SIAM J. SCI. COMPUT. Vol. 33, No. 5, pp. 2169–2198
c 2011 Society for Industrial and Applied Mathematics
AN ONLINE METHOD FOR INTERPOLATING LINEAR PARAMETRIC REDUCED-ORDER MODELS∗ DAVID AMSALLEM† AND CHARBEL FARHAT‡ Abstract. A two-step online method is proposed for interpolating projection-based linear parametric reduced-order models (ROMs) in order to construct a new ROM for a new set of parameter values. The first step of this method transforms each precomputed ROM into a consistent set of generalized coordinates. The second step interpolates the associated linear operators on their appropriate matrix manifold. Real-time performance is achieved by precomputing inner products between the reduced-order bases underlying the precomputed ROMs. The proposed method is illustrated by applications in mechanical and aeronautical engineering. In particular, its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon. Key words. interpolation, matrix manifolds, mode veering, parametric model reduction, realtime computing AMS subject classifications. 37M99, 65L99, 74H45, 74H15, 37N10 DOI. 10.1137/100813051
1. Introduction. Real-time numerical capabilities for the prediction of the behavior of complex systems are desired in many engineering domains to enable routine analysis, practical uncertainty quantification, control, and fast design optimization. High-dimensional computational models—also known as high-fidelity models (HFMs)—currently provide the needed accuracy for many of these applications. However, they are usually computationally intensive and therefore unsuitable for real-time processing. Consequently, sufficiently accurate lower-dimensional computational models, also known as reduced-order models (ROMs), are often sought by practitioners to enable real-time operations. In this context, a “sufficiently accurate” ROM is a lower-dimensional computational model which can faithfully reproduce the essential features of a higher-dimensional model at a fraction of its computational cost. Such a ROM is often constructed by projecting the higher-dimensional counterpart onto a subspace. For many computational engineering applications, the construction of a projectionbased ROM can, however, be computationally expensive, as it typically relies on querying the underlying HFM (see [1] for sample CPU details). For example, the balanced truncation method [2] requires the solution of a high-dimensional Lyapunov equation. The proper orthogonal decomposition (POD) [3] and classical modal truncation methods incur the solution of a large-scale generalized eigenvalue problem. Model reduction methods based on snapshots of the response of the system of interest require ∗ Submitted to the journal’s Computational Methods in Science and Engineering section October 27, 2010; accepted for publication (in revised form) June 3, 2011; published electronically September 1, 2011. This work was partially supported by a research grant from the Academic Excellence Alliance program between King Abdullah University of Science and Technology (KAUST) and Stanford University, by a research grant from King Abdulaziz City for Science and Technology (KACST), by The Boeing Company under contract 45047, and by the Air Force Office of Scientific Research under grant FA9550-10-1-0539. http://www.siam.org/journals/sisc/33-5/81305.html † Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305-4035 (
[email protected]). ‡ Department of Aeronautics and Astronautics, Department of Mechanical Engineering, and Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 943054035 (
[email protected]). 2169
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accumulating solutions of problems formulated using the HFM; they can be computationally expensive, but are tractable for very large-scale systems. Among these methods, one can mention the POD when it is based on the method of snapshots [4], as well as moment-matching methods based on Krylov subspace iterations [5]. Engineering systems are almost always parameterized to allow, for example, variations in shape, material, loading, and boundary and initial conditions during their design or analysis. Consequently, the ROMs for these systems depend on such parameters in two ways: (1) through the dependence on these parameters of the underlying HFMs, and (2) through the dependence on these parameters of the associated reducedorder bases (ROBs). Unfortunately, it was shown that for many applications, ROBs are not robust with respect to variations in boundary conditions [6], initial conditions [7], physical parameters [7, 8], and engineering operating points [9, 10]. Error estimates for the numerical predictions obtained with ROMs built at perturbed values of a set of parameters but a single ROB constructed for a baseline set of values of these parameters can be found in [7, 8]. To address the nonrobustness issue raised above, model reduction methods based on the concept of a global ROB constructed by sampling a wide range of values of the parameters of interest were proposed in [11, 12]. In two cases, such methods lead to simple analytical expressions for describing the dependence of the sought-after ROMs on their parameters: when the underlying HFMs feature an affine [13] or separable [14, 15] dependence on these parameters, and when small perturbations are considered for these parameters in order to enable Taylor expansions of the HFMs around baseline operating points [16, 17]. Both cases correspond to rather limiting assumptions. Moreover, global methods tend to lose the optimal approximation property and as such lead to ROMs of larger dimensions than otherwise possible [18]. Furthermore, for aerodynamic applications, these methods were shown to fail to produce a ROM that can operate in the transonic regime, for example, when the parameter set includes the free-stream Mach number [9]. Most recently, an attempt was made at remedying the nonrobustness of a POD-based ROM of the dynamics of a flow past a cylinder with respect to variations in the shape of the cylinder relied on including snapshot sensitivities in the set of snapshots. However, this approach was shown to lead to unphysical results [19]. In [10], an alternative method based on interpolation on a manifold was proposed to adapt to a new value of a given parameter set, a collection of linearized, equal-dimension, projection-based ROMs that were precomputed for different sampled values of this parameter set. This method consists of two steps: (1) adapting to the new desired value of the parameter set the precomputed projection subspaces and constructing a corresponding ROB, (2) evaluating the underlying HFM at the new desired value of the parameter set and projecting it on the newly constructed ROB. This method generates for the desired value of the parameter set a ROM of the same small dimension as that of its precomputed counterparts. When applied to the subspaces spanned by a set of precomputed ROBs characterized by an optimal approximation property, this method was numerically shown—in the context of challenging three-dimensional computational fluid dynamics (CFD) applications—to generate new ROBs that retain this optimal approximation property, and corresponding ROMs that are as accurate as their precomputed counterparts [10, 20]. The two-step ROM adaptation method outlined above proved to be successful for challenging applications [10, 20]. However, because it requires for each query of the parameter set the offline re-evaluation of the underlying HFM, it does not lead at any stage of the process to an exclusively online computational method, it cannot achieve
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in general real-time processing, and therefore its potential is limited. For example, for complex CFD problems, it was shown to reduce the CPU time associated with rebuilding a linear ROM at a new desired value of the parameter set by about an order of magnitude only [20]. To eliminate the offline component of this method and achieve greater CPU time reduction, a first variant approach in which the precomputed ROMs themselves are directly interpolated on a suitable manifold [21, 22] and another one in which the transfer functions instead are interpolated in Cn×m [23] were subsequently proposed. An additional benefit of the first variant approach is its more practical applicability to nonlinear model reduction methods where linear operators arise during a Newton or Newton-like solution step [24], or when a nonlinear model is piecewise linearized [25, 18]. However, this online approach has so far encountered only mixed success. For example, it performed well for some linear structural dynamics applications [21] and some other linear fluid problems [22]. However, it performed poorly for linearized CFD applications conducted by the authors of this paper. In this work, it is explained that the observed irregular performance of the method of direct interpolation of ROMs on a suitable manifold is due to the fact that this method ignores the parameter dependence of the generalized coordinates system in which the ROM is expressed. To some extent, a similar issue was recently addressed in [26, 27, 28] where two methods for interpolating reduced-order operators expressed in local bases were proposed. One of them relies on the singular value decomposition (SVD) of a global basis matrix that is weighted according to the target value of the parameter set. Hence, the computational complexity of this method scales with the size of the underlying HFM. This prevents it from operating in real time. Furthermore, this method does not lead at any stage of the process to an exclusively online computational strategy either, because for each query of the parameter set, it requires recomputing the aforementioned global basis matrix. The other method proposed in [28] does not require this weighting and has online capabilities. Hence, the main objective of this paper is to present an online method for adapting ROMs to changes in parameter values that can operate in real time. The proposed method is fully developed in the context of linear problems. However, it is equally applicable to the linearized operators arising from the solution of nonlinear problems by a Newton-like method. It is organized around two steps as follows. In the first step, each precomputed ROM of interest is transformed into a consistent set of generalized coordinates by solving a minimization problem analytically, which simplifies numerical implementation. In the second step, the precomputed ROMs of interest are interpolated on their appropriate matrix manifold. The overall computational complexity of the proposed ROM adaptation method scales with the small dimension of the precomputed ROMs. Its real-time performance is achieved by precomputing inner products between the ROBs associated with the precomputed ROMs. Its potential is highlighted by its successful application to the solution of several academic problems involving linear oscillators, a simple mechanical problem featuring a mode veering phenomenon, and a more complex problem from aeronautical engineering. 2. Projection-based model reduction of linear time-invariant parametric dynamical systems. 2.1. Linear time-invariant systems. In this paper, a general linear timeinvariant (LTI) dynamical system of the following form is considered: (2.1) (2.2)
dx (t) = A(μ)x(t) + B(μ)u(t), dt y(t) = C(μ)x(t) + D(μ)u(t),
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where t denotes the time variable, x ∈ Rn the vector of state variables, u ∈ Rp the vector of inputs, and y ∈ Rq the vector of outputs. Here, μ ∈ RNp is a vector of parameters defining an operating point for the system and belonging to a connected set D ⊂ RNp . As such, the system matrices A ∈ Rn×n , B ∈ Rn×p , C ∈ Rq×n , and D ∈ Rq×p depend on this parameter vector. It is assumed in the rest of this paper that this dependence is smooth. The initial conditions for the above dynamical system are defined as x(0) = x0 ∈ Rn .
