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A LOCALLY PARAMETRIZED REDUCED ORDER MODEL FOR THE LINEAR FREQUENCY DOMAIN APPROACH TO TIME-ACCURATE COMPUTATIONAL FLUID DYNAMICS R. ZIMMERMANN∗ Abstract. For transonic flows governed by the time-accurate Navier–Stokes equations, small, approximately periodic perturbations can be calculated accurately by transition to the frequency domain and truncating the Fourier expansion after the first harmonic. This is referred to as the linear frequency domain (LFD) method. In this paper, a parametric trajectory of reduced order models (ROMs) for the LFD solver is presented. To this end, several local projection–based ROMs, which are essentially specified by suitable low-order subspaces, are computed by the method of proper orthogonal decomposition (POD) in an offline stage. The claimed trajectory is obtained locally by interpolating the given local subspaces considered as sample points in the Grassmann manifold. It is shown that the manifold interpolation technique is subject to certain restrictions. Moreover, it turns out that the application of computing accurate ROMs for the LFD solver requires a special choice of underlying inner product, necessitating a non-Euclidean approach. By exploiting a separable parametric dependency, real-time online performance is achieved. Numerical results are presented for emulating an airfoil in the transonic flow regime under a sinusoidal pitching motion. Key words. reduced order model; linear frequency domain; computational fluid dynamics; subspace interpolation; Grassmann manifold AMS subject classifications. 37M99; 37N10; 65F99;

1. Introduction. In many engineering and industrial problems such as design optimization, uncertainty quantification, optimal control or loads prediction, solutions to large-dimensional systems are required not only at a few selected parameter conditions but rather parametric representation of the solutions are sought after, which can be assessed very fast. This objective triggers the need for parametric reduced order models continuously covering a parameter domain of interest, while retaining sufficient accuracy. Arguably the most important subclass of model order reduction techniques is the class of subspacebased methods. Here, the key idea is to construct a low-dimensional subspace of the large dimensional full-order solution space and to subsequently restrict all computations to the low-dimensional subspace by projection. Approaches that subordinate to this scheme are reduced basis methods, proper orthogonal decomposition and Krylov subspace methods, see the collections [30, 32] for a survey. Subspace-based reduced order models (ROMs) may be parametrized by computing a trajectory in the Grassmann manifold of subspaces of a fixed dimension. To this end, each given subspace is considered as a single point in the Grassmann manifold and a trajectory of subspaces is obtained by interpolation. This approach has been introduced in [4], see also [3]. In the paper at hand, the subspace interpolation method is applied to construct a parametric ROM of the linear frequency domain (LFD) approach to unsteady fluid dynamics. The LFD approach applies to time-accurate flows governed by the Navier–Stokes equations under small, approximately periodic perturbations, the key feature being a transition to the frequency domain via a Fourier transformation. The resulting equations are linearized in the frequency domain by a truncation after the first harmonic terms. In this way, the partial differential Navier-Stokes equations are replaced by a large-scale linear equation system. This large-scale system, in turn, may be reduced by projection onto a local low-order subspace, covering a certain local extract of the complete parameter space. The resulting so-called singlepoint ROMs are parametrized by computing a trajectory of projection subspaces. The specific application of the subspace interpolation method to the LFD approach offers challenges as well as opportunities. In this regard, the original contributions of this paper are the following. (1) Following [4], interpolation in a curved manifold is conducted by interpolation in an associated tangent space by a suitable one-to-one mapping. It is exposed theoretically that interpolation in the tangent space requires a special choice of interpolation schemes. (2) The LFD approach makes use of complex valued data and therefore necessitates an adaptation of the manifold interpolation approach to the complex setting. (3) Solutions to the Navier–Stokes equations in the transonic flow regime feature shocks, which are highly nonlinear phenomena. Shock capturing poses a severe challenge for reduced-order models. It will be demonstrated that in the context of the LFD solver, shock capturing succeeds only after considering an appropriate non-Euclidean metric for quantifying the appearing residuals. (4) As a consequence, Grassmann interpolation for subspaces spanned by non-Euclidean orthogonal bases ∗ Institute

’Computational Mathematics’, TU Braunschweig, 38100, Germany ([email protected]). 1

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is considered. (5) An offline-online decomposition of the required computations for the LFD ROM is introduced. It is shown that in the online stage, a separable parameter dependence as considered in [15] is encountered. Numerical results will be presented for emulating an airfoil in the transonic flow regime under a sinusoidal pitching motion, where the objective is to obtain accurately full density distributions at the airfoil’s surface rather than predicting integrated scalar data like lift or drag coefficients. It turns out that shock capturing requires a sufficiently fine sampling with respect to the projection subspaces leading to a parametrized trajectory of single-point ROMs holding locally but not globally. The performance of the subspace interpolation method is compared to ROM predictions using a global spanning subspace. Related work: In regard of related work, we refer to the following non-exhaustive selection of papers. The LFD approach to unsteady Navier–Stokes is described in detail in [34]. Non-parametric single-point POD-based ROMs for the LFD approach have been investigated in [16], see also [21, §4.1] and references therein. The subspace interpolation method has been introduced in [4], where an application to aeroelastics was shown. Interpolation in matrix manifolds with application to time-invariant linear systems have been considered in [10, 25, 5, 24]. In [10], interpolation is conducted for reduced system matrices rather than projection subspaces. In the references [25, 5] the same procedure is enhanced by trying to find compatible coordinates for the local reduced system matrices. As a consequence, all these three approaches require a global basis for mapping the reduced-order solution back onto the full-order space. In [25] it is suggested to use a parametrically weighted global basis, while in [5] a back-mapping to the full-order dimension is not tackled explicitly. A fast interpolation scheme for projection subspaces is developed in [24] for system matrices exhibiting an affine parameter dependency. Non-Euclidean metrics have not been considered in any of the aforementioned papers but prove to be essential for the challenge envisioned in this paper. Matrix manifolds have long been considered in image processing and computer vision, see [22] for a recent review. Apart from matrix interpolation schemes, other approaches to parametric model reduction include the trajectory piecewise linear method [27], and local parametrization via sensitivity information, [17]. The former approach relies on local Taylor approximations to the non-linear system function. In the latter approach, snapshot derivatives are used to compute locally Taylor-like approximation to the POD basis itself rather than to the full-order system. A recent survey of model reduction methods for parametric systems is [8], which also features a variety of methods specifically tailored for linear time-invariant systems. Preliminary discussion of methods: The method proposed in this paper works efficiently in the setting, where the projection subspaces, which define single-point ROMs, exhibit a smooth, yet arbitrary nonlinear parametric dependency, while for evaluating a given local single-point ROM a separable parametric dependency is encountered. (As will be shown in Sections 3, 4 of the work at hand, the LFD method subordinates precisely to this setting.) In contrast, the method introduced in [24] relies on a separable parametric dependency of the complete system on the given parameters, and thus does not apply to the LFD setting. Since in the LFD context, snapshot and/or system sensitivities with respect to the nonlinear parameters are not readily available, the approaches [27], [17] are not an option. The methods suggested in [10, 25, 5] apply to the even more general setting of a smooth nonlinear dependence on all the parameters in the exercise and rely on the approximations of reduced system matrices via interpolation. For the approach proposed here, it will turn out that when utilizing the separable parametric dependency, the local reduced system matrices are directly obtained with respect to any given projection basis, which renders an interpolation of reduced systems unnecessary. Thus, replacing the subspace interpolation scheme in the work at hand by a global basis compares to applying the approaches of [10, 25, 5] but to exact reduced systems instead of interpolated reduced systems. The results of this comparison are discussed in Section 5.3. 2. Parametric reduced order models for linear systems. 2.1. Setting. Let A(p, q)W = b(p, q),

A ∈ RN ×N , b ∈ RN

(2.1)

be a parametrized large-scale linear equation system depending nonlinearly on parameter vectors p ∈ Rdp and q ∈ Rdq of dimensions dp , dq ∈ N, respectively. Borrowing the parlance introduced in [4], the parameter vector q is said to specify an operating point. The parameters collected in p will be referred to as examination parameters. The application scenario envisioned in this paper is that solutions to (2.1) are to be computed at a comparably limited number of operating points, while at each operating point, the system (2.1) may be evaluated for an arbitrarily large number of examination parameters. In the fluid

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A parametric ROM for the LFD approach to CFD

flow examples considered in Section 5, the operating points are specified by the free stream Mach number q = M a∞ ∈ R≥0 (i.e. the relative flow velocity). The frequencies ω, which guide the sinusoidal pitching of the aerodynamic body under consideration play the role of the examination parameters p = ω ∈ R.

