A three-dimensional B-spline wavelet finite element method for structural vibration analysis

Abstract

In this study, a three-dimensional (3-D) B-spline wavelet finite element method (WFEM) is proposed by combining the wavelet theory and 3-D finite element method (FEM). Owing to the semi-orthogonality and multi-resolution analysis of spline wavelet over the solution domain, a cuboid with uniform distribution nodes is established for the 3-D B-spline wavelet on the interval (BSWI) element. The governing equations of vibration for the elastic structures are derived by introducing the 3-D B-spline wavelet theory into the displacement components of each node. A procedure for assembling the BSWI elements is presented by redefining the index element node functions. Numerical studies including the vibration analysis of solid and hollow beams and plates under different boundary conditions are performed to show the performance of the present method. A comprehensive comparison of the predicted vibration characteristics obtained by the present method, traditional FEM, and reference data is presented to verify the accuracy and efficiency of the 3-D B-spline WFEM. The core of the present method is involving the 3-D B-spline wavelet to replace the polynomial interpolation functions in the traditional FEM, which enhances computational accuracy during the vibration analysis of the elastic structures.

Keywords

three-dimensional B-spline wavelet wavelet finite element method three-dimensional vibration analysis dynamic analysis multi-resolution analysis

1. Introduction

Excitation of the resonant modes of elastic structures would lead to a significant vibration and noise. It is attached great importance to keep work frequency of equipment far from their resonant frequencies to guarantee the stable operation of the equipment. Pervious works have introduced several kinematic assumptions about the mid-surface plane and shear deformation of the elastic structures to capture their vibration characteristics, which are convenient to address the vibration problems and it gradually forms classical theories. Nevertheless, those kinematic assumptions involved into the vibration systems are not perfectly harmless for the predicting accuracy of vibration characteristics, due to continuity and three-dimensionality of structures and substances in reality.

Typically, FEM (Zienkiewicz et al., 2005) as one of deterministic methods have been widely applied in static and vibration analysis of the elastic structures. A combination of discretization and numerical approximation ensures the capability of the FEM to deal with the complex structures. To obtain a satisfied accuracy in the high-frequency region, the number of elements or the order of shape functions employed in the conventional FEM would keep an increasing trend, which undoubtedly leads to a dramatic increase in the computation costs and the complexity of the solution. Likewise, there is a heavy and large calculation for using the conventional FEM to cope with large-scale structures (Zou et al., 2022) or complex structures due to the slowness of convergence in shape functions.

One of the common mathematical technologies for signals and signal-processing systems in the frequency domain transformation is known as wavelet transform or wavelet analysis (Lee and Yamamoto, 1994; Mallat, 1999). Haar (Haar, 1911) proposed an orthogonal system of functions constructed by uniformly convergent series on the interval. Besides, various wavelets such as Meyer wavelet (Lin and Qu, 2000; Meyer, 1992), Morlet wavelet, and Daubechies wavelet (Vonesch et al., 2007) emerged in past few decades. Multi-resolution of wavelet provides a foundation to replace the polynomial interpolation of FEM and forms a method known as WFEM.

A great deal of researches over the past decades have focused on the principles and effects of the WFEM (Chen et al., 2004). The WFEM concentrates not only on the static and vibration analysis of the elastic structures but also on the crack detection and identification fields (Li et al., 2005; Xiang et al., 2007, 2013). Ma et al. (2003) set up a wavelet-based beam element by using the Daubechies wavelet to study a beam with an unequal cross-section and local load. Wang et al. (2011) introduced the Daubechies wavelet elements into the Rayleigh beam to settle the problems of pipe transverse crack detection. The static problems of Euler–Bernoulli beams and Mindlin–Reissner plates were taken into consideration via Daubechies wavelet by Díaz et al. (2009). In spite of the employment of Daubechies WFEM in settling the diverse structural problems, computation of connection coefficients brought high difficulties in the implementation and operation of the algorithm. Xue et al. (2016) developed a 2-D Hermite WFEM to wave propagation and load identification problems. Modification and Truncation of pre-processing for the Hermite wavelet was carried out before dynamic computation and analysis leading to an increase of difficulty. Hence, different wavelet selections may incur extra operations and difficulties in the process of introducing wavelet theory, owing to the characteristics and properties of the chosen wavelet.

