Photos placed in horizontal position with even amount

Photos placed in horizontal position with even amount

Photos placed in horizontal position with even amount of white space between photos and header A minimal subspace rotation approach for extreme model reduction in fluid mechanics Irina Tezaur1, Maciej Balajewicz2 1 Extreme Scale Data Science & Analytics Department, Sandia National Laboratories 2 Aerospace Engineering Department, University of Illinois Urbana-Champaign Recent Developments MOR 2016 Paris, France November 7-10, 2016 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energys National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND NO. 2011-XXXXP

SAND2016-10461C Outline 1. Introduction Targeted application POD/Galerkin approach to MOR Extreme model reduction Mode truncation instability in MOR 2. Accounting for modal truncation Traditional linear eddy-viscosity approach New proposed approach via subspace rotation 3. Applications Low Reynolds (Re) number channel driven cavity Moderate Reynolds (Re) number channel driven cavity 4. Extension to Least-Squares Petrov-Galerkin (LSPG) ROMs 5. Summary & future work

Outline 1. Introduction Targeted application POD/Galerkin approach to MOR Extreme model reduction Mode truncation instability in MOR 2. Accounting for modal truncation Traditional linear eddy-viscosity approach New proposed approach via subspace rotation 3. Applications Low Reynolds (Re) number channel driven cavity Moderate Reynolds (Re) number channel driven cavity 4. Extension to Least-Squares Petrov-Galerkin (LSPG) ROMs 5. Summary & future work Targeted Application: Compressible Flow

We are interested in the compressible captive-carry problem. Targeted Application: Compressible Flow We are interested in the compressible captive-carry problem. Of primary interest are long-time predictive simulations: ROM run at same parameters as FOM but much longer in time. Targeted Application: Compressible Flow We are interested in the compressible captive-carry problem.

Of primary interest are long-time predictive simulations: ROM run at same parameters as FOM but much longer in time. QoIs: statistics of flow, e.g., pressure Power Spectral Densities (PSDs) [left]. Targeted Application: Compressible Flow We are interested in the compressible captive-carry problem. Of primary interest are long-time predictive simulations: ROM run at same parameters as FOM but much longer in time.

QoIs: statistics of flow, e.g., pressure Power Spectral Densities (PSDs) [left]. Secondary interest: ROMs robust w.r.t. parameter changes (e.g., Reynolds, Mach number) for enabling uncertainty quantification. POD/Galerkin Method to MOR Snapshot matrix: , , SVD: Truncation: ) Our focus has been primarily on POD/Galerkin ROMs # of dofs in full order model (FOM)

# of snapshots # of dofs in ROM (, ) Extreme Model Reduction Most realistic applications (e.g., high Re compressible cavity): basis that captures 99% snapshot energy is required to accurately reproduce snapshots. leads to >except for toy problems and/or low-fidelity models. Higher order modes are in general unreliable for prediction, so including them in the basis is unlikely to improve the predictive capabilities of a ROM. Figure (right) shows projection error for POD basis constructed using 800 snapshots for cavity problem. Dashed line = end of snapshot

collection period. We are looking for an approach that enables extreme model reduction: ROM basis size is or . 3D Compressible Navier-Stokes Equations [PDEs] We start with the 3D compressible Navier-Stokes equations in primitive specific volume form: (1) 3D Compressible Navier-Stokes Equations

We start with the 3D compressible Navier-Stokes equations in primitive specific volume form: (1) [PDEs] Spectral discretization Galerkin projection applied to (1) yields a system of coupled quadratic ODEs: [ROM] = + + [ (1) + (2) + + () ] where and for all

(2) ROM Instability Problem Stability can be a real problem for compressible flow ROMs! ROM Instability Problem Stability can be a real problem for compressible flow ROMs! A compressible fluid POD/Galerkin ROM might be stable for a given number of modes, but unstable for other choices of basis size (Bui-Tanh et al. 2007). ROM Instability Problem Stability can be a real problem for compressible flow ROMs! A compressible fluid POD/Galerkin ROM might be stable for a given number of modes, but unstable for other choices of basis size (Bui-Tanh et al. 2007). Instability can be due to: ROM Instability Problem

