Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 956793, 10 pages

http://dx.doi.org/10.1155/2015/956793

## A Comparative Study on Optimal Structural Dynamics Using Wavelet Functions

StruHMRS Group, Department of Civil Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received 2 April 2014; Accepted 8 October 2014

Academic Editor: Shuenn-Yih Chang

Copyright © 2015 Seyed Hossein Mahdavi and Hashim Abdul Razak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Wavelet solution techniques have become the focus of interest among researchers in different disciplines of science and technology. In this paper, implementation of two different wavelet basis functions has been comparatively considered for dynamic analysis of structures. For this aim, computational technique is developed by using free scale of simple Haar wavelet, initially. Later, complex and continuous Chebyshev wavelet basis functions are presented to improve the time history analysis of structures. Free-scaled Chebyshev coefficient matrix and operation of integration are derived to directly approximate displacements of the corresponding system. In addition, stability of responses has been investigated for the proposed algorithm of discrete Haar wavelet compared against continuous Chebyshev wavelet. To demonstrate the validity of the wavelet-based algorithms, aforesaid schemes have been extended to the linear and nonlinear structural dynamics. The effectiveness of free-scaled Chebyshev wavelet has been compared with simple Haar wavelet and two common integration methods. It is deduced that either indirect method proposed for discrete Haar wavelet or direct approach for continuous Chebyshev wavelet is unconditionally stable. Finally, it is concluded that numerical solution is highly benefited by the least computation time involved and high accuracy of response, particularly using low scale of complex Chebyshev wavelet.

#### 1. Introduction

Generally, results integrated from dynamic analysis of structures are the main reliable criteria for design of solids and structures. The technique for a solution of general dynamic equilibrium can become very expensive if a complex loading (e.g., an unknown earthquake excitation) is being applied on large-scaled structures [1, 2]. In general, procedure of dynamic analysis is categorized into the two solution methods: first, mode superposition method. Second, direct and indirect integration methods, for example, central differences, family of Newmark-*β*, Houbolt, and Wilson-*θ* fall under direct integration schemes and Fourier transformation and wavelet analysis have been introduced as indirect integration approaches. In addition, the choice as to which method to use for an effective solution is governed by the dynamical problem considered [3, 4]. Theoretically, aforesaid numerical integration methods lie in case of either conditionally stable or unconditionally stable procedures. An integration method is unconditionally stable if the solution for any initial conditions does not grow without a band for any time step while will be conditionally stable if it does grow. In other words, stability of results is achieved at any time interval and there is no restriction on using smaller than a particular value. As a result, optimal solution for dynamic analysis is accomplished using long intervals [5–7].

Nowadays, orthogonal polynomials are being widely implemented as a practical analysis of time dealing problems in engineering, particularly in the form of wavelet analysis. Obvious effectiveness from property of orthogonality is that the repeated components with similar characteristics are neglected in the analytical process [8, 9]. Consequently, computational calculations are being considerably decreased and computation time involved is being therefore reduced, hence, accuracy of responses will be more desirable [10, 11]. Practically, considerable attention has been given for the use of wavelet method for the solution of time dependent problems such as dynamic analysis or identification of systematic problems [12]. Several attractive mathematical characteristics of wavelets such as efficient multiscale decompositions, localization properties in physical and wave-number spaces, and the existence of recursive and fast wavelet transforms have obtained practice of this efficient tool for the numerical solution of partial differential equations (PDEs) and ordinary differential equations (ODEs) [13–15].

Significantly, accuracy of responses is directly related to the basis function of mother wavelet, depending on the kind of objectives. Fundamentally, in the structural dynamics, compatibility of a wavelet basis function is premised upon on not only the degrees of freedom but also the similarity of basic functions to the lateral loading, emphasizing on frequency contents. Considerably, less computational cost of calculations in advanced time history analysis through the compatible wavelet functions makes distinction of this approach over other numerical methods. For instance, for the purpose of time history analysis, a simple basis function of Haar wavelet was indirectly applied on its own free-scaled functions [16]. It is inferred that because of the inherent simple shape function of Haar wavelet accuracy of responses was undesirable even employing large-scaled functions. Furthermore, to improve inadequacy of Haar wavelet known as the simplest and two-dimensional (2D) wavelet basis function, it is indispensable to employ three-dimensional (3D) and adaptive wavelet basis functions [17, 18]. Moreover, there is no consideration reported on the stability of responses calculated using Haar wavelet functions compared with other basis functions.

Mathematically, adaptive wavelets are those that grow in three dimensions, which in the current definition dimensions are time, scale, and frequency, respectively. For instance, Chebyshev, Legender, and Symlet are some basis functions with compatible characteristics [19]. Basically, Chebyshev polynomials are presented as continuous basis function of wavelet [20, 21]. The most popular characteristic of this wavelet is various weight functions of Chebyshev polynomials that directly influence stability and accuracy of responses [22–24]. However, it is reported that stability of results computed by Chebyshev wavelet are independent from initial accelerations. Furthermore, compatibility is being satisfied through the capturing of broad frequency of complex excitations by oscillated shape functions of free-scaled Chebyshev wavelet [25].

