DSpace at VNU: Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds

DSpace at VNU: Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildin...

Reduced-order wavelet-Galerkin solution for the coupled,

nonlinear stochastic response of slender buildings

of a tall structure and (2) a multi-degree-of-freedom reduced-order full building model.Compactly supported Daubechies wavelets are used as orthonormal basis functions inconjunction with the Galerkin projection scheme to decompose and transform the coupled,nonlinear differential equations of the two models into random algebraic equations in thewavelet domain Methodology, feasibility and applicability of the WG method are investigated

in some special cases of stiffness nonlinearity (Duffing type) and damping nonlinearity der-Pol type) for the single-degree-of-freedom model For the reduced-order tall buildingmodel the WG method is used to solve for dynamic coupling, aerodynamics and transient windload effects Computation of“connection coefficients”, effects of boundary conditions, waveletresolution and wavelet order are examined in order to adequately replicate the dynamicresponse Realizations of multivariate stationary and transient wind loads for the buildingmodels are digitally simulated in the numerical computations

(Van-& 2015 Elsevier Ltd All rights reserved

1 Introduction

Tall buildings are prone to wind-induced stochastic vibration and large-amplitude response, which originate fromcomplex fluid–structure interaction, nonlinearity and aerodynamic coupling (e.g.,[1–4]) The large-amplitude wind-inducedresponse of buildings can lead to serious engineering problems, including human discomfort[5–7]and cumulative fatigue

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/jsviJournal of Sound and Vibration

A connection coefficient matrix, associated with

the linear term (sdof model)

direction

^Apðω; zp; tÞ deterministic modulation function

Axx; Axy coefficient matrices associated with the x

B connection coefficient matrix, associated with

nonlinear stiffness term (sdof model)

bx coefficient vector associated with the x

coor-dinate, WG method

by coefficient vector associated with the y

coor-dinate, WG method

C connection coefficient matrix associated with

nonlinear damping term (sdof model)

CD, CL static along-wind and cross-wind force

coefficients

C0D; C0

L first derivatives of static force coefficients CD,

CLwith respect to attack angle

Cohu ;pq
ω; zp; zq

between two sections (floors) of coordinates

zp; zq

cjk detailed wavelet coefficients at very small

scales joj0

cj0k wavelet approximation coefficients at the j0-th

resolution

d1, d2 indices of derivative orders

Fb;rðz; tÞ distributed buffeting force per unit height

Fa ;rðz; t; r; _r; €rÞ distributed self-excited force per

Fa ;xðz; t; _x; _yÞ distributed self-excited force per unit

height in the x coordinate

Fa ;yðz; t; _x; _yÞ distributed self-excited force per unit

height in the y coordinate

ffkg stationary wind force vector, WG method

f0fkg transient wind force vector, WG method

Huðω; zÞ lower triangular matrix, decomposed from

stationary cross spectral matrix

k stiffness coefficients (sdof model)

