Modified Hermitian cubic spline wavelet on interval finite element for wave propagation and load identification

Finite Elements in Analysis and Design 91 (2014) 48–58

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Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel

Modified Hermitian cubic spline wavelet on interval finite element for wave propagation and load identification Xiaofeng Xue, Xingwu Zhang n, Bing Li, Baijie Qiao, Xuefeng Chen State Key Laboratory for Manufacturing System Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 8 January 2014 Accepted 20 July 2014

Accuracy and efficiency are significant factors in wave propagation and load identification of mechanical structure. By introducing modified Hermitian cubic spline wavelets on interval (HCSWI), a multi-scale wavelet-based numerical method is proposed. The present method can avoid the boundary problem of the original Hermitian interpolation wavelet. A modified Hermitian interpolation wavelet base can get transformation matrix, so the modified Hermitian wavelet finite element is proposed in this paper. Positive question-wave propagation and inverse question-load identification is verified by this means. The modified Hermitian wavelet finite element involves wave propagation and load identification in rod and Timoshenko beam which are obtained and then compared with results calculated by traditional finite element method (TFEM) and B-spline wavelet on interval (BSWI) finite element. The results indicate that the present method for wave propagation and load identification has higher precision and costs less time on mechanical structure. & 2014 Elsevier B.V. All rights reserved.

Keywords: Modified Hermitian cubic spline wavelets on interval Transformation matrix Wave propagation Load identification

1. Introduction Wave propagation and load identification are the key subjects of intensive investigations in mechanical engineering over the years [1,2]. Wave propagation techniques providing an efficient and accurate procedure have been of great interest to many researchers in mechanical structures [3–5]. However, a so-called “short wave problem” for finite element based techniques becomes one of the biggest obstacles which must be overcome [6]. Zienkiewicz and Taylor noticed the “rule of thumb” that there should be at least 10 nodes per wave length [6]. Babuska et al. found an effect called “pollution error”, which makes even more astronomical computing requirements than that of the “rule of thumb” [7,8]. Thus, accuracy and efficiency are importantly and greatly required for finite element techniques when they are used in high frequency wave propagation simulation [9]. Manktelow et al. pointed out that nonlinear dispersion through an integrated commercial software environment which enables exploration and optimization of geometrically-complex structures [10]. Pahlavan et al. presented a novel and generic formulation of the wavelet-based spectral finite element approach, which is applicable to elastic wave propagation problems [11]. Yang et al. analyzed elastic wave propagation in arches using a B-spline wavelet on interval finite element [12].

n

Corresponding author. Tel.: þ 86 29 82667963; fax: þ86 29 82663689. E-mail address: [email protected] (X. Zhang).

http://dx.doi.org/10.1016/j.finel.2014.07.011 0168-874X/& 2014 Elsevier B.V. All rights reserved.

Accurate and reliable data on mechanical structure loads are highly necessary not only to design and development but also to strength and rigidity specifications for mechanical structures [13]. But due to the complexity of structure and loading conditions, a traditional way of obtaining the applied force from direct measurement may not always be possible because of difficulties in sensor placement or other practical problems [14]. Rowley described the solved loads from measured responses based on moving force identification algorithm [15]. Therefore, it is crucial to study proper algorithm with load identification. Maes et al. presented an analytical method to estimate the axial force in a beam member. The method accounts for bending stiffness effects and for the rotational inertia and shear deformation of the beam member [16]. Li et al. used system characteristics and responses to calculate loads, which are based on wavelet multi-resolution analysis [17]. Gupta presented a time domain technique for estimating dynamic loads acting on a structure from strain time response [18]. Finite element method (FEM) has been playing an important role in many engineering fields. Wave propagation and load identification techniques adapting to more complex structures became possible and available because of the use of Finite element method [19]. However, for many complicated problems, TFEM has some disadvantages, such as low efficiency, insufficient accuracy, slow convergence to correct solutions etc. Recently, wavelets have been applied to obtain representations of integral and differential operators in many physical problems [20,21]. And because wavelets have the properties of multi-resolution analysis, they provide

X. Xue et al. / Finite Elements in Analysis and Design 91 (2014) 48–58

a natural mechanism for decomposing the solution into a set of coefficients. Wavelet analysis numerical methods can be viewed as those interpolating functions, similar to those used in signal and image processing. Basu indicated that the finite difference and Ritz-type methods had been largely replaced by the FEM, the boundary element method, the meshless method, and in the near future it might be the turn for the wavelet-based numerical method [22]. Since B-spline wavelets have explicit expressions, high degree of accuracy and high efficiency, numerous researchers focus on the B-spline wavelet on interval (BSWI) finite element method [23,24]. But compared with BSWI, which is formed to recalculate the function to improve the precision of the new scales from the original scale function, HCSWI, however, has prominent advantages of improving the precision by adding the appropriate wavelet function. Based on modified HCSWI, this paper presents a multi-scale wavelet-based numerical method, a modified Hermitian interpolation wavelet base, avoiding the boundary problem of the original Hermitian interpolation wavelet [25], and proposing the modified Hermitian wavelet finite element. In the present work, an effective wavelet numerical method is proposed based on wavelet bases of modified Hermitian cubic spline wavelets on interval [26] to analyze wave propagation and load identification of rod and beam. For the orthogonal characteristic of the wavelet bases with respect to the given inner product, the corresponding multi-scale solution equation will be decoupled across levels totally or partially and it suits for the nesting approximation. Some numerical examples indicate that the proposed method has better precision in analyzing mechanical structure [27].

