Time efficiency is a critical concern in Computational Fluid Dynamics (CFD) simulations of industrial applications. Despite the extensive research to improve the underlying numerical schemes to reduce the processing time, many CFD applications still need to improve on this issue. Reduced-Order Models (ROM) appeared as a promising alternative for significantly enhancing cost-effectiveness. The dimension of the initial problem is reduced significantly while keeping an acceptable level of accuracy. Proper Orthogonal Decomposition (POD), a data analysis technique, is widely used to construct a ROM to solve Euler and Navier-Stokes equations in CFD. Many aspects, however, need to be improved. In this paper, first, a general upper-bound error formula is derived, and second, a mesh-adaptivity based algorithm for snapshot locations in parameter space is provided (a sensitive step for POD-based ROM) in the context of Partial Differential Equations (PDE). The method’s efficiency is demonstrated through numerical tests by comparison to exact solutions and numerical results of a finite volume scheme on inviscid flow over a NACA0012.
LoèveM. Probability theory; foundations, random sequences. New York: D. Van Nostrand Company, 1955.
2.
AlgaziVRSakrisonDJ. On the optimality of the karhunen-loeve expansion. IEEE Trans Inf Theor1969; 15: 319–321.
3.
LumleyJL. The structure of inhomogeneous turbulent flows. Atmospheric turbulence and radio wave propagation, 1967; 5: 153–175.
4.
HolmesPLumleyJLBerkoozG. Turbulence, Coherent structurs, dynamical systems and symmetry. Cambridge: Cambridge University Press, 2012.
5.
SirovichL. Turbulence and the dynamics of coherent structures part i: coherent structures. Q Appl Math1987; XLV(3): 561–571.
6.
PinnauR. Model reduction via proper orthogonal decomposition. Model Order Reduction: Theory, Research Aspects and Applications2008; 13: 95–109.
7.
VolkweinS. Proper orthogonal decomposition: theory and reduced-order modelling. Lecture Notes, University of Konstanz2013; 4(4): 100654.
8.
KunischKVolkweinS. Galerkin proper orthogonal decomposition methods for parabolic problems. Numer Math2001; 90(1): 117–148.
9.
KunischKVolkweinS. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J Numer Anal2002; 40(2): 492–515.
10.
RathinamMPetzoldLR. A new look at proper orthogonal decomposition. SIAM J Numer Anal2003; 41(5): 1893–1925.
11.
ChristensenEABronsMSorensenJN. Evaluation of proper orthogonal decomposition–based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J Sci Comput1999; 21(4): 1419–1434.
12.
Bui-ThanhTDamodaranMWillcoxK. Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics. In: 21st AIAA applied aerodynamics conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, pp. 1–11.
13.
Bui-ThanhTWillcoxKGhattasO. Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J Sci Comput2008; 30(6): 3270–3288.
14.
AmssalemDFarhatC. An interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J2008; 46(7): 35374. DOI: 10.2514/1.35374.
15.
LieuTFarhatC. Pod-based aeroelastic analysis of a complete f-16 configuration: rom adaptation and demonstration. Reston, VA: AIAA. DOI: 10.2514/6.2005-2295.
16.
BoyavalSLe BrisCLelièvreT, et al.Reduced basis techniques for stochastic problems. Arch Comput Methods Eng2010; 17(4): 435–454.
17.
BaurUBennerPFengL. Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Arch Comput Methods Eng2014; 21(4): 331–358.
18.
BennerPGugercinSWillcoxK. A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev2015; 57(4): 483–531.
19.
QuarteroniAManzoniANegriF. Reduced basis methods for partial differential equations - an introduction. Salmon: SpringerLink, 2016.
20.
SchildersWHHaVDVRommesJ. Model Order Reduction: Theory, Research Aspects and Applications. Berlin: Springer Science & Business Media, 2008, vol 13.
21.
HesthavenJSRozzaGStammB, et al.Certified reduced basis methods for parametrized partial differential equations. Salmon: Springer, 2016.
22.
QuarteroniARozzaGManzoniA. Certified reduced basis approximation for parametrized partial differential equations and applications. J Math Ind2011; 1(1): 3.
23.
HuynhDRozzaGSenS, et al.A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants. Compt Rendus Math2007; 345(8): 473–478.
24.
RozzaGHuynhDBPateraAT. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Arch Comput Methods Eng2008; 15(3): 229–275.
25.
VeroyKPrud’hommeCRovasD, et al.A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: 16th AIAA computational Fluid Dynamics conference, Orlando, Florida, 23 June 2003 - 26 June 2003, pp. 1–18.
26.
GreplMAPateraAT. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM Math Model Numer Anal2005; 39(1): 157–181.