Excellent characteristics such as inherent symmetry and compact support of B-spline wavelet were combined with the FEM with the potential to break the dilemma of limited convergence speed for polynomial interpolation in the traditional FEM (Unser, 1997; Vadlamani and Arun, 2020). Chen et al. (2012) simulated the wave propagation in the structures and modeled 1-D structures by 1-D B-spline wavelet. The statics and structural dynamic properties of thin plates were evaluated by introducing 2-D B-spline wavelets by Xiang et al. (2008). Similarly, the WFEM with the B-spline wavelet theory as the core was applied to analyze the free vibration and buckling problems of the Mindlin–Reissner plate in recent years (Yang et al., 2013). Despite the application of 2-D B-spline wavelet for the solution of vibration characteristics in the study of both Xiang et al. (2008) and Yang et al. (2013), BSWI elements were built by different lattices and nodes according to different plate and shell theories. Due to the specialty of BSWI elements of thin plates, Geng et al. (2018b) redefined the index element nodes functions to achieve the goal of splicing the BSWI elements and broaden the domain of spectral analysis. For the vibration analysis of plates, a 2-D B-spline wavelet was introduced into laminated shells by Zuo et al. (2021), and they made comparisons between results of BSWI theory and those of Cheyshev polynomials proposed by Ye et al. (2014). Geng et al. (2018a) explored a method using wavelet finite element established by the B-spline wavelet to predict average velocity frequency response functions of thin plates, which required that the mode number in the bandwidth of one-third octave bands must be greater than or equal to 12. Nevertheless, the pervious investigations always cope with vibration problems through simplifying the 3-D spaces into 2-D planes rather than considering 3-D problems directly, which may incur loss of vibration characteristics or differences far from the reality.

In this study, an idea of combing B-spline wavelet and FEM is proposed to release the dilemma of low convergence speed due to polynomial interpolation in the FEM. The contributions of this essay can be summarized as follows: Firstly, the multi-resolution analysis of three-dimensional B-spline wavelet is established and defined to lay the foundations of the completeness in the Lebesgue spaces. Second, 1-D B-spline wavelet is generalized to 3-D B-spline in the Lebesgue spaces to cater for the requirements of 3-D problems. Third, such a bond between 3-D B-spline wavelet and traditional FEM is established and bridged by constructing a novel BSWI element and being introduced into the variational principles to make a free vibration analysis and predict the vibration characteristics of elastic structures.

2. 3-D BSWI functions

1-D and 2-D B-spline wavelets are described explicitly in Chui and Quak (1992); Goswami et al. (1995); Unser et al. (1992). The scaling functions of the B-spline wavelet, are represented as follows

ϕ_{m, k}^{j} (ξ) = {\begin{cases} ϕ_{m, k}^{l} (2^{j - l} ξ), k = - m + 1, \dots, - 1 & (0 boundary scaling functions) \\ ϕ_{m, 2^{j} - m - k}^{l} (1 - 2^{j - l} ξ), k = 2^{j} - m + 1, \dots, 2^{j} - 1 & (1 boundary scaling functions) \\ ϕ_{m, 0}^{l} (2^{j - l} ξ - 2^{- l} k), k = 0, \dots, 2^{j} - m & (inner scaling functions) \end{cases}

(1)

where m indicates the mth order of cardinal B-spline, k expresses the scale of one-dimensional cardinal B-spline, and the scaling functions are shown in Figure 1. Also, B-spline wavelets have several properties (Chui and Quak, 1992), namely,

1. Wavelet functions have properties $V_{0} \subset V_{1} \subset . . . \subset L^{2} [0, 1]$ , where $V_{j} : = span (ψ_{m, k}^{j})$ ;

2. The subspace of j wavelet functions and the subspace of j scaling functions, $W_{j}$ , form the j+1 collection $V_{j}^{[0, 1]} \oplus W_{j}^{[0, 1]} = V_{j + 1}^{[0, 1]}$ , $j \in Ζ_{+}$ .

The definitions for 3-D BSWI functions are given by

Φ = Φ_{1} \otimes Φ_{2} \otimes Φ_{3}

(2)

where

Φ_{1} = {ϕ_{m, - m + 1}^{j} (ξ) ϕ_{m, - m + 2}^{j} (ξ) . . . ϕ_{m, 2^{j} - 2}^{j} (ξ) ϕ_{m, 2^{j} - 1}^{j} (ξ)}

Φ_{2} = {ϕ_{m, - m + 1}^{j} (η) ϕ_{m, - m + 2}^{j} (η) . . . ϕ_{m, 2^{j} - 2}^{j} (η) ϕ_{m, 2^{j} - 1}^{j} (η)}

Φ_{3} = {ϕ_{m, - m + 1}^{j} (ϑ) ϕ_{m, - m + 2}^{j} (ϑ) . . . ϕ_{m, 2^{j} - 2}^{j} (ϑ) ϕ_{m, 2^{j} - 1}^{j} (ϑ)}

, and

\otimes

indicates the Kronecker product. One of the edges of the 3-D B-spline element shown in Figure 2 contains n nodes, where n is numerically equal to

2^{j} + m - 1

. It is worth noting that the study adopts the conditions m = 4 and j = 3 in the construction of B-spline WFEM. The transformation matrices in each direction for the 3-D BSWI element construction are shown below

T_{i}^{e} = R_{i}^{- 1}

(3)

where i = 1, 2, 3 represent the

ξ, η, ϑ

directions of the 3-D BSWI element, respectively. The total transformation matrix of the 3-D BSWI element vibration system is

