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