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...

Journal of Sound and Vibration ] (]]]]) ]]]–]]] Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds Thai-Hoa Le a,b, Luca Caracoglia a,n a b Department of Civil and Environmental Engineering, Northeastern University, Boston, MA 02115, USA Department of Engineering Mechanics, Vietnam National University, Hanoi, Vietnam a r t i c l e i n f o abstract Article history: Received 18 August 2014 Received in revised form January 2015 Accepted January 2015 Handling Editor: W Lacarbonara A tall building is prone to wind-induced stochastic vibration, originating from complex fluid– structure interaction, dynamic coupling and nonlinear aerodynamic phenomena The loading induced by extreme wind events, such as “downburst storms”, hurricanes and tornadoes is naturally transient and nonstationary in comparison with the hypothesis of stationary wind loads, used in both structural engineering research and practice Time-domain integration methods, widely applied for solving nonlinear differential equations, are hardly applicable to the analysis of coupled, nonlinear and stochastic response of tall buildings under transient winds Therefore, the investigation of alternative and computationally-efficient simulation methods is important This study employs the wavelet-Galerkin (WG) method to achieve this objective, by examining the stochastic dynamic response of two tall building models subject to stationary and transient wind loads These are (1) a single-degree-of-freedom equivalent model 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 in conjunction with the Galerkin projection scheme to decompose and transform the coupled, nonlinear differential equations of the two models into random algebraic equations in the wavelet domain Methodology, feasibility and applicability of the WG method are investigated in some special cases of stiffness nonlinearity (Duffing type) and damping nonlinearity (Vander-Pol type) for the single-degree-of-freedom model For the reduced-order tall building model the WG method is used to solve for dynamic coupling, aerodynamics and transient wind load effects Computation of “connection coefficients”, effects of boundary conditions, wavelet resolution and wavelet order are examined in order to adequately replicate the dynamic response Realizations of multivariate stationary and transient wind loads for the building models are digitally simulated in the numerical computations & 2015 Elsevier Ltd All rights reserved Introduction Tall buildings are prone to wind-induced stochastic vibration and large-amplitude response, which originate from complex fluid–structure interaction, nonlinearity and aerodynamic coupling (e.g., [1–4]) The large-amplitude wind-induced response of buildings can lead to serious engineering problems, including human discomfort [5–7] and cumulative fatigue n Corresponding author E-mail addresses: ho.le@neu.edu, lucac@coe.neu.edu (L Caracoglia) http://dx.doi.org/10.1016/j.jsv.2015.01.007 0022-460X/& 2015 Elsevier Ltd All rights reserved Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Nomenclature A connection coefficient matrix, associated with the linear term (sdof model) Ap projected building area in the along-wind direction A^ p ðω; zp ; tÞ deterministic modulation function Axx ; Axy coefficient matrices associated with the x coordinate, WG method Ayx ; Ayy coefficient matrices associated with the y coordinate, WG method a; b scale and translation parameters of the wavelet function ak scaling coefficients of the refinement equation (Eq (9)) B connection coefficient matrix, associated with nonlinear stiffness term (sdof model) B width of the building bx coefficient vector associated with the x coordinate, WG method by coefficient vector associated with the y coordinate, WG method C connection coefficient matrix associated with nonlinear damping term (sdof model) CD, CL static along-wind and cross-wind force coefficients C 0D ; C 0L first derivatives of static force coefficients C D , C L with respect to attack angle À Á Cohu;pq ω; zp ; zq along-wind coherence function between two sections (floors) of coordinates zp ; zq c viscous damping coefficient (sdof model) cjk detailed wavelet coefficients at very small scales joj0 cj0 k wavelet approximation coefficients at the j0-th resolution D depth of the building d1 , d2 indices of derivative orders F b;r ðz; tÞ distributed buffeting force per unit height F a;r ðz; t; r; r_ ; r€ Þ distributed self-excited force per unit height F b;x ðz; t Þ distributed buffeting force per unit height in the x coordinate F b;y ðz; t Þ distributed buffeting force per unit height in the y coordinate F a;x ðz; t; x_ ; y_ Þ distributed self-excited force per unit height in the x coordinate F a;y ðz; t; x_ ; y_ Þ distributed self-excited force per unit height in the y coordinate f fkg stationary wind force vector, WG method f fkg transient wind force vector, WG method Hu ðω; zÞ lower triangular matrix, decomposed from stationary cross spectral matrix H building height h generic variable I identity matrix k stiffness coefficients (sdof model) M number of nodes along the building height M r ; C r ; K r generalized mass, damping and stiffness coefficients of r-th coordinate M x ; C x ; K x generalized mass, damping and stiffness coefficients in the x coordinate M y ; C y ; K y generalized mass, damping and stiffness coefficients in the y coordinate M ðmÞ moment of m-th order of the scaling function j;k m mass (sdof model) mðzÞ distributed mass of the continuous building model per unit height N order or “genus” of the wavelet Nn ; Nx original and extended computational domain, wavelet expansion nr natural frequency Q b;r ðt Þ generalized time-dependent buffeting force Q a;r ðt; r; r_ ; r€ Þ generalized motion-dependent selfexcited force Q xx ; Q yx generalized motion-induced force terms associated with the x coordinate Q b;x ðt Þ generalized turbulent-induced force in the x coordinate Q yy ; Q xy generalized motion-induced force terms associated with the y coordinate Q b;y ðt Þ generalized turbulent-induced force in the y coordinate qb;x;fkg , qb;y;fkg approximate buffeting forces associated with x-, y-coordinates, WG method r mode shape index (r ¼x or r ẳy) S0u ; zị stationary cross spectral matrix Su ðω;À z; t Þ Átransient cross spectral matrix S0u;pp ω; zp along-wind stationary auto-power turbulence À Á spectrum S0u;pq ω; zp ; zq along-wind stationary cross-power turbulence spectrum À Á Su;pp ω; zp ; t along-wind evolutionary auto-power turbulence À Á spectrum Su;pq ω; zp ; zq ; t along-wind evolutionary cross-power turbulence spectrum Uðz; tÞ along-wind time-varying mean wind velocity U HÀ Á reference mean wind velocity at height z¼H U p zp ; t along-wind transient wind velocity at the À Á building floor node p U p 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 uðz; t Þ along-wind zero-mean wind fluctuation À Á up zp ; t along-wind spatially-correlated zero-mean stationary wind fluctuation À Á u0p zp ; t along-wind spatially-correlated zero-mean transient wind fluctuation νðz; tÞ cross-wind zero-mean wind fluctuation νp ðzp ; tÞ cross-wind spatially-correlated zero-mean stationary wind fluctuation À Á ν0p zp ; t cross-wind spatially-correlated zero-mean transient wind fluctuation xfkg , yfkg WG displacement coefficient vectors in the x and y coordinates x_ fkg , y_ fkg WG velocity coefficient vectors in x and y coordinates x€ fkg , y€ fkg WG acceleration coefficient vectors in x and y coordinates Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] x, y along-wind and cross-wind coordinates of the building models xk , yk WG approximation coefficients xðt Þ, x_ ðt Þ, x€ ðt Þ along-wind displacement, velocity and acceleration of the along-wind motion x0 , x_ initial displacement and velocity yðt Þ, y_ ðt Þ, y€ ðt 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 ε nonlinear stiffness parameter ζr damping ratio Λd1 d2 vector of unknown connection coefficients μ nonlinear damping parameter δ0;l À k Kronecker delta φðhÞ father scaling function φj;k ðhÞ dilated and translated scaling function ρ air density Φr ðzÞ continuous mode shape of the building, mode r ϕml random phase angles, generated by Monte Carlo sampling ψ ðhÞ mother wavelet function ψ a;b ðhÞ dilated and translated wavelet function Ω0;0 2-term connection coefficient matrix, containing Ω0;0 lÀk Ω0;1 2-term connection coefficient matrix, containing Ω0;1 lÀk Ω0;2 2-term connection coefficient matrix, contain0;2 ing Ωl À k Ω0;0 , Ω0;1 , Ω0;2 2-term connection coefficient at the lÀk lÀk lÀk derivative order 0, at the derivative orders and 1, and at the derivative orders and Ωdl À1 ;dk2 2-term connection coefficient at the derivative orders d1 ; d2 ;…;dn Ωdl11l2;d…l multiple-term connection coefficient at the n derivative orders d1 ; d2 ; …; dn Subscripts l, k j p translation parameters dilation parameter (resolution level) discrete nodal index (p¼ 1,2,…,M) Operators 〈; 〉 E[] T n inner product operator expectation operator (mean value) transpose operator complex conjugate operator damage [8–10] The wind-induced stochastic structural dynamics of tall vertical structures in general, including tall buildings, slender towers and wind turbines can be investigated by means of either simplified structural models (e.g., [8]) or reduced-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 windinduced stochastic dynamic equations (e.g., [11]) In particular, nonlinearity in the building response has been recently observed as an emerging problem for tall buildings, for example due to the potential nonlinear interaction of the vibrating structure with vortex-shedding effects (e.g., [12]) The coupling of aerodynamic loads is often analyzed in the study of the stochastic response of tall buildings