2.2. Snapshot based reduced order modeling at a single operating point. Let q0 ∈ Rdq specify a fixed operating point. Suppose that m ∈ N solutions W 1 = W (p1 , q0 ), . . . , W m = W (pm , q0 ) ∈ RN to the system A(p, q0 )W = b(p, q0 ) are given. The solution vectors W j , j = 1, . . . , m are referred to as snapshots. The basic idea of snapshot based reduced order modeling is restricting the solution space to the subspace spanned by the snapshots. Let Uq0 ∈ RN ×m be a matrix, whose columns span the snapshot space. The reduced order ansatz space is Uq0 = {Uq0 a| a ∈ Rm } ⊂ RN ,

(2.2)

where the columns of Uq0 are referred to as trial basis. In the following, we will consider orthogonal projections onto Uq0 with respect to inner products induced by diagonal positive definite matrices. ˜ iS = Let S ∈ RN ×N be diagonal positive definite and denote the inner product induced by S by hW, W T T 1/2 ˜ W S W , with corresponding norm kW kS = (W SW ) . The inner product matrix S is assumed to be independent of parametric changes. Let ΠS : RN → span{W 1 , . . . , W m } ⊂ RN denote the orthogonal projection onto the snapshot space with respect to h·, ·iS . Given a snapshot matrix Yq0 = W 1 , . . . , W m ∈ RN ×m corresponding to an operating point q0 , the orthogonal projection onto Uq0 = colspan(Yq0 ) with respect to h·, ·iS is ΠS : RN → RN , W 7→ Uq0 UqT0 SW, where Uq0 ∈ RN ×m is any S-orthogonal matrix featuring colspan(Uq0 ) = colspan(Yq0 ) and UqT0 SUq0 = Im×m , i.e. the columns of Uq0 are orthonormal with respect to h·, ·iS . Such a spanning matrix Uq0 may be computed as stated in Algorithm 2.2.1. The dimension of the ansatz subspace Uq0 = colspan(Uq0 ) Algorithm 2.2.1 Non-Euclidean proper orthogonal decomposition Input: Snapshot matrix Yq0 ∈ RN ×m , S ∈ RN ×N diagonal positive definite √ SV D ˜ N ×m T ˜ , SYq0 := Y˜q0 = U 1: Compute the thin Singular Value Decomposition q0 ΣV , where Uq0 ∈ R m×m m×m Σ∈R √ ,V ∈R . ˜q0 . 2: Uq0 = ( S)−1 U N ×m Output: Uq0 ∈ R ˜q0 , which correspond to the may be reduced by discarding in step 2 of Alg. 2.2.1 those columns of U smallest singular values. More details on orthogonal projections with respect to arbitrary inner products and sketches of proofs may be found in [18, §2] and [36, §2]. The two standard ways to compute reduced order solution to (2.1) based on the approach W = Uq0 a, with Uq0 representing a reduced basis for the solution space valid at the operating point q0 , are the following: • The orthogonal residual approach. Here, the coefficient vector a ∈ Rm is determined by the condition 0 = ΠS (A(p, q0 )Uq0 a − b(p, q0 )) = Uq0 UqT0 S(AUq0 a − b)

⇔

UqT0 SAUq0 a = UqT0 Sb.

(2.3)

• The minimum residual approach. Here, the coefficient vector a ∈ Rm is determined by the condition min kA(p, q0 )Uq0 a − b(p, q0 )k2S

a∈Rm

⇔

UqT0 AT SAUq0 a = UqT0 AT Sb.

(2.4)

This is in analogy to the classification of Krylov subspace methods for linear systems, see [14, §2]. Both the minimum residual approach and the orthogonal residual approach lead to linear equation systems of the dimension of the snapshot space m ≪ N . Summarized, approximate solutions to (2.1) at the operating point q0 are given by W ∗ (p, q0 ) = Uq0 a(p),

(2.5)

where a(p) ∈ Rm is determined by either (2.3) or (2.4). By (2.5) a decoupling of the parameter dependencies by operating parameters and examination parameters is obtained. Obviously, the ROM state

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Fig. 2.1. Grassmann-Interpolation of subspaces. Left: schematic representation of Grassmann-Manifold(curved line) with tangent space attached (straight line). Right: schematic parameter domain with snapshot bases at operating points q0 , q1 , q2 , q3 indicated.

vector W ∗ (p, q0 ) is invariant under change of basis for Uq0 = colspan(Uq0 ). Both approaches (2.3) and (2.4) subordinate to the more general method of Petrov-Galerkin projection, where a low-order solution is searched after inside a trial space with the additional condition that the corresponding residual is forced to be contained in the orthogonal complement of an appropriate test space. The orthogonal residual approach is the special case, where the trial space is equal to the test space, while the minimum residual approach is obtained, when AU is chosen as a test basis corresponding to a trial basis U , just compare (2.3) to (2.4) or see [9] for details. 2.3. Parametric reduced order modeling along a trajectory of operating points. Suppose that for a number of r ∈ N operating points q1 , . . . , qr ∈ Rqd , corresponding (reduced) m-dimensional subspaces Uq1 , . . . , Uqr ⊂ RN in the sense of (2.2) are given by matrix representatives Uq1 , . . . , Uqr ∈ RN ×m , which feature columns that are orthogonal with respect to h·, ·iS , i.e. UqTk SUqk = Im×m , k = 1, . . . , r. In the following, we will consider the projection subspaces Uqk , k = 1, . . . , r as points in the Grassmann manifold G(N, m) of m-dimensional subspaces of RN , [2, §3.4.4], [20, §7.2], G(N, m) = U ⊂ RN | U is subspace, dim(U) = m . The fundamental objective of projection-based parametric reduced order modeling is to compute a trajectory of subspaces in G(N, m) along the operating points U : Rdq → G(N, m),

q 7→ Uq .

(2.6)

Such a trajectory may be obtained by interpolating the points Uq1 , . . . , Uqr on the Grassmann manifold. Interpolation of reduced order subspaces in the Grassmann manifold in the context of engineering applications has first been considered in [4]. Interpolating subspaces on the Grassmannian G(N, m) works by interpolating the corresponding orthogonal matrix representatives. In order to obtain a feasible interpolant represented again by an orthogonal matrix, it is both natural and imperative to take the geometric structure of G(N, m) into account. Excellent references on the numerical treatment of Grassmannians and on the required differential geometric foundations are [11, 2]. For the reader’s convenience, we briefly review the essentials. Attached at each U ∈ G(N, m) is the corresponding tangential space TU G(N, m), which consists of the set of velocity vectors of differentiable curves in G(N, m) passing through U. Let U ∈ RN ×m be a matrix representative of the subspace U such that U T U = Im×m and colspan(U ) = U. Then TU G(N, m) = ∆ ∈ RN ×n | U T ∆ = 0 ⊂ RN ×m , see [11, §2.5]. This characterization does not depend on the chosen matrix representative U . The tangent ˜ = space inherits the standard metric of the embedding Euclidean vector space RN ×m , i.e. g(∆, ∆)


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˜ tr(∆T ∆). By a fundamental result of differential geometry [19, Prop. 8.2], the tangent space at U0 is isometric to a sufficiently small neighborhood ΩU0 ⊂ G(N, m) of U0 . This one-to-one connection is established by the exponential mapping expU0 : TU0 G(N, m) → ΩU0 ⊂ G(N, m), which maps a tangent vector ∆ ∈ TU0 G(N, m) to the endpoint C(1) of a geodesic path C : [0, 1] → G(N, m) starting at U0 ∈ G(N, m) with velocity ∆. An efficient explicit formula for evaluating the Grassmann exponential has been derived in [11, §2.5.1] and is restated here in Algorithm 2.3.1. An analogous expression for nonAlgorithm 2.3.1 [11, §2.5.1] Grassmann Exponential expU0 (∆) = U

Input: base point U0 ∈ G(N, m) represented by U0 ∈ RN ×m orthogonal (Euclidean sense), ∆ ∈ TU0 G(N, m) ⊂ RN ×m SV D 1: ∆ = QΣV T . 2: U = U0 V cos(Σ) + Q sin(Σ). Output: U = colspan(U ) ∈ G(N.m) orthogonal matrix representatives of subspaces is given in [1, Thm. 3.6]. The inverse mapping with respect to the base point U0 ∈ G(N, m), referred to as the logarithm logU0 : G(N, m) ⊃ ΩU0 → TU0 G(N, m) with the property logU0 (U0 ) = 0 ∈ TU0 G(N, m) can numerically be evaluated using Algorithm 2.3.2. Both algorithms 2.3.1 and 2.3.2 can be found in [7], yet the author was not able to track down the origins of Algorithm 2.3.2. The reference [13] features expressions for the Grassmann exponential and the corresponding logarithm based on orthogonal completions U ⊥ of matrix representatives U ∈ Rn×n such that (U0 , U ⊥ ) ∈ Rn×n is orthogonal. The reference [26, §4.3] gives the corresponding mappings after identifying subspaces with orthoprojectors. Algorithm 2.3.2 [7, §3] Grassmann Logarithm logU0 (U) = ∆

Input: base point U0 ∈ G(N, m) represented by U0 ∈ RN ×m orthogonal, U ∈ G(N, m) represented by U ∈ RN ×m orthogonal (Euclidean sense). 1: L = (In×n − U0 U0T )U (U0T U )−1 = U (U0T U )−1 − U0 SV D

L = QΣV T ∆ = Q arctan(Σ)V T Output: ∆ ∈ TU0 G(N, m) 2: 3:

Having the explicit expressions for exp and log at hand, a trajectory of subspaces passing through a given set of subspaces Uq1 , . . . , Uqr ⊂ RN at operating points q1 , . . . , qr as stated in (2.6) may be obtained as follows: First, a base point U0 = Uqk0 , qk0 ∈ {q1 , . . . , qr } is selected. Then, the subspaces Uqk , k 6= k0 are mapped onto the tangential space at U0 via logU0 (Uqk ) =: ∆(qk ). The tangent vectors ∆(q1 ), . . . , ∆(qr ) ∈ TU0 G(N, m) are interpolated as detailed in Section 2.4. The resulting interpolant may be evaluated at any desired operating point q ∗ ∈ Rm , leading to a tangential vector ∆(q ∗ ), which is subsequently mapped back onto the Grassmannian G(N, m) via the exponential expU0 (∆(q ∗ )) = Uq∗ ∈ G(N, m). This procedure of computing a subspace trajectory has been proposed in [4] and is restated in Algorithm 2.3.3. A schematic illustration is given in Fig. 2.1. Algorithm 2.3.3 [4, §IV.B] Grassmann subspace interpolation

Input: Operating points q1 , . . . , qr ; subspaces Uq1 , . . . , Uqr ⊂ RN represented by Uq1 , . . . , Uqr ∈ RN ×m orthogonal (Euclidean sense); trial operating point q ∗ ∈ Rdq 1: select base point qk0 ∈ {q1 , . . . , qr }, set U0 := Uqk0 , ∆(qk0 ) = 0 2: for k ∈ {1, . . . , r} \ {k0 } do 3: ∆(qk ) ← logU0 (Uqk ) 4: end for 5: ∆(q ∗ ) ← interpolant(∆(q1 ), . . . , ∆(qr )) (q ∗ ), see Obervation 1, Section 2.4 6: Uq∗ ← expU0 (∆(q ∗ )) Output: Uq∗ Remark 1. Algorithms 2.3.1 – 2.3.3 require matrices as input, which are orthogonal in the Euclidean sense. Although analogous expressions for non-orthogonal matrix representatives of subspaces exist, [1, Thm. 3.6], [2], for numerical stability and efficiency, only orthogonal representatives are considered in

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−3

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Fig. 2.2. Element-wise interpolation of the entries of a tangent matrix ∆ ∈ {TU0 G(4, 2)} ⊂ R4×2 using MATLAB’s ’pchip’-method. The row (∆4,1 , ∆4,2 ) is zero.

