Lagrangian basis method for dimensionality reduction of convection dominated nonlinear flows
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Lagrangian basis method for dimensionalityreduction of convection dominated nonlinearflows
Rambod Mojgani , Maciej Balajewicz † Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana,IL 61820 USA(Received xx; revised xx; accepted xx)
Foundations of a new projection-based model reduction approach for convection domi-nated nonlinear fluid flows are summarized. In this method the evolution of the flow isapproximated in the Lagrangian frame of reference. Global basis functions are used toapproximate both the state and the position of the Lagrangian computational domain.It is demonstrated that in this framework, certain wave-like solutions exhibit low-rankstructure and thus, can be efficiently compressed using relatively few global basis. Theproposed approach is successfully demonstrated for the reduction of several simple butrepresentative problems.
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1. Introduction
Numerical simulation of nonlinear fluid flows often requires prohibitively large com-putational resources. High–fidelity simulations of high-Reynolds numbers flows, high–speed compressible flows, and combustion often require very fine spatial and temporaldiscretizations to accurately resolve the multi–scale dynamics. There are significantscientific and engineering benefits to developing model reduction techniques that arecapable of delivering physics–based, low–dimensional models.Most existing model order reduction (MOR) approaches are based on projection (Ben-ner et al. et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] J a n R. Mojgani and M. Balajewicz to higher dimensions not straight forward. Moreover, there are many applications whereremaining in the global basis ansatz is desired; for example, global basis generatedvia proper orthogonal decomposition (POD) correspond to coherent structures in aturbulent flow (Lumley 1970). Other attempts have focused on exploiting symmetry, selfsimilarity, and coordinate transformations (Rowley & Marsden 2000; Rowley et al. et al. §
2, the traditional Eulerian MOR approachrecapitulated. In §
3, the new proposed Lagrangian MOR is laid out and in §
4, severalsimple yet representative problems are considered. Finally, in §
5, the main results aresummarized and future prospects are laid out.
2. Traditional Eulerian projection-based model order reduction
We recapitulate the traditional Eulerian projection-based MOR approach as a startingpoint for our innovation in §
3. Consider the following scalar, one-dimensional convection-diffusion equation ∂w ( x, t ) ∂t + f ( x, t, w ) ∂w ( x, t ) ∂x = f ( x, t, w ) ∂ w ( x, t ) ∂x , (2.1)in the domain ( x, t ) ∈ [ x a , x b ] × [0 , T ], equipped with initial conditions w ( x,
0) = w ( x ),and appropriate boundary conditions at x a , and x b . It is assumed throughout thereminder of this paper that (2.1) is discretized uniformly in space x = ( x , . . . , x N ) T usingstandard techniques such as finite-volume or finite-elements. For the sake of simplicity,and without any loss of generality, time discretization is performed using the first-orderimplicit Euler scheme. Hence, if t = 0 < t < · · · < t N t = T denotes a discretization ofthe time interval [0 , T ] and w ( x, t n ) ≈ w n = ( w n , . . . , w nN ) T ∈ R N , for n ∈ { , . . . , N t } ,the discrete counterpart of (2.1) at time-step n is R ( w n ) = w n − w n − + ∆t f n ( w n ) (cid:12) ( D w n ) − ∆t f n ( w n ) (cid:12) ( D w n ) = , (2.2)where (cid:12) denotes the Hadamard product, D ∈ R N × N , and D ∈ R N × N are the discreteapproximations of the first and second spatial derivatives, respectively.In traditional projection-based MOR, the solution is approximated by a global trialsubspace w n ≈ (cid:101) w n = w + U a n , (2.3)where the columns of U ∈ R N × k contain the basis for this subspace, and a n ∈ R k denotesthe generalized coordinates of the vectors in these basis. Substituting (2.3) into (2.2) andprojecting onto test basis Φ ∈ R N × k , yields the square system Φ T R ( w + U a n ) = , (2.4)where Φ = U in the case of a Galerkin projection. imensionality reduction of convection dominated nonlinear flows et al. et al. et al. et al. et al. et al. et al. et al. et al. rank( (cid:101) X )= k (cid:107) X − (cid:102) X (cid:107) F , (2.5)where the snapshot matrix X ∈ R N × K contains the solution of the HFM given by(2.2). In other words, [ X ] : ,i = w i for i = 1 , . . . , K . Here, Matlab row-column subscriptnotation is used. For example, [ A ] a : b, : identifies a matrix formed by extracting rows a to b , and all columns from the matrix A . The main focus of the proposed work is ondynamic problems and on snapshots associated with different time instances for oneset of parameters. However, in general, the snapshot matrix can contain solutions forany combination of parameters, that is, for some specific time t , some specific set offlow parameters and/or some boundary or initial conditions underlying the governingequations.The rank constraint is taken care of by representing the unknown matrix as (cid:102) X = U V where U ∈ R N × k and V ∈ R k × K , so the problem becomesmin U , V (cid:107) X − U V (cid:107) F . (2.6)The solution of the above low-rank approximation problem is given by the Eckart-Young-Mirsky theorem (Eckart & Young 1936; Mirsky 1960) via the singular valuedecomposition (SVD) of X . Specifically, (cid:102) X = U V , where U = [ U ∗ ] : , k and V =[ ΣV ∗ T ] k, : where X = U ∗ ΣV ∗ T .
