z = Y _ W:
w = ( _;_;_
1
v = ( #;';' 1
2)
We de_ne the following norms on X and Z
kuk2X =
Z
kvk2Z =
Z
jr _j 2 +
Z
jr # j 2 +
Z
2
2
Z
d_1
d_2
+ _ 21 (_ 1) +
+ _ 22 (_ 2) ;
dx
[0 ; _ 2] dy
Z
' 21 +
': 22
[0 ; _ 1]
[0 ; _ 1]
[0 ; _ 2]
We then obtain a similar bilinear form a:
a( w;v ) =
Z
Z
r _r # + Bi ext
@
s Bi int
Z
@
s Bi int
1
int
Z
@
2
int
_# + Bi int
Z
@
ext
(_ s _1)'
1
+ F
Z
[0 ;
(_ s _2)'
2
+ _F
_1
2
]
Z
[0 ; _ 2]
1
int
d_1
'
dx
( _ s _ 1 ) # + Bi int
Z
@
1
+ A s _ )F
Z
[
d_2
': 2
dy
_1
2
; _ 1]
2
int
( _ s _2 ) #
d_1
'
dx
1
D.1)
And _nally the weak form is: u 2 X satis_es
a( u;v ) = f ( v)
8v 2 Z:
This channel splitting model preserves thermal energy. For simplicity we consider
the \all-natural" situation, in which case we may choose for our test function _ = 1.
Then the heat source in the system corresponds tof ( _ ), and so we get
15
SCRBE FOR CONJUGATE HEAT TRANSFER
f ( _ ) = a( u;_ )
Z
= Bi ext
D.2)
_+ F
@
= Bi ext
[0 ;
ext
Z
_ + F _1 (
@
= Bi ext
Z
ext
Z
_1
2
]
d_1
+ A s _)F
dx
Z
[
_1
2
; _ 1]
d_1
+ _F
dx
Z
[0 ; _ 2]
d_2
dy
_
_1
_1 _
) + A s _ ) F _ 1 (_ 1) s _ 1 ( ) + _ F ( _ 2 (_ 2) s _ 2 @))
2
2
_ + A s _ ) F _ 1 (_ 1) + _ F _ 2 (_ 2)
@
D.3)
ext
This corresponds exactly to the total heat exchanged by the system, outside the solid
and at the two uid outlets (we recall that the temperature at the uid inlet is zero).
It means that the thermal equilibrium is satisi_ed by a, and that a conserves thermal
energy.
Lemma 4.1. The bilinear form a is inf-sup stable.
Proof. Like in section 2.2, we considera( w;w _ ), and the following new boundary
terms appear, due to the channel splitting:
F
Z
[0 ;
_1
2
]
Z
Z
d_1
d_1
d_2
_1 + A s _ ) F _
_1 + _ F
_2
1
dx
dx
[ 2 ; _ 1]
[0 ; _ 2] dy
_
_1
_1 _
) + A s _ ) F _ 21 (_ 1) s _ 21 ( ) + _ F ~_ 22 (_ 2) s _ 22 @)_
2
2
2
2
= A s _ ) F _ 1 (_ 1) + _ F _ 2 (_ 2) :
= F _ 21 (
D.4)
As can be seen, the \internal" boundary terms cancel out naturally thanks to our
\thermal energy conservation" condition _ 2 @) = _ 1 ( _21 ), and only the boundary
terms at the channel outlets remain, that are part of the X -norm. So the stability
proof is obtained by using exactly the same arguments than in section 2.2.
The rest of the theory (FE, RB) is also very similar and as such we do not get
into further details.
4.2. Channel mixing.
We now consider the case where two uid channels mix
into one, as shown in _gure 4.2. In this case, we de_ne three di_erent variables for the
uid bulk temperature, corresponding to the two incoming channels plus the mixing
channel. _ 1 is de_ned on [0
; _ 1], where _ 1 is the non-dimesional length of the incoming
horizontal channel (blue and _rst half of green), _ 2 is de_ned on [0; _ 2], where _ 2 is
the non-dimesional length of the vertical channel (orange), and
_ 3 is de_ned on [0; _ 3],
where _ 3 is the non-dimesional length of the mixing horizontal channel (second half
of green and red).
We change the de_nition of the following spaces :
V = f _ 1 2 H 1 ([0; _ 1]) ;_ 1 @) = 0 ;_ 2 2 H 1 ([0; _ 2]) ;_ 2 @) = 0 ;
_ 3 2 H 1 ([0; _ 3]) ;_ 3 @) = (1 s _ ) _ 1 (_ 1) + __ 2 (_ 2) g;
W = L 2 ([0; _ 1]) _ L 2 ([0; _ 2]) _ L 2 ([0; _ 3]) :
We also use the notations
u = ( _;_;_;_
1
2
v = ( #;';';'1
2
2 X = Y _ V;
3 ) 2 Z = Y _ W:
3)
S. VALLAGHE_AND A.T. PATERA
16
@
1
int
@
3
int
@
3
int
g
x
@
y
A G _ )F
1
int
F
_F
2
int
@
@
2
int
Fig. 4.2: A channel mixing component. The colors indicate the identity mappings
from @to the uid mixed-mean temperature _lament coordinate.