2. Hermitian cubic splines on interval Wang constructed cubic spline wavelet bases in Sobolev spaces in 1996 [28] and orthogonal multi-wavelets were constructed by Donovan et al. [29]. By L2(R), the linear space of all squareintegrable real-valued functions is denoted on R. The inner product in L2(R) is defined as Z 〈u; v〉 : ¼ uðxÞvðxÞdx; u; v A L2 ðRÞ R

If /u,vS ¼0, then u and v are regarded to beporthogonal. The ffiffiffiffiffiffiffiffiffiffi norm of a function f in L2(R) is given by J f 2 J : ¼ 〈f ; f 〉. Let ϕ1 and ϕ2 be the cubic splines supported on interval [  1, 1], they are given by 8 2 > < ðx þ 1Þ ð1  2xÞ for x A ½  1; 0 2 ð1 þ 2xÞ for x A ½0; 1 ð1  xÞ ϕ1 ðxÞ : ¼ ð1Þ > : 0 for x2 = ½  1; 1 and 8 2 > < ðx þ 1Þ x ϕ2 ðxÞ : ¼ ðx  1Þ2 x > : 0

49

cubic splines ϕ1 and ϕ2 [31]. These wavelets were adapted to the interval [0, 1]. However, their construction for the wavelet basis on interval [0, 1] was quite complicated. Jia et al. constructed wavelet bases of Hermite cubic splines [26]. The corresponding wavelets ψ1 and ψ2 supported on interval [  1, 1] are (

ψ 1 ðxÞ ¼  2ϕ1 ð2x þ 1Þ þ 4ϕ1 ð2xÞ  2ϕ1 ð2x  1Þ  21ϕ2 ð2x þ 1Þ þ 21ϕ2 ð2x  1Þ ψ 2 ðxÞ ¼ ϕ1 ð2x þ 1Þ  ϕ1 ð2x  1Þ þ 9ϕ2 ð2x þ 1Þ þ 12ϕ2 ð2xÞ þ 9ϕ2 ð2x  1Þ

ð3Þ  0 0  ¼ ψ 2 ; ϕm ðd  jÞ ¼ 0; They satisfy the conditions m ¼ 1; 2; 8 j A Z, where symbols d and j denote arbitrary variable and shift parameter, respectively. The shifts of ψ1 and ψ2 generate the wavelet space W. Fig. 2 shows the graphic of ψ1 and ψ2. Obviously, ψ1 is symmetric and ψ2 is anti-symmetric. The above-mentioned wavelets can generate a wavelet basis for the space H 10 ð0; 1Þ. Therefore, the following decomposition of H 10 ð0; 1Þ are 



ψ 01 ; ϕ0m ðd  jÞ

_ W1 þ _ W2 þ _ ⋯ H 10 ð0; 1Þ ¼ V 1 þ

ð4Þ

_ denotes direct sum, V1 is the initial scaling space, and Wj where þ (j¼1, 2, …) is the wavelet space at the different level. The scaling functions ϕ1,k (Fig. 3) are 8 qffiffiffiffiffi 5 > > > ϕ1;1 ðxÞ : ¼ 24 ϕ1 ð2x  1Þ > > ffiffiffiffiffi q > > > > < ϕ1;2 ðxÞ : ¼ 15 4 ϕ2 ð2xÞ qffiffiffiffiffi ð5Þ > 15 > ϕ1;3 ðxÞ : ¼ 8 ϕ2 ð2x  1Þ > > > > qffiffiffiffiffi > > > : ϕ1;4 ðxÞ : ¼ 15 4 ϕ2 ð2x  2Þ Due to the boundary problem, Hermitian interpolation wavelet base cannot get transformation matrix, so it cannot be used as finite element interpolation function independently. A modified Hermitian scaling function is presented in this paper, so as to interpolate the field functions in wavelet finite element, and this modified Hermitian scaling function retains all kinds of the good performance of Hermitian wavelet. The modified scaling functions ϕ01;k are qffiffiffiffiffi 8 5 > ϕ ðxÞ : ¼ > 1;1 24 ϕ1 ð2xÞ > > > qffiffiffiffiffi > > 5 > > > ϕ1;2 ðxÞ : ¼ 24 ϕ1 ð2x  1Þ > > ffiffiffiffiffi q > > > > < ϕ1;3 ðxÞ : ¼ 15 4 ϕ2 ð2xÞ qffiffiffiffiffi ð6Þ > 15 > ϕ1;4 ðxÞ : ¼ 8 ϕ2 ð2x  1Þ > > > > qffiffiffiffiffi > > > > ϕ1;5 ðxÞ : ¼ 15 > 4 ϕ2 ð2x  2Þ > > qffiffiffiffiffi > > > : ϕ1;6 ðxÞ : ¼ 5 ϕ1 ð2x  2Þ 24