Stability can be a real problem for compressible flow ROMs! A compressible fluid POD/Galerkin ROM might be stable for a given number of modes, but unstable for other choices of basis size (Bui-Tanh et al. 2007). Instability can be due to: 1. Choice of inner product: Galerkin projection + L2 inner product is unstable. ROM Instability Problem Stability can be a real problem for compressible flow ROMs! A compressible fluid POD/Galerkin ROM might be stable for a given number of modes, but unstable for other choices of basis size (Bui-Tanh et al. 2007). Instability can be due to: 1. Choice of inner product: Galerkin projection + L2 inner product is unstable. Stable alternatives include: Energy-based inner products: Rowley et al., 2004 (isentropic); Barone et al., 2007 (linear); Serre et al., 2012 (linear);

Kalashnikova et al., 2014 (nonlinear). ROM Instability Problem Stability can be a real problem for compressible flow ROMs! A compressible fluid POD/Galerkin ROM might be stable for a given number of modes, but unstable for other choices of basis size (Bui-Tanh et al. 2007). Instability can be due to: 1. Choice of inner product: Galerkin projection + L2 inner product is unstable. Stable alternatives include: Energy-based inner products: Rowley et al., 2004 (isentropic); Barone et al., 2007 (linear); Serre et al., 2012 (linear); Kalashnikova et al., 2014 (nonlinear). GNAT method/Petrov-Galerkin projection: Carlberg et al., 2014 (nonlinear). ROM Instability Problem

Stability can be a real problem for compressible flow ROMs! A compressible fluid POD/Galerkin ROM might be stable for a given number of modes, but unstable for other choices of basis size (Bui-Tanh et al. 2007). Instability can be due to: 1. Choice of inner product: Galerkin projection + L2 inner product is unstable. Stable alternatives include: Energy-based inner products: Rowley et al., 2004 (isentropic); Barone et al., 2007 (linear); Serre et al., 2012 (linear); Kalashnikova et al., 2014 (nonlinear). GNAT method/Petrov-Galerkin projection: Carlberg et al., 2014 (nonlinear). 2. Basis truncation: destroys balance between energy production & nnndissipation. ROM Instability Problem Stability can be a real problem for compressible flow ROMs!

A compressible fluid POD/Galerkin ROM might be stable for a given number of modes, but unstable for other choices of basis size (Bui-Tanh et al. 2007). Instability can be due to: 1. Choice of inner product: Galerkin projection + L2 inner product is unstable. Stable alternatives include: Energy-based inner products: Rowley et al., 2004 (isentropic); Barone et al., 2007 (linear); Serre et al., 2012 (linear); Kalashnikova et al., 2014 (nonlinear). GNAT method/Petrov-Galerkin projection: Carlberg et al., 2014 (nonlinear). 2. Basis truncation: destroys balance between energy production & hhhdissipation. This talk focuses on remedying mode truncation instability problem for projection-based (POD/Galerkin) compressible flow ROMs. Mode Truncation Instability

Projection-based MOR necessitates truncation. Mode Truncation Instability Projection-based MOR necessitates truncation. POD is, by definition and design, biased towards the large, energy producing scales of the flow (i.e., modes with large POD eigenvalues). Mode Truncation Instability Projection-based MOR necessitates truncation. POD is, by definition and design, biased towards the large, energy producing scales of the flow (i.e., modes with large POD eigenvalues). Truncated/unresolved modes are negligible from a data compression point of view (i.e., small POD eigenvalues) but are crucial for the dynamical equations. Mode Truncation Instability Projection-based MOR necessitates truncation.

POD is, by definition and design, biased towards the large, energy producing scales of the flow (i.e., modes with large POD eigenvalues). Truncated/unresolved modes are negligible from a data compression point of view (i.e., small POD eigenvalues) but are crucial for the dynamical equations. For fluid flow applications, higher-order modes are associated with energy dissipation Mode Truncation Instability Projection-based MOR necessitates truncation. POD is, by definition and design, biased towards the large, energy producing scales of the flow (i.e., modes with large POD eigenvalues). Truncated/unresolved modes are negligible from a data compression point of view (i.e., small POD eigenvalues) but are crucial for the dynamical equations. For fluid flow applications, higher-order modes are associated with energy dissipation

low-dimensional ROMs (Galerkin and Petrov-Galerkin) can be inaccurate and unstable. Mode Truncation Instability Projection-based MOR necessitates truncation. POD is, by definition and design, biased towards the large, energy producing scales of the flow (i.e., modes with large POD eigenvalues). Truncated/unresolved modes are negligible from a data compression point of view (i.e., small POD eigenvalues) but are crucial for the dynamical equations. For fluid flow applications, higher-order modes are associated with energy dissipation low-dimensional ROMs (Galerkin and Petrov-Galerkin) can be inaccurate and unstable. For a low-dimensional ROM to be stable and accurate, the truncated/unresolved subspace must be accounted for.

