2. Wavelet and multiresolution analysis

Wavelets are a class of basic functions that represent functions locally, both in space and time, and allow for the analysis of functions to be carried out at different resolutions (scales) [36]. The wavelet basis emanates from a set of wavelet coefficients associated with a particular location in time and different multiresolution scales. The scaling and wavelet functions stem from multiresolution analysis (MRA), which is a key and desirable property of wavelets, and refers to the simultaneous appearance of multiple scales in function decompositions in the Hilbert space L2 ð Þ R using a sequence of closed subspaces Vj, which is represented mathematically as [36]:

$$\dots \\ \dots \\ V\_{-2} \subset V\_{-1} \subset V\_0 \subset V\_1 \subset V\_2 \subset \dots \\ \dots \tag{1}$$

Therefore in principle, in order for multiresolution to occur, the closed subspaces Vj satisfy the following properties:

$$\overline{\bigcup\_{j \in \mathbb{Z}} V\_j} = L^2(\mathbb{R}) \tag{2}$$

$$\bigcap\_{j \in \mathbb{Z}} V\_j = \{0\} \tag{3}$$

Multiscale Wavelet Finite Element Analysis in Structural Dynamics http://dx.doi.org/10.5772/intechopen.71882 53

$$\begin{aligned} f\_2(\mathbf{x}) &= f(2\mathbf{x}) \forall \mathbf{x} \\ f \in V\_j \Leftrightarrow f\_2 &\in V\_{j+1} \qquad j \in \mathbb{Z} \end{aligned} \tag{4}$$

$$\begin{aligned} f\_n(\mathbf{x}) &= f(\mathbf{x} - n) \\ f \in V\_0 \Leftrightarrow f\_n &\in V\_0 \qquad n \in \mathbb{Z} \end{aligned} \tag{5}$$

The orthogonal complement subspace Wj of Vj contains the additional 'detail' for subspace Vjþ<sup>1</sup> i.e., Vjþ<sup>1</sup> ¼ V<sup>0</sup> ⊕ W<sup>0</sup> ⊕ W<sup>1</sup> ⊕ W2⋯ ⊕ Wj. The union of the subspaces Vj leads to the space L2 ð Þ <sup>R</sup> from the condition in Eq. (2) [36]. The scaling <sup>f</sup>ð Þ<sup>x</sup> <sup>∈</sup>L<sup>2</sup> ð Þ <sup>R</sup> and wavelet <sup>ψ</sup>ð Þ<sup>x</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup> ð Þ R functions correspond to the subspaces Vj and Wj respectively. The difference between current subspace Vj and subsequent subspace Vjþ<sup>1</sup> is represented by the wavelet space Wj which becomes automatically orthogonal to all other Wj for k < j due to the inclusion in and orthogonality to Vj. For the fundamental space V0, the scaling function fð Þx and its translates fð Þ x � k produce an orthonormal basis for V0. The orthonormal basis for the next space V<sup>1</sup> is the rescaled function ffiffiffi 2 <sup>p</sup> <sup>f</sup>ð Þ <sup>2</sup><sup>x</sup> � <sup>k</sup> . Thus, the orthonormal basis of Vj is defined as:

$$\phi\_k^\dagger(\mathbf{x}) = \mathcal{Z}^\dagger \phi \left( \mathbf{2}^\dagger \mathbf{x} - k \right) \qquad k \in \mathbb{Z} \tag{6}$$

Provided Eq. (6) and the above mentioned properties are satisfied, the wavelet orthonormal basis for subspace Wj at scale j is

$$
\psi\_k^\dagger(\mathbf{x}) = \mathcal{D}^\dagger \psi(\mathcal{D}^\dagger \mathbf{x} - k) \qquad k \in \mathbb{Z} \tag{7}
$$

The orthogonal subspaces Wj result from the decomposition of L<sup>2</sup> ð Þ R and subsequently the functions within these subspaces inherit the scale and shift invariance properties from the scaling function subspaces Vj and are orthonormal [8]. The projections of a function f ∈ L<sup>2</sup> ð Þ R at scale j in the subspaces Vj and Wj, defined as Pjf and Qjf respectively, are expressed as:

$$\begin{aligned} P\_{\sharp}f &= \sum\_{k} a\_{k}^{\flat} \phi\_{k}^{\flat}(\mathbf{x}) \\ Q\_{\sharp}f &= \sum\_{k} b\_{k}^{\flat} \psi\_{k}^{\flat}(\mathbf{x}) \end{aligned} \tag{8}$$

where a j <sup>k</sup> and b j <sup>k</sup> are coefficients in the subspaces Vj and Wj respectively. Thus, if all the conditions described above are met, then the scaling and wavelet functions satisfy [8]

$$\begin{cases} \int\_{-\infty}^{\infty} \phi(\mathbf{x}) d\mathbf{x} \neq 0 \\\\ \int\_{-\infty}^{\infty} \psi(\mathbf{x}) d\mathbf{x} = 0 \end{cases} \tag{9}$$

#### 2.1. Daubechies wavelet

was carried out for frame structures using the two wavelet families [27] and the WFEM was compared with an analytical wavelet approach using coiflets for the analysis of vehicle-bridge interaction for fast moving loads [28]. Furthermore, the Daubechies and BSWI wavelets were used to construct a functionally graded beam wavelet finite element under various moving

Other different wavelet families have been selected and employed in the formulation of the WFEM to solve a wide variety of structural analysis problems and research in this field is still ongoing. The trigonometric Hermite wavelet, which can be explicitly expressed, was used to construct beam [31] and thin plate WFEs [32] for static and free vibration analysis. The Hermite Cubic Spline Wavelet on the Interval (HCSWI), polynomial wavelets [33] and the second generation wavelets [34] are other wavelet based approaches that have been introduced and researched on. A more comprehensive synthesis and summary of wavelet based numerical

A generalised Wavelet based Finite Element Method framework is presented based on the BSWI and Daubechies wavelet families to derive rod and beam WFEs for homogenous and functionally graded materials for static and dynamic structural problems. A brief introduction of wavelet analysis is described in Section 2, with emphasis given to the Daubechies wavelets, BSWI, multiresolution and connection coefficients formulations. In Section 3, the wavelet based finite elements for a rod, Euler Bernoulli homogeneous beam and transversely varying functionally graded beam are presented. The evaluation of the element matrices and various load vectors, including the WFEM moving load formulation, are presented. A comparison on the performance of the Daubechies and BSWI WFEMs are highlighted via numerical examples for a variety of static and dynamic structural problems in Section 4 followed by conclusions.

Wavelets are a class of basic functions that represent functions locally, both in space and time, and allow for the analysis of functions to be carried out at different resolutions (scales) [36]. The wavelet basis emanates from a set of wavelet coefficients associated with a particular location in time and different multiresolution scales. The scaling and wavelet functions stem from multiresolution analysis (MRA), which is a key and desirable property of wavelets, and refers to the simultaneous appearance of multiple scales in function decompositions in the Hilbert space

ð Þ R using a sequence of closed subspaces Vj, which is represented mathematically as [36]:

Therefore in principle, in order for multiresolution to occur, the closed subspaces Vj satisfy the

Vj <sup>¼</sup> <sup>L</sup><sup>2</sup>

⋃ j ∈Z

> ⋂ j∈Z

⋯V�<sup>2</sup> ⊂ V�<sup>1</sup> ⊂V<sup>0</sup> ⊂V<sup>1</sup> ⊂V<sup>2</sup> ⊂… (1)

ð Þ R (2)

Vj ¼ f g0 (3)

methods for various engineering problems is presented in [35].

52 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

2. Wavelet and multiresolution analysis

L2

following properties:

load conditions [29, 30].

Daubechies wavelets are compact supported orthonormal wavelets developed by Ingrid Daubechies and for order L, the scaling and wavelet functions are described by the 'two-scale' relation [8]:

$$\phi\_L(\mathbf{x}) = \sum\_{k=0}^{L-1} p\_L(k)\phi\_L(2\mathbf{x} - k) \tag{10}$$

fð Þ <sup>m</sup>

of the scaling functions.

coefficient <sup>a</sup>;<sup>b</sup>Γj, <sup>d</sup>1,d<sup>2</sup>

tion coefficient [30]

Differentiating Eq. (23) m times

k,l <sup>¼</sup> <sup>2</sup><sup>d</sup>1þd2�<sup>1</sup><sup>X</sup>

a;bΓj,d1,d<sup>2</sup>

2.2. Daubechies connection coefficients

a;bΓj, <sup>d</sup>1, <sup>d</sup><sup>2</sup> k,l <sup>¼</sup> <sup>2</sup><sup>j</sup>

<sup>L</sup> ð Þ¼ <sup>x</sup> <sup>2</sup><sup>m</sup><sup>X</sup>

X k¼∞

k¼�∞

k,l and multiscale connection coefficient <sup>Υ</sup>j,m

<sup>X</sup>½ � <sup>0</sup>, <sup>1</sup> ð Þ <sup>ξ</sup> <sup>f</sup>ð Þ <sup>d</sup><sup>1</sup>

<sup>a</sup> 2<sup>j</sup>

where a and b are the orders of the scaling function at multiresolution j, while the values d<sup>1</sup> and

characteristic function. The formulation presented is a modified algorithm of that described in [4] and allows for the evaluation of the connection coefficients for different values of a and b at

p rð Þf<sup>L</sup> <sup>2</sup><sup>j</sup>þ<sup>1</sup>

p rð Þfð Þ <sup>m</sup>

<sup>L</sup> 2<sup>j</sup>þ<sup>1</sup>

pað Þ <sup>r</sup> � <sup>2</sup><sup>k</sup> pbð Þþ <sup>s</sup> � <sup>2</sup><sup>l</sup> pa <sup>r</sup> � <sup>2</sup><sup>k</sup> <sup>þ</sup> <sup>2</sup><sup>j</sup> � �pb <sup>s</sup> � <sup>2</sup><sup>l</sup> <sup>þ</sup> <sup>2</sup><sup>j</sup> � � � � <sup>Γ</sup>j,d1,d<sup>2</sup>

r

Substituting Eq. (24) into Eq. (22) and applying the 'two-scale' relation of the characteristic

where 2 � <sup>a</sup> <sup>≤</sup> k, r <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1 and 2 � <sup>b</sup> <sup>≤</sup> l, s <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1. Eq. (25) can be expressed in matrix form as:

different multiresolution scales j. From the 'two-scale' relation presented in Eq. (10),

r

<sup>ξ</sup> � <sup>k</sup> � �fð Þ <sup>d</sup><sup>2</sup>

<sup>b</sup> <sup>2</sup><sup>j</sup>

ð∞ �∞

d<sup>2</sup> denote the order of the derivative of the scaling functions. X½ � <sup>0</sup>,<sup>1</sup> ð Þ¼ x

<sup>ξ</sup> � <sup>k</sup> � � <sup>¼</sup> <sup>X</sup>

<sup>ξ</sup> � <sup>k</sup> � � <sup>¼</sup> <sup>2</sup>ð Þ <sup>j</sup>þ<sup>1</sup> <sup>m</sup><sup>X</sup>

function, the two-term connection coefficient can be expressed as:

f<sup>L</sup> 2<sup>j</sup>

2jmfð Þ <sup>m</sup> <sup>L</sup> 2<sup>j</sup>

r,s

L�1

p kð Þfð Þ <sup>m</sup>

<sup>L</sup> ð Þ 2x � k (20)

http://dx.doi.org/10.5772/intechopen.71882

55

<sup>k</sup> . We define the two-term connec-

<sup>ξ</sup> � <sup>2</sup><sup>k</sup> � <sup>r</sup> � � (23)

<sup>ξ</sup> � <sup>2</sup><sup>k</sup> � <sup>r</sup> � � (24)

<sup>ξ</sup> � <sup>l</sup> � �d<sup>ξ</sup> (22)

1 0 ≤ x ≤ 1 <sup>0</sup> otherwise �

is the

r,s (25)

<sup>k</sup><sup>m</sup>fð Þ <sup>m</sup> ð Þ¼ <sup>x</sup> � <sup>k</sup> <sup>m</sup>! (21)

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

k¼0

A normalising condition is required to evaluate Eq. (20) which is obtained from the moments

As earlier mentioned, the Daubechies functions cannot be computed analytically and their derivatives are highly oscillatory, particularly at low wavelet orders and/or high order derivatives. Therefore, the integral of the products of the scaling functions and/or derivatives are computed as what is commonly known as connection coefficients [37]. There are two forms of connection coefficients that are of relevance to this study; the multiscale two-term connection

$$\psi\_L(\mathbf{x}) = \sum\_{k=0}^{L-1} q\_L(k) \phi\_L(2\mathbf{x} - k) \tag{11}$$

The scaling and wavelet functions have the supports 0½ � ; <sup>L</sup> � <sup>1</sup> and 1 � <sup>L</sup> 2 ; L 2 � � respectively. The normalised wavelet function filter coefficients qLð Þk and scaling function filter coefficients pLð Þk have the relation qLð Þ¼ � <sup>k</sup> ð Þ<sup>1</sup> <sup>k</sup> pLð Þ 1 � k . The multiresolution scaling and wavelet basis functions corresponding to the subspaces Vj and Wj are defined as:

$$\phi\_{L,k}^{\dagger}(\mathbf{x}) = \mathfrak{D}\_{\mathbf{L}}^{\dagger} \phi\_{L}(\mathfrak{D}^{\dagger}\mathbf{x} - k) \tag{12}$$

$$
\psi\_{L,k}^{\dagger}(\mathbf{x}) = \mathfrak{D}\_{\mathbf{}}^{\dagger} \psi\_{L}(\mathfrak{D}^{\dagger}\mathbf{x} - k) \tag{13}
$$

The scaling and wavelet functions defined in Eqs. (10)–(13) satisfy the following properties [8]:

$$\int\_{-\infty}^{\infty} \phi\_L(\mathbf{x})d\mathbf{x} = 1 \tag{14}$$

$$\int\_{-\infty}^{\infty} \phi\_{L,k}^{\circ}(\mathbf{x}) \phi\_{L,l}^{\circ}(\mathbf{x}) d\mathbf{x} = \delta\_{k,l} \tag{15}$$

$$\int\_{-\infty}^{\infty} \psi\_{L,k}^{\circ}(\mathbf{x}) \psi\_{L,k}^{\circ}(\mathbf{x}) d\mathbf{x} = \delta\_{k,l} \tag{16}$$

$$\int\_{-\infty}^{\infty} \phi\_{L,k}^{j}(\mathbf{x}) \psi\_{L,l}^{j}(\mathbf{x}) d\mathbf{x} = \mathbf{0} \tag{17}$$

$$\int\_{-\infty}^{\infty} x^m \psi\_L(\mathbf{x}) d\mathbf{x} = 0 \qquad m = 0, 1, \dots \\ \frac{L}{2} - 1 \tag{18}$$

Certain wavelet families have no explicit formulation, as is the case with the Daubechies wavelets. Therefore, Eq. (10) gives rise to a system of equations that require a normalising equation obtained from Eq. (14) to evaluate the scaling functions. The Daubechies wavelet of order L has <sup>L</sup> <sup>2</sup> � 1 vanishing moments from property (18) and consequently the scaling functions at scale j can represent a polynomial of order x<sup>m</sup> where 0 ≤ m ≤ <sup>L</sup> <sup>2</sup> � 1, i.e., [37]

$$\mathbf{x}^{m} = \sum\_{k} M\_{k}^{j,m} \boldsymbol{\phi}\_{L,k}^{j}(\mathbf{x}) \tag{19}$$

The coefficients Mj,m <sup>k</sup> denote the moments of the scaling function and it translates at Vj. The derivatives of the Daubechies wavelet scaling functions are evaluated by differentiating the refinement Eq. (10) m times, and are obtained as [12]:

Multiscale Wavelet Finite Element Analysis in Structural Dynamics http://dx.doi.org/10.5772/intechopen.71882 55

$$\phi\_L^{(m)}(\mathbf{x}) = \mathbf{2}^m \sum\_{k=0}^{L-1} p(k) \phi\_L^{(m)}(\mathbf{2x} - k) \tag{20}$$

A normalising condition is required to evaluate Eq. (20) which is obtained from the moments of the scaling functions.

$$\sum\_{k=-\infty}^{k=\infty} k^m \phi^{(m)}(\mathbf{x} - k) = m! \tag{21}$$

#### 2.2. Daubechies connection coefficients

<sup>f</sup>Lð Þ¼ <sup>x</sup> <sup>X</sup> L�1

<sup>ψ</sup>Lð Þ¼ <sup>x</sup> <sup>X</sup> L�1

The scaling and wavelet functions have the supports 0½ � ; <sup>L</sup> � <sup>1</sup> and 1 � <sup>L</sup>

fj

ψj

ð∞ �∞ fj L, <sup>k</sup>ð Þ<sup>x</sup> <sup>f</sup><sup>j</sup> L,l

ð∞ �∞ ψj L, <sup>k</sup>ð Þ<sup>x</sup> <sup>ψ</sup><sup>j</sup>

ð∞ �∞ fj L, <sup>k</sup>ð Þ<sup>x</sup> <sup>ψ</sup><sup>j</sup> L,l

tions at scale j can represent a polynomial of order x<sup>m</sup> where 0 ≤ m ≤ <sup>L</sup>

xm <sup>¼</sup> <sup>X</sup> k

ð∞ �∞

refinement Eq. (10) m times, and are obtained as [12]:

L, <sup>k</sup>ð Þ¼ x 2

L, <sup>k</sup>ð Þ¼ x 2

ð∞ �∞

tions corresponding to the subspaces Vj and Wj are defined as:

54 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

have the relation qLð Þ¼ � <sup>k</sup> ð Þ<sup>1</sup> <sup>k</sup>

order L has <sup>L</sup>

The coefficients Mj,m

k¼0

k¼0

normalised wavelet function filter coefficients qLð Þk and scaling function filter coefficients pLð Þk

j <sup>2</sup>f<sup>L</sup> 2<sup>j</sup>

j <sup>2</sup>ψ<sup>L</sup> <sup>2</sup><sup>j</sup>

The scaling and wavelet functions defined in Eqs. (10)–(13) satisfy the following properties [8]:

xmψLð Þ<sup>x</sup> dx <sup>¼</sup> <sup>0</sup> <sup>m</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, …,

Certain wavelet families have no explicit formulation, as is the case with the Daubechies wavelets. Therefore, Eq. (10) gives rise to a system of equations that require a normalising equation obtained from Eq. (14) to evaluate the scaling functions. The Daubechies wavelet of

> Mj,m <sup>k</sup> <sup>f</sup><sup>j</sup>

derivatives of the Daubechies wavelet scaling functions are evaluated by differentiating the

<sup>2</sup> � 1 vanishing moments from property (18) and consequently the scaling func-

<sup>k</sup> denote the moments of the scaling function and it translates at Vj. The

pLð Þk fLð Þ 2x � k (10)

qLð Þk fLð Þ 2x � k (11)

pLð Þ 1 � k . The multiresolution scaling and wavelet basis func-

2 ; L 2

<sup>x</sup> � <sup>k</sup> � � (12)

<sup>x</sup> � <sup>k</sup> � � (13)

fLð Þx dx ¼ 1 (14)

ð Þx dx ¼ δk,l (15)

L, <sup>k</sup>ð Þx dx ¼ δk,l (16)

ð Þx dx ¼ 0 (17)

<sup>2</sup> � <sup>1</sup> (18)

<sup>2</sup> � 1, i.e., [37]

L, <sup>k</sup>ð Þx (19)

