Model Order Reduction of Dynamic Systems via Proper Orthogonal Decomposition

Abstract

In this chapter, the performance of reduced order modeling of dynamic structural systems based on the proper orthogonal decomposition (POD) technique is investigated. Singular value decomposition and principal component analysis of the so-called snapshot matrix are considered to generate the reduced space, onto which the system equations of motion are projected to speed up the computations. To get intuitions into the achievable computational efficiency and the capability of POD to provide an input-independent reduced model, we consider the 39-story Pirelli tower in Milan-Italy. First, it is assumed that a shear model of the building is excited by the May 18-1940, Mw 7.1, El Centro earthquake, and the ensemble of the data necessary to build the reduced model is acquired. Then, the local and global accuracies of the same reduced model in tracking the dynamics of the building are assessed, if excited by the May 6-1976, Mw 6.4, Friuli earthquake and by the January 17-1995, Mw 6.8, Kobe earthquake, which differ from the El Centro one in terms of excited vibration frequencies. It is shown that POD allows to attain a speedup close to 250, when the reduced order model is required to retain a high fidelity.

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3.1 Introduction

While working with a space discretized system, proper orthogonal decomposition (POD) automatically seeks a dependence structure between the degrees-of-freedom, which are normally assumed to be independent (Buljak 2012). This is accomplished through a set of ordered, orthonormal bases, and through information regarding the relevant energy contents. The POD has been developed independently by various researchers in different fields (see e.g. Kosambi 1943; Karhunen 1947; Obukhov 1954) and has been called by utilizing variety of names. When applied to finite dimensional systems, it is called principal component analysis (PCA) (Jolliffe 1986) which originates in the Pearson’s (1901) study on plane and line fitting to point sets. When working with distributed parameter systems, it is called Karhunen–Loève decomposition (KLD); nevertheless, its discrete representation is introduced as well (Fukunaga 1990). Another POD technique is called singular value decomposition (SVD) (Mees et al. 1978), innovation of such technique is credited to Eckart and Young; where they proposed extension of eigen value decomposition for general non square matrices (Klema and Laub 1980). For a detailed proof of equivalency of PCA, KLD and SVD readers are referred to (Liang et al. 2002a).

As a result of standard numerical tools developed to extract proper orthogonal modes (POMs) of the systems, and as a result of its power in feature extraction and reduced modeling, presently the POD is extensively employed in various engineering fields. To illustrate this issue further, one can see that the POD has been applied for reduced order modeling of heat transfer phenomena (Samadiani and Joshi 2010) and other field such as: computational fluid dynamics (Smith et al. 2005; Tadmor et al. 2006), micro electro mechanical systems (Liang et al. 2002b), various fields of computational physics (Lucia et al. 2004) and aeroelasticity (Thomas et al. 2003). The method of POD has obtain great reputation in the field of structural dynamics, where it is employed for active sensing (Park et al. 2008) and active control of structures (Al-Dmour and Mohammad 2002), damage detection (De Boe and Golinval 2003; Galvanetto et al. 2008; Shane and Jha 2011c), model updating (Lenaerts et al. 2003; Hemez and Doebling 2001), modal analysis (Han and Feeny 2003; Feeny 2002) and model reduction (Steindl and Troger 2001; Buljak and Maier 2011). For a more comprehensive examination of the related literature, readers are referred to (Kerschen et al. 2005). The study carried out in the literature recommends that the POD is a strong tool for model order reduction of structural systems; however, the research lacks a specific study of expediting, computational accuracy of the reduced model and robustness to the change in the source of excitation. The study presented in this chapter acknowledges those aforementioned issues. The author of this monograph has coauthored a journal article on the topics covered in this chapter (Eftekhar Azam and Mariani 2013).

Next, in the Sect. 3.2, structural dynamics of systems are examined and studied in this chapter as well; moreover, their associated set of governing differential equation and the numerical scheme are employed for time discretization. Section 3.3 reviews fundamentals of the POD. Afterwards, the fundamental studies carried out in finding the links between POMs and eigen modes of linear structures are summarized in Sect. 3.4. Furthermore, a reduced model is constructed via Galerkin projection of the set of governing equations onto the reduced space spanned by POMs which is presented in Sect. 3.5. Finally, the results of the numerical assessment of efficiency of POD: speedup and accuracy of reduced models of Pirelli tower, as a case study, are investigated and reported in Sect. 3.6.

3.2 Structural Dynamics and Time Integration

In this study, the POD for reduced order modeling of dynamic systems is exploited. Subsequently, such reduced model will be embedded into a Bayesian filter in the forth-coming chapters. In this section, the differential equations of the governing dynamics of structural systems are reviewed herein; moreover, the numerical integration scheme employed for time discretization of the aforementioned differential equations is briefly discussed.

The dynamic response of the structural system to the external loads is allowed to be described by the following linear equations of motion:

$${\varvec{M}}{\ddot{\varvec{u}}}(t) + {\varvec{D}} \dot{\varvec{u}} (t) + \varvec{Ku}(t) = \varvec{F}(t)$$

(3.1)

where: M is the mass matrix; D is the viscous damping matrix; K is the stiffness matrix; F is the time-dependent external force vector; ${\ddot{\varvec{u}}}$, $\dot{\varvec{u}}$ and $\varvec{u}$ are the time-varying vectors of accelerations, velocities and displacements, respectively. For instance, in a shear model of a building (such as the one adopted in Sect. 2.6), these vectors gather the lateral displacements, velocities and accelerations of the storeys.

