Comput Mech (2015) 56:709–723 DOI 10.1007/s00466-015-1196-4
ORIGINAL PAPER
Surrogate model reduction for linear dynamic systems based on a frequency domain modal analysis T. Kim1
Received: 13 November 2014 / Accepted: 13 August 2015 / Published online: 7 September 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract A novel model reduction methodology for linear dynamic systems with parameter variations is presented based on a frequency domain formulation and use of the proper orthogonal decomposition. For an efficient treatment of parameter variations, the system matrices are divided into a nominal and an incremental part. It is shown that the perturbed part is modally equivalent to a new system where the incremental matrices are isolated into the forcing term. To account for the continuous changes in the parameters, the single-composite-input is invoked with a finite number of predetermined incremental matrices. The frequency-domain Karhunen–Loeve procedure is used to calculate a rich set of basis modes accounting for the variations. For demonstration, the new procedure is applied to a finite element model of the Goland wing undergoing oscillations and shown to produce extremely accurate reduced-order surrogate model for a wide range of parameter variations. Keywords Parameter variation · Reduced-order model (ROM) · Surrogate modeling · POD · Frequency domain · Linear system · Structural dynamics
p q0 q r0 R0 s S u u, v, w U V, W W x x0 x X y α λ
List of symbols Λ A, B, C A0 , B 0 , C 0 AR , B R , C R A, B, C
B 1
Linear system matrices Nominal system matrices Reduced-order system matrices Perturbed incremental system matrices
T. Kim
[email protected] Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore
Λ λ μ φ Φ0 Φ1 Φ
(P × 1) generalized coord. for the second ROSM (R0 × 1) generalized coord. for the nominal ROM (R × 1) generalized coord. for the first ROSM (R0 × 1) statistically uncorrelated signals Fourier transform of r 0 (I × 1) statistically uncorrelated signals Fourier transform of s (I × 1) system inputs nodal displacements Fourier transform of u Right and left eigenvectors of (15) and (19) Fourier transform of the vertical displacement w (N × 1) system states Nominal solution of x Perturbed solution of x Fourier transform of x (L × 1) system outputs Eigenvector of the covariance matrix in (6) Eigenvalue of the covariance matrix in (6) or (15) Diagonal matrix containing eigenvalues of (15) Diagonal matrix containing eigenvalues of (19) Eigenvalue of (19) (H × 1) system parameters (N × 1) mode vector for the first ROSM (N × R0 ) modes set for the nominal system (N × R1 ) modes set for the perturbed system (N × R) modes set for the first ROSM
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ψ Ψ
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(R × 1) mode vector for the second ROSM (R × P) modes set for the second ROSM
1 Introduction In the past two decades, milestones of progress have been made in the field of model reduction as applicable to general areas of engineering encompassing aerospace, mechanical, electrical, nuclear, and chemical engineering. In short, model reduction or reduced-order modeling is achieved by reducing the spatial dimension of large-scaled dynamic systems resulting from finite discretization of continuous systems, e.g., finite element analysis (FEA), computational fluid dynamic (CFD) modeling. These full-order-model (FOM)s, typically in O 105 ∼107 , are impractical to use in daily working environments due to the large computer memories and excessively long CPU times required for running the programs. Moreover, they are unsuitable for design processes such as optimization and control system design due to the complexity of the modeling as well as the uncontrollable size. On the other hand, constructed reduced-order 1 once 2 model (ROM)s in O 10 ∼10 can successfully fulfill the engineering tasks reducing the computational costs significantly and providing compact models for the designing purposes. Kim [1] discusses the variety of model reduction methods available nowadays, especially in aeronautical applications including structural dynamics and aeroelasticity. Despite its major advantages ROM has one serious shortcoming; it normally approximates its parent FOM only for a particular configuration/condition. It cannot account for any changes in the system properties when it undergoes a parameter variation, making it imperative to build a new ROM for the new configuration/condition. Therefore, in recent years researchers have been focusing on this issue trying to make up for the lack of the adaptability in the traditional ROMs. Frangos et al. [2] give a comprehensive overview of all the recent developments in the so called reducedorder surrogate model (ROSM) techniques. ROSM methods broadly fall into two different groups; extended reduced order modeling (EROM) and spanning reduced order modeling (SROM) [3]. The difference between the groups is that the former approximates an updated basis for each change in parameters, while in the latter once the basis vectors are obtained there is no update of the bases as the system parameters change. The EROM requires using Taylor Series expansions and gradients of basis modes [4] which could be computationally expensive. The SROM, on the other hand, does not take into account the local sensitivity of the modes and therefore requires a careful sampling over the parameter space employing, for example, the greedy sampling technique [5] where the maximum error between
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the FOM and ROM models is sought by an optimization through different sample inputs and parameters and the iteration is continued until the desired error value is reached. It is therefore computationally expensive because of the extended amount of sampling of the system response. A third category of the surrogate modeling in reduced dimension is based on multi-dimensional interpolation schemes such as the Grassman manifold [6]. Given a set of ROMs for selected reference conditions, the scheme interpolates the ROMs creating quickly new ROMs for new conditions. Its main limitation is that normally a large number of the reference ROMs are required for the interpolation and the process itself does not give much physical insight overall. In their most recent work, Aguado et al. [7] explored a heat conduction problem using a frequency domain technique in which the reciprocity in the frequency response is utilized and parameter variation in the thermal conductivity is taken into account using the proper generalization decomposition (PGD). In this paper, a novel reduced-order modeling technique for linear systems with parameter variations is presented based on a new frequency domain formulation and use of the proper orthogonal decomposition (POD). For an efficient treatment of parameter variations, the system matrices are divided into a nominal and an incremental part. It will be shown through a matrix manipulation that the perturbed part is modally equivalent to a new system where the incremental matrices are isolated to the right hand side forcing term and no longer appear in the homogeneous part of the equation. Thus, it becomes possible to interpret and analyze the effects and impact of the parameter variations in the context of conventional forced response problems in which the system matrix is that of the nominal system and driven by the incremental matrices. To account for the continuous changes in the system properties, the method of single-composite-input (SCI) [8,9] is invoked with a number of preselected incremental matrices acting as simultaneous inputs. The frequency-domain Karhunen– Loeve (FDKL) procedure [10] is used to calculate the POD modes subject to the SCI. For demonstration, the new procedure is applied to a finite element model of the Goland wing [11,12] and is shown to produce extremely accurate ROSM for a wide range of changes in mass, damping, and stiffness terms of the system. Besides the simplicity and efficiency of the formulation, the new approach does not involve any first-order approximations and is therefore theoretically rigorous to the extent of the use of the SCI. In addition, unlike its predecessors it is cost effective because the computationally expensive oversampling required to cover the parameter space and gradients of mode shapes is avoided and only two set of modes, one for the nominal system, the other for the perturbed systems, need to be obtained.