T^{e} = {[T_{1}^{e} T_{2}^{e} T_{3}^{e}]}^{T}

(4)

and,

R_{i} = \frac{1}{l_{x_{i}}} [\begin{array}{c} Φ (ξ_{1}) \otimes Φ (η_{1}) \otimes Φ (ϑ_{1}) \\ Φ (ξ_{2}) \otimes Φ (η_{1}) \otimes Φ (ϑ_{1}) \\ . . . \\ Φ (ξ_{11}) \otimes Φ (η_{1}) \otimes Φ (ϑ_{1}) \\ Φ (ξ_{1}) \otimes Φ (η_{2}) \otimes Φ (ϑ_{1}) \\ . . . \\ Φ (ξ_{11}) \otimes Φ (η_{11}) \otimes Φ (ϑ_{11}) \end{array}]

where the value of

ξ_{i}

is taken as

ξ_{i} = \frac{i}{2^{j} + m - 1} (i \in

N₀

& i \leq 2^{j} + m - 1)

, and

η_{i}

as well as

ϑ_{i}

are similar to

ξ_{i}

. Furthermore, the coordinate axes x, y, and z denote x₁, x₂, and x₃ to standardize the variable representation.

3. 3-D BSWI functions in free vibration

Whether it is a beam structure, a plate, a shell structure, or even a more complex elastic structure, the vibration characteristics can be coped with within a 3-D coordinate system. To study vibration characteristics, energy balance is observed in the elastic structure. Energy equations for 3-D elastic structures can be constructed, namely

Π = \frac{1}{2} \int_{Ω_{e}} κ^{T} D κ d x_{1} d x_{2} d x_{3} - \int_{Ω_{e}} w q (x_{1}, x_{2}, x_{3}) d x_{1} d x_{2} d x_{3},

(5)

where

Ω_{e}

denotes the field of elements, and

κ

represents the strains of elastic elements

κ = {\frac{\partial u_{1}}{\partial x_{1}} \frac{\partial u_{2}}{\partial x_{2}} \frac{\partial u_{3}}{\partial x_{3}} (\frac{\partial u_{1}}{\partial x_{2}} + \frac{\partial u_{2}}{\partial x_{1}}) (\frac{\partial u_{2}}{\partial x_{3}} + \frac{\partial u_{3}}{\partial x_{2}}) (\frac{\partial u_{3}}{\partial x_{1}} + \frac{\partial u_{1}}{\partial x_{3}})}^{T} .

(6)

where u_i represents the displacement components of the nodes in the x₁, x₂, and x₃ directions, respectively, in the Cartesian coordinate system. Correspondingly, the 3-D element elasticity matrix D is

D = D_{0} [\begin{array}{c} 1 - μ \\ μ & 1 - μ & s y m . \\ μ & μ & 1 - μ \\ 0 & 0 & 0 & \frac{1 - 2 μ}{2} \\ 0 & 0 & 0 & 0 & \frac{1 - 2 μ}{2} \\ 0 & 0 & 0 & 0 & 0 & \frac{1 - 2 μ}{2} \end{array}] D_{0} = \frac{E}{(1 + μ) (1 - 2 μ)}

(7)

where E and

μ

denote Young’s modulus and Poisson’s ratio, respectively. The displacement components of nodes on BSWI elements, namely

u_{i} = {u_{i, 1} u_{i, 2} . . . u_{i, n^{3}}}^{T}

(8)

The equation (2) and equation (3) can be introduced into displacement of each node

u_{i} (x_{1}, x_{2}, x_{3}) = Φ T_{i}^{e} u_{i}^{e} (ξ, η, ϑ) .

(9)

Then, the BSWI element stiffness matrix K ^e is expressed as

K^{e} = D_{0} [\begin{array}{c} K_{11} & K_{12} & K_{13} \\ K_{21} & K_{22} & K_{23} \\ K_{31} & K_{32} & K_{33} \end{array}] .

(10)

The components of the BSWI element stiffness matrix are organized as

$K_{11} = {(T_{1}^{e})}^{T} {(1 - μ) Γ_{1}^{1,1} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{0,0} + \frac{(1 - 2 μ)}{2} Γ_{1}^{0,0} \otimes Γ_{2}^{1,1} \otimes Γ_{3}^{0,0} + \frac{(1 - 2 μ)}{2} Γ_{1}^{0,0} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{1,1}} T_{1}^{e} K_{12} = {(T_{1}^{e})}^{T} {μ Γ_{1}^{1,0} \otimes Γ_{2}^{0,1} \otimes Γ_{3}^{0,0} + \frac{1 - 2 μ}{2} Γ_{1}^{0,1} \otimes Γ_{2}^{1,0} \otimes Γ_{3}^{0,0}} T_{2}^{e}$

$K_{13} = {(T_{1}^{e})}^{T} {μ Γ_{1}^{1,0} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{0,1} + \frac{1 - 2 μ}{2} Γ_{1}^{0,1} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{1,0}} T_{3}^{e}$