˜q as given by this work. To this end, the subspace interpolation procedure is performed for the matrices U k ˜q∗ is then weighted a-posteriori by the inner product step 3 of Algorithm√2.2.1. The resulting interpolant U ˜q∗ . leading to Uq∗ = ( S)−1 U 2.4. Interpolation in tangential spaces - a word of caution. The essential property of the Grassmann interpolation procedure Alg. 2.3.3 is that the interpolation is conducted in a flat Euclidean vector space (the tangent space) rather than in a curved manifold. This fact has led the authors of previous papers on matrix manifold interpolation to claim that the tangent space allows for an entry-by-entry interpolation of the tangent matrices by “any preferred interpolation method”, see [5, §4.1.2], see also [4, §IV.B, Remark 1.] and [10, §3]. It is, however, theoretically possible that the interpolant leaves the tangent space, even if all sample points are contained in it. This may occur even for univariate interpolation. To see this, consider the Grassmann manifold G(4, 2) = {U ≤ R4 | dim(U) = 4} of two-dimensional subspaces in R4 . Points in G(4, 2) may be represented by (4 × 2)-Matrices. Let U0 = colspan((−1, 1, −1, 0)T , (0, 0, 0, −1)T )

with normalized representative U0 = √13 (−1, 1, −1, 0)T , (0, 0, 0, −1)T ∈ R4×2 . The tangent space in U0 is TU0 G(4, 2) = ∆ ∈ R4×2 | U0T ∆ = 0 . Let v1 = (2, 1, −1, 0)T , v2 = (0, 1, 1, 0)T . Apparently, v1 , v2 ⊥colspan(U0 ). Suppose that we have an underlying trajectory U 7→ U (q) sampled at q0 = 0 and qi ∈ {−0.1, −0.056, 0.02, 0.09}, such that U (0) = U0 and U (qi ) = Ui , i = 1, 2, 3, 4 and such that projecting these points onto TU0 G(4, 2) leads to the tangent vectors ∆i = qi2 v1 + 0.1qi v2 , qi3 v1 + 0.01 sin(qi )v2 ∈ TU0 G(4, 2) ⊂ R4×2 . The range for the parameter q has been chosen small enough to ensure that all the ∆i ’s lie in a neighborhood of 0 ∈ TU0 G(4, 2), where the exponential mapping is a diffeomorphism, which has been confirmed numerically. Given the assumed parametric dependency, the numerically exact tangent vector at q∗ = 0.0551 is T 0.0060720 0.0085460 0.0024740 0 ∆(q∗) = . 0.0003346 0.0007180 0.0003834 0 Element-wise interpolation of the sampled tangent matrices ∆i using monotonic piecewise cubic Hermite polynomials as implemented in MATLAB’s [23] ’pchip’ method assessed at q∗ = 0.0551 leads to an interpolant of T 0.0058975 0.0085930 0.0015117 0 0.00068342 0 T ˜ ˜ ∆(q∗) = , ∆(q∗) U0 = . 0.0004357 0.0007887 0.0001892 0 0.00009460 0 T ˜ As can be seen, the tangency constraint ∆(q∗) U0 = 0 does not hold true. Note that neither of the two ˜ ˜ columns of ∆(q∗) is orthogonal to colspan(U0 ). Hence, the pchip-interpolant ∆(q∗) leaves the tangent

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˜ space TU0 G(4, 2). The element-wise interpolants for the matrix entries ∆(q∗) ij ,i ≥ 3, j ≥ 2, are displayed in Fig. 2.2. Another interpolation scheme that may violate the tangency constraint when conducted element-wise is Kriging [12, §2.4], which is a statistically motivated interpolation procedure that is popular in many engineering applications. Without going into details, Kriging relies on estimating optimal correlation lengths for preselected spatial correlation models, e.g. the Gaussian correlation. Hence, the training of separate Kriging predictors for each matrix coordinate may lead to different correlation lengths in each coordinate. Applying coordinate-wise optimized Gaussian Kriging predictors to the sampled entries of the above tangent matrices ∆i leads to an interpolant of T 0.0066941 0.0085465 0.0024735 0 −0.0003587 0 ∆krig (q∗) = , ∆krig (q∗)T U0 = , 0.0006448 0.0007953 0.0002696 0 −0.0000688 0 also not contained in TU0 G(4, 2). However, when keeping the correlation lengths constant for each matrix entry, Kriging will lead to a feasible tangent space interpolation, as will be explained in the next section. In order to ensure that the interpolant is completely contained inside the tangent space, only those interpolation schemes are generally feasible, which lead to a linear combination of the sampled tangent vectors. Observation 1. Interpolation in tangent spaces must be conducted with respect to the coefficients of a linear combination of sampled tangent vectors rather than element by element. More precisely, if operating points q1 , . . . , qr ∈ Rdq are given and if k0 ∈ {1, . . . , r} is the index of the selected base point, (so that ∆(qk0 ) = logU0 (U0 ) = 0), then the interpolant must be of the form ∗

∆(q ) =

r X

k=1,k6=k0

ϕk (q ∗ )∆(qk ),

ϕk (q ∗ ) ∈ R.

(2.7)

Incidentally, this is exactly the scenario, where the complexity reduction method developed in [24] for the special case of subspace representatives depending affinely on the operating points applies. Obviously, in in the univariate case, linear interpolation and Lagrange interpolation fulfill the requirement of (2.7). Here, element-wise interpolation and interpolation of linear expansion coefficients coincide, the corresponding expressions being   r ∗ ∗ ∗ X Y q − q q − q q − q k j j  ∆(qk ) + ∆(qj ), q ∗ ∈ [qj , qk ], ∆(q ∗ ) = ∆(qk ), ∆(q ∗ ) = qk − qj qk − qj qk − qj k=1,k6=k0

k6=j

respectively. These are exactly the interpolation procedures, that have been chosen for the applications presented in [4, 5, 24]. For the sake of completeness, let me note that out of the standard univariate interpolation methods that, say, MATLAB provides, pchip is the only one to fail in the above example. Presumingly, it may be possible to show theoretically that element-wise univariate cubic spline interpolation also cannot leave the tangent space by tracing back the linear dependency of the spline coefficients on the sample points. Other interpolation schemes that qualify for the application in tangent spaces are standard radial basis function interpolation and a suitably restricted version of Kriging [12], as will be shown in the next section. Radial basis function interpolation was also used in [3]. Observation 1 lays the theoretical foundations for the feasibility of these interpolation schemes. In addition to the guaranteed feasibility, interpolation schemes of the form (2.7) reduce the computational costs, since only the interpolation of r expansion coefficients followed by a sum over tangent matrices is required. Observation 1 is not restricted to Grassmann subspace interpolation but also applies to all manifold interpolation techniques relying on projection onto the tangent space.

2.5. Feasible tangent space interpolation schemes: Radial basis functions and limited Kriging. The simplest approach to radial basis function (RBF) interpolation is the following. Given r ∈ N sample points q1 , . . . , qr ∈ Rd in the d-dimensional space and corresponding sample values yk = y(qk ), k = 1, . . . , r, the RBF interpolant at q ∗ reads yˆ(q ∗ ) = (y1 , . . . , yr )Ψ−1 ψ(q ∗ ).

(2.8)

Here, Ψ ∈ Rr×r is the quadratic radial distance matrix with entries Ψi,j = rbf(kqi − qj k), i, j = 1, . . . , r T and ψ(q ∗ ) is the vector ψ(q ∗ ) = (rbf(kq1 − q ∗ k), . . . , rbf(kqr − q ∗ k)) ∈ Rr and rbf: R≥0 → R is a radial basis function. Common choices of RBFs include

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linear cubic multiquadric Gaussian Thin Plate Spline √ , rbf(x) x x3 1 + x2 exp(−x2 ) x2 log(x) see [12, §2.3] for details and enhancements. ∗ −1 ∗ Introducing the RBF weights vector Pr ω(q ∗) := Ψ ψ(q ), equation (2.8) may be written as a weighted ∗ sum of the sample values yˆ(q ) = k=1 ωk (q )yk . Observation 2. Let U (q1 ), . . . , U (qr ) ∈ RN ×m be r samples of a matrix valued function U : Rd → N ×m R . Let ω(q ∗ ) = Ψ−1 ψ(q ∗ ) be the weights vector corresponding to the sample locations q1 , . . . , qr and a chosen radial basis function. Then the RBF interpolant at q ∗ ∈ Rd is U (q ∗ ) =

r X

ωk (q ∗ )U (qk ).