during extreme wind events [13,14] This approach involves the solution of coupled motion equations, combining turbulence-induced buffeting forces and motion-induced forces Nevertheless, the solution of the coupled and nonlinear motion equations in the time domain, necessary in the case of transient wind loads, is still a major challenge for these structures; it is seldom pursued since it may require computationally demanding procedures of analysis (e.g., [15]) Currently, the solution of the wind-induced stochastic dynamics is often preferably carried out under the assumptions 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 Fourier transform-based analysis [1–4]; the equations of motion can subsequently be combined with empirical or experimentallybased power spectral density functions of the stationary turbulent wind field to describe the loading [16–18] This transformation allows the coupled and nonlinear motion equations to be converted to a simpler algebraic form but requires the wind loads to satisfy stationarity conditions The hypothesis on multivariate stationary wind loads is also still preferably used to derive the stochastic response of a tall building due to the need for large computational resources and additional modeling complexity in the case of nonstationary 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 rapidly varying 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, research investigations have been directed towards the modeling of nonstationary winds and the simulation of “time-frequencydependent” response of tall buildings, slender vertical structures [22–24] and long-span bridges [25,26] It has been observed that, if a stationary-wind-based analysis method is employed to study the dynamics, this can cause underestimation in the assessment of the structural response under loads, which originate from a transient/nonstationary wind event [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 Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] adequate simulation method for the analysis of the wind-induced coupled stochastic dynamic response of tall buildings and slender vertical structures is currently not available In recent decades, wavelet transform (WT) and wavelet analysis have gradually emerged as a powerful tool for engineering and scientific computations in the mixed time–frequency domain Wavelets are either piecewise functions or a family 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 oscillatory functions The wavelet transform computes the convolution operation between a “signal” and a family of wavelets The advantage of the WT over the Fourier transform (FT) is that, instead of using a family of complex harmonic functions to decompose an aperiodic signal, a family of wavelets is employed Wavelets can represent not only symmetric, smooth or regular signals, but also asymmetric, sharp or irregular signals, which are typical of nonstationary and nonlinear phenomena, simultaneously on the time–frequency plane and in the context of multi-resolution analysis (MRA) The WT can 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) is employed for signal analysis and processing using only orthogonal wavelets [28] In the context of the MRA, the WT examines high-frequency components of the signal with a “fine” frequency resolution but “coarse” time resolution, and lowfrequency components with “fine” time resolution and “coarse” frequency resolution [28] Although one cannot simultaneously 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 either real or complex Morlet mother wavelets, has been applied to the examination of wind turbulence, pressures and aerodynamic 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 the systematic 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 wavelets allow for a wide range of localization and dilation operations within the multi-resolution framework Daubechies wavelets can describe the analyzed signal at various levels of resolution; this property also makes them particularly attractive for developing approximations to exact solutions in a dynamic problem The Galerkin method belongs to a class of numerical approximation methods for converting a continuous operator and system (such as a differential equation) into a discrete system using basis functions The Galerkin approximation method has been extensively employed for static and dynamic problems in engineering The advantageous properties of orthogonality and aptitude to multi-resolution approximation of the Daubechies wavelets can “empower” the Galerkin approximation method Furthermore, the Daubechies wavelets are more suited for Galerkin approximation than other popular wavelets, like harmonic wavelets, because they are not only piecewise 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 have been integrated with the Galerkin expansion method to solve various kinds of ordinary differential equations [35–37] and dynamical differential equations [38] This wavelet-Galerkin analysis method (WG) is progressively emerging as an approach to build approximate solutions to various engineering problems In structural dynamics, the WG method has been employed to study vibrations of continuous single-degree-of-freedom (sdof), two-degree-of-freedom (2dof) systems with linear and nonstationary parameters [38–40], nonstationary seismic response of sdof systems [41], continuous mechanical systems [42], stochastic response of buildings [43] and long-span bridges [44] This partial review of the existing literature shows that this numerical method is receiving more and more attention from the technical community also because 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 by numerical differentiation are efficient and easy to implement Their efficiency and accuracy is excellent for linear systems with conventional boundary or initial conditions These methods, however, often suffer from a loss of accuracy if they are applied 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 using energy principles and are advantageous for nonlinearly-coupled dynamic problems with natural boundary conditions and complex geometries [45–47] The main application of the WG analysis is to approximate coupled or/and nonlinear motion equations by transforming them into “user-friendly” random algebraic coefficient equations in the wavelet domain, which can be solved numerically in a simple way In particular, the WG method could support the analysis of stochastic structural response due to transient/nonstationary wind loads Nevertheless, limitations and computational challenges of the WG method have also been observed [38–44] These include an accurate treatment of boundary conditions, arbitrary time range, “connection coefficient” estimation, resolution analysis and computational issues in relation to computing time and large memory requirements Among the aforementioned limitations, the management of boundaries and adequate resolution analysis are the main problems Many recent applications of the WG solution to ordinary differential equations have dealt with simple Dirichlet boundary conditions in the unit time [35,38,42] In these applications the resolution analysis of the wavelets can be neglected and the treatment of boundary conditions is drastically simplified These aspects have prevented the application of the WG method to a wide range of problems, until recently Fortunately, the problems of “adaptable” treatment of boundary conditions to better Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] model the two ends of a finite-duration analytical signal along with an improved computation of the wavelet connection coefficients [47] have been recently resolved [40,48–50] Latto et al [47] firstly proposed the methodology for computing the 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 boundary conditions, since it did not provide accurate values at the endpoints of a signal on a bounded domain [47] Subsequent studies examined a “more flexible” management of the boundary conditions in the case of differential equations, and improved the computation of the connection coefficients of the Daubechies wavelets for a bounded domain at various resolution levels [48–50] Recently, the WG method was modified to better model the two ends of a finite-duration analytical signal by extending the original WG method to any boundary conditions and by suitable assemblage of the connection coefficients in matrix form [40,43,44] Furthermore, the open question on the resolution of the computed wavelets 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 the stochastic structural response of two building models, accounting for aerodynamic coupling, system nonlinearity and transient/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 coupled stochastic 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 are considered Realizations of the evolutionary transient wind fields along the height of each structure are artificially simulated from 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 with emphasis 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 WG formulation to arbitrary boundary conditions, the estimation of the wavelet resolution and the selection of the order of Daubechies 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 Galerkin approximation The Galerkin method is a projection method that seeks for an approximating solution through the projection of the exact solution onto a subspace spanned by a basis of suitable functions The Galerkin projection approximates an exact solution xðhÞ to an equation AxðhÞ ¼ f with h being a generic variable The approximating solution xn ðhÞ, based on an inner-product space of Nn finite dimensional functions, is usually defined as [45] xn ðhÞ ẳ Nn X xl l hị; (1) lẳ1 R ỵ1 where φl ðhÞ are basis or shape