Componentbased reduced basis for parametrized symmetric eigenproblems
 Sylvain Vallaghé^{1, 4}Email author,
 Phuong Huynh^{1, 3},
 David J Knezevic^{1, 2, 3},
 Loi Nguyen^{3} and
 Anthony T Patera^{1}
https://doi.org/10.1186/s4032301500210
© Vallaghéet al. 2015
Received: 17 October 2014
Accepted: 14 January 2015
Published: 23 May 2015
Abstract
Background
A componentbased approach is introduced for fast and flexible solution of parameterdependent symmetric eigenproblems.
Methods
Considering a generalized eigenproblem with symmetric stiffness and mass operators, we start by introducing a “ σshifted” eigenproblem where the left hand side operator corresponds to an equilibrium between the stiffness operator and a weighted mass operator, with weightparameter σ>0. Assuming that σ=λ _{ n }>0, the nth real positive eigenvalue of the original eigenproblem, then the shifted eigenproblem reduces to the solution of a homogeneous linear problem. In this context, we can apply the static condensation reduced basis element (SCRBE) method, a domain synthesis approach with reduced basis (RB) approximation at the intradomain level to populate a Schur complement at the interdomain level. In the Offline stage, for a library of archetype subdomains we train RB spaces for a family of linear problems; these linear problems correspond to various equilibriums between the stiffness operator and the weighted mass operator. In the Online stage we assemble instantiated subdomains and perform static condensation to obtain the “ σshifted” eigenproblem for the full system. We then perform a direct search to find the values of σ that yield singular systems, corresponding to the eigenvalues of the original eigenproblem.
Results
We provide eigenvalue a posteriori error estimators and we present various numerical results to demonstrate the accuracy, flexibility and computational efficiency of our approach.
Conclusions
We are able to obtain large speed and memory improvements compared to a classical Finite Element Method (FEM), making our method very suitable for large models commonly considered in an engineering context.
Keywords
Eigenproblems Domain synthesis Reduced basis A Posteriori error estimationBackground
In structural analysis, eigenvalue computation is necessary to find the periods at which a structure will naturally resonate. This is especially important for instance in building engineering, to make sure that a building’s natural frequency does not match the frequency of expected earthquakes. In the case of resonance, a building can endure large deformations and important structural damage, and possibly collapse. The same considerations apply to automobile and truck frames, where it is important to avoid resonance with the engine frequencies. Eigenproblems also appear when considering wind loads, rotating machinery, aerospace structures; in some cases it is also desirable to design a structure for resonance, like certain microelectromechanical systems.
With improvement in computer architecture and algorithmic methods, it is now possible to tackle largescale eigenvalue problems with millions of degrees of freedom; however the computations are still heavy enough to preclude usage in a manyquery context, such as interactive design of a parameterdependent system. In this paper, we present an approach for fast solution of eigenproblems on large systems that present a componentbased structure – such as building structures.
For the numerical solutions of partial differential equations (PDE) in componentbased systems, several computational methods have been introduced to take advantage of the componentbased structure. The main idea of these methods is to perform domain decomposition, and to use a common model order reduction for each family of similar components. The first and classical approach is the component mode synthesis (CMS) as introduced in [1,2]: it uses the eigenmodes of local constrained eigenvalue problems for the approximation within the interior of the component and static condensation to arrive at a (Schur complement) system associated with the coupling modes on the interfaces or ports. One drawback of the CMS approach is the rather slow convergence of eigenmodal expansions. In contrast the reduced basis element (RBE) method [3] employs a reduced basis expansion [4] within each component or subdomain and Lagrange multipliers to couple the local bases and hence compute a global solution of the considered parameter dependent partial differential equation for each admissible parameter. The RBE method thus profits from the fact that RB approximations yield a rapid and in many cases exponential convergence [5].
A combination of RB methods and domain decomposition approaches has for instance also been considered in [6,7]. Similarly RB methods have been employed in the framework of a multiscale finite element method to construct local reduced spaces for the approximation of finescale features on the coarse grid elements in [8,9], where the latter corresponds to the “components” in the RBE method.