Mode Truncation Instability Projection-based MOR necessitates truncation. POD is, by definition and design, biased towards the large, energy producing scales of the flow (i.e., modes with large POD eigenvalues). Truncated/unresolved modes are negligible from a data compression point of view (i.e., small POD eigenvalues) but are crucial for the dynamical equations. For fluid flow applications, higher-order modes are associated with energy dissipation low-dimensional ROMs (Galerkin and Petrov-Galerkin) can be inaccurate and unstable. For a low-dimensional ROM to be stable and accurate, the truncated/unresolved subspace must be accounted for. Turbulence Modeling (traditional approach) Subspace Rotation

(our approach) Outline 1. Introduction Targeted application POD/Galerkin approach to MOR Extreme model reduction Mode truncation instability in MOR 2. Accounting for modal truncation Traditional linear eddy-viscosity approach New proposed approach via subspace rotation 3. Applications Low Reynolds (Re) number channel driven cavity Moderate Reynolds (Re) number channel driven cavity 4. Extension to Least-Squares Petrov-Galerkin (LSPG) ROMs 5. Summary & future work

Traditional Linear Eddy-Viscosity Approach Dissipative dynamics of truncated higher-order modes are modeled using an additional linear term: = + + [ ( 1) + ( 2 ) + + ( ) ] Traditional Linear Eddy-Viscosity Approach Dissipative dynamics of truncated higher-order modes are modeled using an additional linear term:

= + ( + ) + [ ( 1 ) + ( 2) + + ( ) ] Traditional Linear Eddy-Viscosity Approach Dissipative dynamics of truncated higher-order modes are modeled using an additional linear term: = + ( + ) + [ ( 1 ) + ( 2) + + ( ) ] is designed to decrease magnitude of positive eigenvalues and increase magnitude of negative eigenvalues of (for stability).

Traditional Linear Eddy-Viscosity Approach Dissipative dynamics of truncated higher-order modes are modeled using an additional linear term: = + ( + ) + [ ( 1 ) + ( 2) + + ( ) ] is designed to decrease magnitude of positive eigenvalues and increase magnitude of negative eigenvalues of (for stability). Disadvantages of this approach:

Traditional Linear Eddy-Viscosity Approach Dissipative dynamics of truncated higher-order modes are modeled using an additional linear term: = + ( + ) + [ ( 1 ) + ( 2) + + ( ) ] is designed to decrease magnitude of positive eigenvalues and increase magnitude of negative eigenvalues of (for stability).

Disadvantages of this approach: 1. Additional term destroys consistency between ROM and NavierStokes equations. Traditional Linear Eddy-Viscosity Approach Dissipative dynamics of truncated higher-order modes are modeled using an additional linear term: = + ( + ) + [ ( 1 ) + ( 2) + + ( ) ] is designed to decrease magnitude of positive eigenvalues and increase magnitude of negative eigenvalues of (for stability).

Disadvantages of this approach: 1. Additional term destroys consistency between ROM and NavierStokes equations. 2. Calibration is necessary to derive optimal and optimal value is flow dependent. Traditional Linear Eddy-Viscosity Approach Dissipative dynamics of truncated higher-order modes are modeled using an additional linear term: = + ( + ) + [ ( 1 ) + ( 2) + + ( ) ]

is designed to decrease magnitude of positive eigenvalues and increase magnitude of negative eigenvalues of (for stability). Disadvantages of this approach: 1. Additional term destroys consistency between ROM and NavierStokes equations. 2. Calibration is necessary to derive optimal and optimal value is flow dependent. 3. Inherently a linear model cannot be expected to perform well for all classes of problems (e.g., nonlinear). Proposed new approach: basis rotation Instead of modeling truncation via additional linear term, model the truncation a priori by rotating the projection subspace into a more dissipative regime

Proposed new approach: basis rotation Instead of modeling truncation via additional linear term, model the truncation a priori by rotating the projection subspace into a more dissipative regime Illustrative example Standard approach: retain only the most energetic POD modes, i.e., , Proposed approach: add some higher order basis modes to increase dissipation, i.e., Proposed new approach: basis rotation Instead of modeling truncation via additional linear term, model the truncation a priori by rotating the projection subspace into a more dissipative regime

Illustrative example Standard approach: retain only the most energetic POD modes, i.e., , Proposed approach: add some higher order basis modes to increase dissipation, i.e., More generally: approximate the solution using a linear superposition of (with ) most energetic modes: , where is an orthonormal () rotation matrix. (3) Goals of proposed new approach Find such that: 1. New modes remain good approximations of the flow. 2. New modes produce stable and accurate ROMs.