Mr; Cr; Kr generalized mass, damping and stiffness

coefficients of r-th coordinate

Mx; Cx; Kx generalized mass, damping and stiffness

coefficients in the x coordinate

My; Cy; Ky generalized mass, damping and stiffness

coefficients in the y coordinate

MðmÞj;k moment of m-th order of the scaling function

mðzÞ distributed mass of the continuous building

model per unit height

Nn; Nx original and extended computational domain,

wavelet expansion

Qb;rð Þt generalized time-dependent buffeting force

Qa ;rðt; r; _r; €rÞ generalized motion-dependent

self-excited force

Qxx; Qyx generalized motion-induced force terms

asso-ciated with the x coordinate

Qb ;xð Þt generalized turbulent-induced force in the x

coordinate

Qyy; Qxy generalized motion-induced force terms

asso-ciated with the y coordinate

Qb;yð Þt generalized turbulent-induced force in the y

coordinate

qb;x;fkg, qb;y;fkg approximate buffeting forces associated

with x-, y-coordinates, WG method

S0uðω; zÞ stationary cross spectral matrix

Suðω; z; tÞ transient cross spectral matrix

UH reference mean wind velocity at height z¼H

Up
zp; t along-wind transient wind velocity at the

building floor node p

Up
zp; t along-wind time-varying mean wind velocity

at the building floor node p

ufkg stationary wind velocity vector, WG method

u0fkg transient wind velocity vector, WG method

u zð ; tÞ along-wind zero-mean wind fluctuation

up
zp; t along-wind spatially-correlated zero-mean

stationary wind fluctuation

u0p
zp; t along-wind spatially-correlated zero-mean

transient wind fluctuationνðz; tÞ cross-wind zero-mean wind fluctuation

νpðzp; tÞ cross-wind spatially-correlated zero-mean

stationary wind fluctuation

coordinates

€xfkg,€yfkg WG acceleration coefficient vectors in x and y

coordinatesPlease cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear

damage [8–10] The wind-induced stochastic structural dynamics of tall vertical structures in general, including tallbuildings, slender towers and wind turbines can be investigated by means of either simplified structural models (e.g.,[8]) orreduced-order models[1–4], which utilize modal superposition truncated to pre-selected fundamental vibration modes.Structural nonlinearity and aerodynamic coupling of these structures can often be employed in formulating the wind-induced stochastic dynamic equations (e.g., [11]) In particular, nonlinearity in the building response has been recentlyobserved as an emerging problem for tall buildings, for example due to the potential nonlinear interaction of the vibratingstructure with vortex-shedding effects (e.g.,[12]) The coupling of aerodynamic loads is often analyzed in the study of thestochastic response of tall buildings during extreme wind events[13,14] This approach involves the solution of coupledmotion equations, combining turbulence-induced buffeting forces and motion-induced forces Nevertheless, the solution ofthe coupled and nonlinear motion equations in the time domain, necessary in the case of transient wind loads, is still amajor challenge for these structures; it is seldom pursued since it may require computationally demanding procedures ofanalysis (e.g.,[15]) Currently, the solution of the wind-induced stochastic dynamics is often preferably carried out under theassumptions of linear structural response, uncoupled fluid–structure interaction and multivariate stationary wind loading.These assumptions enable to convert the stochastic equations of motion of a tall building to frequency domain via Fouriertransform-based analysis[1–4]; the equations of motion can subsequently be combined with empirical or experimentally-based power spectral density functions of the stationary turbulent wind field to describe the loading [16–18] Thistransformation allows the coupled and nonlinear motion equations to be converted to a simpler algebraic form but requiresthe wind loads to satisfy stationarity conditions.

The hypothesis on multivariate stationary wind loads is also still preferably used to derive the stochastic response of atall building due to the need for large computational resources and additional modeling complexity in the case ofnonstationary wind loads Recent investigations on wind loading environment have also indicated that the wind loads,originating from non-synoptic winds such as thunderstorms, downburst storms, hurricanes and tornadoes are rapidlyvarying Therefore, simulated transient wind loads should be employed to study the stochastic response and for the design

of tall buildings and slender vertical structures under such environmental conditions[19–21] In recent years, researchinvestigations have been directed towards the modeling of nonstationary winds and the simulation of“time-frequency-dependent” response of tall buildings, slender vertical structures [22–24] and long-span bridges [25,26] It has beenobserved that, if a stationary-wind-based analysis method is employed to study the dynamics, this can cause under-estimation in the assessment of the structural response under loads, which originate from a transient/nonstationary windevent [19,20] Moreover it has been agreed that the numerical solution of the dynamic equations with transient/nonstationary loads is still problematical when nonlinearity is included[27] In spite of all the advances, an efficient and

x, y along-wind and cross-wind coordinates of the

building models

xk, yk WG approximation coefficients

x tð Þ, _x tð Þ, €x tð Þ along-wind displacement, velocity and

acceleration of the along-wind motion

x0,_x0 initial displacement and velocity

y tð Þ, _y tð Þ, €y tð Þ cross-wind displacement, velocity and

acceleration of the cross-wind motion

z vertical coordinate of the building

zp vertical coordinate of the generic p-th discrete

floor node

Λd1d2 vector of unknown connection coefficients

δ0;l  k Kronecker delta

φ hð Þ father scaling function

φj ;kðhÞ dilated and translated scaling function

ψ hð Þ mother wavelet function

ψa ;bð Þh dilated and translated wavelet function

Ω0 ;0 2-term connection coefficient matrix,

j dilation parameter (resolution level)

Operators

adequate simulation method for the analysis of the wind-induced coupled stochastic dynamic response of tall buildings andslender vertical structures is currently not available.

In recent decades, wavelet transform (WT) and wavelet analysis have gradually emerged as a powerful tool forengineering and scientific computations in the mixed time–frequency domain Wavelets are either piecewise functions or afamily of functions, containing multiple sub-functions, in which each sub-function can dilate and translate from a basic

“mother” wavelet function within a finite domain [28] Wavelets are localized, rapidly-decaying, zero-mean oscillatoryfunctions The wavelet transform computes the convolution operation between a“signal” and a family of wavelets Theadvantage of the WT over the Fourier transform (FT) is that, instead of using a family of complex harmonic functions todecompose an aperiodic signal, a family of wavelets is employed Wavelets can represent not only symmetric, smooth orregular signals, but also asymmetric, sharp or irregular signals, which are typical of nonstationary and nonlinearphenomena, simultaneously on the time–frequency plane and in the context of multi-resolution analysis (MRA) The WTcan be carried out either as continuous wavelet transform (CWT) or discrete wavelet transform (DWT) The CWT is applied