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optimal modes φ i s are obtained via the so called snap-shot method:
2 Linear system with parameter variations Let us consider a linear dynamic system whose properties are functions of a set of parameters. We will write an equation in state-space form: Linear system with parameters space
M
α ji X ( j) ω j (i = 1, 2, . . . M)
(5)
j=1
where X ( j) ≡ X ω j and α i s are solutions of the reduced dimensional eigenvalue problem,
x˙ (μ, t) = A (μ) x (μ, t) + B (μ) u(t) y (μ, t) = C (μ) x (μ, t)
φi =
(1) 1
1
ω 2 Fω 2 α = λα
(6)
with lower and upper bounds on the parameters where
μi1 ≤ μi ≤ μi2 , nominal (μi ) = μi0 , (i = 1, 2, . . . H ) (2) In the above expression, it is assumed that the parameter changes do not alter the linear dependence on the input.
Fi j ≡ X ∗(i) X ( j) = (M × M) covariance matrix 1 ω 2 ≡ diag ωi = (M × M) frequency weighting matrix
(7)
Objective
Note that ωi ’s include both positive and negative sampling frequencies. Once the POD modes are obtained, the general solution can be approximated by a linear combination of R selected modes as
q˙ (μ, t) = A R (μ) q (μ, t) + B R (μ) u(t)
x(t) ≈
Find basis modes set ≡ φ 1 φ 2 . . . φ R that spans the broad solution space spawned by the parameter variations. Construct the ROSM via Galerkin’s projection:
y (μ, t) = C R (μ) q (μ, t)
(3)
where A R (μ) ≡ T A (μ) (R × R) B R (μ) ≡ T B (μ) (R × I ) C R (μ) ≡ C (μ) (L × R)
(4)
such that (4) accurately reproduces results of the FOM (1) for any combination of the parameters in the range given by (2). Typically, to find the modes we take samples of the response of the system (1) in time or frequency domain, process the output data via the singular value decomposition (SVD) and retain a handful of modes corresponding to the largest singular values. This process known as the Karhunen–Loeve (KL) or POD procedure is well established in both the literature and practice for engineering applications. The following section reviews the frequency domain method briefly.
R
φ i pi (t)
(8)
i=1
It can be shown that when a sufficient number of the POD
2
R
φ i pi
modes are included in (8) the global error x − i=1 becomes the minimum of all possible error bounds and hence (8) proves to be the best possible approximation one can hope for. It is noted that in all of model reductions performed in the work, the FDKL procedure has been used along with the aforementioned SCI method.
4 Full-order expansion of matrices Let’s split the parameters and system equation into the nominal and perturbed parts: μ = μ0 + μ x (μ, t) = x 0 (μ0 , t) + x (μ, t) A (μ) = A0 (μ0 ) + A (μ)
3 Frequency-domain Karhunen–Loeve procedure The FDKL method, also known as the frequency-domain POD, was first formulated by Kim [10] based on the original time-domain version [13] and later it was used for reducedorder modeling of structural dynamics and CFD systems [8,14–16]. Given frequency response of a dynamic system X (ω) at finite sampling frequencies,ω1 , ω2 , . . . , ω M , its
B (μ) = B 0 (μ0 ) + B (μ) C (μ) = C 0 (μ0 ) + C (μ)
(9)
where x 0 , A0 , B 0 , C 0 defined for μ0 satisfy the nominal (or baseline) equation, x˙ 0 = A0 x 0 + B 0 u y0 = C 0 x 0
(10)
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and the perturbed system matrices A, B and C are bounded as per (2):
A such that B = A A− p B. Consequently, one can rewrite Eq. (13) as
A ≤ δ A , B ≤ δ B, , C ≤ δC
X (ω) = ( jω I − A0 − A)−1 A[X 0 (ω)
(11)
+ A− p BU (ω)]
(14)
Expanding the system equation by inserting (9) into (1) yields, after subtracting the nominal Eq. (10),
One can decompose the transfer function as, at a given ω,
x˙ (μ, t) = A0 x (μ, t) + A x (μ, t) + B (μ) u(t)
( jω I − A0 − A)−1 A ≡ V v (ω) Λv (ω) W vT (ω)
(12)
(15)
Obtaining optimal modes set for the nominal system (10) is straightforward. To complete the modes set, however, one needs to calculate additional solutions for the perturbed system (12).