Goals of proposed new approach Find such that: 1. New modes remain good approximations of the flow. 2. New modes produce stable and accurate ROMs. We formulate and solve a constrained optimization problem for : where is the Stiefel manifold. Goals of proposed new approach Find such that: 1. New modes remain good approximations of the flow. 2. New modes produce stable and accurate ROMs. We formulate and solve a constrained optimization problem for : where is the Stiefel manifold. Once is found, the result is a system of the form:

= + + [ ( 1) + ( 2 ) + + ( ) ] with: , (4) Objective function (5) We have considered two objectives in (5): Objective function (5)

We have considered two objectives in (5): Minimize subspace rotation ( )= (+ ) , = tr ( (+ ) ) (6) Objective function (5) We have considered two objectives in (5): Minimize subspace rotation

( )= (+ ) , = tr ( (+ ) ) (6) Maximize resolved turbulent kinetic energy (TKE) ( )=| | (7) Objective function (5) We have considered two objectives in (5):

Minimize subspace rotation ( )= (+ ) , = tr ( (+ ) ) (6) Maximize resolved turbulent kinetic energy (TKE) ( )=| | (7) TKE objective (7) comes from earlier work (Balajewicz et al., 2013) involving

stabilization of incompressible flow ROMs POD modes associated with low KE are important dynamically even though they contribute little to overall energy of the fluid flow. * In (7), denotes the square of second moments of ROM modal coefficients (Balajewicz et al., 2013). Objective function (5) We have considered two objectives in (5): Minimize subspace rotation ( )= (+ ) , = tr ( (+ ) )

(6) Maximize resolved turbulent kinetic energy (TKE) ( )=| | Numerical experiments reveal objective (6) produces better results than objective (7) for compressible flow. (7) Constraint (5) Constraint (5) We use the traditional linear eddy-viscosity closure model ansatz for the

constraint in (5): ( , )=tr ( ) (8) Constraint (5) We use the traditional linear eddy-viscosity closure model ansatz for the constraint in (5): ( , )=tr ( ) Specifically, constraint (8) involves overall balance between linear energy production and dissipation.

proxy for the balance between linear energy production and energy dissipation. (8) Constraint (5) We use the traditional linear eddy-viscosity closure model ansatz for the constraint in (5): ( , )=tr ( ) (8)

Specifically, constraint (8) involves overall balance between linear energy production and dissipation. proxy for the balance between linear energy production and energy dissipation. Constraint comes from property that averaged total power ( energy transfer) has to vanish. Optimization problem summary Minimal subspace rotation: trace minimization on Stiefel manifold (9)

: proxy for the balance between linear energy production and energy dissipation (calculated iteratively using modal energy). is the Stiefel manifold. Equation (9) is solved efficiently offline using the method of Lagrange multipliers (Manopt MATLAB toolbox). See (Balajewicz, Tezaur, Dowell, 2016) and Appendix slide for Algorithm. Remarks on proposed approach

Proposed approach may be interpreted as an a priori consistent formulation of the eddy-viscosity turbulence modeling approach. Remarks on proposed approach Proposed approach may be interpreted as an a priori consistent formulation of the eddy-viscosity turbulence modeling approach. Advantages of proposed approach: Remarks on proposed approach Proposed approach may be interpreted as an a priori consistent formulation of the eddy-viscosity turbulence modeling approach. Advantages of proposed approach: 1.

Retains consistency between ROM and Navier-Stokes equations no additional turbulence terms required. Remarks on proposed approach Proposed approach may be interpreted as an a priori consistent formulation of the eddy-viscosity turbulence modeling approach. Advantages of proposed approach: 1. 2. Retains consistency between ROM and Navier-Stokes equations no additional turbulence terms required. Inherently a nonlinear model should be expected to outperform linear models.

Remarks on proposed approach Proposed approach may be interpreted as an a priori consistent formulation of the eddy-viscosity turbulence modeling approach. Advantages of proposed approach: 1. 2. 3. Retains consistency between ROM and Navier-Stokes equations no additional turbulence terms required. Inherently a nonlinear model should be expected to outperform linear models. Works with any basis and Petrov-Galerkin projection.