$K_{22} = {(T_{2}^{e})}^{T} {(1 - μ) Γ_{1}^{0,0} \otimes Γ_{2}^{1,1} \otimes Γ_{3}^{0,0} + \frac{(1 - 2 μ)}{2} Γ_{1}^{1,1} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{0,0} + \frac{(1 - 2 μ)}{2} Γ_{1}^{0,0} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{1,1}} T_{2}^{e} K_{23} = {(T_{2}^{e})}^{T} {μ Γ_{1}^{0,0} \otimes Γ_{2}^{1,0} \otimes Γ_{3}^{0,1} + \frac{1 - 2 μ}{2} Γ_{1}^{0,0} \otimes Γ_{2}^{0,1} \otimes Γ_{3}^{1,0}} T_{3}^{e}$

$K_{33} = {(T_{3}^{e})}^{T} {(1 - μ) Γ_{1}^{0,0} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{1,1} + \frac{(1 - 2 μ)}{2} Γ_{1}^{0,0} \otimes Γ_{2}^{1,1} \otimes Γ_{3}^{0,0} + \frac{(1 - 2 μ)}{2} Γ_{1}^{1,1} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{0,0}} T_{3}^{e} K_{21} = K_{12}^{T}$

$K_{31} = K_{13}^{T}$

$K_{32} = K_{23}^{T}$

where

$Γ_{1}^{1,0} = \int_{0}^{1} \frac{d Φ_{1}^{T}}{d ξ} Φ_{1} d ξ$

$Γ_{1}^{0,1} = \int_{0}^{1} Φ_{1}^{T} \frac{d Φ_{1}}{d ξ} d ξ$

$Γ_{1}^{0,0} = l_{x_{1}} \int_{0}^{1} Φ_{1}^{T} Φ_{1} d ξ$

$Γ_{1}^{1,1} = \frac{1}{l_{x_{1}}} \int_{0}^{1} \frac{d Φ_{1}^{T}}{d ξ} \frac{d Φ_{1}}{d ξ} d ξ$

Note that regularities in the other directions ( $Γ_{2}^{i, j}$ ; $Γ_{3}^{i, j}$ ) are similar to $Γ_{1}^{i, j}$ . The differences of $Γ_{1}^{i, j}$ , $Γ_{2}^{i, j}$ , and $Γ_{3}^{i, j}$ concentrate on the lengths of BSWI element on three directions in local coordinate systems $l_{x_{i}}$ and the scaling function vectors $Φ_{i}$ . Correspondingly, the mass matrix of BSWI elements is given by

M^{e} = ρ [\begin{array}{c} M_{1}^{e} \\ M_{2}^{e} \\ M_{3}^{e} \end{array}]

(11)

where

ρ

denotes mass density, and the components of BSWI element mass matrix is

$M_{i}^{e} = {(T_{i}^{e})}^{T} {Γ_{1}^{0,0} \otimes Γ_{2}^{0,0} \otimes Γ_{3}^{0,0}} T_{i}^{e}$ .

The governing equations or differential equations of motion which are set up within the framework of Newton mechanics of vibration systems are given as follows

M \ddot{u} + C \dot{u} + K u = F

(12)

where F , C , K , and M denote generalized force vectors, global damping matrix, global stiffness matrix, and global mass matrix, respectively. However, while considering vibration systems as undamped vibration systems and the vibration systems is under free vibration condition ( C = 0, F = 0), then the eigenfunction is expressed as

| K - ω^{2} M | = 0

(13)

where

ω

indicates natural frequency. Simultaneously, the relations between element stiffness matrix (element mass matrix) and the global stiffness matrix (global mass matrix) are given by

K = \overset{e}{\underset{i = 1}{Λ}} K_{i}^{e}

(14)

and

M = \overset{e}{\underset{i = 1}{Λ}} M_{i}^{e}

(15)

where e indicates that the elastic structures are divided into e 3-D BSWI elements. Due to the special characteristics of the arrangement of the BSWI element matrix constructed, it is necessary to construct a different way of assembling the element matrix from the traditional FEM. The arrangement for DOFs of the BSWI element is shown in Table 1.

Table 1.

Arrangement for DOFs of the BSWI element.

	Node 1	Node 2	Node i	Node n³
x ₁	1	2	i	n ³
x ₂	n³+1	n³+2	n³+i	2n³
x ₃	2n³+1	2n³+2	2n³+i	3n³

In the same way, the regularity of the global stiffness matrix (global mass matrix) follows the rules that the number of DOFs in the next direction is not counted until all the DOFs in this direction have been counted. Hence, it is of great importance for index element nodes (IEN) to connect the BSWI element matrix with the global matrix

e_{0, x_{3}} = ⌈ \frac{e_{0}}{e_{x_{1}} e_{x_{2}}} + 1 ⌉ - 1; e_{0, x_{2}} = ⌈ \frac{e - (e_{0, x_{3}} - 1) e_{x_{1}} e_{x_{2}}}{e_{x_{1}}} + 1 ⌉ - 1; e_{0, x_{1}} = e - (e_{0, x_{3}} - 1) e_{x_{1}} e_{x_{2}} - (e_{0, x_{2}} - 1) e_{x_{1}}

(16)

where

e_{x_{1}}

e_{x_{2}}

, and

e_{x_{3}}

represent the number of BSWI elements along x₁, x₂, and x₃, respectively, as well as