We de_ne the following norms on X and Z
kuk2X =
Z
kv k2Z =
Z
jr _j 2 +
Z
jr # j 2 +
Z
2
2
2
Z
Z
d_1
d_2
d_3
+
+
+ _ 23 (_ 3) ;
dx
[0 ; _ 2] dy
[0 ; _ 3] dy
Z
Z
' 21 +
' 22 +
': 23
[0 ; _ 1]
[0 ; _ 1]
[0 ; _ 2]
[0 ; _ 3]
The bilinear form a now reads :
a( w;v ) =
Z
r _r # + Bi ext
Z
_#
@
+ Bi int
Z
@
s Bi int
Z
@
s Bi int
1
int
Z
@
s Bi int
1
int
2
int
Z
@
3
int
ext
Z
( _ s _ 1 ) # + Bi int
@
( _ s _1 ) '
1
2
int
( _ s _ 2 ) # + Bi int
@
Z
+ A s _)F
d_1
'
dx
[0 ; _ 1]
( _ s _2 ) '
2
Z
+ _F
[0 ; _ 2]
( _ s _3 ) '
3
+ F
Z
[0 ; _ 3]
Z
d_2
'
dy
3
int
( _ s _3 ) #
1
2
d_3
': 3
dy
D.5)
And the weak form is: u 2 X satis_es
a( u;v ) = f ( v)
8v 2 Z:
The bilinear form a conserves thermal energy, which ensures that the channel
mixing model is physically well-posed. Denoting _ the test function equal to unity on
17
SCRBE FOR CONJUGATE HEAT TRANSFER
Z , we consider
f ( _ ) = a( u;_ )
Z
= Bi ext
@
= Bi ext
_ + A s _ )F
Z
Z
d_1
d_2
d_3
+ _F
+ F
dx
dx
[0 ; _ 2]
[0 ; _ 3] dy
_ + A s _ ) F _ 1 (_ 1) + _ F _ 2 (_ 2) + F ( _ 3 (_ 3) s _ 3 @))
ext
Z
@
Z
[0 ; _ 1]
ext
Z
@
= Bi ext
D.6)
_ + F _ 3 (_ 3)
D.7)
ext
This corresponds exactly to the total heat exchanged by the system, outside the solid
and at the uid outlet (we recall that the temperature at the uid inlets is zero). It
means that the thermal equilibrium is satisi_ed by a, and that a conserves thermal
energy.
Lemma 4.2. The bilinear form a is inf-sup stable.
Proof. Like in section 2.2, we considera( w;w _ ), and the following new boundary
terms appear, due to the channel mixing :
A s _ )F
Z
[0 ; _ 1]
Z
Z
d_1
d_2
d_3
_1 + _ F
_2 + F
_3
dx
[0 ; _ 2] dy
[0 ; _ 3] dx
= A s _ ) F _ 21 (_ 1) + _ F _ 22 (_ 2) + F ~_ 23 (_ 3) s _ 23 @)_
= _ A s _ ) F ( _ 1 (_ 1) + _ 2 (_ 2))
2
+ F _ 23 (_ 3) ;
D.8)
F _ 23 (_ 3)
where we used the equality _ 3 @) = A s _ ) _ 1 (_ 1) + __ 2 (_ 2). The term
is
directly related to the X -norm, whereas the other term _ A s _ ) F ( _ 1 (_ 1)+ _ 2 (_ 2)) 2
is not, but it's positive, and so it can be neglected in the stability proof.
So once
again the proof follows exactly the same arguments than in section 2.2.
Here also the rest of the theory (FE, RB) is similar to the straight channel case.
Note that for this mixing case, the mathematical model is clearly very approximate.
We will not be able to easily characterize the complexity of the mixing with a constant
heat transfer coe_cient. However, it is not an unrealistic physical limit in which we
assume very good mixing. The same comment also apply to the splitting case.
5. Systemization. We now consider the models presented in the previous sections as components (or subdomains) and we want to build a system composed of
many similar components, connected together by ports. To this end, we will use a
Static Condensation Reduced Basis Element method, which is a domain decomposition approach with the following distinct features: reduced basis approximation of
_nite element bubble functions at the intradomain level; eigenfunction \port" representation at the interface level; static condensation at the interdomain level.
The general framework for static condensation RBE is described in [7], in a rigorous and abstract way. Here we use a di_erent path and we present the method based
on a simple example with two subdomains, focusing on matrix transformations of the
linear system derived from the PDE FE discretization.
We hope this will provide
a complementary view on static condensation RBE. Nevertheless, this section is not
meant to be a complete self contained description and we refer the reader to [7] for
details.
5.1. Static condensation.
We suppose our system domain is composed of
two components 1 and ,2 which both share a part of their boundary P , denoted a
port (_gure 5.1). The PDE _nite element approximation yields the global system
S. VALLAGHE_AND A.T. PATERA
18
P
1
2
Fig. 5.1: A simple rectangular domain with two square subdomains sharing a portP .
2 AP
4 A P; 1
A P; 2
A TP; 1
A 1
0
A TP; 2 3 2 u P 3 2 f P 3
0 5 4u 1 5 = 4f 1 5
u 2
f 2
A 2
where we grouped and reordered to segregate the degrees of freedom on
P from the
degrees of freedom internal to 1 and .2
We now apply static condensation to remove the degrees of freedom internal to
each component. We de_ne the Schur complement matrix A SC and the Schur right
hand side f SC
A SC = A P s A TP; 1 A
f SC = f P s A TP; 1 A
X1
X1
1
1
f
A P;
1
1
s A TP; 2 A X 21 A P;
s A TP; 2 A X 21 f
2
2
:
E.1)
E.2)
Then the vector of port coe_cients u P is solution of the Schur complement system
A SCu P = f SC;
which is of sizeN P , the number of degrees of freedom on
P . We see that computing the
X
quantity A i1 A P; i corresponds to computing N P FE solutions of a PDE on i with
homogeneous Dirichlet conditions onP and N P di_erent source terms. We denote by
\FE bubbles" the solutions to these PDE, each one being associated to one degree of
freedom on P . We denote b ik the vector of coe_cients in the FE space of the bubble
in associated
to the k th degree of freedom on P. Hence we can write
i
A
X1
i
A P;
i
= A
X1
i
_ai1a i2 ___aiN P _ = _b i1b i2 ___b iN P _;
|
{z P }
N_N
i
where aik is the source term for the k th bubble and N i is the number of degrees of
freedom internal to .i
5.2. Static condensation with reduced basis.
Now the key idea of the
method is to replace the FE bubbles by RB bubble approximations (with an RB
space of dimension N) :
b ik s!