Remarks on proposed approach Proposed approach may be interpreted as an a priori consistent formulation of the eddy-viscosity turbulence modeling approach. Advantages of proposed approach: 1. 2. 3. Retains consistency between ROM and Navier-Stokes equations no additional turbulence terms required. Inherently a nonlinear model should be expected to outperform linear models. Works with any basis and Petrov-Galerkin projection.

Disadvantages of proposed approach: Remarks on proposed approach Proposed approach may be interpreted as an a priori consistent formulation of the eddy-viscosity turbulence modeling approach. Advantages of proposed approach: 1. 2. 3. Retains consistency between ROM and Navier-Stokes equations no additional turbulence terms required.

Inherently a nonlinear model should be expected to outperform linear models. Works with any basis and Petrov-Galerkin projection. Disadvantages of proposed approach: 1. Off-line calibration of free parameter is required. Remarks on proposed approach Proposed approach may be interpreted as an a priori consistent formulation of the eddy-viscosity turbulence modeling approach. Advantages of proposed approach: 1. 2.

3. Retains consistency between ROM and Navier-Stokes equations no additional turbulence terms required. Inherently a nonlinear model should be expected to outperform linear models. Works with any basis and Petrov-Galerkin projection. Disadvantages of proposed approach: 1. 2. Off-line calibration of free parameter is required. Stability cannot be proven like for incompressible case.

Outline 1. Introduction Targeted application POD/Galerkin approach to MOR Extreme model reduction Mode truncation instability in MOR 2. Accounting for modal truncation Traditional linear eddy-viscosity approach New proposed approach via subspace rotation 3. Applications Low Reynolds (Re) number channel driven cavity Moderate Reynolds (Re) number channel driven cavity 4. Extension to Least-Squares Petrov-Galerkin (LSPG) ROMs 5. Summary & future work Low Re Channel Driven Cavity Flow over square cavity at Mach 0.6, Re = 1453.9, Pr = 0.72

ROM (91% snapshot energy). Figure 1: Domain and mesh for viscous channel driven cavity problem. Low Re Channel Driven Cavity Minimizing subspace rotation: ( )= (+ ) , = tr ( (+ ) ) -- standard ROM (n=4) stabilized ROM (n=p=4) FOM Figure 2: (a) evolution of modal energy, (b) phase plot of first and second temporal basis and , (c) illustration of stabilizing rotation showing that rotation is small:

Low Re Channel Driven Cavity Minimizing subspace rotation: ( )= (+ ) , = tr ( (+ ) ) -- standard ROM (n=4) stabilized ROM (n=p=4) FOM Figure 3: Pressure power spectral density (PSD) at location ; stabilized ROM minimizes subspace rotation. Low Re Channel Driven Cavity Maximizing resolved TKE:

( )=| | -- standard ROM (n=4) stabilized ROM (n=p=4) FOM Figure 4: Pressure power spectral density (PSD) at location ; stabilized ROM maximizes resolved TKE. Low Re Channel Driven Cavity Minimizing subspace rotation: ( )= (+ ) , = tr ( (+ ) )

FOM Standard ROM () Stabilized ROM () Figure 5: Channel driven cavity Re 1500 contours of -velocity at time of final snapshot. Moderate Re Channel Driven Cavity Flow over square cavity at Mach 0.6, Re = 5452.1, Pr = 0.72 ROM (71.8% snapshot energy). Figure 6: Domain and mesh for viscous channel driven cavity problem. Moderate Re Channel Driven Cavity

Minimizing subspace rotation: ( )= (+ ) , = tr ( (+ ) ) -- standard ROM (n=20) stabilized ROM (n=p=20) FOM Figure 7: (a) evolution of modal energy, (b) illustration of stabilizing rotation showing that rotation is small: Moderate Re Channel Driven Cavity Minimizing subspace rotation: ( )= (+ ) , = tr ( (+ ) )

stabilized ROM (n=p=20) FOM Figure 8: Pressure cross PSD of of and where Power and phase lag at fundamental frequency, and first two super harmonics are predicted accurately using the fine-tuned ROM ( stabilized ROM, FOM) Moderate Re Channel Driven Cavity Minimizing subspace rotation: ( )= (+ ) , = tr ( (+ ) ) FOM