In [10], a static condensation RBE (SCRBE) approach is developed for elliptic problems. It brings together ideas of CMS and RBE by considering standard static condensation at the interdomain level and then RB approximation at the intradomain level. In an Offline stage performed once, the RB space for a particular component is designed to reflect all possible function variations on the component interfaces (which we shall denote “ports”); components are thus completely interchangeable and interoperable. During the Online stage, any system can be assembled from multiple instantiations of components from a predefined library; we can then compute the system solution for different values of the component parameters in a prescribed parameter domain. The Online stage of the SCRBE is much more flexible than both the Online stage for the standard RB method, in which the system is already assembled and only parametric variations are permitted, and the Online stage of the classical (nonstaticcondensation) RBE method, in which the RB intradomain spaces already reflect anticipated connectivity.
In this paper, we present an extension of the SCRBE to eigenproblems. The new aspects are the following. First, the SCRBE normally takes advantage of linearity, which is lost when considering eigenproblems. Hence we begin by reformulating the eigenproblem using a shift σ of the spectrum in order to recover a linear problem. Finding the eigenvalues is then performed at a higher level: using a direct search method, we find the values of the shift σ that correspond to singular systems. Second, we provide a posteriori error estimators of the eigenvalues, not only with respect to RB approximations but also in the context of port reduction.
In the context of CMS approaches for eigenproblems, out method provides some important features: treatment of parameterdependent systems (as explained above), optimal convergence, and port reduction. The classical CMS only achieves a polynomial convergence rate [11,12] with respect to the number of eigenmodes used at the intradomain level. This can be improved to an infinite convergence rate by using overlapping components [12], but at the expense of losing simplicity and flexibility of component connections. Our method somehow provides an optimal tradeoff since it retains the interface treatment of classical CMS – allowing flexibility of component connections – while achieving an exponential convergence rate with respect to the size of RB spaces at the intradomain level.
We also provide port reduction so as to increase even more the speed up. Recent CMS contributions consider several port economizations (or interface reduction strategies): an eigenmode expansion (with subsequent truncation) for the port degrees of freedom is proposed in [11,13]; an adaptive port reduction procedure based on a posteriori error estimators for the port reduction is proposed in [14]; and an alternative port reduction approach, with a different bubble function approximation space, is proposed for timedependent problems in [15]. We can not directly apply CMS port reduction concepts in the parameterdependent context, as the chosen port modes must be able to provide a good representation of the solution for any value of the parameters. In this paper, we adapt to parameterdependent eigenproblems a port approximation and a posteriori error bound framework introduced in [16] for parameterdependent linear elliptic problems.
The paper proceeds as follows. In Section ‘Formulation’, we present the general eigenproblem and its shifted formulation; we then describe the static condensation procedure. In Section ‘Reduced basis static condensation system’, we add reduced basis approximations and develop a posteriori error estimators for the eigenvalues with respect to the corresponding values obtained by the “truth” static condensation of Section ‘Formulation’. In section ‘Port reduction’, we introduce port reduction and provide as well a posteriori error estimators for the eigenvalues. In Section ‘Computational aspects’, we give an overview of the computational aspects of the method. This section somehow brings together all of the previous sections in a compact presentation, and we suggest the reader to often go back to Section ‘Computational aspects’ in order to get a higher level description of the method. Finally, in Section ‘Results and discussion’, we present numerical results to illustrate the computational efficiency of the approach. We first consider simple bridge structures for which we examine the error estimates. We finish with an industrial scale example to show the method’s potential to tackle large systems.
Methods
Formulation
Problem statement
Furthermore, let Y≡L ^{2}(Ω).
We assume that the eigenvalues λ _{ n }(μ) are sorted such that \(0 < \lambda _{1}(\mu) \leq \lambda _{2}(\mu) \ldots \leq \lambda _{\mathcal {N}}(\mu)\), and to each eigenvalue λ _{ n }(μ) we associate a corresponding eigenvector u _{ n }(μ). We can have multiplicities greater than one and hence we can have equal successive eigenvalues λ _{ n }(μ)=⋯=λ _{ n+k }(μ) but each associated to linearly independent eigenvectors.