B ki b~ik ;
where B ki is the matrix of the RB-basis functions FE coe_cients, of size N_i N , and
b~ik are the coe_cients of the bubble in the RB space of dimensionN . We can see that
the matrix B ki is di_erent for each i;k , which means that we construct a di_erent RB
space for each bubble. The RB bubble coe_cientsb~ik are obtained by solving a linear
system of sizeN _ N
19
SCRBE FOR CONJUGATE HEAT TRANSFER
X
~ik ;
b~ik = A~ i1;k a
~ik are the equivalent ofA i ;k and aik projected on the k th RB space.
where A~ i ;k and a
And so in the end we do the following substitution
A
X1
i
A P;
i
hB i A~X 1 a
~i
1
i ;1 1
s!
___ B Ni P A~X i1; N P a
~iN P i
X
~i2
B 2i A~ i1; 2 a
and we obtain an RB approximation
A~SCu~P = ~
f SC
to the original FE truth static condensation systemA SCu P = f SC. By doing so, we only
need to solve 2(N P + 1) linear system of sizeN (to get the RB bubbles) and one linear
system of size N P . If N denotes the size of the complete system FE discretization,
then the complexity is reduced from O ( N 3 ) to O ( N_P N 3 + N P3 ), which is signi_cant
since usually N _N and N_N
. We refer back to section 3.3 for more details on
P
the computational gain obtained with RB.
5.3. Component to system assembly.
The basis functions associated with
the degrees of freedom on the portP are called interface functions. We denote ki the
restriction on i of the interface function associated with the k th degree of freedom
on P . To construct ki , we _rst compute the Laplacian eigenmodes _ k on the port
domain P , and then we lift these eigenmodes in the component domains i such that
they solve the Laplace equation, and such that the restriction of ki to P is equal to
_k .
We denotebik the bubble function on associated
with the k th degree of freedom
i
on P . The general FE solution on a subdomain i , denoted ui , can be expressed with
respect to the port coe_cients u kP as
NP
ui = X u kP (
i
k
+ bik ) :
E.3)
k =1
Using these notations, we can make more explicit the matrices previously introduced.
First of all, solving
b ik = A
X1
ai
i ;k k
corresponds to the bubble equation
i
ah ( b;v
) = s ah (
k
i
k ;v )
8v 2 Z:h
Then, the other matrices are :
1
1
1
1
2 ah (
A P = 6 ah (
4
s
A TP;
+
+
2
1;
2
1;
1
1
1
2
+
+
2
1)
2
2)
ah (
ah (
..
.
1
A
X1
1
A P;
1
2
1
2
+
+
2
2;
2
2;
1
1
1
2
+
+
2
1)
2
2)
..
.
1
2 ah ( b;11
1
= 6 ah ( b;1
..
4
.
1
1)
1
2)
ah ( b;12
ah ( b;12
..
.
1
1)
1
2)
::: 3
::: 7
.. 5
.
::: 3
::: 7
.. 5
.
S. VALLAGHE_AND A.T. PATERA
20
s
A TP;
2
A
X1
2
A P;
2
2 ah ( b;21
2
= 6 ah ( b;1
..
4
.
2
1)
2
2)
ah ( b;22
ah ( b;22
..
.
2
1)
2
2)
::: 3
::: 7
.. 5
.
Since functions with di_erent superscripts do not share support, we see that we can
decompose the static condensation system matrix into two static condensation component matrices, A SC = A 1SC + A 2SC, where :
2 ah (
= 6 ah (
4
1
1
1
1
+ b;11
+ b;11
..
.
1
1)
1
2)
ah (
ah (
1
2
1
2
+ b;12
+ b;12
..
.
1
1)
1
2)
::: 3
::: 7
.. 5
.
2 ah (
2
A SC = 6 ah (
4
2
1
2
1
+ b;21
+ b;21
..
.
2
1)
2
2)
ah (
ah (
2
2
2
2
+ b;22
+ b;22
..
.
2
1)
2
2)
::: 3
::: 7
.. 5
.
A 1SC
So in practice, the assembly of the static condensation system is bottom-up: for each
component, we assemble a local matrix A iSC, and we construct the complete static
condensation matrix A SC by appropriately summing the di_erent component matrices
A iSC. The advantage of this approach is that in many cases, several components share
the same interface and bubble functions, so the assembly ofA iSC has to be performed
only once for a group of components sharing the same parameters.
5.4. Convection treatment.
In [7], only elliptic problems are considered. In
the current framework, the convection term introduces additional complexities and
in particular requires special treatment at the assembly stage. When connecting the
two components 1 and ,2and assuming the ow is from left to right, there is
an interface function and associated bubble at the inlet in
2 but there is none at
the outlet in , 1since we consider pure convection and we do not want an arti_cial
boundary layer at the outlet in . This
would normally break the compatibility of
1
the two components, which should share the same con_guration at the port.
We now describe in more detail the assembly of the two local matrices A 1SC and
2
A SC. We assume that the degrees of freedom onP from 1 to NsP 1 correspond to
the solid domain, and the degree of freedomN P is for the uid inlet in . First,
we
2
consider the interface functions: for indices 1 _ k _Ns P 1, the ki 2 X h = Yh _ Vh
have their support in the solid domain, and as such their component in Vh is null, so
they do as well belong to Z h = Yh _ Wh . These \solid" interface functions can then
2
be used both as trial and test functions. The last interface function
N P (uid inlet
dof in )2 has its support in the uid domain, so it belongs to
X h but not necessarily
to Z h . For these reasons, we test only on the \solid" interface functions; then the
local component matrices are
2 ah (
A 1SC
= 6
4
ah (
1
1
1
1
+ b;11
..