Standard ROM () Stabilized ROM (20) Figure 9: Channel driven cavity Re 5500 contours of -velocity at time of final snapshot. CPU times (CPU-hours) for offline and online computations* Moderate Re Cavity FOM FOM ## of of DOF DOF Time-integration Time-integration of

of FOM FOM 288,250 288,250 72 72 hrs hrs 243,750 243,750 179 179 hrs hrs Basis construction (size ROM)

0.88 0.88 hrs hrs 5.44 5.44 hrs hrs 3.44 3.44 hrs hrs 14.8 14.8 hrs hrs 14 14 sec sec

4 4 170 170 sec sec 20 20 0.16 0.16 sec sec 1.6e6 1.6e6 0.83 0.83 sec

sec 7.8e5 7.8e5 offline Low Re Cavity Galerkin projection (size ROM) online Procedure Time-integration Time-integration of of ROM

ROM Online Online computational computational speed-up speed-up Stabilization Stabilization ROM ROM ## of of DOF DOF * For minimizing subspace rotation. CPU times (CPU-hours) for offline and

online computations* Moderate Re Cavity FOM FOM ## of of DOF DOF Time-integration Time-integration of of FOM FOM 288,250 288,250 72 72 hrs hrs

243,750 243,750 179 179 hrs hrs Basis construction (size ROM) 0.88 0.88 hrs hrs 5.44 5.44 hrs hrs 3.44

3.44 hrs hrs 14.8 14.8 hrs hrs 14 14 sec sec 4 4 170 170 sec sec 20 20

0.16 0.16 sec sec 1.6e6 1.6e6 0.83 0.83 sec sec 7.8e5 7.8e5 offline Low Re Cavity

Galerkin projection (size ROM) online Procedure Time-integration Time-integration of of ROM ROM Online Online computational computational speed-up speed-up Stabilization Stabilization

ROM ROM ## of of DOF DOF Stabilization is fast ((sec) or (min)). * For minimizing subspace rotation. CPU times (CPU-hours) for offline and online computations* Moderate Re Cavity FOM FOM ## of of DOF

DOF Time-integration Time-integration of of FOM FOM 288,250 288,250 72 72 hrs hrs 243,750 243,750 179 179 hrs hrs

Basis construction (size ROM) 0.88 0.88 hrs hrs 5.44 5.44 hrs hrs 3.44 3.44 hrs hrs 14.8 14.8 hrs hrs

14 14 sec sec 4 4 170 170 sec sec 20 20 0.16 0.16 sec sec 1.6e6 1.6e6

0.83 0.83 sec sec 7.8e5 7.8e5 offline Low Re Cavity Galerkin projection (size ROM) online Procedure

Time-integration Time-integration of of ROM ROM Online Online computational computational speed-up speed-up Stabilization Stabilization ROM ROM ## of of DOF DOF Stabilization is fast ((sec) or (min)).

Significant online computational speed-up! * For minimizing subspace rotation. Outline 1. Introduction Targeted application POD/Galerkin approach to MOR Extreme model reduction Mode truncation instability in MOR 2. Accounting for modal truncation Traditional linear eddy-viscosity approach New proposed approach via subspace rotation 3. Applications Low Reynolds (Re) number channel driven cavity Moderate Reynolds (Re) number channel driven cavity

4. Extension to Least-Squares Petrov-Galerkin (LSPG) ROMs 5. Summary & future work Extensions to Least-Squares PetrovGalerkin (LSPG) ROMs Stabilization/enhancement of LSPG ROMs is parallel effort to implementation of LSPG minimal-residual ROMs (GNAT method of Carlberg et al.) in our in-house flow solver, SPARC (see poster by Jeff Fike) Extensions to Least-Squares PetrovGalerkin (LSPG) ROMs Stabilization/enhancement of LSPG ROMs is parallel effort to implementation of LSPG minimal-residual ROMs (GNAT method of Carlberg et al.) in our in-house flow solver, SPARC (see poster by Jeff Fike) FOM is a nonlinear system of the form = (Navier-Stokes discretized in space and in time). Extensions to Least-Squares PetrovGalerkin (LSPG) ROMs Stabilization/enhancement of LSPG ROMs is parallel effort to