functions; xl are unknown coefficients found as xl ¼ 〈xn ; φl 〉 ¼ À xn ðhÞφl ðhÞdh, which satisfy the condition 〈Axn À f ; ψ〉 ¼ with 〈; 〉 the inner product operator This approximation by Eq (1) leads to * + Nn X A xl φl ðhÞ Àf ; k hị ẳ 0; (2) lẳ1 with k ðhÞ being a series of weight or test functions, k ¼ 1; 2; …; Nn If the inner-product operation, is applied to the original equation the following general form is determined: Nn X xl 〈Aφl ; ψ k 〉 ¼ 〈f ; ψ k 〉; (3) l¼1 AG x ¼ f: (4) In Eq (4) a coefficient matrix equation is obtained, where AG is an N n -by- Nn matrix with elements AG;l;k ¼ 〈ψ l ; Aφk , x ẳ x1 ; x2 ; ; xNn ịT , f ¼ ð〈ψ ; f 〉; 〈ψ ; f 〉; …; 〈ψ Nn ; f Nn 〉ÞT The unknown coefficients xl can be estimated from the algebraic equation (4) The basis function can be a simple function like a “hat” function, a cubic function, a spline function, a Hermite polynomial function [45–47] However, it is often preferably composed of multiple piecewise sub-functions Each sub-function is projected onto a given interval of the computational domain The weight functions are chosen to be “collateral” to the basis function; examples are the coincidental basis functions in the Ritz–Galerkin method, the polynomial functions in the “Galerkin finite-element method”, the Dirac Delta function in the “Galerkin collocation method”, proportional functions of Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] the basis function in the “Galerkin least squares method” [45,46] The weight function is orthogonal to the basis function in the WG approximation In this application, they are the Daubechies wavelets, which have the advantageous properties of localization, differentiability, piecewise continuity, orthogonality, multi-resolution, and compact support [28] Daubechies wavelets Wavelets ψ a;b ðhÞ are defined as a family of piecewise functions, generated from a “mother” wavelet by scaling (a) (as the inverse of a Fourier frequency) and translation (b) parameters, as   hb : a;b hị ẳ p a jaj (5) R ỵ1 The wavelets are zero-mean continuous oscillatory functions (i.e hịdh ẳ 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 derivatives R ỵ1 ^ ^ d=jj o1 with ψðωÞ being the Fourier transform decay very fast They also satisfy the admissibility condition À jψðωÞj coefficient 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] Z ỵ1 xhị ẳ WT a;b hị a;b hịdh; where WT ψa;b ðhÞ is wavelet transform coefficient, defined as WT a;b hị ẳ /xhị; a;b hịS ẳ (6) R ỵ1 xhị a;b hịdh: Dyadic wavelets with a ¼ À j ; b ¼ k2 À j (j, k are scale and translation parameters, respectively) are commonly employed The dyadic wavelet ψ j;k can be dilated by À j times or compressed by 2j times depending on the sign of j, and translated by k2 À j units The wavelets in Eq (5) and the projection of signal x(h) in the dyadic form can be written as [34] j;k hị ẳ 2j=2 2j x kị; xhị ẳ ỵ1 X ỵ1 X (7) xhị; j;k hị j;k hị: (8) j ẳ k ẳ À1 In the previous equations the scaling parameter j plays the role of frequency; the translation parameter k plays the role of time or space Since j and k can vary arbitrarily on the time–frequency plane, one obtains the MRA approach Daubechies wavelets are a family of compactly supported orthogonal dyadic wavelets, which are well suited for engineering computations The Daubechies wavelet (D) of order N consists of a pair of “father” scaling function φðhÞ and “mother” wavelet function ψ ðhÞ Construction of the Daubechies wavelet starts from the father scaling function φðhÞ and a set of coefficients, which satisfy the two-scale relation or refinement equation [34]: hị ẳ N X ak 2h kị ẳ kẳ0 N X ak 2hị; (9) kẳ0 where ak denotes scaling coefficients Only a finite number of scaling coefficients ak is non-zero, thus the scaling function φðhÞ is said to have a compact support The mother wavelet function is derived from the father scaling function as P k hị ẳ N k ẳ 1ị ak ỵ 2hỵ kÞ The scaling function φðhÞ and wavelet function ψ ðhÞ are of compact support over the PN À PN k m finite interval suppị ẳ ẵ0; N À 1Š and their coefficients satisfy k ¼ ak ¼ 2, k ¼ ð À1Þ k ak ¼ (m ¼0,1,…N À1) and PN À k ¼ ak ak ỵ l ẳ 0;l (0;l is the Kronecker delta) Correspondingly, the father scaling function and the mother wavelet R ỵ1 R ỵ1 R ỵ1 function satisfy the following conditions: hịdh ẳ 1, h kịh lịdh ẳ 0;k l , h kịh lịdh ẳ 0;k l , R ỵ1 m h hịdh ẳ (l and k are both translation parameters) The scaling function basis is subsequently constructed, in which each member of the basis at the resolution level j is a scaled, dilated and translated version of the underlying scale function as φj;k ðhÞ ¼ 2j=2 φð2j h À kÞ Projection of signal xðhÞ 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] xhị ẳ ỵ1 X kẳ0 cj0 ;k j0 ;k hị ỵ j0 X ỵ1 X cj;k j;k hị; (10) jẳ0kẳ0 Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 2 Scaling function Wavelet function Wavelet function Scaling function Wavelet function 1 Amplitude Amplitude Scaling function -1 -2 Scaling function -1 -2 10 Time(s) 10 Scaling function Scaling function Wavelet function Wavelet function 1 Amplitude Amplitude Time(s) Wavelet function -1 -1 Wavelet function Scaling function -2 Wavelet function Scaling function 10 -2 Time(s) 10 Time(s) Fig Scaling functions and wavelet functions of Daubechies wavelets: (a) D2, (b) D6, (c) D8 and (d) D10 where cj0 ;k denotes the “approximation” coefficients at the scale j0; cj0 ;k ẳ xhị; j0 ;k hị and cj;k ẳ xhị; ψ j;k ðhÞ〉 are “detailed” coefficients at smaller scales j0 oj (higher frequencies) One generalizes that the father scaling function deals with high scales (low frequencies), while the mother wavelet function is used for low scales (high frequencies) If the wavelet expansion in Eq (10) is truncated at the scale j0 (resolution level), after eliminating all components at the scales smaller than P j0 (high frequency components), one has the approximation of a signal x(h) at the scale j0 as xhị % kỵẳ10 cj0 ;k j0 ;k hị Fig shows examples of the pairs of father scaling functions φðhÞ and mother wavelet function ψ ðhÞ of Daubechies wavelets of various orders: D4 (N ¼4), D6 (N ¼6), D8 (N ¼8) and D10 (N ¼10) The support interval of scaling function and wavelet function widens in the time domain and the smoothness increases, when the order of the Daubechies wavelets increases In other words, the Daubechies wavelets are less localized, but smoother with the increase of the order Wavelet-Galerkin analysis In the WG analysis the orthonormal and compactly supported Daubechies wavelets are employed as the basis functions and weight functions in the Galerkin projection to find approximate solutions to dynamic problems If the time variable is denoted by t, a generic motion variable xðt Þ can be expressed, at a pre-selected level of resolution j of the wavelet, as xt ị ẳ Nx X xk j;k tị: (11) kẳ1 In the previous equation k is a translation parameter (time index) defined on a finite-duration computational time domain [1,2,…,Nx] Besides, xk are approximation coefficients, derived from the inner product RN xk ẳ xtị; j;k tị ẳ x xt ịj;k ðtÞdt Similarly, the first and second derivatives x_ ðt Þ; x€ ðt Þ can be approximately determined in the Daubechies wavelet subspace as x_ t ị ẳ Nx X xk _ j;k tị; (12) xk j;k tị; (13) kẳ1 x t ị ẳ Nx X kẳ1 where _ j;k ðtÞ and φ€ j;k ðtÞ 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 À1] The inner products between approximating solutions of the displacement, velocity, acceleration in Eqs (11)–(13) and the weight functions, as Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] indicated in Eq (3), are * φj;l ðtÞ; Nx X + xk j;k tị ẳ kẳ1 * j;l tị; Nx X kẳ1 + xk _ j;k tị ẳ kẳ1 * j;l tị; Nx X Nx X Nx X kẳ1 + xk j;k tị kẳ1 ẳ Nx X kẳ1 xk Ω0;0 ; j;k À l (14) xk Ω0;1 ; j;k À l (15) xk Ω0;2 ; j;k À l (16) with Z Ω0;0 ¼ j;k À l Nx j;l tịj;k tịdt ẳ 0;k l ; Z 0;1 ¼ j;k À l Nx Z Ω0;2 ¼ j;k À l φj;l ðtÞφ_ j;k ðtÞdt; Nx (17) (18) φj;l ðtÞφ€ j;k ðtÞdt; (19) where δ0;k À l is the Kronecker delta; and Ω0;1 , Ω0;1 ; Ω0;2 are 2-term wavelet connection coefficients of the Daubechies lÀk lÀk lÀk wavelets (e.g., [48,49]) The “index of appearance” k–l designates the localized support of each wavelet It is noted that the connection coefficients exclusively depend on the wavelet resolution j and the scaling functions (the order of Daubechies wavelet) within their limit support but not depend on the analytical signal The 2-term connection coefficients [38,39] Ωdj;k1 ;dÀ2l (d1, d2 are derivative orders d1,d2 ¼ 0,1,2) are necessary for computing linear second-order dynamic problems If higher-order derivatives, cross terms and nonlinear terms are present in the dynamics, 3-term connection coefficients RN ðtÞdt or even higher multi-term connection coefficients may be needed The general form of Ωdj;k1 dÀ2 dl;l3À m ¼ x φdj;k1 ðtÞφdj;l2 ðtÞφdj;m the n-term connection coefficients may be found in [48,49] as Z Nx Z d1 d2 dn ;…;dn Ωdj;k1 ;d ¼ φ ð t Þ Â φ ð t Þ Â …  φ t ịdt ẳ j;k1 j;k2 j;kn k2 kn ỵ1 n dj;ki i dt; (20) iẳ1 with Π denoting the product operator The 2-term connection coefficients and 3-term connection coefficients are typically used for second-order linear and nonlinear dynamical systems Estimation of wavelet connection coefficients The 2-term connection coefficients can numerically be computed in the case of unbounded domain ẵ 1; ỵ with the D6 Daubechies wavelet at the resolution level j¼1 only, using the d1, d2-times differentiation of the scaling functions, the moment 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, needed for computing the connection coefficients, can successfully be resolved by modifying the end values of the connection coefficients, 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 WG computations 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 by considering a generic order of differentiation (d1 