.
+
b;11
1
1)
1
NXP 1 )
ah (
ah (
1
2
1
2
+ b;12
..
.
+
b;12
1
1)
1
NXP 1 )
:::
::: a
ah (
h(
1
NXP 1
1
NXP 1
+
+ b1NXP
..
.
b1NXP 1 ;
1;
1
1)
1
NXP 1 )
3
7
5
21
SCRBE FOR CONJUGATE HEAT TRANSFER
+ b;21
..
A 2SC = 6
.
4
ah ( 12 + b;21
2 ah (
2
1
2
1)
+ b;22
..
.
ah ( 22 + b;22
ah (
2
NXP 1 )
2
2
2
1)
:::
2
NXP 1 )
ah (
::: a
h(
2
NP
2
NP
+ b2N P ; 12 ) 3
..
7
.
5
2
+ b2N P ; NX
)
P 1
A 1SC is of size NsP 1 _Ns P 1 and A 2SC is of size NsP 1 _N P . We now need to
restore the compatibility of the two matrices, and also enforce the continuity of the
uid temperature at the port. Let
_ 1 and _k1 be the component of u1 and b1k in Vh ,
then from E.3) and recalling that the \solid" interface functions are zero in
Vh , we
NXP 1
get that _ 1 = P k =1 u kP _:k1 Denoting _ the _lament coordinate of the uid outlet in
NXP 1
,1we then have _ 1 (_) = P k =1 u kP _k1 (_) : So we modify the component matrices as
follows:
2 ah (
A 1SC = 6
6
6 ah (
4
1
1
1
1
+ b;11
..
.
+ b;11
_11 (_)
+ b;21
..
.
A 2SC = 6
6
6 ah ( 12 + b;21
4
0
2 ah (
2
1
1
1)
ah (
1
NXP 1 )
ah (
2
1)
:::
..
.
::: a
:::
2
NXP 1 )
1
2
1
2
+ b;12
..
.
+ b;12
_21 (_)
ah (
h(
1
1)
:::
1
NXP 1 )
2
NXP 1
2
NXP 1
::: a
:::
+ b2NXP
+ b2NXP
0
1;
1
NXP 1
ah (
h(
1;
+ b1NXP
..
.
1
NXP 1
+ b1NXP 1 ;
1
_NXP 1 (_)
2
1)
ah (
2
NXP 1 )
ah (
1;
1
1)
1
NXP 1 )
03
..
.7
7
07
5
0
2
NP
2
NP
+ b2N P ; 12 ) 3
..
7
.
7
2
2
+ bN P ; NXP 1 ) 7
5
s1
Now both A 1SC and A 2SC are of size N_N
P
P , and when summing them to get the
complete system matrix A SC, the last row is
__11 (_)
_21 (_)
::: _
1
NXP 1 (_)
s 1_ :
N
By setting the last coe_cent in the right hand side to zero, f SCP = 0, this will enforce
the value at the uid inlet in
2 to be equal to the value at the uid outlet in . 1
5.5. A Posteriori
Error.
5.5.1. System port error bound.
This section briey describes how to derive
ku P s u~P k2 . The
an error bound for the error in the system level approximation
di_erent steps taken to arrive at this bound are described in details in [7], and we
only present here a few variations. We thus refer the reader interested in technical
details to [7] for a complete description.
The approach presented in [7] exploits standard RBa posteriori error estimators
at the component level to develop a bound for the Frobenius norm kA SC s A~SCkF ;
and then apply matrix perturbation analysis at the system level to arrive at an
a
posteriori bound for ku P s u~P k2 . In the application that we present in this paper,
there are a few variations from the general framework of [7], which a_ect the assembly
of A SC and thus the error bound for kA SC s A~SCkF . We will now present these few
changes.
In [7], the bilinear form a is assumed to be symmetric, and by taking advantage of
this property, the component matrices are symmetrized, meaning the terms of
A iSC are
S. VALLAGHE_AND A.T. PATERA
22
of the form ah ( ki + b;ik ki 0 + bik 0 ). In our application here, since we have a convection
term, the bilinear form a is non-symmetric, and instead we keep the terms ofA iSC to
be of the form ah ( ki + b;ik ki 0 ), as presented in the previous sections. Similarly, the
coe_cients of A~iSC are of the form ah ( ki + ~
b;ik ki 0 ), where ~
bik is the k th RB bubble.
The error bound for kA SC s A~SCkF is derived from the following inequality :
j ah (
i
k
+ b;ik
i
k 0)
s ah (
i
k
+~
b;ik
i
k 0)j
= j ah ( bik s ~
b;ik
i
k 0)j
_
i;k
0_ N
k
i
k 0 kX
E.4)
i;k
where _ N is the standard RB error estimators for thek th bubble. To get the complete
error bound for kA SC s A~SCkF , we also need to take care of the extra row that
we introduced to connect the uid channels. We then have to bound the following
quantity
j _k1 (_) s _~k1 (_) j_k b1k s ~
b1k kX
_ _
1;k
N
E.5)
E.6)
1;k
where _ N is the standard RB error estimator for the k th bubble in .1 This inequality is true since we included the term _ (_) 2 in kuk2X , where u = ( _;_ ).
Then by summing on i ,k and k 0 we get a bound, denoted _2 , for the Frobenius
norm kA SCs A~SCkF . All the terms in this error bound are obviously linear with respect
to the RB error bound. Due to these linear terms, we do not expect an error bound
as sharp as for the symmetric case presented in [7], where all the terms are quadratic
(thanks to the assumption that a is symmetric). Note that a quadratic e_ect can still
be obtained for non-symmetric problems using a primal-dual RB formulation [8], but
the primal-only is computationally more e_cient and hence if adequate accuracy is
obtained in fact preferred. The _nal error bound is of the form:
ku P s u~P k_
2
_1 + _2 ku~P k2
_ _ uP;
p _
~min s _2
~min is the minimum eigenvalue ofA~SC
where _1 is a bound for kf SCs ~
f SCk2 and _
A ~TSC [8].