where Λv is a (v × v) diagonal matrix containing nonzero eigenvalues of ( jω I − A0 − A)−1 A and V v , W v are matrices containing the corresponding right and left eigenvectors, respectively. Denoting K (ω) ≡ ( jω I − A0 − A)−1 and applying the Caley–Hamilton Theorem [18] to (15) yields, given an analytic function f (λ),
5 Modally equivalent perturbed system
f (K (ω) A) = V v (ω) f (Λv (ω)) W vT (ω)
It is clear that Eq. (12) offers little insight into the effect of the variation on the dynamics of the perturbed system. Nor is it efficient to use because the right hand side contains x in x making it necessary to combine A back with the nominal A0 ; if one were to sample the response over an ample space of the parameters, it would be necessary to solve the equation repeatedly with new A (μ) , B (μ) each time μ changes. As an alternative, it is worthwhile to examine closely the eigen-structure of transfer function of (12) whose frequency response is given by
Consider the following special function:
X (ω) = ( jω I − A0 − A)−1 [ AX 0 (ω) + BU (ω)] (13) The first term is driven by the nominal solution, while the second is due to the system input. Without loss of generality, we will assume that rank ( A) is a measure of how far the parameter variation propagates in the domain. If rank ( A) ≡ v = N it implies that the propagation is global. Otherwise, it is limited to a sub-domain. As for the incremental input matrix, if the boundary on which the input is applied is affected by the parameter change, B would be nonzero in those rows representing the boundary. However, this implies that A would also have nonzero elements in the same row and column indices. In fact, A may be nonzero in other rows and columns as well because of the broad propagation of the parameter changes at interior nodes. For instance, if an element in a structural stiffness matrix undergoes a change its symmetric element, i.e., the element with row and column indices switched, will also have the same change due to the Maxwell’s Reciprocal Theorem [17]. In this case, it is possible to find a unique pseudo inverse of
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f (λ) ≡ (1 + λ)−1 λ
(16)
(17)
Rewriting (16) with (17) gives [I + K (ω) A]−1 K (ω) A = V v (ω) [I v + Λv (ω)]−1 Λv (ω) W vT (ω)
(18)
Note that K (ω) and I + K (ω) A are invertible whereas K (ω) A may not be. After some matrix algebra, Eq. (18) can be reduced to ( jω I − A0 )−1 A = V v (ω) Λv W vT (ω)
(19)
where v (ω) ≡ [I v + Λv (ω)]−1 Λv (ω) or λi (ω) λi (ω) ≡ (i = 1, 2, . . . v) 1 + λi (ω)
(20)
Thus, the two eigen decompositions in (15) and (19) share the same set of the v eigenvectors for all ω’s, all that which one has to know about the solution space of Eq. (14). Checking the asymptotic behavior of the eigenvalues of (19) it can be observed that lim
λi (ω)→0
λi (ω) = λi (ω) ,
lim
λi (ω)→∞
λi (ω) = 1
(21)
That is, a small λi (ω) maps into itself while a large one approaches unity. Denoting x as the response of the system whose transfer function is (19) replacing the original
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( jω I − A0 − A)−1 , one can express frequency responses of the two systems as X (ω) = V v (ω) Λ (ω) W T [X 0 (ω) + A− p BU (ω)]
≡ V v (ω) z v (ω)
(22) −p
X (ω) = V v (ω) Λ (ω) W [X 0 (ω) + A ≡
V v (ω) z v
T
(ω)
BU (ω)] (23)
Denoting z v ≡ z 1 z 2 . . . z v T , z v ≡ z 1 z 2 . . . z v T , we recognize that each z i (or z i ) represents a weighting for the i −th eigenvector v i . Revisiting Eq. (21) it is clear that z i ≈ z i for lowest eigenmodes, i.e., v i ’s with smallest ||λi (ω)||’s. Hence, the two frequency responses in (22) and (23) approach each other for the modes with smallest magnitudes of eigenvalues, but otherwise are different because of the deviation in the two sets of the eigenvalues. Since X (ω) and X (ω) are linear combinations of exactly the same eigenvectors, they are modally equivalent; using samples of (23) will lead to modes that span the solution space of (14) [or (22)] provided that they contain the participations from all of the important eigenvectors, i.e., the columns in V v that make up the space. This is a very important property of the transfer function because for the purpose of calculating basis vectors of the perturbed system it allows to use X (ω) in lieu of the original X : X (μ, ω) = ( jω I − A0 )−1 [ A (μ) X 0 (ω) + B (μ) U (ω)]
Fig. 1 Solution space S R1 shared by original (left) and modally equivalent (right) perturbed solutions
modes for the system (See Fig. 1 for the graphical conceptualization of the MEPS). More importantly, no other approximations have been made in deriving the new system equation. It should be emphasized that the eigenvectors and eigenvalues used in the derivation of the modal equivalence between (15) and (19) are dynamic in that they are continuous functions of frequency or time. That they are different from the invariant system eigenvalues and vectors, i.e., the solutions of A0 v = λv or ( A0 + A)v = λv, becomes obvious if one converts the frequency domain equations to time domain. Letting V v (t), Λv (t), v (t), W vT denote the inverse Fourier transforms of V v (ω) , Λv (ω) , v (t), W vT (ω), it can be shown that the time domain solutions to (15) and (19) are
whose differential equation, after dropping the prime and making no distinction with the x in the notation, takes the following form: x˙ (μ, t) = A0 x (μ, t) + A (μ) x 0 (t) + B (μ) u(t) (25) Comparing the modally equivalent perturbed system (MEPS) equation (25) with the original (12), we first note that the new perturbed system has the same system matrix A0 and hence the same eigenvectors, eigenvalues as the nominal system (10). Otherwise, the MEPS is different from the original system in several aspects; it is now excited by the variations in the system matrices A and B. By expanding the original equation in x and manipulating the expression for the transfer function, we have isolated and moved A to the driving term of the new differential equation. Using the formula it is now possible to obtain a modal space for the perturbed solution simply by sampling the response of the conventional forced response problem and applying the POD; while it is wrong to use the new equation to get the solution of the original system, it is right to do so to obtain a valid set of basis
Λv (τ2 − τ1 ) W vT (τ1 ) dτ1 dτ2 (26) 0 0
τ2 t T V v (t − τ2 ) v (τ2 − τ1 ) W v (τ1 ) dτ1 dτ2 (27)
(24)
t
0
V v (t − τ2 )
τ2
0
That is, the system responses are expressed in terms of the time convolutions between the dynamic eigenvectors and eigenvalues and hence include all the causal effects occurred from the beginning to the present time. However, since the two systems share the same dynamic eigenvectors V v (t) it can be again concluded that the two responses are spanned by the same set of basis vectors. In fact, such an argument cannot be made using the conventional system eigenvectors, the solutions of A0 v = λv and ( A0 + A)v = λv. Theoretically, one can expect that the modes based on samples of (25) will approximate the solution of the perturbed system for a rich parameter space with any bounds, δ A , δ B , δC ; in a linear system a magnitude of one is the same as a magnitude of a million as long as the directions are the same. Lastly and most importantly, one can interpret, analyze, and gain insight into the impact and effects of the parameter variations in terms of the conventional eigen analysis and well known properties of the forced response problem.