Equation (3.1) is usually arrived once the structural system has been space discretized (e.g. through finite elements), or once assumptions concerning the behavior of the building (e.g. shear-type deformation) have taken into account. This preliminary stage of the analysis can affect the sparsity of matrices in Eq. (3.1), and can therefore have an impact on the speedup obtained through the POD as well.

In this study, the solution of the vectorial differential equation (3.1) is advanced in time by utilizing the Newmark explicit integration scheme. For details, the reader is referred to Sect. 2.6.

3.3 Fundamentals of Proper Orthogonal Decomposition for Dynamic Structural Systems

The aim of reduced order modeling is to automatically find a solution for the following two conflicting requirements: create the smallest possible numerical model of the original dynamic system; preserve accuracy in the description of the system behavior. Standard techniques attempt to extract fundamental features from the dynamic model; thus the governing equations can be thereafter projected onto a reduced state space or subspace.

The POD, in its snapshot version (Sirovich 1987), is adopted to build the model-specific optimal linear subspace on the basis of an ensemble of system observations in this study. Let us consider the displacement vector $\varvec{u} \in {\mathbb{R}}^{m} ,\;{\mathbb{R}}$ being the set of real numbers and m the dimension of vector u; we assume that u effectively describes system evolution (i.e. it does not need to be supplemented by $\dot{\varvec{u}}$ and ${\ddot{\varvec{u}}}$ to define the full state space), and consider a set of arbitrary orthonormal bases $\left\{ {\varvec{\varphi }_{i} } \right\}$, $i = 1, \ldots ,m$, spanning its vector space ${\mathbb{R}}^{m}$. Such bases satisfy $\varvec{\varphi }_{i}^{\text{T}} \varvec{\varphi }_{j} = \delta_{ij} \;(j = 1, \ldots ,m)$, where $\delta_{ij}$ is the Kronecker’s delta (such that $\delta_{ij} = 1$ if $i = j$, otherwise $\delta_{ij} = 0$). The original vector u can then be written as a linear combination of the aforementioned bases, according to:

$$\varvec{u} = \sum\limits_{i = 1}^{m} {\varvec{\varphi }_{i} y_{i} = {\varvec{\Phi}}\varvec{y}}$$

(3.2)

where $y_{i}$ are the combination coefficients, arranged in the column vector y, and:

$${\varvec{\Phi}} = [\varvec{\varphi }_{1} \varvec{\varphi }_{2} \ldots \varvec{\varphi }_{m} ]$$

(3.3)

is the matrix gathering all the bases.

To ensure computational gain, we define a reduced representation of the state via:

$$\varvec{u}_{l} = \sum\limits_{i = 1}^{l} {\varvec{\varphi }_{i} y_{i} = {\varvec{\Phi}}_{l} \,\varvec{\alpha}}$$

(3.4)

where we enforce $l < m$ or, for large systems, even $l \ll m$. In (3.4), ${\varvec{\Phi}}_{l}$ is the matrix gathering the first l columns of matrix ${\varvec{\Phi}}$ (i.e. the first l bases), and $\varvec{\alpha}$ collects the relevant first l components of vector y. The goal of POD is to provide an ordered sequence of the bases $\varvec{\varphi }_{i}$, so as to satisfy the following extreme value problem:

$$\hbox{min} \left\| {\varvec{u} - \varvec{u}_{l} } \right\|$$

(3.5)

where $||{\blacksquare }||$ represents the L² norm of vector. Given l, Eq. (3.5), thus it is required to find the optimal subspace spanned by the bases $\varvec{\varphi }_{1} , \ldots ,\varvec{\varphi }_{l}$.

We now need to establish l on the basis of the required accuracy of the solution provided by the reduced order model, and to compute the bases gathered by ${\varvec{\Phi}}_{l}$. Both problems can be attacked through the so-called snapshot version of POD. First, since we have to provide a subspace for the state vector u, the characteristic displacements $\varvec{u}^{(k)} = \varvec{u}(t_{k} )\;(k = 1, \ldots ,n)$ at n time instants are computed and collected in an ensemble, or snapshot matrix U, according to:

$$\varvec{U} = \left[ {\varvec{u}^{(1)} \varvec{ u}^{(2)} \ldots \, \varvec{u}^{(n)} } \right].$$

(3.6)

Next PCA and SVD, two POD methods for extracting so-called POMs are briefly discussed.

