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Free vibration analysis of a functionally graded material beam: evaluation of the Haar wavelet method.

1. INTRODUCTION

Accuracy and complexity are two key factors characterizing any numerical method. The Haar wavelet method (HWM) considered in the current study was introduced by Chen and Hsiao in [1,2] almost 20 years ago and up to now it has been applied for solving a wide class of differential and integral equations covering engineering, economic, etc. problems [3-7]. An overview of the applications of the HWM is given in [8]. The wavelet techniques based on the use of an operational matrix of integration are developed for solving ordinal and partial differential equations in [1-10] and for integral equations in [11-13]. All these studies implement the strong formulation based approach of the HWM. The weak formulation based approach of the HWM was introduced in [14].

Most of the authors characterize the HWM as a simple and effective method [3-9]. These estimates cover mainly implementation of the HWM, less its accuracy and convergence results, which are still under development. It is shown in [15] that in the case of function approximation with direct expansion into the Haar wavelet the convergence is of order one. However, according to the HWM approach considered, the highest order derivative included in the differential equation is expanded into a series of Haar functions. Thus, the estimate given in [15] holds good for estimating the accuracy of the highest order derivative, but not the solution of the differential equation. Recently, the convergence theorem of the HWM was proved in [16] for the nth order ordinal differential equations (ODEs) (n [greater than or equal to] 2). It was stated that the order of convergence of the HWM is equal to two. In [17] the accuracy estimates for the extrapolated results in the case of the fourth order ODE are derived, and it is shown that the order of convergence of the extrapolated results is equal to four (Richardson extrapolation is applied).

The application area of new simple methods often includes problems with advanced material models, constitutive laws, etc., which are not yet (well) covered by commercial software.

A new trend in the development of wavelet methods can be outlined as solution of fractional differential and integral equations [15, 18-24], which is an area not yet well covered by commercial software (finite element method (FEM), etc.). It is observed in [21] that in the case of fractional ODE the order of convergence of the HWM is equal to two if higher order derivative [alpha] in the fractional differential equation exceeds one ([alpha] > 1). However, in the case of 0 < [alpha] < 1 the order of convergence of the HWM tends to the value 1 + [alpha].

In [25,26] the HWM was adapted for the analysis of structures of functionally graded material (FGM). In the current study the vibration analysis of the FGM beams is performed and the results obtained by the HWM are compared with the corresponding results obtained by using the finite difference method (FDM) and the differential quadrature method (DQM). Selection of FDM and DQM for comparison of results was motivated by the fact that these methods are widely used numerical methods in engineering and are based on strong formulation (the complexity of implementation is similar). The methods considered are implemented by the authors in the MATLAB code.

In order to verify the obtained results and prepare solution procedures for structures with complex geometry and loading cases, the solid element 3D finite element model was developed.

2. BASICS OF HAAR WAVELETS

The Haar function is defined in [8,9] as

[mathematical expression not reproducible] (1)

In (1) i = m + k + 1, m = [2.sup.j] is the maximum number of square waves that can be sequentially deployed in interval [A, B] and the parameter k indicates the location of the particular square wave,

[mathematical expression not reproducible] (2)

The Haar functions are orthogonal to one another and form a good transform basis

[mathematical expression not reproducible] (3)

Any function f (x) that is square integrable and finite in the interval ([A, B] can be expanded into Haar wavelets as

[mathematical expression not reproducible] (4)

The integrals of the Haar functions (1) of order n can be calculated analytically as [9]

[mathematical expression not reproducible] (5)

The integrals of the Haar functions determined by (5) are continuous functions in the interval [A,B].