for x A ½  1; 0 for x A ½0; 1

ð2Þ

for x2 = ½  1; 1

The graphs of ϕ1 and ϕ2 are depicted in Fig. 1. Clearly, both ϕ1 and ϕ2 belong to C1(R). The main reason for choosing these two spline functions to generate wavelets is that the corresponding wavelets would be equipped with the capability of orthogonal with respect to the inner product /u0 ,v0 S (i.e. /u0 ,v0 S¼0), and this is the main integral term of the numerical method to analyze mechanical structure. Therefore, the multi-scale solution equation will be decoupled accordingly. Heil et al. considered the possibility of construction of wavelets on the basis of Hermite cubic splines [30]. Dahmen et al. constructed biorthogonal multi-wavelets on the basis of the Hermite

and the wavelets ψj,k are 8    j=2 j k > ψ j;k ðxÞ : ¼ p2ffiffiffiffiffiffiffiffiffi ψ 2 x  > 1 2 > 729:6 > >   >  j=2 > > < ψ j;k ðxÞ : ¼ p2ffiffiffiffiffiffiffiffiffiψ 2 2j x  k 2 1 153:6

for k ¼ 2; 4; …; 2j þ 1  2 for k ¼ 3; 5; …; 2j þ 1  1

 j=2 > > ψ ð2j xÞ ψ j;1 ðxÞ : ¼ p2 ffiffiffiffiffiffiffi > > 76:8 2 > >  j=2 > > : ψ j;2j þ 1 ðxÞ : ¼ p2 ffiffiffiffiffiffiffiψ 2 ð2j x  2j Þ 76:8

ð7Þ

All the modified scaling functions ϕ01;k and wavelets functions ψj,k on interval [0, 1] are shown in Figs. 4 and 5. The special properties of wavelet bases of HCSWI are 〈ðϕ01;k Þ0 ; ψ 0j;k 〉 ¼

Z

1 0

ðϕ01;k Þ0 ψ 0j;k dx ¼ 0 for all j and k

ð8Þ

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X. Xue et al. / Finite Elements in Analysis and Design 91 (2014) 48–58

0.16

1.0 0.8

0.08

φ2

φ1

0.6 0.00

0.4 -0.08

0.2 0.0 -1.0

-0.5

0.0

x

0.5

-0.16

1.0

-1.0

-0.5

0.0

x

0.5

1.0

4

1.6

2

0.8

ψ2

ψ1

Fig. 1. Hermite cubic splines on interval [  1,1]: (a) symmetric cubic splines supported and (b) anti-symmetric cubic splines supported.

0

0.0

-0.8

-2

-1.6

-4 -1.0

-0.5

0.0

x

0.5

1.0

-1.0

-0.5

0.0

x

0.5

1.0

Fig. 2. Wavelets ψ1 and ψ2 on interval [  1,1]: (a) symmetric wavelet supported and (b) anti-symmetric wavelet supported.

Fig. 4. Modified scaling functions in scaling space V1.

Fig. 3. Scaling functions in scaling space V1.

and 〈ψ 0j1 ;k ; ψ 0j2 ;k 〉 ¼

Z 0

1

ψ 0j1 ;k ψ 0j2 ;k dx ¼ 0 for j1 a j2 and all k

ð9Þ

partially and it suits for the nesting approximation. Besides that the first derivative of HCSWI is continuity, it is more suitable for analysis of beam, plate, etc. These features make HCSWI as a finite element interpolation function better than that of BSWI.

The wavelet bases in Vj can be written by Φj ¼ fφ1 ; ψ 1; ψ 2 ; …; ψ j  1 g

ð10Þ

where φ1 ¼ fϕ1;1 ; ϕ1;2 ; ϕ1;3 ; ϕ1;4 ; ϕ1;5 ; ϕ1;6 g denotes scaling functions in V1 and ψs (s¼1,2,…,j 1) is the wavelet bases in Ws (i.e. ψ s ¼ fψ s;1 ; ψ s;2 ; …; ψ s;2s þ 1 g). For the orthogonal characteristic of the wavelet bases with respect to the given inner product, the corresponding multiscale solution equation will be decoupled across levels totally or

3. Finite element discretization for differential equation of wave propagation and load identification problem A typical wave propagation problem can be reduced to an ordinary differential equation, which can be expressed as a matrix form: € þ Cu _ þ Ku ¼ FðtÞ Mu

ð11Þ

X. Xue et al. / Finite Elements in Analysis and Design 91 (2014) 48–58

51

Fig. 5. Wavelets in wavelet space W: (a) wavelet in wavelet space W1 and (b) wavelet in wavelet space W2.

Where M is the global mass matrix; C is the global damping matrix; K is global stiffness matrix; F(t) is a global excitation load € u_ and u stand for the vector, which is associated with time; and u; acceleration, velocity and displacement vector, respectively. Usually, the damping matrix C can be expressed as a Rayleigh damping if necessary. Central difference method which assumes € p ¼ ðup þ 1  2u þup  1 Þ=Δt 2 and u _ p ¼ ðup þ 1  up  1 Þ=2Δt is used to u solve Eq. (11); superscript p is the step number of time; and Δt stands for the time interval from the p  1 step to the p step. Then Eq. (11) can be written as up þ 1 ¼



M C þ Δt 2 2Δt

  1 

   2 1 1 M  K up  M C up  1 þ Fðt p Þ 2Δt Δt 2 Δt 2

ð12Þ and Eq. (11) can be written as the load identification equation       M C 2M p M C pþ1 u up  1 Fðt p Þ ¼ þ þ K  þ  ð13Þ u Δt 2 2Δt Δt 2 Δt 2 2Δt Then the load can be calculated. The analysis contained in this work is based on several reasonable simplifications and assumptions: the material is isotropic and homogenous, and the initial displacements and velocities are zero.