k=1

In particular, element-wise interpolation of the entries of the sampled matrices U (qk ) coincides with interpolation of the expansion coefficients of a linear combination of the sampled matrices in the sense of Observation 1. Proof. It is most important to note that the RBF weights vector ω(q ∗ ) does not depend on the sample values but only on the sample locations and the chosen RBF. Interpolating entry (i, j) in the matrix U (q ∗ ) based on the samples values of the corresponding entries in U (q1 ), . . . , U (qr ) leads to U (q ∗ )ij = (U (q1 )ij , . . . , U (qr )ij ) ω(q ∗ ). k Writing U k := U (qk ) and U:,j for the jth column in U k , it holds

U (q ∗ ) := Uij∗

i;j≤N ;m

= =

Uij1 , . . . , Uijr ω(q ∗ ) i;j≤N ;m =

1 r ω(q ∗ ) j≤m U:,j , . . . , U:,j ! r r r X X X k k ωk (q ∗ )U:,1 ,..., ωk (q ∗ )U:,m = ωk (q ∗ )U k , k=1

k=1

k=1

which proofs the observation. Note that ωk (q ∗ ) = eTk Ψ−1 ψ(q ∗ ), i.e. the entries of the RBF weights vector are given by the RBF interpolant using the kth unit vector as a vector of sampled values. Reviewing the Kriging method is beyond the paper. It is sufficient to note that the Prscope of this ∗ ∗ Kriging predictor also takes the form yˆ(q ∗ ) = k=1 ωk (θ, q )yk , where the Kriging weights ωk (θ, q ) additionally depend on a vector θ of so-called correlation lengths. If this vector is not optimized separately for each matrix entry sample data set, but kept constant throughout, then the same argument as outlined above for RBF predictors applies. In this case, the global θ should be chosen by optimizing the Kriging predictor for the sampled expansion coefficients data set ϕk (qi ) according to (2.7). In addition to the simplicity, a major benefit of RBF interpolators and Kriging predictors is that they readily apply to an arbitrary number of input parameters. In contrast, multivariate Lagrange interpolation is much more elaborate, see [29]. 2.6. Transition to complex data. In this section, it is exposed, how the manifold interpolation technique transfers to complex arithmetic. Complex valued snapshots occur for example when applying a Fourier transform to the full-order system, as it is the case for the linear frequency domain approach to the time-accurate Navier-Stokes equations. The exponential mapping and the logarithm as stated in Algortithm 2.3.1 and 2.3.2, respectively, directly transfer to the complex Grassmann manifold GC (N, m) = U ⊂ CN | U is subspace, dim(U) = m ,

as do the expressions for the tangent space by simply replacing the transpose of a matrix U T with the ¯ T , where ¯· denotes complex conjugation and by replacing the term ’orthogonal’ Hermite transpose U H = U by ’unitary’ for subspace representatives U ∈ CN ×m , so that U H U = Im×m . In real life applications, however, complex number arithmetic might not be available due to binding programming guidelines or the use of black-box codes with predefined interfaces. In such cases, it is standard to identify a complex matrix Y by its real-valued analogue as follows ℜ(Y ) −ℑ(Y ) CN ×m ∋ Y = ℜ(Y ) + iℑ(Y ) ←→ = YR ∈ R2N ×2m . (2.9) ℑ(Y ) ℜ(Y )


9

However, care must be taken to perform the proper orthogonal decomposition for YR correctly. Using the so-called method of snapshots, the POD of YR can be conducted as follows: First, compute the eigenvalue decomposition YRT YR = V ΛV T , where Λ = diag(λ1 , . . . , λ2m ) features the eigenvalues sorted by size on the diagonal. Since each eigenvalue of complex multiplicity µ of the Hermitian Y H Y appears as an eigenvalue of real multiplicity 2µ for YRT YR , the eigenvalue matrix Λ is actually of the form Λ = diag(λ1 , λ1 , λ2 , λ2 , . . . , λm , λm ). Therefore, the eigenvalue decomposition is not consistent with the identification (2.9) and a permutation of the columns of Λ and V must be performed so that Λ = diag(λ1 , . . . , λm , λ1 , . . . , λm ) and the columns of V are in corresponding order. It is an easy exercise Vr −Vi required for the identification (2.9), as does Λ. to see, that now indeed V takes the form Vi Vr √ √ By setting Σm = diag( λ1 , . . . , λm ) ∈ Rm×m and Vr Qr Σm , = YR Vi Qi the real and imaginary parts of the left singular vectors (POD eigenmodes) are obtained, which correspond exactly to a complex SVD of Y = QΣRH , where Q = (Qr + iQi ) and R = (Vr + iVi ). Note that the set of real-valued representatives of complex matrices according to (2.9) builds a vector space. Hence, the interpolation schemes described in Section 2.4 automatically preserve this special structure. 3. The time-linearized/small disturbances approach to Navier-Stokes. This section features a condensed review of the time-linearized/small disturbances approach to solving the Navier-Stokes equations following closely the account given in the recent work [34]. As reviewed in the introduction of [34], the small disturbances approach essentially applies to performing flutter analysis, where flutter refers to the amplifying of small oscillations due to fluid-structure interactions. Via flutter analysis, the air speed is determined, where the coupled fluid-structure system becomes unstable. This is essential in aircraft certification. 3.1. The governing equations. Consider the Navier-Stokes equations of fluid dynamics, spatially discretized on a computational grid of size ng ∈ N for an open spatial domain Ω ⊂ R3 . Let the timedependent grid point coordinates be collected in a vector x(t) ∈ R3ng . Let nv ∈ N be the corresponding number of primitive mean flow variables plus the number of primitive variables associated with the turbulence model. The primitive mean-flow variables are the density, ρ, the velocity components in all spatial directions, ux , uy , uz , and the total energy, E. The number of primitive turbulence variables depends on the chosen turbulence model. In the examples of use presented in this paper, the Spalart-Allmaras one equation turbulence model [33] is applied throughout. Here, the additional variable µ models the eddy viscosity. Accordingly, the flow state vector of conservative variables is W = (ρ, ρux , ρuy , ρuz , ρE, ρµ) ∈ Rnv ng , where each entry stands for a discrete vector, e.g. ρ = (ρ1 , ..., ρng ) ∈ Rng . For an unsteady flow motion, the state vector depends on the grid coordinates and the grid point velocities, W = W (x(t), x(t)). ˙ Assuming a finite volume discretization, the unsteady Reynolds-averaged Navier-Stokes (RANS) equations of fluid dynamics in semi-discrete form read d (M (x)W (x, x)) ˙ + R(W (x, x), ˙ x, x) ˙ = 0, (3.1) dt see [34, §II.A] for a derivation. Here, R = R(W (x, x), ˙ x, x) ˙ ∈ Rnv ng denotes the flux residual and nv ng ×nv ng is a diagonal weights matrix featuring nv copies of the grid cell volumes on the M = M (x) ∈ R diagonal, one for each flow variable. The total dimension of the system (3.1) is N = ng nv . The time-linearized/small disturbances approach is based on the fundamental assumption that the flow motion to be simulated is periodic and approximately harmonic. Hence, by assumption, both the flow state vector and the grid coordinates are given by small, time-dependent periodic fluctuations around their respective mean: ¯ +W ˜ (t), x(t) = x ¯ (¯ ˜ (¯ W (t) = W ¯+x ˜(t), M (x) = M x) + M x, x ˜), (3.2) where quantities with a bar denote averages and quantities with a tilde denote the fluctuations. Equation (3.1) is linearized by a first-order Taylor approximation around the mean. Transition to the Fourier space and truncating after the first-order harmonic terms leads to a complex valued linear equation system in the frequency domain, ∂R ¯ W ¯ ∂M c = − ∂R + iω ∂R + W x b =: b1 + iωb2 ∈ CN , + iω M (3.3) ∂W ∂x ∂ x˙ ∂x | | {z } {z } =:AC ∈CN ×N

:=bC ∈CN

10

R. Zimmermann

c ∈ CN and x where boldface i denotes the complex unit, i2 = −1, and W b ∈ C3n denote the Fourier transforms of W and x, respectively. Equation (3.3) can be represented by its real valued analogue ! ! ∂R ¯ ∂M bim − ∂R bre + ω ∂R cre ¯ W −ω M ∂x x ∂ x˙ + W ∂x x ∂W . (3.4) = ∂R ¯ cim ωM ¯ ∂M bre W − ∂R bim − ω ∂R ∂x x ∂ x˙ + W ∂x x | {z ∂W } {z } | =:AR ∈R2N ×2N

=:bR ∈R2N

Hence, the governing time-accurate RANS equation (3.1) is replaced by a real-valued sparse linear equation system of dimension 2N = 2nv ng in the frequency domain. This approach is also termed the linear frequency domain (LFD) method for short. Equations (3.3), (3.4) directly correspond to equations (34), (35) in [34]. 3.2. Parametric dependency. The most common application of the LFD is to compute flow solutions for a range of reduced frequencies, [16, 34]. The frequency ω and the reduced frequency κ are connected in the following way: κ(ω) =

lref ω V∞

⇔

ω(κ) =

V∞ κ. lref

(3.5)

p Here, V∞ = M a∞ γp∞ /ρ∞ is the normalized (dimension-less) free-stream velocity, lref is the reference root cord length and M a∞ , p∞ , ρ∞ and γ = 1.4 are the free-stream Mach number, the free-stream static pressure, the free-stream density and the ratio of specific heats, respectively. In reference to Section 2.1, the reduced frequencies act as examination parameters p = κ, while operating points are specified by the free-stream Mach number q = M a∞ . For a given reduced frequency κ, the corresponding full-order flow state solution ! cre W (ω(κ), M a∞ ) = A−1 R (ω(κ), M a∞ )bR (ω(κ), M a∞ ) cim W

is obtained by solving the sparse linear system (3.4). This partitioning of parameters comes natural in the LFD-context, since the free-stream Mach number determines the mean flow solution, which is used as point of origin for the linearization in (3.2). The same partitioning was also considered in [16], the difference being that only single-point ROMs without interpolation across operating points have been considered in this reference and a separable parametric dependency as will be exposed in Section 4 was not exploited. Note that the flow state vector also depends on a number of additional parameters, which specify the basic flow conditions, e.g. the Reynolds number (measuring the viscosity of a fluid), the free-stream pressure, the free-stream density and so forth. As in the LFD references [34, 21, 16], these parameters are all considered as fixed throughout 4. A parametric ROM of the Linear Frequency Domain solver.

