POD Reduced-Order Modeling of Complex Fluid Flows Zhu Wang - PowerPoint PPT Presentation

POD Reduced-Order Modeling of Complex Fluid Flows Zhu Wang Department of Mathematics University of South Carolina ICERM - Algorithms for Dimension and Complexity Reduction March 25, 2020 Collaborated with Jeff Borggaard, Traian Iliescu (VT), Xuping Xie (NYU) Z. Wang, USC POD-ROM of complex flows ICERM 2020 1 / 41

Turbulent Flows ◮ flow control and optimization computer aided design Photographer Christian Steiness. http://www.ict-aeolus.eu ◮ challenges many-query simulations of large-scale, time-dependent, nonlinear systems. However, short time, even real-time evaluation is needed ◮ what to do? computational efficient and reliable surrogate – reduced-order models Z. Wang, USC POD-ROM of complex flows ICERM 2020 2 / 41

Outline POD-ROM for Incompressible Fluid Flows 1 Closure Methods for POD-ROM 2 Implementation Improvements 3 Conclusions 4 Z. Wang, USC POD-ROM of complex flows ICERM 2020 3 / 41

Galerkin Projection-Based POD-ROM Representative ¡ Data ¡ POD ¡ Optimal ¡Basis ¡ Galerkin ¡ POD-­‑G-­‑ROM ¡ Z. Wang, USC POD-ROM of complex flows ICERM 2020 4 / 41

Galerkin Projection-Based POD-ROM ◮ Navier-Stokes equations (NSE) Representative ¡ � Data ¡ ∂ u 1 ∂ t + ( u · ∇ ) u − Re ∆ u + ∇ p = 0 POD ¡ ∇ · u = 0 ◮ non-parametric case in following discussion Optimal ¡Basis ¡ ◮ FE, FD, FV ⇒ snapshots u ( · , t i ) Galerkin ¡ ◮ V = span { u ( · , t 1 ) , u ( · , t 2 ) , . . . , u ( · , t n s ) } POD-­‑G-­‑ROM ¡ POD ◮ S n dof × n s ⇒ POD basis φ 1 , . . . , φ r , φ r + 1 , . . . , φ d = Z. Wang, USC POD-ROM of complex flows ICERM 2020 4 / 41

Galerkin Projection-Based POD-ROM ◮ Proper Orthogonal Decomposition (POD) Representative ¡ Data ¡ � � 2 � � � M � r 1 � � POD ¡ min � u ( · , t j ) − � ( u ( · , t j ) , φ i ( · )) H φ i ( · ) � n s � � φ i � 2 H = 1 j = 1 i = 1 H Optimal ¡Basis ¡ ◮ R φ ( x ) = λ φ ( x ) Galerkin ¡ R k , j = 1 n s ( u ( · , t j ) , u ( · , t k )) H POD-­‑G-­‑ROM ¡ ◮ method of snapshots Z. Wang, USC POD-ROM of complex flows ICERM 2020 4 / 41

Galerkin Projection-Based POD-ROM Representative ¡ Data ¡ POD ¡ Optimal ¡Basis ¡ 0.1 0.08 0.06 0.04 Galerkin ¡ 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 POD-­‑G-­‑ROM ¡ φ 1 Z. Wang, USC POD-ROM of complex flows ICERM 2020 5 / 41

Galerkin Projection-Based POD-ROM Representative ¡ Data ¡ POD ¡ Optimal ¡Basis ¡ 0.1 0.08 0.06 0.04 Galerkin ¡ 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 POD-­‑G-­‑ROM ¡ φ 3 Z. Wang, USC POD-ROM of complex flows ICERM 2020 5 / 41

Galerkin Projection-Based POD-ROM � r ◮ u ( x , t ) ≈ u r = u c ( x ) + a i ( t ) φ i ( x ) Representative ¡ i = 1 Data ¡ ◮ POD-G ROM POD ¡ � ∂ u r � � 2 � Re D ( u r ) , ∇ φ k + (( u r · ∇ ) u r , φ k ) + = 0 ∂ t , φ k Optimal ¡Basis ¡ k = 1 , . . . , r ◮ r ∼ O(10) << n dof Galerkin ¡ d a d t = A + B a + C ( a ⊗ a ) ◮ POD-­‑G-­‑ROM ¡ A r × 1 , B r × r , C r × r 2 precomputed Z. Wang, USC POD-ROM of complex flows ICERM 2020 6 / 41