3.1. Hermitian rod element The generalized function of potential energy for rod element is Z Π p ðuÞ ¼

b a

 2 Z b EA du dx  f ðxÞu dx ∑ pi uðxi Þ 2 dx a i

ð15Þ

where the element stiffness matrix K can be expressed as EA e T 1;1 e ðT Þ Γ T le

Considering the influence of shear deformation, the generalized function of potential energy for Timoshenko beam element can be given as  2 2 Z b  Z b Z b EI dθ GA dw Π p ðw; θÞ ¼   θ dx  dx þ f ðxÞw dx dx a 2 a 2k dx a ð17Þ  ∑ pi wðxi Þ þ ∑ M l θðxl Þ i

ð16Þ

R1 among integral Γ1;1 ¼ 0 ðdΦT =dξÞðdΦ=dξÞdξ. The physical DOFs column vector of HCSWI axial rod element is shown in Fig. 6.

l

where θ is cross section rotation; G is shear modulus; Ml is the lump bending moment and k is the shear deformation coefficient. The other symbols are the same with Eq. (14). According to the Timoshenko beam theory, the displacement w(x) and the rotation θ(x) can be independently interpolated by scaling functions as wðξÞ ¼ ΦT e we ;

θðξÞ ¼ ΦT e θe

ð18Þ

Similar to what we have done to construct the HCSWI rod element, an element solving equation can be formulated: " # " Pe # we wi K e;1 K e;2 ð19Þ ¼ P eθl θe K e;3 K e;4 K e;1 ¼

e

Ke ¼

3.2. Hermitian Timoshenko beam element

ð14Þ

where E is the modulus of elasticity of the rod; A is the area of cross-section; f(x) is the function of distributed loading; Pi is lump loading; xi is the acting position and element length is le ¼ b a. Then Eq. (14) can be solved by interpolating u, u(ξ)¼ ΦTeue. Eq. (14) can be mapped to the interval [0,1] based on the Jacobi matrix, then we can submit u(ξ)¼ ΦTeue into Eq. (14). According to variational principle, let δΠp ¼0, the following equation can be obtained: K e Ue ¼ P e

Fig. 6. DOFs of HCSWI axial rod element.

GA e T 1;1 e ðT Þ Γ T kle

K e;2 ¼ 

GA e T 1;0 e ðT Þ Γ T k

K e;3 ¼ ðK e;2 ÞT

ð20Þ

ð21Þ ð22Þ

EI e T 1;1 e GAle e T 0;0 e ðT Þ Γ T ðT Þ Γ T þ ð23Þ le k R1 R1 among integral Γ 1;1 ¼ 0 ðdΦT =dξÞðdΦ=dξÞdξ, Γ 1;0 ¼ 0 ðdΦT =dξÞ R1 T 0;0 Φdξ, Γ ¼ 0 Φ Φ dξ. The physical DOFs column vector of HCSWI Timoshenko beam element is shown in Fig. 7. K e;4 ¼

4. Lifting scale scheme Fig. 8 shows the lifting scheme for the wavelet based numerical method. V1 is the initial scaling function space and Wl 1(l¼1, 2, j 1)

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X. Xue et al. / Finite Elements in Analysis and Design 91 (2014) 48–58

From Eq. (29), only the sub-matrices Bx,y, (x ¼ψ1,…,ψj  1; y¼ψl) will be calculated while the scale l is lifted to l þ1 and other submatrices will be preserved. This can increase the calculating efficiency. R1 For the integral Γ 0;0 ¼ 0 ΦT Φ dξ, the multi-scale constant matrix is 2 3 C φ1 ;φ1 C φ1 ;ψ 1 ⋯ C φ1 ;ψ j  1 6 C ψ 1 ;ψ 1 ⋯ C ψ 1 ;ψ j  1 7 6 7 7 ð30Þ Γ 0;0 ¼ 6 6 sym 7 ⋱ ⋮ 4 5 C ψ j  1 ;ψ j  1

Fig. 7. DOFs of HCSWI Timoshenko beam element.

where the sub-matrix of Γ0,0 can be calculated by Z 1 xT y dξ; x; y ¼ φ1 ; ψ 1 ; …; ψ j  1 C x;y ¼

ð31Þ

0

Fig. 8. Lifting scheme of the wavelet-based numerical method.

is the wavelet space. The scaling function ϕ01 in V1 is used as initial coarse approximation space. Then the wavelet bases are inductively added. The multi-resolution properties of the wavelet bases of HCSWI are used, that is _ W1 þ _ W2 þ _ ⋯þ _ Wj1 Vj ¼ V1 þ

5. Numerical examples

ð24Þ

It should be pointed out that the wavelets spaces by themselves form a complete space. Therefore, the unknown function could be entirely expanded in terms of the wavelets. However, to retain only a finite number of terms in the expansion, the scaling function space V1 must be included. The key to realizing the multi-scale solution of mechanical structures is to compute the integral. For the integral R1 Γ 1;1 ¼ 0 ðdΦT =dξÞðdΦ=dξÞdξ, the multi-scale constant matrix is 2 3 Aφ1 ;φ1 Aφ1 ;ψ 1 ⋯ Aφ1 ;ψ j  1 6 Aψ 1 ;ψ 1 ⋯ Aψ 1 ;ψ j  1 7 6 7 7 ð25Þ Γ 1;1 ¼ 6 6 sym 7 ⋱ ⋮ 4 5 Aψ j  1 ;ψ j  1 where the sub-matrix of Γ1,1 can be calculated by Z 1 T dx dy dξ; x; y ¼ φ1 ; ψ 1 ; …; ψ j  1 Ax;y ¼ 0 dξ dξ

From Eq. (31), only the sub-matrices Cx,y, (x ¼ψ1,…,ψj  1; y¼ ψl) will be calculated when the scale l is lifted to lþ1 and other submatrices will be preserved. This can also increase the calculating efficiency.