~min .
Of course this bound makes sense only if_2< p _
5.5.2. System output error bound.
We now consider an output of our system
which is a quantity de_ned on the port domain, and which can be de_ned as a linear
functional of the system solution u P . Such outputs can be simply the value of the
solution at a particular point in the port domain, or the solution average over the
port. We can write the output as
s = m Tu
P;
with m 2 RN P , and the corresponding RB output
s~ = m Tu~P :
We introduce z 2 RN P solution of
A~TSCz = s m :
SCRBE FOR CONJUGATE HEAT TRANSFER
23
Recalling that A SCu P = f SC and A~SCu~P = ~
f SC, we obtain the following matrix perturbation equation:
A~SC
_ u P = _f SC s _ASCu~P s _ASC
_ u P;
E.7)
where _u P = u P s u~P , _f SC = f SC s ~
f SC and _ASC = A SC s A~SC. Now multiplying E.7)
T ~X 1
T
s
on the left with m A SC = z we get
s s s~ = s zT _f SC + zT _ASCu~P + zT _ASC
_ u P:
E.8)
Now we need to bound each term on the right hand side of E.8) to get a bound for
j s s s~j . First of all, using E.4), we directly obtain
X
j zT _ASCu~P j_
j zi jj _AijSCj bound j u jP j_ " A SC ;
i;j over port dof
ij
where j _ASCj bound
refers to the bound on the Schur complement entry RB error given
by E.4) and E.6). A similar bound " f SC can be obtained for j zT _f SCj but we do not
give details about it here since in our particular application _f SC = f SC = ~
f SC = 0. So
we are left with the last term on the right hand side, and we can simply bound it by
reusing the system port error bound presented in the previous section:
j zT _ASC
_ u P j_k zk2 _2 _ u P = " quad :
So our _nal output error bound is
_ s = " f SC + " A SC + " quad :
This output error bound is much sharper than simply using km k2 _ u P . First, _ u P
is presumable pessimistic because the Frobenius norm in _2 is too strong. On the
contrary, the " A SC term should be relatively sharp { we miss only sign cancellation.
Second, as opposed to _u P , the factor p ~ 1
does not appear in the terms " f SC
_ min X _ 2
~min
and " A SC . This is important as this quantity is generally large because, _rst
_
corresponds to the conditioning of the matrix A~SC, and second _2 can be close to
p _
~min , especially in our case where _2 has only a linear e_ect in the RB bubble
error bound. The bound _ u P is still a factor of " quad , but premultiplied by _2 which
mitigates the impact.
The term " quad is hence quadratic in the error bounds since it involves the product
_2 _ u P . As a consequence, it is expected to become negligible compared "tof SC + " A SC
for a su_ciently good RB approximation. We can then consider the following simple
output error estimator (although not rigorous)
s
_ = " f SC + " A SC :
This error estimator is interesting from a computational point of view because by
dropping " quad we do not have to compute _u P , which requires the computation of the
minimum eigenvalue ofA~SC
A ~TSC. For large systems whereN P is large (typically N P >
6
10 ), the minimum eigenvalue computation time can become prohibitive, especially
s
in a many query or real time context. The error estimator _
is then an interesting
alternative to _ s when considering large systems.
24
S. VALLAGHE_AND A.T. PATERA
Fig. 6.1: Graphs of _ (left) and _ (right) for the simple test case on [0 ; 4]. Boundary
conditions are homogeneous Neumann for_ and _ @) = 0.
6. Numerical Examples.
All the results presented in this section were obtained using rbOOmit [11] and libMesh [10].
6.1. Model problem AD).
We _rst consider a 1D version of our problem
which has an analytical solution, so that we can compare our method to a ground
truth. Towards this end, we set the solid domain to be 1D, meaning = [0
; _] .
In this case, we can reduce our problem B.1) to a system of ODE (see appendix A)
where both _ and _ are de_ned on [0; _] :
( s _00+ Bi ext _ + Bi int ( _ s _ ) = f
F _ 0 s Bi int ( _ s _ ) = 0 :
F.1)
We impose the following boundary conditions: _0@) = 0, _0(_) = 0 and _ @) = 0.
For the values Bi ext = 1 ;Bi int = 65 ; F = 1 ;f = 1, a particular solution of the
system 6.1 without boundary conditions is _ = 1 ;_ = 1, and the set of solutions to
the homogeneous system is
p
p
2+
19
2q
19
8
X 2x
+ 45Be 5 x + 45Ce 5 x
< _( x ) = 2 Ae
p
p
2+
19
X 2x
+ ( s 6 19)Be 5 x + ( + 6
: _ ( x ) = s 3Ae
p
2q
19)Ce
p
5
19
x
A;B;C 2 R
The values for A;B;C are then chosen such that the solution satis_es the boundary
conditions. We show in _gure 6.1 the graphs of _ and _ for _ = 4.