(2.3)
A transfer function describing the input-output behavior of the above system can be defined in the frequency domain as (2.4)
H(s; μ) = C(μ) (sIn − A(μ))
−1
B(μ) + D(μ), s ∈ C.
2.2. Petrov–Galerkin projection-based model reduction. The goal of model reduction is to generate a dynamical system of dimension k that is much lower than the dimension n of the HFM defined by (2.1)–(2.2), while retaining the main dynamical properties of this higher-order model. One approach to achieve this objective is to project the high-dimensional set of equations on a lower-dimensional subspace using a well-suited Petrov–Galerkin projector. This approach can be described in two steps as follows: 1. Define a trial basis V(μ) ∈ Rn×k having full-column rank and describing a trial subspace SV (μ) belonging to the Grassmann manifold G(k, n) [3, 10]. The state vector x(t) will be subsequently approximated as a linear combination of the column vectors of V(μ)—that is, x(t) ≈ V(μ)xr (t),
(2.5)
where the reduced size vector xr ∈ Rk defines the components of x in the trial basis V(μ). The linear system (2.1)–(2.2) becomes (2.6)
V(μ)
(2.7)
dxr (t) = A(μ)V(μ)xr (t) + B(μ)u(t) + r(t), dt yr (t) = C(μ)V(μ)xr (t) + D(μ)u(t),
where r(t) denotes the residual resulting from the state vector approximation. 2. Define a test basis W(μ) ∈ Rn×k having full-column rank and describing a test subspace SW (μ) belonging also to the Grassmann manifold G(k, n), and constrain r(t) to be orthogonal to this basis. Premultiply (2.6) by the test basis to obtain (2.8) (2.9)
W(μ)T V(μ)
dxr (t) = W(μ)T A(μ)V(μ)xr (t) + W(μ)T B(μ)u(t), dt yr (t) = C(μ)V(μ)xr (t) + D(μ)u(t).
Assuming that the matrix W(μ)T V(μ) is nonsingular, the reduced dynamical system is subsequently obtained in the same form as that of (2.1)–(2.2), (2.10) (2.11)
dxr (t) = Ar (μ)xr (t) + Br (μ)u(t), dt yr (t) = Cr (μ)xr (t) + Dr (μ)u(t),
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where
(2.13)
−1 Ar (μ) = W(μ)T V(μ) W(μ)T A(μ)V(μ) ∈ Rk×k , −1 W(μ)T B(μ) ∈ Rk×p , Br (μ) = W(μ)T V(μ)
(2.14) (2.15)
Cr (μ) = C(μ)V(μ) ∈ Rq×k , Dr (μ) = D(μ) ∈ Rq×p .
(2.12)
An initial condition for the reduced system can then be defined by projecting the initial condition (2.5) as follows: −1 −1 (2.16) xr (0) = W(μ)T V(μ) W(μ)T x(0) = W(μ)T V(μ) W(μ)T x0 ∈ Rk . The reduced system (2.10)–(2.11) corresponds to an oblique projection of the highdimensional system (2.1)–(2.2) onto SV (μ) orthogonally to SW (μ). A transfer function Hr (s; μ) can be defined for the reduced dynamical system as (2.17)
Hr (s; μ) = Cr (μ) (sIk − Ar (μ))−1 Br (μ) + Dr (μ), s ∈ C.
Since Dr (μ) is identical to D(μ), this term can be dropped in the following study without any loss of generality. In the remainder of this paper, R1 (p, k, q) is defined as the set of triplets defining ROMs with an input-space, a state-space, and an output-space of dimensions p, k, and q, respectively. Hence, (2.18) R1 (p, k, q) = (Ar , Br , Cr ) | Ar ∈ Rk×k , Br ∈ Rk×p , Cr ∈ Rq×k . 2.3. Equivalent classes of LTI ROMs. The triplet of reduced-order linear operators (Ar (μ), Br (μ), Cr (μ)) determined by (2.12)–(2.14) defines the ROM in the representative bases (V(μ), W(μ)). The generation of such a ROM through a projection process leads to invariance properties of the reduced system [5]. For instance, the input-output behavior of a ROM is independent from the choice of the bases defining the trial and test subspaces. This property justifies focusing in the remainder of this paper on trial and test bases characterized by orthogonality properties. Indeed, any basis can be transformed into an orthogonal one by, for example, a Gram–Schmidt procedure. These orthogonal bases belong to the compact Stiefel manifold ST (k, n) defined as (2.19)
ST (k, n) = {Φ ∈ Rn×k , such that ΦT Φ = Ik },
where Ik denotes the identity matrix of dimension k. The following proposition demonstrates how the reduced-order operators are transformed when different ROBs are chosen inside the trial and test subspaces. Proposition 2.1. Considering the reduced-order system defined by (2.10)–(2.14), ˜ ˜ and let (V(μ), W(μ)) denote another doublet of trial and test bases defining the same subspaces as the doublet (V(μ), W(μ)). Since all bases considered here belong to the compact Stiefel manifold and as such have orthogonal columns, there exist two orthogonal matrices P and Q in the set of orthogonal matrices of size k, O(k), such that (2.20)
˜ ˜ W(μ) = W(μ)P, V(μ) = V(μ)Q.
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˜ r (μ), B ˜ r (μ), C ˜ r (μ)) denote the ROM defined by the Petrov–Galerkin projection Let (A ˜ ˜ of the HFM (2.1)–(2.2) using (V(μ), W(μ)). Then ˜ r (μ) = QT Ar (μ)Q, A ˜ r (μ) = QT Br (μ), B
(2.21)
˜ r (μ) = Cr (μ)Q. C The proof of this proposition directly follows from the substitution of the new expressions (2.20) of the ROBs in (2.12)–(2.14). The above result is important because it demonstrates that transforming the ROBs using the matrices P and Q results in a congruence transformation of the ROM that is only based on Q. Consequently, one can consider the following group action α1 [29]: (2.22)
α1 : O(k) × R1 (p, k, q) −→ R1 (p, k, q) (Q, (Ar , Br , Cr )) −→ (QT Ar Q, QT Br , Cr Q).
¯ r, C ¯ r , ) belonging to R1 (p, k, q) is then ¯ r, B The orbit of an element (A (2.23)
¯ r, B ¯ r, B ¯ r, C ¯ r = α1 Q, (A ¯ r, C ¯ r ) | Q ∈ O(k) F1 A ¯ r Q, QT B ¯ r, C ¯ r Q) | Q ∈ O(k) , = (QT A
and the corresponding equivalence relation ∼ between elements R1 and R2 of R1 (p, k, q) is (2.24)
R1 ∼ R2 ⇔ ∃Q ∈ O(k) | α1 (Q, R1 ) = R2 .
To summarize, any given ROM can be expressed in a variety of orthogonal bases. The resulting set of expressions of its equivalent ROMs forms its orbit F1 (·). Therefore Nµ are to be compared—as is the case, for example, in when several ROMs {Ri }i=1 an interpolation process—a real challenge is to be able to express these ROMs in consistent sets of generalized coordinates. This is equivalent to finding for each ROM Ri a consistent representative element inside its orbit F1 (Ri ). 2.4. Role of the ROBs. Oftentimes, reduced operators of the form Ar (μ) = (W(μ)T V(μ))−1 W(μ)T A(μ)V(μ) are computed and compared for various values of the parameter vector μ. When the reduced bases do not depend on μ, these can be written in terms of their components as (2.25)
V(μ) = V =
v1 , . . . ,
vk
, W(μ) = W =
w1 , . . . , wk
.
In this case, the comparison of the linear ROMs is straightforward because all operators are written in the same bases, and therefore each entry of a reduced linear operator has the same interpretation for all values of the parameter set. For example, the (i, j)-entry of Ar (μ) is (2.26)
arij (μ) = zTi A(μ)vj ,
where zi denotes the ith column vector of Z = W(VT W)−1 .