The parametric dependence of the problem usually takes the form of variable PDE coefficients or variable geometry. For instance, in linear elasticity, the vector μ can contain the different Young’s modulus values of different subdomains, as well as the parameters of some mapping function describing the geometrical variability.
is our “shifted” bilinear form. Note that we change the bilinear form on the right hand side from m(·,·) to a(·,·), which corresponds to a different norm. This choice is motivated by error estimation, presented later in the paper, as it permits to derive relative error estimates for the eigenvalue λ _{ n }(μ).
for \(n = 1,\ldots,\mathcal {N}\).
Remark 2.1.
Static condensation
We denote by \(\mathcal {N}_{p}^{\Gamma }\) the dimension of the port space \(Z_{p}^{\mathcal {N}}\) associated with global port p, and we say that the global port p has \(\mathcal {N}_{p}^{\Gamma }\) degrees of freedom (dof). For each component i, we denote by k ^{′} a local port dof number, and K _{ i } the total numbers of dof on its local ports, such that 1≤k ^{′}≤K _{ i }. We then introduce the map \(\mathcal {P}_{i}(k')=(p,k)\) which associate a local port dof k ^{′} in component i to its global port representation: global port p and dof k, \(1\leq k \leq \mathcal {N}_{p}^{\Gamma }\).
where the U _{ p,k }(μ,σ) are interface function coefficients corresponding to the port p, and \(b_{i}(\mu,\sigma) \in X_{i;0}^{\mathcal {N}}\). Here χ _{ n } is independent of σ, but we shall see shortly that we will need b _{ i } and U _{ k,p } to be σdependent in general.
for all \(v \in X^{\mathcal {N}}_{i;0}\).
the bilinear form \(\mathcal {B}_{i}(\cdot,\cdot ;\mu ;\sigma)\) is coercive on \(X_{i;0}^{\mathcal {N}}\) if σ<λ _{ i,1}(μ), where λ _{ i,1}(μ) is the smallest eigenvalue of (17). Hence (16) has a unique solution under this condition. Note that we expect that λ _{ i,1}(μ)>λ _{1}(μ), and even λ _{ i,1}(μ)>λ _{ n }(μ) for n=2 or 3 or 4 — of course in practice the balance between λ _{ n } and \(\lambda _{i,n^{\prime }}\) will depend on the details of a particular problem.
This space is of dimension \(n_{\text {sc}}=\sum _{p=1}^{n^{\Gamma }} \mathcal {N}_{p}^{\Gamma }\).
Remark 2.2.
Note that the interface functions are intermediate quantities that are completed with bubble functions. Although the interface functions are the result of a simple harmonic lifting with the homogeneous Laplace operator, the subsequent bubble functions are computed based on the problemdependent a and m bilinear forms, hence they capture the possible heterogeneities intrinsic to the problem. Hence the skeleton space \(X_{\mathcal {S}}(\mu,\sigma)\) is suitable for approximation.
It is important to note that this new eigenproblem (21) (22) differs from (3) (4) in two ways: first, we consider a subspace \(X_{\mathcal {S}}(\mu,\sigma)\) of \(X^{\mathcal {N}}\), and as a consequence \(\overline {\tau }_{n}(\mu,\sigma) \geq \tau _{n}(\mu,\sigma)\); second, the subspace \(X_{\mathcal {S}}(\mu,\sigma)\), unlike \(X^{\mathcal {N}}\), depends on σ, and furthermore only for σ=λ _{ n } does the subspace \(X_{\mathcal {S}}(\mu,\sigma)\) reproduce the eigenfunction χ _{ n }(μ). We now show
Proposition 2.1.
 (i)
\(\overline {\tau }_{n}(\mu,\sigma) \geq \tau _{n}(\mu,\sigma)\), \(n=1,\ldots,\text {dim}(X_{\mathcal {S}}(\mu,\sigma))\),
 (ii)
τ _{ n }(μ,σ)=0 if and only if σ=λ _{ n }(μ),
 (iii)
σ=λ _{ n }(μ) if and only if there exists some n ^{′} such that \(\overline {\tau }_{n'}(\mu,\sigma) = 0\).
Proof.
