Residual Data-Driven Variational Multiscale Reduced Order Models for Convection-Dominated Problems

Residual Da a-D i en Va ia ional
Mul iscale Reduced O de Models o
Con ec ion-Domina ed P oblems ?
Bi gul Koc ∗Samuele Rubino ∗∗ Tom´as Chac´on ∗∗∗
T aian Iliescu ∗∗∗∗
∗Uni e si y o Se ille, EDAN, Spain (e-mail: bko[email p o ec ed]).
∗∗ Uni e si y o Se ille, EDAN, Spain (e-mail: [email p o ec ed])
∗∗∗ Uni e si y o Se ille, EDAN, Spain (e-mail: chac[email p o ec ed])
∗∗∗∗ Vi ginia Tech, USA (e-mail: [email p o ec ed]du)
1. INTRODUCTION
As a ma hema ical model, we use Na ie -S okes equa ions
(NSE) (1)-(2):
∂u
∂ −Re−1∆u+u· ∇u+∇p=0,(1)
∇ · u= 0 ,(2)
whe e uis he eloci y, p he p essu e, he con inuous
ime ins an , and Re he Reynolds numbe . Fu he mo e,
we use homogeneous Di ichle bounda y condi ions.
We use p ope o hogonal decomposi ion (POD) o ob ain
he educed o de model (ROM) basis and ope a o s o
all ROMs. Thanks o he o hogonali y o he ROM basis
unc ions, we can decompose he ROM space in o la ge
and small spaces as ollows: Xd=XL⊕XS, whe e
Xd:= span{ϕ1, ..., ϕd},XL:= span{ϕ1, ..., ϕL}, and
XS:= span{ϕL+1, ..., ϕd}.
When all he ROM modes a e used, he ROM app oxima-
ion ud, i.e.,
ud=
d
X
j=1
(ad)jϕj(3)
is he mos accu a e ROM app oxima ion o he ull o de
model (FOM) solu ion wi h he gi en da a in he POD
sense.
Fo lamina lows, a low-dimensional ROM solu ion uL,
wi h small Ld, yields an accu a e app oxima ion
o he FOM solu ion. In he esol ed egime, he mos
s aigh o wa d model o ROMs, Gale kin ROM (G-ROM)
can be used o ob ain he ROM solu ion uL:
˙
aL=ALL aL+aL
>BLLL aL,(4)
whe e (ALL)ij := −Re−1(∇ϕi,∇ϕj) and (BLLL)ijk :=
−(ϕi,ϕj·∇ϕk), espec i ely, ∀i, j, k = 1, ..., L. The de i a-
ion o he G-ROM (4) is buil by eplacing uin (1)-(2)
wi h uLand p ojec ing he esul ing sys em on o he ROM
space XL.
Howe e , o u bulen lows, he low-dimensional ROM
solu ion aLo (4) is no an accu a e app oxima ion o he
FOM solu ion. To inc ease he nume ical accu acy o aL
?
wi hou signi ican ly inc easing he compu a ional cos ,
one needs o add a low-dimensional ROM closu e e m
o he G-ROM (4).
2. ROM CLOSURE MODELS
The ROM closu e modeling aims o model he closu e
e m which is de i ed om a a ia ional mul iscale (VMS)
se ing (see Mou e al. (2021) and Balla in e al. (2020)).
To cons uc he ROM closu e e m, i s , we need o de ine
he la ge and sub-scale solu ions o he mos accu a e
ROM solu ion, ud, as ollows:
uL:=
L
X
j=1
(aL)jϕj,uS:=
d
X
j=L+1
(aS)jϕj.(5)
Then, we ob ain he la ge and sub-scale equa ions: (i)
eplace he uin (1)-(2) wi h ud=uL+uSand p ojec
he esul ing sys em on o he ROM spaces XLand XS,
espec i ely. Then, he la ge and sub-scale equa ions a e:
˙
aL=ALLaL+ALSaS+a>
LBLLLaL
+a>
LBLLSaS+a>
SBLSLaL+a>
SBLSSaS,(6a)
˙
aS=ASSaS+ASLaL+a>
SBSSS aS
+a>
SBSSLaL+a>
LBSLSaS+a>
LBSLLaL.(6b)
In his wo k, we use wo di e en ROM closu e con-
s uc ions, which yield wo di e en ROM model: he
coe icien -based da a-d i en a ia ional mul iscale ROM
(C-D2-VMS-ROM) and he esidual-based da a-d i en
a ia ional mul iscale ROM (R-D2-VMS-ROM).