3.3.1 Principal Component Analysis

To detect the main dependence structure in an ensemble of data, PCA looks for the subspace which is able to keep the maximum variability in the data. A very naïve justification of this procedures is expressed as: in the state-space, the directions along which data vary are important, since the dynamics of the system is actually occurring along those directions, whereas the directions featuring no variations are redundant in the dynamic representation, and computational cost will be spent in calculating something that we already know if they were retained in the analysis. Consider the aforementioned vector $\varvec{u} \in {\mathbb{R}}^{m}$; suppose $y_{1} ,\,y_{2} , \ldots ,\,y_{m} \in {\mathbb{R}}$ are the first, second,… and $m{\rm{th}}$ principal components, respectively. Let the first principal component $y_{1}$ be a linear combination of each element of the original vector, i.e.:

$$y_{1} = \sum\limits_{i = 1}^{m} {\xi_{i1} u_{i} =\varvec{\xi}_{1}^{T} \varvec{u}}$$

(3.7)

where: $\varvec{\xi}_{1} = \{ \xi_{11} ,\xi_{21} , \ldots ,\xi_{m1} \}^{T}$. The variance of $y_{1} ,$ assumed to be a random variable, is then:

$$S_{{y_{1} }}^{2} =\varvec{\xi}_{1}^{T} \;{\varvec{Z}}_{u} \;\varvec{\xi}_{1}$$

(3.8)

where ${\varvec{Z}}_{u}$ is the covariance of the variable u, assumed to be random as well. To find the direction in which the maximum variability of data is captured, we look for the direction in which the projection of the samples onto it yields the maximum variance. The maximum of $S_{{y_{1} }}^{2}$ will not be achieved for a finite value of $\varvec{\xi}_{1}$, thus a constraint have to be imposed and reads:

$$\mathop { \hbox{max} }\limits_{{\varvec{\xi}_{1} }} \;(\varvec{\xi}_{1}^{T} \varvec{Z}_{u}\varvec{\xi}_{1} ),\;s.t.\,(\varvec{\xi}_{1}^{T}\varvec{\xi}_{1} ) = 1 .$$

(3.9)

Introducing the Lagrangian multiplier $\lambda_{1}$, from (3.8) and (3.9) we obtain:

$$L(\varvec{\xi}_{1} ,\lambda_{1} ) =\varvec{\xi}_{1}^{T} {\varvec{Z}}_{u}\varvec{\xi}_{1} + \lambda_{1} (1 -\varvec{\xi}_{1}^{T}\varvec{\xi}_{1} )$$

(3.10)

where $L\left( \blacksquare \right)$ is Lagrangian operator. After differentiation (3.10) gives:

$$\frac{{\partial L(\varvec{\xi}_{1} ,\lambda_{1} )}}{{\partial\varvec{\xi}_{1} }} = 2({\varvec{Z}}_{u} - \lambda_{1} \varvec{I})\varvec{\xi}_{1} = 0\, \Rightarrow \,{\varvec{Z}}_{u}\varvec{\xi}_{1} = \lambda_{1}\varvec{\xi}_{1}$$

(3.11)

where $\lambda_{1}$ and $\varvec{\xi}_{1}$ are the eigenvalue and the corresponding eigenvector of the covariance matrix $\varvec{Z}_{u}$, respectively.

Applying the same procedures, the objective function to be maximized in order to extract the principal components of a random variable which is written as:

$$\mathop { \hbox{max} }\limits_{{\varvec{\xi}_{i} }} \,(\sum\limits_{i = 1}^{m} {\varvec{\xi}_{i}^{T} \varvec{Z}_{\varvec{u}}\varvec{\xi}_{i} ),\;s.t.\,(\varvec{\xi}_{i}^{T}\varvec{\xi}_{j} ) = \delta_{ij} }$$

(3.12)

and the approximation error due to a representation by its first l principal components, $u \approx \mathop \sum \limits_{i = 1}^{l} y_{i}\varvec{\xi}_{i} ,$ would be:

$$\varepsilon^{2} (l) = E(\left\| {\varvec{u} - \varvec{u}(l)} \right\|^{2} ) = \sum\limits_{i = l + 1}^{m} {E(y_{i}^{2} ) = \sum\limits_{i = l + 1}^{m} {S_{{y_{i} }}^{2} } }.$$

(3.13)

One has to handle the covariance matrix of the random vectorial variable in order to compute the principal components. However, since in practical problems, it is usually impossible to determine this covariance matrix, it is a common practice to use the correlation matrix as an acceptable approximation of it (Schilders 2008). To approximate the covariance matrix with the required accuracy, one needs an appropriately chosen ensemble of the samples; such a seed of samples is the so-called snapshot matrix, wherein each snapshot represents the state of the system at a specific time instant (see Fig. 3.1).

Fig. 3.1

Building the matrix of snapshots

The covariance of the data set, allocated in a snapshot matrix U, is then calculated as (Schilders 2008):

$$\varvec{Z}_{u} = \mathop { \lim }\limits_{n \to \infty } \left( {\tilde{\varvec{Z}}_{u} = \frac{1}{n}\varvec{UU}^{T} } \right).$$

(3.14)

3.3.2 Singular Value Decomposition

Exploiting the singular value decomposition of the snapshot matrix U we obtain (Liang et al. 2002a):

$$\varvec{U} = \varvec{L}\,{\varvec{\Sigma}}\varvec{R}^{\text{T}}$$

(3.15)

where: L is a $m \times m$ orthonormal matrix, whose columns are the left singular vectors of U; ${\varvec{\Sigma}}$ is a $m \times n$ pseudo-diagonal and semi-positive definite matrix, whose pivotal entries $\varSigma_{ii}$ are the singular values of U; R is a $n \times n$ orthonormal matrix, whose columns are the right singular vectors of U.