Subsequently, the main contributions of this study (which received little attention in the literature) are composed of the following: (i) a numerical assessment of structural dynamic problems using free-scaled simple and complex wavelet functions, with the emphasis on the large scales of wavelets, (ii) numerical stability analysis of an indirect algorithm using family of discrete Haar wavelets, established as 2D wavelet functions, (iii) stability analysis of continuous Chebyshev wavelet functions with respect to the third-ordered derivation of time, (iv) a comparative evaluation of computational efficiency of simple and complex wavelets for smooth and wide-band frequency contents of loading, and (v) capability evaluation of Haar and Chebyshev wavelets in linear and nonlinear structural dynamic problems. Accordingly, to achieve the proposed objectives of this paper, the second-ordered differential equation of motion is indirectly solved by free scale of Haar wavelet and later, free-scaled Chebyshev wavelet functions are directly implemented to compute responses. For this aim, a brief background of wavelet is discussed in Section 2 of this study. In addition, coefficient matrices of wavelet and operation matrices of integration corresponding to complex scales of efficient Chebyshev wavelet are formulated and presented in this section. In Section 3, the computational procedure is developed for an optimal dynamic analysis. Section 4 is allocated to numerical stability analysis of responses, corresponding to the indirect method proposed for Haar and direct scheme proposed for Chebyshev wavelet functions. Accordingly, stability of responses has been comparatively presented. Section 5 is devoted to investigation of the validity and effectiveness of results. For this purpose, various scales of Chebyshev and Haar wavelet functions are considered for dynamic analysis. Finally, efficiency and accuracy of results have been compared in this section.

#### 2. Fundamental of Wavelet

##### 2.1. Haar Wavelet

The simple family of Haar wavelet was presented by Alfred Haar in 1910 for as follows [16, 18]: where where denotes the order of wavelet; is the value of transition. In (1), and indicate scale function and mother wavelet of Haar, respectively. In this study, indicates the number of segmentations in each global time interval regarding the scale of proposed wavelet which refers to segmentation method (SM). For example, in the case of Haar wavelet denotes the th scale of Haar wavelet [18].

Basically, signal can be expanded in Haar series as [16]:

Accordingly, Haar coefficients () are defined by

Hence, is a square matrix , including the first scale of Haar wavelet; Haar coefficients are directly given as

Equivalently, signal may be rewritten as

Subsequently, integration of is obtained by Haar series with new square coefficient matrix of integration as [16, 18]

Pointed out that local times are calculated relatively to the scale of wavelet as

Finally, the local time divisions are adapted to the global domain. Assumption of as global time interval we have [16, 18]

##### 2.2. Chebyshev Wavelet

In mathematics, families of continuous wavelet transforms (CWT) are considered as follows [22, 24]: where CWT denotes corresponding wavelet transform. As long as wavelet function is supposed as mother wavelet, the continuous wavelet transform of a signal is obtained as where indicate the transition and the scale of wavelet and indicates the complex conjugate form of , respectively.

In general, the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials defined as two kinds. The general expression for Chebyshev polynomials of the first kind () is defined as follows [20, 21]:

In addition, the variable weight functions of is defined as [25]

Mathematically, the Chebyshev wavelet arguments are defined as where positive integer denotes the value of transition, indicates the time, is degree of Chebyshev polynomials for the first kind, and denotes the considered scale of wavelet. Chebyshev wavelets are formulated with substituting the first kind with relevant weight functions for each scale and transition in (11) as follows [22, 25]: where in (15) is obtained as

Thus, weight functions corresponding to different scales are obtained as

Regarding the idea of SM method is number of partitions in the global time, is the order of Chebyshev polynomials that is derived as

Numerically, the signal can be expanded with Chebyshev wavelets as [19, 22]

Here, and are obtained as two vectors: where

Chebyshev coefficients matrix is defined as where where .

Local time for collocation points are considered as follows [25]:

Subsequently, a -dimensional square matrix of is derived as

For instance, assumption of and ; lies on the first four equations of Chebyshev wavelet corresponding to eight collocation points (in referring to the SM method). Accordingly, coefficient matrix of free-scaled Chebyshev wavelet of corresponding to local times is improved as follows:where

Local times for and are calculated on points as

As can be seen from the matrix of (26), column of th refers to and integration of in the local time is obtained as where -dimensional and denote the operation matrix of integration and local time, respectively. Accordingly, operation matrices of integration [24] are improved as follows: where

And, respectively,

Hence, the operation matrix of integration is obtained as

#### 3. Dynamic Analysis of Equation of Motion

##### 3.1. Haar Wavelet

Theoretically, (1) reveals one major shortcoming of Haar wavelet where at the point of 0.5 there is no continuity. In other words, the second derivation is not existed at this point. As a result, it is not possible to use this simple wavelet function, directly. Consequently, to utilize free scale of Haar wavelet functions for solution of second-ordered* ODE* of motion, an alternative procedure is implemented, indirectly.

After discretization of external loading to the equal partitions, the dynamic equilibrium governing on corresponding mass (), damping (), and stiffness () is converted to the local time analysis as follows (related to the points of SM method) [16]: where terms of velocity and acceleration are considered as

Or equivalently are expanded as

Here and are row vectors with dimension of , for multiplying with* aPH or bPH* a unit vector is suffixed as . Finally, are initial and boundary conditions at , that are obtained with linear interpolation of in the current interval and calculated from previous interval. -dimensional and also represent operation of integral matrix of Haar and coefficient matrix of Haar relevant to the collocation points, respectively. Assumption of , it gives vector of as [16]
where .

Substituting (38) into (37) vector of is developed as

Here denotes -dimensional identity matrix. Finally, substituting (38) and (39) into (36) and (37) vectors of acceleration, velocity, and displacement being calculated in each collocation point. A schematic view of results calculated with Haar wavelet is depicted in Figure 1. This figure shows a stairs-shaped plot for responses corresponding to all collocation points of Haar wavelet.