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On the other hand, when the bases V(μ) and W(μ) depend on the parameter μ, the ROBs may not correspond to the same generalized coordinates system, and the direct comparison of the linear ROMs may become incorrect. In fact, this may also occur in cases where the subspaces SV (μ) and SW (μ) spanned by the column vectors of V(μ) and W(μ), respectively, do not depend on the vector of parameters μ, as demonstrated by the following example. Example 1. Full-order operators A ∈ Rn×n depending continuously on μ are considered here for two values μl , l = 1, 2, of the parameter vector. Let {e1 , . . . , en } denote the canonical basis of Rn . Each entry aij (μ1 ) = eTi A(μ1 )ej defined for a given parameter value μ1 can be compared to its counterpart aij (μ2 ) since both operators are expressed in the same generalized coordinates system. Consider now two sets of ROBs defined by (2.27) W(μ1 ) = e1 e2 , V(μ1 ) = e1 √12 (e2 + e3 ) and (2.28)
W(μ2 ) =
e1
e2
, V(μ2 ) =
√1 (e2 2
−e1
+ e3 )
.
The projection subspaces associated with the two parameter values are in this case SW (μ1 ) = SW (μ2 ) = span{e1 , e2 } and SV (μ1 ) = SV (μ2 ) = span{e1 , e2 + e3 }. However,
a11 (μ1 ) √12 (a12 (μ1 ) + a13 (μ1 )) (2.29) Ar (μ1 ) = a21 (μ1 ) √12 (a22 (μ1 ) + a23 (μ1 )) and
(2.30)
Ar (μ2 ) =
−a11 (μ2 ) −a21 (μ2 )
√1 (a12 (μ ) 2 2 √1 (a22 (μ ) 2 2
+ a13 (μ2 )) + a23 (μ2 ))
,
which shows that in this case a direct comparison of the entries of the first columns of Ar (μl ), l = 1, 2, is ill-advised. This is a consequence of the fact that the reduced operators are in this case written in different sets of generalized coordinates systems. When the subspaces SW (μ) and SV (μ) depend continuously on the parameter μ (i.e., they are not invariant), it is possible however to interpolate the reducedorder operators as long as they are expressed in consistent generalized coordinate systems. This is illustrated by the following example, in which the entries of the reduced operators depend continuously on the parameters as well, resulting in a welldefined problem for interpolation. Example 2. Let W(μ) = e1 cos( μ 2 )e2 + sin( μ 2 )e3 , (2.31) V(μ) = e1 cos( μ 2 + π4 )e2 + sin( μ 2 + π4 )e3 . In this case, (2.32)
Ar (μ) =
ar11 (μ) ar12 (μ) ar21 (μ) ar22 (μ)
,
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with
(2.33)
ar11 (μ) =a11 (μ), π π ar12 (μ) = cos μ 2 + a12 (μ) + sin μ 2 + a13 (μ), 4 4 √ ar21 (μ) = 2 (cos( μ 2 )a21 (μ) + sin( μ 2 )a31 (μ)) , √ π (cos( μ 2 )a22 (μ) + sin( μ 2 )a32 (μ)) ar22 (μ) = 2 cos μ 2 + 4 π (cos( μ 2 )a23 (μ) + sin( μ 2 )a33 (μ)) . + sin μ 2 + 4
Hence, when A depends continuously on μ, Ar also depends continuously on this parameter vector. This justifies interpolating precomputed operators such as Ar (μl ), l = 1, . . . , Nµ . However, this does not prevent a mode veering or mode crossing phenomenon from occurring. Therefore, attention must be paid to this issue when solving the main problem addressed in this paper, formulated in the following section. 3. Problem formulation. Let N
N
µ µ = {(Ar (μl ), Br (μl ), Cr (μl ))}l=1 {(Arl , Brl , Crl )}l=1
denote a sequence of triplets defining ROMs for several values of the parameter vector μ. The ROMs are assumed to have the same dimension k and to be obtained by Petrov–Galerkin projection of their HFM counterparts as derived in (2.12)–(2.14). The physical systems associated to these HFMs are assumed to share the same topology. Furthermore, it is assumed that the underlying high-dimensional operators as well as the subspaces SW (μ) and SV (μ) depend continuously on μ. However, neither the high-dimensional operators (A(μl ), B(μl ), C(μl )) nor the sets of ROBs W(μl ) and V(μl ) are assumed to be known in the online phase.1 Online problem. Let μNµ +1 = μl , l = 1, . . . , Nµ , define a new value of the paramNµ Nµ = {V(μi )T V(μj )}i,j=1 have been eter set. Assume that the quantities {Pi,j }i,j=1 Nµ precomputed. Interpolate the triplets {(Arl , Brl , Crl )}l=1 to compute a new triplet (ArNµ +1 , BrNµ +1 , CrNµ +1 ) defining a ROM at the new operating point μNµ +1 .2 4. Proposed solution method. The solution method proposed for solving the interpolation problem formulated above is organized around two steps, denoted in the remainder of this paper by Step A and Step B. In Step A, an element of the orbit of each precomputed ROM is chosen so that all elements associated with all sampled values of μ are expressed in consistent sets of generalized coordinates systems. Hence, Step A determines a set of congruence transformations of the precomputed ROMs. In Step B, the transformed ROMs are interpolated to compute a ROM for the target value of the parameter set. Although higher-order linear dynamical systems can be recast in the form of a first-order ordinary differential equation (ODE), a distinction is made here between second-order and first-order systems. The reason is that linear second-order systems— which represent linear oscillators in general and linear structural dynamics systems in 1 Hence, the respective dimensions of the quantities of interest do not depend on the dimension of the high-dimensional state space n, thereby guaranteeing the real-time property of the proposed method. 2P i,j ∀i, j ∈ 1, . . . , Nµ are small-size k×k matrices, and therefore the complexity of any algorithm for solving the online problem should scale with k n.
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particular—are characterized by linear operators with distinct mathematical properties and specialized numerical algorithms. Hence, first-order linear dynamical systems are considered first in section 4.1. Then second-order linear dynamical systems are considered in section 4.2. 4.1. First-order dynamical systems. 4.1.1. Step A: Congruence transformations. As shown in section 2.3, a given ROM can be expressed in a variety of equivalent bases. However, it was explained in section 2.4 why the validity of an interpolation may crucially depend on the choice of the representative element within each equivalent class. Hence, a discriminative criterion must be introduced to choose this representative element. As stated in the formulation of the online problem, it is assumed here that the following matrices associated with the precomputed ROMs are also precomputed: Pi,j = V(μi )T V(μj ).
(4.1)
These matrices provide information about the relative configurations of the precomputed ROBs and the subspaces they define. Therefore, they enable re-expressing the precomputed ROMs into consistent sets of generalized coordinates. Let l0 ∈ {1, . . . , Nµ } designate the parameter set μ = μl0 for which the precomputed ROM Rl0 is chosen as a reference ROM.3 The first idea here is to recognize that Rl differs from Rl0 when l = l0 for two different reasons, namely, μl = μl0 and Rl and Rl0 may not have been precomputed in the same generalized coordinates system. The second idea is to eliminate the second reason by solving the following sequence of minimization problems: (4.2)
min
Rl ∈F1 (Arl ,Brl ,Crl ),l=l0
V(Rl ) − V(Rl0 ) 2F ,
where V(Rl ) ∈ ST (k, n) denotes a possible choice of test basis, that is, a possible choice of generalized coordinates, associated with the ROM Rl ∈ R1 (p, k, q). Hence if Rl belongs to F1 (Arl , Brl , Crl ), there exists a matrix Q ∈ O(k) such that Rl = α1 (Q, (Arl , Brl , Crl )) and (4.3)
V(Rl ) = V (Arl , Brl , Crl ) Q = V(μl )Q.
Therefore, each minimization problem (4.2) can also be written as min V(μl )Q − V(μl0 ) 2F ,
(4.4)
Q∈O(k)
or equivalently as (4.5)
max tr QT V(μl )T V(μl0 ) = max Q, Pl,l0 ,
Q∈O(k)
Q∈O(k)
where (4.6)
M, N = tr(MT N), M, N ∈ Rm×n
denotes the scalar product associated with the Frobenius norm. 3 An
optimal choice of the reference configuration, if it exists, remains an open problem.
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Proposition 4.1. An analytical solution of the maximization problem (4.5) is given by Ql,l0 = Ul,l0 ZTl,l0 ,
(4.7)
where Ul,l0 Σl,l0 ZTl,l0 is an SVD of Pl,l0 . Problem (4.4) is the classical orthogonal Procrustes problem. A derivation of its solution can be found in [30]. In summary, Step A of the ROM interpolation method proposed in this paper is carried out by Algorithm 1 described below. Algorithm 1. Step A Algorithm (identification of congruence transformation). Input: Nµ matrices P1,l0 , . . . , PNµ ,l0 belonging to Rk×k Output: Nµ rotations matrices Q1,l0 , . . . , QNµ ,l0 1: for l = 1, . . . , Nµ do 2: Compute Pl,l0 = Ul,l0 Σl,l0 ZTl,l0 (SVD). 3: Compute Ql,l0 = Ul,l0 ZTl,l0 . ˜ l = α1 Q , (Arl , Brl , Crl ) . 4: Set R l,l0
5:
end for
Step A described above is related to mode tracking procedures based on the modal assurance criterion (MAC) [31]. The MAC between two eigenmodes φ and ψ is defined as MAC(φ, ψ) =
(4.8)
|φT ψ|2 . (φT φ)(ψ T ψ)
When φ and ψ are normalized, MAC(φ, ψ) = |φT ψ|2 . Hence, Pl,l0 is the matrix of square roots of the MACs between the modes contained in V(μl ) and those contained in V(μl0 ). It pertains to the modal assurance criterion square root (MACSR) [32]. For the case of a simple mode crossing exemplified by (4.9)
V(μl0 ) = [v1 , . . . , vi−1 , vi , vi+1 , vi+2 , . . . , vk ]
and (4.10)
V(μl ) = [v1 , . . . , vi−1 , vi+1 , vi , vi+2 , . . . , vk ],
the reader can check that ⎡
(4.11)
MAC V(μl ), V(μl0 ) = Pl,l0
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎤
1 ..