L

� � respectively. The

As earlier mentioned, the Daubechies functions cannot be computed analytically and their derivatives are highly oscillatory, particularly at low wavelet orders and/or high order derivatives. Therefore, the integral of the products of the scaling functions and/or derivatives are computed as what is commonly known as connection coefficients [37]. There are two forms of connection coefficients that are of relevance to this study; the multiscale two-term connection coefficient <sup>a</sup>;<sup>b</sup>Γj, <sup>d</sup>1,d<sup>2</sup> k,l and multiscale connection coefficient <sup>Υ</sup>j,m <sup>k</sup> . We define the two-term connection coefficient [30]

$$\mathcal{I}\_{a,b} \Gamma\_{k,l}^{j,d\_1,d\_2} = 2^j \int\_{-\infty}^{\infty} \mathcal{X}\_{[0,1]}(\xi) \phi\_a^{(d\_1)} \left( 2^j \xi - k \right) \phi\_b^{(d\_2)} \left( 2^j \xi - l \right) d\xi \tag{22}$$

where a and b are the orders of the scaling function at multiresolution j, while the values d<sup>1</sup> and d<sup>2</sup> denote the order of the derivative of the scaling functions. X½ � <sup>0</sup>,<sup>1</sup> ð Þ¼ x 1 0 ≤ x ≤ 1 <sup>0</sup> otherwise � is the characteristic function. The formulation presented is a modified algorithm of that described in [4] and allows for the evaluation of the connection coefficients for different values of a and b at different multiresolution scales j. From the 'two-scale' relation presented in Eq. (10),

$$\phi\_L(2^j \xi - k) = \sum\_r p(r) \phi\_L\left(2^{j+1}\xi - 2k - r\right) \tag{23}$$

Differentiating Eq. (23) m times

$$2^{jm} \phi\_L^{(m)} \left( 2^j \xi - k \right) = 2^{(j+1)m} \sum\_{r} p(r) \phi\_L^{(m)} \left( 2^{j+1} \xi - 2k - r \right) \tag{24}$$

Substituting Eq. (24) into Eq. (22) and applying the 'two-scale' relation of the characteristic function, the two-term connection coefficient can be expressed as:

$$\Gamma\_{a,b} \Gamma\_{k,l}^{i,d\_l,d\_2} = 2^{d\_1+d\_2-1} \sum\_{r\_l s} \left[ p\_a(r-2k) p\_b(s-2l) + p\_a(r-2k+2^l) p\_b(s-2l+2^l) \right] \Gamma\_{r,s}^{i,d\_l,d\_2} \tag{25}$$

where 2 � <sup>a</sup> <sup>≤</sup> k, r <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1 and 2 � <sup>b</sup> <sup>≤</sup> l, s <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1. Eq. (25) can be expressed in matrix form as:

$$\begin{aligned} \mathbf{P}^{\{\{a+2^{j}-2\}\{b+2^{j}-2\}\times 1\}} \left\{{}\_{a,b}\mathbf{P}^{\{\}}\right\} &= \\ \mathbf{P}^{\{a+2^{j}-2\}\{b+2^{j}-2\}\times \{a+2^{j}-2\}\{b+2^{j}-2\}} \left[ {}\_{a,b}\mathbf{P} \right] \left\{ \left( {}\_{a+2^{j}-2} \begin{matrix} \mathbf{P} \\ \end{matrix} \right) \left\{{}\_{a,b}\mathbf{P}^{\{\}} \right\} \right\} \end{aligned} \tag{26}$$

where the square matrix <sup>a</sup>;<sup>b</sup>P h i contains the filter coefficients as expressed in Eq. (25) and <sup>a</sup>;<sup>b</sup>Γ<sup>j</sup> n o contains the connection coefficients. To uniquely determine the connection coefficients, normalising conditions are required to generate a sufficient number of inhomogeneous equations via the multiscale moment condition from Eq. (19)

$$\xi^{m} = 2^{\frac{j}{2}} \sum\_{k} {}\_{L} \mathcal{M}\_{k}^{j,m} \phi\_{L} \left( 2^{j} \xi - k \right) \tag{27}$$

t j k; t j <sup>k</sup>þ<sup>1</sup>;…; <sup>t</sup> j kþm

h i

and have support suppB<sup>j</sup>

The basis B<sup>j</sup>

where f<sup>j</sup>

B<sup>j</sup>þ1,ð Þ <sup>m</sup>

multiresolution j [38]:

t

Bj m, <sup>k</sup>ð Þ¼ x

Bj <sup>1</sup>, <sup>k</sup>ð Þ¼ x

respect to variable t. The general B-splines take the form

t j <sup>k</sup>þm�<sup>1</sup> � <sup>t</sup>

m, <sup>k</sup>ð Þ¼ x t

t j <sup>k</sup> ¼

ness is unaffected. For the knot sequence on [0,1], t

fj

corresponding B-wavelet with support suppψ<sup>j</sup>

ψj m, <sup>k</sup>ð Þ¼ x

explicitly from Eq. (34). Given that the requirement j > j

m, <sup>k</sup>ð Þ¼ <sup>x</sup> <sup>B</sup><sup>j</sup>

x � t j k

> j k; t j kþm

j k Bj

1 k ≤ x ≤ k þ 1

2�<sup>j</sup>

8 >><

>>:

<sup>m</sup>�1, <sup>k</sup>ð Þþ <sup>x</sup>

the unit interval and m-tuple knots at 0 and 1, as expressed in Eq. (33). The knots at 0 ad 1 coalesce and form multiple knots for BSWI while the internal knots are simple hence smooth-

j

0 � m þ 1 ≤ k < 1

k 1 ≤ k < 2<sup>j</sup> 1 2<sup>j</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

m, <sup>k</sup>ð Þ<sup>x</sup> from the inner knots corresponds to the <sup>m</sup>th cardinal B-splines, Nmð Þ<sup>x</sup> , at

m, <sup>k</sup>ð Þx is the BSWI scaling function which can be differentiated m times. The

<sup>2</sup><sup>j</sup> ; <sup>k</sup>þ2m�<sup>1</sup> 2j

<sup>N</sup>2mð Þ <sup>l</sup> <sup>þ</sup> <sup>1</sup> Bjþ1,ð Þ <sup>m</sup>

m, <sup>k</sup>ð Þ¼ <sup>x</sup> <sup>k</sup>

ð Þ �<sup>1</sup> <sup>l</sup>

<sup>2</sup>m, <sup>k</sup> ð Þ<sup>x</sup> is the <sup>m</sup>th derivative for the B-spline of order 2<sup>m</sup> and scale <sup>j</sup> <sup>þ</sup> 1 and can be evaluated

The number of inner scaling functions present in the formulation of BSWI is determined by the scale j. There must be at least one inner scaling function on the interval [0,1] and this gives rise

Nmð Þ¼ <sup>x</sup> <sup>m</sup>½ � <sup>0</sup>; <sup>1</sup>;…; <sup>m</sup> ð Þ <sup>t</sup> � <sup>x</sup> <sup>m</sup>�<sup>1</sup>

to the minimum value of j necessary to ensure this condition is met and is defined as j

2j

m, <sup>k</sup>ð Þ¼ <sup>x</sup> Nm <sup>2</sup><sup>j</sup>

1 2<sup>m</sup>�<sup>1</sup>

is present, the scaling and wavelet function of the BSWI are obtained as [39]:

2 X<sup>m</sup>�<sup>2</sup> l¼0

, is the <sup>m</sup>th divided difference of the truncated power function ð Þ <sup>t</sup> � <sup>x</sup> <sup>m</sup>�<sup>1</sup>

t j <sup>k</sup>þ<sup>m</sup> � <sup>x</sup>

<sup>m</sup>�1, <sup>k</sup>þ<sup>1</sup>ð Þ<sup>x</sup>

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

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<sup>0</sup> ≥ 2m � 1 (36)

<sup>x</sup> � <sup>k</sup> � �<sup>0</sup> <sup>≤</sup> <sup>k</sup> <sup>&</sup>lt; <sup>2</sup><sup>j</sup> � <sup>m</sup> <sup>þ</sup> <sup>1</sup> (38)

h i is expressed as:

2m,2iþl

<sup>þ</sup> (37)

ð Þx (39)

<sup>0</sup> ensures at least one inner B-wavelet

<sup>0</sup> otherwise ( (34)

h i. The B-spline basis function has simple knots inside

<sup>k</sup> is given as [38]:

t j <sup>k</sup>þ<sup>m</sup> � <sup>t</sup> j kþ1 Bj <sup>þ</sup> with

57

(35)

0:

Defining the second form of the connection coefficient

$$\Upsilon\_k^{j,m} = 2^{\frac{j}{2}} \int\_0^1 \mathbf{x}^m \phi\_L(2^j \xi - k) d\xi = 2^{\frac{j}{2}} \int\_{-\infty}^\infty \mathcal{X}\_{[0,1]}(\xi) \xi^m \phi\_L(2^j \xi - k) d\xi \tag{28}$$

Substituting Eq. (27) into (28)

$$\Upsilon\_k^{j,m} = 2^j \sum\_l M\_l^{j,m} \int\_{-\infty}^{\infty} \mathcal{X}\_{[0,1]}(\mathbf{x}) \phi\_L \left( 2^j \mathbf{x} - l \right) \phi\_L \left( 2^j \mathbf{x} - k \right) d\mathbf{x} \tag{29}$$

However,

$$\mathcal{R}\_{L,L}\Gamma\_{k,l}^{j,0,0} = 2^j \int\_{-\infty}^{\infty} \mathcal{X}\_{[0,1]}(\mathbf{x}) \phi\_L \left( 2^j \mathbf{x} - l \right) \phi\_L \left( 2^j \mathbf{x} - k \right) d\mathbf{x} \tag{30}$$

Thus

$$\Upsilon\_k^{j,m} = \sum\_l \mathcal{M}\_l^{j,m} \llcorner\_{L,L} \Gamma\_{k,l}^{j,0,0} \tag{31}$$

where <sup>L</sup>;<sup>L</sup>Γj,0,<sup>0</sup> k,l are the two-term connection coefficients with a ¼ b ¼ L and d<sup>1</sup> ¼ d<sup>1</sup> ¼ 0 and Mj,m <sup>l</sup> are the moments earlier described.

#### 2.3. B-spline wavelets on the interval [0,1] (BSWI)

The BSWI are a family of wavelets that emanate from Basis splines functions (B-Splines) and the basic functions in subspace Vj of order m and scale j > 0 are expressed as [20]

$$B\_{m,k}^{j}(\mathbf{x}) = \left(t\_{k+m}^{j} - t\_k^{j}\right) \left[t\_k^{j}, \dots, t\_{k+m}^{j}\right]\_{f} (t - \mathbf{x})\_{+}^{m-1} \tag{32}$$

with the knot sequence

$$\begin{aligned} \left\{ \left. t\_k^j \right\} \right\}\_{k=-m+1}^{2^j+m-1} \\ \mathbf{t}\_k^j \le \mathbf{t}\_{k+1}^j \end{aligned} \tag{33}$$

t j k; t j <sup>k</sup>þ<sup>1</sup>;…; <sup>t</sup> j kþm h i t , is the <sup>m</sup>th divided difference of the truncated power function ð Þ <sup>t</sup> � <sup>x</sup> <sup>m</sup>�<sup>1</sup> <sup>þ</sup> with respect to variable t. The general B-splines take the form

<sup>a</sup><sup>þ</sup> <sup>2</sup><sup>j</sup> ð Þ � <sup>2</sup> <sup>b</sup><sup>þ</sup> <sup>2</sup><sup>j</sup> ð Þ ð Þ � <sup>2</sup> � <sup>1</sup> <sup>a</sup>; <sup>b</sup>Γ<sup>j</sup> n o

2<sup>d</sup>1þd2�<sup>1</sup>

where the square matrix <sup>a</sup>;<sup>b</sup>P

Υj,m <sup>k</sup> ¼ 2 j 2 ð1 0

> Υj,m <sup>k</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup>

<sup>l</sup> are the moments earlier described.

2.3. B-spline wavelets on the interval [0,1] (BSWI)

Bj

m, <sup>k</sup>ð Þ¼ x t

L;LΓj,0,<sup>0</sup> k,l <sup>¼</sup> <sup>2</sup><sup>j</sup>

Substituting Eq. (27) into (28)

a;bΓ<sup>j</sup> n o

However,

Thus

Mj,m

where <sup>L</sup>;<sup>L</sup>Γj,0,<sup>0</sup>

with the knot sequence

¼

56 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

h i

tions via the multiscale moment condition from Eq. (19)

Defining the second form of the connection coefficient

xmf<sup>L</sup> 2<sup>j</sup>

X l

Mj,m l ð∞ �∞

> ð∞ �∞

> > Υj,m <sup>k</sup> <sup>¼</sup> <sup>X</sup> l

<sup>a</sup><sup>þ</sup> <sup>2</sup><sup>j</sup> ð Þ � <sup>2</sup> <sup>b</sup><sup>þ</sup> <sup>2</sup><sup>j</sup> ð Þ � <sup>2</sup> � <sup>a</sup><sup>þ</sup> <sup>2</sup><sup>j</sup> ð Þ � <sup>2</sup> <sup>b</sup><sup>þ</sup> <sup>2</sup><sup>j</sup> ð Þ ð Þ � <sup>2</sup> <sup>a</sup>; <sup>b</sup><sup>P</sup>

<sup>ξ</sup><sup>m</sup> <sup>¼</sup> <sup>2</sup> j 2 X k

<sup>ξ</sup> � <sup>k</sup> � �d<sup>ξ</sup> <sup>¼</sup> <sup>2</sup>

h i

contains the connection coefficients. To uniquely determine the connection coefficients,

normalising conditions are required to generate a sufficient number of inhomogeneous equa-

LMj,m <sup>k</sup> <sup>f</sup><sup>L</sup> <sup>2</sup><sup>j</sup>

> j 2 ð∞ �∞

<sup>X</sup>½ � <sup>0</sup>;<sup>1</sup> ð Þ<sup>x</sup> <sup>f</sup><sup>L</sup> <sup>2</sup><sup>j</sup>

Mj,m <sup>l</sup> <sup>L</sup>;<sup>L</sup>Γj,0,<sup>0</sup>

The BSWI are a family of wavelets that emanate from Basis splines functions (B-Splines) and

t j <sup>k</sup>;…; t j kþm h i

þm�1 k¼�mþ1

the basic functions in subspace Vj of order m and scale j > 0 are expressed as [20]

t j k n o<sup>2</sup><sup>j</sup>

> t j <sup>k</sup> ≤ t j kþ1

j <sup>k</sup>þ<sup>m</sup> � <sup>t</sup> j k � �

k,l are the two-term connection coefficients with a ¼ b ¼ L and d<sup>1</sup> ¼ d<sup>1</sup> ¼ 0 and

f

ð Þ <sup>t</sup> � <sup>x</sup> <sup>m</sup>�<sup>1</sup>

<sup>X</sup>½ � <sup>0</sup>;<sup>1</sup> ð Þ<sup>x</sup> <sup>f</sup><sup>L</sup> <sup>2</sup><sup>j</sup>

<sup>a</sup><sup>þ</sup> <sup>2</sup><sup>j</sup> ð Þ � <sup>2</sup> <sup>b</sup><sup>þ</sup> <sup>2</sup><sup>j</sup> ð Þ ð Þ � <sup>2</sup> � <sup>1</sup> <sup>a</sup>; <sup>b</sup>Γ<sup>j</sup> n o

<sup>ξ</sup> � <sup>k</sup> � � (27)

<sup>ξ</sup> � <sup>k</sup> � �d<sup>ξ</sup> (28)

<sup>x</sup> � <sup>k</sup> � �dx (29)

<sup>x</sup> � <sup>k</sup> � �dx (30)

<sup>þ</sup> (32)

(33)

k,l (31)

contains the filter coefficients as expressed in Eq. (25) and

<sup>X</sup>½ � <sup>0</sup>;<sup>1</sup> ð Þ <sup>ξ</sup> <sup>ξ</sup><sup>m</sup>f<sup>L</sup> <sup>2</sup><sup>j</sup>

<sup>x</sup> � <sup>l</sup> � �f<sup>L</sup> <sup>2</sup><sup>j</sup>

<sup>x</sup> � <sup>l</sup> � �f<sup>L</sup> <sup>2</sup><sup>j</sup>

(26)

$$\begin{aligned} B\_{m,k}^{j}(\mathbf{x}) &= \frac{\mathbf{x} - \mathbf{t}\_{k}^{j}}{\mathbf{t}\_{k+m-1}^{j} - \mathbf{t}\_{k}^{j}} B\_{m-1,k}^{j}(\mathbf{x}) + \frac{\mathbf{t}\_{k+m}^{j} - \mathbf{x}}{\mathbf{t}\_{k+m}^{j} - \mathbf{t}\_{k+1}^{j}} B\_{m-1,k+1}^{j}(\mathbf{x}) \\ B\_{1,k}^{j}(\mathbf{x}) &= \begin{cases} 1 & k \le \mathbf{x} \le k+1 \\ 0 & \text{otherwise} \end{cases} \end{aligned} \tag{34}$$

and have support suppB<sup>j</sup> m, <sup>k</sup>ð Þ¼ x t j k; t j kþm h i. The B-spline basis function has simple knots inside the unit interval and m-tuple knots at 0 and 1, as expressed in Eq. (33). The knots at 0 ad 1 coalesce and form multiple knots for BSWI while the internal knots are simple hence smoothness is unaffected. For the knot sequence on [0,1], t j <sup>k</sup> is given as [38]:

$$t\_k^j = \begin{cases} 0 & -m+1 \le k < 1\\ 2^{-j}k & 1 \le k < 2^j\\ 1 & 2^j \le k \le 2^j + m - 1 \end{cases} \tag{35}$$

The number of inner scaling functions present in the formulation of BSWI is determined by the scale j. There must be at least one inner scaling function on the interval [0,1] and this gives rise to the minimum value of j necessary to ensure this condition is met and is defined as j 0:

$$2^{j\_0} \ge 2m - 1\tag{36}$$

The basis B<sup>j</sup> m, <sup>k</sup>ð Þ<sup>x</sup> from the inner knots corresponds to the <sup>m</sup>th cardinal B-splines, Nmð Þ<sup>x</sup> , at multiresolution j [38]:

$$N\_m(\mathbf{x}) = m[0, 1, \dots, m](t - \mathbf{x})\_+^{m-1} \tag{37}$$

$$\boldsymbol{\phi}\_{m,k}^{j}(\mathbf{x}) = \boldsymbol{B}\_{m,k}^{j}(\mathbf{x}) = \mathbf{N}\_{m}(2^{j}\mathbf{x} - k)\mathbf{0} \le k < 2^{j} - m + 1\tag{38}$$

where f<sup>j</sup> m, <sup>k</sup>ð Þx is the BSWI scaling function which can be differentiated m times. The corresponding B-wavelet with support suppψ<sup>j</sup> m, <sup>k</sup>ð Þ¼ <sup>x</sup> <sup>k</sup> <sup>2</sup><sup>j</sup> ; <sup>k</sup>þ2m�<sup>1</sup> 2j h i is expressed as:

$$\psi\_{m,k}^{j}(\mathbf{x}) = \frac{1}{2^{m-1}} \sum\_{l=0}^{2m-2} (-1)^{l} \mathbf{N}\_{2m}(l+1) B\_{2m,2i+l}^{j+1,(m)}(\mathbf{x}) \tag{39}$$

B<sup>j</sup>þ1,ð Þ <sup>m</sup> <sup>2</sup>m, <sup>k</sup> ð Þ<sup>x</sup> is the <sup>m</sup>th derivative for the B-spline of order 2<sup>m</sup> and scale <sup>j</sup> <sup>þ</sup> 1 and can be evaluated explicitly from Eq. (34). Given that the requirement j > j <sup>0</sup> ensures at least one inner B-wavelet is present, the scaling and wavelet function of the BSWI are obtained as [39]:

$$\phi\_{m,k}^{j}(\mathbf{x}) = \begin{cases} B\_{m,k}^{j\_0}(2^{j-j\_0}\mathbf{x}) & -m+1 \le k \le -1 \\\\ B\_{m,0}^{j\_0}(2^{j-j\_0}\mathbf{x} - 2^{-j\_0}k) & 0 \le i \le 2^j - m \\\\ B\_{m,2^j-k-m}^{j\_0}(1-2^{j-j\_0}\mathbf{x}) & 2^j \le i \le 2^j + m - 1 \end{cases} \tag{40}$$
 
$$\psi\_{m,k}^{j}(\mathbf{x}) = \begin{cases} \psi\_{m,k}^{j\_0}(2^{j-j\_0}\mathbf{x}) & -m+1 \le k \le -1 \\\\ \psi\_{m,0}^{j\_0}\left(2^{j-j\_0}\mathbf{x} - 2^{-j\_0}k\right) & 0 \le i \le 2^j - m \\\\ \psi\_{m,2^j-k-2m+1}^{j\_0}\left(1 - 2^{j-j\_0}\mathbf{x}\right) & 2^j \le i \le 2^j + m - 1 \end{cases} \tag{41}$$

and the scaling function derivatives can be evaluated directly by differentiating Eq. (40).