3. Lagrangian projection-based model order reduction
In this section, the proposed new Lagrangian dimensionality reduction approach is laidout. Technical details are outlined in § § § § Lagrangian formulation of high-fidelity model
For the purpose of the proposed dimensionality reduction approach, the governingequations (2.1) are formulated in the Lagrangian frame of reference dxdt = f ( x, t, w ) , (3.1 a ) ∂w∂t = f ( x, t, w ) ∂ w∂x . (3.1 b )The discrete counterpart of (3.1) at time-step n is R x ( x n ) = x n − x n − − ∆t f n ( w n ) = , (3.2 a ) R w ( w n ) = w n − w n − − ∆t f n ( w n ) (cid:12) ( D n w n ) = , (3.2 b ) R. Mojgani and M. Balajewicz where x n = ( x n , . . . , x nN ) T denotes the locations of the Lagrangian computational gridat time level n , and D n denotes the discrete approximation of the second derivative onthe Lagrangian grid at time level n .3.2. Nonlinear model reduction
In the proposed new dimensionality reduction approach, the Lagrangian solution isapproximated by a global trial subspace x n ≈ (cid:101) x n = x + U x a nx , (3.3 a ) w n ≈ (cid:101) w n = w + U w a nw , (3.3 b )where the columns of U x ∈ R N × k and U w ∈ R N × k contain the basis for the correspondingsubspace, and a nx ∈ R k and a nw ∈ R k denote the generalized coordinates of the vectors inthese basis. Substituting (3.3) into (3.2) and projecting onto test basis Φ x ∈ R N × k and Φ w ∈ R N × k , yields the square system Φ Tx R x ( x + U x a nx ) = , (3.4 a ) Φ Tw R w ( w + U w a nw ) = , (3.4 b )where Φ x = U x and Φ w = U w in the case of a Galerkin projection.3.3. Construction of optimal Lagrangian global basis functions
For cases where the HFM is formulated in the Lagrangian frame of reference, thatis, when the governing equations are in the form of (3.2), construction of Lagrangianbasis follows a procedure very similar to traditional POD. Specifically, we solve thelow-rank approximation problem given by (2.6), for a snapshot matrix X ∈ R N × K containing solution snapshots computed by (3.2). In other words, [ X ] : ,i = (cid:20) x i w i (cid:21) for i = 1 , . . . , K . Therefore, the optimal Lagrangian basis corresponds to U x = [ U ] N, k and U w = [ U ] N +1:2 N, k , where U are the left singular vectors of the snapshot matrix X .For cases where the HFM is formulated in the Eulerian frame of reference, that is,when the governing equations are in the form of (2.1), Lagrangian basis cannot beconstructed by solving the standard low-rank approximation problem because EulerianHFMs typically do not provide the grid deformation x i . Thus, it is not possible to formthe snapshot matrix [ X ] : ,i = (cid:20) x i w i (cid:21) . For these cases, it is proposed here to constructLagrangian basis by solving a modified low-rank approximation problemmin U x , V x , U w , V w (cid:107) X − P xU x V x ( U w V w ) (cid:107) F , (3.5)where P xU x V x is the interpolation from the Lagrangian grid x i = U x [ V x ] : ,i to the Euleriangrid x , and the snapshot matrix X ∈ R N × K contains the Eulerian snapshots on thestationary Eulerian computational grid, x . Unlike problem (2.6), problem (3.5) does nothave a closed form solution. Consequently, it must be solved using an iterative method.In this work, (2.6) is solving in Matlab using the lsqnonlin unconstrained optimizationalgorithm. Forward finite differences are used to approximate the gradients and the linear interp1 algorithm is used for the interpolation, P xU x V x .3.4. Lagrangian grid entanglement