in signal analysis using both orthogonal and non-orthogonal wavelets, while the discrete wavelet transform (DWT) isemployed for signal analysis and processing using only orthogonal wavelets [28] In the context of the MRA, the WTexamines high-frequency components of the signal with a“fine” frequency resolution but “coarse” time resolution, and low-frequency components with “fine” time resolution and “coarse” frequency resolution [28] Although one cannotsimultaneously characterize both the“fine” time resolution and the “fine” frequency resolution in each frequency band

of a signal, the features of the WT are adequate for analyzing practical signals In wind engineering, the CWT, using eitherreal or complex Morlet mother wavelets, has been applied to the examination of wind turbulence, pressures andaerodynamic forces [29–31], and to the detection of correlation and coherence in the pressure or wind loads [32,33].Nevertheless, no research studies exist on the application of the WG method for the modeling of wind fields and thesystematic simulation and analysis of the stochastic response of civil structures

Wavelets have attracted more and more interest in engineering computations since Daubechies[34]derived the theory

of compactly supported wavelets in orthonormal bases, known as the Daubechies wavelet family Daubechies waveletsallow for a wide range of localization and dilation operations within the multi-resolution framework Daubechies waveletscan describe the analyzed signal at various levels of resolution; this property also makes them particularly attractive fordeveloping approximations to exact solutions in a dynamic problem The Galerkin method belongs to a class of numericalapproximation methods for converting a continuous operator and system (such as a differential equation) into a discretesystem using basis functions The Galerkin approximation method has been extensively employed for static and dynamicproblems in engineering The advantageous properties of orthogonality and aptitude to multi-resolution approximation ofthe Daubechies wavelets can“empower” the Galerkin approximation method Furthermore, the Daubechies wavelets aremore suited for Galerkin approximation than other popular wavelets, like harmonic wavelets, because they are not onlypiecewise and orthogonal functions, but also compactly supported functions, which accelerate the Galerkin approximation

In recent years, the orthogonal-basis and multi-resolution features of the compactly-supported Daubechies wavelets havebeen integrated with the Galerkin expansion method to solve various kinds of ordinary differential equations[35–37]anddynamical differential equations [38] This wavelet-Galerkin analysis method (WG) is progressively emerging as anapproach to build approximate solutions to various engineering problems In structural dynamics, the WG method hasbeen employed to study vibrations of continuous single-degree-of-freedom (sdof), two-degree-of-freedom (2dof) systemswith linear and nonstationary parameters [38–40], nonstationary seismic response of sdof systems [41], continuousmechanical systems[42], stochastic response of buildings[43]and long-span bridges[44] This partial review of the existingliterature shows that this numerical method is receiving more and more attention from the technical community alsobecause of technology and computer power evolution

It is noteworthy that the approximate solutions to differential equations often rely on numerical differentiation methods,which employ the Taylor expansion of the function to be sought as an approximation Solutions to differential equations bynumerical differentiation are efficient and easy to implement Their efficiency and accuracy is excellent for linear systemswith conventional boundary or initial conditions These methods, however, often suffer from a loss of accuracy if they areapplied to large and nonlinear systems and, especially, they cannot effectively deal with coupled dynamical systems (e.g.,[45]) On the other hand, Galerkin approximation-based methods, and the WG method in particular, are derived usingenergy principles and are advantageous for nonlinearly-coupled dynamic problems with natural boundary conditions andcomplex geometries[45–47] The main application of the WG analysis is to approximate coupled or/and nonlinear motionequations by transforming them into“user-friendly” random algebraic coefficient equations in the wavelet domain, whichcan be solved numerically in a simple way In particular, the WG method could support the analysis of stochastic structuralresponse due to transient/nonstationary wind loads

Nevertheless, limitations and computational challenges of the WG method have also been observed [38–44] Theseinclude an accurate treatment of boundary conditions, arbitrary time range,“connection coefficient” estimation, resolutionanalysis and computational issues in relation to computing time and large memory requirements Among the aforemen-tioned limitations, the management of boundaries and adequate resolution analysis are the main problems Many recentapplications of the WG solution to ordinary differential equations have dealt with simple Dirichlet boundary conditions inthe unit time[35,38,42] In these applications the resolution analysis of the wavelets can be neglected and the treatment ofboundary conditions is drastically simplified These aspects have prevented the application of the WG method to a widerange of problems, until recently Fortunately, the problems of “adaptable” treatment of boundary conditions to betterPlease cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear

model the two ends of a finite-duration analytical signal along with an improved computation of the wavelet connectioncoefficients[47]have been recently resolved[40,48–50] Latto et al.[47]firstly proposed the methodology for computingthe connection coefficients with unbounded domain, using the D6 Daubechies wavelet, at the resolution level j¼1 only.However, this initial attempt to compute the connection coefficients did not adequately address the issue of boundaryconditions, since it did not provide accurate values at the endpoints of a signal on a bounded domain [47] Subsequentstudies examined a “more flexible” management of the boundary conditions in the case of differential equations, andimproved the computation of the connection coefficients of the Daubechies wavelets for a bounded domain at variousresolution levels [48–50] Recently, the WG method was modified to better model the two ends of a finite-durationanalytical signal by extending the original WG method to any boundary conditions and by suitable assemblage of theconnection coefficients in matrix form [40,43,44] Furthermore, the open question on the resolution of the computedwavelets has been lately solved as a result of a fitting with the resolution of analytical signal[40].