.
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(0) 1 0 1 1 0 1
(0)
..
. 1
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which is simply a permutation matrix. In references [33] and [34], the authors ex ploited the matrix MAC V(μl ), V(μl0 ) in order to address the impact of mode crossing on their applications. More specifically, they permuted the vectors in V(μl ) to obtain a MAC matrix that is the identity matrix, thereby achieving a perfect correlation between the two sets of modes at hand. For the case where the modes depend on the parameter μ, they proposed to explicitly permute the vectors in V(μl ) to obtain a MAC matrix that is as close to the identity matrix as possible. The algorithm associated with Step A of the proposed ROM interpolation method is based on a more general concept than permutation. It is also capable of automatically detecting situations where mode crossing occurs. Indeed, in the case where Pl,l0 is a permutation matrix, Σl,l0 is equal to identity and, as such, Ql,l0 is also equal to the permutation matrix Pl,l0 . As a result, the basis V(μl ) becomes V(μl )Ql,l0 , which corresponds to the desired permutation of the columns in V(μl ). More importantly, however, the present algorithm can also address the more frequent situations where reference modes are not only permuted but also combined as in mode veering phenomena [35]. For such problems, the transformations of the bases do not correspond to simple permutations, but to rotations. Consequently, the procedure proposed in [33, 34] is inapplicable to such problems, as demonstrated in [36]. However, the algorithm associated with Step A of this work performs well in such cases (see section 5.3). 4.1.2. Step B: Interpolation on matrix manifolds. Let M denote a manifold contained in RM×N whose elements are characterized by properties such as nonsingularity, symmetry, positive definiteness, or orthogonality. The interpolation of these elements must be performed on their manifold in order to preserve their distinctive properties. To this effect, this section summarizes a suitable interpolation algorithm that was first proposed in [10]. Let X ∈ M denote a reference point on M and Γ an element of the tangent space TX M at X. Let also Y ∈ M denote a point on M belonging to a neighborhood of X on M. The following mappings can be defined: (i) The exponential mapping ExpX from the tangent space TX M to M. (ii) The logarithm mapping LogX from a subset UX ⊆ M to the tangent space TX M. Such a subset is characterized by the property that for any of its elements Y, the equation ExpX (Γ) = Y has a unique solution Γ satisfying LogX (Y) = Γ. For example, Figure 4.1 graphically depicts the above mappings in the case where the manifold is a circle. In this work, the following matrix manifolds are of interest: (i) The manifold of nonsingular matrices of size k, GL(k), or the manifold of ˜ rl . square matrices of size k, Rk×k , for A ˜ rl . (ii) The manifold of k × p real matrices, Rk×p , for B ˜ rl . (iii) The manifold of q × k real matrices, Rq×k , for C Table 4.1, where exp and log denote the matrix exponential and logarithm, respectively, gives the expressions of their exponential and logarithm mappings. Step B of the ROM interpolation method proposed in this paper is carried out by Algorithm 2 described below. Here, the main idea is to first map all precomputed elements to the tangent space to the matrix manifold of interest at a point using the logarithm mapping, interpolate the mapped data in this linear space, and finally map the interpolated result back to the manifold of interest using the associated exponential map. Remark. In step 5 of Algorithm 2, any preferred interpolation method can be used, as TYi0 M is a vector space.
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Fig. 4.1. Schematic representation of the exponential and logarithm mappings associated with a circle. Table 4.1 Exponential and logarithm mappings for matrix manifolds of interest. Manifold
RM ×N
Nonsingular matrices
SPD matrices
ExpX (Γ)
X+Γ
exp(Γ)X
X1/2 exp(Γ)X1/2
LogX (Y)
Y−X
log(YX−1 )
log X−1/2 YX−1/2
Algorithm 2. Step B Algorithm (interpolation on a manifold M). Input: Nµ matrices Y1 , . . . , YNµ belonging to M Output: Interpolated matrix YNµ+1 1: Choose i0 ∈ 1, . . . , Nµ . {The interpolation process takes place in the linear space TYi0 M} 2: for i = 1, . . . , Nµ do 3: Compute Γi = LogYi0 (Yi ). 4: end for 5: Interpolate each entry of the matrices Γi , i = 1, . . . , Nµ , independently to obtain ΓNµ +1 . 6: Compute Y(μNµ +1 ) = YNµ +1 = ExpYi (ΓNµ +1 ). 0
4.2. Second-order dynamical systems. Second-order parametric ODEs such as those arising from the discretization of parametric linear structural dynamics systems4 can be written as (4.12) (4.13) 4 Such
M(μ)
dx d2 x (t) + D(μ) (t) + K(μ)x(t) = B(μ)u(t), dt2 dt y(t) = C(μ)x(t),
ODEs also arise in the context of electronic systems.
ONLINE INTERPOLATION OF LINEAR PARAMETRIC ROMs
2181
where x(t) ∈ Rn is the time-dependent field vector of interest, M is a symmetric positive definite (SPD) matrix known as the mass matrix, D is an SPD damping matrix, and K is another SPD matrix known as the stiffness matrix. Hence, all three n × n real matrices M, D, and K belong to the matrix manifold SPD(n). The entities B ∈ Rn×p , u ∈ Rp , C ∈ Rq×n , and y ∈ Rq are defined as in section 2.1. The matrices M, D, K, B, and C are parameterized by μ ∈ RNp . Here, (4.12)–(4.13) are orthogonally projected by the Galerkin method onto a subspace S(μ) ∈ G(k, n) spanned by the columns of a full-column rank ROB matrix V(μ) ∈ ST (k, n). Such a projection is a special case of the Petrov–Galerkin method presented in section 2.2, with W(μ) = V(μ). The extension of this approach to a projection based on different test and trial bases for second-order ODEs is straightforward. Then x(t) is approximated as in (2.5), which transforms the dynamical system (4.12), (4.13) into (4.14) (4.15)
Mr (μ)
d2 xr dxr (t) + Kr (μ)xr (t) = Br (μ)u(t), (t) + Dr (μ) dt2 dt yr (t) = Cr (μ)xr (t),
where Mr (μ) = V(μ)T M(μ)V(μ), Dr (μ) = V(μ)T D(μ)V(μ), Kr (μ) = V(μ)T K(μ)V(μ), Br (μ) = V(μ)T B(μ), and Cr (μ) = C(μ)V(μ). The matrices Mr (μ), Dr (μ), and Kr (μ) belong to the manifold of SPD matrices of size k, SPD(k). In the remainder of this paper, R2 (p, k, q) is defined as the set of quintuplets describing second-order ROMs with an input-space, a state-space, and an outputspace of dimensions p, k, and q, respectively. Hence, this can be written as R2 (p, k, q) = (Mr , Dr , Kr , Br , Cr ), | Mr ∈ SPD(k), Dr ∈ SPD(k), (4.16) Kr ∈ SPD(k), Br ∈ Rk×p , Cr ∈ Rq×k . As in the case of first-order systems, a group action α2 is considered, and an orbit is associated to each element (Mr , Dr , Kr , Br , Cr ) ∈ R2 (p, k, q). This can be written as (4.17) ¯ r, D ¯ r, D ¯ r, K ¯ r, B ¯ r, C ¯ r = α2 Q, (M ¯ r, K ¯ r, B ¯ r, C ¯ r ) | Q ∈ O(k) F2 M ¯ r Q, QT D ¯ r Q, QT K ¯ r Q, QT B ¯ r, C ¯ r Q) | Q ∈ O(k) . = (QT M Then, the online problem defined in section 3 can be naturally extended to secondorder systems by substituting the triplets (Arl , Brl , Crl ) belonging to R1 (p, k, q) by the quintuplets (Mrl , Drl , Krl , Brl , Crl ), l = 1, . . . , Nµ + 1, belonging to R2 (p, k, q). Similarly, the interpolation method for ROMs is extended as follows. 4.2.1. Step A: Congruence transformations. Here, the minimization problem to be solved is the same as that formulated for first-order systems—that is, (4.18)
min V(μl )Q − V(μl0 ) 2F .