implementation of LSPG minimal-residual ROMs (GNAT method of Carlberg et al.) in our in-house flow solver, SPARC (see poster by Jeff Fike) FOM is a nonlinear system of the form = (Navier-Stokes discretized in space and in time). Solving ROM amounts to solving non-linear least-squares problem: )||22 Extensions to Least-Squares PetrovGalerkin (LSPG) ROMs Stabilization/enhancement of LSPG ROMs is parallel effort to implementation of LSPG minimal-residual ROMs (GNAT method of Carlberg et al.) in our in-house flow solver, SPARC (see poster by Jeff Fike) FOM is a nonlinear system of the form = (Navier-Stokes discretized in space and in time). Solving ROM amounts to solving non-linear least-squares problem: )||22 Equivalent to Petrov-Galerkin projection with test basis where is the Jacobian

of . Extensions to Least-Squares PetrovGalerkin (LSPG) ROMs Stabilization/enhancement of LSPG ROMs is parallel effort to implementation of LSPG minimal-residual ROMs (GNAT method of Carlberg et al.) in our in-house flow solver, SPARC (see poster by Jeff Fike) FOM is a nonlinear system of the form = (Navier-Stokes discretized in space and in time). Solving ROM amounts to solving non-linear least-squares problem: )||22 Equivalent to Petrov-Galerkin projection with test basis where is the Jacobian of . POD/LSPG ROMs are more stable than POD/Galerkin ROMs. Extensions to Least-Squares PetrovGalerkin (LSPG) ROMs Stabilization/enhancement of LSPG ROMs is parallel effort to

implementation of LSPG minimal-residual ROMs (GNAT method of Carlberg et al.) in our in-house flow solver, SPARC (see poster by Jeff Fike) FOM is a nonlinear system of the form = (Navier-Stokes discretized in space and in time). Solving ROM amounts to solving non-linear least-squares problem: )||22 Equivalent to Petrov-Galerkin projection with test basis where is the Jacobian of . POD/LSPG ROMs are more stable than POD/Galerkin ROMs. Nevertheless, low-dimensional LSPG ROMs can benefit from basis stabilization. Stabilization of Inviscid Pulse in Uniform Flow Low Order LSPG ROM 1. 2.

3. 4. Preliminary Workflow Run LSPG ROM in SPARC output POD basis. Use POD/Galerkin ROM code Spirit to produce , , and matrices in (2). Stabilize POD basis using stabilization approach described in this talk. Run LSPG ROM in SPARC with stabilized basis. Stabilization of Inviscid Pulse in Uniform Flow Low Order LSPG ROM 1. 2. 3. 4. Preliminary Workflow

Run LSPG ROM in SPARC output POD basis. Use POD/Galerkin ROM code Spirit to produce , , and matrices in (2). Stabilize POD basis using stabilization approach described in this talk. Run LSPG ROM in SPARC with stabilized basis. Figure (left) shows generalized coordinates for mode 2 compared to FOM projection. Our approach effectively stabilizes LSPG ROM. Stabilization of Inviscid Pulse in Uniform Flow Low Order LSPG ROM 1. 2. 3. 4. Preliminary Workflow

Run LSPG ROM in SPARC output POD basis. Use POD/Galerkin ROM code Spirit to produce , , and matrices in (2). Stabilize POD basis using stabilization approach described in this talk. Run LSPG ROM in SPARC with stabilized basis. Figure (left) shows generalized coordinates for mode 2 compared to FOM projection. Our approach effectively stabilizes LSPG ROM. Preliminary approach needs improvement, as there are inconsistencies between SPARC and Spirit codes. We are currently working on extending our stabilization/enhancement approach to ROMs with generic nonlinearities. Outline 1. Introduction

Targeted application POD/Galerkin approach to MOR Extreme model reduction Mode truncation instability in MOR 2. Accounting for modal truncation Traditional linear eddy-viscosity approach New proposed approach via subspace rotation 3. Applications Low Reynolds (Re) number channel driven cavity Moderate Reynolds (Re) number channel driven cavity 4. Extension to Least-Squares Petrov-Galerkin (LSPG) ROMs 5. Summary & future work Summary We have developed a non-intrusive approach for stabilizing and finetuning projection-based ROMs for compressible flows.

The standard POD modes are rotated into a more dissipative regime to account for the dynamics in the higher order modes truncated by the standard POD method. The new approach is consistent and does not require the addition of empirical turbulence model terms unlike traditional approaches. Mathematically, the approach is formulated as a quadratic matrix program on the Stiefel manifold.

The constrained minimization problem is solved offline and small enough to be solved in MATLAB. The method is demonstrated on several compressible flow problems and shown to deliver stable and accurate ROMs. Future work Future work Application to higher Reynolds number problems.