ð26Þ

Considering Eqs. (8) and (9), the non-diagonal sub-matrix of Eq. (25) will become zeros, which will be given by 2 3 Aφ1 ;φ1 0 ⋯ 0 6 0 7 0 Aψ 1 ;ψ 1 ⋯ 6 7 7 ð27Þ Γ 1;1 ¼ 6 6 0 7 0 ⋱ ⋮ 4 5 0 0 0 Aψ j  1 ;ψ j  1

5.1. Wave propagation It is well known that the vibration structures are multiple in engineering structures. It is very meaningful and necessary that the excitation in different position produces fluctuation on the mechanical structures. In this paper, we assume two kinds of working conditions on the rod and Timoshenko beam. One applies load on the left side; another applies load on the left side and middle side with the same load at the same time. The solution of the displacement of the rod or beam left side and middle side verifies the advantage of the HCSWI. Wave propagation is done with undamped condition. The dimensions of rods are as follows: length 1 m, height 0.02 m and width 0.02 m, and the dimensions of beams are as follows:length 1 m, height 0.06 m and width 0.06 m. The material is assumed as aluminum, with Young’s modulus 70 GPa, Poisson ratio 0.3 and density 2730 kg/m3. The excitation signal used in this section is the product of a sinusoidal signal with frequency 100 kHz modulated and amplitude 2 by Hanning window in the left side of rod or beam (in Fig. 9). In all examples, a computer with an Intel CPU 3.1 GHz and memory 4 G bytes is used, and we select Matlab and 64 bit Windows 7 operating system. A rod or Timoshenko beam model is divided into 30 HCSWI rod or beam elements. The actuating point locates at the left end or at the middle side of the rod or beam, the actuating signal is a sinusoidal signal with frequency 100 kHz modulated by Hanning window. By comparing the precision with different scales of

From Eq. (27), only the sub-matrices Ax,y, (x¼ y¼ψ1,…,ψj  1) will be calculated when the scale 1 is lifted to j 1 and other submatrices will remain zeros. This can enormously improve the calculating efficiency, compared with the general used multi-scale wavelet numerical method. R1 For the integral Γ 1;0 ¼ 0 ðdΦT =dξÞΦ dξ, the multi-scale constant matrix is 2 3 Bφ1 ;φ1 Bφ1 ;ψ 1 ⋯ Bφ1 ;ψ j  1 6 Bψ 1 ;ψ 1 ⋯ Bψ 1 ;ψ j  1 7 6 7 7 ð28Þ Γ 1;0 ¼ 6 6 An  sym 7 ⋱ ⋮ 4 5 Bψ j  1 ;ψ j  1 where the sub-matrix of Γ1,0 can be calculated by Z Bx;y ¼ 0

1

dxT y dξ; dξ

x; y ¼ φ1 ; ψ 1 ; …; ψ j  1

ð29Þ Fig. 9. Excitation signal used in numerical examples.

X. Xue et al. / Finite Elements in Analysis and Design 91 (2014) 48–58

53

Fig. 10. Comparison of axial rod responses on the left side: (a) HCSWI scale promotion and TFEM and (b) HCSWI, BSWI and TFEM.

Fig. 11. Comparison of axial rod responses on the middle side: (a) HCSWI scale promotion and TFEM and (b) HCSWI, BSWI and TFEM.

HCSWI elements with that of 30 BSWI elements, the superiority of HCSWI is verified. Here 600 TFEM results are applied as surrogate experimental data. 30 HCSWI wavelet elements at level j¼ 1, 2, 3; 30 BSWI elements and 600 TFEM are adopted respectively. Fig. 10(a) shows the left side wave propagation of rod with different scales. As the scale ascension of HCSWI, the precision of wave propagation is higher than that of the low scale of HCSWI. The results of scale at j¼ 3 of HCSWI elements are close to those of the 600 TFEM. Fig. 10(b) shows the left side wave propagation of rod with different method. It can be seen that 30 HCSWI elements with j¼ 3 have higher precision than 30 BSWI elements in 4 order 3 scale of rod. Similar to Fig. 10, Fig. 11 shows the comparison of axial rod responses on the middle side with HCSWI elements, BSWI elements and TFEM. Fig. 11(a) shows the middle side wave propagation of rod with different scales of HCSWI. As the scale ascension of HCSWI, the precision of wave propagation is higher than that of the low scale of HCSWI. Compared with Fig. 10(a), the accuracy of 30 HCSWI elements with j¼1 is very low. The results of scale at j¼ 3 of HCSWI elements are close to that of 600 TFEM. Fig. 11(b) shows the middle side wave propagation of rod with different method. It can be seen that 30 HCSWI elements with j¼3 have higher precision than BSWI elements in 4 order 3 scale of rod. Compared with the Fig. 10(b), the accuracy of 30 BSWI elements is very low. Table 1 shows wave propagation time consumption at the left side and the middle side of rod with different methods. In combination with Figs. 10 and 11, it can be seen that the precision of wave propagation is higher than low scale of HCSWI. 30 HCSWI