### 3. The wavelet finite element method

#### 3.1. Axial rod wavelet finite element

Assume each WFE is divided into equal segments, ns, connected by r ¼ ns þ 1 elemental nodes, as shown in Figure 1, with axial deformation ui. The total number of degrees of freedom (DOFs) within each WFE is denoted by n ¼ r for n, r∈ N. Vector f g ue ¼ fu1u<sup>2</sup> <sup>⋯</sup>ur�<sup>1</sup>urg<sup>T</sup> contains all the axial DOFs in physical space, as illustrated in Figure 2(a), where ui ¼ u xð Þ<sup>i</sup> represents the elemental node axial deformation DOF at node i corresponding to coordinate position xi. The nodal natural coordinates is <sup>ξ</sup><sup>i</sup> <sup>¼</sup> xi�x<sup>1</sup> Le (0 ≤ ξ<sup>i</sup> ≤ 1, 1 ≤ i ≤ r). The Daubechies and BSWI scaling functions f<sup>j</sup> z, <sup>k</sup>ð Þx are used as the interpolating functions and for a family of order z at multiresolution scale j, the axial deformation

$$\mu(\xi) = \sum\_{k=h}^{2'-1} a\_{z,k}^{j} \phi\_{z,k}^{j}(\xi) \tag{42}$$

contains the unknown wavelet coefficients a

The matrix Rw

The matrix T<sup>w</sup>

r � � <sup>¼</sup> <sup>Φ</sup><sup>j</sup>

elemental nodes and f g ae ¼ a

r � � <sup>¼</sup> Rw

evaluated as f g Nr, <sup>e</sup>ð Þ <sup>ξ</sup> <sup>¼</sup> <sup>Φ</sup><sup>j</sup>

ing function vectors Φ<sup>j</sup>

axial deformations at all elemental nodes in physical space.

Figure 2. (a) Axial rod and (b) Euler Bernoulli beam wavelet finite element layout.

<sup>z</sup>ð Þ ξ<sup>1</sup> � � Φ<sup>j</sup>

<sup>z</sup>ð Þ ξ<sup>i</sup>

any point along the rod element can be generalised as:

r

<sup>Π</sup><sup>a</sup> <sup>¼</sup> ðl 0 EA 2

of length Le is expressed in natural coordinates as:

derivative of the scaling functions and is symmetric.

f g ue <sup>T</sup> ð1 0 T<sup>w</sup> r � �<sup>T</sup> <sup>d</sup>Φ<sup>j</sup>

The stiffness matrix of the rod element in wavelet space, kw

Ua <sup>e</sup> <sup>¼</sup> <sup>1</sup> 2 EA Le

j z, <sup>h</sup> a j

<sup>z</sup>ð Þ <sup>ξ</sup> � � <sup>T</sup><sup>w</sup>

<sup>u</sup>ð Þ¼ <sup>ξ</sup> ð Þ <sup>1</sup>�<sup>n</sup> <sup>Φ</sup><sup>j</sup>

r

du xð Þ dx � �<sup>2</sup>

the potential energy within the axial rod Π<sup>a</sup> can be generally expressed as [40]:

Therefore, given the relation highlighted in Eq. (44), the axial stain energy U<sup>a</sup>

j

r � �

> j z, 2<sup>j</sup> �<sup>2</sup> <sup>a</sup> j z,2<sup>j</sup> �1

� � h i<sup>T</sup>

h i<sup>T</sup>

<sup>z</sup>ð Þ <sup>ξ</sup> � �ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup>

w denoting rod and wavelet respectively. The wavelet based axial rod shape functions can be

� � within each element.

Suppose the axial rod is subjected to nodal point loads f xi and distributed loading f <sup>d</sup>ð Þx , then

dx �<sup>X</sup> i

where E is the Young's modulus, A is the cross-sectional area and l is the length of the rod.

<sup>z</sup>ð Þ ξ dξ

� �<sup>T</sup> dΦ<sup>j</sup>

z,hþ<sup>1</sup> <sup>⋯</sup> <sup>a</sup>

<sup>z</sup>ð Þ ξ<sup>r</sup>�<sup>1</sup> � � Φ<sup>j</sup>

> r � �

u xð Þ<sup>i</sup> f xi �

ðl 0

<sup>z</sup>ð Þ ξ dξ � � <sup>T</sup><sup>w</sup>

r, e

r

� ��<sup>1</sup> is the axial rod wavelet transformation matrix with the scripts r and

� � approximating the axial deformation at the corresponding

ð Þ <sup>n</sup>�<sup>1</sup> f g¼ ue ð Þ <sup>n</sup>�<sup>n</sup> <sup>R</sup><sup>w</sup>

<sup>z</sup>ð Þ ξ<sup>2</sup> � � ⋯ Φ<sup>j</sup>

z, <sup>k</sup>. This gives rise to the vector f g ue containing the

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

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59

<sup>z</sup>ð Þ ξ<sup>r</sup>

ð Þ <sup>n</sup>�<sup>1</sup> f g ae (43)

contains the scal-

. The axial deformation at

ð Þ <sup>n</sup>�<sup>1</sup> f g ue (44)

f <sup>d</sup>ð Þx u xð Þ dx (45)

� �dξf g ue (46)

h i is computed using the first

<sup>e</sup> within each WFE

Figure 1. Wavelet finite element layout.

Figure 2. (a) Axial rod and (b) Euler Bernoulli beam wavelet finite element layout.

fj m, <sup>k</sup>ð Þ¼ x

ψj m, <sup>k</sup>ð Þ¼ x

3. The wavelet finite element method

3.1. Axial rod wavelet finite element

Daubechies and BSWI scaling functions f<sup>j</sup>

Figure 1. Wavelet finite element layout.

Bj 0 m, <sup>k</sup> <sup>2</sup><sup>j</sup>�<sup>j</sup> <sup>0</sup> x � �

8 >>>>>><

58 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

>>>>>>:

8 >>>>>><

>>>>>>:

Bj 0 m,<sup>0</sup> <sup>2</sup><sup>j</sup>�<sup>j</sup>

Bj 0 m,2<sup>j</sup>

ψj 0 m, <sup>k</sup> <sup>2</sup><sup>j</sup>�<sup>j</sup> <sup>0</sup> x � �

ψj 0 m,<sup>0</sup> <sup>2</sup><sup>j</sup>�<sup>j</sup>

ψj 0 m,2<sup>j</sup>

coordinate position xi. The nodal natural coordinates is <sup>ξ</sup><sup>i</sup> <sup>¼</sup> xi�x<sup>1</sup>

a family of order z at multiresolution scale j, the axial deformation

<sup>u</sup>ð Þ¼ <sup>ξ</sup> <sup>X</sup> 2j �1

k¼h a j z, <sup>k</sup>f<sup>j</sup>

<sup>0</sup> <sup>x</sup> � <sup>2</sup>�<sup>j</sup> <sup>0</sup> k � � �m þ 1 ≤ k ≤ � 1 <sup>0</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>2</sup><sup>j</sup> � <sup>m</sup>

(40)

(41)

<sup>2</sup><sup>j</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

�m þ 1 ≤ k ≤ � 1 <sup>0</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>2</sup><sup>j</sup> � <sup>m</sup>

<sup>2</sup><sup>j</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

Le (0 ≤ ξ<sup>i</sup> ≤ 1, 1 ≤ i ≤ r). The

z, <sup>k</sup>ð Þx are used as the interpolating functions and for

z, <sup>k</sup>ð Þ ξ (42)

�k�<sup>m</sup> <sup>1</sup> � <sup>2</sup><sup>j</sup>�<sup>j</sup> <sup>0</sup> x � �

<sup>0</sup> <sup>x</sup> � <sup>2</sup>�<sup>j</sup> <sup>0</sup> k � �

�k�2mþ<sup>1</sup> <sup>1</sup> � <sup>2</sup><sup>j</sup>�<sup>j</sup>

and the scaling function derivatives can be evaluated directly by differentiating Eq. (40).

Assume each WFE is divided into equal segments, ns, connected by r ¼ ns þ 1 elemental nodes, as shown in Figure 1, with axial deformation ui. The total number of degrees of freedom (DOFs) within each WFE is denoted by n ¼ r for n, r∈ N. Vector f g ue ¼ fu1u<sup>2</sup> <sup>⋯</sup>ur�<sup>1</sup>urg<sup>T</sup> contains all the axial DOFs in physical space, as illustrated in Figure 2(a), where ui ¼ u xð Þ<sup>i</sup> represents the elemental node axial deformation DOF at node i corresponding to

<sup>0</sup> x � �

contains the unknown wavelet coefficients a j z, <sup>k</sup>. This gives rise to the vector f g ue containing the axial deformations at all elemental nodes in physical space.

$$\mathbf{u}\_{\left(n\times1\right)}\left\{\mathbf{u}\_{\mathbf{e}}\right\} = {}\_{\left(n\times n\right)}\left[\mathbf{R}\_{\mathbf{r}}^{\mathbf{w}}\right]\_{\left(n\times1\right)}\left\{\mathbf{a}\_{\mathbf{e}}\right\}\tag{43}$$

The matrix Rw r � � <sup>¼</sup> <sup>Φ</sup><sup>j</sup> <sup>z</sup>ð Þ ξ<sup>1</sup> � � Φ<sup>j</sup> <sup>z</sup>ð Þ ξ<sup>2</sup> � � ⋯ Φ<sup>j</sup> <sup>z</sup>ð Þ ξ<sup>r</sup>�<sup>1</sup> � � Φ<sup>j</sup> <sup>z</sup>ð Þ ξ<sup>r</sup> � � h i<sup>T</sup> contains the scaling function vectors Φ<sup>j</sup> <sup>z</sup>ð Þ ξ<sup>i</sup> � � approximating the axial deformation at the corresponding elemental nodes and f g ae ¼ a j z, <sup>h</sup> a j z,hþ<sup>1</sup> <sup>⋯</sup> <sup>a</sup> j z, 2<sup>j</sup> �<sup>2</sup> <sup>a</sup> j z,2<sup>j</sup> �1 h i<sup>T</sup> . The axial deformation at any point along the rod element can be generalised as:

$$\mu(\xi) = {}\_{(1\times n)}\left\{ \mathbf{op}\_z^j(\xi) \right\} {}\_{(n\times n)}\left[T\_r^w\right] {}\_{(n\times 1)}\left\{ \mathfrak{u}\_\mathfrak{e} \right\} \tag{44}$$

The matrix T<sup>w</sup> r � � <sup>¼</sup> Rw r � ��<sup>1</sup> is the axial rod wavelet transformation matrix with the scripts r and w denoting rod and wavelet respectively. The wavelet based axial rod shape functions can be evaluated as f g Nr, <sup>e</sup>ð Þ <sup>ξ</sup> <sup>¼</sup> <sup>Φ</sup><sup>j</sup> <sup>z</sup>ð Þ <sup>ξ</sup> � � <sup>T</sup><sup>w</sup> r � � within each element.

Suppose the axial rod is subjected to nodal point loads f xi and distributed loading f <sup>d</sup>ð Þx , then the potential energy within the axial rod Π<sup>a</sup> can be generally expressed as [40]:

$$
\Pi^{\mathfrak{a}} = \int\_0^l \frac{EA}{2} \left( \frac{du(\mathbf{x})}{d\mathbf{x}} \right)^2 d\mathbf{x} - \sum\_i u(\mathbf{x}\_i) f\_{\mathbf{x}i} - \int\_0^l f\_d(\mathbf{x}) u(\mathbf{x}) \, d\mathbf{x} \tag{45}
$$

where E is the Young's modulus, A is the cross-sectional area and l is the length of the rod. Therefore, given the relation highlighted in Eq. (44), the axial stain energy U<sup>a</sup> <sup>e</sup> within each WFE of length Le is expressed in natural coordinates as:

$$\mathbf{U}\_{\epsilon}^{\mathfrak{s}} = \frac{1}{2} \frac{EA}{L\_{\epsilon}} \left\{ \mathfrak{u}\_{\epsilon} \right\}^{T} \int\_{0}^{1} \left[\mathbf{T}\_{r}^{w}\right]^{T} \left\{ \frac{d\mathbf{\varPhi}\_{z}^{j}(\xi)}{d\xi} \right\}^{T} \left\{ \frac{d\mathbf{\varPhi}\_{z}^{j}(\xi)}{d\xi} \right\} \left[\mathbf{T}\_{r}^{w}\right] d\xi \left\{ \mathfrak{u}\_{\epsilon} \right\} \tag{46}$$

The stiffness matrix of the rod element in wavelet space, kw r, e h i is computed using the first derivative of the scaling functions and is symmetric.

$$\mathbb{E}\_{(n\times n)}\left[\mathbb{k}\_{r,\epsilon}^{w}\right] = \int\_{0}^{1} \left\{\boldsymbol{\Phi}\_{z}^{\circ j}(\boldsymbol{\xi})\right\}^{T} \left\{\boldsymbol{\Phi}\_{z}^{\circ j}(\boldsymbol{\xi})\right\} d\boldsymbol{\xi} \tag{47}$$

nodes can be tailored according to the desired requirements and this in turn will affect the total number of elemental segments and nodes present in each element. In this case the internal WFE nodes only have the transverse displacement present and the total number of DOFs within each beam element is n as illustrated in Figure 2(b). Therefore, there are n � 2 displacement DOFs and 2 rotation DOFs in total for each WFE and consequently r ¼ n � 2 elemental nodes and ns ¼ n � 3 elemental segments. Let the vector {veg ¼ <sup>f</sup> <sup>v</sup><sup>1</sup> <sup>θ</sup><sup>1</sup> <sup>v</sup><sup>2</sup> <sup>v</sup><sup>3</sup> <sup>⋯</sup> vr�<sup>2</sup> vr�<sup>1</sup> vrθrg<sup>T</sup> denote all the physical DOFs within the beam element. The displacement and rotation DOFs corresponding to coordinate position xi ∈ x1; xr ½ � i ∈ N and 1ð Þ ≤ i ≤ r in local coordinates are denoted as vi ¼ v xð Þ<sup>i</sup>

rotation at any point of the wavelet based beam finite element can be approximated by applying

k¼h b j z, <sup>k</sup>f<sup>j</sup> z, <sup>k</sup>ð Þ ξ

> k¼h b j z, k ∂f<sup>j</sup> z, <sup>k</sup>ð Þ ξ ∂ξ

b � �

h � � i<sup>T</sup>

From Eq. (54), the transverse displacement and rotation at any point of the beam element can

<sup>z</sup>ð Þ <sup>ξ</sup> � �ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup>

<sup>z</sup> ð Þ <sup>ξ</sup> � �ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup>

<sup>Π</sup><sup>b</sup> within a Euler Bernoulli beam subjected to concentrated forces <sup>f</sup> yi, distributed force <sup>f</sup> <sup>d</sup>ð Þ<sup>x</sup>

f yiv xð Þ�<sup>i</sup>

where E is the Young's modulus, I is the moment of inertia and l is the length of the beam. The

imation of the transverse displacement via scaling functions as highlighted in Eq. (55).