3. FREE VIBRATION ANALYSIS OF THE FGM BEAM

In the following the free vibration analysis of the FGM beam is considered [27-29]. It is assumed that the material properties of the beam of length L vary axially. The governing differential equation of the beam can be written as

[mathematical expression not reproducible] (6)

The varying properties of the bending stiffness EI(x) and the distributed mass per unit length [rho]A(x) are described by exponential functions as

[mathematical expression not reproducible] (7)

The reference values of the bending stiffness and distributed mass per unit length at x = 0 are denoted by EI(0) and [rho]A(0), respectively. Relation (7) is used in a number of papers [27-29]. The volume fractions of the material corresponding to relation (7) can be derived as

[mathematical expression not reproducible] (8)

The wavelet method approach considered can be applied for a wide range of functions describing properties of FGM. In the following a more general power law relation for describing FG materials is considered:

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

Here k is the non-negative power-law exponent describing the material variation profile along the length of the beam and the indexes L and R stand for the values of the material properties on the left and right support of the beam, respectively. Relations (9)-(10) seem to be the most widely used relations for describing FGM properties found in the literature [30].

In the following the solution of the partial differential equation (6) is assumed in the form

w(x,t) = W (x)sin([omega]t). (11)

Considering Eqs (7) and (11), the governing differential equation (6) can be rewritten in a non-dimensional form as

[mathematical expression not reproducible] (12)

where

[mathematical expression not reproducible] (13)

As a result, the vibration analysis problem of the FGM beam considered above is converted to solving the ordinal differential equation (12). The particular boundary conditions are introduced in Section 7.

4. THE HAAR WAVELET DISCRETIZATION METHOD

Herein the most commonly used approach of the HWM is employed. According to this method, the highest order derivative existing in a differential equation is expanded into Haar wavelets. Thus, Eq. (12) implies that the fourth order derivative should be expanded into Haar wavelets as

[mathematical expression not reproducible] (14)

where N = 2M is the resolution used.

The solution of the differential governing equation (12) W(X) can be obtained by integrating the expansion (14) four times with respect to X as

[mathematical expression not reproducible] (15)

In (15) the operational matrix of integration [P.sup.4] is defined by formulas (5) and [[alpha].sup.T] is a vector of coefficients. The integration constants [c.sub.0],...,[c.sub.3] can be determined for each particular boundary condition separately. Corresponding expressions of the integration constants are omitted for conciseness sake.

Inserting the solution of (15) in the differential equation (12) and assuming uniform grid points in the form

[mathematical expression not reproducible] (16)

one obtains a linear system of algebraic equations, which can be solved with respect to coefficient vector [a.sup.T]. Finally, substituting the values of [a.sup.T] in (15) gives the solution of the posed problem in an analytical form.

5. CONVERGENCE AND ACCURACY ESTIMATES

The convergence theorem for the HWM is given in [16] for the nth order ODE (n[greater than or equal to]2) as

THEOREM: Let us assume that [mathematical expression not reproducible] is a continuous function on [0,1] and its first derivative is bounded

[mathematical expression not reproducible] (17)

Then the HWM, based on the approach in [1,2], will be convergent, i.e. [E.sub.M] will vanish as the number of collocation points approaches N infinity. The convergence is of the order two

[mathematical expression not reproducible] (18)

The proof of the theorem is given in [16]. Furthermore, the quadrate of the [L.sup.2] -norm of the error function can be estimated as

[mathematical expression not reproducible] (19)

In the case of the considered problem the highest order derivative in differential equation equals four (n = 4) and formula (19) reduces to

[mathematical expression not reproducible] (20)

Furthermore, it is proved in [17] that in the case of the general fourth-order ODE the accuracy of the results of the HWM can be improved from two to four by applying Richardson's extrapolation method. The theoretical estimates pointed out above are validated numerically in the following section.

6. FEM SIMULATION MODEL

Commercial analysis software Mechanical APDL 16.0 was used to develop a 3D finite element simulation model for free vibration analysis of an axially functionally graded beam. The FGM beam was partitioned through its length into a number of strips with constant material properties inside the strip (see Fig. 1).

Figure 1 shows the mesh of the zoomed right-hand side of the beam corresponding to the third row of Table 1 (5 elements in the thickness and width directions and 500 elements in the length direction). The elements considered were cubical 3D 8-Node Homogeneous Structural Elements SOLID185. The detailed mesh values used are given in column 1 of Table 1.