We now solve the same problem numerically by using static condensation and a
system of four components each of unity length. We show in _gure 6.2a the error
in the L 2 norm between the FE static condensation and the analytical solution. We
observe that the error is O ( h2 ), which is optimal for our P1 FE discretization. We
now consider RB bubble approximations. We considerBi
( ext ; F ) 2 [0:33; 3]2 to be the
parameters of the system of ODE F.1), and we build RB spaces using the standard
Greedy algorithm [14] up to N max = 15. These RB approximations are built on a
\truth" FE approximation with a uniform mesh of step size equal to 0
:002. The
error at the ports degrees of freedom between the FE static condensation and the RB
static condensation is shown in _gure 6.2b as well as the system solution error bound
presented in section 5.5.1. The error bound has an e_ectivity of about 10 2, which is
25
SCRBE FOR CONJUGATE HEAT TRANSFER
10
10
10
10
10
10
10
10
10
(a)
0
true error
error bound
-1
-2
-3
-4
-5
-6
-7
-8
4
5
6
7
8
9
RB space dimension
10
11
12
13
(b)
Fig. 6.2: (a) Static condensation FE error with respect to the analytical solution;
blue: error in the solid domain; green: error in the uid domain. (b) scRBE error
with respect to static condensation FE.
similar to the results reported in [7]. Although we predicted that the error bound
would be less sharp than in [7], this result can be simply explained by the fact that
we are considering here a very simple 1D problem with few components (only four),
whereas in [7] the numerical results are obtained for more components and 2D/3D
problems.
6.2. Simple automotive radiator.
We now consider the library of components
shown in _gure 6.3. All these components are based on the various models presented
in the previous sections. We can actually assemble many such components to model
a car radiator shown in _gure 6.4.
Based on physical considerations for real car radiators, we selected the following
domains for the parameters: 2_F_ 40, 0:01 _ Bi ext _ 0:1 and 0:025 _ Bi int _ 0:4.
The range for Bi ext is actually higher than the physical values, but since in our
examples we consider small radiators with far less _ns than actual car radiators,
we use correspondingly larger external heat transfer coe_cients to ensure nontrivial
behavior.
In all the following, the RB was trained on the previously de_ned range of parameters, except forBi int which was _xed to 0.1. It means that Online computations can be
performed for any parametersF[ ;Bi ext ;Bi int ] 2 D with D = [2 ; 40]_ [0:01; 0:1]_f 0:1g.
We use RB spaces of maximum dimensionN max = 30.
We will now consider a set of examples on a radiator with _ve coolant tubes and
_ve _ns per tube _ components), as shown in _gure 6.5a. As a boundary condition,
we will always set the (non-dimensional) uid bulk temperature at the inlet equal
to 1. As the ouput of interest s, we will consider the uid exit temperature at the
radiator outlet. We will consider di_erent scenarios by varying the parametersF and
Bi ext to show the design exibility of our approach, as well as the accuracy of the
error bound for the output. Note that in practice the ow distribution in the coolant
tubes would be determined from some simple \head loss" hydraulic model but here we
directly specify the owrates in the di_erent channels: we conserve mass and invoke
as appropriate symmetry or homogeneity.
S. VALLAGHE_AND A.T. PATERA
26
(a) Single channel + thermal _n.
F
_F
A G _ )F
(b) Square angle.
A G _ )F
(c) Channel splitting.
_F
F
(d) Channel mixing.
Fig. 6.3: The component library. Components can be connected at the ports shown
in red.
inlet from engine
coolant tubes
air _ns
outlet to engine
Fig. 6.4: Car radiator.
For the _rst scenario, we choose Bi ext = 0 :02 and Bi int = 0 :1 throughout the
whole system. The ow rate at the entrance is F = 15, and then splits so that it is
equal to 3 in each tube. The temperature in the solid domain is shown in Figure 6.5a.
In the inlet channel (_gure 6.5b), the uid bulk temperature decreases more rapidly
after each channel splitting, because the ow rate in the inlet channel diminishes after
each splitting. In the outlet channel (_gure 6.5d), the uid bulk temperature jumps
after each channel mixing, which is expected due to our mixing model described in
section 4.2. In the coolant tubes (_gure 6.5c), the uid bulk temperature decreases
27
SCRBE FOR CONJUGATE HEAT TRANSFER
1.005
1.000
0.995
0.990
0.985
flu
id
bu
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m
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rat
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e
0.980
0.975
0.970
0.965
0.960
0
(a) solid
2
4
6
inlet abscissa
8
10
12
(b) uid in the inlet channel
1.00
0.945
1st channel
2nd channel
3rd channel
4th channel
5th channel
0.99
0.98
0.940
0.97
0.935
0.96
0.930
flu
id
bu
lk
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m
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rat
ur
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flu
id
bu
lk
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m
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rat
ur
e
0.95
0.94
0.925
0.93
0.920
0.92
0.91
0
1
2
3
channel abscissa
(c) uid in the coolant tubes
4
5
0.915
0
2
4
6
8
outlet abscissa
10
12
14
(d) uid in the outlet channel
Fig. 6.5: Temperature _eld in the system. Bi ext = 0 :02 and Bi int = 0 :1. F = 15
at the inlet, F = 3 in each coolant tube. The red error bar corresponds to the error
bound for the uid outlet temperature.
at the same rate for all tubes, because the parameters Bi ext and Bi int are the same
everywhere, and so the overall heat transfer coe_cient is the same for all tubes.
j s s s~j
Finally, at the outlet, the uid bulk temperature is 0.918. The absolute error
X6
s
_
_
of the uid exit temperature is 1 :3 10 , and the error bound _ is 1:6 10X 3 . It
means that the temperature at the outlet is certi_ed to have an error of at most 0.17
% with respect to the one obtained with FE.
It is important to note the gain obtained with the new output error bound _ . s
Indeed, the system port error bound _ u P of section 5.5.1 is 0.19, whereas the actual
sytem port error ku P s u~P k2 is 2 :4 _ 10X 5 , so the e_ectivity of the system port
error bound is 10 .5 Comparatively, the system port error bounds reported in [7]
have e_ectivities of 10 2. This huge di_erence in e_ectivities as been predicted in
section 5.5.1, and is due to the linear versus quadratic e_ect in the RB bubble error
bounds. But thanks to the new output error bound _ , a sdecent e_ectivity can be
recovered when considering an output of the solution at the ports (in this case 103).