Future work Application to higher Reynolds number problems. Extension of the proposed approach to problems with generic nonlinearities, where the ROM involves some form of hyper-reduction (e.g., DEIM, gappy POD). Future work Application to higher Reynolds number problems.

Extension of the proposed approach to problems with generic nonlinearities, where the ROM involves some form of hyper-reduction (e.g., DEIM, gappy POD). Extension of the method to minimal-residual-based nonlinear ROMs. Future work Application to higher Reynolds number problems. Extension of the proposed approach to problems with generic nonlinearities, where the ROM involves some form of hyper-reduction (e.g., DEIM, gappy POD).

Extension of the method to minimal-residual-based nonlinear ROMs. Extension of the method to predictive applications, e.g., problems with varying Reynolds number and/or Mach number. Future work Application to higher Reynolds number problems. Extension of the proposed approach to problems with generic nonlinearities,

where the ROM involves some form of hyper-reduction (e.g., DEIM, gappy POD). Extension of the method to minimal-residual-based nonlinear ROMs. Extension of the method to predictive applications, e.g., problems with varying Reynolds number and/or Mach number. Selecting different goal-oriented objectives and constraints in our optimization problem: e.g.,

Maximize parametric robustness: . ODE constraints: References [1] M. Balajewicz, E. Dowell. Stabilization of projection-based reduced order models of the Navier-Stokes equation. Nonlinear Dynamics 70(2),1619-1632, 2012. [2] M. Balajewicz, E. Dowell, B. Noack. Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier-Stokes equation. Journal of Fluid Mechanics 729, 285-308, 2013. [3] M. Balajewicz, I. Tezaur, E. Dowell. Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations. J. Comput. Phys., 321, 224241, 2016. [4] M. Barone, I. Kalashnikova, D. Segalman, H. Thornquist. Stable Galerkin reduced order models for linearized compressible flow. J. Computat. Phys. 228(6), 1932-1946, 2009.

[5] K. Carlberg, C. Farhat, J. Cortial, D. Amsallem. The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Computat. Phys. 242, 623-647, 2013. [6] I. Kalashnikova, S. Arunajatesan, M. Barone, B. van Bloemen Waanders, J. Fike. Reduced order modeling for prediction and control of large-scale systems. Sandia Tech. Report, 2014. [7] C. Rowley, T. Colonius, R. Murray. Model reduction for compressible flows using POD and Galerkin projection. Physica D: Nonlinear Phenomena. 189(1) 115-129, 2004. [8] G. Serre, P. Lafon, X. Gloerfelt, C. Bailly. Reliable reduced-order models for timedependentlinearized euler equations. J. Computat. Phys. 231(15) 5176-5194, 2012. [9] N. Aubry, P. Holmes, J. Lumley, E. Stone The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192(115) 115-173, 1988. [10] J. Osth, B. Noack, R. Krajnovic, D. Barros, J. Boree. On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high Reynolds number flow over an Ahmed body. J. Fluid Mech. 747 518-544, 2004. [11] T. Bui-Thanh, K. Willcox, O. Ghattas, and B. van Bloemen Waanders. Goal-oriented, model constrained optimization for reduction of large-scale systems. J. Comput. Phys., 224, 880896, 2007. Appendix: Accounting for modal truncation

Stabilization algorithm: returns stabilizing rotation matrix . Targeted Application: Compressible Flow We are interested in the compressible captive-carry problem. Majority of fluid MOR approaches in the literature are for incompressible flow. Targeted Application: Compressible Flow We are interested in the compressible captive-carry problem. Majority of fluid MOR approaches in the literature are for incompressible flow. Desired numerical properties of ROMs: Consistency (w.r.t. the continuous PDEs).

Stability: if full order model (FOM) is stable, ROM should be stable. Convergence: requires consistency and stability. Accuracy (w.r.t. FOM). Efficiency. Robustness (w.r.t. time or parameter changes). Targeted Application: Compressible Flow We are interested in the compressible captive-carry problem. Majority of fluid MOR approaches in the literature are for incompressible flow. Desired numerical properties of ROMs: Consistency (w.r.t. the continuous PDEs). Stability: if full order model (FOM) is stable, ROM should be stable. Convergence: requires consistency and stability. Accuracy (w.r.t. FOM).

Efficiency. Robustness (w.r.t. time or parameter changes). Stability can be a real problem for compressible flow ROMs!

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