elements with j¼3 have higher precision than BSWI elements in 4 order 3 scale of rod. HCSWI has the good performance of the lifting scheme observed when the wavelets are added step by step to realize multi-scale approximation of the rod and beam, but the ascending scale way of BSWI is more troublesome. Compared with the 600 TFEM, the accuracy of 30 HCSWI elements with j¼3 is close to that of 600 TFEM, but time is less than that of 600 TFEM. Fig. 12 illustrates the wave propagation process in the rod response. Wave propagation spreads continuously from left to right, when wave reaches the rod fixed end reversely. The process is consistent with the actual process of wave propagation [5]. The result shows that HCSWI in simulating wave propagation is very effective. Fig. 13 shows different methods compared with axial rod responses on the left side, which is loaded on the left side and the middle at the same time. The first wave propagation is produced by the excitation signal at the left side. The second wave is propagated over the respond by the excitation signal at the middle side. Fig. 13(a) shows the left side wave propagation of rod with different scales of HCSWI. As the scale ascension of HCSWI, the precision of wave propagation is higher than that of low scale of HCSWI. The results of scale at j¼3 of HCSWI elements are close to that of 600 TFEM. Fig. 13(b) shows the left side wave propagation of rod with 30 HCSWI elements, 30 BSWI elements and 600 TFEM, 30 HCSWI elements with j¼3 have higher precision than that of BSWI elements in 4 order 3 scale of rod. Fig. 14 shows the wave propagation process in the axial rod responses with double loads, which are carried on the left side and the middle at the same time. The first wave propagation spread

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X. Xue et al. / Finite Elements in Analysis and Design 91 (2014) 48–58

continuously from left to right, when it reaches the rod fixed end reversely. The second wave is spread on both sides, when reaching the rod fixed end reversely. The process is consistent with the actual process of wave propagation. The result shows that HCSWI elements are very effective in simulating wave propagation. Fig. 15 shows the comparison of Timoshenko beam responses on the left side with 30 BSWI elements, 30 HCSWI elements and 600 TFEM. Fig. 15 shows before 30 μs BSWI elements and HCSWI elements are both highly precision, close to the precision of 600 TFEM elements, but after 30 μs, the BSWI elements are a little disordered. Therefore HCSWI elements have a higher accuracy. Fig. 16 shows the responses of different methods compared with Timoshenko beam on the left side and on the middle side, which is loaded on the left side and the middle at the same time. Fig. 16(a) shows the left side wave propagation of Timoshenko beam with different methods, 30 HCSWI elements with j¼ 3 have higher precision than scale j¼2 and BSWI elements in 4 order

3 scale of Timoshenko beam, HCSWI has the good performance of the lifting scheme observed when the wavelets are added step by step to realize multi-scale approximation of the Timoshenko beam, but the ascending scale way of BSWI is more troublesome. The results of scale at j¼3 of HCSWI elements are close to that of 600 TFEM. Different from the axial rod, the wave propagation of Timoshenko beam spread velocity is slower than that of rod. Similar to Fig. 16(a), (b) shows the middle side wave propagation of Timoshenko beam with different methods, 30 HCSWI elements with j¼3 have higher precision than scale j¼2 and BSWI elements in 4 order 3 scale of Timoshenko beam, the accuracy of 30 HCSWI

Table 1 Time contrast of different method in axial rod. Element

30 HCSWI j¼1 j¼2 j¼3 30 BSWI 600 TFEM

Time (s) The left side

The left and middle side

14.56 14.36 15.69 14.97 63.92

27.86 27.76 28.13 27.76 77.72

Fig. 12. Wave propagation in the rod.

Fig. 14. Wave propagation in the rod with double load.

Fig. 15. Comparison of Timoshenko beam responses on the left side.

Fig. 13. Comparison of axial rod responses on the left side with double load: (a) HCSWI scale promotion and TFEM and (b) HCSWI, BSWI and TFEM.

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Fig. 16. Comparison of different methods of Timoshenko beam responses with double load: (a) responses in the left side and (b) responses in the middle side.

elements with j¼ 3 is close to that of 600 TFEM. Only excitation wave at the middle side on the Timoshenko beam can be seen. Table 2 shows the left side and the middle side wave propagation with time consumption of Timoshenko beam with different methods. In combination with Figs. 15 and 16, it can be seen that as the scale ascension of HCSWI, the precision of wave propagation is higher than that of the low scale of HCSWI. 30 HCSWI elements with j ¼3 have higher precision than that of BSWI elements in 4 order 3 scale of rod. HCSWI has the good performance of the lifting scheme observed when the wavelets are added step by step to realize multi-scale approximation of the Timoshenko beam, but the ascending scale way of BSWI is more troublesome. Compared with the 600 TFEM, the accuracy of 30 HCSWI elements with j¼ 3 is close to that of 600 TFEM, but time is less than that of 600 TFEM.