The geometrical parameters of the beam considered are width (b), height (h), and length (L). The material properties of the steel and aluminium used in the FEM analysis are given in Table 2. The boundary conditions considered correspond to a cantilever beam. The results obtained from FEM analysis were originally in the dimensional form, i.e. computed for a particular beam with the given geometry, rigidity, and mass per unit length values. In order to compare these results with the results of the FDM, DQM, and HWM, the frequency parameter was converted into the non-dimensional form using the following formula:

[mathematical expression not reproducible] (21)

where f stands for natural/dimensional frequency parameter value (in Hz). The FEM results are discussed in detail in the following section.

7. NUMERICAL RESULTS

In the following five different boundary conditions of the FGM beam are considered (see Fig. 2) and the results obtained by applying HWM, FDM, and DQM are compared (two symmetric and three non-symmetric conditions).

The first two values of the fundamental frequency parameter [OMEGA] are presented in Tables 3 and 4 for a pinned-pinned beam, in Tables 5 and 6 for a clamped--clamped beam, in Tables 7 and 8 for a clamped-pinned beam, in Tables 9 and 10 for a pinned-clamped beam, and in Tables 11 and 12 for a clamped-free beam).

Note that in the FE model all supports with pinned boundary conditions (a, c, and d) have the ability to move in the horizontal direction ([u.sub.y] = [u.sub.z] = 0,[u.sub.x] [not equal to] 0).

In Tables 1-10 the properties of the beam are considered to vary according to formula (7), i.e. by exponential functions.

In Tables 3-10 the value of the parameter [beta] is taken equal to 2. The exact solutions computed based on transcendental algebraic equations derived in [27] are given in the headings of Tables 3-10. Obviously, the convergence of the HWM (also of the FDM and DQM) to the exact solution can be observed in all these tables.

The numerical rates of the convergence, computed for the solutions presented in Table 3, are presented in Table 13.

In Table 13 the values of N start from 16 because each rate of convergence was computed on the basis of three consecutive values of the solution [16]. The rate of the convergence of the HWM and FDM obviously tends to two, but the DQM has an ultrafast rate for N [less than or equal to] 32 and a negative rate for N > 32 (loss of accuracy).

In Table 14, the convergence rates of the extrapolated results of the HWM are given for four different boundary conditions considered above. The Richardson extrapolation method was applied, and it can be seen from Table 14 that the order of the convergence of extrapolated results tends to four in the case of all boundary conditions considered.

Based on results given in Tables 3-12, it can be concluded that in the case of the posed problem the highest accuracy was achieved by applying the DQM, also in most cases the accuracy of the results obtained by the HWM is higher than that obtained by the FDM (there fundamental frequencies in Tables 3 and 7 are exceptions). Detailed analysis of DQM results shows that the maximum accuracy was achieved extremely quickly with N = 16 or N = 32; thereafter the accuracy of the solution decreased with increasing resolution. These results are in agreement with the theoretical concept of the DQM (it is based on the use of high order polynomials whose denominator vanishes for large N) and results found in the literature.

It can be seen from Fig. 3 that in the case of the parameter value k = 1.5 the functions of the elasticity modulus corresponding to the exponential and power law functions (7) and (9) are close (here a steel/aluminium cantilever beam with [beta]= -0.549306 is considered).

In Tables 11 and 12 the FG material properties corresponding to steel/aluminium are considered with steel in the left and aluminium in the right support. The particular values of the material used are presented in Table 2.

The exponential model (7) does not include directly material properties at the right end of the beam. The required values of the material are obtained by determining the value of the parameter [beta]([beta]= -0.549306).

The first four mode shapes for the above-considered FG steel/aluminium cantilever beam are depicted in Fig. 4. The corresponding mode shapes obtained by a FEM are shown in Figs 5-8.