The whole basis set ${\varvec{\Phi}},$ i.e. the set of all the so-called POMs, is given by L, i.e. by the left singular vectors of the snapshot matrix (Kerschen and Golinval 2002). If singular values $\Sigma_{ii}$ are sorted decreasingly, and the columns of L and R are accordingly arranged; the decomposition (3.15) is such that the first l columns (with l given) of ${\varvec{\Phi}} = \varvec{L}$ represent the optimal basis subset that fulfills (3.5). Moreover, it is known (see, e.g. Kerschen and Golinval 2002) that the ith singular value squared (i.e. $\Sigma_{ii}^{2}$) represents the maximum of the relevant oriented energy¹; this means that the ith oriented energy is maximized, among all the possible unit vectors, by the basis $\varvec{\varphi }_{i}$. Since we are looking for the most informative subspace, which should be able to furnish as much insight as possible into the dynamics of the original system and therefore, into how energy fluxes take place inside, we retain in the reduced order model the proper modes $\varvec{\varphi }_{i}$ that feature the highest singular values. Additional proper modes, featuring less energy contents, will be redundant in the reduced order representation, and add computational costs with marginal enhancement in the accuracy.

Now, having established a method to sort bases $\varvec{\varphi }_{i} ,$ and the link between the singular value $\Sigma_{ii}$ and the energy content of the proper mode $\varvec{\varphi }_{i},$ we need to set l. According to (Kerschen and Golinval 2002), we assign the required accuracy p of the reduced order solution, intended as a fraction of the total oriented energy of the full model, and select the dimension l of the subspace by fulfilling:

$$\frac{{\mathop \sum \nolimits_{i = 1}^{l} {\Sigma_{ii}}^{2} }}{{\mathop \sum \nolimits_{i = 1}^{m} {\Sigma_{ii}}^{2} }} \ge p$$

(3.16)

hence, on the basis of the ratio between the sum of the singular values of the kept modes and the sum of all the singular values.

3.4 Physical Interpretation of Proper Orthogonal Modes

It is known that the POD is a statistical technique which extracts POMs from the response of the system. However, a close relationship has been established between the POMs and natural eigen-modes of a mechanical system (Feeny and Kappagantu 1998; Kerschen and Golinval 2002). The effort toward establishing a link between the POMs and eigen-modes of the system intends in making the POD a modal identification tool (Yadalam and Feeny 2011). To accomplish this task, theoretical and experimental study has been carried out to link the POMs with eigen-modes of a linear (Feeny 2002) and nonlinear (Georgiou 2005) mechanical systems. In this section, we do not discuss the details offered by the existing literature and only mean to summarize interesting findings published therein.

Free vibrations of an undamped linear system with mass matrix proportional with identity matrix (e.g. a shear building with equal masses at each storey) results in a set of POMs that asymptotically converge to eigen-modes of the system. POMs of a lightly damped similar system are reasonable approximations of eigen-modes of the system (Kerschen and Golinval 2002); however in case of forced harmonic vibration, there is no guarantee that POMs converge to eigen-modes.

When the system resonates at a certain frequency, independently of mass matrix entries, the POMs coincides with the respective eigen-modes of that frequency (Kerschen et al. 2005). It has been shown that POMs coincide with eigen-modes for many noise driven oscillators (Preisendorfer 1979); moreover, North has established a general criterion for symmetry of POMs and eigen-modes of the mechanical systems excited by noise (North 1984).

3.5 Galerkin Projection

Once POD has furnished the required subspace, the displacement vector can be approximated through ${\mathbf{u}}_{l}$. Since matrix ${\varvec{\Phi}}_{l}$ is a function of the position vector only, and defines the shapes of POMs for the structure, while $\varvec{\alpha}$ governs the evolution in time of the structural response, it follows that:

$$\begin{array}{*{20}l} {{\ddot{\varvec{u}}} \approx {\varvec{\Phi}}_{l} {\ddot{\varvec{\alpha}}}} & {\dot{\varvec{u}} \approx{\varvec{\Phi}}_{l} \dot{\varvec{\alpha}}} & {\varvec{u} \approx{\varvec{\Phi}}_{l}\varvec{\alpha}} \\ \end{array} .$$

(3.17)

The equations of motion (3.1), allowing for (3.17), can now be approximately stated as:

$$\varvec{M}{\varvec{\Phi}}_{l} {\ddot{\varvec{\alpha}}}(t) + \varvec{D}{\varvec{\Phi}}_{l} \dot{\varvec{\alpha}}(t) + \varvec{K}{\varvec{\Phi}}_{l}\varvec{\alpha}(t) \cong \varvec{F}(t) .$$

(3.18)

By defining the residual r of such approximation as:

$$\varvec{r} = \varvec{F}(t) - \left({\varvec{M}{\varvec{\Phi}}_{l} {\ddot{\varvec{\alpha}}}(t) + \varvec{D}{\varvec{\Phi}}_{l} \dot{\varvec{\alpha}}(t) + \varvec{K}{\varvec{\Phi}}_{l}\varvec{\alpha}(t)} \right)$$

(3.19)

within a Galerkin projection frame (Steindl and Troger 2001), we enforce it to be orthogonal to the subspace ${\varvec{\Phi}}_{l}$ spanned by the solution, i.e.:

$${\varvec{\Phi}}_{l}^{\text{T}} \varvec{r} = 0 .$$

(3.20)