⌈ \cdot ⌉

denotes the ceiling function. It is found that

e_{0, x_{1}}

e_{0, x_{2}}

, and

e_{0, x_{3}}

are a position of an arbitrary BSWI element

e_{0}

. In addition to determining the position of an arbitrary BSWI element, it is also necessary to need the local position of an arbitrary point in order to determine the IEN, and the local position information is expressed as

a_{0, x_{3}} = ⌈ \frac{a_{0}}{n^{2}} + 1 ⌉ - 1

a_{0, x_{2}} = ⌈ \frac{a_{0} - (a_{0, x_{1}} - 1) n^{2}}{n} + 1 ⌉ - 1

(17)

a_{0, x_{1}} = a_{0} - (a_{0, x_{1}} - 1) n^{2} - (a_{0, x_{1}} - 1) n

where

a_{0, x_{1}}

a_{0, x_{2}}

, and

a_{0, x_{3}}

denote the position along

x_{1}

x_{2}

, and

x_{3}

in local coordinate systems. Then, the IEN function is set up in the following

\begin{array}{l} I E N (a_{0}, e_{0}) = [(e_{0, x_{3}} - 1) n - (e_{0, x_{3}} - 1)] (e_{x_{1}} n - e_{x_{1}} + 1) (e_{x_{2}} n - e_{x_{2}} + 1) + \\ (a_{0, x_{3}} - 1) (e_{0, x_{1}} n - e_{0, x_{1}} + 1) (e_{x_{2}} n - e_{x_{2}} + 1) + \\ [(e_{0, x_{2}} - 1) n - (e_{0, x_{2}} - 1)] (e_{x_{1}} n - e_{x_{1}} + 1) + \\ (a_{0, x_{2}} - 1) (e_{0, x_{1}} n - e_{0, x_{1}} + 1) + (e_{0, x_{1}} - 1) n - (e_{0, x_{1}} - 1) + a_{0, x_{1}} \end{array}

(18)

The IEN function successfully links the local node numbering to the global node numbering. Hence, the three-dimensional WFEM is similar to the traditional FEM, which can assemble elements to make computation with higher precision and make the foundations for exploring the vibrational properties of medium and high frequencies. Then, a series of natural frequencies and mode shapes can compute via equation (13), namely

ω = [ω_{1} ω_{2} ω_{3} . . . ω_{N}]

(19)

and

Θ = [Θ_{1} Θ_{2} Θ_{3} . . . Θ_{N}]

(20)

where

Θ

denotes the mode shape of structural vibrations.

4. Numerical results

Several numerical studies and investigations have been performed to check the validity of present method and summarize the features in terms of precision, costs, and stability.

4.1 Free vibration analysis of plates

The vibration characteristics due to free vibration are presented for a square plate of length l_x1 = l_x2 = 1 m and thickness l_x3 = 0.1 m with the material parameters of Young’s modulus E = 2.1×10¹¹ Pa, Poisson’s ratio $μ$ = 0.3, and mass density $ρ$ = 7850 kg/m³.

There are natural frequencies of the plate under three kinds of boundary conditions shown in Table 2, where S, C, and F refer to simply supported, clamped, and free boundary conditions for a face of elastic structures, respectively. Note that the type of elements for the FEM is the hexahedron element, which can form a mesh of elastic structures with identical node-number, node-arrangement, and DOFs. Also, several mode shapes under C-C-C-F boundary condition of the plate are collected through present methods and drawn in Figure 3(a).

Table 2.

The first five natural frequencies of the square plate under three boundary conditions.

Boundary conditions	Layout and Elem. (DOFs)	Natural frequency (Hz)
Boundary conditions	Layout and Elem. (DOFs)	N = 1	N = 2	N = 3	N = 4	N = 5
S-S-S-S	1 BSWI (3993)	457.4	1110.8	1110.8	1688.9	2003.4
	10×10×10 H8^a (3993)	457.79	1112.3	1112.3	1691.6	2003.7
	15×15×10 H8 (8448)	457.24	1110.7	1110.7	1688.5	2003.5
	20×20×10 H8 (14553)	457.09	1110.4	1110.4	1687.8	2003.2
	25×25×10 H8 (22308)	457.04	1110.3	1110.3	1687.5	2003.2
C-C-C-C	1 BSWI (3993)	819.68	1565.6	1565.6	2196.0	2591.4
	10×10×10 H8 (3993)	820.99	1570.0	1570.0	2201.9	2602.9
	15×15×10 H8 (8848)	818.34	1563.7	1563.7	2193.6	2587.9
	20×20×10 H8 (14553)	817.34	1561.6	1561.6	2190.9	2583.9
	25×25×10 H8 (22308)	816.84	1560.7	1560.7	2189.6	2582.2
C-C-C-F	1 BSWI (3993)	556.30	895.75	1367.6	1641.0	1691.8
	10×10×10 H8 (3993)	557.41	897.13	1372.4	1644.7	1696.3
	15×15×10 H8 (8848)	555.37	894.57	1365.9	1639.7	1689.8
	20×20×10 H8 (14553)	554.61	893.66	1363.8	1638.2	1687.7
	25×25×10 H8 (22308)	554.24	893.22	1362.9	1637.5	1686.7

^aH8 denotes hexahedron element with eight nodes.