<sup>z</sup>ð Þ ξ<sup>r</sup>�<sup>1</sup> � � Φ<sup>j</sup>

> b � �

> > b � �

� ��<sup>1</sup> is the beam wavelet transformation matrix which is used to obtain the

ðl 0

<sup>e</sup> within each beam element of length Le can expressed in terms of the approx-

ð Þ <sup>n</sup>�<sup>1</sup> f g ve

<sup>z</sup>ð Þ <sup>ξ</sup> � � <sup>T</sup><sup>w</sup>

<sup>f</sup> <sup>d</sup>ð Þ<sup>x</sup> vdx �<sup>X</sup>

b

k m k

j

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup> Le X 2j �1

Therefore, the DOFs present within the entire beam element can be represented as ð Þ <sup>n</sup>�<sup>1</sup> f g ve <sup>¼</sup> ð Þ <sup>n</sup>�<sup>n</sup> Rw

> <sup>z</sup>ð Þ ξ<sup>2</sup> � � ⋯ Φ<sup>j</sup>

ð Þ <sup>1</sup>�<sup>n</sup> <sup>Φ</sup>0<sup>j</sup>

dx �<sup>X</sup> i

<sup>v</sup>ð Þ¼ <sup>ξ</sup> ð Þ <sup>1</sup>�<sup>n</sup> <sup>Φ</sup><sup>j</sup>

<sup>v</sup>ð Þ¼ <sup>ξ</sup> <sup>X</sup> 2j �1

θ ξð Þ¼ <sup>∂</sup>vð Þ <sup>ξ</sup>

Le (0 ≤ ξ<sup>i</sup> ≤ 1, 1 ≤ i ≤ r). The deflection and

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

http://dx.doi.org/10.5772/intechopen.71882

ð Þ <sup>n</sup>�<sup>1</sup> f g be (54)

Le <sup>Φ</sup>0<sup>j</sup> <sup>z</sup> ð Þ ξ<sup>r</sup>

z, <sup>k</sup> representing the beam wavelet

ð Þ <sup>n</sup>�<sup>1</sup> f g ve (55)

dv xð Þ<sup>k</sup>

� �. The potential energy

dx (56)

<sup>z</sup>ð Þ ξ<sup>r</sup> � � <sup>1</sup> (53)

61

and

z, <sup>k</sup>ð Þx of order z at multiresolution scale j as interpolating functions.

and <sup>θ</sup><sup>i</sup> <sup>¼</sup> <sup>θ</sup>ð Þ xi . The nodal natural coordinate <sup>ξ</sup><sup>i</sup> <sup>¼</sup> xi�x<sup>1</sup>

the wavelet scaling functions f<sup>j</sup>

<sup>z</sup>ð Þ ξ<sup>1</sup> � � <sup>1</sup>

b

<sup>Π</sup><sup>b</sup> <sup>¼</sup> ðl 0 EI 2

Le <sup>Φ</sup>0<sup>j</sup> <sup>z</sup> ð Þ ξ<sup>1</sup> � � Φ<sup>j</sup>

vector f g be contains the unknown wavelet coefficients b

θ ξð Þ¼ <sup>1</sup> Le

wavelet based shape functions for the beam Nb, <sup>e</sup>ð Þ <sup>ξ</sup> � � <sup>¼</sup> <sup>Φ</sup><sup>j</sup>

and bending moments m <sup>i</sup> can be generally expressed as [40]:

d2 v dx<sup>2</sup> � �<sup>2</sup>

R<sup>w</sup> b � � <sup>¼</sup> <sup>Φ</sup><sup>j</sup>

space DOFs.

where T<sup>w</sup> b � � <sup>¼</sup> <sup>R</sup><sup>w</sup>

strain energy U<sup>b</sup>

be expressed as:

In order for one to obtain the stiffness matrix in physical space, the element properties and transformation matrix T<sup>w</sup> r � � are applied to the wavelet space stiffness matrix in Eq. (47).

$$\mathbf{r}\_{(n\times n)}\left[\mathbf{k}\_{\mathbf{r},\epsilon}^{\mathbf{p}}\right] = \frac{EA}{L\_{\epsilon}}\_{(n\times n)}\left[\mathbf{T}\_{\mathbf{r}}^{w}\right]^{T}{}\_{(n\times n)}\left[\mathbf{k}\_{\mathbf{r},\epsilon}^{w}\right]\_{(n\times n)}\left[\mathbf{T}\_{\mathbf{r}}^{w}\right] \tag{48}$$

The load vector containing the axial point loads of the WFE in physical space is obtained as:

$$\mathbf{r}\_{(n\times 1)}\left\{\mathbf{f}\_{\mathbf{r},\mathbf{e}}^{n,p}\right\} = \sum\_{i} \left[\mathbf{T}\_{\mathbf{r}}^{w}\right]^{T} \left\{\boldsymbol{\Phi}\_{\mathbf{z}}^{j}(\xi\_{i})\right\}^{T} \boldsymbol{f}\_{xi} \tag{49}$$

and the equivalent nodal load vector for the distributed load f <sup>d</sup>ð Þx in physical space is

$$\mathcal{L}\_{(n\times 1)}\left\{\boldsymbol{f}\_{r,\varepsilon}^{d,p}\right\} = \boldsymbol{L}\_{\varepsilon}\int\_{0}^{1} \boldsymbol{f}\_{d}(\boldsymbol{\xi}) \left[\boldsymbol{T}\_{r}^{w}\right]^{T} \left\{\boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi})\right\}^{T} d\boldsymbol{\xi} \tag{50}$$

When applying the Daubechies wavelet family, the WFE has a total of <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>L</sup> � 2 DOFs. The wavelet space stiffness matrix is evaluated from the multiscale two-term connection coefficients <sup>a</sup>;<sup>b</sup>Γj, <sup>d</sup>1, <sup>d</sup><sup>2</sup> k,l a ¼ b ¼ L and d<sup>1</sup> ¼ d<sup>1</sup> ¼ 1 and is given as:

$$\mathbb{E}\left(\left(2^{j}+L-2\right)\mathbf{x}\left(2^{j}+L-2\right)\right)^{D}\left[\mathbf{k}\_{\mathbf{r},\mathbf{e}}^{w}\right]=2^{2j}\left[\Gamma^{j,1,1}\right]\tag{51}$$

where 22<sup>j</sup> � � is the normalising factor and the matrix <sup>Γ</sup>j,1,<sup>1</sup> � � has the entries <sup>L</sup>;<sup>L</sup>Γj,1,<sup>1</sup> k,l for the limits <sup>2</sup> � <sup>L</sup> <sup>≤</sup> k, l <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1. Similarly, the distributed forces acting on the element require the form <sup>Υ</sup>j,m k for limits 2 � <sup>L</sup> <sup>≤</sup> k, l <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1 of connection coefficients and the value of <sup>m</sup> depends on the order of the function f <sup>d</sup>ð Þx of the forces. In the case of the BSWI formulations, the total DOFs is <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � 1 and the condition <sup>j</sup> <sup>≥</sup> <sup>j</sup> <sup>0</sup> must be satisfied. Therefore, the wavelet space stiffness matrices of the BSWI axial rod are computed as:

$$\mathbb{E}\left(\left(2^{j}+m-1\right)\times\left(2^{j}+m-1\right)\right)\left[\mathbf{k}\_{r,\epsilon}^{\mathbf{w}}\right]=\int\_{0}^{1}\left\{\boldsymbol{\varPhi}\_{\mathbf{m}}^{\circ j}(\boldsymbol{\xi})\right\}^{T}\left\{\boldsymbol{\varPhi}\_{\mathbf{m}}^{\circ j}(\boldsymbol{\xi})\right\}d\boldsymbol{\upxi}\tag{52}$$

#### 3.2. Euler Bernoulli beam wavelet finite element

According to Euler Bernoulli beam theory, it is assumed that the shear deformation effects are neglected because before and after bending occurs, the plane cross-sections remain plane and perpendicular to the axial centroidal axis of the beam. The beam WFE of length Le, is divided into ns equally spaced elemental segments connected by r elemental nodes at coordinate values xi ∈ x1; xr ½ � and i∈ N as illustrated in Figure 1. The WFE has the transverse displacement v and rotation θ taken into account, with corresponding transverse forces f <sup>y</sup> and moments m respectively. The transverse displacement and rotation DOFs must be present at each elemental end node to ensure inter-element compatibility [4–6]. However, the DOFs at the internal elemental nodes can be tailored according to the desired requirements and this in turn will affect the total number of elemental segments and nodes present in each element. In this case the internal WFE nodes only have the transverse displacement present and the total number of DOFs within each beam element is n as illustrated in Figure 2(b). Therefore, there are n � 2 displacement DOFs and 2 rotation DOFs in total for each WFE and consequently r ¼ n � 2 elemental nodes and ns ¼ n � 3 elemental segments. Let the vector {veg ¼ <sup>f</sup> <sup>v</sup><sup>1</sup> <sup>θ</sup><sup>1</sup> <sup>v</sup><sup>2</sup> <sup>v</sup><sup>3</sup> <sup>⋯</sup> vr�<sup>2</sup> vr�<sup>1</sup> vrθrg<sup>T</sup> denote all the physical DOFs within the beam element. The displacement and rotation DOFs corresponding to coordinate position xi ∈ x1; xr ½ � i ∈ N and 1ð Þ ≤ i ≤ r in local coordinates are denoted as vi ¼ v xð Þ<sup>i</sup> and <sup>θ</sup><sup>i</sup> <sup>¼</sup> <sup>θ</sup>ð Þ xi . The nodal natural coordinate <sup>ξ</sup><sup>i</sup> <sup>¼</sup> xi�x<sup>1</sup> Le (0 ≤ ξ<sup>i</sup> ≤ 1, 1 ≤ i ≤ r). The deflection and rotation at any point of the wavelet based beam finite element can be approximated by applying the wavelet scaling functions f<sup>j</sup> z, <sup>k</sup>ð Þx of order z at multiresolution scale j as interpolating functions.

ð Þ <sup>n</sup>� <sup>n</sup> kw r, e h i ¼ ð1 0 Φ0<sup>j</sup> <sup>z</sup> ð Þ <sup>ξ</sup> � �<sup>T</sup> <sup>Φ</sup>0<sup>j</sup>

60 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

<sup>¼</sup> EA

ð Þ <sup>n</sup>� <sup>1</sup> <sup>f</sup> <sup>n</sup>,<sup>p</sup> r, e n o

ð Þ <sup>n</sup>� <sup>1</sup> <sup>f</sup> <sup>d</sup>,<sup>p</sup> r, e n o

Le ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup> r � �<sup>T</sup>

> <sup>¼</sup> <sup>X</sup> i

and the equivalent nodal load vector for the distributed load f <sup>d</sup>ð Þx in physical space is

¼ Le ð1 0

k,l a ¼ b ¼ L and d<sup>1</sup> ¼ d<sup>1</sup> ¼ 1 and is given as:

<sup>2</sup><sup>j</sup> ð Þ <sup>þ</sup> <sup>L</sup>� <sup>2</sup> <sup>x</sup> <sup>2</sup><sup>j</sup> ð Þ ð Þ <sup>þ</sup> <sup>L</sup>� <sup>2</sup> <sup>k</sup><sup>w</sup>

where 22<sup>j</sup> � � is the normalising factor and the matrix <sup>Γ</sup>j,1,<sup>1</sup> � � has the entries <sup>L</sup>;<sup>L</sup>Γj,1,<sup>1</sup>

BS

r, e h i ¼ ð1 0 Φ0<sup>j</sup> <sup>m</sup>ð Þ <sup>ξ</sup> � �<sup>T</sup> <sup>Φ</sup>0<sup>j</sup>

According to Euler Bernoulli beam theory, it is assumed that the shear deformation effects are neglected because before and after bending occurs, the plane cross-sections remain plane and perpendicular to the axial centroidal axis of the beam. The beam WFE of length Le, is divided into ns equally spaced elemental segments connected by r elemental nodes at coordinate values xi ∈ x1; xr ½ � and i∈ N as illustrated in Figure 1. The WFE has the transverse displacement v and rotation θ taken into account, with corresponding transverse forces f <sup>y</sup> and moments m respectively. The transverse displacement and rotation DOFs must be present at each elemental end node to ensure inter-element compatibility [4–6]. However, the DOFs at the internal elemental

<sup>2</sup><sup>j</sup> ð Þ <sup>þ</sup> <sup>m</sup>� <sup>1</sup> � <sup>2</sup><sup>j</sup> ð Þ ð Þ <sup>þ</sup> <sup>m</sup>� <sup>1</sup> kw

r

ð Þ <sup>n</sup>� <sup>n</sup> <sup>k</sup><sup>p</sup> r, e h i

transformation matrix T<sup>w</sup>

coefficients <sup>a</sup>;<sup>b</sup>Γj, <sup>d</sup>1, <sup>d</sup><sup>2</sup>

<sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � 1 and the condition <sup>j</sup> <sup>≥</sup> <sup>j</sup>

matrices of the BSWI axial rod are computed as:

3.2. Euler Bernoulli beam wavelet finite element

In order for one to obtain the stiffness matrix in physical space, the element properties and

The load vector containing the axial point loads of the WFE in physical space is obtained as:

T<sup>w</sup> r � �<sup>T</sup> Φ<sup>j</sup>

<sup>f</sup> <sup>d</sup>ð Þ <sup>ξ</sup> <sup>T</sup><sup>w</sup> r � �<sup>T</sup> Φ<sup>j</sup>

When applying the Daubechies wavelet family, the WFE has a total of <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>L</sup> � 2 DOFs. The wavelet space stiffness matrix is evaluated from the multiscale two-term connection

D

<sup>2</sup> � <sup>L</sup> <sup>≤</sup> k, l <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1. Similarly, the distributed forces acting on the element require the form <sup>Υ</sup>j,m

for limits 2 � <sup>L</sup> <sup>≤</sup> k, l <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1 of connection coefficients and the value of <sup>m</sup> depends on the order of the function f <sup>d</sup>ð Þx of the forces. In the case of the BSWI formulations, the total DOFs is

r, e h i

� � are applied to the wavelet space stiffness matrix in Eq. (47).

ð Þ <sup>n</sup>� <sup>n</sup> kw r, e h i

> <sup>z</sup>ð Þ ξ<sup>i</sup> � �<sup>T</sup>

> > <sup>z</sup>ð Þ <sup>ξ</sup> � �<sup>T</sup>

ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup> r

<sup>z</sup> ð Þ <sup>ξ</sup> � �d<sup>ξ</sup> (47)

� � (48)

f xi (49)

dξ (50)

k,l for the limits

k

<sup>¼</sup> 22<sup>j</sup> <sup>Γ</sup>j, <sup>1</sup>, <sup>1</sup> � � (51)

<sup>m</sup>ð Þ <sup>ξ</sup> � �d<sup>ξ</sup> (52)

<sup>0</sup> must be satisfied. Therefore, the wavelet space stiffness

$$\begin{aligned} v(\xi) &= \sum\_{k=h}^{2^j - 1} b\_{z,k}^j \phi\_{z,k}^j(\xi) \\ \theta(\xi) &= \frac{\partial v(\xi)}{\partial \mathbf{x}} = \frac{1}{L\_t} \sum\_{k=h}^{2^j - 1} b\_{z,k}^j \frac{\partial \phi\_{z,k}^j(\xi)}{\partial \xi} \end{aligned} \tag{53}$$

Therefore, the DOFs present within the entire beam element can be represented as

$$\mathbf{r}\_{(n\times1)}\{\mathbf{v}\_{\mathbf{e}}\} = {}\_{(n\times n)}\left[\mathbf{R}^{w}\_{\mathbf{b}}\right]\_{(n\times1)}\{\mathbf{b}\_{\mathbf{e}}\} \tag{54}$$

R<sup>w</sup> b � � <sup>¼</sup> <sup>Φ</sup><sup>j</sup> <sup>z</sup>ð Þ ξ<sup>1</sup> � � <sup>1</sup> Le <sup>Φ</sup>0<sup>j</sup> <sup>z</sup> ð Þ ξ<sup>1</sup> � � Φ<sup>j</sup> <sup>z</sup>ð Þ ξ<sup>2</sup> � � ⋯ Φ<sup>j</sup> <sup>z</sup>ð Þ ξ<sup>r</sup>�<sup>1</sup> � � Φ<sup>j</sup> <sup>z</sup>ð Þ ξ<sup>r</sup> � � <sup>1</sup> Le <sup>Φ</sup>0<sup>j</sup> <sup>z</sup> ð Þ ξ<sup>r</sup> h � � i<sup>T</sup> and vector f g be contains the unknown wavelet coefficients b j z, <sup>k</sup> representing the beam wavelet space DOFs.

From Eq. (54), the transverse displacement and rotation at any point of the beam element can be expressed as:

$$\boldsymbol{\sigma}(\boldsymbol{\xi}) = {}\_{\left(1 \times n\right)} \left\{ \boldsymbol{\Phi}\_{\boldsymbol{z}}^{j}(\boldsymbol{\xi}) \right\}\_{\left(n \times n\right)} \left[ \boldsymbol{T}\_{\boldsymbol{b}}^{w} \right]\_{\left(n \times 1\right)} \left\{ \boldsymbol{v}\_{\boldsymbol{e}} \right\}$$

$$\boldsymbol{\Theta}(\boldsymbol{\xi}) = \frac{1}{L\_{\boldsymbol{e}}} {}\_{\left(1 \times n\right)} \left\{ \boldsymbol{\Phi}\_{\boldsymbol{z}}^{j}(\boldsymbol{\xi}) \right\}\_{\left(n \times n\right)} \left[ \boldsymbol{T}\_{\boldsymbol{b}}^{w} \right]\_{\left(n \times 1\right)} \left\{ \boldsymbol{v}\_{\boldsymbol{e}} \right\} \tag{55}$$

where T<sup>w</sup> b � � <sup>¼</sup> <sup>R</sup><sup>w</sup> b � ��<sup>1</sup> is the beam wavelet transformation matrix which is used to obtain the wavelet based shape functions for the beam Nb, <sup>e</sup>ð Þ <sup>ξ</sup> � � <sup>¼</sup> <sup>Φ</sup><sup>j</sup> <sup>z</sup>ð Þ <sup>ξ</sup> � � <sup>T</sup><sup>w</sup> b � �. The potential energy <sup>Π</sup><sup>b</sup> within a Euler Bernoulli beam subjected to concentrated forces <sup>f</sup> yi, distributed force <sup>f</sup> <sup>d</sup>ð Þ<sup>x</sup> and bending moments m <sup>i</sup> can be generally expressed as [40]:

$$
\delta I \Gamma^{\emptyset} = \int\_0^l \frac{EI}{2} \left( \frac{d^2 v}{d\mathbf{x}^2} \right)^2 d\mathbf{x} - \sum\_i f\_{yi} v(\mathbf{x}\_i) - \int\_0^l f\_d(\mathbf{x}) v d\mathbf{x} - \sum\_k \acute{m}\_k \frac{dv(\mathbf{x}\_k)}{d\mathbf{x}} \tag{56}
$$

where E is the Young's modulus, I is the moment of inertia and l is the length of the beam. The strain energy U<sup>b</sup> <sup>e</sup> within each beam element of length Le can expressed in terms of the approximation of the transverse displacement via scaling functions as highlighted in Eq. (55).

$$\mathbf{U}\_{\varepsilon}^{\boldsymbol{b}} = \frac{1}{2} \frac{\boldsymbol{E} \, \boldsymbol{I}}{\boldsymbol{L}\_{\varepsilon}^{3}} \left\{ \boldsymbol{\upsilon}\_{\varepsilon} \right\}^{\boldsymbol{T}} \int\_{0}^{1} \left[ \boldsymbol{T}\_{\boldsymbol{b}}^{\boldsymbol{w}} \right]^{\boldsymbol{T}} \left\{ \frac{d^{2} \boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi})}{d\boldsymbol{\xi}^{2}} \right\}^{\boldsymbol{T}} \left\{ \frac{d^{2} \boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi})}{d\boldsymbol{\xi}^{2}} \right\} \left[ \boldsymbol{T}\_{\boldsymbol{b}}^{\boldsymbol{w}} \right] d\boldsymbol{\xi} \, \{\boldsymbol{\upsilon}\_{\varepsilon}\} \tag{57}$$

This gives rise to the beam WFE stiffness matrix in wavelet space

$$\mathbf{1}\_{(n\times n)}\left[\mathbf{k}\_{b,\epsilon}^{w}\right] = \int\_{0}^{1} \left\{\boldsymbol{\Phi}\_{z}^{\prime \eta^{i}}(\boldsymbol{\xi})\right\}^{T} \left\{\boldsymbol{\Phi}\_{z}^{\prime \eta^{i}}(\boldsymbol{\xi})\right\} d\boldsymbol{\xi} \tag{58}$$

In various engineering problems, the loading conditions analysed vary in location and/or magnitude with respect to time, e.g., a train travelling over a track, and this is generally referred to as moving load problems. Assume a moving load of magnitude P travels across a

represented by the function ð Þ¼ x; t Pδð Þ x � x<sup>0</sup> [41]. δð Þx is the Dirac Delta function and x<sup>0</sup> is the distance travelled by the moving load at time t. The potential work of the load at this

<sup>P</sup>δ ξð Þ � <sup>ξ</sup><sup>0</sup> <sup>v</sup>ð Þ <sup>ξ</sup> <sup>d</sup><sup>ξ</sup> <sup>¼</sup> <sup>P</sup> f g ve <sup>T</sup> <sup>T</sup><sup>w</sup>

b � �<sup>T</sup> <sup>t</sup> Φ<sup>j</sup>

Assuming the moving load transverses to a new position ξ<sup>0</sup> within the same WFE, the numerical values of the shape functions, and consequently load vector, will change accordingly. All other WFEs representing the system with no loading present have zero entries within the load vectors at that particular time t. When the moving load is acting on a new WFE, the scaling functions corresponding to the WFE subjected to the moving load are used to obtain

When applying the Daubechies wavelet family of order L at multiresolution j, the total DOFs within a single element is <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>L</sup> � 2 and for this specific layout, the total number of elemental nodes is <sup>r</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>L</sup> � 4 and corresponding elemental segments ns <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>L</sup> � 5. The Daubechies wavelet space stiffness and mass matrices of the Euler Bernoulli beam WFE are

D

D

evaluated the distributed loads and the value of m is based on the load function f <sup>d</sup>ð Þx . For the

b, e

b, e

<sup>2</sup><sup>j</sup> ð Þ <sup>þ</sup> <sup>L</sup>� <sup>2</sup> � <sup>2</sup><sup>j</sup> ð Þ ð Þ <sup>þ</sup> <sup>L</sup>� <sup>2</sup> kw

<sup>2</sup><sup>j</sup> ð Þ <sup>þ</sup> <sup>L</sup>� <sup>2</sup> � <sup>2</sup><sup>j</sup> ð Þ ð Þ <sup>þ</sup> <sup>L</sup>� <sup>2</sup> mw

b � �<sup>T</sup> Φ<sup>j</sup>

<sup>z</sup>ð Þ ξ<sup>0</sup>

<sup>z</sup>ð Þ ξ<sup>0</sup>

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

http://dx.doi.org/10.5772/intechopen.71882

� �<sup>T</sup> (65)

h i<sup>¼</sup> <sup>24</sup><sup>j</sup> <sup>Γ</sup>j,2,<sup>2</sup> � � (66)

h i<sup>¼</sup> <sup>Γ</sup>j,0,<sup>0</sup> � � (67)

<sup>k</sup> for 2 � <sup>L</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1 are used to

� �<sup>T</sup> (64)

�<sup>1</sup> and is

63

beam element, as illustrated in Figure 3, from the left at a constant speed of c m�s

Le in natural coordinates is [1, 30]:

instant at position <sup>ξ</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>0</sup>

Ωb <sup>e</sup>ð Þ¼ ξ<sup>0</sup>

the load vector for that particular element.

