( ) ( ) (2) h t h t dt numerical results - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS WAVELET BASED DISCRETIZATION TECHNIQUE FOR ANALYSIS AND DESIGN OF COMPOSITE STRUCTURES J. Majak 1 , J. Kers 2 * , M. Pohlak 1 , M.Eerme 1 , K.Luiga 1 1 Department of Machinery, Tallinn University of Technology, Estonia, 2 Department of Material Engineering, Tallinn University of Technology, Estonia, * Corresponding author(jmajak@staff.ttu.ee) Keywords : discretization method, Wavelets, decomposition of the solution compared with the results given in [8] (cas study 1). 1. Introduction During last decade, the Haar wavelet theory has An analysis of the corresponding discrete systems been applied to various problems including image of algebraic equations has been performed.The compression, signal processing, solution of possibilities to increase an accuracy of the solution differential and integral equations etc. Chen and are pointed out. The higher order approximation is Hsiao [1] derived a Haar operational matrix for the proposed. Later approximation is based on integrals of the Haar function vector, which is a decomposition of the solution introduced by authors fundamental result for the Haar wavelet analysis of of the current study in [5]. Recently, the Haar the dynamic systems. In [2] Hsiao introduced a Haar wavelet techniques have been treated for the solution product matrix and a coefficient matrix. In [1-2] the of the PDE-s [9-11]. Numerical results are given for integral method for solution of the differential a linearly tapered plate. equations is used. The highest order derivative included in the differential equation is expanded into 2. Haar wavelet family the Haar series. Latter approach allows to overcome The set of Haar functions is defined as a group of � in some intervals problems with computing derivatives in the points of 1 square waves with magnitude discontinuities of the Haar function. The higher and zero elsewhere order operational matrices and the properties of the corresponding integrals of the Haar functions need � � � � 0 . 5 k k � 1 , � still examination. An approach suggested by Chen � for t � � � m m � and Hsiao [1] is successfully applied for solving � � � � � 0 . 5 1 k k , (1) integral and differential equations in several papers � � � � � ( ) 1 , h i t for t � � � [3-6]. Latter approach is assumed also in the current � m m � 0 study. Both, weak and strong formulatin based Haa elsewhere � wavelet discretization methods are discussed. The � weak formulation based Haar wavelet discretization method has been introduced by authors of the � � � � 2 j , { 0 , 1 . , }, 0 , 1 , , 1 . where � � m j J k m current study in [5]. Three case studies are The integer J determines the maximal level of considered: free transverse vibrations of the resolution and the index i is calculated from the orthotropic rectangular plates of variable thickness � � � 1 in one direction, transverse vibrations of Bernoulli- formula . The Haar functions are i m k Euler beam, vibration analysis of wint turbine orthogonal to each other and form a good transform towers. In order to estimate the accuracy of the basis obtained numerical solution more adequately, the case studies are chosen so that closed form � � � � � 1 2 j 2 j i l k analytical solution exists in special case. The � � ( ) ( ) � (2) h t h t dt numerical results corresponding to the special case � i l 0 � i l 0 where the thickness of the plate is constant(case study 1) has been validated against closed form analytical results [7]. The numerical results are

The Haar matrix is defined through Haar functions 3. Case studies as 3.1 Free transverse vibrations of the orthotropic rectangular plates of variable thickness � � T Classical deformation theory is employed. It is � . (3) ( ) ( ) ( ) ..... ( ) H t h t h t h m t 1 2 assumed that the principal directions of orthotropy coincide with natural co-ordinate ( t ) Any function that is square integrable and y system. The equation of motion governing finite in the interval [0,1) can be expanded into Haar natural vibration of a thin orthotropic wavelets. It follows from (1) that the integration of rectangular plate is given by Haar wavelets results in triangular functions. These functions can be expanded into Haar series as � 4 � 4 � 4 � � 3 � � 3 � � 3 w w w T w T w D w � � � � � � 2 2 2 2 x D D T x y � � � t � 4 � 4 � 2 � 2 � � 2 � � 2 � 3 x y x x y x y x y y x x � � � � . (4) H ( ) d P H ( t ) N N N 2 � � � 3 � 2 � 2 � 2 � 2 � 2 � 2 � 2 D D w D w w D w D w � y � � y � � � 0 x 2 2 2 � � 3 � 2 � 2 � 2 � 2 � 2 � 2 � 2 � 2 y y x x y y y x x y The operational matrix of integration is � 2 P 2 2 � � D w w N xy � � �� � 4 , kw determined by equalizing the left and right sides of � � � � 2 � x y y x t the relation (4) in the collocation points , (7) t 1 ,... t N � 2 � � where 1 2 , 1 , t l ( l ) /( N ) l ,..., N where � � � T and t t ,.., t 1 C N D � * � 3 , � * � 3 , � � 3 , / 12 / 12 / 12 D E D G E x x y y xy xy t � � � d � , (5) Q ( t ) H ( ) E � 1 D � * � 3 N N , � � , * x , / 12 2 E E T D D xy � � � x 0 � � x y � T where . Q ( t ) q ( t ),..., q ( t ) 1 E N N � 1 y � � � � * * * * , . (8) E E E E As pointed out above, the higher order operational � � � y y x x y matrices and the properties of the corresponding x y In (7)-(8) � , D and E stand for the Poisson’s integrals of the Haar functions need still examination. Let us denote the first order ratio, flexural rigidity and modulus of elasticity, operational matrix and corresponding (first) integrals respectively, with subscript corresponding to co- w � P 1 1 ( ) ( ) ordinate axis. is the transverse of the Haar functions by and , w ( t , x , y ) Q ( t ) N N deflection, � is the mass density, � � � is the � ( x ) 1 1 ( ) ( ) respectively ( , = ). The P P Q ( t ) Q N ( t ) N N N variable plate thickness and k is the modulus of a second and higher order integrals of the vector of Winkler type foundation. In-plane dimensions of the ( i ) Haar functions are defined as Q ( t ) plate are denoted by a and b .It is assumed that the N � 0 � edges of the plate along are simply y , y b t � supported, and the other two edges each � � � d � �� 1 ( i ) ( i ) , 2 . (6) (7) Q ( t ) Q ( ) i � 0 � N N ( ) have clamped or simply supported x , x a 0 boundary conditions. The time-harmonic-dependent solution and the Lévi approach are considered. The ( i ) The vector functions can be expanded into Q ( t ) transverse deflection w can be assumed as N Haar series similarly to (5). The higher order ( i ) operational matrices can be evaluated by � � � P i t w ( t , x , y ) w ( x ) sin( n y / b ) e , (9) N n discretization of integrals of the Haar functions ( i ) . Q ( t ) N