Hence, the equations of motion of the reduced order model turn out to be:

$${\varvec{\Phi}}_{l}^{\text{T}} \varvec{M}{\varvec{\Phi}}_{l} {\ddot{\varvec{\alpha}}}(t) + {\varvec{\Phi}}_{l}^{\text{T}} \varvec{D}{\varvec{\Phi}}_{l} \dot{\varvec{\alpha}}(t) + {\varvec{\Phi}}_{l}^{\text{T}} \varvec{K}{\varvec{\Phi}}_{l}\varvec{\alpha}(t) = {\varvec{\Phi}}_{l}^{\text{T}} \varvec{F}(t)$$

(3.21)

or, equivalently:

$$\varvec{M}_{l} {\ddot{\varvec{\alpha}}}(t) + \varvec{D}_{l} \dot{\varvec{\alpha}}(t) + \varvec{K}_{l}\varvec{\alpha}(t) = \varvec{F}_{l} (t) .$$

(3.22)

Once the solution of (3.22) is obtained, the full state of the system can be computed by utilizing (3.17).

3.6 Results: Reduced-Order Modeling of a Tall Building Excited by Earthquakes

For linear systems, it will be beneficial if POMs ${\varvec{\varphi}}_{i}$ depend only on physical and geometrical properties of the structure, with marginal effects of the kind of loading considered in the phase of construction of the snapshot matrix. Since different loading conditions may excite a different set of structural vibration modes, what claimed here above does not necessarily hold true. Although a thorough analysis of theoretical aspects of the POD, when applied to structural systems, has been carried out in the literature, only a handful of studies are available on several practical points including the load-dependency of the POMs. Such issue may become crucial especially when the structure is subject to seismic loadings, which are difficult to predict in nature.

The performance of POD has been already assessed in defining reduced models for multi-support structures subject to seismic excitation (Tubino et al. 2003); furthermore, the POD has been applied for efficient reduced modeling of high-rise buildings subject to earthquake loads (Gutiérrez and Zaldivar 2000; Aschheim et al. 2002). However, its efficiency for high fidelity reduced order modeling of multi-storey buildings trained by a certain seismic load and excited by another one has not been done yet. In this section, we investigate whether a reduced order model, built by considering a specific input while constituting the snapshot matrix, can be used to represent with a similar level of accuracy the dynamics of the full structure in case of different excitation, in terms of e.g.: frequency content and therefore, excited vibration modes.

In the forthcoming numerical examples, we will set $p > 0.99$ to ensure accuracy. As a case study, we investigate the capability of POD in speeding up the computations by considering the Pirelli Tower in Milan, see Fig. 3.2. The building features 39 stories, and its total height is about 130 m. The plan dimensions of the standard floor are approximately 70 × 20 m. The structure is entirely made of CIP reinforced concrete. The structure is assumed to behave elastically, with lumped masses at each storey that basically undergo horizontal displacements. Such an assumption may be far from reality if the rigid diaphragm assumption does not hold true for vertical displacements of all the nodes at the same floor.

Fig. 3.2

The Pirelli tower in Milan, Italy

We start with a three-dimensional finite element discretization of the whole building featuring 6219 DOFs (Barbella et al. 2011). For the sake of simplicity, we have neglected the damping effect; hence, in a relative frame moving with the basement of the tower, the undamped equations of motion of the structure are written as:

$$\varvec{M}\ddot{\varvec{u}} + \varvec{Ku} = - a(t)\varvec{MB}\,$$

(3.23)

where $a(t)$ denotes the earthquake-induced acceleration time history, whereas B is a Boolean matrix of appropriate dimension which defines the shacked DOFs. To simplify the problem, static condensation has been adopted to keep out the vertical displacements of the floors. By partitioning the nodal displacements u into horizontal $\varvec{u}_{h}$ and vertical $\varvec{u}_{v}$ components, we can write:

$$\left[{\begin{array}{*{20}c} {\varvec{M}_{h}} & 0 \\ 0 & {\varvec{M}_{v}} \\ \end{array}} \right]\left[{\begin{array}{*{20}c} {{\ddot{\varvec{u}}}_{h}} \\ {{\ddot{\varvec{u}}}_{v}} \\ \end{array}} \right] + \left[{\begin{array}{*{20}c} {\varvec{K}_{hh}} & {\varvec{K}_{hv}} \\ {\varvec{K}_{vh}} & {\varvec{K}_{vv}} \\ \end{array}} \right]\left[{\begin{array}{*{20}c} {\varvec{u}_{h}} \\ {\varvec{u}_{v}} \\ \end{array}} \right] = - a(t)\left[{\begin{array}{*{20}c} {\varvec{M}_{h} \varvec{B}_{h}} \\ {\varvec{M}_{v} \varvec{B}_{v}} \\ \end{array}} \right] .$$

(3.24)

Keeping only the horizontal DOFs only in the equations of motion, to be thereafter managed by POD, we arrive at:

$$\varvec{M}_{h} {\ddot{\varvec{u}}}_{h} + \left[{\varvec{K}_{hh} - \varvec{K}_{hv} \varvec{K}_{vv}^{- 1} \varvec{K}_{vh}} \right]\varvec{u}_{h} = - a(t)\varvec{M}_{h} \varvec{B}_{h}$$

(3.25)

where now $\varvec{u}_{h} \in {\text{R}}^{39}$.

