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

Residual Data-Driven Variational Multiscale Reduced Order Models for Convection-Dominated Problems ? Birgul Koc ∗Samuele Rubino ∗∗ Tom´as Chac´on ∗∗∗ Traian Iliescu ∗∗∗∗ ∗University of Seville, EDAN, Spain (e-mail: bko[email protected]). ∗∗ University of Seville, EDAN, Spain (e-mail: [email protected]) ∗∗∗ University of Seville, EDAN, Spain (e-mail: chac[email protected]) ∗∗∗∗ Virginia Tech, USA (e-mail: [email protected]du) 1. INTRODUCTION As a mathematical model, we use Navier-Stokes equations (NSE) (1)-(2): ∂u ∂t −Re−1∆u+u· ∇u+∇p=0,(1) ∇ · u= 0 ,(2) where uis the velocity, pthe pressure, tthe continuous time instant, and Re the Reynolds number. Furthermore, we use homogeneous Dirichlet boundary conditions. We use proper orthogonal decomposition (POD) to obtain the reduced order model (ROM) basis and operators for all ROMs. Thanks to the orthogonality of the ROM basis functions, we can decompose the ROM space into large and small spaces as follows: Xd=XL⊕XS, where Xd:= span{ϕ1, ..., ϕd},XL:= span{ϕ1, ..., ϕL}, and XS:= span{ϕL+1, ..., ϕd}. When all the ROM modes are used, the ROM approximation ud, i.e., ud= d X j=1 (ad)jϕj(3) is the most accurate ROM approximation of the full order model (FOM) solution with the given data in the POD sense. For laminar flows, a low-dimensional ROM solution uL, with small Ld, yields an accurate approximation of the FOM solution. In the resolved regime, the most straightforward model of ROMs, Galerkin ROM (G-ROM) can be used to obtain the ROM solution uL: ˙ aL=ALL aL+aL >BLLL aL,(4) where (ALL)ij := −Re−1(∇ϕi,∇ϕj) and (BLLL)ijk := −(ϕi,ϕj·∇ϕk), respectively, ∀i, j, k = 1, ..., L. The derivation of the G-ROM (4) is built by replacing uin (1)-(2) with uLand projecting the resulting system onto the ROM space XL. However, for turbulent flows, the low-dimensional ROM solution aLof (4) is not an accurate approximation of the FOM solution. To increase the numerical accuracy of aL ? without significantly increasing the computational cost, one needs to add a low-dimensional ROM closure term to the G-ROM (4). 2. ROM CLOSURE MODELS The ROM closure modeling aims to model the closure term which is derived from a variational multiscale (VMS) setting (see Mou et al. (2021) and Ballarin et al. (2020)). To construct the ROM closure term, first, we need to define the large and sub-scale solutions of the most accurate ROM solution, ud, as follows: uL:= L X j=1 (aL)jϕj,uS:= d X j=L+1 (aS)jϕj.(5) Then, we obtain the large and sub-scale equations: (i) replace the uin (1)-(2) with ud=uL+uSand project the resulting system onto the ROM spaces XLand XS, respectively. Then, the large and sub-scale equations are: ˙ 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 this work, we use two different ROM closure constructions, which yield two different ROM model: the coefficient-based data-driven variational multiscale ROM (C-D2-VMS-ROM) and the residual-based data-driven variational multiscale ROM (R-D2-VMS-ROM). The C-D2-VMS-ROM (Mou et al. (2021)) is derived from the large-scale equation (6a) by defining the closure term as ”closure term =ALS aS+a> LBLLSaS+a> SBLSLaL+ a> SBLSSaS”. Since the closure term is not in a closed form, to close it, we use a quadratic coefficient-based ansatz (Mou et al. (2021)), which depends on the largescale solution aL: ”ansatz =e ALL aL+a> Le BLLLaL”. In the new R-D2-VMS-ROM, we define the closure term and residual-based ansatz from the sub-scale equation (6b) as ”closure term =aS” and ”ansatz = e ASS ResS(aL)+ResS(aL)>e BSSS ResS(aL)”, where the residual is ResS(aL) := ASLaL+a> LBSLLaL. 