The C-D2-VMS-ROM (Mou e al. (2021)) is de i ed om
he la ge-scale equa ion (6a) by de ining he closu e e m
as ”closu e e m =ALS aS+a>
LBLLSaS+a>
SBLSLaL+
a>
SBLSSaS”. Since he closu e e m is no in a closed
o m, o close i , we use a quad a ic coe icien -based
ansa z (Mou e al. (2021)), which depends on he la ge-
scale solu ion aL: ”ansa z =e
ALL aL+a>
Le
BLLLaL”.
In he new R-D2-VMS-ROM, we de ine he closu e
e m and esidual-based ansa z om he sub-scale equa-
ion (6b) as ”closu e e m =aS” and ”ansa z =
e
ASS ResS(aL)+ResS(aL)>e
BSSS ResS(aL)”, whe e he
esidual is ResS(aL) := ASLaL+a>
LBSLLaL.
11
DOI: 10.34726/9005
To ind he unknown ope a o s e
A,e
B, we use a da a-d i en
(D2) app oach (Rebollo and Co onil (2024)). We ob ain
he D2 ope a o s by sol ing he ollowing minimiza ion
p oblem:
min
e
A,e
B
M
X
k=1
closu e e m(ak
L,ak
S)−ansa z(ak
L)
2
L2,(7)
whe e M ep esen s he numbe o snapsho s. By using he
closu e e ms and ansa zes o he C-D2-VMS-ROM and
R-D2-VMS-ROM, we sol e (7) o ob ain he co esponding
D2 ope a o s, i.e. e
Aand e
B. Then, by plugging he
esul ing ansa zes in o (6a), C-D2-VMS-ROM and R-D2-
VMS-ROM ead as ollows:
˙
aL= (ALL +e
ALL)aL+a>
L(BLLL +e
BLLL)aL,(8a)
˙
aL=ALLaL+a>
LBLLLaL+ALSa∗
S+a>
LBLLSa∗
S
+ (a∗
S)>BLSLaL+ (a∗
S)>BLSSa∗
S,(8b)
whe e app oxima ed sub-scale coe icien a∗
Sis compu ed
as a∗
S:= e
ASS ResS(aL) + ResS(aL)>e
BSSS ResS(aL).
3. NUMERICAL RESULTS
We in es iga e he nume ical accu acy o G-ROM, C-D2-
VMS-ROM, and new R-D2-VMS-ROM in he nume ical
simula ion o a 2D channel low pas a ci cula cylinde a
Reynolds numbe s Re = 1000. We p esen he nume ical
accu acy o he ROM models o wo di e en egimes:
(i) econs uc i e egime: we build he ROM basis and
ope a o s, and D2 ope a o s by using he FOM snapsho s
om = 13 o = 16. Then, we es ROMs o e he
same ime in e al. (ii) p edic i e egime: we build he
ROM basis and ope a o s by using he FOM snapsho s
om = 13 o = 16, and D2 ope a o s by using he
FOM snapsho s om = 13 o = 13.134. Then, we es
ROMs o e he longe ime in e al, = 16 o = 23.
Fu he mo e, in ou nume ical accu acy in es iga ion o
he ROMs, we use he a e age L2p ojec ion e o :
Ea gL2p oj =1
M
M
X
k=1




uL( k)−
L
X
i=1
uF OM ( k),ϕiϕi



L2
.