ð1 0

Therefore, the element load vector in physical space is evaluated as

f p,p <sup>b</sup>, <sup>e</sup> ð Þt n o <sup>¼</sup> <sup>P</sup> <sup>T</sup><sup>w</sup>

obtained from the connection coefficients and are expressed as:

Correspondingly, the connection coefficients of the form Υj,m

Figure 3. Layout of a beam WFE subjected to a moving point load.

The vector <sup>Φ</sup><sup>00</sup><sup>j</sup> <sup>z</sup>ð Þ ξ n o <sup>¼</sup> <sup>f</sup><sup>00</sup><sup>j</sup> z, <sup>h</sup>ð Þ <sup>ξ</sup> <sup>f</sup><sup>00</sup><sup>j</sup> z, <sup>h</sup>þ<sup>1</sup>ð Þ <sup>ξ</sup> <sup>⋯</sup> <sup>f</sup><sup>00</sup><sup>j</sup> z, 2<sup>j</sup> �2 ð Þ <sup>ξ</sup> <sup>f</sup><sup>00</sup><sup>j</sup> z, 2<sup>j</sup> �1 ð Þ ξ n o contains the second derivative of the scaling functions. Taking into account the material properties of the beam, the wavelet space stiffness matrix is transformed into physical space via the transformation matrix T<sup>w</sup> b � �.

$$\mathbf{E}\_{\left(\boldsymbol{n}\times\boldsymbol{n}\right)}\left[\mathbf{k}\_{\mathbf{b},\mathbf{e}}^{\mathbf{p}}\right] = \frac{E}{L\_{\mathbf{e}}}\frac{I}{\left(\boldsymbol{n}\times\boldsymbol{n}\right)}\left[\mathbf{T}\_{\mathbf{b}}^{\mathbf{w}}\right]^{T}{}\_{\left(\boldsymbol{n}\times\boldsymbol{n}\right)}\left[\mathbf{k}\_{\mathbf{b},\mathbf{e}}^{\mathbf{w}}\right]\_{\left(\boldsymbol{n}\times\boldsymbol{n}\right)}\left[\mathbf{T}\_{\mathbf{b}}^{\mathbf{w}}\right] \tag{59}$$

The transverse kinetic energy of the beam element is expressed as

$$\Lambda^b\_\epsilon = \frac{1}{2} \rho A L\_\epsilon \int\_0^1 \dot{\boldsymbol{v}}(\boldsymbol{\xi})^T \dot{\boldsymbol{v}}(\boldsymbol{\xi}) d\boldsymbol{\xi} \tag{60}$$

where <sup>v</sup>\_ð Þ¼ <sup>ξ</sup> <sup>∂</sup>vð Þ <sup>ξ</sup> <sup>∂</sup><sup>t</sup> , r is the density and A is the cross-sectional area of the beam. Applying the scaling functions to approximate the displacements within the beam, the kinetic energy becomes

$$A\_{\varepsilon}^{b} = \left\{ \dot{\boldsymbol{\sigma}}\_{\varepsilon} \right\}^{T} \frac{1}{2} \rho A L\_{\varepsilon} \int\_{0}^{1} \left[\boldsymbol{T}\_{b}^{w}\right]^{T} \left\{\boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi})\right\}^{T} \left\{\boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi})\right\} \left[\boldsymbol{T}\_{b}^{w}\right] d\boldsymbol{\xi}, \left\{\dot{\boldsymbol{\sigma}}\_{\varepsilon}\right\} \tag{61}$$

The mass matrix in physical space of the Euler Bernoulli beam element, mp b, e h i, can be evaluated as:

$$
\rho \begin{bmatrix} \boldsymbol{m}\_{b,\epsilon}^{p} \end{bmatrix} = \rho A \boldsymbol{L}\_{\epsilon} \begin{bmatrix} \boldsymbol{T}\_{b}^{w} \end{bmatrix}^{T} \int\_{0}^{1} \left\{ \boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi}) \right\}^{T} \left\{ \boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi}) \right\} d\boldsymbol{\xi} \begin{bmatrix} \boldsymbol{T}\_{b}^{w} \end{bmatrix} \tag{62}
$$

The vectors containing the element concentrated point loads, bending moments and equivalent distributed loads in physical space respectively are subsequently evaluated as:

$$f\_{(n\times 1)}\left\{f\_{b,\epsilon}^{\mathfrak{a},p}\right\}=\sum\_{i=1}^{r}\left(\_{(n\times n)}\left[T\_{\mathfrak{b}}^{\mathfrak{w}}\right]^{T}\right)\_{(n\times 1)}\left\{\mathbf{O}\_{z}^{j}(\xi\_{i})\right\}^{T}f\_{yi}$$

$$f\_{(n\times 1)}\left\{f\_{b,\epsilon}^{\mathfrak{w},p}\right\}=\sum\_{k}\left(\_{(n\times n)}\left[T\_{\mathfrak{b}}^{\mathfrak{w}}\right]^{T}\right)\_{(n\times 1)}\left\{\mathbf{O}\_{z}^{j}(\xi\_{k})\right\}^{T}\dot{\mathfrak{m}}\_{k}$$

$$f\_{(n\times 1)}\left\{f\_{b,\epsilon}^{d,p}\right\}=L\_{\epsilon}\int\_{0}^{1}f\_{d}(\xi)\_{(n\times n)}\left[T\_{\mathfrak{b}}^{\mathfrak{w}}\right]^{T}\left\{\mathbf{O}\_{z}^{j}(\xi)\right\}^{T}d\xi\tag{63}$$

In various engineering problems, the loading conditions analysed vary in location and/or magnitude with respect to time, e.g., a train travelling over a track, and this is generally referred to as moving load problems. Assume a moving load of magnitude P travels across a beam element, as illustrated in Figure 3, from the left at a constant speed of c m�s �<sup>1</sup> and is represented by the function ð Þ¼ x; t Pδð Þ x � x<sup>0</sup> [41]. δð Þx is the Dirac Delta function and x<sup>0</sup> is the distance travelled by the moving load at time t. The potential work of the load at this instant at position <sup>ξ</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>0</sup> Le in natural coordinates is [1, 30]:

$$
\Omega\_{\varepsilon}^{b}(\xi\_{0}) = \int\_{0}^{1} P\delta(\xi - \xi\_{0})\upsilon(\xi)d\xi = P\left\{\mathfrak{v}\_{\varepsilon}\right\}^{T}\left[\mathsf{T}\_{b}^{w}\right]^{T}\left\{\mathfrak{O}\_{z}^{t}(\xi\_{0})\right\}^{T} \tag{64}
$$

Therefore, the element load vector in physical space is evaluated as

Ub <sup>e</sup> <sup>¼</sup> <sup>1</sup> 2 E I Le <sup>3</sup> f g ve <sup>T</sup>

<sup>z</sup>ð Þ ξ n o

Λb

<sup>e</sup> <sup>¼</sup> <sup>v</sup>\_ f g<sup>e</sup> <sup>T</sup> <sup>1</sup>

mp b, e h i

ð Þ <sup>n</sup>� <sup>1</sup> <sup>f</sup>

ð Þ <sup>n</sup>� <sup>1</sup> <sup>f</sup>

ð Þ <sup>n</sup>� <sup>1</sup> <sup>f</sup>

n,p b, e n o

m,p b, e n o

d,p b, e n o

2 rALe ð1 0 T<sup>w</sup> b � �<sup>T</sup> Φ<sup>j</sup>

The mass matrix in physical space of the Euler Bernoulli beam element, mp

<sup>¼</sup> <sup>r</sup>ALe <sup>T</sup><sup>w</sup> b � �<sup>T</sup> ð1 0 Φj <sup>z</sup>ð Þ <sup>ξ</sup> � �<sup>T</sup> <sup>Φ</sup><sup>j</sup>

<sup>¼</sup> <sup>f</sup><sup>00</sup><sup>j</sup>

ð Þ <sup>n</sup>� <sup>n</sup> kp b, e h i

The vector <sup>Φ</sup><sup>00</sup><sup>j</sup>

where <sup>v</sup>\_ð Þ¼ <sup>ξ</sup> <sup>∂</sup>vð Þ <sup>ξ</sup>

becomes

ated as:

T<sup>w</sup> b � �. ð1 0 T<sup>w</sup> b � �<sup>T</sup> <sup>d</sup><sup>2</sup>

62 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

This gives rise to the beam WFE stiffness matrix in wavelet space

ð Þ <sup>n</sup>� <sup>n</sup> <sup>k</sup><sup>w</sup> b, e h i ¼ ð1 0 Φ00<sup>j</sup> <sup>z</sup> ð Þ <sup>ξ</sup> � �<sup>T</sup> <sup>Φ</sup>00<sup>j</sup>

z, <sup>h</sup>ð Þ <sup>ξ</sup> <sup>f</sup><sup>00</sup><sup>j</sup>

The transverse kinetic energy of the beam element is expressed as

Λb <sup>e</sup> <sup>¼</sup> <sup>1</sup> 2 rALe ð1 0 <sup>v</sup>\_ð Þ <sup>ξ</sup> <sup>T</sup>

<sup>¼</sup> E I Le

Φj <sup>z</sup>ð Þ ξ dξ<sup>2</sup> � �<sup>T</sup> d<sup>2</sup>

z, <sup>h</sup>þ<sup>1</sup>ð Þ <sup>ξ</sup> <sup>⋯</sup> <sup>f</sup><sup>00</sup><sup>j</sup>

<sup>3</sup> ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup> b � �<sup>T</sup>

derivative of the scaling functions. Taking into account the material properties of the beam, the wavelet space stiffness matrix is transformed into physical space via the transformation matrix

the scaling functions to approximate the displacements within the beam, the kinetic energy

The vectors containing the element concentrated point loads, bending moments and equiva-

ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup> b � �<sup>T</sup>

ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup> b � �<sup>T</sup>

<sup>f</sup> <sup>d</sup>ð Þ <sup>ξ</sup> ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup>

ð Þ <sup>n</sup>�<sup>1</sup> <sup>Φ</sup><sup>j</sup>

ð Þ <sup>n</sup>�<sup>1</sup> <sup>Φ</sup>0<sup>j</sup>

b � �<sup>T</sup> Φ<sup>j</sup>

lent distributed loads in physical space respectively are subsequently evaluated as:

<sup>¼</sup> <sup>X</sup><sup>r</sup> i¼1

<sup>¼</sup> <sup>X</sup> k

¼ Le ð1 0

n o

z, 2<sup>j</sup> �2

ð Þ <sup>n</sup>� <sup>n</sup> <sup>k</sup><sup>w</sup> b, e h i

<sup>∂</sup><sup>t</sup> , r is the density and A is the cross-sectional area of the beam. Applying

<sup>z</sup>ð Þ <sup>ξ</sup> � � <sup>T</sup><sup>w</sup>

<sup>z</sup>ð Þ <sup>ξ</sup> � �d<sup>ξ</sup> <sup>T</sup><sup>w</sup>

<sup>z</sup>ð Þ ξ<sup>i</sup> � �<sup>T</sup>

<sup>z</sup> ð Þ ξ<sup>k</sup> � �<sup>T</sup>

<sup>z</sup>ð Þ <sup>ξ</sup> � �<sup>T</sup>

b

f yi

m� k

b

<sup>z</sup>ð Þ <sup>ξ</sup> � �<sup>T</sup> <sup>Φ</sup><sup>j</sup>

Φj <sup>z</sup>ð Þ ξ dξ<sup>2</sup> � �

> ð Þ <sup>ξ</sup> <sup>f</sup><sup>00</sup><sup>j</sup> z, 2<sup>j</sup> �1 ð Þ ξ

> > ð Þ <sup>n</sup>�<sup>n</sup> <sup>T</sup><sup>w</sup> b

T<sup>w</sup> b

<sup>z</sup> ð Þ <sup>ξ</sup> � �d<sup>ξ</sup> (58)

v\_ð Þ ξ dξ (60)

� �dξf g ve (57)

contains the second

� � (59)

� �d<sup>ξ</sup> <sup>v</sup>\_ f g<sup>e</sup> (61)

, can be evalu-

b, e h i

� � (62)

dξ (63)

$$\left\{ f\_{b,\epsilon}^{p,p}(t) \right\} = P \left[ T\_b^w \right]^T \left\{ \Phi\_z^j(\xi\_0) \right\}^T \tag{65}$$

Assuming the moving load transverses to a new position ξ<sup>0</sup> within the same WFE, the numerical values of the shape functions, and consequently load vector, will change accordingly. All other WFEs representing the system with no loading present have zero entries within the load vectors at that particular time t. When the moving load is acting on a new WFE, the scaling functions corresponding to the WFE subjected to the moving load are used to obtain the load vector for that particular element.

When applying the Daubechies wavelet family of order L at multiresolution j, the total DOFs within a single element is <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>L</sup> � 2 and for this specific layout, the total number of elemental nodes is <sup>r</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>L</sup> � 4 and corresponding elemental segments ns <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>L</sup> � 5. The Daubechies wavelet space stiffness and mass matrices of the Euler Bernoulli beam WFE are obtained from the connection coefficients and are expressed as:

$$\mathbb{E}\left(\left(2^{j}+L-2\right)\times\left(2^{j}+L-2\right)\right)^{D}\left[\mathbf{k}\_{b,\epsilon}^{w}\right]=2^{4j}\left[\Gamma^{j,2,2}\right]\tag{66}$$

$$\mathbb{E}\left(\left(2^{j}+L-2\right)\times\left(2^{j}+L-2\right)\right)\left[\mathfrak{m}\_{b,\mathfrak{e}}^{\mathfrak{w}}\right]=\left[\Gamma^{j,0,0}\right] \tag{67}$$

Correspondingly, the connection coefficients of the form Υj,m <sup>k</sup> for 2 � <sup>L</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> <sup>2</sup><sup>j</sup> � 1 are used to evaluated the distributed loads and the value of m is based on the load function f <sup>d</sup>ð Þx . For the

Figure 3. Layout of a beam WFE subjected to a moving point load.

BSWI family of order <sup>m</sup> and at scale <sup>j</sup>, there are <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � 1 total DOFs, <sup>r</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � <sup>3</sup> elemental nodes and ns <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � 4 elemental segments within the each WFE for this layout. The stiffness and mass matrices in wavelet space can be evaluated directly and are obtained as:

$$\mathbb{E}\left(\left(2^{j}+m-1\right)\times\left(2^{j}+m-1\right)\right)\left[\mathbb{A}\_{b,\epsilon}^{\text{uv}}\right]=\int\_{0}^{1}\left\{\boldsymbol{\Phi}\_{m}^{\eta j}(\boldsymbol{\xi})\right\}^{T}\left\{\boldsymbol{\Phi}\_{m}^{\eta j}(\boldsymbol{\xi})\right\}d\boldsymbol{\xi}\tag{68}$$

exponent, n. Pratio is the ratio of the upper and lower surface material properties Pu and Plo

The beam WFE has axial deformation ui and transverse deflection vi DOFs at all elemental nodes and rotation θ<sup>i</sup> DOFs only present at elemental end nodes with corresponding axial forces f xi, transverse forces f yi and bending moments θ<sup>i</sup> as illustrated in Figure 4(c). The wavelet scaling functions are implemented as interpolating functions and the axial deformation, deflection and rotation at any point of the beam element are described by Eqs. (42) and (53) respectively. However, in order to ensure that the defined DOFs are positioned correctly, the layout of the element determines the order of scaling functions selected. In this case, the order of the scaling functions selected to approximate the axial displacement is z � 2 if the scaling function order approximating the bending DOFs is z. The vector containing the total number of DOFs, s, present in the functionally graded beam element is f g he ¼

f g <sup>u</sup><sup>1</sup> <sup>v</sup><sup>1</sup> <sup>θ</sup><sup>1</sup> <sup>u</sup><sup>2</sup> <sup>v</sup><sup>2</sup> <sup>u</sup><sup>3</sup> <sup>v</sup><sup>3</sup> <sup>⋯</sup> ur�<sup>1</sup> vr�<sup>1</sup> ur vr <sup>θ</sup><sup>r</sup> <sup>T</sup> and subsequently

ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup><sup>j</sup>

ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup><sup>j</sup>

∂vð Þ ξ <sup>∂</sup><sup>ξ</sup> <sup>¼</sup> <sup>1</sup> Le t ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup>0<sup>j</sup>

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o

<sup>z</sup>ð Þ <sup>ξ</sup> � �ð Þ <sup>s</sup>�<sup>1</sup> f g <sup>c</sup><sup>e</sup>

<sup>z</sup>�2, <sup>h</sup>þ<sup>1</sup>ð Þ <sup>ξ</sup> <sup>0</sup> <sup>⋯</sup> <sup>0</sup> <sup>f</sup><sup>j</sup>

n o

n o

p h i

p h i

> p h i

ð Þ <sup>s</sup>�<sup>1</sup> f g he

ð Þ <sup>s</sup>�<sup>1</sup> f g he

ð Þ <sup>ξ</sup> 0 0 n o

z,iþ<sup>1</sup>ð Þ <sup>ξ</sup> <sup>0</sup> <sup>⋯</sup> <sup>0</sup> <sup>f</sup><sup>j</sup>

z,iþ<sup>1</sup>ð Þ <sup>ξ</sup> <sup>0</sup> <sup>⋯</sup> <sup>0</sup> <sup>f</sup><sup>0</sup><sup>j</sup>

p h i

<sup>z</sup>ð Þ <sup>ξ</sup> � �ð Þ <sup>s</sup>�<sup>s</sup> <sup>T</sup><sup>w</sup>

<sup>z</sup>ð Þ <sup>ξ</sup> � �ð Þ <sup>s</sup>�<sup>s</sup> <sup>T</sup><sup>w</sup>

<sup>z</sup> ð Þ <sup>ξ</sup> � �ð Þ <sup>s</sup>�<sup>s</sup> <sup>T</sup><sup>w</sup>

p h i�<sup>1</sup> ð Þ <sup>s</sup>�<sup>1</sup> f g <sup>c</sup><sup>e</sup>

<sup>z</sup> ð Þ <sup>ξ</sup> � �ð Þ <sup>s</sup>�<sup>1</sup> f g <sup>c</sup><sup>e</sup> (71)