The results given in Table 15 were obtained by applying the HWM with the general power law function (9)-(10).

The steel/aluminium FGM with properties given in Table 2 is considered and the value of the exponent is taken equal to 1.5 (k = 1.5). The boundary conditions for a clamped-clamped beam are applied.

The FEM results computed for all boundary conditions considered above are given in the last row of each table. The number of elements used is 10 x 10 x 1000. Obviously, the values of the frequency parameters computed using 3D FEM analysis are in excellent agreement with those given in Tables 3-12, obtained by applying the HWM, DQM, and FDM.

An example of the results of more detailed FEM analysis is given in Table 15. In this table the first free frequencies for a pinned-pinned beam are presented. The convergence of the solution with increasing mesh can be observed. The results are in agreement with corresponding results obtained by applying the HWM, FDM, and DQM given in Tables 3 and 4.

8. CONCLUSIONS

Three strong formulation based numerical methods (HWM, FDM, and DQM) were applied for the analysis of the FGM beam and the obtained results were compared. The algorithms for all methods were coded by the authors in MATLAB. Good performance was observed in the case of all three methods used.

It can be concluded that in the case of the considered problem the accuracy of the solutions obtained by applying the HWM and FDM was in the same range. However, in most cases the accuracy of the results of the HWM outperformed that of the FDM. The accuracy of the DQM appears to be higher than that of the HWM and FDM. The convergence results presented in Table 13 confirm the accuracy of the HWM, FDM, and DQM. Similar accuracy was observed also for cylindrical shells in [16].

The obtained numerical results were validated with the solid element 3D finite element model developed for analysing more complex FGM structures. The results obtained with applying the 3D FEM and HWM were found to be in good (rather excellent) agreement.

Our future studies will focus on the application of the HWM for the analysis of nanostructures and solving fractional differential equations, which are not yet well covered by commercial software solutions. An interesting subtopic, whose research is underway, is adaption of global optimization methods and techniques, developed by the workgroup of design composite structures [31-36] to design nano- and graphene structures.

ACKNOWLEDGEMENTS

The research was supported by the Estonian Research Council (grant PUT1300); Estonian Centre of Excellence in Zero Energy and Resource Efficient Smart Buildings and Districts, ZEBE, TK146 funded by the European Regional Development Fund (grant 2014-2020.4.01.15-0016); Innovative Manufacturing Engineering Systems

Competence Centre IMECC (supported by Enterprise Estonia and co-financed by the European Union Regional Development Fund, project EU48685). The publication costs of this article were covered by Tallinn University of Technology and the Estonian Academy of Sciences.

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Funktsionaalgradientmaterjalist tala vabavonkumised: Haari lainikute meetodi evalveerimine

Maarjus Kirs, Kristo Karjust, Imran Aziz, Erko Ounapuu ja Ernst Tungel

Uurimistoos on keskendutud Haari lainikute meetodi evalveerimisele. Haari lainikute meetodi abil saadud tulemusi on vorreldud insenerirakendustes laialdaselt kasutatavate tugeval formulatsioonil pohinevate meetodite, nagu loplike vahede meetodi ja diferentsiaalkvadratuuride meetodi tulemustega. Vaadeldava ulesande korral on Haari lainikute meetod loplike vahede meetodist tapsem. Diferentsiaalkvadratuuride meetod osutus vaiksema kollokatsioonipunktide arvu korral Haari lainikute meetodist tapsemaks, kuid selle rakendamine suurema kollokatsioonipunktide arvu korral on komplitseeritud. Samuti on loodud 3D loplike elementide meetodil pohinev mudel ja selle rakendamisel saadud tulemused on eeltoodud meetodite tulemustega kooskolas. Haari lainikute meetodi abil saadud lahendi ja ekstrapoleeritud tulemuste koonduvuskiirus on kooskolas vastavate koonduvusteoreemidega toestatud teoreetiliste tulemustega.