Stepwise comparisons are established by the different meshes of the FEM in the table, and the present results match well with those of the FEM. It shows that the convergence of BSWI element is always better than H8 elements for all the boundary conditions under the same DOFs. The present results are compared with those obtained by using Eigenfrequency Analysis module of COMSOL Multiphysics, which is based on ARPACK solver. The relative error is defined as follows

ε = \frac{f_{i}^{'} - f_{i}^{F E M_{s t .}}}{f_{i}^{F E M_{s t .}}} \times 100 %

(21)

where

f_{i}^{'}

and

f_{i}^{F E M_{s t .}}

represent the natural frequency predicted by the present method and those provided by the FEM, respectively.

On the basis of equation (21), the relative errors of the first nine natural frequencies for the plate with different boundary conditions are illustrated in Figure 4. The average relative error obtained by a single BSWI element under three different boundary conditions is 0.070%, 0.278%, and 0.278%, while the corresponding average relative errors calculated by H8 elements with the same DOFs are 0.193%, 0.536%, and 0.543%. This phenomenon in the histogram proves that multi-resolution analysis of BSWI is beneficial for the traditional FEM to improve its convergent ratio.

Table 3 presents a stepwise increase in the element number and correlative predictions of natural frequencies. There is also a clear decreasing trend in the results of natural frequencies as the number of BSWI elements increases. Note that the case BSWI 2×2×1 have the same DOFs with the case H8 20×20×10, and the elastic plates are arranged with the same nodes along each edge of the considered plate. Comparing the results of the two cases, the convergence level of BSWI 2×2×1 is better than that of H8 20×20×10.

Table 3.

The natural frequencies of the plate via assembling BSWI elements.

Boundary conditions	Layout and Elem. (DOFs)	Natural frequency (Hz)
Boundary conditions	Layout and Elem. (DOFs)	N = 1	N = 2	N = 3	N = 4	N = 5
C-C-C-C	1×2×1 BSWI (7623)	818.31	1561.68	1564.70	2192.82	2586.00
	1×3×1 BSWI (11253)	817.97	1560.71	1564.47	2192.05	2584.86
	2×2×1 BSWI (14553)	816.95	1560.75	1560.75	2189.66	2581.95

To figure out the effects and accuracy of assembling BSWI elements, the numerical study provides average relative error level $[ε]$ of various arrangements for BSWI and H8 elements as drawn in Figure 5. We can find that the case BSWI 2×2×1 has the lowest relative error in all cases, and the average relative error of first nine modes reaches 0.007%. Except for the above case, the average relative errors for other cases in the sequence of Table 3 are 0.276%, 0.130%, and 0.098%, respectively. Correspondingly, the average relative error of the FEM reaches 0.536%, 0.165%, and 0.051%, respectively. Standard deviation of the relative errors for the natural frequencies has downward trend with increasing DOFs, which suggests that WFEM becomes more stable for each natural frequency.

4.2 Free vibration analysis of beams

In this numerical example, two cases of beams are taken into consideration and analysis. The geometric parameters and material properties of the beam structures are listed in Table 4. Comparison of the bending frequencies of Case 1 obtained through present method, traditional FEM, CUF-FEM, and quasi-3D vibration solutions is displayed in Table 5.

Table 4.

The geometric parameters and material properties of beams.

Case	Geometric parameters			Material properties
Case	Length (m)	Width (m)	Thickness (m)	Density (kg/m³)	Poisson’s ratio	Young’s modulus (GPa)
1	l_x1 = 2	l_x2 = 0.2	l_x3 = 0.2	2700	0.33	75
2	l_x1 = 3	l_x2 = 0.3	l_x3 = 0.2	7850	0.3	210

Table 5.

The bending natural frequency of Case 1 beam under S-S boundary condition.

Parameters and Approach		Natural frequency (Hz)
Parameters and Approach		N = 1	N = 2	N = 3	N = 4	N = 5
No. and Elem.	DOFs	CUF-FEM (Carrera et al., 2011)
10 B3	189	117.704	451.971	958.91	1592.39	2317.45
20 B3	369	117.701	451.846	957.68	1586.64	2299.29
30 B3	729	—^a	451.838	957.61	1586.26	2298.06
M	DOFs	Quasi-3D solutions (Chen et al., 2019)
20	207	117.746	452.084	958.34	1587.84	2300.83
25	252	117.746	452.084	958.34	1587.83	2300.81
30	297	117.746	452.084	958.34	1587.83	2300.80
No. and Elem.	DOFs	FEM
10×10×10 H8	3993	117.6	450.25	951.68	1574.5	2284
15×15×10 H8	8848	117.58	449.79	949.12	1565.6	2260.5
20×20×10 H8	22308	117.57	449.71	948.66	1564	2256.2
No. and Elem.	DOFs	Present method
1 BSWI	3993	117.57	449.70	949.13	1570	2297.1

^aThe results are not mentioned in the reference.