and d2 times), can be expressed as [49] Z NÀ1 Z NÀ1 d2 ¼ Ωdj;k1 dÀ2 l ¼ φdj;k1 h kịdj;l2 h lịdx ẳ dj;01 xịdj;k2 l h k ỵ lịdx; (21) dj;k;l dj;k1 ẳ d1 h kị ẳ 2d1 dj;l2 ẳ d2 h lị ¼ 2d2 N À1 X ap φdp1 ð2h À 2k ỵ pịị; (22) aq dq2 2h 2l ỵ qịị: (23) p¼0 N À1 X q¼0 Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] By using the recursive definition of wavelet scaling function, the integration can be simplified as follows [49,50], noting d2 that Ωdj;k;l ¼ Ωdj;k1 dÀ2 l , d2 ¼ 2d Ωdj;k;l N À1 N À1 X X Z p¼0q¼0 d2 Ωdj;k;l ¼ 2d À 1 d2 Ωdj;k;l ¼ 2d À NÀ1 ap aq N À1 N À1 X X Z (Z þ NÀ1 ap aq p¼0q¼0 Z 2N À ap aq p¼0q¼0 N À1 N À1 X X φdp1 2h 2k ỵpịịdq2 2h 2l ỵ qịịdx; 2N N1 dp1 h 2k ỵpịịdq2 h 2l þ qÞÞdx:; (24) (25) φdp1 ðhÀ ð2k þpÞÞφdq2 ðh À 2l ỵ qịịdx ỵ ) dp1 h 2k ỵ pịịdq2 h 2l ỵqịịdx ; (26) or d2 ẳ 2d À Ωdj;k;l N À1 N À1 X X pẳ0qẳ0 d2 d2 ap aq fdj;2k ỵ dj;2k g; ỵ p;2l ỵ q ỵ p N 1ị;2l ỵ q N 1ị (27) where d ẳd1 ỵd2 and with ap and aq being the coefficients of the refinement equation (Eq (9)) Since the scaling functions are compactly supported on the [0, N À 1] interval (0 rk; l rN À 1), the support of the connection coefficients with the index (k–l) is on the [0,2N À2] interval (2 ÀN r k À l rN À2 or r k Àl r2N À 2) Therefore, Eq (27) can be written in a matrix form of linear equations: " ( )# N À1 N À1 X X d1 I ap aq d1 d2 ẳ ẵd1 d2 ¼ 0: (28) p¼0q¼0 In the previous equation I is the identity matrix; Λd1 d2 is a vector containing the unknown connection coefficients n o P À PN À of dimension 2N–3 the quantity within ¼ Ωdj;k1 dÀ2 l with rk; l r N À1; in the matrix ϒ ¼ I À 2d À N p¼0 q ¼ ap aq d2 Ωdj;k;l the braces is a “recursion matrix” obtained from Eq (27) The connection coefficients can be computed once the values of d2 Ωdj;k;l ¼ Ωdj;k1 dÀ2 l with À N rk; l r0 are known (recursively) No unique nonzero solution exists since Eq (28) is homogeneous Therefore, one additional nonhomogeneous equation must be supplemented to normalize and enable the solution R ỵ1 m ẳ h j;k hịdh ẳ with m¼0,1,…, The moments at the m-th degree of the scaling function are defined as [34]: M ðmÞ j;k ðmÞ N 1; this definition implies M 0ị j;0 ẳ 0, by the property of the scaling functions The moments M j;k on the interval [0,N À 1] can be computed as [47] ẳ M mị j;k m X m 22 1ị p ẳ m p m Therefore, an expression for the monomial term h m h ¼ ! mÀp k p À1 X q¼0 p ! q M ðqÞ j;0 N À1 X pÀq ak k : (29) k¼1 on the interval [0,N À 1] can be obtained [49] N X k ẳ 2N Mmị ðhÞ: j;k j;k (30) If Eq (30) is differentiated d1 or d2 times with respect to the variable h, on the interval [0,N À 1], it is found that, with the symbol “!” designating factorial product: d1 ! ¼ N À2 X k ¼ 2ÀN d2 ! ¼ N À2 X l ẳ 2N d1 ị d1 M j;k j;k ðhÞ; (31) ðd2 Þ d2 M j;l φj;l ðhÞ: (32) After multiplying Eqs (31) and (32) together and integrating them over the interval [0,N À1], the normalization equation d2 can be obtained, noting that Ωdj;k;l ¼ Ωdj;k1 dÀ2 l , as [48] N 1ịd1 !d2 ! ẳ N À2 X N À2 X k ¼ 2ÀN l ¼ 2ÀN ðd1 Þ ðd2 Þ d1 d2 M j;k M j;l Ωj;k;l : (33) Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 10 Table of the Daubechies wavelets D4, D6 and D8 at the resolution level j¼ 2-Term connection coefficients Ωdj;k1 ;d Àl Finally, solutions to the connection coefficients are found by using the first 2N À rows from Eqs (28) and (33) as [48,49] " # ϒð2N À 4ÞÂð2N À 3Þ ( ) dd N À2 N À2 X X ẩ ẫ ẳ : (34) d ị ðd Þ Mj;k1 M j;l ðN À 1ịd1 !d2 ! k ẳ Nl ẳ À N A total of (2N À 3) connection coefficients are computed; their values only depend on the pre-selection of the Daubechies wavelet (the order of the wavelet) and the resolution levels For example, the D6 Daubechies wavelet requires nine 2-term connection coefficients with indices ranging for (k–l)¼ À 4, À3, À 2,À 1,0,1,2,3,4 The fictitious interval approach is used herein, in which the computational domain of the original signal (Nn discrete points) is expanded by adding (N À 1) points to the left of the original computational domain (before initial time) and (N À 1) points to the right of the original domain (beyond final time) The modified domain has (Nx ẳNn ỵ2N 1) wavelet expansion points in the interval [1 N, Nn ỵN 1] For the implementation of WG analysis, Nx-by-Nx sparse square matrices can be used for collecting the compactly supported connection coefficients The construction of the connection coefficient matrices are presented in a later section Table shows, as an example, the 2-term connection coefficients Ωdj;k1 dÀ2 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,d2 ¼0,1 and For example, the scaling function D4 requires 5¼(2N À 3) 2-term connection coefficients with the support indices (k–l) on [ À 2,À 1,0,1,2]; the D6 has connection coefficients according to the index (l–k), supported on [ À4, À 3,À 2, À 1,0,1,2,3,4]; there are 13 connection coefficients with (l–k), evaluated on [ À 6, À5, À 4,À 3, À 2,À 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 from Table 1: Ω0;1 ¼ ÀΩ1;0 , Ω0;1 ¼ ÀΩ0;1 , Ω0;2 ¼ Ω2;0 , Ω0;2 ¼ Ω2;0 , Ω1;2 ¼ À Ω2;1 , Ω1;2 ¼ À Ω1;2 j;k À l j;k À l j;l À k j;k À l j;l À k j;l À k j;l À k j;k À l j;l À k j;l À k j;l À k j;k À l 0;1 0;2 1;2 At the “center point” k Àl ¼0 we have Ω0;0 j;0 ¼ 1, Ωj;0 ¼ 0, Ωj;0 a 0, Ωj;0 ¼ Treatment of wavelet resolution and boundary conditions Adequate treatment of wavelet resolution and boundary conditions is essential to the computation of connection coefficients and the implementation of WG analysis The WG analysis “expands” a time-varying signal at the pre-selected resolution j Therefore, the initial choice of wavelet resolution level (j) is required for the computation of the connection coefficients The resolution parameter of the Daubechies wavelets is 2j at a scale j; the resolution must be determined so that the scaling function is centered, given the number of discretization points, i.e., the sampling time of the original signal The wavelet resolution (j) can be approximately found from the number of samples per unit time of the signal (Nx ) or the sampling rate of the signal from the following relationship: N x ¼ 2j As a result, the resolution level is fitted with the original signal as j ¼ log2 N x : (35) In this study we followed the assumption, common to most numerical analysis and integration methods, that the sampling time Nx should be taken as a fraction of the main vibration period of the structure Therefore, initial selection was made to sample the wind load and the wavelet in both input and output at intervals corresponding to frequency N x ¼100 Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 16 À Á Ayy ẳ M y 0;2 ỵ C y Q yy 0;1 ỵ K y 0;0 ; 0;2 0;1 (72) 0;0 in which Ω , Ω , Ω are 2-term connection coefficient matrices at various derivatives The two coupled algebraic matrix equations in Eq (68) must be solved simultaneously to find the WG approximating random solutions in terms of generalized coordinates, xfkg and yfkg The resultant random velocities and accelerations in the x and y coordinates can be found from xfkg and yfkg as x_ fkg ¼ Ω0;1 xfkg ; y_ fkg ¼ Ω0;1 yfkg ; (73) x€ fkg ¼ Ω0;2 xfkg ; y€ fkg ¼ Ω0;2 yfkg ; (74) in which x_ fkg , y_ fkg , x€ fkg and y€ fkg are the WG approximations of the velocities and accelerations in x and y The global responses of the tall building can be later reconstructed from xfkg and yfkg , etc Description of the numerical examples 9.1 Simplified sdof model of a tall structure The sdof model represents a simplified lumped-mass vertical tall structure In this example, the height is H ¼35 m; the projected area used for wind load calculations has dimensions B ¼5 m and D¼ m The mass m ¼6000 kg is lumped at the top node at height z ¼H¼35 m The dynamic response of the support structure is simulated by an sdof equivalent model, representing the bending vibration in the x-along-wind direction, with a natural frequency fx ¼0.5 Hz (the mode shape is normalized as x H ị ẳ at the top node) The damping ratio of the sdof model is ζx ¼0.01 The static coefficient of the alongwind drag force, normalized to the projected area BD, is assumed as C D ¼ 1:1 More details may be found in Fig 9.2 Full model of a tall building The model is derived from the CAARC benchmark building [52] Dimensions are B ¼30.5 m (width), D ¼45.7 m (depth) and H ¼183 m (height) The mass is constant and uniformly distributed along the building height; the mass per unit height is m(z)¼220,800 kg/m The two fundamental building modes in the x-along-wind and y-cross-wind directions have natural frequencies equal to fx ¼0.20 Hz and fy ¼0.22 Hz, respectively The damping ratios of the two fundamental modes are equal, with ζx ¼ζy ¼0.01 The mode shapes of the two fundamental modes are determined as x zị ẳ y zị ẳ z=Hị Torsional modes and response are neglected Aerodynamic static coefficients and their first-order derivatives per unit height are assumed as constant, independent of the building height and mean wind velocity with C D ¼ 1:2; C L ¼ À2:2; C 0D ¼ À1:1; C 0L ¼ À0:1 [4] Layout and symbolic definitions are shown in Fig For the estimation of the generalized quantities in Eqs (60)–(65), which include the lateral wind load, the building is discretized into 41 equally-spaced nodes (floors) at a distance of 4.575 m along the height 9.3 Empirical wind field model An empirical wind field model is employed to simulate the wind turbulence fluctuations and the stochastic wind forces at the discretized nodes along the building height The mean wind velocity profile UðzÞ of the stationary winds is independent of time t and varies with z only This hypothesis is also used, as a first approximation, to represent a transient/ nonstationary wind event A power-law model with a factor α ¼ 0:25, acceptable for urban terrain category, is used to describe the profile UðzÞ [52] The mean wind velocity at the rooftop node of the building z¼H is