Q∈O(k)
Therefore, the Step A algorithm outlined in section 4.1.1 is applicable here as is. 4.2.2. Step B: Interpolation on matrix manifolds. In this case, the matrix manifolds of interest are as follows: ˜ rl , (i) The manifold of symmetric positive matrices of size k, SPD(k), for M ˜ ˜ Drl , and Krl .
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˜ rl . (ii) The manifold of k × p real matrices, Rk×p , for B q×k ˜ (iii) The manifold of q × k real matrices, R , for Crl . Expressions of the exponential and logarithm mappings for the above matrix manifolds M are given in Table 4.1, where the notation has the same meaning as in section 4.1.2. Using these expressions, the Step B algorithm (Algorithm 2) described in section 4.1.2 ˜ l ), l = 1, . . . , Nµ . can be applied to interpolate the elements of the sets R(μ 5. Applications. The online method for interpolating linear parametric ROMs proposed in this paper is illustrated here with many examples of increasing complexity. Sections 5.1 and 5.2 focus on sample first-order dynamical problems. Sections 5.3 and 5.4 focus on sample second-order dynamical problems. 5.1. A simple academic problem. This first example problem focuses on an academic single-input single-output system characterized by a single parameter μ ∈ R. Its objective is to highlight Steps A and B of the proposed interpolation method individually. Consider the high-dimensional operators (5.1)
A11 (μ) A12 (μ) b1 (μ) A(μ) = , B(μ) = , C(μ) = c1 (μ) c2 (μ) , A21 (μ) A22 (μ) b2 (μ) where A11 (μ) ∈ R3×3 , A12 (μ) ∈ R3×(n−3) , A21 (μ) ∈ R(n−3)×3 , A22 (μ) ∈ R(n−3)×(n−3) , b1 (μ) ∈ R3 , b2 (μ) ∈ Rn−3 , c1 (μ) ∈ R1×3 , and c2 (μ) ∈ R1×(n−3) . Here, the full statespace dimension n is an arbitrary integer. The matrix blocks associated with the first three coordinates are ⎡ ⎤ 1 (5.2) A11 (μ) = A011 + μA111 ∈ R3×3 , b1 (μ) = ⎣ 0 ⎦ , c1 (μ) = 1 1 1 , 0 where A011 and A111 are randomly generated as ⎡ ⎡ ⎤ ⎤ 0.54 0.86 −0.43 2.77 0.73 −0.21 0.32 0.34 ⎦ , A111 = ⎣ −1.35 −0.06 −0.12 ⎦ . (5.3) A011 = ⎣ 1.83 −2.26 −1.31 3.58 3.03 0.71 1.49 The subspaces spanned by the ROBs are set to be the same as those considered in section 2.4—that is, (5.4)
SW (μ) = span {e1 , cos(μ)e2 + sin(μ)e3 } , π π e2 + sin μ + e3 . SV (μ) = span e1 , cos μ + 4 4
Three Petrov–Galerkin ROMs {(Arl , Brl , Crl )}3l=1 based on the ROBs W(μl ) and V(μl ), l = 1, 2, 3, spanning SW (μl ) and SV (μl ), l = 1, 2, 3, are precomputed for μ1 = 0, μ2 = 0.5, and μ3 = 1. The aforementioned ROBs can be written as W(μl ) = e1 cos(μ)e2 + sin(μ)e3 P(μl ), (5.5) V(μl ) = e1 cos μ + π4 e2 + sin μ + π4 e3 Q(μl ), where P(μl ), Q(μl ) ∈ O(2). For the sake of simplicity, it is assumed here that both P(μ2 ) and Q(μ2 ) are equal to the identity matrix I2 . Therefore, the bases of the reference system are set here to (5.6) e1 cos(μ)e2 + sin(μ)e3 , e1 cos μ + π4 e2 + sin μ + π4 e3 , μ ∈ R.
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ONLINE INTERPOLATION OF LINEAR PARAMETRIC ROMs 4
0.4
3.5
0.2
3
0
1.5
−0.2
br1
2.5
ar 1 2
ar 1 1
?
0.5
2
−0.4
1.5
−0.6
1
−0.8
0
−0.5
0
0
0.2
0.4
μ
0.6
0.8
−1 0
1
0.2
(a) ar11 (μ)
0.4
0.6
μ
0.8
−1 0
1
0.2
(b) ar12 (μ)
0.6
μ
0.8
1
(c) br1 (μ)
7
3
0.4
0.05 ??
6
2
−0.05 5 −0.01
1
0
−0.15
br 2
ar 2 2
ar 2 1
4 3
−0.2 −0.25
2
−1
−0.3 1 −0.35
−2 0
−3 0
0.2
0.4
0.6
μ
0.8
−0.4
−1 0
1
0.2
(d) ar21 (μ)
0.4
0.6
μ
0.8
−0.45 0
1
(e) ar22 (μ)
1.5
0.2
0.4
μ
0.6
0.8
1
(f) br2 (μ)
1.6
1.4
?
1.2
0.5
cr 2
cr 1
1
0
0.8
−0.5 0.6
−1
−1.5 0
0.4
0.2
0.4
μ
0.6
0.8
(g) cr1 (μ)
1
0.2 0
0.2
0.4
μ
0.6
0.8
1
(h) cr2 (μ)
Fig. 5.1. Academic problem: Performance of Step A and Step B of the proposed ROM interpolation method.
The matrices P(μl ) and Q(μl ), l = 1, 3, are randomly generated as
−0.6672 −0.7449 −0.9675 −0.2527 , Q(μ1 ) = (5.7) P(μ1 ) = −0.7449 0.6672 −0.2527 0.9675 and (5.8)
P(μ3 ) =
−0.9864 −0.1645 −0.1645 0.9864
, Q(μ3 ) =
−0.9097 −0.4152 −0.4152 0.9097
.
In Figure 5.1, the entries of the matrices defining the precomputed ROMs are represented by squares. Their counterparts associated with the exact expression of these matrices in the single reference system specified by (5.6) are represented by stars. Their counterparts obtained by applying Step A of the proposed ROM interpolation method to the precomputed ROM matrices are represented by triangles. The superposition of the stars and triangles demonstrate the accuracy of the Step A algorithm. In the same chosen reference system (5.6), the exact variations with μ of the entries of the sought-after ROMs are shown in Figure 5.1 by dotted lines. Their coun-
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DAVID AMSALLEM AND CHARBEL FARHAT
Fig. 5.2. Schematic representation of a mass-damper-spring system. Table 5.1 Parameterized mechanical properties of a mass-damper-spring system. Masses
Dampers
(kg) m1 m2 m3 m4
125 25 5 1
Springs
(Ns/m) c1 c2 c3 c4
μ 1.6 0.4 0.1
(N/m) k1 k2 k3 k4 k5 k6
2 + 2μ 1 3 9 27 1 + 2μ
terparts obtained by the direct application of the Lagrange interpolation method to the precomputed matrix entries are shown by solid lines; therefore, these are found to be very inaccurate approximations. On the other hand, the interpolation on the ˜ rl }3 , {B ˜ rl }3 , and matrix manifolds R2×2 , R2×1 , and R1×2 of the matrices {A l=1 l=1 3 ˜ {Crl }l=1 , respectively, obtained after Step A of the proposed method leads to variations with μ of the matrix entries which, as shown by the dashed lines in Figure 5.1, are in excellent agreement with their exact counterparts; these illustrate the accuracy of the Step B algorithm. 5.2. A mass-damper-spring system. Here, the mass-damper-spring system previously studied in [26] is considered. This system is schematically represented in Figure 5.2. Each of its operating points is defined by a single parameter μ, as shown in Table 5.1, where the masses {mj }4j=1 , springs {kj }6j=1 , and dampers {cj }4j=1 are specified as constants or functions of μ. First, two different predictions of the input-output behavior of this dynamical system are considered for μ = 0.25: (a) that obtained by the HFM of dimension n = 8, and (b) that obtained by a ROM of dimension k = 4, built using the two-sided Krylov moment matching technique described in [5] and referred to in the figures and remainder of this section as the “direct ROM” (meaning that it is computed directly for the operating point of interest). The amplitude Bode plots reported in Figure 5.3 show that the magnitudes of the transfer functions associated with these two models are in excellent agreement in the frequency band [0.01, 1] (note that the variables of both axes for these plots are represented in the logarithmic scale). This conclusion is confirmed by the amplitude Bode plot of the corresponding error system reported in Figure 5.4. Here, an “error system” is defined by its transfer function, which in turn is defined as the difference between the transfer function of the HFM and that of a corresponding ROM.