Galerkin Projection-Based POD-ROM d a d t = A + B a + C ( a ⊗ a ) Representative ¡ Data ¡ → a handful DOF POD ¡ → low CPU time Optimal ¡Basis ¡ → same order accuracy Galerkin ¡ POD-­‑G-­‑ROM ¡ Z. Wang, USC POD-ROM of complex flows ICERM 2020 7 / 41

Galerkin Projection-Based POD-ROM d a d t = A + B a + C ( a ⊗ a ) Representative ¡ Data ¡ √ → a handful DOF POD ¡ → low CPU time Optimal ¡Basis ¡ → same order accuracy Galerkin ¡ POD-­‑G-­‑ROM ¡ Z. Wang, USC POD-ROM of complex flows ICERM 2020 7 / 41

Galerkin Projection-Based POD-ROM d a d t = A + B a + C ( a ⊗ a ) Representative ¡ Data ¡ √ → a handful DOF POD ¡ √ → low CPU time Optimal ¡Basis ¡ → same order accuracy Galerkin ¡ POD-­‑G-­‑ROM ¡ Z. Wang, USC POD-ROM of complex flows ICERM 2020 7 / 41

Galerkin Projection-Based POD-ROM d a d t = A + B a + C ( a ⊗ a ) Representative ¡ Data ¡ √ → a handful DOF POD ¡ √ → low CPU time Optimal ¡Basis ¡ → same order accuracy ? Galerkin ¡ POD-­‑G-­‑ROM ¡ Z. Wang, USC POD-ROM of complex flows ICERM 2020 7 / 41

Galerkin Projection-Based POD-ROM d a d t = A + B a + C ( a ⊗ a ) Representative ¡ Data ¡ √ → a handful DOF POD ¡ √ → low CPU time Optimal ¡Basis ¡ → same order accuracy Galerkin ¡ ◮ simple flow POD-­‑G-­‑ROM ¡ ◮ complex flow Z. Wang, USC POD-ROM of complex flows ICERM 2020 7 / 41

Galerkin Projection-Based POD-ROM d a d t = A + B a + C ( a ⊗ a ) Representative ¡ Data ¡ √ → a handful DOF POD ¡ √ → low CPU time Optimal ¡Basis ¡ → same order accuracy Galerkin ¡ √ ◮ simple flow POD-­‑G-­‑ROM ¡ ◮ complex flow Z. Wang, USC POD-ROM of complex flows ICERM 2020 7 / 41

Galerkin Projection-Based POD-ROM d a d t = A + B a + C ( a ⊗ a ) √ → a handful DOF √ → low CPU time → same order accuracy √ ◮ simple flow ◮ complex flow × Z. Wang, USC POD-ROM of complex flows ICERM 2020 7 / 41

Outline POD-ROM for Incompressible Fluid Flows 1 Closure Methods for POD-ROM 2 Implementation Improvements 3 Conclusions 4 Z. Wang, USC POD-ROM of complex flows ICERM 2020 8 / 41

Energy Cascade ◮ to model the effect of truncated POD modes ◮ POD and Fourier are connected Holmes, Lumley, Berkooz, 1996 ◮ energy cascade in solutions of NSE Richardson, 1922 Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity. Kolmogorov, 1941 • at high Reynolds numbers 2 3 k − 5 in inertial range, E ( k ) = α � ǫ � 3 beyond it, kinetic energy is negligible Pope, 2000 Z. Wang, USC POD-ROM of complex flows ICERM 2020 9 / 41

Energy Cascade ◮ to model the effect of truncated POD modes ◮ POD and Fourier are connected Holmes, Lumley, Berkooz, 1996 ◮ energy cascade in solutions of NSE 1 energy is input into the largest scales of the flow; 2 there is an intermediate range in which nonlinearity drives this energy into smaller and smaller scales and conserves the global energy because dissipation is negligible; 3 at small enough scales, dissipation is non negligible and the energy in those smallest scales decays to zero exponentially fast. Layton, 2008 Z. Wang, USC POD-ROM of complex flows ICERM 2020 9 / 41