set to U H ¼ 20 m/s in most simulations; U H ¼ 10 /s and U H ¼ 30 m/s are on occasion utilized for comparison purposes Power spectra and spatial correlations of the two stationary turbulence components, uðz; t Þ and νðz; tÞ respectively in the x-along-wind and y-crosswind axes of the building, are empirically formulated using Harris' power spectral function and Davenport's coherence function [52]; more details are given in Appendix C The turbulence intensities of u and v components are Iu ¼Iv ¼15% [52] and independent of z 9.4 Digital simulation of stationary and transient wind time histories In order to simulate the stochastic wind forces on the buildings, the stationary and transient wind fields must be digitally generated at the discrete nodes (41 nodes) along the building height for the tall building, and at the reference top node for the simplified sdof model (refer to Figs and for the position of the nodes) The empirical wind model is used for this purpose, as presented in Section 9.3 The digital simulation of the multivariate wind fields, used for generating the lateral loads on the two models, is illustrated in Appendix C It is also noted that Appendix C describes the general procedure for wind field simulation in the case of multivariate turbulence processes, needed at the discrete nodes along the height of the tall building (Section 9.2) The wind turbulence “field”, required at the single node of the sdof model in Section 9.1, is a special simplified case of the general procedure in Appendix C Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 105 Simulated Harris spectrum 2.5 Suu (m2/s) Amplitude (m/s) u-stationary-wind 100 -2.5 10-5 -2 10 -5 50 100 150 200 250 300 Time (s) 10-1 100 101 Frequency n(Hz) 105 u-transient-wind Harris spectrum Simulated 2.5 Suu (m2/s) Amplitude(m/s) 17 100 -2.5 -5 50 100 150 Time (s) 200 250 300 10-2 10-1 100 101 Frequency(Hz) Fig Time histories of simulated wind velocity fluctuations at the roof-top node of the simplified vertical structure model at a reference mean wind speed U H ¼ 20 m/s: (a) stationary wind velocity fluctuation, (b) PSD of stationary wind velocity fluctuation, (c) transient wind velocity fluctuation and (d) equivalent PSD of transient wind velocity fluctuation The procedure for the simulation of the transient wind fields can be summarized as follows: (i) realizations of multivariate stationary wind turbulence are digitally generated using the spectral representation approach, and (ii) stationary wind turbulence realizations are transformed to multivariate transient turbulence records by utilizing the amplitude modulation function A simplified cosine modulation function is employed in this study for illustration purposes [43] The cosine modulation function is designed to modulate amplitudes of the stationary turbulence processes by filtering out the signal outside the modulation window and by maintaining the signal magnitudes inside the modulation window The duration and “center” of modulation window with respect to the original stationary process must be defined The definition of the cosine modulation function is included in Appendix C Other examples of modulation functions, used in wind engineering for replicating the main features of a downburst or a thunderstorm, may be found in [23,24] In the numerical simulations, random realizations of the time histories of wind turbulence with 300 s duration, 100 Hz sampling rate and 10 Hz upper cut-off frequency are generated These synthetic realizations of the turbulence components uðz; tÞ and vðz; tÞ are converted to u0 ðz; tÞ and v0 ðz; tÞ and later used to compute time-histories of the generalized wind forces 9.5 Utilization of Daubechies wavelets The Daubechies wavelet of order N ¼6 (D6) is predominantly used for the WG analysis in this study; the Daubechies wavelets D2, D4, D8 and D10 are on occasion employed for comparison purposes only The recursive construction of the scaling function and the wavelet function of the D6 are given in Fig The 2-term connection coefficients, associated with different derivatives, are also computed in Table 1, at the resolution level j ¼6, as an example It can be noticed that the D6 wavelet has very good compactness compared to higher-order wavelets, but the function is less “smooth” since it rapidly fluctuates about the zero axis 10 Numerical results and discussion 10.1 Simplified sdof model of a tall structure For the WG analysis of the nonlinear stochastic transient-wind response of the sdof model in Fig 2, univariate transient turbulence realizations (wind speed fluctuations only) have been digitally generated at the top reference node in Fig Fig 4a and c illustrate examples of 300 s realizations of the stationary wind speed fluctuations and corresponding transient wind speed fluctuations In both cases, the power spectral densities of the synthetic wind fluctuations are verified against Harris' empirical power spectrum Adequate agreement can be noted between the empirical spectrum model and the power Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 18 spectrum of the digitally simulated realizations (Fig 4b), while the spectral ordinates of the digitally simulated transient record are lower than Harris' power spectrum (Fig 4d) This reduction is a consequence of the average energy diminution (over the entire duration of the 300 s signal) in comparison with the stationary process, after application of the amplitude modulation function The WG-based approximate solutions of displacement and velocity are shown in Fig for the simplified sdof tall structure model The WG displacement (in Fig 5a) and velocity (in Fig 5c), after reconstruction of the signal, are compared with corresponding responses, obtained by Newmark-beta (NM) numerical integration method with parameters α ¼ 1=2 ; β ¼ 1=4 The relative error function, between the WG-based responses and the NM-based “exact” responses, is defined as E%ị ẳ 100xNM xWG ị2 =x2NM , in which the x variable denotes resultant response The comparison is also extended to “energy density” by examining the PSD functions of the WG solutions and NM solutions The error functions and the PSD functions are illustrated in Fig 5b and d Adequate agreement between the two methods can be observed The influence of the selection of Daubechies wavelets (i.e., the order of the wavelet) and the resolution levels have also been investigated by using the wavelets from D2 to D20 to derive the WG-based displacement of the sdof model Fig 6a shows that the solutions based on D2 and D4 are totally inaccurate, whereas Daubechies wavelets D6–D20 provide adequate results with very similar WG solutions The authors believe that the discrepancy, evident for D2 and D4, depends on the limited number of discrete points on the support and the small number of connection coefficients, which are unable to adapt to the rapidly varying changes in the wind load and the dynamics of the system This fact yields a poor resolution in the analysis of the response The comparison among D6–D20 in terms of computing time is described in Fig 6b; the various computing times are normalized with respect to the computing time of a D6 analysis (T D6 ) as RDi %ị ẳ 100T Di =T D6 , in which T Di is the computing time of D6–D20 As anticipated, a D6 analysis is the fastest Also, the computing time tends to increase with the increment of the wavelet order, i.e., the decrement of compactness As an example, the computing time of D8 is 170 percent longer than with D6, becoming 370 percent longer with D10 and 1300 percent longer with D20 From the examination of these results, the use of Daubechies wavelet D6 is recommended in the WG analysis, since it preserves acceptable accuracy by limiting the computational burden The next investigation analyzes the effect of resolution level on the accuracy of the WG approximation The resolution level is initially estimated as j¼6.64, by fitting the resolution with the digital realization of the wind speed fluctuation, corresponding to the time resolution of 0.01 s for the signal Fig 6c shows the WG-based displacement at the top node of the building model, obtained at j¼6.64 but also at other wavelet resolutions; these include higher resolutions (j¼ 6.77 and j¼9.96) and lower resolutions (j¼6.38 and j¼6.50) As a result, the time resolutions of the signal are respectively higher (Δt ¼ 0:008 s and Δt ¼ 0:009 s) and lower (Δt ¼ 0:011 s and Δt ¼ 0:012 s) than the time resolution of the original signal (Δt ¼ 0:010 s) Fig 6c 0.2 0.1 Error function (%) Displacement (m) Wavelet-Galerkin Newmark-beta -0.1 -0.2 50 100 150 200 250 Displacement 300 x 10-6 50 100 Time (s) 150 200 300 x 10-3 1.5 Wavelet-Galerkin 0.5Hz Acceleration Newmark-beta Error function (%) Acceleration (m/s2) 250 Time (s) -1 -2 50 100 150 Time (s) 200 250 300 1 0.5 0 50 100 150 200 250 300 Time(s) Fig Time histories of dynamic displacement and acceleration of the linear simplified sdof tall structure model due to stationary wind loads at a reference mean wind velocity U H ¼ 20 m/s, reconstructed by the WG method and error analysis in comparison with the Newmark-beta integration method: (a) dynamic displacement (mean wind force effect not included), (b) error function of the displacement, (c) acceleration and (d) error function of acceleration Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 1500 D6 - D20 D4 D2 0.1 Normalized time ratios (%) Displacement (m) 0.2 -0.1 -0.2 20 40 60 80 1000 500 100 D6 100 D8 D10 Time (s) -0.1 20 40 D16 D18 D20 40 20 Error line to Newmark-beta E=5.7E-6 (%) j=6.96 ∆ = 0.008 D14 60 j=6.64 ∆ = 0.01 j=6.38 j=6.79 ∆ = 0.012 ∆ = 0.009 j=6.50 0.1 ∆ = 0.011 D12 Daubechies wavelets Error function E(%) Displacement (m) 0.2 -0.2 19 j=6.64 (fitted) 60 Time (s) 80 100 -20 6.5 7.5 Resolution parameter j Fig Influence of the selection