This paper builds on the recent advances in WG analysis by expanding and adapting the existing method to compute thestochastic structural response of two building models, accounting for aerodynamic coupling, system nonlinearity andtransient/nonstationary wind loads The WG analysis method is used to simulate the nonlinear stochastic dynamics of a

35 m-high equivalent vertical structural model with lumped masses and aerodynamic properties (sdof), and the coupledstochastic dynamics of a reduced-order model examining the response of the 183 m-high benchmark CAARC building[52]with coupling between turbulent-induced forces and motion-induced forces Both stationary and transient wind loading areconsidered Realizations of the evolutionary transient wind fields along the height of each structure are artificially simulatedfrom stationary wind processes by employing the amplitude modulation function technique[56,57]

In summary, the main objectives of this study are to

1) Employ the latest developments of wavelet analysis to advance the study of stochastic structural dynamics withemphasis on the coupled, nonlinear response of vertical structures due to transient stochastic wind loads

2) Investigate fundamental features and shortcomings of the WG analysis method, such as the extension of the WGformulation to arbitrary boundary conditions, the estimation of the wavelet resolution and the selection of the order ofDaubechies wavelets

3) Generalize the computation of the wavelet connection coefficients in a matrix form for multi-dof dynamic simulations.4) Verify the feasibility of the WG method for numerically solving the stochastic response of both simplified tall structures(sdof) and reduced-order tall building models due to transient wind loads

2 Galerkin approximation

The Galerkin method is a projection method that seeks for an approximating solution through the projection of the exactsolution onto a subspace spanned by a basis of suitable functions The Galerkin projection approximates an exact solutionxðhÞ to an equation Ax hð Þ ¼ f with h being a generic variable The approximating solution xnðhÞ, based on an inner-productspace of Nnfinite dimensional functions, is usually defined as[45]

“Galerkin finite-element method”, the Dirac Delta function in the “Galerkin collocation method”, proportional functions of

the basis function in the“Galerkin least squares method”[45,46] The weight function is orthogonal to the basis function inthe WG approximation In this application, they are the Daubechies wavelets, which have the advantageous properties oflocalization, differentiability, piecewise continuity, orthogonality, multi-resolution, and compact support[28].

The wavelets are zero-mean continuous oscillatory functions (i.e.Rþ 1

 1ψ hð Þdh ¼ 0), which can be n-times differentiable.The wavelets are well localized both in scale and time domains, which means that they and their n-th-order derivativesdecay very fast They also satisfy the admissibility conditionRþ 1

 1j^ψðωÞj2dω=jωjo1 with ^ψðωÞ being the Fourier transformcoefficient of the wavelet[28] Wavelets dilate and translate in the time and frequency domains, depending on the scaling(a) and translation (b) Projection of any signal x(h) onto a subspace of the wavelet at scale a is expressed as[28]

where ak denotes scaling coefficients Only a finite number of scaling coefficients ak is non-zero, thus the scaling function

φ hð Þ is said to have a compact support The mother wavelet function is derived from the father scaling function as

Projection of signal x hð Þ on the N-order Daubechies wavelet, based on the pair of scaling function and wavelet function at

a pre-selected scaling level j0(resolution) can be expressed as[34]

where cj

0 ;kdenotes the“approximation” coefficients at the scale j0; cj

0 ;k¼ 〈x hð Þ; φj0;kðhÞ〉 and cj;k¼ 〈x hð Þ; ψj ;kðhÞ〉 are “detailed”coefficients at smaller scales j0oj (higher frequencies) One generalizes that the father scaling function deals with highscales (low frequencies), while the mother wavelet function is used for low scales (high frequencies) If the waveletexpansion in Eq.(10)is truncated at the scale j0(resolution level), after eliminating all components at the scales smaller than

j0(high frequency components), one has the approximation of a signal x(h) at the scale j0as x hð Þ Pþ 1

k ¼ 0cj0;kφj0;kð Þ.h

wavelets of various orders: D4 (N ¼4), D6 (N ¼6), D8 (N ¼8) and D10 (N ¼10) The support interval of scaling function andwavelet function widens in the time domain and the smoothness increases, when the order of the Daubechies waveletsincreases In other words, the Daubechies wavelets are less localized, but smoother with the increase of the order