The present results are compared and contrasted with those obtained by quadratic (B3) finite elements in the CUF-FEM solutions and different truncation configurations M = 20, 25, 30 in the quasi-3D solutions. There are nearly identical results from four approaches in the first and second bending modes of the beam. However, the differences in the results from these approaches emerge and happen from the third bending mode. Few quantities of DOFs and truncation configurations may account for the deviations of bending natural frequency obtained by CUF-FEM and quasi-3D solutions.

The free vibration analysis of Case II contains three types of structural constraints, S-S, C-C, and C-F. The mode shapes of considered beam under C-C boundary condition are painted in Figure 3(b), whereas the results of Case II are represented in Table 6.

Table 6.

The natural frequencies of Case II beam.

Boundary conditions	Layout and Elem. (DOFs)	Natural frequency (Hz)
Boundary conditions	Layout and Elem. (DOFs)	N = 1	N = 2	N = 3	N = 4	N = 5
S-S	1 BSWI (3993)	51.750	171.27	202.83	438.92	442.60
	10×10×10 H8 (3993)	51.780	171.34	203.29	439.55	444.64
	15×15×10 H8 (8448)	51.756	171.28	202.90	436.75	442.55
	20×20×10 H8 (14553)	51.752	171.27	202.83	435.46	442.16
	25×25×10 H8 (22308)	51.751	171.26	202.81	434.82	442.05
C-C	1 BSWI (3993)	116.58	168.62	310.03	432.34	460.72
	10×10×10 H8 (3993)	116.92	168.95	311.83	433.87	461.01
	15×15×10 H8 (8848)	116.28	168.17	309.38	431.43	460.27
	20×20×10 H8 (22308)	116.05	167.87	308.63	430.61	459.98
	25×25×10 H8 (20280)	115.94	167.72	308.30	430.22	459.83
C-F	1 BSWI (3993)	18.656	27.837	114.56	166.91	228.78
	10×10×10 H8 (3993)	18.666	27.850	114.78	167.12	228.85
	15×15×10 H8 (8848)	18.628	27.793	114.43	166.70	228.67
	20×20×10 H8 (14553)	18.611	27.769	114.31	166.54	228.60
	25×25×10 H8 (22308)	18.603	27.756	114.25	166.47	228.57

It is significant that the WFEM has a satisfactory convergency, and the present results for the considered beam with all boundary conditions converge better than those of H8 element with identical DOFs. Then, a single BSWI element has average relative errors of 0.224%, 0.580%, and 0.272%, whereas those of H8 with the identical DOFs are 0.423%, 0.908%, and 0.468%. The relative errors rise a high point 1.30% and peak in the 9^th mode of considered beam with C-C boundary conditions in Figure 6.

The boundary condition C-C is chosen as the object of study, and Table 7 presents two types of the BSWI element layouts. In regard to the first natural frequency, the results highly agree with the results of 25×25×10 H8 while splicing into two BSWI elements along the x₁ axis. In contrast with 25×25×10 H8, the third case in the table has remarkable convergences and saves a nearly 50 percent DOFs. Furthermore, better convergences of natural frequencies for Case II of the beam are shown as more BSWI elements are split along the x₁ axis.

Table 7.

The natural frequencies of Case II beam via assembling BSWI elements.

Boundary conditions	Layout and Elem. (DOFs)	Natural frequency (Hz)
Boundary conditions	Layout and Elem. (DOFs)	N = 1	N = 2	N = 3	N = 4	N = 5
C-C	2×1×1 BSWI (7623)	115.98	167.79	308.37	430.37	459.88
C-C	3×1×1 BSWI (11253)	115.83	167.59	307.99	429.91	459.68

The average relative error level of natural frequencies for two categories of element with different DOFs is taken into account and drawn in Figure 7. The average relative errors of 2×1×1 BSWI and 3×1×1 BSWI for the first nine natural frequencies are 0.024% and 0.081%, respectively. Comparison of Table 4 and Table 5 shows that the average relative errors of 3×1×1 BSWI perform increasing tendency instead of declining tendency as 3×1×1 BSWI has better convergences than the 25×25×10 H8. As the number of BSWI elements increases, there is a significant decreasing trend in the standard deviation.

4.3 Free vibration analysis of the hollow beams

In the following discussion, the present numerical calculation involves a complex structure of a hollow beam with a cross-section with a hollow square center in the middle, an external edge of 0.3 m, and an internal edge of 0.1 m. This numerical calculation is classified into four cases relying on the different lengths of the hollow beam, ranging from Case I to Case IV whose lengths are 5 m, 8 m, 10 m, and 15 m, respectively. The distribution of BSWI elements is illustrated and the vertices of BSWI elements are emphasized in Figure 8.

To investigate the characteristics of the present method intuitively, there are several executions for node layout and element arrangement of the FEM developing as follows: 1. Keep the node layout aligned with the BSWI layout; 2. The quantity of H8 elements in the x₁ direction is twice as many as the equivalent element inside the BSWI elements; 3. The amount of H8 elements is raised two times as against equivalent elements of BSWI elements; and 4. There is three-fold growth in the number of H8 elements than the equivalent elements of the BSWI elements in the mesh of cross-section and is consistent with the node layout in the x₁ direction of the BSWI elements.