To obtain the reduced model, the building has been assumed to be shaken by the well-known El Centro earthquake, whose time versus acceleration record, together with its relevant fast Fourier transform, is reported in Fig. 3.3. To give an idea about the number of vibration modes that may be excited by such earthquake, the first natural eigen-frequencies of the structure (see also Table 3.1) are denoted by red vertical lines in Fig. 3.3b. It can be deduced that only the first five eigen-modes of structure can be effectively excited as the power of the spectra of the accelerogram is intuitively seen to be small for the frequencies higher than the 6th natural frequency of the structure.

Fig. 3.3

Top May 18-1940, El Centro accelerogram (east–west direction); bottom relevant FFT

Table 3.1 First natural frequencies of the building

A comparison among the dynamics of the original 39-DOF system and the responses of reduced order models at varying accuracy index p (see Eq. (3.20)) has been performed. The link between p and the retained DOFs in the reduced systems is reported in Table 3.2. The result reported in Figs. 3.4 and 3.5 compare the time histories of (lateral) displacements, velocities and accelerations of the 20th and 39th (roof) floors, respectively, with the target values which are available from the simulations. In these plots, the blue vertical line indicates the end of the time window within which the snapshots are collected; hence, only around t = 4 s all the reduced order analyses start departing from the full model response.

Table 3.2 Outcomes achieved through POD, in terms of accuracy p and speedup as functions of the number of DOFs retained in the reduced order model

Fig. 3.4

Time histories of the horizontal, displacement (top), velocity (middle) and acceleration (bottom) of the 20th floor, as induced by the El Centro earthquake

Fig. 3.5

Time histories of the horizontal displacement (top), velocity (middle) and acceleration (bottom) of the 39th floor, as induced by the El Centro earthquake

To have a more clear view of the time histories, a close up of the last 5 s of the time histories of 20th floor is presented in Fig. 3.6. By making a comparison between time histories of displacements, velocities and accelerations, it can be observed that two POMs are sufficient for a reduced model to accurately reproduce displacements of the full model; however, at least four POMs are necessary to feature the same level of accuracy for velocities and accelerations as well. By investigating the FFTs of the aforementioned time histories (see Figs. 3.7, 3.8 and 3.9), it is shown that in the FFT of the displacement time histories, only the two first natural modes are effectively excited. Instead, in the velocity and acceleration time histories, looking at the FFTs, it is observed that the six and seven first natural frequencies are effectively excited. Such a trend suggests that a reduced model that retains a few POMs may feature a better accuracy in reconstruction of the displacements of the system, when compared with velocities and acceleration responses.

Fig. 3.6

Close up of the time histories of the horizontal displacement (top), velocity (middle) and acceleration (bottom) of the 20th floor, as induced by the El Centro earthquake

Fig. 3.7

FFTs of the horizontal displacements of the storeys as induced by the El Centro earthquake at a 20th and b 39th floors

Fig. 3.8

FFTs of the horizontal velocities of the storeys as induced by the El Centro earthquake at a 20th and b 39th floors

Fig. 3.9

FFTs of the horizontal accelerations of the storeys as induced by the El Centro earthquake, a 20th (top), and b 39th floor

Moving to the speedup obtained by reducing the order of the full model, the results here discussed have been obtained with a personal computer featuring and Intel Core 2 Duo CPU E8400, with 4 Gb of RAM, running Windows 7 × 64 as operating system and performing the simulations with MATLAB version 7.6.0.324. The speedup values reported in Table 2.1 testify the dramatic decrease of the computing time obtained through POD, and reveal how powerful this methodology can be to approach real-time computing.

Previous figures have reported the results concerning time histories of two representative storeys of the structure: 20th storey is the mid floor and 39th storey is the last floor (roof) of the building. To further test the efficiency of the reduced models in reconstructing snapshots of the system, and therefore assess the capacity of the methodology in tracking the dynamics of all the system DOFs, two time instants are selected to assess the accuracy: Fig. 3.10a and b show snapshots taken in t = 10 s and t = 30 s of the analysis. At t = 30 s, the deformation of the structure is rather similar to a line with constant slope that is the reduced model with two POMs can reconstruct the relevant snapshot; however more POMs are required to appropriately approximate snapshot taken at t = 10 s, since the shape of the building is more complicated and higher modes are playing more significant role when compared to t = 30 s.

Fig. 3.10

Snapshots of the horizontal storey displacements as induced by the El Centro earthquake. Top t = 10 s and bottom t = 30 s

Another global feature of the reduced model which may be of interest for design practice is the envelope of the displacement, which is reported in Fig. 3.11. It is observed that even the reduced model with a single POM has an acceptable performance in reconstructing the envelope, even though it underestimates the envelope itself. By increasing the flexibility of the reduced model through additional POMs, as the higher POMs are retained in the analysis, it is observed that the envelope of the reduced model nearly matches that of full one.