and resolution of the Daubechies wavelet on the WG solution of the linear simplified sdof tall structure model due to stationary wind loads at a reference mean wind speed U H ¼ 20 m/s: (a) selection of Daubechies wavelets (D2, D4, D6 to D20), (b) comparison of normalized computing time for D2–D20, (c) effect of wavelet resolution (D6 wavelet), (d) error function as a function of the resolution j for D6 wavelet illustrates that the WG solution can drastically differ with small changes of wavelet resolution The relative error function between the WG-based displacement obtained by “fitted” wavelet resolution and by other resolutions is shown in Fig 6d; the relative error is defined as Ej %ị ẳ 100xj xj0 ị2 =x2j0 , where xj0 is the maximum displacement (in absolute values) at the “fitted” wavelet resolution and xj is the maximum (“peak”) response at other resolutions The analysis of the error function suggests that the WG-based displacement at resolutions other than the “fitted” resolution can be highly inaccurate Even though no consistent tendency is observed, there appears that higher resolution levels tend to produce a much larger peak response than lower resolution levels (Fig 6d) As a consequence, adequate estimation of the resolution level (j) becomes essential for the accuracy of the WG analysis and the numerical approximation of the stochastic dynamic response In the case of a nonlinear stochastic response with the Duffing-type stiffness nonlinearity, the WG approximation results are summarized in Fig This figure depicts the time histories of the WG-based displacements and accelerations at the reference node of the model due to a transient wind load with mean wind speed U H ¼ 20 m/s It is worth reminding that the secondorder nonlinear stochastic equation of motion has been transformed by the WG approximation to an easily-solvable nonlinear algebraic system of equations with cubic nonlinearity in the wavelet space (Eq (44)) In this study, the Levenberg–Marquardt numerical method has been utilized to solve the nonlinear equations The WG-based displacement and acceleration are shown in the case of “weak” stiffness nonlinearity (ε ¼ 0:1) in Fig 7a and Fig 7b, and “strong” stiffness nonlinearity (ε ¼ 10) in Fig 7c and Fig 7d Fig shows the time histories of the WG-based displacements and accelerations at the top node of the sdof model with the Van-der-Pol damping nonlinearity at the reference mean wind velocity U H ¼ 20 m/s due to transient wind loads The WG solutions are illustrated for two special cases of “weak” damping nonlinearity (μ ¼ 0:1) and “strong” damping nonlinearity (μ ¼ 10) in Fig 8a–d The WG solutions are determined after solving the nonlinear algebraic equations of power three in wavelet space as in Eq (47) 10.2 Full building model In the application of the WG analysis for the coupled stochastic wind-induced response of the reduced-order CAARC tall building (Fig 3), realizations of multivariate stationary and transient wind velocity fluctuations must be digitally constructed at each discrete node along the building height Fig shows, as an example, 300 s digitally simulated time histories of the stationary wind fluctuations uðz; tÞ and the transient wind fluctuations u0 ðz; tÞ in the x-along-wind direction at nodes 41 (rooftop node), 30, 20 and 10 at the reference mean wind velocity U H ¼ 20 m/s The stationary and transient fluctuating wind velocity realizations at other building nodes and for the y-cross-wind direction (v turbulence component) are not shown for the sake of brevity Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 0.1 0.05 0.5 Acceleration (m/s2) Displacement (m) 20 -0.05 -0.1 50 100 150 200 250 -0.5 -1 300 50 100 Time (s) 200 250 300 200 250 300 Acceleration (m/s2) Displacement (m) 0.1 0.05 -0.05 -0.1 150 Time (s) 50 100 150 200 250 0.5 -0.5 -1 300 50 100 Time (s) 150 Time (s) 0.1 0.05 0.5 Acceleration (m/s2) Displacement (m) Fig Time histories of the dynamic response at the roof-top node of the simplified sdof tall structure model with Duffing-type stiffness nonlinearity due to a transient wind load at the reference mean wind velocity U H ¼ 20 m/s: (a) dynamic displacement (mean removed), weak nonlinearity (ε¼ 0.1); (b) acceleration, weak nonlinearity (ε¼ 0.1); (c) dynamic displacement (mean removed), strong nonlinearity (ε¼ 10); and (d) acceleration, strong nonlinearity (ε¼ 10) -0.05 -0.1 50 100 150 200 250 -0.5 -1 300 50 100 0.1 0.05 0.5 -0.05 -0.1 50 100 150 Time (s) 150 200 250 300 200 250 300 Time (s) Acceleration (m/s2) Displacement (m) Time (s) 200 250 300 -0.5 -1 50 100 150 Time (s) Fig Response time histories of the simplified sdof tall structure model with Van-der-Pol-type damping nonlinearity due to a transient wind load at the reference mean wind velocity U H ¼ 20 m/s: (a) dynamic displacement (mean removed), weak nonlinearity (μ¼0.1), (b) acceleration, weak nonlinearity (μ¼ 0.1), (c) dynamic displacement (mean removed), strong nonlinearity (μ¼ 10), and (d) acceleration, strong nonlinearity (μ¼ 10) The WG analysis is subsequently applied to approximate the x-along-wind and the y-cross-wind global displacements at the discrete building nodes, induced by the stationary and transient wind velocity fluctuations Fig 10 illustrates a typical 300 s realization of the global response in the x-along-wind and the y-cross-wind directions at the rooftop node 41 when Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 21 the reference mean wind velocity is U H ¼ 20 m/s The maximum dynamic response (mean removed) at the building nodes can be extracted from the collection of a statistical ensemble of the response time histories at all building nodes, which can later be used to construct the response envelopes of the global displacements It is also of interest to study how the WG approximation can be used to examine the global response in terms of “energy distribution” by time–frequency representation, obtained under both stationary and transient wind loads Fig 11 shows the wavelet power spectra of the x-along-wind and the y-cross-wind global dynamic displacements at the rooftop node 41 It is observed that the global dynamic response “energy” is primarily concentrated along the time axis at certain frequencies, which correspond to the natural structural frequencies (fx ¼0.20 Hz and fy ¼0.22 Hz) of the fundamental modes in the x and y planes (directions) High “energies” in the global dynamic displacements due to stationary wind loads are equally distributed along the time axis, whereas those corresponding to the transient winds are much more concentrated, especially intensively localized around 185 s point in the example shown in Fig 11 10 10 Node 41 Node 41 Ampl.(m/s) Ampl.(m/s) -5 -5 -10 0 50 100 150 200 250 -10 300 50 100 Time (s) 150 200 Node 30 Node 30 Ampl.(m/s) Ampl.(m/s) -5 -5 50 100 150 200 250 -10 300 50 100 Time (s) 150 200 Node 20 Ampl.(m/s) Ampl.(m/s) -5 -5 50 100 150 200 250 -10 300 50 100 Time (s) 150 200 250 300 Time (s) 10 10 Node 10 Node 10 5 Ampl.(m/s) Ampl.(m/s) 300 10 Node 20 -5 -10 250 Time (s) 10 -10 300 10 10 -10 250 Time (s) -5 50 100 150 Time (s) 200 250 300 -10 50 100 150 200 250 300 Time (s) Fig Synthetic realizations of u along-wind velocity fluctuations at nodes 41, 30, 20, 10 for reference mean wind velocity U H ¼ 20 m/s: (a) stationary wind and (b) transient wind Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 22 Advantages of the WG analysis for the solution of aerodynamically and dynamically coupled stochastic problems have been examined Fig 12 illustrates, as an example, the global displacements at the rooftop node 41 at U H ¼ 20 m/s by including or neglecting the coupling terms in the equations, i.e., the effect of motion-induced wind forces In this particular example, there appears to be limited difference between the two cases with and without aerodynamic coupling For the 0.2 0.04 y-cross-wind 0.1 Amplitude (m) Amplitude (m) x-along-wind 0 50 100 150 Time (s) 200 250 -0.04 300 50 100 150 200 250 300 200 250 300 0.02 x-along-wind y-cross-wind 0.1 Amplitude (m) Amplitude (m) Time (s) 0.2 -0.1 -0.2 -0.02 -0.1 -0.2 0.02 0.01 -0.01 50 100 150 Time (s) 200 250 300 -0.02 50 100 150 Time (s) Fig 10 Time histories of x-along-wind and y-cross-wind global displacements at top node 41 for reference mean wind velocity U H ¼ 20 m/s: (a) x-alongwind displacement due to stationary winds, (b) y-cross-wind displacement due to stationary winds, (c) x-along-wind displacement due to transient winds, (d) y-cross-wind displacement due to transient winds Fig 11 Time–frequency representation of global displacements at top node 41 for reference mean wind velocity U H ¼ 20 m/s: (a) x-along-wind displacement due to stationary winds, (b) y-cross-wind displacement due to stationary winds, (c) x-along-wind displacement due to transient winds, and (d) y-cross-wind displacement due to transient winds Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 23 selected realization, the difference between the two cases is observed at the beginning of the interval in the computational domain It is observed that the WG analysis efficiently solves the coupled stochastic response of the tall building due to transient wind forces and aerodynamic coupling forces Two additional examples of WG-based maximum dynamic displacements of the CAARC building at rooftop node 41 are computed at mean wind speeds U H ¼ 10 m/s and U H ¼ 30 m/s for comparison purposes Table summarizes the comparison of the maximum rooftop displacements in the x-along-wind and y-cross-wind directions at various wind speeds, with (“Yes”) and without (“No”) coupling effect of the motion-induced aeroelastic forces It can be seen that the addition of coupling effect in the motion-induced forces produces larger global dynamic displacements in comparison with the case