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6 Treatment of discrete matrices: single composite input Recall that in Eq. (25) the driving terms A, B are continuous functions of the parameters μi ’s. In practice, however, it may be intuitive to look into the effects of the changes by taking a few predetermined but representative Ai ’s and B i ’s. The challenge is to know how they would behave to simulate the effects of the continuous variation. Towards this end, we resort to the SCI method [8,9]. The underlying idea is, given a linear system subject to multiple driving inputs one can calculate response-based modes by having all the inputs activated simultaneously with statistically independent signals. When applied to a large-scaled computational model such as a CFD program, the SCI can result in a significant saving in the CPU time because it avoids the expensive input-by-input excitations. The SCI can be implemented either in frequency domain [8,15] or in time-domain [9,19,20]. Although it was originally aimed at reducing the computational cost, here we also seek to use it as a mean to capture the essence of the continuous variation by considering the discrete variations. Let us assume that we have selected a rich ensemble collection of the perturbed matrices: Ai ≡ A μi , B i ≡ B μi , C i ≡ C μi (i = 1, 2, . . . , K )
(28)
Then, any perturbed solution of the MEPS must satisfy for the i − th variation, x˙ i μi , t = A0 x i μi , t + Ai μi x 0 (t) + B i μi u(t)
(29)
In order to simulate the continuous variation, one must let the chosen parameter permutations to take action collectively with their own time functions. More specifically, let us consider the following system driven simultaneously by the multiplicity of the incremental matrices and input signals: K Ai μi x 0i (t) x˙ (μ, t) = A0 x (μ, t) + i=1 K B i μi ui (t) + i=1
(30)
Two comments are in order here. First, x 0i ’s and ui ’s are not to be confused with x 0 , u in Eq. (29), the nominal solution and system input that drives the nominal system. Rather, they are to be interpreted as arbitrary but independent time functions multiplying the different incremental matrix inputs. Second, x is considered a function of μ with the expectation that having the multiple discrete μi ’s act simultaneously would create accumulatively the effect of simulating the continuous parameters. Note that the solution of the nominal
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system would be in general rank deficient, i.e., x 0 ∈ S R0 where R0 < N . In other words, the states in x 0 are confined to remain in the reduced dimensional space. It is therefore desirable to express the nominal solution as a linear combination of basis vectors of the sub-dimensional space S R0 . Otherwise, using Eq. (30) might lead to modes that are redundant and spurious. Let (N × R0 ) 0 represent the reduced set of basis modes for the nominal solution: x0 ≈ Φ0 q0
(31)
Inserting (31) into (30) and using random inputs according to the SCI scheme, we get K Ai μi Φ 0 r 0i (t) x˙ (μ, t) = A0 x (μ, t) + i=1 K + B i μi si (t) (32) i=1
r 0i ≡ (R0 × 1) random uncorrelated signal vector
(33)
si ≡ (I × 1) random uncorrelated signal vector
(34)
Note that r 0i ’s and si ’s themselves are uncorrelated each other.
7 Convergence A main question is now how many Δ Ai ’s and ΔB’s must be included in the right hand side of (32) to produce a rich set of basis vectors that can sufficiently approximate the perturbed system for a broad spectrum of the parameters. Before addressing this question one must be reminded that the equation is essentially a linear system with a finite dimension subject to multiple forcing inputs that consist of the columns of [ A1 0 A2 0 . . . A K 0 ] and the columns of [B i B 2 . . . B K ]. Naturally, these inputs can impact the system in so much as the columns are linearly independent, that is, rank([ A1 0 A2 0 . . . A K 0 ]) and rank([B i B 2 . . . B K ]) are full. In estimating the ranks it helps to understand that the elements of Ai ’s and B i ’s would not vary completely arbitrarily but in a structured manner because of their functional dependency on the parameters. It is reasonable to expect that the ranks will converge as more of the discrete matrices are added in, but when they will converge will depend on how bounded functionally the elements of the matrices are to the parameters. Numerically, estimating the ranks might be nontrivial due to the potentially large size of the system matrices. For all the practical purposes, it seems reasonable to assume that the rank([ A1 0 A2 0 . . . A K 0 ]) would be in the order of R0 . Otherwise, it is upper bounded by max (rank( Ai )) = v which it would approach if all of
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the elements of Ai were to change independently of each other. Also noteworthy from the perspective of linear system theory is that applying different magnitudes of Δ Ai ’s, ΔB i ’s will not change the ranks and therefore will not make qualitative differences on the final POD modes set. This is another advantage of using the MEPS formulation because, although we set out to seek for solutions with certain parameter bounds the modes set obtained based on (32) will cover the full parameter space without such bounds as long as the convergence is met.