<sup>z</sup>�2, <sup>2</sup><sup>j</sup> �1

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

http://dx.doi.org/10.5772/intechopen.71882

65

ð Þ <sup>ξ</sup> <sup>f</sup><sup>j</sup> z,2<sup>j</sup> �1 ð Þ ξ

(72)

ð Þ <sup>ξ</sup> <sup>f</sup><sup>0</sup><sup>j</sup> z,2<sup>j</sup> �1 ð Þ ξ

<sup>s</sup>�<sup>1</sup>f g ce (73)

ð Þ <sup>s</sup>�<sup>1</sup> f g he (74)

. The strain energy of the functionally

z,2<sup>j</sup> �2

z,2<sup>j</sup> �2

<sup>u</sup>ð Þ¼ <sup>ξ</sup> <sup>a</sup>

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup> Le

<sup>z</sup>�2,hð Þ <sup>ξ</sup> 0 0 <sup>f</sup><sup>j</sup>

z,i ð Þ <sup>ξ</sup> <sup>f</sup><sup>j</sup>

z,i

<sup>u</sup>ð Þ¼ <sup>ξ</sup> <sup>a</sup>

<sup>v</sup>ð Þ¼ <sup>ξ</sup> <sup>t</sup>

θ ξð Þ¼ <sup>1</sup> Le t ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup>0<sup>j</sup>

The wavelet transformation matrix T<sup>w</sup>

graded beam element, Ue, is defined as

θ ξð Þ¼ <sup>∂</sup>vð Þ <sup>ξ</sup>

<sup>z</sup>ð Þ <sup>ξ</sup> � �<sup>¼</sup> <sup>0</sup> <sup>f</sup><sup>j</sup>

<sup>z</sup> ð Þ <sup>ξ</sup> � �<sup>¼</sup> <sup>0</sup> <sup>f</sup><sup>0</sup><sup>j</sup>

<sup>v</sup>ð Þ¼ <sup>ξ</sup> <sup>t</sup>

where the vector ce f g contains the unknown wavelet space element DOFs and

ð Þ <sup>ξ</sup> <sup>f</sup><sup>0</sup><sup>j</sup>

Therefore, the DOFs present within the entire beam element can be represented as

ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup><sup>j</sup>

ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup><sup>j</sup>

p h i <sup>¼</sup> Rw

<sup>s</sup>�<sup>1</sup>f g¼ he <sup>s</sup>�<sup>s</sup> <sup>R</sup><sup>w</sup>

respectively.

a <sup>1</sup>�<sup>s</sup> <sup>Φ</sup><sup>j</sup>

and consequently

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o<sup>¼</sup> <sup>f</sup><sup>j</sup>

> t <sup>1</sup>�<sup>s</sup> <sup>Φ</sup><sup>j</sup>

t <sup>1</sup>�<sup>s</sup> <sup>Φ</sup>0<sup>j</sup>

$$\left( \left( 2^{j} + m - 1 \right) \times \left( 2^{j} + m - 1 \right) \right) \left[ m\_{b, \epsilon}^{w} \right] = \int\_{0}^{1} \left\{ \boldsymbol{\Phi}\_{m}^{j} (\boldsymbol{\xi}) \right\}^{T} \left\{ \boldsymbol{\Phi}\_{m}^{j} (\boldsymbol{\xi}) \right\} d\boldsymbol{\xi} \tag{69}$$

### 3.3. Transversely varying functionally graded Euler Bernoulli beam wavelet finite element

Functionally graded materials are a recent evolution of composite materials where the material constituents, hence properties, vary continuously in the desired spatial directions. The need for such revolutionary materials arose to overcome limitations of conventional composite materials, for instance, desirable properties would diminished when applied to highly intense thermal environments or material debonding due to increased stress concentration at material interfaces [42]. In the formulation of the wavelet based functionally grade beam as presented in Figure 4(a), of height h, length l and width b, the material distribution is modelled based on the power law of transverse gradation [43]

$$P(y) = P\_{lo} \left( [P\_{ratio} - 1] \left( \frac{y}{h} + \frac{1}{2} \right)^n + 1 \right) \tag{70}$$

As illustrated in Figure 4(b), the transverse variation of the effective material properties P(y) (Young's modulus) can be infinitely altered via the non-negative volume fraction power law

Figure 4. (a) Cross-section of transversely varying functionally graded beam. (b) Effective Young's modulus variation of steel-alumina functionally graded beam for different n. (c) Functionally graded beam layout.

exponent, n. Pratio is the ratio of the upper and lower surface material properties Pu and Plo respectively.

The beam WFE has axial deformation ui and transverse deflection vi DOFs at all elemental nodes and rotation θ<sup>i</sup> DOFs only present at elemental end nodes with corresponding axial forces f xi, transverse forces f yi and bending moments θ<sup>i</sup> as illustrated in Figure 4(c). The wavelet scaling functions are implemented as interpolating functions and the axial deformation, deflection and rotation at any point of the beam element are described by Eqs. (42) and (53) respectively. However, in order to ensure that the defined DOFs are positioned correctly, the layout of the element determines the order of scaling functions selected. In this case, the order of the scaling functions selected to approximate the axial displacement is z � 2 if the scaling function order approximating the bending DOFs is z. The vector containing the total number of DOFs, s, present in the functionally graded beam element is f g he ¼ f g <sup>u</sup><sup>1</sup> <sup>v</sup><sup>1</sup> <sup>θ</sup><sup>1</sup> <sup>u</sup><sup>2</sup> <sup>v</sup><sup>2</sup> <sup>u</sup><sup>3</sup> <sup>v</sup><sup>3</sup> <sup>⋯</sup> ur�<sup>1</sup> vr�<sup>1</sup> ur vr <sup>θ</sup><sup>r</sup> <sup>T</sup> and subsequently

<sup>u</sup>ð Þ¼ <sup>ξ</sup> <sup>a</sup> ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup><sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n oð Þ <sup>s</sup>�<sup>1</sup> f g <sup>c</sup><sup>e</sup> <sup>v</sup>ð Þ¼ <sup>ξ</sup> <sup>t</sup> ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup><sup>j</sup> <sup>z</sup>ð Þ <sup>ξ</sup> � �ð Þ <sup>s</sup>�<sup>1</sup> f g <sup>c</sup><sup>e</sup> θ ξð Þ¼ <sup>∂</sup>vð Þ <sup>ξ</sup> <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup> Le ∂vð Þ ξ <sup>∂</sup><sup>ξ</sup> <sup>¼</sup> <sup>1</sup> Le t ð Þ <sup>1</sup>�<sup>s</sup> <sup>Φ</sup>0<sup>j</sup> <sup>z</sup> ð Þ <sup>ξ</sup> � �ð Þ <sup>s</sup>�<sup>1</sup> f g <sup>c</sup><sup>e</sup> (71)

where the vector ce f g contains the unknown wavelet space element DOFs and

$$\begin{aligned} \label{eq:Wilson\_{\tiny{z}}} \quad \left\{ \boldsymbol{\Phi}\_{z-2}^{j}(\boldsymbol{\xi}) \right\} &= \left\{ \boldsymbol{\phi}\_{z-2,h}^{j}(\boldsymbol{\xi}) \quad \boldsymbol{0} \quad \boldsymbol{0} \quad \boldsymbol{\phi}\_{z-2,h+1}^{j}(\boldsymbol{\xi}) \quad \boldsymbol{0} \quad \cdots \quad \boldsymbol{0} \quad \boldsymbol{\phi}\_{z-2,\boldsymbol{2}'-1}^{j}(\boldsymbol{\xi}) \quad \boldsymbol{0} \quad \boldsymbol{0} \right\} \\ \boldsymbol{1}\_{\times s}^{\prime} \Big{(}\boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi}) \Big{)} &= \left\{ \boldsymbol{0} \quad \boldsymbol{\phi}\_{z,i}^{j}(\boldsymbol{\xi}) \quad \boldsymbol{\phi}\_{z,i+1}^{j}(\boldsymbol{\xi}) \quad \boldsymbol{0} \quad \cdots \quad \boldsymbol{0} \quad \boldsymbol{\phi}\_{z,2^{\prime}-2}^{j}(\boldsymbol{\xi}) \quad \boldsymbol{\phi}\_{z,2^{\prime}-1}^{j}(\boldsymbol{\xi}) \right\} \\ \boldsymbol{1}\_{\times s}^{\prime} \Big{(}\boldsymbol{\Phi}\_{z}^{j}(\boldsymbol{\xi}) \Big{)} &= \left\{ \boldsymbol{0} \quad \boldsymbol{\phi}\_{z,i}^{j}(\boldsymbol{\xi}) \quad \boldsymbol{\phi}\_{z,i+1}^{\prime j}(\boldsymbol{\xi}) \quad \boldsymbol{0} \quad \cdots \quad \boldsymbol{0} \quad \boldsymbol{\phi}\_{z,2^{\prime}-2}^{j}(\boldsymbol{\xi}) \quad \boldsymbol{\phi}\_{z,2^{\prime}-1}^{j}(\boldsymbol{\xi}) \end{aligned} \tag{72}$$

Therefore, the DOFs present within the entire beam element can be represented as

$$\mathbf{h}\_{s \times 1} \{ \mathbf{h}\_{\mathbf{e}} \} = \_{s \times s} \begin{bmatrix} \mathbf{R}\_{\mathbf{p}}^{w} \\ \mathbf{p} \end{bmatrix}\_{s \times 1} \{ \mathbf{c}\_{\mathbf{e}} \} \tag{73}$$

and consequently

BSWI family of order <sup>m</sup> and at scale <sup>j</sup>, there are <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � 1 total DOFs, <sup>r</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � <sup>3</sup> elemental nodes and ns <sup>¼</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>m</sup> � 4 elemental segments within the each WFE for this layout. The stiffness and mass matrices in wavelet space can be evaluated directly and are obtained as:

<sup>m</sup>ð Þ <sup>ξ</sup> � �d<sup>ξ</sup> (68)

<sup>m</sup>ð Þ <sup>ξ</sup> � �d<sup>ξ</sup> (69)

(70)

BS

BS

3.3. Transversely varying functionally graded Euler Bernoulli beam wavelet

P yð Þ¼ Plo ½ � Pratio � <sup>1</sup> <sup>y</sup>

b, e h i ¼ ð1 0 Φ00<sup>j</sup> <sup>m</sup>ð Þ <sup>ξ</sup> � �<sup>T</sup> <sup>Φ</sup>00<sup>j</sup>

b, e h i ¼ ð1 0 Φj <sup>m</sup>ð Þ <sup>ξ</sup> � �<sup>T</sup> <sup>Φ</sup><sup>j</sup>

Functionally graded materials are a recent evolution of composite materials where the material constituents, hence properties, vary continuously in the desired spatial directions. The need for such revolutionary materials arose to overcome limitations of conventional composite materials, for instance, desirable properties would diminished when applied to highly intense thermal environments or material debonding due to increased stress concentration at material interfaces [42]. In the formulation of the wavelet based functionally grade beam as presented in Figure 4(a), of height h, length l and width b, the material distribution is modelled based on the

> h þ 1 2 � �<sup>n</sup>

� �

As illustrated in Figure 4(b), the transverse variation of the effective material properties P(y) (Young's modulus) can be infinitely altered via the non-negative volume fraction power law

Figure 4. (a) Cross-section of transversely varying functionally graded beam. (b) Effective Young's modulus variation of

steel-alumina functionally graded beam for different n. (c) Functionally graded beam layout.

þ 1

<sup>2</sup><sup>j</sup> ð Þ <sup>þ</sup> <sup>m</sup>� <sup>1</sup> � <sup>2</sup><sup>j</sup> ð Þ ð Þ <sup>þ</sup> <sup>m</sup>� <sup>1</sup> <sup>k</sup><sup>w</sup>

64 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

<sup>2</sup><sup>j</sup> ð Þ <sup>þ</sup> <sup>m</sup>� <sup>1</sup> � <sup>2</sup><sup>j</sup> ð Þ ð Þ <sup>þ</sup> <sup>m</sup>� <sup>1</sup> <sup>m</sup><sup>w</sup>

finite element

power law of transverse gradation [43]

$$u(\xi) = \begin{matrix} \mu\_{(1\times s)}\left\{\boldsymbol{\Phi}\mathbf{D}\_{z}^{j}(\xi)\right\}\boldsymbol{\}\_{(s\times s)}\left[\boldsymbol{T}\_{\mathcal{p}}^{w}\right]\_{(s\times 1)}\left\{\boldsymbol{h}\_{\varepsilon}\right\} \\ \boldsymbol{v}(\xi) = \boldsymbol{t}\_{(1\times s)}\left\{\boldsymbol{\Phi}\mathbf{D}\_{z}^{j}(\xi)\right\}\boldsymbol{\}\_{(s\times s)}\left[\boldsymbol{T}\_{\mathcal{p}}^{w}\right]\_{(s\times 1)}\left\{\boldsymbol{h}\_{\varepsilon}\right\} \end{matrix}$$

$$\boldsymbol{\Theta}(\xi) = \frac{1}{L\_{\boldsymbol{\varepsilon}}}\boldsymbol{t}\_{(1\times s)}\left\{\boldsymbol{\Phi}\mathbf{D}\_{z}^{j}(\xi)\right\}\boldsymbol{\}\_{(s\times s)}\left[\boldsymbol{T}\_{\mathcal{p}}^{w}\right]\_{(s\times 1)}\left\{\boldsymbol{h}\_{\varepsilon}\right\} \tag{74}$$

The wavelet transformation matrix T<sup>w</sup> p h i <sup>¼</sup> Rw p h i�<sup>1</sup> . The strain energy of the functionally graded beam element, Ue, is defined as

$$\begin{split} \mathcal{U}\_{\varepsilon} &= \frac{b}{2} \Big|\_{-\frac{b}{2}}^{\frac{b}{2}} \int\_{0}^{1} E(\boldsymbol{\eta}) \left[ \frac{1}{L\_{\varepsilon}} \left( \frac{\partial \boldsymbol{u}(\boldsymbol{\xi})}{\partial \boldsymbol{\xi}} \right)^{T} \left( \frac{\partial \boldsymbol{u}(\boldsymbol{\xi})}{\partial \boldsymbol{\xi}} \right) - \frac{\boldsymbol{y}}{L\_{\varepsilon}^{2}} \left( \frac{\partial^{2} \boldsymbol{v}(\boldsymbol{\xi})}{\partial \boldsymbol{\xi}^{2}} \right)^{T} \left( \frac{\partial \boldsymbol{u}(\boldsymbol{\xi})}{\partial \boldsymbol{\xi}} \right) \\ & - \frac{\boldsymbol{y}}{L\_{\varepsilon}^{2}} \left( \frac{\partial \boldsymbol{u}(\boldsymbol{\xi})}{\partial \boldsymbol{\xi}} \right)^{T} \left( \frac{\partial^{2} \boldsymbol{v}(\boldsymbol{\xi})}{\partial \boldsymbol{\xi}^{2}} \right) + \frac{\boldsymbol{y}^{2}}{L\_{\varepsilon}^{3}} \left( \frac{\partial^{2} \boldsymbol{v}(\boldsymbol{\xi})}{\partial \boldsymbol{\xi}^{2}} \right)^{T} \left( \frac{\partial^{2} \boldsymbol{v}(\boldsymbol{\xi})}{\partial \boldsymbol{\xi}^{2}} \right) \Bigg] d\boldsymbol{\xi} \, d\boldsymbol{y} \end{split} \tag{75}$$

<sup>A</sup> r<sup>e</sup> ¼ ð h 2 � h 2

<sup>B</sup>r<sup>e</sup> <sup>¼</sup> ð h 2 � h 2

<sup>C</sup>r<sup>e</sup> <sup>¼</sup> ð h 2 � h 2

A ð Þ <sup>s</sup>�<sup>s</sup> mw e � �<sup>¼</sup>

> B ð Þ <sup>s</sup>�<sup>s</sup> mw e � �<sup>¼</sup>

> C ð Þ <sup>s</sup>�<sup>s</sup> mw e � �<sup>¼</sup>

D ð Þ <sup>s</sup>�<sup>s</sup> <sup>m</sup><sup>w</sup> e � �<sup>¼</sup>

E ð Þ <sup>s</sup>�<sup>s</sup> mw e � �<sup>¼</sup>

4. Numerical examples

P xð Þdx <sup>¼</sup> <sup>1</sup>

E0A<sup>0</sup> Ð q0 x2

u xð Þ¼ <sup>1</sup> EA Ð x o

ð Þ <sup>s</sup>�<sup>s</sup> <sup>m</sup><sup>p</sup> e � �<sup>¼</sup> <sup>A</sup> mp

rð Þy dy ¼

yrð Þy dy ¼

rð Þy dy ¼

The wavelet based physical space elemental mass matrix of the beam, mp

b<sup>A</sup>reLe T<sup>w</sup> p h iTa <sup>Φ</sup><sup>j</sup>

b<sup>B</sup>r<sup>e</sup> T<sup>w</sup> p h iTa <sup>Φ</sup><sup>j</sup>

b<sup>B</sup>r<sup>e</sup> T<sup>w</sup> p h iTt ∂Φ<sup>j</sup>

bCr<sup>e</sup> T<sup>w</sup> p h iTt ∂Φ<sup>j</sup>

e � � � <sup>B</sup> <sup>m</sup><sup>p</sup>

bAreLe T<sup>w</sup> p h iTt <sup>Φ</sup><sup>j</sup>

y2

ð1 0

> ð1 0

> ð1 0

> > ð1 0

> > > ð1 0

ð h 2 � h 2

ð h 2 � h 2

ð h 2 � h 2

r<sup>u</sup> � r<sup>l</sup> � � y

y r<sup>u</sup> � r<sup>l</sup> � � y

> <sup>y</sup><sup>2</sup> <sup>r</sup><sup>u</sup> � <sup>r</sup><sup>l</sup> � � y

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o<sup>T</sup> <sup>a</sup> <sup>Φ</sup><sup>j</sup>

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o<sup>T</sup> <sup>t</sup> ∂Φ<sup>j</sup>

<sup>z</sup>ð Þ ξ ∂ξ � �<sup>T</sup>

<sup>z</sup>ð Þ ξ ∂ξ � �<sup>T</sup>

> <sup>z</sup>ð Þ ξ n o<sup>T</sup> <sup>t</sup> <sup>Φ</sup><sup>j</sup>

> > e � � <sup>þ</sup> <sup>D</sup> mp

e � � � <sup>C</sup> mp

Example 1: A uniform axial cantilever rod (free-fixed) subjected to linear varying load q xð Þ¼�q<sup>0</sup> x has a uniform cross sectional area, A ¼ A0, Young's Modulus, E ¼ E<sup>0</sup> and length l. The exact solution for displacement at a particular point x can be obtained by solving