As a second scenario, we consider the fact that in real practice, it is unlikely that
S. VALLAGHE_AND A.T. PATERA
28
1.00
0.99
0.98
flu
id
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m
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rat
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e
0.97
0.96
0.95
0.94
0
(a) solid
2
4
6
inlet abscissa
8
10
12
(b) uid in the inlet channel
1.00
0.97
0.98
0.96
0.96
0.95
0.94
0.94
0.93
1st channel
2nd channel
3rd channel
4th channel
5th channel
0.90
0.88
0.86
flu
id
bu
lk
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flu
id
bu
lk
te
m
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rat
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0.92
0
1
0.92
2
3
channel abscissa
(c) uid in the coolant tubes
4
5
0.91
0
2
4
6
8
outlet abscissa
10
12
14
(d) uid in the outlet channel
Fig. 6.6: Temperature _eld in the system. Bi ext = 0 :02 and Bi int = 0 :1. F = 15 at
the inlet, F = 5 in the _rst coolant tube, F = 4 in the second one, F = 2 in all other
coolant tubes. The red error bar corresponds to the error bound for the uid outlet
temperature.
all coolant tubes have the same ow rate, especially if some get obstructed, corroded
or bent. We keep Bi ext = 0 :02 and Bi int = 0 :1 throughout the whole system, but we
now modify the ow rates in the coolant tubes: the ow rate at the entrance is still
F = 15, but it then splits as F = 5 in the _rst coolant tube, F = 4 in the second one,
F = 2 in all other coolant tubes. As a consequence, we can see in _gure 6.6c that
the uid bulk temperature decreases more in coolant tubes with the lowest ow rate.
After the mixing of all coolant tubes though, the exit temperature is almost the same
as in the _rst scenario @.919). The ow distribution in the coolant tubes seems not
to have any signi_cant e_ect on the exit temperature. Due to the fact that we use
the same RB spaces, the output error and output error bound are almost exactly the
same than for the _rst scenario.
For the third scenario, we consider the common situation when the _ns get dirty
with mud or bugs and the heat transfer coe_cient decreases. So in this example we
take Bi ext = 0 :02 and Bi int = 0 :1 throughout the whole system, except for the three
29
SCRBE FOR CONJUGATE HEAT TRANSFER
1.005
1.000
0.995
0.990
flu
id
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rat
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0.985
0.980
0.975
0.970
0
(a) solid
1.00
0.950
0.948
0.98
0.946
0.97
0.944
0.940
0.92
1st channel
2nd channel
3rd channel
4th channel
5th channel
0
1
6
inlet abscissa
8
10
12
flu
id
bu
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0.942
0.95
flu
id
bu
lk
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0.96
0.93
4
(b) uid in the inlet channel
0.99
0.94
2
0.938
0.936
2
3
channel abscissa
(c) uid in the coolant tubes
4
5
0.934
0
2
4
6
8
outlet abscissa
10
12
14
(d) uid in the outlet channel
Fig. 6.7: Temperature _eld in the system. Bi int = 0 :1. F = 15 at the inlet, F = 3
in each coolant tube. Bi ext = 0 :01 for the middle coolant tube (walls and _ns),
Bi ext = 0 :02 everywhere else. The red error bar corresponds to the error bound for
the uid outlet temperature.
middle \dirty" coolant tubes where we choose a smaller value of Bi ext = 0 :01. The
ow rate at the entrance is F = 15, and then splits so that it is equal to 3 in each tube.
We can see in _gure 6.7c that the uid bulk temperature decreases less in the three
middle coolant tubes, due to a smaller overall heat transfer coe_cient. This time the
outlet temperature is signi_cantly higher @.938), showing that the heat exchanger
does not perform as well as in the previous cases.
For the fourth and last scenario, we illustrate the many-parameter and heterogeneous capability of our method. We take Bi int = 0 :1 throughout the whole system,
but we select random values ofBi ext for each component independently, in the range
[0:01; 0:1]. The ow rate at the entrance is F = 15, and then splits so that it is equal
to 3 in each tube. We can see in _gure 6.8a the heterogeneity of the temperature
distribution in the solid domain.
In _gure 6.8c, we also see that the temperature
in the coolant tubes decreases at a variable rate, due to the variability of the heat
transfer coe_cient.
S. VALLAGHE_AND A.T. PATERA
30
1.00
0.99
0.98
0.97
0.96
flu
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e
0.95
0.94
0.93
0.92
0
(a) solid
2
4
6
inlet abscissa
8
10
12
(b) uid in the inlet channel
1.00
0.91
0.98
0.90
0.96
0.89
0.94
0.92
0.88
0.87
1st channel
2nd channel
3rd channel
4th channel
5th channel
0.88
0.86
0.84
flu
id
bu
lk
te
m
pe
rat
ur
e
flu
id
bu
lk
te
m
pe
rat
ur
e
0.90
0
1
0.86
2
3
channel abscissa
(c) uid in the coolant tubes
4
5
0.85
0
2
4
6
8
outlet abscissa
10
12
14
(d) uid in the outlet channel
Fig. 6.8: Temperature _eld in the system. Bi int = 0 :1. F = 15 at the inlet, F = 3
in each coolant tube. Bi ext is selected randomly in the range [0 :01; 0:1], for each
component independently. The red error bar corresponds to the error bound for the
uid outlet temperature.
Finally, we show in _gure 6.9 a zoom on the temperature distribution in the base
of a _n. We can see the restriction e_ect near the base of the _n where the temperature
_eld is 2D, then followed by the low-Biot largely 1D temperature within the _n itself.
It emphasizes why our detailed 2D PDE model in the solid is important, as opposed
to a simpli_ed 1D non-PDE model, like a lumped capacitance model.