8 New algorithm for reduced-order surrogate model The new proposed method for ROSM proceeds as follows. 1. Calculate R0 POD modes 0 ≡ φ 10 φ 20 . . . φ 0R0 for the nominal system:
Fig. 2 Statistically uncorrelated random signals
X (μ, ω) = ( jω I − A0 )−1 K × Ai μi 0 R0i (ω) i=1
x˙ 0 = A0 + B 0 u y0 = C 0 x 0
To account for the multiple system inputs u i (t)’s and minimize the computation, execute the following equation with the SCI: x˙ 0 = A0 +
+
(35)
I
b0i ri
(36)
i=1
where b0i ’s are the columns of B 0 and ri (t)’s are statistically uncorrelated time signals. The POD calculation can be conveniently done using the FDKL procedure and the frequency domain formula of (36): X 0 (ω) = ( jω I − A0 )−1
I
b0i Ri (ω)
(37)
i=1
Any statistically uncorrelated real numbers can be assigned toRi (ω). For instance, these numbers can be generated easily on MATLAB (see Fig. 2). Alternatively, assumed modes could be used in lieu of the POD modes, especially in the field of structural dynamics based on the Rayleigh-Ritz Method [17]. 1 2 2. Next, calculate R1 POD modes 1 ≡ φ 1 φ 1 . . . φ 1R1 for the MEPS (32) with a simultaneous excitation of a finite number of matrix variations, with the nominal generalized coordinates acting as the new driving inputs. As before, the POD calculation can be done in the frequency domain:
K
B i μi S i (ω)
(38)
i=1
Note that, as in the case of the nominal system, only one execution of the computation is necessary since the multiple driving inputs are applied at the same time. As discussed earlier, checking rank([ A1 0 A2 0 . . . A K 0 ]) and rank([B i B 2 . . . B K ]) will help determine K . 3. Once 0 and 1 are calculated, the total R POD modes set is ≡ [ 0 1 ]
(39)
Alternatively, one can calculate from the compo(M) (1) (2) site samples of X 0 X 0 . . . X 0 X (1) X (2) . . . X (M) by the FDKL procedure. Recall, however, X (ω) = X 0 (ω) + X (ω). R i Finally, expanding x (μ, t) ≈ i=1 φ qi (μ, t) and executing the Galerkin’s projection leads to the desired ROSM: q˙ (μ, t) = A R (μ) q (μ, t) + B R (μ) u(t) y (μ, t) = C R (μ) q (μ, t)
(40)
where A R (μ) ≡ T A (μ) (R × R) B R (μ) ≡ T B (μ) (R × I ) C R (μ) ≡ C (μ) (L × R)
(41)
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3. (Second ROSM) Although the ROM (40) is valid for the broad parameter space for which it was obtained, it is not optimal for a specific combination of the parameters, μs . If desired, however, it can be further reduced by performing the FDKL to (40) with μ ≡ μs . Assuming that we have obtained a secondary modes set
≡ ψ 1 ψ 1 . . . ψ P for approximation of q μs , t ≈ P i i=1 ψ pi μs , t , the ROSM can be reduced to: p˙ μs , t = A P μs p μs , t + B P μs u(t) y μs , t = C P μ s p μ s , t
(42)
where A P μs ≡ T A R μs (P × P) B P μs ≡ T B R μs (P × I ) C P μs ≡ C R μs (L × P)
(43)
bar elements connecting and with nonuniformly distributed concentrated masses at the grid points. The skins (shells) and bars provide torsional and bending rigidity. 40 shell elements (20 top, 20 bottom) are used resulting in total of 180 degrees of freedom. For details of the modeling, see Meyer [11]. For the purpose of the surrogate model reduction, the wing is divided into two zones, one quarter of the structure consisting of the areas 12, 14, 16, 18, 20, and the rest. The first zone is subject to variations in all of the parameters except for the geometry of the structure, whereas the second zone retains its nominal properties. A structural damping matrix B is also introduced, given a structural damping coefficient g, by the approximate formula for proportional damping [21]: B = αM + β K
Hence, x(t) is obtained by two back transformations, followed by :
where
x μs , t ≈ p μs , t
α=
(44)
9 Numerical results and discussion 9.1 Finite element model of the Goland wing For demonstration, the new ROSM scheme is applied to a finite element model of the Goland wing [11,12] that undergoes vibrations in bending and torsional degrees of freedom (Fig. 3). The wing herein is derived from the “heavy” version of the original Goland wing structurally represented by a box structure but with additional nonstructural mass. It consists of two layers of shell elements plus Fig. 3 Goland wing finite element model subject to a point load P and parameter variation in areas 12, 14, 16, 18, 20
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gω1 ω2 ω1 + ω2 g β= ω1 + ω2 ω1,2 ≡ min. and max. natural frequencies
(45)
(46)
For calculation of the POD modes a point load is applied at the forward edge of 20 % of the span from the root as shown in Fig. 3, and frequency responses of the three displacements u, v, w at all nodes are obtained in the frequency domain. It is mentioned that since the structural dynamic equations are second order in time, the modes calculations can be done more economically using a variation of the nominal and perturbed equations of the first-order formulation, (37) and (38). See Appendix for details of the formulation.
P
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For a validation of the finite element modeling, natural frequencies of the first four natural modes are computed and compared to the results reported previously [22]. See Table 1 and Fig. 4. It can be seen that comparison of the natural frequencies between the present and the previous results is excellent. As stated earlier, the wing is subject to parameter variations in the one quarter of the area. Since the finite element modeling allows sectional properties to vary it is possible to reflect the parameter changes by updating K , M, B matrices accordingly. The four parameters in this zone considered to vary are
Table 1 Natural frequencies of the cantilevered beam ω (Hz)
1st
2nd
3rd
4th
...
Present
1.98
4.05
9.69
13.49
...
Ref. [22]
1.97
4.05
9.65
13.40
...