Daubechies and BSWI WFEM approaches and the results are compared with the exact, h-FEM and p-FEM formulations. The governing equation of the system for FEM and WFEM is

where Kr ½ � is the system stiffness matrix, f g Ur is the system vector containing the DOFs and f g Fr is the loading vector of the system. The axial deformation of the rod is analysed at the arbitrary point 0.1l and the rate of convergence of the different approaches is compared in Figure 5. The plot shows the absolute relative error of the axial deformation and corresponding number of DOFs. The FEM (h-FEM) solution involves increasing the number of elements, p-FEM involves increasing the order of the polynomials (one element only) and

h þ 1 2 � �<sup>n</sup>

h þ 1 2 � �<sup>n</sup>

� �dy

h þ 1 2 � �<sup>n</sup>

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o <sup>T</sup><sup>w</sup>

<sup>z</sup>ð Þ ξ ∂ξ � � <sup>T</sup><sup>w</sup>

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o <sup>T</sup><sup>w</sup>

<sup>z</sup>ð Þ ξ n o <sup>T</sup><sup>w</sup>

> e � � <sup>þ</sup> <sup>E</sup> <sup>m</sup><sup>p</sup>

<sup>2</sup> dx [40]. One WFE is used to represent the rod using

Kr ½ �f g Ur ¼ f g Fr (81)

<sup>a</sup> Φ<sup>j</sup>

<sup>t</sup> ∂Φ<sup>j</sup> <sup>z</sup>ð Þ ξ ∂ξ � � <sup>T</sup><sup>w</sup>

þ rldy

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

http://dx.doi.org/10.5772/intechopen.71882

67

þ r<sup>l</sup>

þ r<sup>l</sup> � �dy (79)

> e � �, is

p h i<sup>T</sup> dξ

p h i<sup>T</sup> dξ

p h i<sup>T</sup> dξ

p h i<sup>T</sup> dξ

p h i<sup>T</sup> dξ

e

� � (80)

where Le is the length of the element and E yð Þ the effective Young's modulus obtained from Eq. (70). Let

$$\begin{aligned} \,^A E\_e &= \begin{bmatrix} \,^h & h\\ -\,^h & E(y)dy = \int \frac{\hbar}{2} \left[ E\_u - E\_l \right] \left( \frac{y}{h} + \frac{1}{2} \right)^n + E\_l dy\\ -\frac{h}{2} & \frac{h}{2} \end{bmatrix} \\ \,^B E\_e &= \begin{bmatrix} \,^h & h\\ -\frac{h}{2} & \frac{h}{2} \end{bmatrix} \,^B \begin{pmatrix} \left[ E\_u - E\_l \right] \left( \frac{y}{h} + \frac{1}{2} \right)^n + E\_l \end{pmatrix} \, dy \\ \,^C E\_e &= \begin{bmatrix} \,^h & h\\ -\frac{h}{2} & \frac{h}{2} \end{bmatrix} \,^B \begin{pmatrix} y^2 [E\_u - E\_l] \left( \frac{y}{h} + \frac{1}{2} \right)^n + E\_l \end{pmatrix} \, dy \end{aligned} \tag{76}$$

AEe, BEe and CEe denote axial, axial-bending coupling and bending stiffness of the WFE respectively. The wavelet based physical space elemental stiffness matrix of the beam, k<sup>w</sup> e � �, is

A ð Þ <sup>s</sup>�<sup>s</sup> <sup>k</sup><sup>w</sup> e � �<sup>¼</sup> ð1 0 bAEe Le T<sup>w</sup> p h iTa <sup>∂</sup>Φ<sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> ∂ξ ( )<sup>T</sup> <sup>a</sup> <sup>∂</sup>Φ<sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> ∂ξ ( ) T<sup>w</sup> p h id<sup>ξ</sup> B ð Þ <sup>s</sup>�<sup>s</sup> <sup>k</sup><sup>w</sup> e � �<sup>¼</sup> ð1 0 bBEe Le <sup>2</sup> <sup>T</sup><sup>w</sup> p h iTt ∂<sup>2</sup> Φj <sup>z</sup>ð Þ ξ ∂ξ<sup>2</sup> � �<sup>T</sup> <sup>a</sup> <sup>∂</sup>Φ<sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> ∂ξ ( ) T<sup>w</sup> p h i <sup>d</sup><sup>ξ</sup> C ð Þ <sup>s</sup>�<sup>s</sup> <sup>k</sup><sup>w</sup> e � �<sup>¼</sup> ð1 0 bBEe Le <sup>2</sup> <sup>T</sup><sup>w</sup> p h iTa <sup>∂</sup>Φ<sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> ∂ξ ( )<sup>T</sup> <sup>t</sup> ∂<sup>2</sup> Φj <sup>z</sup>ð Þ ξ ∂ξ<sup>2</sup> � � <sup>T</sup><sup>w</sup> p h i <sup>d</sup><sup>ξ</sup> D ð Þ <sup>s</sup>�<sup>s</sup> kw e � �<sup>¼</sup> ð1 0 bCEe Le <sup>3</sup> <sup>T</sup><sup>w</sup> p h iTt ∂<sup>2</sup> Φj <sup>z</sup>ð Þ ξ ∂ξ<sup>2</sup> � �<sup>T</sup> <sup>t</sup> ∂<sup>2</sup> Φj <sup>z</sup>ð Þ ξ ∂ξ<sup>2</sup> � � <sup>T</sup><sup>w</sup> p h i <sup>d</sup><sup>ξ</sup> ð Þ <sup>s</sup>�<sup>s</sup> <sup>k</sup><sup>p</sup> e � �<sup>¼</sup> <sup>A</sup> kp e � � � <sup>B</sup> <sup>k</sup><sup>p</sup> e � � � <sup>C</sup> kp e � � <sup>þ</sup> <sup>D</sup> kp e � � (77)

The kinetic energy of the functionally graded beam element, Λe, is defined as

$$\begin{split} \Lambda\_{\varepsilon} = \frac{1}{2} \int\_{0}^{b} dz \int\_{-\frac{b}{2}}^{\frac{b}{2}} \int\_{0}^{1} \rho(y) \left( L\_{\varepsilon}(\dot{u}(\xi,t)\dot{u}(\xi,t)) - y \left( \dot{u}(\xi,t) \frac{\partial \dot{v}(\xi,t)}{\partial \mathbf{x}} \right) - y \left( \frac{\partial \dot{v}(\xi,t)}{\partial \xi} \dot{u}(\xi,t) \right) \right. \\ \left. + \frac{y^{2}}{L\_{\varepsilon}} \left( \frac{\partial \dot{v}(\xi,t)}{\partial \mathbf{x}} \frac{\partial \dot{v}(\xi,t)}{\partial \mathbf{x}} \right) + L\_{\varepsilon}(\dot{v}(\xi,t)\dot{v}(\xi,t)) \right) d\xi \, dy \end{split} \tag{78}$$

rð Þy is the effective density also obtained from Eq. (70). Let the inertial coefficients be denoted as:

Multiscale Wavelet Finite Element Analysis in Structural Dynamics http://dx.doi.org/10.5772/intechopen.71882 67

$$\begin{aligned} \prescript{A}{}{\rho}\_{\epsilon} &= \begin{cases} \frac{h}{2} \\ \frac{h}{2} \rho(y) dy = \int\_{-h}^{\overline{\epsilon}} \frac{1}{2} \left[ \rho\_{u} - \rho\_{l} \right] \left( \frac{y}{h} + \frac{1}{2} \right)^{n} + \rho\_{l} dy \\ \frac{h}{2} \rho\_{\epsilon}(y) dy = \int\_{-\frac{h}{2}}^{\overline{\epsilon}} y \left( \left[ \rho\_{u} - \rho\_{l} \right] \left( \frac{y}{h} + \frac{1}{2} \right)^{n} + \rho\_{l} \right) dy \\ \frac{h}{2} \rho\_{\epsilon} = \int\_{-\frac{h}{2}}^{h} y^{2} \rho(y) dy = \int\_{-\frac{h}{2}}^{\overline{\epsilon}} \left( y^{2} \left[ \rho\_{u} - \rho\_{l} \right] \left( \frac{y}{h} + \frac{1}{2} \right)^{n} + \rho\_{l} \right) dy \end{aligned} \tag{79}$$

The wavelet based physical space elemental mass matrix of the beam, mp e � �, is

A ð Þ <sup>s</sup>�<sup>s</sup> mw e � �<sup>¼</sup> ð1 0 b<sup>A</sup>reLe T<sup>w</sup> p h iTa <sup>Φ</sup><sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o<sup>T</sup> <sup>a</sup> <sup>Φ</sup><sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o <sup>T</sup><sup>w</sup> p h i<sup>T</sup> dξ B ð Þ <sup>s</sup>�<sup>s</sup> mw e � �<sup>¼</sup> ð1 0 b<sup>B</sup>r<sup>e</sup> T<sup>w</sup> p h iTa <sup>Φ</sup><sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o<sup>T</sup> <sup>t</sup> ∂Φ<sup>j</sup> <sup>z</sup>ð Þ ξ ∂ξ � � <sup>T</sup><sup>w</sup> p h i<sup>T</sup> dξ C ð Þ <sup>s</sup>�<sup>s</sup> mw e � �<sup>¼</sup> ð1 0 b<sup>B</sup>r<sup>e</sup> T<sup>w</sup> p h iTt ∂Φ<sup>j</sup> <sup>z</sup>ð Þ ξ ∂ξ � �<sup>T</sup> <sup>a</sup> Φ<sup>j</sup> <sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> n o <sup>T</sup><sup>w</sup> p h i<sup>T</sup> dξ D ð Þ <sup>s</sup>�<sup>s</sup> <sup>m</sup><sup>w</sup> e � �<sup>¼</sup> ð1 0 bCr<sup>e</sup> T<sup>w</sup> p h iTt ∂Φ<sup>j</sup> <sup>z</sup>ð Þ ξ ∂ξ � �<sup>T</sup> <sup>t</sup> ∂Φ<sup>j</sup> <sup>z</sup>ð Þ ξ ∂ξ � � <sup>T</sup><sup>w</sup> p h i<sup>T</sup> dξ E ð Þ <sup>s</sup>�<sup>s</sup> mw e � �<sup>¼</sup> ð1 0 bAreLe T<sup>w</sup> p h iTt <sup>Φ</sup><sup>j</sup> <sup>z</sup>ð Þ ξ n o<sup>T</sup> <sup>t</sup> <sup>Φ</sup><sup>j</sup> <sup>z</sup>ð Þ ξ n o <sup>T</sup><sup>w</sup> p h i<sup>T</sup> dξ ð Þ <sup>s</sup>�<sup>s</sup> <sup>m</sup><sup>p</sup> e � �<sup>¼</sup> <sup>A</sup> mp e � � � <sup>B</sup> <sup>m</sup><sup>p</sup> e � � � <sup>C</sup> mp e � � <sup>þ</sup> <sup>D</sup> mp e � � <sup>þ</sup> <sup>E</sup> <sup>m</sup><sup>p</sup> e � � (80)

### 4. Numerical examples

Ue <sup>¼</sup> <sup>b</sup> 2 ðh 2 �h 2

Eq. (70). Let

ð1 0

� y Le 2

BEe <sup>¼</sup> ð h 2 � h 2

CEe <sup>¼</sup> ð h 2 � h 2

A ð Þ <sup>s</sup>�<sup>s</sup> <sup>k</sup><sup>w</sup> e � �<sup>¼</sup>

B ð Þ <sup>s</sup>�<sup>s</sup> <sup>k</sup><sup>w</sup> e � �<sup>¼</sup>

C ð Þ <sup>s</sup>�<sup>s</sup> <sup>k</sup><sup>w</sup> e � �<sup>¼</sup>

D ð Þ <sup>s</sup>�<sup>s</sup> kw e � �<sup>¼</sup>

> ð1 0

<sup>Λ</sup><sup>e</sup> <sup>¼</sup> <sup>1</sup> 2 ðb o dz ðh 2 �h 2 E yð Þ <sup>1</sup> Le

66 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

∂uð Þ ξ ∂ξ � �<sup>T</sup> ∂<sup>2</sup>

<sup>A</sup> Ee <sup>¼</sup>

y2

ð1 0

ð1 0

ð1 0

> ð1 0

ð Þ <sup>s</sup>�<sup>s</sup> <sup>k</sup><sup>p</sup> e � �<sup>¼</sup> <sup>A</sup> kp

þ y2 Le

bAEe Le

bBEe Le <sup>2</sup> <sup>T</sup><sup>w</sup> p h iTt ∂<sup>2</sup>

bBEe Le <sup>2</sup> <sup>T</sup><sup>w</sup> p h iTa <sup>∂</sup>Φ<sup>j</sup>

bCEe Le <sup>3</sup> <sup>T</sup><sup>w</sup> p h iTt ∂<sup>2</sup>

T<sup>w</sup> p h iTa <sup>∂</sup>Φ<sup>j</sup>

ð h 2 � h 2

yE yð Þdy ¼

E yð Þdy ¼

∂uð Þ ξ ∂ξ

vð Þ ξ ∂ξ<sup>2</sup> � �

E yð Þdy ¼

ð h 2 � h 2

ð h 2 � h 2

� �<sup>T</sup> <sup>∂</sup>uð Þ <sup>ξ</sup>

∂ξ � �

þ y2 Le 3 ∂2 vð Þ ξ ∂ξ<sup>2</sup> � �<sup>T</sup> ∂<sup>2</sup>

where Le is the length of the element and E yð Þ the effective Young's modulus obtained from

ð h 2 � h 2

AEe, BEe and CEe denote axial, axial-bending coupling and bending stiffness of the WFE respectively. The wavelet based physical space elemental stiffness matrix of the beam, k<sup>w</sup>

> Φj <sup>z</sup>ð Þ ξ ∂ξ<sup>2</sup> � �<sup>T</sup>

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> ∂ξ ( )<sup>T</sup>

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> ∂ξ ( )<sup>T</sup>

Φj <sup>z</sup>ð Þ ξ ∂ξ<sup>2</sup> � �<sup>T</sup>

> e � � � <sup>C</sup> kp

e � � � <sup>B</sup> <sup>k</sup><sup>p</sup>

The kinetic energy of the functionally graded beam element, Λe, is defined as

rð Þy Leðu\_ð Þ ξ; t u\_ð Þ ξ; t Þ � y u\_ð Þ ξ; t

� �

∂v\_ð Þ ξ; t ∂x

rð Þy is the effective density also obtained from Eq. (70). Let the inertial coefficients be denoted as:

∂v\_ð Þ ξ; t ∂x

� y Le 2 ∂2 vð Þ ξ ∂ξ<sup>2</sup>

Eu � El ½ � <sup>y</sup>

y Eu � El ½ � <sup>y</sup>

<sup>y</sup><sup>2</sup> Eu � El ½ � <sup>y</sup>

h þ 1 2 � �<sup>n</sup>

h þ 1 2 � �<sup>n</sup>

h þ 1 2 � �<sup>n</sup>

<sup>a</sup> <sup>∂</sup>Φ<sup>j</sup>

<sup>a</sup> <sup>∂</sup>Φ<sup>j</sup>

<sup>t</sup> ∂<sup>2</sup> Φj <sup>z</sup>ð Þ ξ ∂ξ<sup>2</sup> � �

<sup>t</sup> ∂<sup>2</sup> Φj <sup>z</sup>ð Þ ξ ∂ξ<sup>2</sup> � �

e � � <sup>þ</sup> <sup>D</sup> kp

� �

� � �

∂v\_ð Þ ξ; t ∂x

þ Leð Þ v\_ð Þ ξ; t v\_ð Þ ξ; t

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> ∂ξ ( )

<sup>z</sup>�<sup>2</sup>ð Þ <sup>ξ</sup> ∂ξ ( )

� �

� �

" � �

� �<sup>T</sup> <sup>∂</sup>uð Þ <sup>ξ</sup>

vð Þ ξ ∂ξ<sup>2</sup> � �# ∂ξ

dξ dy

þ Eldy

dy

dy

T<sup>w</sup> p h i dξ

T<sup>w</sup> p h i dξ

T<sup>w</sup> p h i dξ

T<sup>w</sup> p h i dξ

� � (77)

<sup>∂</sup><sup>ξ</sup> <sup>u</sup>\_ð Þ <sup>ξ</sup>; <sup>t</sup>

(78)

∂v\_ð Þ ξ; t

e

� y

� dξ dy

þ El

þ El

(75)

(76)

e � �, is

> Example 1: A uniform axial cantilever rod (free-fixed) subjected to linear varying load q xð Þ¼�q<sup>0</sup> x has a uniform cross sectional area, A ¼ A0, Young's Modulus, E ¼ E<sup>0</sup> and length l. The exact solution for displacement at a particular point x can be obtained by solving u xð Þ¼ <sup>1</sup> EA Ð x o P xð Þdx <sup>¼</sup> <sup>1</sup> E0A<sup>0</sup> Ð q0 x2 <sup>2</sup> dx [40]. One WFE is used to represent the rod using Daubechies and BSWI WFEM approaches and the results are compared with the exact, h-FEM and p-FEM formulations. The governing equation of the system for FEM and WFEM is

$$\left[\mathbf{K}\_{\mathbf{r}}\right]\{\mathbf{U}\_{\mathbf{r}}\} = \{\mathbf{F}\_{\mathbf{r}}\} \tag{81}$$

where Kr ½ � is the system stiffness matrix, f g Ur is the system vector containing the DOFs and f g Fr is the loading vector of the system. The axial deformation of the rod is analysed at the arbitrary point 0.1l and the rate of convergence of the different approaches is compared in Figure 5. The plot shows the absolute relative error of the axial deformation and corresponding number of DOFs. The FEM (h-FEM) solution involves increasing the number of elements, p-FEM involves increasing the order of the polynomials (one element only) and

Figure 5. Comparison of the convergence of the axial deformation at point x = 0.1l.

both Daubechies and BSWI WFEMs have the order and/or multiresolution scale j increased. The results show that although the rates of convergence of all the methods are similar, the WFEM approaches have a slightly improved rate with only one element employed.

Example 2: A simply supported two-stepped beam of length 2l has non-uniform flexural stiffness represented by the unequal cross sections; the bending stiffness of the right and left half is given as E1I<sup>1</sup> ¼ E0I<sup>0</sup> and E2I<sup>2</sup> ¼ 4 E0I<sup>0</sup> respectively. The entire beam is subjected to a uniformly distributed load q(x) = 1. The flexural stiffness function is expressed as [44]:

$$E(\mathbf{x})I(\mathbf{x}) = E\_0 I\_0 \left[ 1 - \gamma \,\hat{H}(\mathbf{x} - \mathbf{x}\_0) \right] \tag{82}$$

of order 9 (p-FEM-9; 2 elements), Daubechies WFEM of order L ¼ 10 and scale j ¼ 1 (D101; 2 elements) and the BSWI WFEM of order m ¼ 3 and scale j ¼ 3 (BSWI33; 2 elements) are selected for comparison with the exact solution governed by Eq. (83). Each approach has a total of 18 DOFs within the beam. The deflection and rotation across the beam is presented in Figure 6(a) and (b) respectively. The percentage errors of the deflections are compared for the different approaches and presented in Figure 6(c). All numerical approaches describe the deflection and rotation across the beams very accurately. However, given that both the Daubechies and BSWI WFEM deflection solutions have a maximum error of 1.28% in comparison to 3.82% from the h-FEM and p-FEM approaches, the WFEMs exhibit better convergence to the exact solution. Furthermore, improved accuracy is attained with fewer elements implemented than the h-FEM and p-FEM and this results in reduced computational time.