 (i)The case n=1 follows from the Rayleigh quotients$$ \tau_{1}(\mu,\sigma) = \inf_{w \in X^{\mathcal{N}}} \frac{\mathcal{B}(w,w;\mu;\sigma)}{a(w,w;\mu)}, $$(23)and$$ \overline{\tau}_{1}(\mu,\sigma) = \inf_{w \in X_{\mathcal{S}}(\mu,\sigma)} \frac{\mathcal{B}(w,w;\mu;\sigma)}{a(w,w;\mu)}, $$(24)
and fact that \(X_{\mathcal {S}}(\mu,\sigma) \subset X^{\mathcal {N}}\).
For n>1, the CourantFischerWeyl minmax principle [17] states that for an arbitrary ndimensional subspace of \(X^{\mathcal {N}}\), S _{ n }, we have$$ \eta_{n}(\mu,\sigma) \equiv \max_{w \in S_{n}} \frac{\mathcal{B}(w,w;\mu;\sigma)}{a(w,w;\mu)} \geq \tau_{n}(\mu,\sigma). $$(25)Let \(S_{n} \equiv \text {span}\{ \overline {\chi }_{m}(\mu,\sigma), m=1,\ldots,n\} \subset X_{\mathcal {S}}(\mu,\sigma)\). Then \(\eta _{n}(\mu,\sigma) = \overline {\tau }_{n}(\mu,\sigma)\), and the result follows.
 (ii)
This equivalence is due to (8).
 (iii)
(⇐) Suppose σ=λ _{ n }(μ) for some n, then by construction \(\chi _{n}(\mu,\sigma) \in X_{\mathcal {S}}(\mu,\sigma)\). Since the same operator appears in both (19) and (21), it follows that χ _{ n }(μ,σ) is also eigenmode for (21), (22) with corresponding eigenvalue 0. That is, for some n ^{′}, \(\overline {\tau }_{n'}(\mu,\sigma) = 0\) is an eigenvalue of (21), (22).
(⇒) Suppose \(\overline {\tau }_{n'}(\mu,\sigma) = 0\) for some index n ^{′}. Then \(\overline {\chi }_{n'}(\mu,\sigma)\) satisfies (19), (20), or equivalently, (3), (4) for τ _{ n }(μ,σ)=0. From part (ii) of this Proposition, this implies that σ=λ _{ n }(μ).
Remark 2.3.
Regarding our method, the main result is 2.1(iii), which informs on how to recover eigenvalues of the original problem (3), (4) from the shifted and condensed problem (21), (22): we look for the values of σ such that (21), (22) has a zero eigenvalue. Note that in 2.1(iii), the equivalence between \(\overline {\tau }_{n'}(\mu,\sigma) = 0\) and σ=λ _{ n }(μ) possibly happens for n ^{′}≠n. In practice though, we always have n ^{′}=n and there is a onetoone correspondence between the original problem and the shifted and condensed system which make the eigenvalues much easier to track. We are not able to demonstrate that n ^{′}=n in all cases, but assuming that property, we can demonstrate some stronger properties (see 4) that we will use to derive error estimates.
As explained above, in order to find the eigenvalues of the original problem (3), (4), we need to find the values of σ for which (28), (29) has a zero eigenvalue. When performing this search, for each new value of σ that is considered, we need to perform the assembly of the static condensation system (28), which involves many finite element computations at the component level in order to get the bubble functions (16), and is potentially costly. Note that we also need to reassemble (28) when the parameters μ of the problem change. In order to dramatically reduce the computational cost of this assembly, we will use reduced order modeling techniques as described in the next Sections ‘Reduced basis static condensation system’ and ‘Port reduction’.
Reduced basis static condensation system
Reduced basis bubble approximation
(Note that \(\widetilde {X}_{\mathcal {S}}(\mu,\sigma) \not \subset X_{\mathcal {S}}(\mu,\sigma)\)).
Remark 3.1.
As opposed to CMS where the static condensation space is built from local component natural modes, the RB static condensation space \(\widetilde {X}_{\mathcal {S}}(\mu,\sigma)\) is built from RB bubbles that can accommodate for any global mode shape thanks to their (μ,σ) parametrization. The only restriction is due to condition (32) which means that we only ensure to capture global modes for which the wavelength is typically greater than a component’s size.
Reduced basis error estimator
First, since \(\widetilde {X}_{\mathcal {S}}(\mu,\sigma) \subset X^{\mathcal {N}}\), by the same argument as part (i) of Proposition 2.1, we have
Corollary 3.1.