without the coupling Also, from the results in Table it is concluded that maximum displacements due to the transient wind loads tend to be smaller than those induced by stationary wind loads In particular, at the mean wind speeds U H ¼ 10 m/s, U H ¼ 20 m/s and U H ¼ 30 m/s the maximum dynamic displacements with coupling effect due to stationary wind loads are percent, 4.9 percent, 7.8 percent (in the x-along-wind direction) and 11 percent, 6.2 percent, 2.9 percent (the y-cross-wind direction) higher than those observed for a transient wind, respectively Finally, the relative difference between the maximum displacements in the two cases is defined as DR %ị ẳ 100x}Yes} x}No} Þ2 =x2}Yes} , where x}Yes} , x}No} respectively denote maximum dynamic displacements at node 41 with and without coupling effect It is observed that the differences in the WG-based global displacements in the x-along-wind, the y-cross-wind axes between the two cases vary from 0.28 percent (x-along-wind, transient wind, U H ¼ 20 m/s) to 13.2 percent (y-cross-wind, stationary wind, U H ¼ 20 m/s) More details are in Table Fig 13 examines the influence of the selection of the Daubechies wavelet order (from D2 to D20) on the WG-based dynamic response in the case of the full building model by considering the x-along-wind dynamic displacement at the rooftop node 41 Similar observations as those derived for the simplified sod model (Fig 6a) are found for the full building model The WG analysis with D6–D20 provides very similar and adequate solutions, while the use of D2 and D4 wavelets leads to underestimation of the response The reason for this issue is again related to the high compactness and small number of supported points of the D2 and D4 wavelets Finally, the simulations confirm that the D6 wavelet is also optimal for WG approximation in this second example since it provides adequate results without compromising the computing time (Fig 13b) 11 Conclusions The WG analysis has been explored to estimate the nonlinear coupled stochastic dynamics of both a simplified tall structure and a reduced-order full building model due to transient wind loads It has been demonstrated that the WG 0.3 No coupling Coupling 0.1 -0.1 -0.2 No coupling Coupling Amplitude (m) Amplitude (m) 0.2 x 10-3 Coupling -2 Coupling -4 No coupling 50 100 150 200 250 -6 300 No coupling 50 100 02 Coupling No coupling Amplitude (m) Amplitude (m) 0.2 0.1 -0.1 -0.2 150 200 250 300 Time (s) Time (s) Coupling No coupling 0.01 -0.01 Coupling No coupling 50 100 150 200 Time (s) 250 300 -0.02 Coupling No coupling 50 100 150 200 Time (s) 250 300 Fig 12 Coupling effect on global displacements at top node 41 for reference mean wind velocity U H ¼ 20 m/s: (a) x-along-wind displacement due to stationary winds, (b) y-cross-wind displacement of stationary winds, (c) x-along-wind displacement due to transient winds, and (d) y-cross-wind displacement due to transient winds Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 24 Table Maximum dynamic displacements of CAARC building at the top node 41 for reference wind velocities U H ¼ 10 m/s, 20 m/s, 30 m/s (mean wind force effect not included) Stationary wind load Transient wind load U H ¼ 10 m/s Yes U H ¼ 20 m/s No Yes U H ¼ 30 m/s U H ¼ 10 m/s U H ¼ 20 m/s U H ¼ 30 m/s No Yes No Yes No Yes No Yes No x-Along-wind displacement (m) 0.027 0.026 0.192 DR(%) 7.2 0.189 1.6 0.389 0.385 1.1 0.023 0.023 0.46 0.157 0.156 0.28 0.311 0.310 0.32 y-Cross-wind displacement (m) 0.004 0.004 0.020 DR(%) 7.1 0.018 13.2 0.058 0.054 8.8 0.003 0.003 2.3 0.016 0.015 1.1 0.055 0.054 0.6 Note: “Yes”: with coupling effect; “No”: no coupling effect; DR(%): difference function in percentage 1500 0.2 D2 D6 - D20 Normalized time ratios (%) D4 Amplitude (m) 0.1 -0.1 -0.2 50 100 150 Time (s) 200 250 300 1000 500 100 D6 D8 D10 D12 D14 D16 D18 D20 Daubechies wavelets Fig 13 Effect of Daubechies wavelets on global displacements at top node 41 for reference mean wind velocity U H ¼ 20 m/s: (a) effect of Daubechies wavelets and (b) comparison of normalized computing time analysis in the Daubechies wavelet domain can successfully transform a nonlinear coupled differential dynamic problem into a much simpler system of random algebraic equations The study also examined and successfully resolved some of the existing shortcomings of the WG analysis, such as the evaluation of the connection coefficients and the treatment of arbitrary initial conditions Influence of the Daubechies wavelets and the wavelet resolution levels on the WG solutions has been investigated The main findings can be summarized as  Daubechies wavelets D6 to D20 should be employed in the WG analysis to ensure adequate accuracy of the results The    D6 wavelet appears to be the most efficient selection for the WG analysis due to a combined accuracy of the dynamic solution and limited computing time; Estimation of the wavelet resolution is crucial for the accuracy of the WG approximation, since a slightly incorrect wavelet resolution, either higher or lower than the “fitted value”, can significantly overestimate the stochastic response; The use of 2-term connection coefficients allows resolving Duffing-type stiffness nonlinearities and Van-der-Pol-type damping nonlinearities in second-order dynamical systems; Aerodynamic coupling has a secondary influence on the stochastic response of the CAARC building with winds orthogonal to either vertical face of the structure Finally, even though it is widely accepted that time-step integration methods are quite effective in solving wind-induced response of structures, it is believed that the WG method will become advantageous in the context of probability-based wind engineering [60] For example, if the simulation of the structural response is affected by wind loading uncertainty [60], the WG method will transform the differential problem into an equivalent algebraic problem This algebraic problem could be easily solved by Monte Carlo sampling by generating a suitable set of downburst wind simulations, even if the structure is nonlinear or if a detailed model of the downburst wind is employed [20,61,62] In contrast, time-domain integration may become challenging since the computational complexity may significantly increase and Monte Carlo simulation may become practically unfeasible Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 25 Acknowledgments This material is based upon work supported in part by the National Science Foundation (NSF) of the United States under CAREER Award CMMI-0844977 in 2009–2014 Any opinions, findings and conclusions or recommendations are those of the authors and not necessarily reflect the views of the NSF The authors would also like to acknowledge the partial support of the NSF Award CMMI-1434880; the study, described in this document, constitutes a preliminary investigation on computer-based estimation of transient wind loading on tall buildings, which is being considered as part of the current and future research activities Appendix A Connection coefficient matrices of D6 wavelet The matrices of connection coefficients, used 0;2 ΩB 1;1 ⋮ Ω0;2 ¼ Ω0;2 BL;1 in the structural models, are assembled as 0;1 ΩB1;1 ⋯ Ω0;1 ⋯ Ω0;2 B1;L B1;L 7 ⋱ ⋮ 7; Ω0;1 ¼ ⋮ ⋱ ⋮ 7; 5 0;2 0;1 0;1 ⋯ ΩBL;L ΩBL;1 ⋯ ΩBL;L Ω0;0 B1;1 ⋮ Ω0;0 ¼ Ω0;0 BL;1 ⋮ 7¼4⋮ Ω0;0 BL;L Ω0;0 B1;L ⋯ ⋱ ⋯ ⋯ ⋱ ⋮ 5: ⋯ As an example, the block matrices along the diagonal of Ω0;2 in Eq (A.1) are Ω0;2 Ω0;2 Ω0;2 Ω0;2 Ω0;2 0 6 Ω0;2 Ω0;2 Ω0;2 Ω0;2 Ω0;2 Ω0;2 À1 0;2 0;2 0;2 6Ω Ω0;2 Ω0;2 Ω0;2 Ω0;2 À Ω À Ω0 0;2 0;2 0;2 0;2 0;2 0;2 0;2 6Ω Ω Ω Ω Ω Ω Ω À2 À1 À3 0;2 0;2 0;2 0;2 0;2 0;2 6Ω Ω À Ω À Ω À Ω0 Ω1 Ω0;2 Ω0;2 B1;1 ¼ À 60 Ω0;2 Ω0;2 Ω0;2 Ω0;2 Ω0;2 Ω0;2 À À À À 1 6 0;2 0;2 0;2 0;2 60 Ω0;2 Ω Ω Ω Ω À4 À3 À2 À1 6 0 Ω0;2 Ω0;2 Ω0;2 Ω0;2 60 À À À À1 0;2 0;2 0 0 Ω0;2 Ω Ω À4 À3 À2 0;2 0;2 0;2 0;2 Ω0;2 Ω Ω Ω Ω 0 6 Ω0;2 Ω0;2 Ω0;2 Ω0;2 Ω0;2 Ω0;2 À1 0;2 0;2 0;2 0;2 0;2 0;2 0;2 6Ω Ω Ω Ω Ω Ω Ω À2 À1 0;2 0;2 0;2 0;2 6Ω Ω0;2 Ω0;2 Ω0;2 À Ω À Ω À Ω0 0;2 0;2 0;2 0;2 0;2 0;2 Ω À Ω0;2 Ω Ω Ω Ω Ω ¼ Ω0;2 À3 À2 À1 BL;L 6 0;2 0;2 0;2 0;2 0;2 Ω À Ω À Ω À Ω À Ω0 Ω0;2 6 0 Ω0;2 Ω0;2 Ω0;2 Ω0;2 Ω0;2 À À À À 6 0;2 0;2 0;2 0 Ω0;2 Ω Ω Ω À4 À3 À2 À1 0 0 Ω0;2 Ω0;2 Ω0;2 À4 À3 À2 Similarly, the block matrices along the Ω0;1 6 Ω0;1 À1 0;1 6Ω À2 0;1 6Ω À3 0;1 0;1 ΩB1;1 ¼ 6 ΩÀ4 6 6 6 6 (A.1) 0 0 7 7 0 7 7 Ω0;2 0;2 0;2 Ω3 Ω4 7 7 Ω0;2 Ω0;2 0;2 Ω0;2 Ω 7 0;2 Ω0;2 Ω Ω0;2 Ω0;2 À1 0 07 7 07 7 Ω0;2 07 0;2 7 Ω0;2 Ω 0;2 7 Ω0;2 Ω 0;2 0;2 Ω1 Ω2 7 Ω0;2 Ω0;2 Ω0;2 Ω0;2 À1 diagonal of Ω0;1 in Eq (A.1) are Ω0;1 Ω0;1 Ω0;1 Ω0;1 0 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 0 Ω0;1 À1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À2 Ω0;1 À3 Ω0;1 À4 Ω0;1 À1 Ω0;1 À2 Ω0;1 À3 Ω0;1 À4 0 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 À3 Ω0;1 À4 0 Ω0;1 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 À3 Ω0;1 À4 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 À3 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À1 (A.2) 07 7 07 7 07 7 Ω0;1 0;1 Ω3 7 Ω0;1 7 0;1 Ω1 Ω0;1 Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 26 Ω0;1 BL;L Ω0;1 6 Ω0;1 À1 0;1 6Ω À2 0;1 6Ω À3 0;1 ¼6 ΩÀ4 6 6 6 6 Ω0;1 Ω0;1 Ω0;1 Ω0;1 0 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 0 Ω0;1 À1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À2 Ω0;1 À3 Ω0;1 À4 Ω0;1 0 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 À3 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 À3 Ω0;1 À4 Ω0;1 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 À3 Ω0;1 À4 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 À3 Ω0;1 À4 Ω0;1 Ω0;1 Ω0;1 À1 Ω0;1 À2 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À1 07 7 07 7 07 7 Ω0;1 0;1 Ω3 7 Ω0;1 7 Ω0;1 Ω0;1 (A.3) Appendix B Treatment of initial conditions in the connection coefficient matrices The initial conditions can be taken into account by suitable modification of the