9.2 Nominal parameter values and bounds Nominal specs of the Goland wing are given as follows: l ≡ span = 20 ft c ≡ chord = 4 ft
μ ≡ ρ E ν g T
t ≡ thickness = .33ft ρ ≡ density = .0001 slugs/ft 3
(47)
The nominal or baseline coefficients selected are (as given above)
E ≡ Young’s Modulus = 1.4976 × 102 slugs/ft2 ν ≡ Poisson Ratio = .33
μ0 ≡ ρ0 E 0 ν0 g0 T = .0001 1.4976 × 102 .33 .03 T
g ≡ structural damping coefficient = .03
0.25
Vertical Displ. (ft)
Vertical Displ. (ft)
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25
0
0 2
2 0
5
10
15
20
0
4
5
Mode 1: 1.98 Hz
15
20
4
Mode 2: 4.05 Hz
0.25
0.25
0.2
0.2
0.15
0.15
Vertical Displ. (ft)
Vertical Displ. (ft)
10
0.1 0.05 0 −0.05 −0.1 −0.15
0.1 0.05 0 −0.05 −0.1 −0.15
−0.2
−0.2
−0.25
0
−0.25
0 2
2 0
5
10
15
20
4
Mode 3: 9.69 Hz
0
5
10
15
20
4
Mode 4: 13.49 Hz
Fig. 4 Vibrational modes of Goland wing with nominal parameter μ0
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used), which produces total of 100 modes, 50 for the positive frequencies, 50 for the negative frequencies. The covariance matrix in the FDKL procedure yields 24 linearly independent eigenvectors hence when all of them are taken 0 becomes (180 × 24) (R0 = 24). It must be noted that the sampling range is arbitrary and up to the user but there must be enough sample points such that the frequency responses are always oversampled within the selected range. That is, the rank of the covariance matrix should be less than or equal to twice the number of the frequency samples. Otherwise, it will lead to a set of POD modes that are not complete or inaccurate. In Ref. [23], an iterative technique was implemented to guarantee this condition for the oversampling while keeping the number of frequency samples to a minimum. Figure 5 shows eigenvalues and frequency response of the vertical displacement at the mid wing tip for the nominal FOM and ROM. As expected, the two models match extremely well in the sampling range. Next, the mass, damping, stiffness in the noted quarter of the wing are changed from the nominal values to see the effects of the parameter variations. For the calculation of POD modes φ i1 ’s the MEPS, Eq. (53) (or Eq. (38) if the stat-space format is used) is driven by a simultaneous excitation of a few parametrically varied system matrices. Table 3 contains eight such combinations of the parameters for the possible choices. Note that the last case has severe ranges of changes that are outside the bounds specified in Table 2. Figure 6 is a plot of rank (R B K i ) ¯ 1 0 K¯ 1 0 B ¯ 2 0 K¯ 2 0 . . . B ¯ i 0 ≡ rank([ B K¯ i 0 ]) versus the number of the discrete matrices. It is ¯ i ’s clear that the rank converges after including a few B ¯ ¯ ¯ and K i ’s. Hence, six sets of B i , K i resulting from
Table 2 Parameters in areas 12, 14, 16, 18, 20: mean values and bounds E slugs/ft 2 ν g Parameter ρ slugs/ft 3 Nominal
.0001
1.4976 × 102
.33
.03
Upper bound (%)
+40
+60
+30
+60
Lower bound (%)
−60
−40
−40
−20
Note that the structural damping coefficient g is assumed to affect the entire structure. Lower and upper bounds on these nominal parameters considered in the current study are given in Table 2 (numbers in parenthesis are percentage changes from the nominal values). The discrete matrices used in the calculations of POD modes are all within these bounds, hence the resulting SROMs are expected to perform well within the bounds. 9.3 Surrogate ROMs: results As seen in the Fig. 3, there are 40 shell elements on the top and bottom sides of the wing resulting in total of 180 degrees of freedom for the three displacements defined at each node. These variables exclude 18 displacements that are fixed at the root. Hence, M, B, and K are (180 × 180), f is (180 × 1). First, POD modes are calculated for the nominal system according to the Step 1. described in the previous section. The frequency range of interest is chosen as (0, 500) rad/sec. The two frequency components used in the structural damping approximation (46) are ω1 = 12.4, ω2 = 500 rad/sec, respectively. Total of 50 sampling frequencies evenly placed in this range are used to get the frequency responses of the system (51) (or Eq. (37) if the first-order state-space format is 4
2
x 10
−4
x 10
1.5
FOM (μ , 180)
wmag.