27.
PateraARozzaG. Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. version 1.0, copyright mit 2006, to appear in (tentative rubric) mit pappalardo graduate monographs in mechanical engineering. Tentative Rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering2006; 125: 243–251.
LassOVolkweinS. Adaptive pod basis computation for parametrized nonlinear systems using optimal snapshot location. Comput Optim Appl2014; 58: 645–677.
30.
AkkariELAHamdouniNJazarMLibergeE, et al.A mathematical and numerical study of the sensitivity of a reduced order model by pod (rom–pod), for a 2d incompressible fluid flow. J Comput Appl Math2014; 270: 522–530.
31.
Berkooz PHGLumleyJ. The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech1993; 25: 539–575.
32.
Qs ZhangYLWangS. The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition. J Fluid Struct2014; 49: 53–72.
33.
BertsimasDTsitsiklisJN. Introduction to linear optimization. Belmont: Athena Scientific Belmont, 1997, vol volume 6.
34.
VeroyKPateraA. Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced-basis a posteriori error bounds. Int J Numer Methods Fluid2005; 47(8-9): 773–788.
35.
O’ConnorRGreplM. Offline error bounds for the reduced basis method. In: Model reduction of parametrized systems. Berlin: Springer, 2017, pp. 121–136.
36.
FortinM. Scientific computing and applications. Chapter anisotropic mesh adaptation through hierarchical error estimators. Commack. NY, USA: Nova Science Publishers, Inc., 2001, pp. 53–65.
37.
DolejsiVFelcmanJ. Anisotropic mesh adaptation for transonic and supersonic flow simulation. In Proceedings of ALGORITMY conference on scientific computing. New York, NY: Association for Computing Machinery, 78–852002.
38.
HabashiWGDompierreJBourgaultY, et al.Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD solutions: Part I: general principles. Int J Numer Methods Fluid2000; 32: 725–744.
39.
RemakiLZhongqiangXHassanO, et al.A high order finite volume-hllc solver and anisotropic delaunay mesh adaptation. In: 47th AIAA aerosp. sci. Meet. Incl. New horizons forum aerosp. expo. Aerospace sciences meetings. Reston, VA: American Institute of Aeronautics and Astronautics, 2009. DOI: 10.2514/6.2009-1498.
40.
RemakiLHabashiWG. 3-D mesh adaptation on multiple weak discontinuities and boundary layers. SIAM J Sci Comput2006; 28(4): 1379–1397.
41.
PeraireJVahdatiMMorganK, et al.Adaptive remeshing for compressible flow computations. J Comput Phys1987; 72(2): 449–466.
42.
VerfürthR. A posteriori error estimation and adaptive mesh-refinement techniques. J Comput Appl Math1994; 50(1): 67–83.
43.
BakerTJ. Mesh adaptation strategies for problems in fluid dynamics. Finite Elem Anal Des1997; 25(3): 243–273.
44.
KoperaMAGiraldoFX. Analysis of adaptive mesh refinement for imex discontinuous galerkin solutions of the compressible euler equations with application to atmospheric simulations. J Comput Phys2014; 275: 92–117.
45.
ZhanZHabashiWGFossatiM. Local reduced-order modeling and iterative sampling for parametric analyses of aero-icing problems. AIAA J2015; 53(8): 2174–2185.
46.
NairNJBalajewiczM. Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter-dependent shocks. Int J Numer Methods Eng2019; 117(12): 1234–1262.
47.
EconomonTDPalaciosFCopelandSR, et al.Su2: an open-source suite for multiphysics simulation and design. AIAA J2016; 54(3): 828–846. DOI: 10.2514/1.J053813.
48.
PiniMVitaleSColonnaP, et al.SU2: the open-source software for non-ideal compressible flows. In: 1ST international seminar on non-ideal compressible-fluid dynamics for propulsion & power, journal of physics conference series. Varenna, ITALY: 1st International Seminar on Non-Ideal Compressible-Fluid Dynamics for Propulsion and Power (NICFD), vol volume 821.
49.
GoriGZoccaMCammiGet al.Experimental assessment of the open-source SU2 CFD suite for ORC applications, DossenaVGuardoneAAstolfiM (ed.) 4TH international seminar on orc power systems, energy procedia, 4th International Seminar on Organic Rankine Cycle Power Systems (ORC), Politecnico Milano Bovisa Campus, Milano, ITALY, volume 129. pp. 256–263.
50.
PallabSMHGSouvickCKoushikG, et al.Analysis of natural convection in different geometries using proper orthogonal decomposition. Proceedings of the 22nd National and 11th International ISHMT-ASME Heat and Mass Transfer Conference2013; 87(2): 1–29.