The exhibition of natural frequencies and statistical analysis for absolute errors of the hollow beams are displayed in Table 8 and Figure 9, respectively. With an increase in the length of the structures, height-to-length ratios (l_x3/l_x1 = 1/50 1/80 1/100 and 1/150) of BSWI elements behave inappropriately and uncoordinatedly, which may incur instability and imprecision of the present method. The average absolute errors are obtained from the BSWI elements which range from Case I to Case IV for the natural frequency 0.66 Hz, 0.74 Hz, 0.78 Hz, and 1.06 Hz. Correspondingly, the average absolute errors of H8 29040 are listed as 3.86 Hz, 3.98 Hz, 4.00 Hz, and 4.08 Hz.

Table 8.

The first five natural frequencies of the hollow beam.

Boundary conditions	Situation	Elem. (DOFs)	Natural frequency (Hz)
Boundary conditions	Situation	Elem. (DOFs)	N = 1	N = 2	N = 3	N = 4	N = 5
Bottom clamped	Case I	BSWI (29040)	1487.7	1495.2	1521.7	1568.5	1639.2
		H8 (29040)	1490.9	1498.4	1524.8	1571.7	1642.5
		H8 (55440)	1490.7	1498	1524.3	1570.9	1640.6
		H8 (81840)	1490.6	1497.7	1524.1	1570.7	1640.2
		H8 (110880)	1486.9	1494.5	1520.9	1567.9	1638.8
	Case Ⅱ	BSWI (29040)	1488.2	1490.9	1500.9	1518.0	1543.4
		H8 (29040)	1491.4	1494.2	1504.1	1521.2	1546.7
		H8 (55440)	1491.2	1493.8	1503.8	1520.8	1545.8
		H8 (81840)	1491.1	1493.6	1503.6	1520.6	1545.6
		H8 (110880)	1487.4	1490.2	1500.1	1517.3	1542.7
	Case Ⅲ	BSWI (29040)	1488.4	1490.1	1496.3	1507.0	1522.7
		H8 (29040)	1491.6	1493.3	1499.5	1510.2	1526
		H8 (55440)	1491.4	1493	1499.3	1509.9	1525.4
		H8 (81840)	1491.3	1492.8	1499.2	1509.8	1525.2
		H8 (110880)	1487.5	1489.3	1495.5	1506.3	1522
	Case IV	BSWI (29040)	1485.5	1489.2	1492.0	1496.6	1503.2
		H8 (29040)	1491.7	1492.5	1495.5	1499.8	1506.5
		H8 (55440)	1491.6	1492.3	1495	1499.6	1506.2
		H8 (81840)	1491.5	1492.1	1495	1499.6	1506.1
		H8 (110880)	1487.7	1488.4	1491.2	1495.8	1502.5

The extreme BSWI outlier of box plot emerges at first natural frequency in Case IV and imply that the present method may be instable under this height-to-length ratio. The mean values of the absolute errors for the four scenarios in Figure 9 are sequentially arranged as 0.810 Hz, 3.980 Hz, 3.535 Hz, and 3.375 Hz. Overall accuracy of the present method performs well and are influenced little by the height-to-length ratio, despite the present method with the lowest number of DOFs. Furthermore, the interquartile ranges of box for BSWI element are lowest in four situations, which means that there is a most compact sample distribution of BSWI elements.

5. Conclusions

Herein was elaborated a WFEM study combing the B-spline wavelet and the FEM to investigate the 3-D vibration characteristics for elastic structures. Multi-resolution analysis of 3-D B-spline wavelet determined BSWI element possessing a group of uniformly distributed nodes. The way based on characteristics of 3-D BSWI element for assembling elements was redefined by reorganizing the IEN functions to bridge the connections of BSWI elements in global coordinate system. The investigations of the WFEM and traditional FEM were performed in the hollow structures and solid structures with various boundary conditions. Semi-orthogonality and symmetry of BSWI brought satisfactory convergence and considerable precision in comparison with the FEM (e.g., the average error of the plate by using two different methods is 0.070%, 0.278%, 0.278% and 0.193%, 0.536%, 0.543%). A significant trend of decreasing in the average value and standard deviation of relative errors was found clearly as the number of BSWI element grown. Furthermore, verification and investigation on accuracy and steadiness of present methods for hollow beam were performed by changing aspect ratio of BSWI element. The statistics of absolute error shows the strong stability and good accuracy of the present method due to the high-order smoothness of the B-spline wavelet theory.

Footnotes

Acknowledgment

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (nos. 52225109 & 52271309) and Natural Science Foundation of Heilongjiang Province of China (nos. YQ2022E104).

Declaration of conflicting interests

The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (nos. 52225109 & 52271309) and Natural Science Foundation of Heilongjiang Province of China (nos. YQ2022E104).

ORCID iD

Ze-xi Sun

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