4.1. Prerequisites. Suppose that a number of r ∈ N operating points q1 = M a∞,1 , . . . , qr = c 1 , ..., W cm ∈ M a∞,r have been selected and that at each operating point, full-order snapshot solutions W N C to the linear system (3.3) at reduced frequencies of κ1 , . . . , κm are precomputed. Suppose furN ×N ther that an inner product . Let Ybqk = is induced by a positive definite diagonal matrix S ∈ R N ×m N×m c (κ1 , qk ), ..., W c (κm , qk ) ∈ C W be the snapshot matrix at qk and let Uq ∈ C be an S-orthogonal k

representative of the snapshot subspace, i.e. k = 1, . . . , r.

colspan(Uqk ) = colspan(Yqk ) and UqHk SUqk = Im×m ,

4.2. Online predictions at a fixed single operating point. Following the orthogonal residual approach (2.3), the single-point reduced order system corresponding to (3.3) at qk reads Aorth (ω(κ), qk ) = B(qk ) + iω(κ)D(qk ), borth (ω(κ), qk ) = βorth,1 (qk ) + iω(κ)βorth,2 (qk ), where B(qk ) =

UqHk S

∂R Uq ∂W k

¯ Uq ∈ Cm×m , D(qk ) = UqHk S M k

(4.1)

11


and βorth,1 (qk ) = UqHk Sb1 , βorth,2 (qk ) = UqHk Sb2 ∈ Cm , while the minimum residual approach (2.4) yields Amin (ω(κ), qk ) = E(qk ) + ω 2 (κ)G(qk ) + iω(κ)H(qk ), bmin (ω(κ), qk ) = βmin,1 (qk ) + ω 2 (κ)βmin,2 (qk ) + iω(κ)βmin,3 (qk ),

(4.2)

where E(qk ) =

UqHk

∂R T ∂R ¯ 2 SUq , H(qk ) = UqH S Uq , G(qk ) = UqHk M k k ∂W ∂W k

∂R T ¯ ¯ S ∂R SM − M ∂W ∂W

!

Uqk ∈ Cm×m

and βmin,1 (qk ) =

UqHk

∂R T ¯ Sb2 , βmin,3 (qk ) = Sb1 , βmin,2 (qk ) = UqHk M ∂W

UqHk

∂R T ¯ Sb1 Sb2 − UqHk M ∂W

!

∈ Cm .

∂R ¯ and the inner product matrix S are real, the system Although the Jacobian ∂W , the volume matrix M matrices in (4.1), (4.2) are not written as decomposed in real and imaginary part. The sorting of terms is rather with respect to parts depending on the frequency ω and parts independent of it, so that the (m × m) matrix summands B(qk ), D(qk ), E(qk ), G(qk ), H(qk ) and the corresponding right hand side vectors βorth,j (qk ), j = 1, 2, βmin,j (qk ), j = 1, 2, 3 are fixed at an operating point. Hence, a separable parameter dependence as considered in [15] is encountered here. Precomputing the frequency-independent data by conducting Alg. 4.2.1 allows for real-time online predictions according to Alg. 4.2.2 for predicting the periodic perturbations in the frequency domain at an operating point. Algorithm 4.2.1 Semi-online preparations at operating point qk ∂R ¯ at qk , ∈ RN ×N (sparse) at qk , diagonal volume weights M Input: Operating point qk , Jacobian ∂W N ×m N ×N subspace matrix Uqk ∈ C ,S∈R diagonal, symmetric positive definite 1: if orthogonal residual then 2: Compute B(qk ), D(qk ) ∈ Cm×m . (cf. (4.1)) 3: Compute βorth,j (qk ) ∈ Cm , j = 1, 2. (cf. (4.1)) 4: else if minimum residual then 5: Compute E(qk ), G(qk ), H(qk ) ∈ Cm×m . (cf. (4.2)) 6: Compute βmin,j (qk ) ∈ Cm , j = 1, 2, 3. (cf. (4.2)) 7: end if Computational costs: N vm + O(N m2 ), where v is the average number of non-zero entries in ∂R ∂R ∂W per row. Note that only the matrix product ∂W Uqk involves the Jacobian, which has to be computed only once in order to obtain B, D and βorth,1 , βorth,2 or E, G, H and βmin,1 , βmin,2 , βmin,3 , respectively

Algorithm 4.2.2 Online predictions at operating point qk Input: Subspace matrix Uqk ∈ CN ×m , Reduced frequency κ, matrices B(qk ), G(qk ), E(qk ) ∈ Cm×m , vectors βmin,j (qk ) ∈ Cm , j = 1, 2, 3. p V 1: ω ← l∞,k κ, where V∞,k = qk γp∞ /ρ∞ ref 2: Amin ← E(qk ) + ω 2 G(qk ) + iωH(qk ) 3: bmin ← βmin,1 (qk ) + ω 2 βmin,2 (qk ) + iωβmin,3 (qk ) 4: Solve Amin a = bmin for a ∈ Cm cROM ← Uq a 5: W k cROM = W cROM (ω(κ), qk ) Output: W Computational costs: N m + O(m3 ) (the costs in m are dominated by solving the (m × m)-system in Step 4.)

12

R. Zimmermann

4.3. Parametric construction of ROM subspaces at untried operating points. Suppose that r ∈ N reduced-order bases Uqk ∈ CN ×m are given for operating points qk = M a∞,k , k = 1, . . . , r. ∗ A reduced-order basis at an untried operating point / {q1 , . . . , qk } is obtained by applying Algorithm √q ∈ ˜q = SUq appearing in step 2 of Alg. 2.2.1. 2.3.3 to the Euclidean orthogonal matrices U k k In summary, the online stages consist of • computing the trial subspace Uq∗ via subspace interpolation (semi-online) • storing the (frequency-independent) full-order system data (semi-online) • computing the (frequency-independent) reduced-order system data (Alg. 4.2.1, semi-online) • (For back transition to the time domain:) computing the mean flow at q ∗ (semi-online), e.g. via full-order RANS or any preferred ROM approach • conducting Alg. 4.2.2 for as many frequencies as desired (online) The semi-online stages have to be performed only once at each interpolated operating point.

1

50

0.5

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Z

100

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-50

-100

-1 -50

0

50

100

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X

0

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1

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X

Fig. 5.1. Computational grid for the NACA 64A010 airfoil. Right hand side: Detailed view close to the surface.

5. Results. In this section, the parametric ROM approach as proposed in Section 4 will be demonstrated for the standard transonic NACA 64A010 airfoil. This test case was also considered in [34]. The computational grid features ng = 10, 727 points and is displayed in Fig. 5.1. The total complex dimension of the full-order system is 5 · 10, 727 = 53, 635, yielding a real dimension of 107, 270, as nv = 5 flow variables are to be considered. As in [34], a sinusoidal pitching motion is simulated, governed by α(τ ) = α ¯+α ˆ sin(κτ ), where α ¯ is the mean flow angle of incidence, α ˆ is the pitching amplitude, κ is the reduced frequency, see (3.5) and τ is the nondimensional time, τ = tV∞ /lref , cf. (3.5). The Reynolds number is fixed at 12.5 · 106 and the mean flow angle of incidence and the pitching amplitude are fixed throughout at values of α ¯ = 0.0◦ and α ˆ = 0.5◦ , respectively. As operating points, M a∞,1 = 0.8, M a∞,2 = 0.802, M a∞,3 = 0.804, M a∞,4 = 0.806, M a∞,5 = 0.808, M a∞,6 = 0.81 have been chosen in the transonic flow regime, where shock phenomena are encountered, which appear as quasi-discontinuities in the density distributions. Numerical experiments performed by the author have shown that shock capturing requires such a fine sampling in the M a∞ operating points in order to obtain acceptable results. A uniform sampling in M a∞ in steps of 0.005 turned out to be insufficient, even though the injectivity radius of the Grassmann exponential was (numerically) not exceeded. This behavior was to be partly expected, since the full-order LFD-approach is already based on a local linearization of the non-linear Navier-Stokes equations. To the best of the author’s knowledge, the work at hand is the first to evaluate the Grassmann interpolation approach not only based on its prediction capabilities for scalar quantities like lift or drag, but for complete density distributions including sonic shocks. c (κ1 , M a∞,k ), . . . , W c (κ5 , M a∞,k ) At each operating point, five full-order state solution snapshots W have been computed at reduced frequencies of κ1 = 0.1, κ2 = 0.3, κ3 = 0.5, κ4 = 0.7, κ5 = 0.9.

13


0.9

0.8

reduced frequency κ

0.7

0.6

0.5

0.4

0.3

0.2

0.1 0.8

0.801

0.802

0.803

0.804

0.805 Mach

0.806

0.807

0.808

0.809

0.81

Fig. 5.2. Sample locations for parametric LFD ROM (full squares). The Mach numbers define operating points, the reduced frequencies κ are considered as examination parameters. The dashed lines indicate locations of operating points for which subspace interpolation is performed.

Steady mean flow at Ma = 0.805: CFD reference (contour + dotted), ROM (dashed)

Steady mean flow at Ma = 0.805 0.6

0.5

density

Z

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ROM CFD Reference

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Fig. 5.3. Reduced-order approximation of the steady mean flow at operating point Mach 0.805 based on CFD flux residual minimization following [37]. Left: density field close to the airfoil, full-order CFD (contour map and dotted lines) vs. ROM approximation (dashed). Right: density distribution at the airfoil’s surface, full-order CFD (solid) vs. ROM (dashed, square markers). Full-order CFD and ROM solutions virtually coincide. The surface density distributions match up to a relative error < 10−8 .