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ONLINE INTERPOLATION OF LINEAR PARAMETRIC ROMs 40
20
Magnitu de (dB)
0
–20
–40
–60
–80
HFM Direct ROM GPOD-Bas ed ROM Interpo la ted ROM (Panzer et al.) – No Weights Interpo la ted ROM (Panzer et al.) – Weights Interpo la ted ROM (Amsallem and Farhat)
–100 –2 10
–1
0
10
10
1
10
Frequency (rad/sec)
Fig. 5.3. Mass-damper-spring system: Bode plots of the HFM and various ROMs for μ = 0.25. 50
0
Magnitu de (dB)
–50
–100
–150
–200
Direct ROM GPOD-Based ROM Interpolated ROM (Panzer et al.) – No Weights
–250
Interpolated ROM (Panzer et al.) – Weights Interpolated ROM (Amsallem and Farhat)
–300 –2 10
–1
0
10
10
1
10
Frequency (rad/sec)
Fig. 5.4. Mass-damper-spring system: Bode plots of various error systems for μ = 0.25.
In the remainder of this section, the frequency band [0.01, 1] is chosen as the frequency interval of reference for assessing the quality of a ROM for the mass-damperspring system considered herein. Next, the accuracy of the proposed ROM interpolation method is assessed by its application to the numerical prediction of the frequency response of the above mass-damper-spring system—that is, the evaluation of the magnitude of its transfer function—for μ = 0.25, using precomputed reduced-order information for μ1 = 0, μ2 = 0.5, and μ3 = 1. It is also contrasted with the accuracy of the two following ROM approaches applied to the solution of the same problem: (i) A ROM based on global ROBs computed by the global POD (GPOD) method of [12]. More specifically, this ROM is computed by constructing left and
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DAVID AMSALLEM AND CHARBEL FARHAT
right global ROBs of size k = 4 and projecting the HFM assembled at μ = 0.25 onto them. In general, the evaluation of this ROM cannot be performed in real time for two reasons. First, the corresponding HFM needs to be recomputed for the target parameter (here μ = 0.25). Second, the computation of the left and right global ROBs requires the SVD of a matrix whose size scales with that of the HFM. (ii) The weighted ROM interpolation method proposed in [28]. This method too involves in the online phase the SVD of a matrix whose size scales with that of the HFM. Therefore, it is not capable in general of real-time processing. (iii) The nonweighted version of the above method which is amenable to online computations. Table 5.2 summarizes the computational characteristics of algorithms (ii) and (iii), as well as those of the proposed method. Excluding the precomputation of the ROBs and ROMs which is required by all compared algorithms, the computational complexity of the proposed method is approximated by (Nµ −1)k 2 (2n+26k +2(p+q)) operations, and that of the nonweighted algorithm proposed in [28] by Nµ k 2 (6Nµ n + 11Nµ2 k + 2k(2k + p + q)). Both methods require storing Nµ k(k + p + q) words. The weighted algorithm exposed in [28] does not require additional precomputations, but requires storing Nµ k(2n + k + p + q) words. During the online phase, this algorithm incurs, in addition to the interpolation cost, Nµ k 2 (6Nµ n + 11Nµ2 k + 2k(2k + p + q)) operations, while the other two methods do not incur any additional cost. To summarize, only the method proposed in this paper and the nonweighted method proposed in [28] have online capabilities. The proposed method requires, however, less preprocessing overhead. The results reported in Figure 5.3 show that, for this problem, all considered adaptation methods deliver the same excellent performance. Regarding the ROM built using the GPOD technique [12], it should be mentioned, however, that Figure 5.5 reveals that this ROM predicts an erroneous peak of the frequency response of the mass-damper-spring system considered here. Similarly, Figure 5.5 shows that the eigenvalues with the largest real part of all interpolated ROMs match those of the HFM, but the counterpart eigenvalues of the GPOD-based ROM do not. This suggests that ROM interpolation, when properly performed, is a better alternative to a global ROM approach. Finally, the performance of the ROM approaches considered in this section is assessed in the full parameter range μ ∈ [0, 1]. For this purpose, the relative H2 - and H∞ -norms of their error systems are computed and reported in Figures 5.6 and 5.7. The associated average errors in the same norms are reported in Table 5.3. Again, the largest error is observed for the global method. Both considered interpolation methods deliver ROMs whose performances are comparable to that of the directly computed ROM. However, the reader is reminded once again that among these three interpolation methods, only that proposed in this paper and the nonweighted method presented in [28] have an online computational complexity which scales with the size of the ROM and not that of the HFM. The method presented in this paper requires less preprocessing than its counterparts proposed in [28]. Furthermore, this method has a geometric interpretation that the two other methods presented in [28] lack. For the latter methods, the construction of operators of the form (WlT R)−1 Ar,l (RT Vl )−1 depends on the well-posedness of RT Vl and WlT R, which is not necessarily guaranteed. 5.3. A dynamical system exhibiting mode veering. A simple mechanical system presenting nevertheless a mode veering phenomenon [35] is considered here.
ONLINE INTERPOLATION OF LINEAR PARAMETRIC ROMs
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Table 5.2 Offline and online computational steps associated with the considered interpolation algorithms. Weighted method [28]
Nonweighted method [28] Nµ Precompute Vl , Wl
Proposed method
l=1
Nµ Precompute Arl , Brl , Crl
l=1
Offline
Compute
V = V1 · · · VNµ and its thin SVD
phase
V = UΣZT .
Nµ Compute Pl,l0 l=1 . Apply Step A and compute Nµ ˜ rl , C ˜ rl ˜ rl , B . A l=1
Let R = U(:, 1 : k). Compute for l = 1, . . . , Nµ : ˜ rl = A (WlT R)−1 Arl (RT Vl )−1 , ˜ rl = (WT R)−1 Brl , B l ˜ rl = Crl (RT Vl )−1 . C Stored quantities
N
µ {Vl , Wl }l=1 Nµ Arl , Brl , Crl
Nµ ˜ rl , C ˜ rl ˜ rl , B A
l=1
l=1
Choose µNµ +1
Online
phase
Compute Vω = ω1 V1 · · · ωNµ VNµ and its thin SVD Vω = UΣZT . Let R = U(:, 1 : k). Compute for l = 1, . . . , Nµ : ˜ rl = A (WlT R)−1 Arl (RT Vl )−1 , ˜ rl = (WT R)−1 Brl , B l ˜ rl = Crl (RT Vl )−1 . C Interpolate
Nµ ˜ rl , C ˜ rl ˜ rl , B A
l=1
This system consists of two masses coupled by three springs as shown in Figure 5.8. It was previously studied in [37]. It is parameterized by the spring stiffness k1 as all of its other mechanical properties are fixed as follows: m1 = m2 = k2 = 1, k˜ = 0.05.
(5.9)
The mass and stiffness matrices of this system are (5.10)
M=
m1 0
0 m2
, K=
k1 + k˜ −k˜
−k˜ k2 + k˜
.
A reference operating point of this system is chosen as that corresponding to k11 = 0.1. Therefore, the system is reparameterized by (5.11)
μ = k1 − k11 ,
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DAVID AMSALLEM AND CHARBEL FARHAT 8
6
HFM Direct ROM GPOD-Based ROM Interpolat ed ROM (Panzer et al.) – No Weights
4
Interpolat ed ROM (Panzer et al.) – Weights
Imagin ary Part
Interpolat ed ROM (Amsallem and Farhat) 2
0
–2
–4
–6
–8 –0.06
–0.05
–0.04
–0.03
–0.02
–0.01
0
Real Part
Fig. 5.5. Mass-damper-spring system: Eigenvalues of the HFM and various ROMs for μ = 0.25. 0
10
–1
Relative Error
10
–2
10
Direct ROM GPOD-Based ROM Interpolated ROM (Panzer et al.) – No Weights Interpolated ROM (Panzer et al.) – Weights Interpolated ROM (Amsallem and Farhat) –3
10
0
0.1
0.2
0.3
0.4
0.5
μ
0.6
0.7
0.8
0.9
1
Fig. 5.6. Ratio of the H2 -norm of various error systems and H2 -norm of the HFM.
and the domain of variation for μ is chosen as D = [0, k12 − k11 ], where k12 = 3. The eigenvalues of this dynamical system are given by (5.12) ˜ ˜ 2 − (k2 + k)m ˜ 1 )2 + 4m1 m2 k˜2 k1 m2 + k2 m1 + k(m1 + m2 ) ∓ ((k1 + k)m λ1,2 = . 2m1 m2 For the chosen values of the mechanical properties (5.9), the above eigenvalues become μ + 1.2 ∓ ((μ − 0.9)2 + 0.01 (5.13) λ1,2 (μ) = . 2
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ONLINE INTERPOLATION OF LINEAR PARAMETRIC ROMs 0
10
–1
Relative Error
10
–2
10
Direct ROM GPOD-Based ROM Interpolat ed ROM (Panzer et al.) – No Weights Interpolat ed ROM (Panzer et al.) – Weights Interpolat ed ROM (Amsallem and Farhat) –3
10
0
0.1
0.2
0.3
0.4
0.5
μ
0.6
0.7
0.8
0.9
1
Fig. 5.7. Ratio of the H∞ -norm of various error systems and H∞ -norm of the HFM. Table 5.3 Comparison of the errors obtained with various adaptation methods over the μ-interval [0, 1]. Method
Average relative error in the H2 -norm
Average relative error in the H∞ -norm
Direct ROM GPOD-based ROM Interpolated ROM (Panzer et al.) - no weights Interpolated ROM (Panzer et al.) - with weights Interpolated ROM (Amsallem and Farhat)
1.81% 41.0% 2.54%
1.96% 51.7% 2.92%
2.38%
2.69%
1.86%
1.96%
Fig. 5.8. A two-mass three-spring mechanical system.