4 Wavelet-Galerkin analysis

In the WG analysis the orthonormal and compactly supported Daubechies wavelets are employed as the basis functionsand weight functions in the Galerkin projection to find approximate solutions to dynamic problems If the time variable isdenoted by t, a generic motion variable x tð Þ can be expressed, at a pre-selected level of resolution j of the wavelet, as

x tð Þ ¼ XN x

k ¼ 1

In the previous equation k is a translation parameter (time index) defined on a finite-duration computational time

where _φj ;kðtÞ and €φj ;kðtÞ are the first-order and second-order derivatives of the scaling function, respectively

The derivatives of the wavelets can be obtained in the limit support, i.e., in the interval [0, N 1] The inner productsbetween approximating solutions of the displacement, velocity, acceleration in Eqs.(11)–(13) and the weight functions, as

-2

-1 0 1 2

Time(s)

Scaling function Wavelet function Scaling function

Wavelet function

-2 -1 0 1 2

Time(s)

Scaling function Wavelet function Wavelet function

Scaling function

-2 -1 0 1 2

Time(s)

Scaling function Wavelet function

Scaling function Wavelet function

-2 -1 0 1 2

Time(s)

Scaling function Wavelet function

Scaling function Wavelet function

Fig 1 Scaling functions and wavelet functions of Daubechies wavelets: (a) D2, (b) D6, (c) D8 and (d) D10.

indicated in Eq.(3), are

5 Estimation of wavelet connection coefficients

The 2-term connection coefficients can numerically be computed in the case of unbounded domain ½1; þ1 with theD6 Daubechies wavelet at the resolution level j¼1 only, using the d1, d2-times differentiation of the scaling functions, themoment equations and the normalization equation[48] Nevertheless, the connection coefficients must also be calculated

on a limited and bounded domain The issue of the boundary conditions at the two ends of the supported domain, neededfor computing the connection coefficients, can successfully be resolved by modifying the end values of the connectioncoefficients, obtained from unbounded domain[49], or by adding“fictitious intervals” at two ends of the supported domain[50,51] It seems that the latter approach is more suitable for estimating the connection coefficients in the context of WGcomputations involving systems with initial conditions[40] The analytical development is taken from previous studies[49,50], which are briefly summarized in this section

The general form of the 2-term connection coefficients, constructed for the scaling function at the j-th resolution level byconsidering a generic order of differentiation (d1and d2times), can be expressed as[49]

By using the recursive definition of wavelet scaling function, the integration can be simplified as follows[49,50], notingthatΩd1d2

 1hmφj;kð Þdh ¼ 0 with m¼0,1,…,h

N 1; this definition implies Mð0Þj;0¼ 0, by the property of the scaling functions The moments MðmÞ

j ;k on the interval [0,N  1]can be computed as[47]

MðmÞj;k ¼ 12ð2m1Þ

Xm

p ¼ 1

mp

!

km  pp  1X

q ¼ 0

pq

!

MðqÞj;0N  1X

k ¼ 1

akkp  q: (29)Therefore, an expression for the monomial term hmon the interval [0,N  1] can be obtained[49]

Finally, solutions to the connection coefficients are found by using the first 2N  4 rows from Eqs.(28)and(33)as[48,49]

375Λd1 d 2¼ ðN 1Þd0

The fictitious interval approach is used herein, in which the computational domain of the original signal (Nndiscretepoints) is expanded by adding (N  1) points to the left of the original computational domain (before initial time) and (N  1)points to the right of the original domain (beyond final time) The modified domain has (Nx¼Nnþ2N1) wavelet expansionpoints in the interval [1  N, NnþN1]

For the implementation of WG analysis, Nx-by-Nxsparse square matrices can be used for collecting the compactlysupported connection coefficients The construction of the connection coefficient matrices are presented in a later section

j ;k  l of D4, D6 and D8 with orders N ¼4,6 and 8,computed for pre-selected wavelet resolution level j¼ 6, and for the derivative orders d1,d2¼0,1 and 2 For example, thescaling function D4 requires 5¼(2N  3) 2-term connection coefficients with the support indices (k–l) on [2,1,0,1,2]; theD6 has 9 connection coefficients according to the index (l–k), supported on [4,3,2,1,0,1,2,3,4]; there are 13connection coefficients with (l–k), evaluated on [6,5,4,3,2,1,0,1,2,3,4,5,6] for the D8 It must be noted that the

WG analysis relies on the initial choice of the wavelet resolution (j) Some properties of the connection coefficients can also

be observed fromTable 1:Ω0 ;1

6 Treatment of wavelet resolution and boundary conditions

Adequate treatment of wavelet resolution and boundary conditions is essential to the computation of connectioncoefficients and the implementation of WG analysis The WG analysis“expands” a time-varying signal at the pre-selectedresolution j Therefore, the initial choice of wavelet resolution level (j) is required for the computation of the connectioncoefficients The resolution parameter of the Daubechies wavelets is 2jat a scale j; the resolution must be determined so thatthe scaling function is centered, given the number of discretization points, i.e., the sampling time of the original signal Thewavelet resolution (j) can be approximately found from the number of samples per unit time of the signal (Nx) or thesampling rate of the signal from the following relationship: Nx¼ 2j