Computations were conducted using the linear frequency domain solver developed at the DLR (German Aerospace Center), which was recently presented in [34]. Full-order RANS solutions are computed with the DLR flow solver TAU [31]. The sample locations are displayed in Fig. 5.2. In summary, the offline stage consists of • computing the mean flow solution at each operating point using full-order RANS. • solving the full-order real-valued linear system (3.4) for the selected 5 reduced frequencies κ ∈ {0.1, 0.3, 0.5, 0.7, 0.9} at each of the chosen operating points. • Compute a trial subspace by performing complex proper orthogonal decomposition at each operating point. The computation time for the offline stage is displayed in Table 5.1. Based on the given five fullorder snapshots, a projection subspace Uqk of complex dimension 5 is computed at each operating point qk = M a∞,k , k = 1, . . . , 5 by Algorithm 2.2.1. The associated computation time is also listed in Table 5.1. As underlying inner products for performing orthogonal projection and proper orthogonal decomposition, the choices displayed in Table 5.2 are considered. Here, the descriptor L2 –metric corresponds to

14

R. Zimmermann

Ma 0.805, κ =0.2 metric: 1/vols

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pROM LFD Ref Proj LFD Ref

0.04

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pROM LFD Ref proj LFD Ref

0.03

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Fig. 5.4. Parametric ROM predictions (pROM, dashed) of the real part first-harmonic density distribution at Mach 0.805 using the minimum residual approach (2.4) and the non-descriptor L2 –metric, see Table 5.2, compared to the fullorder reference (LFD Ref, solid) and the full-order reference projected onto the interpolated subspace (LFD Ref Proj, dash-dotted). full-order LFD building subspaces (complex POD) total ≈ 5 · 35s ≈ 0.26s ≈ 475s(8min) Table 5.1 Average timing results for offline computations for constructing a single operating point. The timing for computing the trial subspaces decomposes in 0.25s for reading the snapshots from disk and 0.01s for the actual POD on average. Timing results were all obtained on a Dell desktop computer endowed with eight processors of type Intel(R) Core(TM) i7-3770 [email protected]. CPU time

mean flow (RANS) ≈ 300s

measuring the residual in descriptor form in the discrete L2 -sense, while the non-descriptor L2 –metric corresponds to measuring the residual in non-descriptor form in the discrete L2 -sense. The non-descriptor L2 –metric puts special emphasis on the small grid cells in the nearfield grid close to the airfoil. By omitting the information on the grid cell volumes, the Euclidean metric also accentuates the residuals in the small grid cells, simply because they are large in number, but not as much as the non-descriptor L2 –metric. The descriptor L2 –metric puts equal weighting on all subdomains of the computational grid and thus corresponds to looking at the computational domain from a distance as displayed in Fig. 5.1, left hand side. The non-descriptor L2 –metric has been found advantageous both in [35] and [6], where in the latter reference, it is recovered as “hybrid form” with respect to the Euclidean metric, see also ¯ = diag(Ω1 , . . . , Ωng , . . . , . . . , Ω1 , . . . , Ωng ) ∈ Rng nv ×ng nv appearing in Table Appendix A. Recall that M 5.2 is the diagonal matrix of mean grid cell volumes per flow variable associated with the underlying

15



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Fig. 5.5. Same as Fig. 5.4, but at Mach 0.807. The shock location has moved downstream. Metric ˜ iS = W H S W ˜ hW, W

Euclidean descriptor L2 non-descriptor L2 ¯ ¯ −1 S = IN×N S=M S=M Table 5.2 Selected choices of positive definite diagonal matrices S ∈ RN×N inducing an inner product.

¯W ˜. ˜ iL2 = W H M computational grid, so that the discrete L2 –metric is hW, W For validating the parametric ROM approach, trial subspaces Uq∗ are computed by interpolation in the Grassmann manifold via Algorithm 2.3.3 at operating points q ∗ ∈ {M a∞ = 0.805, M a∞ = 0.807}, see Fig. 5.2. As a base point for attaching the tangent space, the subspace at M a∞ = 0.804 was chosen. At the trial operating points, no full-order snapshots enter the subspace computation process. Preceding the actual online-predictions, Algorithm 4.2.1 has to be executed. Here, the dominating part is to read the full-order system matrix (eq. (3.3), left-hand side), which has been assembled during the snapshot computations in the offline stage, followed by a down-projection onto the trial space according to either the orthogonal residual approach (2.3) or the minimum residual approach (2.4). The DLR LFD solver outputs the system matrix in the real number arithmetic form of (3.4) in CSRF compressed sparse row format, [28, §3.4]. As explained in Section 4.2, the full-order system matrix must be stored only once at each operating point. Also, Algorithm 4.2.1 for downsizing the frequency-independent system parts must be performed only once at each operating point. Back transition to the time domain eventually requires a single steady RANS solution for computing the mean flow at each untried operating point. Note that mean flow snapshots are already assembled during the offline stage. Hence, without

16

R. Zimmermann

Ma 0.805, κ =0.2 metric: Euclid

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Fig. 5.6. Same as Fig. 5.4, but for the Euclidean metric, see Table 5.2. This approach is non-competitive. Semi-online CPU time

(ROM approximation of the steady RANS mean flow solution < 2.0s.) Grassmann interpolation Reading of system matrix Projecting system (Alg. 4.2.1) ≈ 0.66s ≈ 10.0s ≈ 0.15s

(Frequency domain predictions only.) Set up ROM (Alg. 4.2.2 (1.-3.)) solve ROM (Alg. 4.2.2 (4.)) build state approximation (2.5) below measurement accuracy below measurement accuracy < 0.01s CPU time Table 5.3 Average timing results for semi-online computations for constructing a trial subspace at an untried operating point and for online predictions. Back transition to the time domain additionally requires a single steady RANS solution for computing the mean flow at an untried operating point (here < 2.0s via ROM). Timing results were all obtained on a Dell desktop computer endowed with eight processors of type Intel(R) Core(TM) i7-3770 [email protected]. Online

any additional full-order computations, a nonlinear POD-based ROM for predicting the mean flow is constructed and assessed at q ∗ . In this work, the nonlinear POD-based ROM is obtained via CFD flux residual minimization following [37]. Since the interpolation of trial subspaces requires a much finer sampling than actually necessary for a high-quality nonlinear POD-based mean flow prediction (see the results in [37]), the mean flow here is approximated to a very high accuracy, see Fig. 5.3. (At both trial points q ∗ ∈ {M a∞ = 0.805, M a∞ = 0.807}, the ROM mean flow density prediction matches the full-order CFD density function up to a relative error less than 4.0 · 10−5 over the complete field. The surface density distributions match even up to a relative error less than 10−8 !) The timing results for this semi-online stage are listed in Table 5.3.

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Ma 0.805, κ =0.4 orthogonal res

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Fig. 5.10. ROM predictions using a global POD basis including all available full-order snapshots at M a∞ = 0.805, kappa = 0.4. Inner products are measured according to the non-descriptor L2 –metric. Left: minimum residual residual approach (2.4). Right: orthogonal residual approach (2.3). This approach is non-competitive.

Having the trial subspace Uq∗ at hand, online predictions are performed for reduced frequencies κ∗ ∈ {0.2, 0.4, 0.6, 0.8}. Note that no snapshots corresponding to this set of examination parameters have entered the parametric ROM at any of the sampled operating points. Hence, the selected trial parameters (p∗ , q ∗ ) = (κ∗ , M a∞,∗ ) are intermediate points with respect to both the sampled operating points and the sampled examination parameters, as can be seen in Fig. 5.2. Online predictions are obtained by executing Algorithm 4.2.2. Here, all operations scale in the trial subspace dimension m = 5 (resp. m = 10, if real degrees of freedom are counted) and thus are negligible. In fact, the associated computational costs are dominated by finally building the approximate ROM state vector according to (2.5), which is the only step during the online predictions, that scales in the grid size N . The timing results for the online stage are also included in Table 5.3. The accuracy of the parametric LFD ROM is quantified by comparison to the full-order LFD solver. The full-order LFD-solver, in turn, being a linearized approximation to the full-order time-accurate RANS equations, has been validated against full-order RANS as well as againts experimental data in [34]. In the next section, results obtained via the minimum residual approach (2.4) are discussed, while results obtained via the orthogonal residual approach (2.3) are stated in the section after next. Section 5.3