Figures 5.9 and 5.10 graphically depict the behavior of these eigenvalues when μ is varied in the interval D = [0, 2.9]. In Figure 5.10, the first eigendirection is shown by a full line, and the second one by a dashed line. For the critical value μcr = 0.9—that is, k1 = 1—a mode veering phenomenon can be observed. The modes undergo a rotation of π2 between μ = 0 and μ = 2.9. For μcr = 0.9, the angle of this rotation is equal to π4 , and the eigenvalues of the dynamical system are (5.14)
cr λcr 1 = 1.0, λ2 = 1.1.
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DAVID AMSALLEM AND CHARBEL FARHAT 3.5 3 2.5
λ
2 1.5 1 0.5 0 0
0.5
1
1.5
2
μ
2.5
3
Fig. 5.9. Eigenvalues loci for the two-mass three-spring mechanical system parameterized by μ.
0
0.5
1
1.5
μ
2
2.5
3
Fig. 5.10. Variation of the orientations of the eigenmodes of the two-mass three-spring mechanical system with μ.
Next, the two operating points of the above system defined by (5.15)
μ1 = 0, μ2 = 2.9
are considered and the following corresponding information is precomputed: (i) Diagonalized matrices (and therefore eigenvalues for each operating point)
0.1472 0 , (5.16) Mr (μ1 ) = I2 , Kr (μ1 ) = Λ(μ1 ) = 0 1.0528
(5.17)
Mr (μ2 ) = I2 , K2 (μ2 ) = Λ(μ2 ) =
1.0488
0
0
3.0512
.
(ii) Inner products between the eigenmodes for each operating point
0.0802 0.9968 T . (5.18) P1,2 = X(μ1 ) X(μ2 ) = −0.9968 0.0802
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ONLINE INTERPOLATION OF LINEAR PARAMETRIC ROMs 3.5
3
2.5
λ
2
1.5
1
0.5
0
0
0.5
1
1.5
μ
2
2.5
3
Fig. 5.11. Eigenvalues loci of the systems interpolated by Step B of the proposed method and the complete (Steps A and B) method. Squares designate precomputed points, x’s designate exact eigenvalues, triangles designate eigenvalues obtained by Step B only, and circles designate eigenvalues computed by the proposed method.
The objective is set to numerically predict the variations of the eigenvalues of the two-mass three-spring system when μ is varied in D = [0, 2.9]—that is, to reproduce the results shown in Figure 5.9—by interpolating the precomputed information described above using the method proposed in this paper. Because of the mode veering phenomenon exhibited in D = [0, 2.9], this objective is rather difficult. At this point, it is noted that whereas the precomputed matrices Mr (μl ) and Kr (μl ), l = 1, 2, are not exactly reduced-order matrices, they derive from the classical modal reduction technique. However, no truncation is performed in this case because the considered mechanical system has already a very small size. Therefore, despite the lack of model reduction per se, the problem stated above is a good problem for illustrating the performance of the proposed ROM interpolation method for parametric problems featuring a curve veering phenomenon. More specifically, the interpolation method proposed in this paper and the variant consisting of Step B alone are applied to the solution of the above problem. In both cases, the chosen manifold is SPD(2) because the matrices obtained after Step A of the proposed interpolation method and the precomputed matrices (5.16), (5.17) are SPD. The obtained results are reported in Figure 5.11. The reader can observe that the variations with μ of the eigenvalues computed by the proposed interpolation method are found to be indistinguishable from the exact counterparts. However, those computed by Step B alone are erroneous. For the critical value μcr = 0.9, performing Step B of the proposed interpolation method alone returns
0.427 0 B B (5.19) Mr (μcr ) = I2 , Kr (μcr ) = , 0 1.6730
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Fig. 5.12. Finite element structural model of the wing-store configuration.
which is consistent with the mathematical result established in [21], and stating that the interpolation in a tangent space to the manifold SPD(k) preserves the pattern of a diagonal matrix. On the other hand, performing both Step A and Step B of the proposed interpolation method returns
1.0438 0.0243 AB ˆ ˆ AB (μ ) = I , K (μ ) = . (5.20) M cr 2 cr r r 0.0243 1.0735 As can be expected, Step B of the proposed method alone can neither capture nor address the effect of the veering phenomenon. Therefore, as can be seen from (5.14) and the diagonal entries of KB r (μcr ) (5.19)—or from Figure 5.11—Step B delivers erroneous eigenvalues of the dynamical system at the critical parameter value μcr = 0.9. Step A of the proposed interpolation method chooses the first operating point as reference and returns
0.0802 −0.9968 . (5.21) Q2 = 0.9968 0.0802 This reveals that the proposed interpolation method correctly matches the first and second eigenmodes for the second operating point of the dynamical system with the second and first eigenmodes for its first operating point, respectively. This explains how it captures in this case the veering curve phenomenon. 5.4. Aircraft wing with store configuration. Finally, the following example is chosen to illustrate the application of the proposed ROM interpolation method to a more realistic, larger-scale mechanical system. This system consists of the AGARD Wing 445.6 [38] equipped with a fuel tank (the store). The wing is discretized by 800 composite shell elements, and the store by 1408 shell elements. The two subsystems are connected by two sets of linear multipoint constraints, as illustrated in Figure 5.12. The finite element model of the assembled system has 6834 degrees of freedom and therefore a dimension n = 6834. It is referred to in the remainder of this section as the HFM of this realistic system. The wing-store configuration outlined above is parameterized by the percentage of fill of the store by JP-8 fuel (the fill level). This percentage is denoted here by μ and is varied between 14.6% and 99.9%. It is also referred to in this section as the
ONLINE INTERPOLATION OF LINEAR PARAMETRIC ROMs
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Table 5.4 Variations with the fill level of the first eight eigenfrequencies of the HFM of the wing-store system.
Mode #
14.6% fill
30.9% fill
1 2 3 4 5 6 7 8
11.508 38.312 43.151 55.650 95.833 128.06 156.36 163.06
11.421 35.518 37.935 53.609 93.001 126.35 146.70 162.37
Frequency (Hz) 50.0% fill 69.1% fill 11.293 31.508 33.578 52.482 90.383 124.11 130.06 161.51
11.177 28.162 31.245 51.871 88.478 110.54 125.86 158.57
60
80
99.9% fill 11.048 25.706 29.504 51.668 85.900 91.711 125.49 152.42
180 160
Frequency (Hz)
140 120 100 80 60 40 20 0
20
40
μ (in %)
100
Fig. 5.13. Variations with the fill level μ of the first eight eigenfrequencies of the wing-store system.
operating point. JP-8 is considered to be an incompressible fluid and is modeled by a hydroelastic added mass [35]. As a result, for all possible fill levels, the HFM of the wing-store system has the same number of degrees of freedom and therefore the same dimension, but represents a different dynamic behavior (further details about this computational modeling procedure can be found in [39]). Studying this dynamic behavior is an arduous task because for each new fill level, this requires regenerating a mesh for the region of space enclosed by the tank and occupied by the fuel, and reassembling the HFM with the new added mass. For this reason, the fast prediction of the dynamic behavior of this parametric system without remeshing is desirable. Consequently, the problem is defined here as that of precomputing the eigenvalues of the wing-store structural system for two different operating points and interpolating them to quickly determine their values at three other operating points—that is, three other fill levels. For reference, Table 5.4 reports and Figure 5.13 plots the eigenfrequencies f — that is, the square roots of the eigenvalues divided by 2π—of the wing-store system computed for various fill levels using the HFM. The reader can observe that modes 2 and 3 on one hand and modes 6 and 7 on the other hand see their eigenfrequencies
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Table 5.5 Variations with the fill level of the first eight eigenfrequencies of the wing-store system computed by Step B alone.