Maarjus Kirs (a*), Kristo Karjust (a), Imran Aziz (b), Erko Ounapuu (a), and Ernst Tungel (a)

(a) Department of Mechanical and Industrial Engineering, School of Engineering, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia

(b) Department of Mathematics, University of Peshawar, Peshawar, Pakistan

(*) Corresponding author, Maarjus.Kirs@ttu.ee

Received 1 December 2016, revised 14 March 2017, accepted 10 May 2017, available online 17 August 2017
Table 1. FEM model. First three values of frequency parameter,
pinned-pinned beam

N               [[ohm].sub.1]  [[ohm].sub.2]  [[ohm].sub.3]

  2700            8.4522         41.2579        91.5302
(3 x 3 x 300)
  6400            8.4366         41.1807        91.3549
(4 x 4 x 400)
 12500            8.4276         41.1363        91.2543
(5 x 5 x 500)
100000            8.4136         41.0672        91.0984
(10 x 10 x 1000)

Table 2. Material properties of FG steel/aluminium material

Property  Unit          Steel  Aluminium

E         GPa            210     70
[rho]     Kg/[m.sup.3]  7800   2600

Table 3. Fundamental frequency parameter [[OMEGA].sub.1] ([beta]= 2,
exact solution 8.41047573)

N              HWM       FDM        DQM

  4          7.235577  8.157141
  8          8.118951  8.332735   8.40822662
 16          8.337942  8.390010   8.41047574
 32          8.392365  8.405292   8.41047574
 64          8.405950  8.409176   8.41047568
128          8.409344  8.410150   8.41047711
256          8.410193  8.410394   8.41047514
FEM results (10 x 10 x 1000 el.)  8.4136

Table 4. Second natural frequency parameter [[OMEGA].sub.2] ([beta]= 2,
exact solution 41.07055822)

N             HWM         FDM        DQM

  4        36.595239   34.453858
  8        39.999094   39.252370  41.07596761
 16        40.805039   40.603112  41.07055821
 32        41.004320   40.952833  41.07055821
 64        41.054008   41.041072  41.07055830
128        41.066421   41.063183  41.07055855
256        41.069524   41.068714  41.07056029
FEM results (10 x 10 x 1000 el.)  41.0672

Table 5. Fundamental frequency parameter [[OMEGA].sub.1] ([beta]= 2,
exact solution 24.78955023)

N         HWM           FDM         DQM

  4     21.242723     21.212781
  8     24.016796     23.517337   24.24277247
 16     24.602325     24.432036   24.78954915
 32     24.743104     24.697286   24.78955023
 64     24.777961     24.766296   24.78955023
128     24.786654     24.783725   24.78955013
256     24.788826     24.788093   24.78955092
FEM results (10 x 10 x 1000 el.)  24.8074

Table 6. Second natural frequency parameter [[OMEGA].sub.2] ([beta]= 2,
exact solution 64.70943426)

N            HWM        FDM         DQM

  4        57.202697
  8        62.966887  57.405312   65.21032574
 16        64.287324  62.612504   64.70946202
 32        64.604874  64.163748   64.70943427
 64        64.683357  64.571582   64.70943427
128        64.702919  64.674880   64.70943434
256        64.707806  64.700790   64.70943804
FEM results (10 x 10 x 1000 el.)  64.7032

Table 7. Fundamental frequency parameter [[OMEGA].sub.1] ([beta]= 2,
exact solution 11.18278324)

N       HWM             FDM         DQM

  4    8.764599        9.455451
  8   10.647800       10.572089   11.17258124
 16   11.052596       11.012971   11.18278324
 32   11.150448       11.139111   11.18278324
 64   11.174712       11.171786   11.18278327
128   11.180766       11.180029   11.18278302
256   11.182279       11.182094   11.18278285
FEM results (10 x 10 x 1000 el.)  11.1901

Table 8. Second natural frequency parameter [[OMEGA].sub.2] ([beta]= 2,
exact solution 48.26066843)