□
Port reduction
Empirical mode construction
We can truncate the Laplacian eigenmode expansion in order to reduce \({\mathcal {N}}_{p}^{\Gamma }\) – often without any significant loss in accuracy of the method. However, we can obtain better results by tailoring the port basis functions to a specific class of problems. A strategy for the construction of such empirical port modes is presented in [16]. We briefly describe this strategy here and refer the reader to [16] for further detail.
A key observation is that, in a system of components, the solution on any given interior global port is “only” influenced by the parameter dependence of the two components that share this port and the solution on the nonshared ports of these two components. We shall exploit this observation to explore the solution manifold associated with a given port through a pairwise training algorithm.
To construct the empirical modes we first identify groups of local ports on the components which may interconnect; the port spaces for all ports in each such group must be identical. For each pair of local ports within each group (connected to form a global port Γ _{ p }), we execute Algorithm (1): we sample this I=2 component system many (N _{samples}) times for random (typically uniformly or loguniformly distributed) parameters over the parameter domain and for random boundary conditions on nonshared ports. For each sample we extract the solution on the shared port Γ _{ p }; we then subtract its average and add the resulting zeromean function to a snapshot set S _{pair}. Note that by construction all functions in S _{pair} are thus orthogonal to the constant function.
Upon completion of Algorithm 1 for all possible component connectivity within a library, we form a larger snapshot set S _{group} which is the union of all the snapshot sets S _{pair} generated for each pair. We then perform a data compression step: we invoke proper orthogonal decomposition (POD) [19] (with respect to the L ^{2}(Γ _{ p }) inner product). The output from the POD procedure is a set of mutually L ^{2}(Γ _{ p })orthonormal empirical modes that have the additional property that they are orthogonal to the constant mode.
Portreduced system
as the number of total active and inactive port modes, respectively; and n _{SC}=n _{A}+n _{I} is the total number of port modes in the nonreduced system.
in which we may discard the (presumably large) \(\mathbb {B}_{\text {II}}(\mu,\sigma)\) and \(\mathbb {A}_{\text {II}}(\mu,\sigma)\) blocks; however the \(\mathbb {B}_{\text {IA}}(\mu,\sigma)\)block is required later for residual evaluation in the context of a posteriori error estimation.
Port reduction error estimator
It is important to note that \(\widehat \Delta (\mu,\sigma _{n})\) will only decrease linearly in the residual, whereas the actual eigenvalue error is expected to decrease quadratically in the residual. This is due to the fact that port reduction can be viewed as a Galerkin approximation over a subspace of the skeleton space \(X_{\mathcal {S}}(\mu,\sigma)\), and in that framework several a priori and a posteriori error results demonstrate the quadratic convergence of the eigenvalue [20]. As a consequence the effectivity of the error estimator \(\widehat \Delta (\mu,\sigma _{n})\) is expected to degrade as n _{A,p } gets larger.
Computational aspects
In this section, we summarize the main steps of the method from a computational point of view. There are two clearly separated stages. The “Offline” stage involves heavy precomputations and is performed only once. The “Online” stage corresponds to the actual solution of the eigenproblem and can be performed many times for various parameters μ and different eigenvalue targets. The “Online” computations are very fast thanks to our approach and allow to solve eigenproblems in a many query context such as model optimization or design.
Offline computations
In the Offline stage, we already have some knowledge about the class of eigenproblems we will have to solve. We know the bilinear forms a and m corresponding to the stiffness and mass operators. We have a predefined library of archetype components that will be allowed to be connected together at compatible ports to form bigger systems that will be considered in the Online stage. See Figure 3 for an example of library, and Figure 6 for an example of system obtained from component assembly. Note that each archetype component in the library is allowed to have some parametric variability.

Compute a set of port modes, possibly empirical modes as described in Section ‘Empirical mode construction’.

Compute the harmonic extension of the port modes inside the archetype component reference domain to get the interface functions.

For each interface function, compute a reduced basis space for the bubble Eq. 16. Each RB space is tuned for the stiffness and mass operators, as well as the component parametric variability and the shift σ variability.

Precompute some component quantities used in (35), (36), that will be ready in the Online stage for system assembly.