We now move on to a much bigger system with 20 coolant tubes and 20 _ns per
tube components) to illustrate the computational savings of the Static Condensation Reduced Basis Element method. The parameters Bi ext and Bi int are chosen
to be 0.02 and 0.1 respectively throughout the whole system. The ow rate at the
entrance is F = 40, and then splits so that it is equal to 2 in each tube. The temperature _eld for this system is shown in _gure 6.10. The outlet temperature is 0.679
and the error bound is 2 :4 _ 10X 2 , which corresponds to an error of at most 3.5 %
between the RB and FE outlet temperature. The quality of the error bound decreases
compared to the previous examples, which is expected due to the fact that we are
SCRBE FOR CONJUGATE HEAT TRANSFER
31
Fig. 6.9: Zoom on the base of a _n
summing the RB bubble error bounds over many more components instead of
35).
The computational timings are the following: the assembly of the FE static condensation system takes 12 minutes whereas the assembly of the RB static condensation
system takes 1.3 seconds, corresponding to a speedup factor of 500. Then the static
condensation system is of dimension N P = 8500 and is solved in 0.06 seconds. It is
important to mention that although the Schur complement matrix has dense blocks
between coupled ports, globally there is much sparsity, which is taken advantage of
when solving the static condensation system. In comparison, the total system FE
space is of dimension 3_ 105 . To get an estimate of the cost of the system FE, we
solved a 2D Laplacian on a square with FE and 3_ 105 dofs: the assembly time is 1.5
seconds and the solving time is 4 minutes. So compared to the system FE, the speed
up factor is 180.
It should be noted that other computational savings can be obtained. First, in
the previous timings we did not take advantage of repeated subdomains which share
the same parameters, like the 400 _n components in the present example. But it is
possible to perform the RB evaluation only once for all these _n components, and in
this case the RB static condensation assembly time drops to less than 0.01 s. Second,
we use a Laplacian eigenmode decomposition for the degree of freedoms at the ports,
NP
and we can drop the non relevant modes (typically the high frequencies) to reduce
without any signi_cant loss in accuracy of the method. These computational aspects
are discussed in details in [7].
Finally, we show in _gure 6.11 the outlet temperature in the same system when
the parameter Bi ext is varied from 0.01 to 0.1 (for all components of the system).
The error bars correspond to the error bound, which is not shown for Bi ext = 0 :01
~min , as explained
because for this particular value we encountered the case_2> p _
in section 5.5.1.
Acknowledgement. This work was supported by ONR Grant N00014-11-0713.
We would like to acknowledge helpful discussions with Dr David Knezevic of Harvard
and Dr Phuong Huynh of MIT.
Appendix A. 1D problem.
S. VALLAGHE_AND A.T. PATERA
32
If is a rectangular domain, and we take the limit as W! 0 (the width of the
solid domain), then Bi ext ! 0, Bi int ! 0 and the solution of B.1)B.2) becomes
independent of the y coordinate in the solid. In fact, as W! 0, the PDE B.1) tends
to the following ODE where both _ and _ are de_ned on [0; _]:
( s _00+ Bi ext _ + Bi int ( _ s _ ) = f
F _ 0 s Bi int ( _ s _ ) = 0
(A.1)
We can hence consider (A.1) as the 1D version of our original 2D problem B.1). Since
this 1D limit is valid as Bi tends to zero, we would then take _ large in order to see
any appreciable temperature decay. But for the purpose of our simple numerical test
in section 6.1, we consider order unityBi and _. We will summarize in this appendix
the few di_erences of the 1D compared to the 2D problem.
First we denote I = [0 ; _]. The weak form reads: u 2 X satisfy a( u;v ) = f ( v)
8v 2 Z where
a( w;v ) =
Z d_d#
Z
Z
Z
Z d_
+ Bi ext
_# + Bi int
( _ s _ ) # s Bi int
( _ s _) ' + F
':
dx
dx
I
I
I
I
I dx
(A.2)
Looking at the stability, we get
_
a( w;w ) _
Z d_ 2
Z
1 Bi 2 Z
1
+ ( Bi ext s _ int ) _2 + _F
F
2
2
I dx
I
@
int
d_ 2 1
+ ( F + _Bi int ) _ 2 (_)
dx
2
_ K kwk2X
(A.3)
for _ small enough. We can take
Bi ext s
1 Bi 2int
_
= 0 ()
2 F
_=
2Bi ext F
Bi 2int
(A.4)
so that the second term in the right hand side of (A.3) vanishes.
a is then inf-sup
stable using the same arguments than in the 2D case, and the lower bound for the
inf-sup constant _0 is the same except for the new value of _. a is also bounded and
an upper bound for the continuity constant is
p
0 = 1 + Bi ext _ + 2 Bi int _ + 2 _ Bi int + F :
Using these new de_nitions for_;_;0 0 , the methodology for FE and RB approximations is exactly the same than in the 2D case.
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S. VALLAGHE_AND A.T. PATERA
34
(a) solid
1.00
0.715
0.710
0.98
0.705
0.700
0.96
0.695
0.94
flu
id
bu
lk
te
m
pe
rat
ur
e
flu
id
bu
lk
te
m
pe
rat
ur
e
0.690
0.685
0.92
0.680
0.90
0
10
20
30
inlet abscissa
40
(b) uid in the inlet channel
50
60
0.675
0
10
20
30
outlet abscissa
40
50
60
(c) uid in the outlet channel
Fig. 6.10: Temperature _eld in the system. Bi ext = 0 :02 and Bi int = 0 :1. F = 40 at
the inlet, F = 2 in each coolant tube.
35
SCRBE FOR CONJUGATE HEAT TRANSFER
0.80
0.75
0.70
0.65
0.60
O
utl
et
flu
id
bu
lk
te
m
pe
rat
ur
e
0.55
0.50
0.45
0.40
0.00
0.02
0.04
0.06
Biot number (ext)
0.08
0.10
0.12
Fig. 6.11: Outlet temperature with respect to Bi ext for a system with 20 coolant
tubes and 20 _ns per tube. The error bound is shown as red error bars.