In the proposed Lagrangian MOR approach, the evolution of the Lagrangian spatialgrid is approximated in a low-dimensional subspace, x i ≈ U x a ix . Unfortunately, this imensionality reduction of convection dominated nonlinear flows b ), the state basis U w are interpolated from the Lagrangian to thestationary Eulerian grid at every time level n and the projection is performed in theEulerian frame. Therefore, (3.4 b ), is replaced with the following (cid:98) Φ Tw (cid:98) R w ( (cid:98) w + (cid:98) U w a nw ) = , (3.6)where (cid:98) Φ w = P xΦ x a nx ( U w ), (cid:98) U w = P xU x a nx ( U w ), and (cid:98) w = P xU x a nx ( w ), are the interpolatedbasis and initial conditions and (cid:98) R w is the diffusion step in the Eulerian frame, definedas (cid:98) R ( w n ) = w n − w n − − ∆t f n ( w n ) (cid:12) ( D w n ) = . (3.7)
4. Applications
In this section, the proposed approach is applied to several canonical one-dimensionalproblems. In § § Convection-diffusion equation
The proposed approach is first applied to the reduction of the scalar linear convectionequation and a high P´eclet number convection-diffusion equation. Specifically, we consider(2.1) with f ( x, t, w ) = 1, f ( x, t, w ) = 1 / Pe , w ( x,
0) = 0 . e − ( x − . / . , w (0 , t ) = 0,for ( x, t ) ∈ [0 , . × [0 , Pe = ∞ and Pe = 10 .Two HFMs are constructed for this case; one in the Eulerian frame, as in (2.2), andone in the fully Lagrangian frame, as in (3.2). For both models, a second-order centralfinite difference discretization is used. N = 2000 grid points are used to discretizedthe domain 0 (cid:54) x (cid:54) .
5. A total of K = 2000 Eulerian and Lagrangian snapshots arecollected. Eulerian and Lagrangian basis are constructed by solving (2.6). Eulerian ROMsare solved in the form of (2.4) and Lagrangian ROMs are solved in the fully Lagrangianframe, as in Eq.(3.4 a ) and Eq.(3.4 b ). Galerkin projection is used in all cases so Φ = U and Φ x = U x , Φ w = U w .ROM solutions for the convection equation and the high P´eclet number convection-diffusion equation are illustrated in Fig. 1 and Fig. 2, respectively. In both cases,traditional Eulerian ROMs and the new, Lagrangian ROMs, are illustrated in in the(a) and (b) subfigures. Solutions are plotted for t = 0 , / , / , k = 2, and k = 5 ROMs, respectively.Convergence of Eulerian and Lagrangian ROMs of the high P´eclet number convection-diffusion are illustrated in Fig. 4a, where error is defined as Frobenius distance betweenHFM and its ROM. For both cases considered, lagrangian ROMs significantly outperformthe Eulerian ROMs in all cases considered. R. Mojgani and M. Balajewicz . . . . . . x w (a) Traditional Eulerian MOR. . . . . . . x w (b) Lagrangian MOR. Figure 1: Model order reduction of scalar convection equation; HFM (thick grey), k =2 ROMs (dashed green), and k = 5 ROMs (solid red). Solutions are plotted for t =0 , / , / , . . . . . . x w (a) Traditional Eulerian MOR. . . . . . . x w (b) Lagrangian MOR. Figure 2: Model order reduction of scalar convection-diffusion equation with Pe = 10 ;HFM (thick grey), k = 2 ROMs (dashed green), and k = 5 ROMs (solid red). Solutionsare plotted for t = 0 , / , / ,
1. 4.2.
Burger’s equation
The proposed approach is next applied to the reduction of a convection-dominatedBurger’s equation. Specifically, we consider (2.1) with f ( x, t, w ) = w ( x, t ), f ( x, t, w ) =1 / Re , w ( x,
0) = 0 . . e − ( x − . / . , w (0 , t ) = 0, for ( x, t ) ∈ [0 , . × [0 , Re = 10 . As before, two HFM are constructed, one in the Eulerian frame, as in (2.2), andone in the fully Lagrangian frame, as in (3.2). For both models, a second-order centralfinite difference discretization is used. N = 2000 grid points are used to discretized thedomain 0 (cid:54) x (cid:54) .