11 DOI: 10.34726/9005 To find the unknown operators e A,e B, we use a data-driven (D2) approach (Rebollo and Coronil (2024)). We obtain the D2 operators by solving the following minimization problem: min e A,e B M X k=1 closure term(ak L,ak S)−ansatz(ak L) 2 L2,(7) where Mrepresents the number of snapshots. By using the closure terms and ansatzes for the C-D2-VMS-ROM and R-D2-VMS-ROM, we solve (7) to obtain the corresponding D2 operators, i.e. e Aand e B. Then, by plugging the resulting ansatzes into (6a), C-D2-VMS-ROM and R-D2VMS-ROM read as follows: ˙ 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) where approximated sub-scale coefficient a∗ Sis computed as a∗ S:= e ASS ResS(aL) + ResS(aL)>e BSSS ResS(aL). 3. NUMERICAL RESULTS We investigate the numerical accuracy of G-ROM, C-D2VMS-ROM, and new R-D2-VMS-ROM in the numerical simulation of a 2D channel flow past a circular cylinder at Reynolds numbers Re = 1000. We present the numerical accuracy of the ROM models for two different regimes: (i) reconstructive regime: we build the ROM basis and operators, and D2 operators by using the FOM snapshots from t= 13 to t= 16. Then, we test ROMs over the same time interval. (ii) predictive regime: we build the ROM basis and operators by using the FOM snapshots from t= 13 to t= 16, and D2 operators by using the FOM snapshots from t= 13 to t= 13.134. Then, we test ROMs over the longer time interval, t= 16 to t= 23. Furthermore, in our numerical accuracy investigation of the ROMs, we use the average L2projection error: EavgL2proj =1 M M X k=1     uL(tk)− L X i=1 uF OM (tk),ϕiϕi    L2 . (9) In Tables 1-2, we list the average L2projection errors of G-ROM, C-D2-VMS-ROM, and R-D2-VMS-ROM for the reconstructive and predictive regimes, respectively. In Table 1, we observe that C-D2-VMS-ROM and R-D2VMS-ROM yield much better accuracy (for some values, the improvement is more than 2 orders of magnitude) than G-ROM in the reconstructive regime. C-D2-VMS-ROM and R-D2-VMS-ROM have similar accuracy behavior. In Table 2, we still observe that C-D2-VMS-ROM and R-D2VMS-ROM yield much better accuracy (for some values, the improvement is more than 1 order of magnitude) than G-ROM in the predictive regime. Furthermore, R-D2VMS-ROM yields better accuracy than C-D2-VMS-ROM. In Figures 1-2, we plot the kinetic energy of the FOM projection, G-ROM, C-D2-VMS-ROM, and R-D2-VMS-ROM for the reconstructive and predictive regimes, respectively. We fix the large-scale ROM dimension L= 6, to compare the kinetic energy behavior of C-D2-VMS-ROM and RD2-VMS-ROM. We observe that R-D2-VMS-ROM is sigFig. 1. Reconstructive regime; kinetic energy of ROMs. Fig. 2. Predictive regime; kinetic energy of ROMs. nificantly more accurate than C-D2-VMS-ROM, especially in the predictive regime. Acknowledgments: Research is partially funded by the Spanish Research Agency Juan de la Cierva 2022 with 2023/1061 and PID2021-123153OB-C21 - Feder Fund Grants. REFERENCES Ballarin, F., Rebollo, T.C., Avila, E.D., M´armol, M.G., and Rozza, G. (2020). Certified reduced basis vmssmagorinsky model for natural convection flow in a cavity with variable height. Computers & Mathematics with Applications, 80(5), 973–989. Mou, C., Koc, B., San, O., Rebholz, L.G., and Iliescu, T. (2021). Data-driven variational multiscale reduced order models. Computer Methods in Applied Mechanics and Engineering, 373, 113470. Rebollo, T.C. and Coronil, D.F. (2024). Data-driven stabilized finite element solution of advection-dominated flow problems. Mathematics and Computers in Simulation, 226, 540–559. Table 1. Reconstructive regime; average L2 projection error (9) for different Lvalues. 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. Predictive regime; average L2projection error (9) for different Lvalues. 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