Online stage
System assembly. In the Online stage, we form a component assembly by instantiating I components from our library of archetype components, and connecting them together. Several instantiated components can correspond to the same archetype component, but with possibly different parameter values. Each instantiated component i has a set of parameter values μ _{ i }, and the whole system has a set of parameters μ=∪_{ i=1..I } μ _{ i }. We also define a value of σ for the whole system.
Eigenvalue computation. At this point, we now need to find the values of σ for which the system (33) has a zero eigenvalue. We proceed by fixing an eigenvalue number n and we then follow Algorithm 2 with tolerance δ≪1.
Applying this algorithm for n=1,2,3,… we can recover the first eigenvalues of the component assembly. In practice Brent’s method [21] applied to the search of σ such that \(\widetilde {\overline {\tau }}_{n}(\mu,\sigma)=0\) converges in about 10 iterations, and there is only a single root for the function \(\sigma \mapsto \widetilde {\overline {\tau }}_{n}(\mu,\sigma)\).
Once an approximation \(\widetilde {\lambda _{n}}(\mu)=\sigma \) of the eigenvalue has been found, we obtain an associated eigenvector following (33). Note that we use a standard eigensolver from the SLEPc library [22] as a black box, hence we have no control on the eigenvector computation, especially when the eigenvalue multiplicity is two or more.
Remark 5.1.
The parametric dependence comes into play in the Online stage when the RB bubble functions are computed, as they depend on (μ,σ). As a consequence, the resulting shifted system depends on (μ,σ), and also its eigenvalues \(\widetilde {\overline {\tau }}_{n}(\mu,\sigma)\). The vector of parameters μ is chosen by the user for the whole system (material properties of the different components, geometry), while σ is automatically updated at each step of Algorithm 2: as a result, the RB bubble functions have to be recomputed at each step of Algorithm 2. In the end though, we obtain an approximation \(\widetilde {\lambda _{n}}(\mu)\) that depends only on μ, the “natural” parameters of the original system. The user is then free to modify the system by choosing a different vector of parameters μ ^{′}, and restart Algorithm 2.
Results and discussion
Linear elasticity
and a similar expression could be obtained for the stiffness bilinear form a(·,·;μ).
Note that λ(μ)=ω ^{2}(μ) – the eigenvalue is the frequency squared.
Simple component library
We consider a linear elasticity library of two components shown in Figure 3: a beam and a connector. The FE hexahedral meshes are shown in Figure 3, and in all the following we use first order approximation with trilinear elements. The components can connect at square ports of dimension 1×1 with \(\mathcal {N}_{p}^{\Gamma }=3\times 36=108\) degrees of freedom. The beam has two parameters: the Young’s modulus E∈[0.5,2] and the length scaling s∈[0.5,2], where the beam is of length 5s. The connector has one parameter, the Young’s modulus E∈[0.5,2]. Finally, for the shift parameter σ, we consider the range [0,0.01], based on the fact that the local minimum eigenvalues of the two components are larger than 0.01 for the previous E and s parameter ranges. For each component, we build RB bubble spaces of size N=10 using a Greedy algorithm [23], for the parameter ranges previously defined. See [4] for a detailed example of reduced basis applied to linear elasticity. We also perform a pairwise training for the component pair beamconnector to build empirical port modes as described in Section ‘Empirical mode construction’; and we build a parameter independent preconditioner (necessary for the computation of \(\widehat \Delta \)) using parameter values E=0.5 and s=0.5.
Simple beam
We first present a simple example where we compare with beam theory to demonstrate that the FE resolution is adequate and that we capture the different modes. We connect eight beam components together, corresponding to a system with a vector of parameters μ of dimension 16. By using the same values of s=1 and E=1 for all beam components – or equivalently \(\mu =\mathbb {1}\in \mathbb {R}^{16}\) – we obtain a system corresponding to a uniform beam of square section, with thickness d=1 and length L=40, and Young’s modulus E=1. As boundary conditions, we clamp this beam on both ends.