As a result, the resolution level is fitted with the originalsignal as

In this study we followed the assumption, common to most numerical analysis and integration methods, that thesampling time Nxshould be taken as a fraction of the main vibration period of the structure Therefore, initial selection wasmade to sample the wind load and the wavelet in both input and output at intervals corresponding to frequency Nx¼100

Table 1

2-Term connection coefficients Ω d 1 ;d 2

j;k  l of the Daubechies wavelets D4, D6 and D8 at the resolution level j¼ 6.

Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear

Moreover, initial displacement and velocity at initial time t ¼0 are usually required as the boundary conditions of adynamic problem; in the case of a sdof problem, they can be defined as xð0Þ ¼ x0, _xð0Þ ¼ _x0 The initial conditions aretransformed into the wavelet domain using Eqs.(11) and (12); this leads to

N x

k ¼ 1

xkΩ0 ;1 j;l  k¼_x0: (37)

If Eq.(37)is expanded at the initial time t¼0 (backwards in time) in the computational domain [ N þ1,NnþN1], thecondition at k–l¼0 is needed Therefore, the initial conditions in Eq.(37)can be assembled in the first row and the last row

of the connection coefficient matrices, in which a unit value is placed at the index k–l¼0 on the first row, while Ω0 ;1

j ;0 must beplaced in the column with the same index, k–l¼0, on the last row The values x0and_x0are assembled as part of the vectorcollecting the loading or forcing terms in the wavelet domain; more details will be provided in the next sections A similarapproach enables to apply any arbitrary boundary conditions to any dynamical problem with arbitrary number of degrees offreedom

7 Wavelet-Galerkin method applied to the solution of nonlinear stochastic response of simplified tall structures

In this section, three stochastic dynamic problems along with their WG solution in the wavelet domain are illustrated.The problems describe the response of a simplified tall structure due to stationary and transient random lateral wind forces

In many cases, for pre-calculation and design feasibility studies of vertical structures, simplified sdof models can be used inboth static and dynamic problems (e.g., dynamics dominated by first-mode response [2]) The structural model is asimplified sdof system, in which lumped mass and wind force are concentrated at the top node of the structure Definitionand layout of the simplified building model are shown inFig 2

7.1 Linear stochastic dynamics and WG solution due to stationary random wind load

The governing motion equation of the building model in the x-along-wind direction, subjected to stationary randomalong-wind force, can be written as

Fig 2 Simplified tall structure model (schematic): (a) elevation and (b) cross-section.

velocity at the reference top node The effect associated with the mean wind speed UH, i.e., the static wind load is notinvestigated but could be readily included in the analyses Also, the effect of fluid–structure interaction is neglected at thisstage The along-wind fluctuating velocity uðtÞ at the top node is digitally simulated in this study in relation to a pre-selectedwind model Initial conditions are assumed as xð0Þ ¼ 0, _xð0Þ ¼ 0.

Eq.(38)is first converted to the wavelet domain Subsequently, the WG solution of the linear equation can be obtained bythe following steps: (i) time-varying system responses u; _u; €u and the wind force are approximated in the wavelet domainusing Eqs.(11)–(13); (ii) the inner product operations are employed in the wavelet domain using Eqs.(14)–(16); (iii) the WGsolution is found from the approximating functions of the responses Concretely, Eq.(38)is converted to the wavelet domainas

mΩ0 ;2xfkg þ cΩ0 ;1xfkgþkΩ0 ;0xfkg¼ ρUHApCDufkg: (39)

In the previous equation Ω0 ;2, Ω0 ;1, Ω0 ;0 are Nx-by-Nx connection coefficient matrices with Ω0 ;2¼ ½Ω0 ;2

p ;qNx by  Nx,

Ω0 ;1¼ ½Ω0 ;1

p ;qNx by  Nx,Ω0 ;0¼ I ¼ ½δp;qNx by  Nx; xfkgis an unknown Nx-element vector of the unknown wavelet coefficients,

xk; the subscript within braces“{k}” is used to describe the fact that the elements of this vector span the whole domain ofinvestigation Similarly, the vector ufkgis derived from the WG expansion of the turbulence realization and is an Nx-elementwind velocity vector; p, q are indices with 1  Nrp,qrNx–(N1) One could write Eq.(39)in a matrix form with compactnotation as

where A is Nx-by-Nx system connection coefficient matrix A ¼ mΩ0 ;2þcΩ0 ;1þkΩ0 ;0 Each element of this matrix isdetermined as Ap ;q¼ mΩ0;2