Mode 1 2 3 4 5 6 7 8
Frequency (Hz) Relative error with respect to HFM-provided reference solutions shown in parentheses 14.6% fill 30.9% fill 50.0% fill 69.1% fill 99.9% fill 11.508 38.312 43.151 55.65 95.833 128.06 156.36 163.06
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
11.420 35.911 40.552 54.892 93.941 121.14 150.48 161.03
(0.01) (1.11) (6.90) (2.39) (1.01) (−4.13) (2.58) (−0.82)
11.316 33.060 37.465 53.991 91.694 112.92 143.50 158.63
(0.21) (4.92) (11.65) (2.87) (1.45) (−9.02) (10.33) (−1.79)
11.212 30.208 34.378 53.090 89.448 104.69 136.52 156.22
(0.32) (7.27) (10.03) (2.35) (1.09) (−5.29) (8.47) (−1.48)
11.048 25.706 29.504 51.668 85.900 91.711 125.49 152.42
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
getting closer to each other for μ4 = 50%, thereby suggesting the possibility of yet another veering phenomenon. The two configurations to be precomputed are set to be the “end point” configurations defined by μ1 = 14.6% and μ2 = 99.9%. For each of them, eight eigenmodes of the HFM are computed and exploited to construct reduced-order mass and stiffness matrices similar to those described in (5.16) and (5.17), but of dimension k = 8. Each pair of these matrices defines a ROM counterpart of the HFM for the sampled fill level. The inner products between the eigenmodes of the two precomputed operating points are also computed and stored in the matrix P12 of dimension k = 8. The three intermediate operating points are chosen as those defined by the fill levels μ3 = 30.9%, μ4 = 50.0%, and μ5 = 69.1%. As in section 5.3, the interpolation method proposed in this paper and the variant consisting of Step B alone are applied to the solution of the problem formulated above. In both cases, the chosen manifold is SPD(8) for the same reason stated in the case of the problem of section 5.3, but in this case with k = 8. Table 5.5 reports the eigenfrequencies associated with the intermediate operating points computed using only Step B of the interpolation method proposed in this paper and their relative errors with respect to the HFM-provided reference solutions. It also includes the data for the two precomputed operating points. In general, the obtained results are found to be tainted by less than 12% relative error and therefore are acceptable as fast numerical predictions. However, the reader can observe that Step B of the proposed method alone does not reproduce the tendency of the eigenfrequencies of modes 2 and 3 on one hand and modes 6 and 7 on the other hand to get closer for μ4 = 50% than they are for μ1 = 14.6% and μ2 = 99.9%. Table 5.6 reports the eigenfrequencies associated with the intermediate operating points computed by both steps of the proposed interpolation method, their relative errors with respect to the HFM-provided reference solutions, and the similar data for the two precomputed operating points. The accuracy of the obtained results is found to be in general at least as good as that obtained with Step B alone of the proposed method. Interestingly, it can be observed that the relative errors associated with the predictions of the eigenfrequencies of modes 2 and 3 (roughly 10%) are much larger than those for modes 6 and 7 (roughly 1%). This may be due to a strong nonlinear behavior of the second and third eigenfrequencies when μ is varied between μ1 and μ2 that cannot be properly captured using two precomputed operating points only. More importantly, however, whereas Step B of the proposed ROM interpolation method
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ONLINE INTERPOLATION OF LINEAR PARAMETRIC ROMs
Table 5.6 Variations with the fill level of the first eight eigenfrequencies of the wing-store system computed by the proposed ROM interpolation method. Frequency (Hz) Relative error with respect to HFM-provided reference solutions shown in parentheses 14.6% fill 30.9% fill 50.0% fill 69.1% fill 99.9% fill
Mode 1 2 3 4 5 6 7 8
11.508 38.312 43.151 55.650 95.833 128.06 156.36 163.06
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
11.436 38.496 40.524 52.854 95.136 125.25 142.71 159.69
(0.13) (14.67) (6.83) (−1.41) (2.30) (−0.87) (−2.72) (−1.65)
11.340 36.130 38.283 50.907 93.773 121.29 129.07 156.68
(0.42) (8.38) (14.01) (−3.00) (3.75) (−2.27) (−0.76) (−2.99)
11.234 32.346 35.437 50.668 91.722 110.57 124.51 154.53
(0.51) (14.86) (13.42) (−2.32) (3.67) (0.02) (−1.07) (−2.55)
11.048 25.706 29.504 51.668 85.900 91.711 125.49 152.42
0.9
0.9 1
1 0.8
0.8 0.8
0.7
0.6 0.4
0.6
0.2
0.5
MAC Value
0.8 MAC Value
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
0.6
0.7
0.4
0.6
0.2
0.5
0
0
0.4
0.4
8
8 7
0.3
7
0.3 6
6 5
0.2
0.2 4
4 3 2 1 Mode Number
5
1
2
3
4
5
6
7
8
3
0.1
2 1
Mode Number
(a) before Step A transformation
Mode number
1
2
3
4
5
6
7
8
0.1
Mode Number
(b) after Step A transformation
Fig. 5.14. Three-dimensional representation of the MAC values of the bases of natural modes precomputed for μ = 14.9% and μ = 99.9% before (left) and after (right) transformation by the Step A algorithm of the basis precomputed for μ = 99.9%.
alone was found to be incapable of detecting the narrowing of the frequency gaps between modes 2 and 3 on one hand and modes 6 and 7 on the other hand for μ4 = 50%, the results reported in Table 5.6 reveal that the proposed ROM interpolation method with both of its steps reproduces qualitatively this frequency gap narrowing. This is because, as shown in Figure 5.14, the precomputed natural bases are not consistent (there is no reason why they should be consistent a priori), but Step A of the proposed ROM interpolation method transforms the natural basis of dimension 8 associated with the second operating point into one that is almost perfectly consistent with that associated with the first operating point. For convenience, a summary of all of the results discussed above is given in graph form in Figure 5.15. 6. Summary and conclusions. A method for interpolating parametric reducedorder models (ROMs) on matrix manifolds has been presented. This method accounts for the fact that parameterization typically results in ROMs that are naturally constructed in inconsistent reduced-order bases—that is, in different generalized coordinates systems. It requires relatively little storage offline. It is also characterized by a computational complexity which scales with the small size of the ROM. Therefore, it is
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220 200
Frequency (Hz)
180 160 140 120 100 80 60 40 20 0
20
40
60
μ (in %)
80
100
Fig. 5.15. Variations with the fill level μ of the eigenfrequencies of the wing-store system computed by the proposed ROM interpolation method and its Step B alone.
suitable for real-time processing. Like any other interpolation method, the dimension of the parameter space on which it can operate is limited by practical considerations. Therefore for high-dimensional parameter spaces, it should be applied together with a parameter space reduction scheme [40]. The results of the application of this ROM interpolation method to both academic and realistic dynamical systems pertaining to mechanical and aerospace engineering suggest that it is robust, capable of detecting and handling mode crossing and veering, and can deliver in general a good accuracy. REFERENCES [1] T. Lieu, C. Farhat, and M. Lesoinne, Reduced-order fluid/structure modeling of a complete aircraft configuration, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 5730–5742. [2] B.C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26 (1981), pp. 17–32. [3] M. Rathinam and L.R. Petzold, A new look at proper orthogonal decomposition, SIAM J. Numer. Anal., 41 (2003), pp. 1893–1925. [4] L. Sirovich, Turbulence and the dynamics of coherent structures. Part 1: Coherent structures, Quart. Appl. Math., 45 (1987), pp. 561–571. [5] A.C. Antoulas, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control 6, SIAM, Philadelphia, 2005. [6] B.I. Epureanu, A parametric analysis of reduced order models of viscous flows in turbomachinery, J. Fluid. Struct., 17 (2003), pp. 971–982. [7] C. Homescu, L.R. Petzold, and R. Serban, Error estimation for reduced-order models of dynamical systems, SIAM J. Numer. Anal., 43 (2005), pp. 1693–1714. [8] R. Serban, C. Homescu, and L.R. Petzold, The effect of problem perturbations on nonlinear dynamical systems and their reduced-order models, SIAM J. Sci. Comput., 29 (2007), pp. 2621–2643. [9] T. Lieu and C. Farhat, Adaptation of aeroelastic reduced-order models and application to an F-16 configuration, AIAA J., 45 (2007), pp. 1244–1269. [10] D. Amsallem and C. Farhat, Interpolation method for adapting reduced-order models and application to aeroelasticity, AIAA J., 46 (2008), pp. 1803–1813.
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[35] H.J.-P. Morand and R. Ohayon, Fluid-Structure Interaction, Wiley, New York, 1995. [36] G.H.K. Heirman, F. Naets, and W. Desmet, A system-level model reduction technique for the efficient simulation of flexible multibody systems, Internat. J. Numer. Methods Engrg., 85 (2011), pp. 330–354. [37] N.G. Stephen, On veering of eigenvalue loci, J. Vib. Acoust., 131 (2009), 054501. [38] E.C. Yates, Agard standard aeroelastic configurations for dynamic response, candidate configuration I. Wing 445.6, NASA TM-100462, Langley Research Center, Hampton, VA, 1987. [39] E.K. Chiu and C. Farhat, Effects of Fuel Slosh on Flutter Prediction, AIAA Paper 2009-2682, AIAA, Reston, VA, 2009. [40] C. Lieberman, K. Willcox, and O. Ghattas, Parameter and state model reduction for largescale statistical inverse problems, SIAM J. Sci. Comput., 32 (2010), pp. 2523–2542.