Example 3: A steel-alumina functionally graded beam of length l and uniform cross-sectional area <sup>A</sup> <sup>¼</sup> <sup>0</sup>:36 m2 (height <sup>h</sup> <sup>¼</sup> <sup>0</sup>:9 m and width <sup>b</sup> <sup>¼</sup> <sup>0</sup>:4 m) is fully alumina at the upper surface and fully steel at the lower surface with material properties; Eu <sup>¼</sup> <sup>3</sup>:<sup>9</sup> � 1011 Pa, <sup>r</sup><sup>u</sup> <sup>¼</sup> <sup>3</sup>:<sup>96</sup> � 103

r denote the Young's modulus and density respectively. The slenderness ratio for the beam is

=<sup>h</sup> ¼ 100. The free vibration of the steel-alumina beam is analysed for the boundary conditions pinned-pinned (PP), pinned-clamped (PC), clamped-clamped (CC) and clamped-free (CF), for

Figure 6. (a) Deflection and (b) rotation (c) comparison of the deflection percentage error across a simply supported

stepped beam subjected to a uniformly distributed load q(x) = 1.

El

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

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; <sup>r</sup>ratio <sup>¼</sup> <sup>r</sup><sup>u</sup> rl

). E and

69

kg�m�<sup>3</sup> and El <sup>¼</sup> <sup>2</sup>:<sup>1</sup> � 1011 Pa, <sup>r</sup><sup>l</sup> <sup>¼</sup> <sup>7</sup>:<sup>8</sup> � 103 kg�m�<sup>3</sup> respectively (Eratio <sup>¼</sup> Eu

l

where γ ¼ 0:75 is defined as the decrement of discontinuity intensity and satisfies the condition 0 ≤ γ ≤ 1 to ensure positivity of the flexural stiffness. H x b ð Þ � x<sup>0</sup> is the Heaviside function for 0 ≤ x<sup>0</sup> ≤ 2l. The general analytical governing equation is

$$\left\{E\_0 I\_0 \left[1 - \gamma \,\widehat{H}(\mathbf{x} - \mathbf{x}\_0)\right] \boldsymbol{\widehat{v}}\,'(\mathbf{x})\right\}\,' = q(\mathbf{x})\tag{83}$$

The FEM and WFEM governing equation is summarised as:

$$[\mathbf{K}\_b] \{ V\_b \} = \{ F\_b \} \tag{84}$$

The vector f g Vb contains the system DOFs within the entire beam, ½ � Kb is the beam stiffness matrix and f g Fb is the equivalent system load vector. The h-FEM (FEM-8; 8 elements), p-FEM of order 9 (p-FEM-9; 2 elements), Daubechies WFEM of order L ¼ 10 and scale j ¼ 1 (D101; 2 elements) and the BSWI WFEM of order m ¼ 3 and scale j ¼ 3 (BSWI33; 2 elements) are selected for comparison with the exact solution governed by Eq. (83). Each approach has a total of 18 DOFs within the beam. The deflection and rotation across the beam is presented in Figure 6(a) and (b) respectively. The percentage errors of the deflections are compared for the different approaches and presented in Figure 6(c). All numerical approaches describe the deflection and rotation across the beams very accurately. However, given that both the Daubechies and BSWI WFEM deflection solutions have a maximum error of 1.28% in comparison to 3.82% from the h-FEM and p-FEM approaches, the WFEMs exhibit better convergence to the exact solution. Furthermore, improved accuracy is attained with fewer elements implemented than the h-FEM and p-FEM and this results in reduced computational time.

Example 3: A steel-alumina functionally graded beam of length l and uniform cross-sectional area <sup>A</sup> <sup>¼</sup> <sup>0</sup>:36 m2 (height <sup>h</sup> <sup>¼</sup> <sup>0</sup>:9 m and width <sup>b</sup> <sup>¼</sup> <sup>0</sup>:4 m) is fully alumina at the upper surface and fully steel at the lower surface with material properties; Eu <sup>¼</sup> <sup>3</sup>:<sup>9</sup> � 1011 Pa, <sup>r</sup><sup>u</sup> <sup>¼</sup> <sup>3</sup>:<sup>96</sup> � 103 kg�m�<sup>3</sup> and El <sup>¼</sup> <sup>2</sup>:<sup>1</sup> � 1011 Pa, <sup>r</sup><sup>l</sup> <sup>¼</sup> <sup>7</sup>:<sup>8</sup> � 103 kg�m�<sup>3</sup> respectively (Eratio <sup>¼</sup> Eu El ; <sup>r</sup>ratio <sup>¼</sup> <sup>r</sup><sup>u</sup> rl ). E and r denote the Young's modulus and density respectively. The slenderness ratio for the beam is l =<sup>h</sup> ¼ 100. The free vibration of the steel-alumina beam is analysed for the boundary conditions pinned-pinned (PP), pinned-clamped (PC), clamped-clamped (CC) and clamped-free (CF), for

both Daubechies and BSWI WFEMs have the order and/or multiresolution scale j increased. The results show that although the rates of convergence of all the methods are similar, the

Example 2: A simply supported two-stepped beam of length 2l has non-uniform flexural stiffness represented by the unequal cross sections; the bending stiffness of the right and left half is given as E1I<sup>1</sup> ¼ E0I<sup>0</sup> and E2I<sup>2</sup> ¼ 4 E0I<sup>0</sup> respectively. The entire beam is subjected to a

where γ ¼ 0:75 is defined as the decrement of discontinuity intensity and satisfies the condi-

b ð Þ � x<sup>0</sup> h i

The vector f g Vb contains the system DOFs within the entire beam, ½ � Kb is the beam stiffness matrix and f g Fb is the equivalent system load vector. The h-FEM (FEM-8; 8 elements), p-FEM

n o<sup>00</sup>

v 00 ð Þx

b ð Þ � x<sup>0</sup> h i

(82)

b ð Þ � x<sup>0</sup> is the Heaviside function for

¼ q xð Þ (83)

½ � Kb f g Vb ¼ f g Fb (84)

WFEM approaches have a slightly improved rate with only one element employed.

uniformly distributed load q(x) = 1. The flexural stiffness function is expressed as [44]:

E xð ÞI xð Þ¼ E0I<sup>0</sup> 1 � γ H x

tion 0 ≤ γ ≤ 1 to ensure positivity of the flexural stiffness. H x

Figure 5. Comparison of the convergence of the axial deformation at point x = 0.1l.

68 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

The FEM and WFEM governing equation is summarised as:

E0I<sup>0</sup> 1 � γ H x

0 ≤ x<sup>0</sup> ≤ 2l. The general analytical governing equation is

Figure 6. (a) Deflection and (b) rotation (c) comparison of the deflection percentage error across a simply supported stepped beam subjected to a uniformly distributed load q(x) = 1.

different values of n in Eq. (70). The free vibration of the functionally graded beam is governed by [45]

$$\left[ \left[ \mathbf{K} \right] - \omega^2 \mathbf{[M]} \right] \left\{ \mathbf{\acute{U}} \right\} = 0 \tag{85}$$

observed that all approaches give highly accurate results with respect to the reference (BSWI55), particularly for the fundamental frequencies. Furthermore, the BSWI WFEM solution exhibits better levels of accuracy than the Daubechies WFEM and h-FEM solutions for the higher frequencies. Both WFEM solutions achieve high levels of accuracy with the described layout of having the rotation DOFs present at elemental and nodes and using fewer elements that the

Table 1. The non-dimensional frequencies of a steel-alumina FG beam for different transverse varying distributions and

n = 0 n = 0.1 n = 0.5 n = 1 n = 5 n = 10 n = 104

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

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71

BSWI54 9.77473 9.4365 8.65967 8.22977 7.58512 7.41335 7.06849

FEM 10.8604 10.4848 9.62387 9.14808 8.43044 8.238 7.85355 D120 10.8636 10.4877 9.62433 9.14655 8.43009 8.23918 7.8559 BSWI54 10.8597 10.4839 9.62083 9.14322 8.42702 8.23619 7.85305

FEM 6.49138 6.2668 5.75131 5.46615 5.03778 4.92342 4.69417 D120 6.49134 6.26673 5.75084 5.46535 5.03723 4.92316 4.69415 BSWI54 6.49133 6.26671 5.75083 5.46534 5.03722 4.92315 4.69413

FEM 13.0334 12.5826 11.5489 10.9774 10.1166 3.2951 3.33258 D120 13.0461 13.5947 11.5578 10.984 10.1237 3.29504 3.33251 BSWI54 13.0317 12.5808 11.545 10.9719 10.1125 3.29504 3.33251

FEM 14.1201 13.6318 12.5131 11.895 10.9617 10.711 10.2108 D120 14.1444 13.655 12.5308 11.9088 10.9761 10.7275 10.2284 BSWI54 14.1176 13.629 12.507 11.8861 10.9551 10.7071 10.209 CC BSWI55 15.2034 14.6773 13.4689 12.8003 11.7977 11.5306 10.9942

FEM 15.2071 14.6813 13.4779 12.8135 11.8074 11.5364 10.9968 D120 15.2662 14.7379 13.5247 12.8533 11.8466 11.5783 11.0396 BSWI54 15.2034 14.6773 13.469 12.8003 11.7978 11.5306 10.9942

FEM 10.8618 10.4861 9.62437 9.14797 8.4307 8.23872 7.85458 D120 10.8667 10.4907 9.62702 9.1491 8.43249 8.24155 7.85814 BSWI54 10.8611 10.4853 9.62206 9.14438 8.42815 8.2373 7.8541

PC BSWI55 10.8611 10.4853 9.62205 9.14437 8.42814 8.2373 7.85409

CC BSWI55 10.8597 10.4839 9.62083 9.14322 8.42702 8.23619 7.85305

CF BSWI55 6.49133 6.26671 5.75083 5.46534 5.03722 4.92315 4.69413

λ<sup>3</sup> PP BSWI55 13.0317 12.5808 11.545 10.9719 10.1125 3.29504 3.33251

PC BSWI55 14.1176 13.629 12.507 11.8861 10.9551 10.7071 10.209

Assume the same beam, with simply supported boundary conditions and length l ¼ 20 m, is

of the beam is described using Euler Bernoulli beam theory and is assumed to be

�1

. The behaviour

subjected to a moving load of magnitude <sup>P</sup> <sup>¼</sup> <sup>1</sup> � 105 N travelling across at <sup>c</sup> <sup>m</sup>�<sup>s</sup>

h-FEM approach.

boundary conditions.

The matrices ½ � K and ½ � M are the stiffness and mass matrices for the functionally graded beam, <sup>ω</sup> is the natural frequency and <sup>U</sup> n o is the vector containing the DOFs within the entire beam. The i th non-dimensional frequency λ<sup>i</sup> of the FGM beam is evaluated from the relation λi <sup>2</sup> <sup>¼</sup> <sup>ω</sup>il 2 12 r<sup>l</sup> Elh<sup>2</sup> � �<sup>1</sup> 2 . The functionally graded beam is modelled for the different approaches using 2 Daubechies WFEs (L ¼ 12; j ¼ 0; 37 DOFs); one BSWI (m ¼ 5; j ¼ 4; 38 DOFs) WFE and 12 h-FEM elements (39 DOFs). The results of the first 3 non-dimensional natural frequencies of the beam are presented in Table 1 for different boundary conditions and material distributions. It is


Multiscale Wavelet Finite Element Analysis in Structural Dynamics http://dx.doi.org/10.5772/intechopen.71882 71


different values of n in Eq. (70). The free vibration of the functionally graded beam is governed

The matrices ½ � K and ½ � M are the stiffness and mass matrices for the functionally graded beam,

2 Daubechies WFEs (L ¼ 12; j ¼ 0; 37 DOFs); one BSWI (m ¼ 5; j ¼ 4; 38 DOFs) WFE and 12 h-FEM elements (39 DOFs). The results of the first 3 non-dimensional natural frequencies of the beam are presented in Table 1 for different boundary conditions and material distributions. It is

λ<sup>1</sup> PP BSWI55 4.34462 4.1943 3.84903 3.65795 3.37139 3.29504 3.33251

th non-dimensional frequency λ<sup>i</sup> of the FGM beam is evaluated from the relation

. The functionally graded beam is modelled for the different approaches using

n = 0 n = 0.1 n = 0.5 n = 1 n = 5 n = 10 n = 104

FEM 4.34463 4.19431 3.84912 3.65811 3.3715 3.2951 3.33258 D120 4.34462 4.1943 3.84903 3.65795 3.37139 3.29504 3.33251 BSWI54 4.34462 4.1943 3.84903 3.65795 3.37139 3.29504 3.33251

FEM 5.43024 5.24238 4.81112 4.57253 4.21419 4.11856 3.92682 D120 5.43023 5.24234 4.8108 4.57197 4.21382 4.11839 3.92681 BSWI54 5.43022 5.24233 4.81079 4.57197 4.21381 4.11839 3.92681

FEM 6.54137 6.31509 5.79585 5.50867 5.07685 4.96145 4.73028 D120 6.54132 6.31498 5.79514 5.50745 5.07601 4.96106 4.73028 BSWI54 6.54131 6.31498 5.79514 5.50745 5.07601 4.96105 4.73028

FEM 2.59318 2.50346 2.2974 2.18337 2.01232 1.96673 1.87523 D120 2.59318 2.50345 2.29737 2.18333 2.01229 1.96671 1.87523 BSWI54 2.59318 2.50345 2.29737 2.18333 2.01229 1.96671 1.87523

FEM 8.68894 8.38834 7.69844 7.31684 6.74338 6.59024 6.66537 D120 8.68968 8.389 7.6984 7.31623 6.74313 6.59043 6.66535 BSWI54 8.68871 8.38806 7.69754 7.31541 6.74238 6.58969 6.66461

FEM 9.77513 9.43702 8.66145 8.23263 7.58713 7.41442 7.06879 D120 9.7765 9.43821 8.66124 8.23127 7.5865 7.4147 7.06977

PC BSWI55 9.77473 9.4365 8.65966 8.22977 7.58512 7.41335 7.06849

PC BSWI55 5.43022 5.24233 4.81079 4.57197 4.21381 4.11838 3.92681

CC BSWI55 6.54131 6.31498 5.79514 5.50745 5.07601 4.96105 4.73028

CF BSWI55 2.59318 2.50345 2.29737 2.18333 2.01229 1.96671 1.87523

λ<sup>2</sup> PP BSWI55 8.68871 8.38806 7.69754 7.31541 6.74237 6.58969 6.66461

¼ 0 (85)

is the vector containing the DOFs within the entire beam.

½ �� <sup>K</sup> <sup>ω</sup><sup>2</sup> ½ � <sup>M</sup> � � <sup>U</sup> n o

70 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

by [45]

The i

λi <sup>2</sup> <sup>¼</sup> <sup>ω</sup>il

<sup>ω</sup> is the natural frequency and <sup>U</sup> n o

2 12 r<sup>l</sup> Elh<sup>2</sup> � �<sup>1</sup> 2

> Table 1. The non-dimensional frequencies of a steel-alumina FG beam for different transverse varying distributions and boundary conditions.

> observed that all approaches give highly accurate results with respect to the reference (BSWI55), particularly for the fundamental frequencies. Furthermore, the BSWI WFEM solution exhibits better levels of accuracy than the Daubechies WFEM and h-FEM solutions for the higher frequencies. Both WFEM solutions achieve high levels of accuracy with the described layout of having the rotation DOFs present at elemental and nodes and using fewer elements that the h-FEM approach.

> Assume the same beam, with simply supported boundary conditions and length l ¼ 20 m, is subjected to a moving load of magnitude <sup>P</sup> <sup>¼</sup> <sup>1</sup> � 105 N travelling across at <sup>c</sup> <sup>m</sup>�<sup>s</sup> �1 . The behaviour of the beam is described using Euler Bernoulli beam theory and is assumed to be

undamped. The governing equation describing the dynamic behaviour of the system is given by [45]:

$$\{\mathbf{M}\}\{\ddot{\mathbf{U}}(t)\} + [\mathbf{K}]\{\mathbf{U}(t)\} = \{\mathbf{F}(t)\}\tag{86}$$

5. Conclusions

boundary conditions.

n Critical velocity c m∙s

�1

Author details

Mutinda Musuva and Cristinel Mares\*

\*Address all correspondence to: cristinel.mares@brunel.ac.uk

and Physical Sciences, Brunel University, London, England, UK

A generalised formulation framework for the construction of an axial rod, Euler Bernoulli beam and functionally graded two-dimensional wavelet based finite elements is presented. The Daubechies and BSWI families are selected due to their desirable properties, particularly compact support, 'two-scale' relation and multiresolution. It is illustrated via a set of numerical examples that the WFEMs perform exceptionally well when compared to conventional h-FEM and p-FEM where high levels of accuracy are achieved with fewer elements required and the approaches converge more rapidly to the exact solution. Furthermore, the methods are able to accurately describe the behaviour of static and dynamic systems with singularities, variation in material properties and loading conditions present. This exhibits the vast potential of the method in the analysis of more complicated systems and the ability to alter the multiresolution scales without affecting the original mesh allows effective and efficient avenues solution accuracy improvement.

Table 2. The non-dimensional frequencies of a steel-alumina FG beam for different transverse varying distributions and

Max <sup>v</sup> <sup>l</sup> <sup>2</sup> ð Þ;<sup>t</sup> v0 

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Ref. [46] FEM D120 BSWI43 Ref. [46] FEM D120 BSWI43

0 252 252 252 252 0.9328 0.9322 0.9323 0.9322 0.1 – 235 235 235 – 0.9863 0.9864 0.9863 0.2 222 222 222 222 1.0344 1.0340 1.0340 1.0340 0.5 198 198 198 198 1.1444 1.1435 1.1437 1.1436 1 179 178 178 178 1.2503 1.2491 1.2495 1.2493 2 164 164 164 164 1.3376 1.3363 1.3368 1.3365 3 – 157 158 158 – 1.3747 1.3751 1.3748 5 – 151 151 152 – 1.4217 1.422 1.4218 7 – 148 148 148 – 1.4567 1.4570 1.4568 10 – 145 145 145 – 1.4974 1.4976 1.4974 104 132 132 132 132 – 1.7308 1.7309 1.7308

Department of Mechanical, Aerospace and Civil Engineering, College of Engineering, Design

where <sup>U</sup>€ð Þ<sup>t</sup> n o and f g <sup>U</sup>ð Þ<sup>t</sup> represent the system acceleration and displacement vectors at time <sup>t</sup>. f g Fð Þt is the moving load vector. The deflection of the beam v xð Þ ; t , as the moving load travels across, is normalised as a non-dimensional parameter v xð Þ ; <sup>t</sup> <sup>=</sup>v<sup>0</sup> where <sup>v</sup><sup>0</sup> <sup>¼</sup> Pl<sup>3</sup> <sup>48</sup>ElI is the deflection at the centre of the simply supported functionally graded beam when subjected to a static load of magnitude P at the centre. The maximum normalised deflection mid-span of the beam is analysed over a moving load velocity range 0 < c ≤ 300 m�s �<sup>1</sup> at increments of 1 m�<sup>s</sup> �<sup>1</sup> to identify the critical velocity for the different variations of the constituent materials as illustrated in Figure 7. The results present are obtained from the BSWI (2 element; m ¼ 4; j ¼ 3; 37 DOFs) WFEM solution. The h-FEM (12 elements; 39 DOFs) and Daubechies (2 elements; L ¼ 12; j ¼ 0; 37 DOFs) WFEM solution gives similar results. The values of the critical moving load velocity and corresponding maximum non-dimensional displacement are presented in Table 2 for the different values of n for all approaches. The results are compared with those presented in [46]. Both the Daubechies and BSWI WFE M solutions very accurately yield the correct values.

Figure 7. Variation of the maximum non-dimensional vertical displacement at the centre of a simply supported steelalumina beam with respect to moving load velocities, for different n.


Table 2. The non-dimensional frequencies of a steel-alumina FG beam for different transverse varying distributions and boundary conditions.