0
1
ROM (μ , 24)
2
0
1
Imag[λ]
0.5 0
FOM (μ , 180) 0
0
0
10
20
30
40
50
60
70
80
50
60
70
80
Freq.(Hz)
ROM (μ0, 24)
200
wphase
−0.5
−1
100 0 −100
−1.5
−200
−2 −2
−1.5
−1
Real[λ]
−0.5
0
0
10
20
30
40
Freq.(Hz)
4
x 10
Fig. 5 FOM versus ROM with nominal parameter μ0 : eigenvalues (left), frequency response (right) of mid w at 100 % span
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719
Table 3 Percentage variations in parameters in areas 12, 14, 16, 18, 20 Variation
ρ (%)
E (%)
ν (%)
g (%)
μ1
+40
−40
−30
+20
μ2
−40
+40
0
−20
μ3
−30
+40
+30
0
μ4
−40
+20
−40
+20
μ5
+30
−30
+10
−10
μ6
−10
+30
−20
+10
μ7
−60
+60
+30
+60
μ8
+80
−50
+40
−40
Structural damping g affects all areas
100
90
Rank(RKBi)
80
70
60
50
40
1
2
3
4
5
6
7
8
μ Index i
Fig. 6 rank (R B K i ) versus number of μi ’s when the parameters are varied in elements 12, 14, 16, 18, 20
the first six variations μ1 , μ2 , ..., μ6 are used in the SCI. Using the six sets of the discrete matrices generates total of 6 × 24 = 144 input columns in the MEPS (53) requiring 144 random signals, or equivalently, six sets of R0i ’s in (38). If preferred the MEPS could be run six times with the individual sets of the discrete matrices with just 24 input columns each, but the point is that by using the SCI a great deal of CPU time can be saved. As before, 50 evenly placed sampling frequencies in (0, 500) rad/sec are used. The FDKL procedure with SCI described earlier produces 49 linearly independent φ i1 ’s yielding (180 × 49) 1 . Thus, total of 73 linearly independent modes are available to build ROSMs accounting for all the parameter changes consid¯ i ’s and ered. It is reported that other randomly generated B ¯ K i ’s were also tried and not surprisingly they all led to the same ranks yielding the same and hence the same ROSM. All of the comparisons against FOMs are made using the second ROSMs that are obtained following the procedure described in the previous section. The sizes of these six ROMs are (25 × 25), (23 × 23), (23 × 23), (23 × 23), (24 × 24), (24 × 24) for μ1−6 , respectively. In order to check the efficiency of the modes in capturing parameters other than the six cases used in the SCI, two additional SROMs for the remaining μi ’s (i = 7, 8) are also generated. The sizes of the additional ROMs are (22 × 22), (27 × 27), respectively. Figure 7 is plots of eigenvalues for the two parameter combinations μ1 and μ8 . It is seen that all eigenvalues of the ROMs are stable staying in the left hand side of the complex domain. As a measure of the accuracy of the ROMs the following accumulative error is defined and estimated for all of the frequency response components of the vertical displacement w:
4
4
2
x 10
2.5
x 10
2
1.5
1.5 1 1
FOM (μ )
Imag[λ]
Imag[λ]
0.5
1
0
ROM (μ , 25) 1
FOM (μ )
0 −0.5
0
−0.5
0.5
FOM (μ ) 8
ROM (μ , 27) 8
FOM (μ ) 0
−1 −1 −1.5 −1.5 −2 −2
−2
−1.5
−1
Real[λ]
−0.5
0
−2.5 −2
−1.5
4
x 10
−1
Real[λ]
−0.5
0 4
x 10
Fig. 7 Eigenvalues for parameters μ1 (left) and μ8 (right) where the ROMs are constructed based on μ1−6
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ROMs based on μ1−6
7
ROMs based on μ
ROMs based on μ1,3,6 3
Max(ε ) (%)
6
i
5 4 3 2 1 0 0
2
4
6
8
μi Index
Fig. 8 Maximum cumulative errors occurring in any of w frequency responses where the ROMs are constructed based on μ1−6 , μ1,3,6 , and μ3 , respectively
M k=1 |[Wi
εi ≡
(ωk )] F − [Wi (ωk )] R |2
M k=1 |[Wi
than .14 %. In fact, within the bounds (case 1 through case 7) all the errors are less than .06 % no matter how many discrete system matrices are used in the SCI. Based on this comparison, one can conclude that the errors are not too sensitive to the number of the discrete matrices and including only ¯ i , K¯ i ) leads to a ROM that is sufficiently one set of ( B accurate for most variations and reasonably fine for extreme variations. Figure 9 through Fig. 10 are frequency responses of the lead w at 20 % and mid w at 100 % span locations for two selected parameters, μ1 and μ8 , showing an excellent agreement between the FOMs and ROMs as confirmed by the error estimates in Fig. 8. In particular, despite the significant deviation from the nominal case, the frequency response plots for μ8 match very well with the FOM. As speculated earlier, as far as rank (R B K i ) and rank R f i are full there seems to be no limit in the extent of the parameter variation that the new ROSM cannot account for. Figure 11 shows transient responses of lead w at 20 % and mid w at 100 % span locations for the case 4 subject to a step forcing input. Once again, despite the large amount of deviation from the nominal parameters, the time histories are in a great agreement between FOM and ROM.
(ωk )] F |2
(i = 1, 2, . . . , N )
(48)
Figure 8 illustrates maximum cumulative errors occurring in any of the 60 frequency responses for all the 8 parameter variations examined. Note that three different cases where the ROMs are constructed based on μ1−6 , μ1,3,6 , and μ3 , respectively, are illustrated. As expected, in general the less number of the discrete matrices are used in the SCI the less accurate the resulting ROM becomes. However, except for the case 8 with the second and third ROM all the errors are less
9.4 ROSM based on nominal POD modes Finally, to demonstrate how poorly the nominal POD (or any other) modes could approximate the FOM undergoing a parameter variation, a ROM is built for the case 8 using the nominal modes 0 , i.e., ≡ 0 and after the second reduction the ROM becomes (24 × 24). Figure 12 shows frequency responses of lead w at 20 % and mid w at 100 % span locations. From the figure, it is evident that using the nominal modes instead of the surrogate modes could lead to −4
−4
x 10
x 10
1
1
ROM (μ1, 25) FOM (μ0)
1
0
wmag.
mag.
w
FOM (μ )
FOM (μ )
2
0
10
20
30
40
50
60
70
FOM (μ0) 1 0
80
ROM (μ1, 25)
2
0
10
20
30
200
−50
100
wphase
0
−100
50
60
70
80
50
60
70
80
0 −100
−150 −200
40
Freq.(Hz)
w
phase
Freq.(Hz)
0
10
20
30
40
Freq.(Hz)
50
60
70
80
−200
0
10
20
30
40
Freq.(Hz)
Fig. 9 Frequency responses of lead w (left) at 20 % and mid w (right) at 100 % span for μ1 where the ROM is constructed based on μ1−6
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721 −4
−4
x 10
4
6
x 10
FOM (μ )
FOM (μ ) ROM (μ , 27)
wmag.