Eigenvalues for a clampedclamped uniform beam of square section, with thickness d =1 and length L =40
λ _{ 1 }  λ _{ 2 }  λ _{ 3 }  λ _{ 4 }  

Euler Bernoulli  1.6294e05  1.2381e04  4.7583e04  1.3003e03 
Timoshenko  1.6204e05  1.2224e04  4.6524e04  1.2560e03 
Global FEM  1.6612e05  1.2489e04  4.7327e04  1.2708e03 
SCRBE n _{A,p}=108  1.6612e05  1.2489e04  4.7327e04  1.2708e03 
SCRBE n _{A,p}=20  1.6612e05  1.2489e04  4.7327e04  1.2708e03 
\(\widetilde \Delta \)  1.4418e06  2.0695e07  7.9612e08  9.6913e08 
\(\widehat \Delta \)  5.5488e03  7.3845e03  8.4207e03  7.4811e03 
λ _{ 5 }  λ _{ 6 }  λ _{ 7 }  λ _{ 8 }  
Euler Bernoulli  —  2.9016e03  5.6603e03  — 
Timoshenko  —  2.7622e03  5.2991e03  — 
Global FEM  2.0732e03  2.7775e03  5.2916e03  6.1912e03 
SCRBE n _{A,p}=108  2.0732e03  2.7775e03  5.2916e03  6.1912e03 
SCRBE n _{A,p}=20  2.0732e03  2.7775e03  5.2916e03  6.1912e03 
\(\widetilde \Delta \)  5.4576e09  4.1418e07  1.0262e06  8.8249e09 
\(\widehat \Delta \)  3.3180e02  8.3262e03  8.9995e03  4.7761e03 
Bridge structure
We are now ready to consider larger systems with more complicated connections which will better exercise the RB and port reduction capabilities. Towards this end, we consider a system of 30 components, corresponding to a bridge structure. It is composed of 22 beam components and 8 connectors, hence the vector of parameters μ for this system is of dimension 52.
We first set the vector of parameters μ such that E=0.5 and s=1 for all components, and we show in Figure 6 the second and third eigenmodes for the corresponding system. In the following, we will provide systematic analysis of the RB and port reduction convergence and also performance of the a posteriori error estimates.
Industrial example
In this last section, we apply our approach to a large industrial structure. In the following, we will first focus on computational performance (without using the error estimators that have already been presented in the previous section), and then we will illustrate the parametric variability offered by our approach. Note that we will now consider linear elasticity in its dimensional form.
Comparison between SCRBE and FE
Global FEM  SCRBE  Relative error  

Frequency 1  2.526 Hz  2.532 Hz  0.3% 
Frequency 2  2.775 Hz  2.792 Hz  0.6% 
Frequency 3  4.984 Hz  5.028 Hz  0.9% 
Frequency 4  6.597 Hz  6.688 Hz  1.4% 
Frequency 5  7.372 Hz  7.501 Hz  1.7% 
RAM usage  12 GB  100 MB  
Solving time  350 s  0.5 s 
The second and most important goal of this section is to show the parametric advantage of our method with respect to CMS. We demonstrated that SCRBE has a computational advantage relative to FE, but the same decrease in computational time could in theory be obtained with CMS. One crucial advantage of SCRBE with respect to CMS (in addition to convergence rate) is its flexibility with respect to parameter variations. Thanks to the RB approximations at the component level, we can modify the component parameters and directly recompute the eigenproblem solution “Online”. In the case of CMS, any change of the component parameters (especially geometrical) would require some “Offline” work to recompute the modal decomposition of each component, hence precluding its use in a many query context with parametric variability. Although we did not implement the CMS method for direct comparison as we did for FE previously, we hope the following examples of Online parametric variations will convince the reader of the crucial advantage of SCRBE with respect to CMS.