(9)
In Tables 1-2, we lis he a e age L2p ojec ion e o s
o G-ROM, C-D2-VMS-ROM, and R-D2-VMS-ROM o
he econs uc i e and p edic i e egimes, espec i ely. In
Table 1, we obse e ha C-D2-VMS-ROM and R-D2-
VMS-ROM yield much be e accu acy ( o some alues,
he imp o emen is mo e han 2 o de s o magni ude) han
G-ROM in he econs uc i e egime. C-D2-VMS-ROM
and R-D2-VMS-ROM ha e simila accu acy beha io . In
Table 2, we s ill obse e ha C-D2-VMS-ROM and R-D2-
VMS-ROM yield much be e accu acy ( o some alues,
he imp o emen is mo e han 1 o de o magni ude)
han G-ROM in he p edic i e egime. Fu he mo e, R-D2-
VMS-ROM yields be e accu acy han C-D2-VMS-ROM.
In Figu es 1-2, we plo he kine ic ene gy o he FOM p o-
jec ion, G-ROM, C-D2-VMS-ROM, and R-D2-VMS-ROM
o he econs uc i e and p edic i e egimes, espec i ely.
We ix he la ge-scale ROM dimension L= 6, o compa e
he kine ic ene gy beha io o C-D2-VMS-ROM and R-
D2-VMS-ROM. We obse e ha R-D2-VMS-ROM is sig-
Fig. 1. Recons uc i e egime; kine ic ene gy o ROMs.
Fig. 2. P edic i e egime; kine ic ene gy o ROMs.
ni ican ly mo e accu a e han C-D2-VMS-ROM, especially
in he p edic i e egime.
Acknowledgmen s: Resea ch is pa ially unded by he
Spanish Resea ch Agency Juan de la Cie a 2022 wi h
2023/1061 and PID2021-123153OB-C21 - Fede Fund
G an s.
REFERENCES
Balla in, F., Rebollo, T.C., A ila, E.D., M´a mol, M.G.,
and Rozza, G. (2020). Ce i ied educed basis ms-
smago insky model o na u al con ec ion low in a
ca i y wi h a iable heigh . Compu e s & Ma hema ics
wi h Applica ions, 80(5), 973–989.
Mou, C., Koc, B., San, O., Rebholz, L.G., and Iliescu, T.
(2021). Da a-d i en a ia ional mul iscale educed o de
models. Compu e Me hods in Applied Mechanics and
Enginee ing, 373, 113470.
Rebollo, T.C. and Co onil, D.F. (2024). Da a-d i en s abi-
lized ini e elemen solu ion o ad ec ion-domina ed low
p oblems. Ma hema ics and Compu e s in Simula ion,
226, 540–559.
Table 1. Recons uc i e egime; a e age L2
p ojec ion e o (9) o di e en L alues.
L G-ROM C-D2-VMS-ROM R-D2-VMS-ROM
2 4.94e-01 4.00e-03 5.06e-03
3 5.11e-01 3.09e-03 4.17e-03
4 5.98e-01 2.89e-03 1.45e-03
5 6.58e-01 6.07e-03 1.31e-03
6 1.50e-01 2.62e-03 9.83e-04
7 1.36e-01 2.76e-03 4.42e-03
8 7.08e-02 3.14e-03 1.32e-03
Table 2. P edic i e egime; a e age L2p ojec-
ion e o (9) o di e en L alues.
L G-ROM C-D2-VMS-ROM R-D2-VMS-ROM
2 1.15e+00 4.11e-01 3.59e-01
3 9.22e-01 5.51e-01 6.16e-02
4 7.21e-01 1.98e-01 1.18e-01
5 7.28e-01 5.81e-01 3.11e-01
6 3.54e-01 1.48e-01 4.36e-02
7 3.02e-01 2.81e-01 5.29e-02
8 1.59e-01 9.44e-02 2.53e-02
12
DOI: 10.34726/9005