Table 2 Time contrast of different method in Timoshenko beam. Element

30 HCSWI j¼1 j¼2 j¼3 30 BSWI 600 TFEM

Time (s) The left side

The left and middle side

30.36 30.90 32.62 56.76 357.71

56.48 56.86 58.58 70.41 390.83

5.2. Load identification Load identification is carried out by considering an undamped condition. The result of wave propagation obtained by using FEM is used as input, added white Gaussian noise and reverse solving excitation as signal. The excitation signals are used as exact solutions. In this section of the article, load identification based wavelet finite method will be presented to illustrate the performance of axial rod and Timoshenko beam. The excitation signal identified in this section is the product of a sinusoidal signal with frequency 100 kHz modulated by Hanning window in the left side (Fig. 9) and a sinc function form of wave with frequency 100 kHz modulated in the middle side of rod or beam (Fig. 17). A rod or Timoshenko beam model is divided into 30 HCSWI rod or beam elements, the actuation point locates at the left end with a sinusoidal signal with frequency 100 kHz modulated by Hanning window and the actuation point locates at the middle with a sinc function form of wave with frequency 100 kHz modulated of axial rod or Timoshenko beam. The rod or beam right end is clamped. Compared with the 30 BSWI elements and 600 TFEM, HCSWI elements are proved to be more superior in time consumption and the precision. 30 HCSWI wavelet bases with scale j¼1,2,3, 30 BSWI elements and 600 TFEM are used respectively. Fig. 18(a) shows the left side load identification of rod under different scales, as the scale ascension of HCSWI, the precision of load identification is higher than that of low scale. The results of scale j ¼3 of HCSWI elements are close to the exact solution. Fig. 18(b) shows the left side load identification of rod with 30 BSWI elements and 600 TFEM, 30 HCSWI elements with j¼3 have higher precision than BSWI elements in 4 order 3 scale of rod. Similar to Fig. 18, Fig. 19(a) shows the middle side load identification of rod with different scales of HCSWI elements. As the scale ascension of HCSWI, the precision of load

Fig. 17. The excitation signal identified.

identification is higher than that of low scale of HCSWI. The accuracy of scale j¼ 3 of HCSWI elements is close to that of the exact solution. Fig. 19(b) displays the middle side load identification of rod with 30 HCSWI elements, 30 BSWI elements and 600 TFEM. 30 HCSWI elements with j¼3 have higher precision than BSWI elements in 4 order 3 scale of rod. Table 3 shows the load identification error and time consumption of rod with different methods, when the signal to noise ratio is 250 dB. With the scale ascension of HCSWI, the precision of load identification becomes higher. 30 HCSWI elements with j ¼3 have higher precision and less time consumption than that of BSWI elements in 4 order 3 scale of rod. HCSWI has the good performance of the lifting scheme observed when the wavelets are added step by step to realize multi-scale approximation of the rod and beam, but the ascending scale way of BSWI is more troublesome. Compared with the 600 TFEM, the accuracy of 30 HCSWI elements with j¼3 is close to that of 600 TFEM, but time consumption is less than that of TFEM.

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Fig. 18. Comparison of load identification on the left side of axial rod: (a) HCSWI scale promotion and TFEM and (b) HCSWI, BSWI and TFEM.

Fig. 19. Comparison of load identification on the middle side of axial rod: (a) HCSWI scale promotion and TFEM and (b) HCSWI, BSWI and TFEM.

Table 3 Error and time contrast of rod with different method (the signal to noise ratio is 250 dB). Element

30 HCSWI j¼1 j¼2 j¼3 30 BSWI 600 TFEM

Time (s)

Error The left side

The middle side

0.30 0.18 0.13 0.17 0.10

0.14 0.07 0.05 0.07 0.03

28.08 28.33 28.36 28.23 77.98

Table 4 shows the load identification error and time consumption of rod with different methods, when the signal to noise ratio is 240 dB. Table 4 shows that with the increase of noise, the ability of FEM for load identification is reduced. Similar to Table 3, with the ascension of the scale of HCSWI, the precision of load identification becomes higher. 30 HCSWI elements with j¼3 have higher precision and less time consumption than BSWI elements in 4 order 3 scale of rod. Fig. 20(a) shows the left side load identification of beam with different scales of HCSWI. As the scale ascension of HCSWI, the precision of load identification is higher than that of low scale of HCSWI. The result of 30 HCSWI elements with j¼ 3 is close to that of the exact solution. Fig. 20(b) shows the left side load identification of rod with 30 HCSWI elements, 30 BSWI elements and 600

Table 4 Error and time contrast of rod with different method (the signal to noise ratio is 240 dB). Element

30 HCSWI j¼1 j¼2 j¼3 30 BSWI 600 TFEM

Time (s)

Error The left side

The middle side

1.11 0.62 0.37 0.64 0.16

0.41 0.25 0.16 0.25 0.06

29.26 29.32 29.64 29.96 79.25

TFEM. 30 HCSWI elements with j ¼3 have higher precision than that of BSWI elements in 4 order 3 scale of rod. Similar to Fig. 20, Fig. 21(a) shows the left side load identification of beam with different scales of HCSWI. As the scale ascension of HCSWI, the precision of load identification is higher than that of low scale of HCSWI. The result of 30 HCSWI elements with j¼3 is close to that of the exact solution. Fig. 21(b) shows the left side load identification of rod with 30 HCSWI elements, 30 BSWI elements and 600 TFEM. 30 HCSWI elements with j¼ 3 have higher precision than that of BSWI elements in 4 order 3 scale of rod. Table 5 shows when the signal to noise ratio is 250 dB, the left side and the middle side load identification error and time consumption of beam with different methods. The precision of load identification is higher than that of low scale of HCSWI, as the

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Fig. 20. Comparison of load identification on the left side of Timoshenko beam: (a) HCSWI scale promotion and TFEM and (b) HCSWI, BSWI and TFEM.