The parametric variability of each component used in the shuttle assembly
Component type  Number of parameters  Parameters and ranges 

Young’s modulus (frame, panel) ∈[60,220]G P a  
Side panel  4  Mass density (frame, panel) ∈[1000,8000]k g.m ^{−3} 
Young’s modulus ∈[60,220]G P a  
Truss joint  2  Mass density ∈[1000,8000]k g.m ^{−3} 
Young’s modulus ∈[60,220]G P a  
Diagonal truss  4  Mass density ∈[1000,8000]k g.m ^{−3} 
Length, shear  
Young’s modulus ∈[60,220]G P a  
Mass density ∈[1000,8000]k g.m ^{−3}  
Vertical prebending v∈[−1.5,1.5]  
Horizontal truss  4  Horizontal prebending h∈[−1,1] 
The first and fifth natural frequencies of the shuttle for various configurations
Shuttle configuration  First natural frequency  Fifth natural frequency 

Pristine, all steel  2.53 Hz  7.50 Hz 
Pristine, steel frame, aluminium panels  3.13 Hz  8.10 Hz 
Pristine, all aluminium  2.54 Hz  7.51 Hz 
Prebent trusses, all steel  2.53 Hz  7.07 Hz 
We observe almost no change in the frequencies when the shuttle is either all steel or all aluminium. This is because the homogeneous Young modulus and mass density can be factored out of the stiffness and mass matrices, and the ratio between these two quantities is almost the same for steel and aluminium. On the opposite, if we mix both materials, as in the case of a steel frame and aluminium panels, then the natural frequencies significantly change. Now if we prebend the horizontal trusses, we observe that the first frequency is unchanged whereas the fifth frequency is affected. This is because the first eigenmode corresponds to a bending along the principal axis of the shuttle, and does not involve any deformation of the horizontal trusses, whereas the fifth eigenmode corresponds to a lateral bending which involves some horizontal trusses (see Figure 12).
The first natural frequency for the system composed of two prebent truss components (steel)
No prebending  Horizontal prebending  Vertical prebending  Horizontal and vertical  

prebending  
SCRBE  12.608 Hz  11.868 Hz  12.246 Hz  11.553 Hz 
FE  12.606 Hz  11.852 Hz  12.241 Hz  11.534 Hz 
Conclusions
We extended the SCRBE approach – originally introduced for parametrized linear problems – to parametrized symmetric eigenproblems, in order to analyze largescale componentbased structures. Thanks to the componentinterior reduced basis and the port reduction, we are able to compute fast accurate approximations for any component parameter values, as well as providing a posteriori error estimates.
We presented an application to a large structure – a shiploader shuttle used in mining – in the context of threedimensional linear elasticity. Compared to a finite element method, we obtain a speed up of 700 and a reduction of 120 in memory consumption. We are also able to explore the parametric variability of the shuttle – a vector μ of dimension 136 – and recompute the solution at the same speed for every new value of μ.
We obviously presented a limited number of cases, but the parametric variability of the shuttle, to which we can add variable clamping conditions, allows to consider thousands of designs, and in very little time thanks to the computational speed of SCRBE. Moreover, the small memory requirements of the method would allow to consider even larger structure, such as a full shiploader model. For these reasons we think our method can be very valuable in an engineering context where design optimization and multiscenario analysis of large models is common practice.
Endnotes
^{a} The tolerance applies to \(\widetilde {\overline {\tau }}_{n}(\mu,\sigma)=0\), hence it corresponds to a relative tolerance for \(\widetilde {\lambda }_{n}(\mu)\).
^{b} The global FEM eigenvalue is not expected to be exactly the same as what would be obtained with FE static condensation – in theory they should be the same, but the different computational paths lead to different numerical results – hence it explains why \(\frac {\lambda _{\textit {FE}}\widetilde {\lambda }}{\lambda _{\textit {FE}}}\) does not converge to zero, and why \(\widetilde {\Delta }\) gets smaller than \(\frac {\lambda _{\textit {FE}}\widetilde {\lambda }}{\lambda _{\textit {FE}}}\) for N big enough.
Appendix A
Properties used for error estimates
Hypothesis A.1.
σ=λ _{ n }(μ) if and only if \(\overline {{\tau }}_{n}(\mu,\sigma) = 0\)
Lemma A.1.
for each \(n=1,2,\ldots,\mathcal {N}\).
Proof.
However, the result does not apply to \(\overline {\tau }_{n}(\mu,\sigma)\): we cannot apply the argument from the proposition to (21), (22) since in general \(\overline {\chi }_{n}(\mu,\sigma)\) depends on σ. We can still state the following
Proposition A.1.
Proof.
Declarations
Acknowledgements
This work was supported by OSD/AFOSR/MURI Grant FA95500910613, by ONR Grant N000141110713, and by a grant from the MIT Deshpande Center for Technological Innovation. This work was also supported by the Commission for Technology and Innovation CTI of the Swiss Confederation.
Authors’ Affiliations
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