19


finally features results obtained by a global POD-based ROM rather than a parametric one. The various approaches are compared based on their capability to predict the surface density distributions including the sonic shock, while the overall accuracy of the predicted flow field is of minor importance. This restricted objective poses a special challenge, since residual vectors are available only for the complete flow field. For better judging the results presented in this section, I will record the following Observation 3. 1. At a single operating point, the trial subspace constructed via snapshot POD is the same regardless of the underlying metric, see Algorithm 2.2.1. Only the spanning basis may differ. Moreover, both the orthogonal residual approach (2.3) and the minimum residual approach (2.4) are invariant under changes of basis for the trial subspace. As a consequence, untruncated Euclidean POD and untruncated non-Euclidean POD will produce exactly the same results, apart from the following subtlety: If a non-Euclidean metric induced by S ∈ RN ×N diagonal, positive definite is considered, then the Grassmann subspace interpolation is conducted for representatives span√ ning Scolspan(Yqk ) rather than spanning the snapshot space colspan(Yqk ). It is only after the interpolation, that the interpolant Uq∗ is weighted by S. Hence, the interpolated subspaces may differ, while the subspaces at the operating points are the same. Note, however, that the choice of metric affects, which POD modes are discarded when truncating the basis according to the relative information content. 2. In contrast, the objective functions (2.3) and (2.4) do depend on the selected metric. More precisely, the orthogonal residual equation UqT0 SAUq0 a = UqT0 Sb, (2.3) will lead to the same state vector W (p, q0 ) = U a for all U satisfying colspan(U ) = colspan(Uq0 ) but will give different results for different positive definite matrices S. The same holds true for the minimum residual equation (2.4). OP q ∗ = M a∞ = 0.805 reduced frequency κ = 0.2 k · k2 Error ℜ (pROM/Projection) k · k2 Error ℑ (pROM/Projection) k · k2 Error ℜ (pROM surface) residual (pROM/Projection) reduced frequency κ = 0.4 k · k2 Error ℜ (pROM/Projection) k · k2 Error ℑ (pROM/Projection) k · k2 Error ℜ (pROM surface) residual (pROM/Projection) reduced frequency κ = 0.6 k · k2 Error ℜ (pROM/Projection) k · k2 Error ℑ (pROM/Projection) k · k2 Error ℜ (pROM surface) residual (pROM/Projection) reduced frequency κ = 0.8 k · k2 Error ℜ (pROM/Projection) k · k2 Error ℑ (pROM/Projection) k · k2 Error ℜ (pROM surface) residual (pROM/Projection) OP q ∗ = M a∞ = 0.807 reduced frequency κ = 0.8 k · k2 Error ℜ (pROM/Projection) k · k2 Error ℑ (pROM/Projection) k · k2 Error ℜ (pROM surface) residual (pROM/Projection)

minimum residual predictions for Euclidean metric descriptor L2 –metric 3.9341 / 0.1023 3.2330 / 1.8268 2.0556 / 0.0646 1.7132 / 2.5370 0.2733 0.2225 0.0028 / 0.0222 0.0160 / 0.0522

the various metrics non-descriptor L2 –metric 0.2267 / 0.1168 0.1269 / 0.0818 0.0155 0.0212 / 0.0769

2.7407 / 0.0952 1.6894 / 0.0582 0.1827 0.0027 / 0.0146

0.9199 / 0.9512 0.9706 / 1.8748 0.0547 0.0104 / 0.0671

0.4195 / 0.1400 0.2144 / 0.1217 0.0299 0.0198 / 0.1031

1.7700 / 0.0843 1.5858 / 0.0849 0.1175 0.0024 / 0.0158

0.7971 / 0.5964 0.4569 / 0.6632 0.0523 0.0047 / 0.0374

0.3552 / 0.1780 0.3867 / 0.1880 0.0241 0.0200 / 0.1387

1.1086 / 0.0461 1.3129 / 0.0854 0.0777 0.0022 / 0.0193

0.2372 / 0.4577 0.3071 / 0.2101 0.0144 0.0042 / 0.0248

0.2075 / 0.1061 0.2935 / 0.1367 0.0138 0.0181 / 0.1212

1.1353 / 0.0414 0.2550 / 0.4540 0.1299 / 0.0846 1.3068 / 0.0879 0.3980 / 0.1873 0.2301 / 0.1195 0.0803 0.0164 0.0078 0.0022 / 0.0223 0.0039 / 0.0248 0.0119 / 0.1030 Table 5.4 Approximation errors associated with the minimum residual parametric ROM approach compared to the approximation errors of the reference solutions projected onto the interpolated subspaces.

5.1. Examining the minimum residual approach. This section features the results of the parametric LFD ROM (pROM) at the interpolated operating points M a∞,1 = 0.805 and M a∞,2 = 0.807, when imposing the minimum residual constraint (2.4) on the ROM solutions in the trial subspace. Figure 5.4 shows the real part of the first-harmonic pROM density distributions at the surface of the NACA 64A010 airfoil at the operating point q ∗ = M a∞.∗ = 0.805 for all considered reduced frequencies compared to the full-order LFD solutions. The graphs displayed in the aforementioned figures were obtained using the non-descriptor L2 –metric (Table 5.2) as an underlying inner product for both the POD and the

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minimum residual objective function (2.4). The corresponding results when using the Euclidean metric and the descriptor L2 –metric are displayed in Figs. 5.6 and 5.7, respectively. Included in all the figures are the first-harmonic density distributions of the the full-order LFD snapshots projected onto the trial subspace, where the orthogonal projection is conducted in consistency with the selected metric. This serves two purposes; one is that the projection of the full-order solution onto the trial subspace is expected to be the best possible solution contained in the trial subspace with respect to the selected metric. If the projection almost coincides with the full-order solution, then this fact indicates that the trial subspace is rich enough to produce very accurate ROM solutions. Secondly, in contrast, if the projection itself is of unacceptable accuracy, then this fact is a strong indicator the chosen metric is unfeasible for the problem at hand. The errors with respect to the full-order LFD references are listed in Table 5.4. As can be seen both from the figures and the error table, the minimum residual approach based on the non-descriptor L2 – metric outperforms its competitors at all parameter combinations considered here. Roughly speaking, the errors of the pROM predictions associated with the non-descriptor L2 –metric are one order of magnitude smaller than the corresponding results for the Euclidean metric. When compared to the results obtained by using the descriptor L2 inner product, the gap between the errors exhibits a larger undulation as it ranges from one order of magnitude at M a∞ = 0.805, κ = 0.2 to a very comparable accuracy at M a∞ = 0.805, κ = 0.8, see Table 5.4. However, the large deviation in the errors for the descriptor L2 – metric and the fact, that even the orthogonal projection of the full-order solution onto the trial subspace with respect to this metric may be considerably off, see Fig. 5.7, renders the L2 –metric an inappropriate measure for the challenge tackled here. The parametric LFD ROM has also been evaluated at the interpolated operating point M a∞,2 = 0.807 for the same reduced frequencies κ ∈ {0.2, 0.4, 0.6, 0.8} and to a very comparable outcome. Due to space limitations, only the first-harmonic density distributions at κ = 0.8 are displayed in Fig. 5.8; the associated errors are included in Table 5.1.4. Again, the non-descriptor L2 –metric proves to be the best choice featuring errors about one order of magnitude below the Euclidean results and lower by a factor of two than the descriptor L2 results. OP q ∗ = M a∞ = 0.805 reduced frequency κ = 0.4 k · k2 Error ℜ (pROM/Projection) k · k2 Error ℑ (pROM/Projection) k · k2 Error ℜ (pROM surface) residual (pROM/Projection) OP q ∗ = M a∞ = 0.807 reduced frequency κ = 0.8 k · k2 Error ℜ (pROM/Projection) k · k2 Error ℑ (pROM/Projection) k · k2 Error ℜ (pROM surface) residual (pROM/Projection)

orthogonal residual predictions for Euclidean metric descriptor L2 –metric 1.0946 / 0.0952 3.6449 / 0.9512 0.7242 / 0.0582 2.5101 / 1.8748 0.0728 0.2451 0.0294 / 0.0146 0.0180 / 0.0671

the various metrics non-descriptor L2 –metric 2.2937 / 0.1399 3.7122 / 0.1217 0.1456 0.1251 / 0.1031

0.3121 / 0.0414 2.0644 / 0.4540 2.7706 / 0.0846 0.2413 / 0.0879 2.5390 / 0.1873 4.0137 / 0.1195 0.0206 0.1466 0.2140 0.0095 / 0.0223 0.0165 / 0.0248 0.2996 / 0.1030 Table 5.5 Approximation errors associated with the orthogonal residual parametric ROM approach compared to the approximation errors of the reference solutions projected onto the interpolated subspaces.

5.2. Examining the orthogonal residual approach. This section features the results of the parametric LFD ROM (pROM) at the interpolated operating points M a∞,1 = 0.805 and M a∞,2 = 0.807, when imposing the orthogonal residual constraint (2.3) on the ROM solutions in the trial subspace. Figure 5.9 shows the real part of the first-harmonic pROM density distributions at the surface of the NACA 64A010 airfoil for a reduced frequency of κ = 0.4 at the operating point M a∞,1 = 0.805 as well as for a reduced frequency of κ = 0.8 at the operating point M a∞,2 = 0.807 compared to the respective full-order LFD solutions. The corresponding errors are listed in Table 5.5. Even though the orthogonal residual approach when conducted with respect to the Euclidean metric at M a∞,2 = 0.807, κ = 0.8 leads to a surprisingly good result, the outcome is inferior to the minimum residual approach using the non-descriptor L2 –metric at all tested parameter combination and appears to be much less robust. 5.3. LFD-ROM predictions using a global POD subspace. As an alternative to the subspace interpolation based parametric LFD ROM method, a global proper orthogonal decomposition-based ROM may be constructed by collecting all available 30 snapshots corresponding to the sampling plan displayed in Fig. 5.2 in a single snapshot matrix Yg ∈ CN ×30 . After performing an SVD Yg = Ug Σg VgH , the fullorder LFD system (3.3) is projected onto the subspace colspan(Ug ) either according to the orthogonal

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A parametric ROM for the LFD approach to CFD OP q ∗ = M a∞ = 0.805 reduced frequency κ = 0.4 k · k2 Error ℜ (gROM/Projection) k · k2 Error ℑ (gROM/Projection) k · k2 Error ℜ (gROM surface) residual (gROM/Projection) OP q ∗ = M a∞ = 0.805 reduced frequency κ = 0.4 k · k2 Error ℜ (gROM/Projection) k · k2 Error ℑ (gROM/Projection) k · k2 Error ℜ (gROM surface) residual (gection)

minimum residual predictions for the various metrics Euclidean metric descriptor L2 –metric non-descriptor L2 –metric 3.5910 / 0.0491 24.299 / 2.2489 1.4721 / 0.1256 3.1327 / 0.0401 41.110 / 2.9935 1.1856 / 0.1078 0.2302 1.5240 0.0937 0.0011 / 0.0164 0.0016 / 0.0569 0.0104 / 0.0454 orthogonal residual predictions for the various metrics

5.9013 / 0.0491 5.7739 / 2.2489 5.1806 / 0.1256 5.0194 / 0.0401 7.0721 / 2.9935 4.1377 / 0.1078 0.3193 0.2301 0.3494 0.0022 / 0.0164 0.0070 / 0.0569 0.2166 / 0.0454 Table 5.6 Approximation errors associated with the global basis ROM approach compared to the approximation errors of the reference solutions projected onto the global subspace.