Fig. 3.11

Envelope of horizontal storey displacements, as induced by the El Centro earthquake

To assess the efficiency of the reduced models in retaining the energy of the system, we now compare the resulting time histories of kinetic and potential energies of the system (see Fig. 3.12), respectively defined as: $E_{k} = \frac{1}{2}\dot{\varvec{u}}^{T} {\varvec{M}}{\dot{\varvec{u}}}$ and $E_{p} = \frac{1}{2}\varvec{u}^{T} \varvec{Mu}$ for the full model; $E_{kl} = \frac{1}{2}\dot{\varvec{\alpha }}^{T} \varvec{M}_{\varvec{l}} \dot{\varvec{\alpha }}$ and $E_{pl} = \frac{1}{2}\varvec{\alpha}^{T} \varvec{K}_{\varvec{l}}\varvec{\alpha}$ for ROMs. The cumulative discrepancy of the energies of the reduced models from the target one is considered as well (see Fig. 3.13). It is seen that the energy time histories of the 4-DOF reduced model well match those of the full model. To have more insight into the ability of the models to preserve energy of the system, the cumulative discrepancies of kinetic and potential energies are reported as well. It is seen that as the number of the DOFs of the reduced model increases the slope of the relevant line decreases, it means the rate of accumulation of the discrepancy decreases. Besides, it is observed that the accumulation of the discrepancies features a line with an almost constant slope implies that at different time intervals of the analysis, the amount of energy loss is the same. It means that the rate of energy loss is constant; hence, the accuracy of the reduced model in terms of energy preservation is constant over the interval of the analysis.

Fig. 3.12

Time histories of a kinetic and b potential energies

Fig. 3.13

Time evolution of cumulative discrepancy between full model and reduced order model, in terms of a kinetic, b potential energies

From this point on, we examine the accuracy of the reduced models that are built via snapshots resulting from excitation by the El Centro earthquake, when the building are shaken by other seismic records. In this regard, as an instance, we consider the May-1976 Friuli earthquake which time history of its acceleration records along with the relevant FFT are shown in Fig. 3.14. To have an idea concerning the number of natural frequencies that are covered by this seismic action, again the red vertical lines (as indicator of the natural frequencies of the structure) are drawn in the figure to allow for understanding the number of eigen-modes which is excited by accelerogram of the relevant earthquakes. By an intuitive comparison of Figs. 3.3 and 3.14, it is observed that a different amount of eigen-modes of the structure are excited by the two earthquake records.

Fig. 3.14

Top May 6-1976, Friuli earthquake and bottom relevant FFT

Let us now consider the time histories of displacement, velocity and acceleration of the 39th storey (see Fig. 3.15). It is seen that, while a 2-DOF reduced model satisfactorily mimics the behavior of the full model in terms of displacement, a 4-DOF reduced model is required to match the full model in terms of velocity and acceleration time histories. The number of POMs required for reconstructing the whole state of the structure, when it is shaken by Friuli earthquake, is the same as if it was shaken by El Centro earthquake. This fact shows that a reduced model built by the POD may be robust to change in the excitation source.

Fig. 3.15

Time histories of the horizontal floor displacement (top), velocity (middle) and acceleration (bottom) of the 39th floor, as induced by the Friuli earthquake

By investigating the FFTs of the aforementioned reported time histories (see Fig. 3.16a), the trend seen in the time histories of the state reconstruction is corroborated: one can see there are several peaks in the displacement response of the structure, when shaken by Friuli record; similarly to the FFTs of the structure when subjected to El Centro record, moving from displacement to velocity and acceleration FFTs, the number of peaks increases. Therefore, the number of POMs is required to match the FFT of the response of the structure increases.

Fig. 3.16

FFTs of the 39th floor, displacement (top), velocity (middle) and acceleration (bottom) as induced by the Friuli earthquake

In Fig. 3.17 (top), it is observed that out of the several spikes in FFT of the displacement, four are coincident with the system natural frequencies. As DOF reduced model is able to match only the first spike, however a two DOF reduced model matches the two of the spikes relevant to natural frequencies, the reduced models featuring three and four DOF models are matched up to third and the fourth spikes coinciding with the third forth natural frequency of the system, respectively. Considering the velocities and accelerations, the same trend is observed; however in latter cases, more natural vibration modes are effectively excited. Hence, the accuracy of a reduced model in reconstructing the acceleration responses of the system is not the same as the velocities and displacements.

Fig. 3.17

Close up of FFT of the horizontal displacement (a), velocity (b) and acceleration (c) of 39th floors, as induced by the Friuli earthquake

The performance of the reduced models in approximating snapshots of the system are once again tested at t = 10 s and t = 30 s. Looking at Fig. 3.18, it is seen that at t = 10 s the state of the system is similar to a line with constant slope; hence, all reduced models feature more or less similar accuracy; however, at t = 30 s the state of the structure is more complicated and at least four POMs are required to approximate the considered snapshot.

Fig. 3.18

Snapshots of the horizontal storey displacements at (top) t = 10 s, and (bottom) t = 30 s, as induced by the Friuli earthquake

Concerning the envelope of the displacements (see Fig. 3.19), it is observed that even a two DOF reduced model is matched with the envelope featured by the full model. It is observed that, in the vicinity of the 25th floor, there is a break in the envelope of the structure, while in the envelope of floor displacements relevant to the El Centro earthquake such a break is not observed. This is due to the fact that the range of frequency content of Friuli earthquake is wider than that of El Centro earthquake, see Figs. 3.3b and 3.14b, it results in excitation, and therefore contribution of higher natural modes in the response of the structure and as a consequence the shape of the structure may become more complicated.