5. Total of K = 2000 Eulerian and Lagrangian snapshots are collected.Eulerian and Lagrangian basis are constructed by solving (2.6). Eulerian ROMs are solvedin the form of (2.4). Due to the significant Lagrangian grid entanglement caused by thenonlinear convection term in the Burger’s equation, the Lagrangian ROMs are solvedusing the modified diffusion step; i.e. Eq.(3.4 b ) is replaced with (3.6). Galerkin projectionis used in both cases.Solutions at t = 0 , / , / , k = 1Lagrangian ROM has approximately the same error as a k = 20 Eulerian ROM. Note thatLagrangian ROMs only up to k = 5 are considered. After k = 5, some of the interpolated imensionality reduction of convection dominated nonlinear flows (cid:98) U w become linearly dependent and thus, no further performance gaincan be expected. 4.3. Quasi-1D Euler equation
Finally, the proposed approach is applied to the parametric, non-linear, quasi-one-dimensional Euler equations modeling a flow in a variable-area stream tube A ( x ) on afinite domain x ∈ [0 , ∂ρ∂t + ∂∂x ( ρu ) = − A dAdx ρu, (4.1 a ) ∂ρu∂t + ∂∂x ( ρu + p ) = − A dAdx ρu , (4.1 b ) ∂ρE∂t + ∂∂x ([ ρE + p ] u ) = − A dAdx ( ρE + p ) u, (4.1 c )where ρ is the fluid density, u is the fluid velocity, p is the thermodynamic pressure, and ρE = ρe + 12 ρu , (4.2)is the total energy density. The pressure is related to ρE by the equation of state p = ( γ − (cid:18) ρE − ρu (cid:19) , (4.3)for a perfect gas with ratio of specific heats γ = 1 .
4. The x = 0 boundary modelsa reservoir with specified total stagnation pressure p t = 101325Pa, and stagnationtemperature T t = 300T, while the right boundary at x = 10 enforces a specific staticback pressure, p b = 73145. The variable-area stream tube is defined as follows A ( x ) = 1 .
398 + µ tanh(1 . x − , (4.4)where µ ∈ [0 . , N = 200 grid points are used to discretized the domain0 (cid:54) x (cid:54)
10. The solution is marched to steady state using the implicit Euler timeintegration scheme.Solution snapshots are computed using a fully Eulerian solver for 10 instances of theparameter µ i = 0 . . i − i = 1 , . . . ,
10. Lagrangian basis are constructed bysolving the modified low-rank approximation problem, (3.5). As demonstrated in Fig. 5,the solution compressed using k = 2 Lagrangian basis is indistinguishable from theHFM, while the k = 2 Eulerian approximation contains large amplitude oscillations inthe vicinity of the shock. 4.4. Computational speed-up
Solving (3.4), or (3.6) requires the computation of the projection of high-dimensionalvectors and matrices on the reduced basis U x and U w . The complexity of this compu-tation scales with the size of the HFM, N . Therefore, while MOR reduces the size ofthe computational model from N to k , part of the computational cost associated withsolving the reduced problem still scales with the size of the HFM. For general nonlinearsystems, an additional level of approximation is required to achieve the desired speed-up.During the last decade, several methods, occasionally referred to as “hyper-reduction”methods, have been developed for reducing the computational complexity of projection-based ROMs (Chaturantabut & Sorensen 2010; Carlberg et al. et al. R. Mojgani and M. Balajewicz . . . . . . x w (a) Traditional Eulerian MOR. . . . . . . x w (b) Lagrangian MOR. Figure 3: Model order reduction of scalar Burger’s equation with Re = 10 ; HFM (thickgrey), k = 2 ROMs (dashed green), and k = 5 ROMs (solid red). Solutions are plottedfor t = 0 , / , / , − − k E rr o r (a) Pe = 10 k E rr o r (b) Re = 10 Figure 4: ROM convergence for scalar convection-diffusion equation and Burger’sequation. Traditional Eulerian ROMs (dashed red lines with filled markers) andLagrangian ROMs (solid black lines with empty markers). . . . . x ρ µ µ µ (a) HFM . . . . x ρ HFME–ROML–ROM (b) k = 2 ROM solutions for µ Figure 5: The density along a converging/diverging nozzle, (5a): some different cases( µ , µ , µ ) comprising the snapshot matrix (5b): comparison of HFM (thick grey) andtheir reduced order representation in Eulerian (dashed blue) and Lagrangian (solid red)framework for µ case. imensionality reduction of convection dominated nonlinear flows
5. Conclusions and future directions
A new Lagrangian projection-based model reduction approach has been introduced forthe reduction of nonlinear, convection dominated flows. Global basis functions are used toapproximate both the state and the location of the Lagrangian grid. In this framework, wedemonstrate that certain wave-like solutions, or solutions characterized by moving shocks,discontinuities and sharp gradients exhibit low-rank structure and thus, admit efficientreduction using only a handful of global basis. The proposed approach was applied toseveral canonical one-dimensional problems for which the traditional Eulerian approachis known to fail. Lagrangian reduced order models are demonstrated to significantlyoutperform traditional, Eulerian-based reduced order models. An unexplored opportunityof our approach is the generalization to an arbitrary Lagrangian-Eulerian framework inorder to avoid Lagrangian grid entanglement issues.
Acknowledgments
This material is based upon work supported by the National Science Foundation underGrant No. NSF-CMMI-14-35474.
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