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residual condition (2.3) or according to the minimum residual condition (2.4). As in the above examples, the best results are again obtained when the non-descriptor L2 –metric from Table 5.2 is used. The resulting first-harmonic surface density distributions at the parameter condition M a∞,1 = 0.805, κ = 0.4 compared to the respective full-order LFD solutions are displayed in Fig. 5.10. The corresponding errors are listed in Table 5.6. Both the figure and the table suggest that the global POD approach is noncompetitive, since the errors are larger by about one order of magnitude than those associated with the parametric ROM approach. One possible explanation is that the 30-dimensional global ROM may suffer from snapshot-overfitting. Finally, the prediction quality of the LFD-ROM is assessed when reducing the global POD basis of dimension 30 to the five most dominant eigenmodes. Note that precisely five local snapshots are available at each operating point, and thus the dimension of the parametric subspaces also equals five in the above example. Hence, this approach corresponds to the proposal considered in [25], the only difference being that the ROM coordinate vector for returning to the full dimension according to (2.5) here is not obtained by solving an interpolated low-order system but rather by projecting the full-order system onto the order-5 POD basis according to (2.4). Therefore, the results presented here are expected to be of even higher quality as when employing interpolated low-order systems as in [25]. It turns out that the non-descriptor L2 –metric is again key to obtain acceptable results. Using this metric, the global ROB approach achieves a good overall prediction accuracy that at some parameter conditions even equals the quality of the parametric ROM approach, see Fig. 5.10. However, it is inherent in the nature of any global ROB approach, that the resulting ROM is not interpolating, while the parametric LFD-ROM exactly reproduces the sampled subspaces. As a consequence, the parametric ROM also exactly reproduces the snapshots used in the construction and thus features at the sampled

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operating points the same prediction quality as a local single point ROM. This can be seen from Fig. 5.11, where the errors corresponding to both approaches are displayed for the four operating points M a ∈ {0.804, 0.805, 0.806, 0.807}, two of them at sampled locations, two of them interpolated, and each examined at the reduced frequencies κ ∈ {0.1, . . . 0.9}. As a rule of thumb, for the example at hand, the prediction quality of the global reduced-basis approach is about the same as the worst-case prediction accuracy of the parametric ROM approach. 5.4. Quantification of the ROM’s speed-up factor. In order to quantify the speed-up obtained via the ROM procedure, consider the following application example. Suppose that as in the above setting, local trial subspaces have been computed for a number of six operating points. Now assume that for a number of, say, 20 reduced frequencies, the periodic disturbances are to be predicted at an untried operating point q ∗ . Then, the full-order LFD approach would require a single full-order RANS solution at q ∗ and 20 solutions to the full-order linear LFD system, summing up according to Table 5.1 to a total of about 300s + 20 · 35s = 1, 000s ≈ 17min. In contrast, the parametric LFD-ROM approach requires a reduced-order mean-flow prediction at q ∗ , an interpolated trial subspace and 20 solutions to the reduced linear system, summing up according to Table 5.3 to a total of less than 2s + 11s + 20 ∗ 0.01s = 13.2s. Obviously, the ROM becomes the more efficient, the more online computations are to be performed. In terms of a single frequency domain prediction, the speed-up factor is larger than 35.0s/0.01s = 3500. In terms of a single mean flow prediction, the ROM’s speed-up factor is larger than 300s/2s = 150. 6. Summary and conclusions. A parametric reduced order model for the linear-frequency domain (LFD) approach to solving the time-accurate Navier-Stokes equations has been presented. The method is essentially an adaptation of the subspace interpolation method introduced in [4] to the special context of the LFD solver. It was exposed that the associated interpolation problems in tangent spaces, that appear also in all the parametric ROM methods considered in [4, 10, 5, 24] require a special choice of interpolation schemes. Moreover, it was shown how the subspace interpolation method transfers to complex-valued snapshot data and arbitrary underlying inner products. In regard of the envisioned application, it was demonstrated that the projected LFD system exhibits a separable parameter dependency in the sense of [15, §3]. The separable nature of the parameters under consideration at an operating point allows for fast real-time online predictions in the frequency domain and renders a local interpolation of reduced system matrices as investigated in [10, 25, 5] unnecessary. The proposed parametric LFD-ROM was assessed by emulating the full-order LFD-solver for predicting the unsteady flow around an airfoil under a sinusoidal pitching motion in the transonic flow regime. This challenging problem features shocks, which appear as quasi-discontinuities in the density distributions. The findings when tackling this problem can be summarized under four headings. The first conclusion is that a sufficiently fine operator point sampling is required. The second conclusion is that the parametric ROM has lead to acceptable results only after imposing an appropriate metric. This observation is completely in line with the results of [6] and [35]. This fact motivated generalizing the theory to arbitrary inner products in the first place. The third conclusion is that the minimum residual approach must be followed. The orthogonal residual approach turned out to be non-competitive. The last conclusion is that a reduced global basis approach may also lead to acceptable results but lacks the interpolation property and underperforms when compared to the parametric LFD-ROM. The global basis approach is explicitly or implicitly inherent in all the methods [10, 25, 5]. Acknowledgments. The work presented in this paper uses and extends the Surrogate Modeling for Aero-data Toolbox (SMART), a code package developed by the Aero-Loads Prediction Group of the German Aerospace Center (DLR), Braunschweig, with contributions from the Institute ’Computational Mathematics’ (ICM), TU Braunschweig. The work was initially motivated by research questions considered in the Future Fast Aeroelastic Simulation Technologies (FFAST) project funded by the European Communitys Seventh Framework Programme (FP7 / 2007-2013, grant agreement number 233665), in which the author was partly involved when employed at DLR. The author would like to thank the anonymous referees for their valuable input and criticism. Appendix A. The non-descriptor L2 - metric. The residual corresponding to (3.3) in descriptor form is rd (W ) = AC W − bC . Assume that a spatial discretization with associated diagonal cell volume ¯ and corresponding discrete L2 norm kW k2 = W T M ¯ W is given. The non-descriptor residual matrix M L2 −1 ¯ is rnd (W ) = M rd . In this work, the use of the residual norm krd (W )kM¯ −1 is deemed essential. It holds ¯ −1 rd (W ) = rd (W )H M ¯ −1/2 M ¯ −1/2 rd (W ) = rd (W )H M ¯ −1 M ¯M ¯ −1 rd (W ) krd (W )k2M¯ −1 = rd (W )H M ¯ −1/2 rd (W )k2 = kM ¯ −1 rd (W )k2 = krnd (W )k2 . = kM 2

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Therefore, the residual choice krd (W )kM¯ −1 corresponds to the standard L2 norm of the residual in nondescriptor form krnd (W )kL2 , which was advocated in the nonlinear CFD context in [35, 37]. It equals also ¯ −1/2 rd (W )k22 recently recovered as beneficial in the hybrid form with respect to the Euclidean metric kM [6, eq. (20)]. The findings of this work are thus perfectly in line with the above papers. REFERENCES [1] P.-A. Absil, R. Mahony, and R. Sepulchre, Riemannian geometry of Grassmann manifolds with a view on algorithmic computation, Acta Applicandae Mathematica, 80 (2004), pp. 199–220. , Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, New Jersey, 2008. [2] [3] D. Amsallem, Interpolation on Manifolds of CFD-based Fluid and Finite Element-based Structural Reduced-order Models for On-line Aeroelastic Prediction, PhD thesis, Stanford University, 2010. [4] D. Amsallem and C. Farhat, Interpolation method for adapting reduced-order models and application to aeroelasticity, AIAA Journal, 46 (2008), pp. 1803–1813. , An online method for interpolating linear parametric reduced-order models, SIAM J. Sci. Comput., 33 (2011), [5] pp. 2169–2198. [6] D. Amsallem, C. Farhat, and M. J. Zahr, On the robustness of residual minimization for constructing pod-based reduced-order cfd models, no. 2013-2447 in AIAA, San Diego, CA, June 26-29 2013, 43rd AIAA Fluid Dynamics Conference and Exhibit. [7] Evgeni Begelfor and Michael Werman, Affine invariance revisited, 2012 IEEE Conference on Computer Vision and Pattern Recognition, 2 (2006), pp. 2087–2094. [8] P. Benner, S. Gugercin, and K. Willcox, A survey of model reduction methods for parametric systems, Tech. Report MPIMD/13-14, MPI, Max Planck Institute, Magdeburg, August 2013. [9] T. Bui-Thanh, K. Willcox, and O. 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Greenbaum, Iterative Methods for solving linear systems, vol. 17 of Frontiers in applied mathematics, SIAM Society for Industrial and Applied Mathematics, Philadelphia, 1997. [15] B. Haasdonk and M. Ohlberger, Efficient reduced models and a-posteriori error estimation for parametrized dynamical systems by offline/online decomposition, Mathematical and Computer Modelling of Dynamical Systems, 17 (2011), pp. 145–161. [16] K. C. Hall, J. P. Thomas, and E. H. Dowell, Proper orthogonal decomposition technique for transonic unsteady aerodynamic flows, AIAA Journal, 38 (2000), pp. 1853–1862. [17] A. Hay, J. T. Borggaard, and D. Pelletier, Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition, J. Fluid Mech., 629 (2009), pp. 41–72. [18] M. Hinze and S. 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