Fig. 3.19

Envelope of horizontal storey displacements, as induced by the Friuli earthquake

To evaluate the accuracy of the reduced models concerning the energies, accumulated discrepancies has been considered; as previously, the time histories feature the same features of those related to El Centro record. Figure 3.20 shows the accumulated discrepancies of kinetic and potential energies for two scenarios: the continuous lines represent the case in which snapshots are related to the El Centro excitation, instead the dot lines stand for the case in which snapshots are related to Friuli record. It is worth recalling that in both cases, the reduced and full models are shaken by Friuli record. It is seen that, despite the fact that the reduced models are constructed by different inputs in simulations, the accumulated discrepancies nearly coincide. However in this case, the accumulated discrepancies appears to be bilinear: the graphs look similar to an straight line which changes its slope at t = 30 s. This is due to the fact that the amplitude of the excitations increases at the vicinity of the t = 30 s, the increase in the energy of input excitation therefore changes the rate of accumulation of the discrepancies changes.

Fig. 3.20

Time evolution of cumulative discrepancy between full model and reduced order model, in terms of (a) kinetic, and (b) potential energies. Comparison between outcomes of the reduced order models trained by El Centro and Friuli earthquakes

To conclude this section, we further assess the performance of the already built reduced models when excited by January 17-1995 Kobe acceleration record. In Fig. 3.21, the acceleration time history and its relevant FFT is presented. Once more, one can observe that the frequency content of this record is different from those of El Centro and Friuli. Figure 3.22 presents the time histories of displacement, velocity and acceleration of 39th storey. The situation is rather similar to the two previous cases: concerning displacements, reduced models retaining two or more DOFs nearly coincide with the full model, whereas dealing with velocity and acceleration a four DOF model is necessary to fully capture the dynamics of the system (Fig. 3.23).

Fig. 3.21

a January 17-1995, Kobe earthquake and b relevant FFT

Fig. 3.22

Time histories of the horizontal displacement (top), velocity (middle) and acceleration (bottom) of the 39th floor, as induced by the Kobe earthquake

Fig. 3.23

Snapshots of the horizontal storey displacements at (top) t = 10 s, and (bottom) t = 30 s, as induced by the Kobe earthquake

Considering snapshots and envelope of the displacements of the system (see Figs. 3.24 and 3.25), a reduced model consisting of a single DOF is not able to feature the dynamics of the system similar to the case shaken by El Centro and Friuli earthquake. To assess the global efficiency of the reduced model when subject to Kobe record (see Fig. 3.25), once more, one can observe that the ability to retain energy is independent of the training stage. The reduced models have the same number of DOFs, no matter how many snapshots are collected from simulation of El Centro or Kobe earthquake simulations, which nearly feature the same level of accuracy.

Fig. 3.24

Envelope of horizontal storey displacements, as induced by the Kobe earthquake

Fig. 3.25

Time evolution of cumulative discrepancy between full model and reduced order model, in terms of (top) kinetic, and (bottom) potential energies. Comparison between outcomes of the reduced order models trained by El Centro and Kobe earthquakes

Through the results shown in this Section, it has been revealed that prediction capabilities of POD-based reduced order models when dealing with different seismic excitations along with their high speed-up in computation makes them suitable candidates for models used in online and real-time structural health monitoring.

3.7 Summary and Conclusion

In this chapter, we have investigated the capability and efficiency of the POD in reducing the order of dynamic structural systems. In its SVD description, the POD is expected to find the directions in which retain the maximum energy of the system, whereas its PCA explanation is based on the search for the directions which preserve the maximum variability of the set of samples, which are collected into the so-called matrix of snapshots. Handling snapshots collected in an initial time window, we have built the reduced model through a coupling of POD and Galerkin projection.

To assess the performance of the studied methodology, the Pirelli Tower in Milan has been assumed to be shaken by an earthquake. Concerning accuracy issues, time histories of the state of the system (storey displacements, velocities and accelerations), together with their associated Fourier transform, have been compared with their real values available through the simulations. The power of the order reduction method in preserving the energies of the system is tested via a comparison of their time histories with those of full model. It has been observed that energy time histories of a 4-DOF reduced model nearly coincided with target values.

When dealing with accuracy versus sped-up, it has been shown that the POD can decrease the number of DOFs from the original 39 (one at each storey) to just 1, guaranteeing an accuracy of 0.99 (1 being featured by the original model) according to what is explained in this study, and leading to a speedup in the computations higher than 500. We have also shown that, to further match higher order frequency oscillations (accuracy of 0.99999), the retained degrees of freedom result to be increased to 4, still obtaining a speedup higher than 200.

It has been shown that the POD based reduced models are robust to a change of loading as well; moreover, the models built by snapshots resulting from simulations of the full model subject to El Centro record feature the same level of accuracy when are shaken by Kobe and Friuli record.

In following chapters, the reduced model built by POD will be incorporated into Bayesian filters to assess the capabilities of such an approach in state estimation of non-damaging and dual estimation of damaging structures, possibly detecting and locating the occurring damage.