Fig. 21. Comparison of load identification on the middle side of Timoshenko beam: (a) HCSWI scale promotion and TFEM and (b) HCSWI, BSWI and TFEM.

Table 5 Error and time contrast of Timoshenko beam with different method (the signal to noise ratio is 250 dB). Element

30 HCSWI j¼ 1 j¼ 2 j¼ 3 30 BSWI 600 TFEM

Time (s)

Error The left side

The middle side

1.26 0.83 0.31 0.48 0.48

0.47 0.27 0.14 0.25 0.20

29.91 29.64 29.83 29.23 117.30

scale ascension of HCSWI. 30 HCSWI elements with j¼ 3 have higher precision and less time than BSWI elements in 4 order 3 scale of rod. HCSWI has the good performance of the lifting scheme observed when the wavelets are added step by step to realize multi-scale approximation of the rod and beam, but the ascending scale way of BSWI is more complex. Compared with the 600 TFEM, the accuracy of 30 HCSWI elements with j¼ 3 is higher than that of 600 TFEM, and time consumption is less than that of 600 TFEM. It has vital significance in engineering that after correction, the FEM can identify the immeasurable load. Table 6 shows the left side and the middle side load identification error and time consumption of beam with different methods when the signal to noise ratio is 240 dB. This table also indicates the increase of noise makes finite element’s ability is reduced to

Table 6 Error and time contrast of Timoshenko beam with different method (the signal to noise ratio is 240 dB). Element

30 HCSWI j¼1 j¼2 j¼3 30 BSWI 600 TFEM

Time (s)

Error The left side

The middle side

4.24 2.11 1.42 1.76 1.52

1.65 0.79 0.42 0.72 0.64

29.68 29.98 29.33 28.17 113.80

identify load. But similar to Table 5, as the scale ascension of HCSWI, the precision of load identification is higher than that of low scale of HCSWI. 30 HCSWI elements with j¼3 have higher precision and less time consumption than that of BSWI in 4 order 3 scale of rod.

6. Conclusion The modified Hermitian interpolation wavelet base can get transformation matrix, so the modified Hermitian wavelet finite element is proposed in this paper. Compared with BSWI, HCSWI has the prominent advantage of improving precision by adding the appropriate wavelet function, but the BSWI is formed to recalculate

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the function to improve the precision of the new scales from the original scale function. By using modified HCSWI, a multi-scale wavelet-based numerical method is proposed. The present method can avoid the boundary problem of the original Hermitian interpolation wavelet. The modified Hermitian wavelet finite element method is verified through positive question-wave propagation and inverse question-load identification. By comparing modified HCSWI wavelet bases at level j¼1, 2, 3 with BSWI elements in 4 order 3 scale and TFEM, the superiority of HCSWI elements is verified. As the scale ascension of HCSWI, the precision of wave propagation and load identification is improved. HCSWI is illustrated as having the good performance of the lifting scheme observed when the wavelets are added step by step to realize multi-scale approximation of rod and beam, but the ascending scale of BSWI is known to be more troublesome. And compared with the TFEM, the precision and the time of HCSWI are dominant. In practical engineering applications the modified finite element model can be used to simulate wave propagation and load identification. The given numerical examples testify the correctness and effectiveness of numerical performance of the modified HCSWI method. Acknowledgments This paper was supported by the National Natural Science Foundation of China (Nos. 51225501 and 51175401), the Research Fund for the Doctoral Program of Higher Education of China (No. 20120201110028) and the Program for Changjiang Scholars and Innovative Research Team in University. References [1] W. Witkowski, M. Rucka, J. Chróścielewski, et al., On some properties of 2D spectral finite elements in problems of wave propagation [J], Finite Elem. Anal. Des. 55 (2012) 31–41. [2] Z. Li, Z.P. Feng, F.L. Chu., A load identification method based on wavelet multiresolution analysis, J. Sound Vib. 333 (2) (2014) 381–391. [3] C.B. Pol, S. Banerjee, Modeling and analysis of propagating guided wave modes in a laminated composite plate subject to transient surface excitations, Wave Motion 50 (5) (2013) 964–978. [4] A. Chakraborty, S. Gopalakrishnan, Wave propagation in inhomogeneous layered media: solution of forward and inverse problems [J], Acta Mech. 169 (2004) 153–185. [5] N. Cretu, G. Nita, M.l. Pop, Wave transmission approach based on modal analysis for embedded mechanical systems, J. Sound Vib. 332 (2013) 4940–4947. [6] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, Fluid Dynamics [M], Butter worth Heine Mann, Oxford, 2000. [7] I. Babuska, F. Ihlenburg, T. Stroubolis, et al., A posteriori error estimation for finite element solutions of Helmholtz equations. Part I: the quality of local indicators and estimators, Int. J. Numer. Methods Eng. 40 (1997) 3443–3462.

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