p;qþ cΩ0;1 p;qþkδp ;q; ffkgis the wind force vector ffkg¼ ρUhApCDufkg.The time-varying second-order stochastic equation has been transformed into first-order algebraic equation in thewavelet space, the solution of which is much simpler and computationally advantageous As outlined above, construction ofthe connection coefficient matrices, estimation of the resolution level and treatment of the boundary conditions are keyissues of the WG method The connection coefficient matrices are sparse matrices, in which only a few elements,appropriately located, are the non-zero connection coefficients, whereas the rest of the elements are zeroes The D6 wavelet(N¼ 6) is used for demonstrating the assemblage of the connection coefficient matrices The computational domain is[ N þ1,NxNþ1], with a total of Nxdiscrete points It utilizes Nxmoving D6 scaling functions at discretized points of thecomputational domain The 2-term connection coefficients of D6 wavelet,Ω0 ;2

k  l,Ω0 ;1

k  l, are a series of (2N–3)¼9 coefficients atthe pre-determined resolution level j, respectively, with the moving indices (k–l)¼ 4,3,…,3,4 (see Table 1) The 9connection coefficients are placed on each row of the connection coefficient matrices, in which the scalar termsΩ0 ;2

0 ; and

Ω0;1

0 with index (k–l)¼0 are located at the “center” of the series at a moving point on the computational domain As a result,the Nx-by-Nxconnection coefficient matrices can be divided in 9-by-9 blocks; the two key upper corner and lower cornerblocks,ΩB1;1,ΩBL;L, are presented inAppendix Afor clarification

The initial conditions, xð0Þ ¼ 0 and _xð0Þ ¼ 0, are transformed into the wavelet space as in Eq.(37) As discussed above,initial conditions are assembled in the first row and the last row of the connection coefficient matrices, by placing a unitvalue at the index (l–k)¼0, i.e., at the column of the first row corresponding to t¼0 (on the real signal domain), while Ω0 ;1

0

must be placed in the same column on the last row of the matrices At the same time, two zero values of the initialconditions xð0Þ ¼ 0 and _xð0Þ ¼ 0 are placed in the first and the last rows of the force vector ffkgon the right hand side of Eq.(40) These two operations are illustrated inAppendix B

Any available numerical technique can be applied for the solution of the linear algebraic equation system in Eq.(40).Since ffkgis random, the vector xfkgis also random After xfkgis found, the random velocities and accelerations of the sdofsystem can be determined as

in which _xfkgand €xfkgare resultant velocity and acceleration vectors of the wavelet coefficients, respectively

7.2 Stiffness-nonlinear stochastic dynamics and WG solution due to stationary random wind load

The nonlinear dynamic equation of the sdof model in the x-along-wind direction with cubic-stiffness nonlinearity(Duffing type), subjected to a stationary random along-wind fluctuating force, is

Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear

A similar procedure as in the previous section is used and the motion equation is transformed into the Daubechieswavelet space as

mΩ0 ;2xfkgþ cΩ0 ;1xkfkgþkΩ0 ;0xfkgþkεΩ0 ;0x3

fkg¼ ρUHApCDufkg: (43)

It is noted that the nonlinear stiffness term kεxðtÞ3

in Eq (42) has been approximated in the wavelet space as

Existing numerical methods, such as the Newton method, the Gauss–Newton method, the Levenberg–Marquardt method

or the“trust region” method (e.g.,[59]), can be applied for the solution of the nonlinear algebraic equation system.7.3 Damping-nonlinear stochastic dynamics and WG solution due to transient random wind load

The WG method is used in this sub-section to investigate the nonlinear stochastic dynamic response of a sdof system due

to downburst-type transient wind load The along-wind motion equation of the sdof stochastic system with dampingnonlinearity (Van-der-Pol type), subjected to downburst-like along-wind transient wind force, is expressed as

After applying the WG transformation, the stochastic nonlinear dynamic equation in Eq.(45)can be rewritten in theDaubechies wavelet space in a matrix form, as

mΩ0;2xfkg þ cΩ0;1xkfkg þkΩ0;0xfkgþcμΩ0;1x3

fkg¼ ρUhApCDu0fkg; (46)

AxfkgþCx3

The two previous equations are the same equation, with Eq.(47)being a compact and generalized representation of the

“expanded” system in Eq.(46) In Eq.(46)u0fkgdenotes the transient random wind velocity vector in the wavelet space; thequantities A, f0fkgdepend on the linear terms of Eq.(45)and are formed from the connection coefficient matrices and thesystem force vector, as outlined inSections 7.1and7.2and formulated in detail inAppendix A The quantity C ¼ cμΩ0 ;1is anew matrix, which contains the connection coefficients of the first-order derivative; it depends on the nonlinear dampingterm in Eq.(45) The nonlinear damping term in Eq.(47)is approximated, in the wavelet domain, by converting x tð Þ2_x tð Þ to