Energy Cascade ◮ to model the effect of truncated POD modes ◮ POD and Fourier are connected Holmes, Lumley, Berkooz, 1996 ◮ energy cascade for solutions of NSE in POD setting ◮ Couplet, Sagaut, Basdevant, J. Fluid Mech., 2003 Z. Wang, USC POD-ROM of complex flows ICERM 2020 9 / 41

POD Filter ◮ ( u − u r , φ ) = 0 ( POD projection ) ⇐ ⇒ ( u − u , φ ) = 0 ( Filter ) ∂ t + ∇ · ( u u ) − Re − 1 ∆ u = 0 ◮ POD-ROM ∂ u ◮ NSE ∂ u ∂ t + ∇ · ( u u ) − Re − 1 ∆ u = 0 ∂ u ∂ t + ∇ · ( u u ) + ∇ · τ − Re − 1 ∆ u = 0 τ = u u − u u ◮ to close POD-ROM ∇ · τ := − ν ∗ ∆ u 1 functional closure ∇ · τ := ∇ · ( u ∗ u ∗ − u u ) 2 structural closure Z. Wang, USC POD-ROM of complex flows ICERM 2020 10 / 41

POD Filter ◮ ( u − u r , φ ) = 0 ( POD projection ) ⇐ ⇒ ( u − u , φ ) = 0 ( Filter ) ∂ t + ∇ · ( u u ) − Re − 1 ∆ u = 0 ◮ POD-ROM ∂ u ◮ NSE ∂ u ∂ t + ∇ · ( u u ) − Re − 1 ∆ u = 0 ∂ u ∂ t + ∇ · ( u u ) + ∇ · τ − Re − 1 ∆ u = 0 τ = u u − u u ◮ to close POD-ROM ∇ · τ := − ν ∗ ∆ u 1 functional closure ∇ · τ := ∇ · ( u ∗ u ∗ − u u ) 2 structural closure Z. Wang, USC POD-ROM of complex flows ICERM 2020 10 / 41

POD Filter ◮ ( u − u r , φ ) = 0 ( POD projection ) ⇐ ⇒ ( u − u , φ ) = 0 ( Filter ) ∂ t + ∇ · ( u u ) − Re − 1 ∆ u = 0 ◮ POD-ROM ∂ u ◮ NSE ∂ u ∂ t + ∇ · ( u u ) − Re − 1 ∆ u = 0 ∂ u ∂ t + ∇ · ( u u ) + ∇ · τ − Re − 1 ∆ u = 0 τ = u u − u u ◮ to close POD-ROM ∇ · τ := − ν ∗ ∆ u 1 functional closure ∇ · τ := ∇ · ( u ∗ u ∗ − u u ) 2 structural closure Z. Wang, USC POD-ROM of complex flows ICERM 2020 10 / 41

POD Filter ◮ ( u − u r , φ ) = 0 ( POD projection ) ⇐ ⇒ ( u − u , φ ) = 0 ( Filter ) ∂ t + ∇ · ( u u ) − Re − 1 ∆ u = 0 ◮ POD-ROM ∂ u ◮ NSE ∂ u ∂ t + ∇ · ( u u ) − Re − 1 ∆ u = 0 ∂ u ∂ t + ∇ · ( u u ) + ∇ · τ − Re − 1 ∆ u = 0 τ = u u − u u ◮ to close POD-ROM ∇ · τ := − ν ∗ ∆ u 1 functional closure ∇ · τ := ∇ · ( u ∗ u ∗ − u u ) 2 structural closure Z. Wang, USC POD-ROM of complex flows ICERM 2020 10 / 41

ML-POD ◮ mixing-length model � ∂ u r � �� � � ν ML + 2 D ( u r ) , ∇ φ k ∂ t , φ k + (( u r · ∇ ) u r , φ k ) + = 0 , Re k = 1 , . . . , r ◮ ν ML := α U ML L ML ◮ Aubry, Holmes, Lumley, Stone, 1988 Podvin, Lumley, 1998 . . . . . . Z. Wang, USC POD-ROM of complex flows ICERM 2020 11 / 41