block matrices, described in Appendix A This leads to " # 0 0 0 0 Ω0;2 ¼ B1;1 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0;2 (B.1) ΩBL;1 ¼ Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 Ω0;1 À4 À3 À2 À1 " Ω0;1 B1;1 Ω0;1 BL;1 ¼4 ⋮ Ω0;1 À4 ¼ ⋮ Ω0;1 À3 ¼4 ⋮ ⋮ ⋮ Ω0;1 À4 ⋮ ⋮ ⋮ Ω0;1 À3 ⋮ ⋮ Ω0;1 À2 Ω0;0 B1;1 ¼ ⋮ ⋮ " Ω0;0 BL;1 ⋮ ⋮ Ω0;1 À1 ⋮ ⋮ Ω0;1 À1 ⋮ ⋮ ⋮ ⋮ ⋮ # ⋮ Ω0;1 ⋮ Ω0;1 À2 ⋮ Ω0;1 ⋮ ⋮ Ω0;1 ⋮ ⋮ Ω0;1 ⋮ ⋮ ⋮ Ω0;1 Ω0;1 (B.2) # ⋮ ⋮ Ω0;1 ⋮ Ω0;1 ⋮ Ω0;1 3 ⋮ Ω0;1 For the stochastic force vector (wind speed fluctuations) the augmented vector becomes > < x0 > = ⋮ ufkg ¼ > : x_ > ; (B.3) (B.4) Appendix C Digital simulation of transient wind velocity records, used to simulate the wind load at the nodes of the building models The digital simulation of multivariate transient wind velocity fields can be implemented by spectral representation and by considering either of the two following approaches: (i) combination of stationary power spectral density functions (PSD) and modulation functions; (ii) evolutionary power spectral density functions (EPSD) In the former case, the multivariate stationary winds are first synthetically generated at discrete nodes using the cross PSD matrix of the wind turbulence; the stochastic realizations of the stationary wind turbulence components are subsequently modulated by amplitude or frequency modulation functions to construct the multivariate transient winds (e.g., [53]) In the latter case, multivariate transient winds fields are directly simulated by the EPSD matrix, which can be obtained from available transient wind records (e.g., [54,55]) In this study, the first approach is followed In particular, a synthetic realization of the stationary wind speed process can be digitally generated using either the Cholesky decomposition (e.g., [56]) or the proper orthogonal decomposition (e.g., [57]) Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 27 The multivariate transient wind velocity fields at the discrete nodes in the x-along-wind and the y-cross-wind directions of the building can be expressed (e.g., [53]): U p ðzp ; tị ẳ U p zp ; tị ỵu0p zp ; tÞ and ν0p ðzp ; tÞ; (C.1) where p is the index designating a generic node with p ¼1,2,…,M; M is the number of building nodes; zp is the vertical coordinate of the node position along an axis corresponding to the building model vertical axis; U p ðzp ; tÞ is the time-varying “mean” wind velocity; u0p ðzp ; tÞ and ν0p ðzp ; tÞ are the transient/nonstationary wind velocity fluctuations The quantities u0p ðzp ; tÞ and ν0p ðzp ; tÞ are found by combining two multivariate zero-mean stationary random turbulence processes with a slowlyvarying deterministic modulation function A^ p ðω; zp ; tÞ (e.g., [58]): u0p ðzp ; tị ẳ A^ p ; zp ; tịup zp ; tị; (C.2) 0p zp ; tị ẳ A^ p ; zp ; tÞνp ðzp ; tÞ; (C.3) where up ðzp ; tÞ and νp ðzp ; tÞ are multivariate zero-mean stationary turbulence processes at generic discrete node p The modulation function A^ p ðω; zp ; tÞ is established in a general form, which depends both on time t and circular frequency ω It is noted that the time-varying mean velocity term U p ðzp ; tÞ in a “downburst-like” transient wind is either determined from direct observations of the extreme wind events [21,22] or a phenomenological model [23,24] In this study, the examination of time-independent mean wind velocity profile U p ðzp ; tÞ % U p ðzp Þ was employed as a first approximation and for the primary purpose of examination of the WG analysis method The evolutionary power spectral theory of a transient random process determines the elements of the evolutionary cross spectral matrix Su ðω; z; tÞ of the wind velocity fluctuations as  2   Su;pp ; zp ; tị ẳ A^ p ; zp ; tÞ S0u;pp ðω; zp Þ; (C.4) Tn Su;pq ; zp ; zq ; tị ẳ A^ p ; zp ; tÞA^ q ðω; zq ; tÞS0u;pq ðω; zp ; zq Þ; (C.5) S0u;pp ðω; zp Þ and matrix S0u ðω; zÞ; where “T” and “n” symbols denote the transpose and complex conjugate operators The quantities S0u;pq ðω; zp ; zq Þ are the stationary auto-spectral and cross-spectral elements of the stationary cross spectral p and q are the two generic nodal indices; zp and zq are the vertical coordinates of the nodes The above-indicated equations are valid for x-along-wind fluctuating velocity component, up ðzp ; tÞ, but similar relationships can be established for the y-cross-wind turbulence component, νp ðzp ; tÞ The cross-spectral elements of the stationary cross spectral matrix can be estimated by standard empirical spatial coherence function and PSD model of the stationary wind turbulence (e.g., [56,57]): q (C.6) S0u;pq ; zp ; zq ị ẳ Cohu;pq ðω; zp ; zq Þ S0u;pp ðω; zp ÞS0u;qq ; zq ị; Cohu;pq ; zp ; zq ị ẳ exp À ωS0u;pp ðω; zp Þ 2πσ u ðzp Þ2 ¼ C u ωjzp À zq j πðU p ðzp ị ỵ U q zq ịị 0:6Xzp ị ỵXzp Þ2 Þ5=6 ! ; : (C.7) (C.8) In Eq (C.7), Cohu;pq ðω; zp ; zq Þ is Davenport's spatial coherence function between node p and node q with Cu being a turbulence decay factor The PSD model in Eq (C.8) is Harris' spectrum with u zp ị ẳ I u;p U p ðzp Þ being the standard deviation of the stationary wind turbulence at zp (I u;p , turbulence intensity), and with Xzp ị ẳ 1600=2U p zp ịị depending on the stationary mean wind speed UðzÞ at the generic node p with vertical coordinate zp It is noted in this study that constant turbulence intensity was used at various elevations z as a first approximation The Cholesky decomposition is applied to decompose the evolutionary cross-PSD matrix of the transient winds, Su ðω; z; tÞ, as (e.g., [53]) ^ z; tÞj2 Hu ðω; zÞHTun ðω; zÞ; Su ðω; z; tị ẳ jA; (C.9) in which Hu ; zị is a lower triangular matrix, obtained from the decomposition of the stationary cross spectral matrix of the turbulence, which is assembled from Eqs (C.6)–(C.8) As a result, realizations of multivariate transient wind turbulence in the x-along-wind direction at the discrete node p can be generated as (e.g., [53]) M X n pffiffiffiffiffiffiffi X À Á À Á À Á  u0p zp ; tị ẳ jA^ p ωl ; zp ; t jjH u;pm ωl ; zp j cos ωl t À ϑu;pm ωl ; zp ; t ỵ u;ml (C.10) mẳ1lẳ1 In the previous equation is a circular frequency step with Δω ¼ ωup =n; the quantity ωup is the upper cut-off circular frequency; n is the number of circular frequencies, used by the wave-superposition method [56,57]; ωl is a generic circular Please cite this article as: T.-H Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/ j.jsv.2015.01.007i T.-H Le, L Caracoglia / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 28 10 Fluctuating velocity (m / s) Cosine modulation 0.8 0.6 0.4 0.2 0 50 100 150 200 Time (s) 250 300 -5 -10 50 100 150 200 250 300 Time (s) Fig C1 Effect of variation in the parameters of the modulation function on simulated transient wind velocity: (a) cosine modulation function with variable parameters and (b) synthetic transient wind velocity records À Á frequency (ωl ¼ lΔω) The matrix decomposition of the wind spectrum leads to H u;pm ω; zp ¼ jH u;pm ðω; zp Þjeiϑu;pm ðω;tÞ , with ϑu;pm ẳ tan ImẵH u;pm ; tị=ReẵH u;pm ; tÞŠÞ and ϕu;ml being independent phase angles, uniformly distributed in the interval [0,2π] Realizations of the random phase angles ϕu;ml can be generated by Monte Carlo sampling Similarly, we obtain for the multivariate realizations of y-cross-wind turbulence component: M X n pffiffiffiffiffiffiffi X À Á À Á ν0p zp ; t ¼ Δω jA^ p ωl ; zp ; t jjH ;pm l ; zp ịj cos ẵl t ;pm l ; z; t ị ỵ;ml (C.11) m¼1l¼1 Finally, for generating the transient wind speed records, a cosine-type modulation function is utilized [43]: ! À Á cos ð2πt=T Þ η : A^ p ωl ; zp ; t % A^ p tị ẳ (C.12) In the previous equation η is the “width” parameter of the window (a positive even integer); T0 is the “location” parameter of the maximum amplitude The cosine modulation function is strictly positive and its maximum amplitude is the unit value Fig C1 illustrates the effect of a variation in the parameters of the cosine modulation function on the simulated transient fluctuating velocity Three cases are considered by varying width parameter η and location T : T ¼ 50 s; η ¼ 0:5; T ¼ 100 s; η ¼ 1; T ¼ 150 s; η ¼ 1:5 (see Fig C1a) In this figure the width of the modulating window reduces with an increase in the parameter η (see Fig C1b) A short-duration transient wind velocity realization can be obtained from a stationary process realization by widening and sharpening the modulating window through η The dominant peaks of the original stationary process are unaltered by the modulating windows Moreover, the cosine modulation function symmetrically sharpens the original signal on both ends (Fig C1.b) In this study, η ¼ and T ¼ 300 s have been employed to create the transient fluctuating wind velocities, used in the main sections, since these approximately correspond to the duration of a microburst (e.g., Andrews Air Force Base downburst; [61]) It is also noted that the thunderstorm downburst winds are non-synoptic transient wind processes and some shortcomings still exist in the models for the digital simulation of transient thunderstorm downburst winds, especially for vertical structures such as tall buildings These are reflected in Eq (C.1), which is a simplification of the physical phenomenon in a downburst wind Concretely, challenges in the simulation of realistic thunderstorm downburst wind velocities are related to: (i) mean wind velocities of the thunderstorm downburst at building heights are slowly varying and time-dependent; (ii) an additional translational velocity effect due to the downburst motion is 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to formulate... (EPSD) In the former case, the multivariate stationary winds are first synthetically generated at discrete nodes using the cross PSD matrix of the wind turbulence; the stochastic realizations of the. .. Le, & L Caracoglia, Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/