8
FOM (μ )
2
0
w
mag.
8
8
3
ROM (μ , 27)
4
8
FOM (μ ) 0
2
1 0
0 0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
50
60
70
80
Freq.(Hz) 200
−50
100 phase
0
0
w
−100
w
phase
Freq.(Hz)
−100
−150 −200
−200 0
10
20
30
40
50
60
70
80
0
10
20
Freq.(Hz)
30
40
Freq.(Hz)
Fig. 10 Frequency responses of lead w (left) at 20 % and mid w (right) at 100 % span for μ8 where the ROM is constructed based on μ1−6 0.06
0.07
FOM (μ4)
FOM (μ4)
0.06
ROM (μ4, 23)
0.05
ROM (μ4, 23)
0.05 0.04
w (ft)
w (ft)
0.04 0.03
0.03 0.02
0.02 0.01 0.01 0 0
0
2
4
6
8
Time (sec)
−0.01
0
2
4
6
8
Time (sec)
Fig. 11 Time histories of lead w (left) at 20 % and mid w (right) at 100 % span for μ4 subject to a step force input where the ROM is constructed based on μ1−6
the ROMs whose results can deviate significantly from those of the FOMs.
10 Conclusions In this work, a novel reduced-order surrogate modeling methodology for linear systems has been explored based on a new frequency domain formulation and use of the POD method. Its goal is, given a range of parameter values, to obtain a rich set of basis modes that can adaptively project the solutions into a sub-dimensional space accounting for all the possible changes, thereby bypassing the expensive and time-consuming parameter-by-parameter calculations of the modes. Two sets of basis vectors are obtained separately for the nominal and perturbed parts and are collected together to
form a total bases set. It was shown that the perturbed equation can be transformed into the so called MEPS in which the incremental matrices are separated into the forcing term and do not show up in the homogeneous equation. To account for the continuity in the parameter variation, the SCI is invoked with a few predetermined incremental matrices acting as simultaneous inputs. The FDKL procedure is then employed to calculate the POD modes in the frequency domain. Since it allows all the inputs to act simultaneously, the full-order system needs to be executed only once rather than multiple times saving a significant computing time during the modes calculation. The new algorithm does not involve any first-order approximations and therefore its formal derivation is valid to the extent that the MEPS is represented with a rich modal participation from all of the important eigenvectors that make up the solution space of the original system, and the use of the
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Comput Mech (2015) 56:709–723 −4
−4
x 10
ROM (μ8, 24)
2
w
mag.
FOM (μ8)
FOM (μ8)
3
1 0
x 10
6
wmag.
4
0
10
20
30
40
50
60
70
8
2 0
80
ROM (μ , 24)
4
0
10
20
30
200
−50
100
wphase
0
−100
50
60
70
80
50
60
70
80
0 −100
−150 −200
40
Freq.(Hz)
w
phase
Freq.(Hz)
0
10
20
30
40
50
60
70
80
−200
0
10
20
30
Freq.(Hz)
40
Freq.(Hz)
Fig. 12 Frequency responses of lead w (left) at 20 % and mid w (right) at 100 % span for μ8 where the ROM is constructed based on the nominal μ0
SCI. Most importantly, with the new formulation it becomes possible to analyze and interpret the impact and effect of the parameter variation using the traditional eigenmode/value and forced response analysis. The proposed method has been applied to a finite element model of the Goland wing undergoing bending and torsional oscillations, and shown to produce extremely accurate ROSM for a wide range of the variations in the mass, damping, and stiffness. Through this application, it was demonstrated that the convergence of the ROM can be guaranteed by assuring the convergence of the rank of the incremental matrices used in the SCI. Obviously, the more parameters are varied the more pronounced their effects will be, but it can be also handled by following the convergence criterion established earlier in Sect. 7. These features should be very appealing for future engineering applications where reduced-order modeling with parameter variations in a simple but efficient manner is in great need. The main object of the present paper has been the theoretical development with the application to a representative aeronautical system. For its full validation, applications to large-scaled systems such as Finite Element models of realistic structural dynamic systems should be demonstrated with a detailed estimation of the convergence and errors. Acknowledgments The author is grateful to Yu Qijing, a graduate research assistant, who kindly prepared the Goland wing finite element model using NASTRAN.
Expanding the solution and system matrices in nominal and perturbed parts μ = μ0 + μ x (μ, t) = x 0 (μ0 , t) + x (μ, t) M (μ) = M 0 (μ0 ) + M (μ) B (μ) = B 0 (μ0 ) + B (μ) K (μ) = K 0 (μ0 ) + K (μ)
(50)
it can be shown that inserting (50) into (49) and following the previous modal analysis results in the following nominal, perturbed, and MEPS equations analogous to the equations (10), (12), and (25): Nominal system ¯ 0 x˙ 0 + K¯ 0 x 0 = ¯f 0 x¨ 0 + B
(51)
Perturbed system ¯ ( x˙ 0 + x) + K¯ (x 0 + x) ¯ 0 x˙ + K¯ 0 x + B x¨ + B = ¯f (52) MEPS ¯ x˙ 0 + K¯ x 0 = ¯f ¯ 0 x˙ + K¯ 0 x + B x¨ + B
(53)
Appendix: MEPS for structural dynamic system where Given a structural dynamic equation of motion with mass, damping, and stiffness matrices subject to changes in parameters: M (μ) x¨ + B (μ) x˙ + K (μ) x = f
123
(49)
¯ 0 ≡ M −1 B 0 , B 0 K¯ 0 ≡ M −1 0 K 0, ¯f 0 ≡ M −1 f , 0
¯ ≡ M −1 B − B ¯0 B K¯ ≡ M −1 K − K¯ 0 ¯f ≡ M −1 f − ¯f 0
(54)
Comput Mech (2015) 56:709–723
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