N             HWM        FDM        DQM

  4         41.945882  35.930268
  8         46.799191  43.888087  48.31401606
 16         47.903565  47.016589  48.26066840
 32         48.171942  47.938093  48.26066844
 64         48.238522  48.179262  48.26066848
128         48.255134  48.240268  48.26066852
256         48.259285  48.255565  48.26066842
FEM results (10 x 10 x 1000 el.)  48.2589

Table 9. Fundamental frequency parameter [[OMEGA].sub.1] ([beta]= 2,
exact solution 20.777978)

N         HWM          FDM          DQM

  4     18.799637    19.770973
  8     20.309972    20.506995    20.77852832
 16     20.662461    20.708729    20.77797932
 32     20.749190    20.760567    20.77797932
 64     20.770788    20.773620    20.77797931
128     20.776182    20.776889    20.77797921
256     20.777530    20.777707    20.77797837
FEM results (10 x 10 x 1000 el.)  20.7897

Table 10. Second natural frequency parameter [[OMEGA].sub.2] ([beta]=
2, exact solution 56.294438)

N             HWM        FDM        DQM

  8         54.965168  52.599124
 16         55.969009  55.338806  56.09705480
 32         56.213501  56.053355  56.29443879
 64         56.274230  56.234028  56.29443858
128         56.289388  56.279327  56.29443857
256         56.293176  56.290660  56.29443848
FEM results (10 x 10 x 1000 el.)  56.2907

Table 11. Fundamental frequency parameter [[OMEGA].sub.1]
([beta]= -0.549306)

N        HWM         FDM            DQM

  8    4.884627    4.842031       4.87118515
 16    4.874540    4.863858       4.87119849
 32    4.872033    4.869360       4.87119848
 64    4.871407    4.870739       4.87119797
128    4.871251    4.871084       4.87120621
256    4.871212    4.871170       4.87220829
FEM results (10 x 10 x 1000 el.)  4.8758

Table 12. Second natural frequency parameter [[OMEGA].sub.2]
([beta]= - 0.549306)

N         HWM          FDM          DQM

  8     24.798280    23.143597    24.41704668
 16     24.517676    24.092313    24.42645172
 32     24.449153    24.342014    24.42645172
 64     24.432120    24.405285    24.42645167
128     24.427868    24.421156    24.42645138
256     24.426806    24.425128    24.42665633
FEM results (10 x 10 x 1000 el.)  24.4397

Table 13. Rates of convergence corresponding to results given in
Table 3

N    HWM     FDM     QDM

 16  2.0122  1.6163  -
 32  2.0086  1.9060    23.8789
 64  2.0023  1.9766  - 8.6258
128  2.0006  1.9942  - 4.6331
256  2.0001  1.9985  - 0.4590

Table 14. Fundamental frequency parameter [[ohm].sub.1], convergence
rates of extrapolated results

N    Pinned-pinned    Clamped-clamped  Clamped-pinned  Pinned-clamped

 32    2.514821         4.2685           4.3015         4.1711
 64    3.916921         4.0516           4.0774         4.0425
128    3.984539         4.0120           4.0191         4.0105
256    3.996411         4.0029           4.0047         4.0026

Table 15. First three values of fundamental frequency parameter (k =
1.5, power law)

N    [[OMEGA].sub.1]  [[OMEGA].sub.2]  [[OMEGA].sub.3]

  8    22.573818        62.648322        124.802042
 16    22.559805        62.097576        122.045771
 32    22.552926        61.958469        121.384345
 64    22.548905        61.920592        121.216644
128    22.546296        61.908898        121.172322
256    22.544517        61.904408        121.159519
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Title Annotation:INDUSTRIAL ENGINEERING
Author:Kirs, Maarjus; Karjust, Kristo; Aziz, Imran; Ounapuu, Erko; Tungel, Ernst
Publication:Proceedings of the Estonian Academy of Sciences
Article Type:Report
Date:Mar 1, 2018
Words:5032
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