8 Wavelet-Galerkin solution of coupled stochastic dynamics of tall buildings (full model)

8.1 Coupled stochastic dynamics due to transient wind forces

Coupled stochastic dynamics and WG solution of a multi-degree-of-freedom reduced-order tall building, subjected tostochastic transient wind forces, are formulated in this section The first two fundamental bending modes of the tall building

in the“x” and “y” directions are taken into account.Fig 3illustrates the layout and typical cross-section (floor plan) of thetall building, and schematic information on wind loading A linearized expansion of the quasi-steady wind forces, acting atthe generic cross section of the building, is used to formulate the coupled aerodynamic problem in terms of both motion-induced and turbulent-induced forces, as well as the two fundamental bending modes The modal stochastic dynamicequation of the r-th mode is represented in the generic form of generalized coordinates, as[1–4]

Mr€r tð Þ þ Cr_r tð Þ þ Krr tð Þ ¼ Qb ;rð ÞþQt a ;rðt; r; _r; €rÞ; (48)where Mr; Cr; Kr are the generalized mass, damping and stiffness of the r-th mode, respectively; r ¼ fx; yg are dynamicmotion variables, associated with the generalized coordinates of the fundamental modes, designated as“x” or predomi-nantly along-wind and “y” as cross-wind The quantities Qb ;rð Þ and Qt a ;rðt; r; _r; €rÞ are, respectively, a time-dependentbuffeting force and motion-dependent aeroelastic force The mean wind force and consequent mean static response are

eliminated from Eq.(48)since the main objective of the analysis is the stochastic dynamic response The quantities in Eq.(48)can be found from the equations below:

The distributed lateral wind forces Fb ;rðz; tÞ and Fa ;rðz; t; r; _r; €rÞ are separately derived for the two designated coordinates,

r ¼x or r ¼y, by quasi-steady linearized aerodynamic theory in turbulent winds, assuming a rectangular prismatic crosssection (floor plan)– seeFig 3, as[3,4,43]

Fb;xðz; tÞ ¼ 1

2

 ρUðzÞ2B 2CD

u0ðz; tÞUðzÞ þ C0

u0ðz; tÞUðzÞ þ C
0LCDv0ðz; tÞ

In the previous equationsρ and U zð Þ are the air density and the mean-wind velocity profile; B is a reference width

CD; CL; C0D; C0L are the static force coefficients, normalized with respect to width B, and their first-order derivatives withrespect to the mean wind angle of attack; _x and _y are the “x-along-wind” and “y-cross-wind” motion velocities; thequantities u0ðz; tÞ; v0ðz; tÞ are, respectively, the transient wind velocity fluctuations in the x-along-wind and the y-cross-winddirections at the height z

The transient wind fluctuations u0ðz; tÞ; v0ðz; tÞ at various elevations of the building axis can be digitally simulated usingthe modulation function technique, which is presented inAppendix C The distributed lateral wind forces in Eqs.(55)–(57)are coupled due to the presence of the motion-induced aeroelastic forces The effects of vortex shedding and nonlinear

Fig 3 The reduced-order tall building model (schematic): (a) elevation and (b) cross-section.

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second-order aerodynamics, unresolved by the quasi-steady approximation in the lateral building wind forces, are neglected

at this stage but can be included in a future study, e.g., by using a Van-der-Pol type model similar to Eq.(45) [11].8.2 WG response solution of the reduced-order full building model

The numerical solution of the coupled stochastic dynamics in Eq.(48)in the time domain by numerical integration isoften extremely complex, not very accurate, and even impossible in the case of large systems In this section, the coupledmotion equations in generalized modal form are transformed into the wavelet space Eq.(48)can be explicitly written forthe x-along-wind and y-cross-wind generalized coordinates as follows:

Mx€x tð Þþ½CxQxx_x tð ÞQyx_yðtÞþ Kxx tð Þ ¼ Qb;xð Þ;t (58)

My€y tð Þþ½CyQyy_y tð ÞQxy_xðtÞþ Kyy tð Þ ¼ Qb ;yð Þ;t (59)where Mx; My; Cx; Cy; Kx and Ky are derived from Eqs.(49)–(51); Qxx; Qyxare the generalized motion-induced loadingterms, related to the x-coordinate, which linearly depend on the velocities _x and _y; Qyy, Qxy are motion-induced loadingterms, related to the y-coordinate; Qb ;xð Þ is the turbulence-induced generalized force in the x-coordinate and Qt b ;yð Þ is thetturbulence-induced generalized force in the y-coordinate as a function of time t These quantities are

The coupled generalized stochastic equations in the wavelet space are

qb ;x;fkgand qb ;y;fkg are the WG approximations to the time-dependent generalized buffeting forces The system of two Eqs.(66) and (67) can be re-organized as a set of two algebraic matrix equations: