Predicting dynamic response of stiffened-plate composite structures in a wide-frequency domain based on Composite B-spline Wavelet Elements Method (CBWEM)

Highlights

• The new numerical scheme is derived for coupling the c₁ type substructures.

• CBWEM is proposed for modeling stiffened-plate composite structures.

• CPU time to predict the dynamic response in wide frequency domain is 3.5 s.

• The CBWEM can be taken as a new technique to solve mid-frequency problem.

Abstract

Due to the mid-frequency problem, the Hybrid Statistical Energy Analysis (SEA) / Traditional Finite Element Methods (TFEMs) methods still cannot provide the dynamic responses of the stiffened-plate composite structures in a wide frequency domain. The main reason is that all of the SEA and TFEMs cannot provide the reliable numerical solutions in the middle frequency domain when simulating the thin plate substructures. In order to solve the problem, this paper proposes the Composite B-spline Wavelet Elements Method (CBWEM) based on the c₁ type wavelet plate and beam elements for modeling the stiffened-plate composite structures and predicting its dynamic responses in a wide frequency domain. Unfortunately, due to the complex interpolation functions and transformation matrices of these c₁ type wavelet elements, the existing numerical scheme to construct the constraint matrix will be invalid when modeling the coupling relationship between the c₁ type wavelet plate and beam elements. To solve the problem, this study deduces and gives the formulas of the new numerical scheme for constructing the constraint matrix and modeling the stiffened-plate composite structures based on the CBWEM. Besides, the numerical and experimental studies are carried out to verify the CBWEM, respectively. On the one hand, the numerical study displays that the proposed method can solve the mid-frequency problem and provide reliable dynamic responses in a wide frequency domain within an acceptable computational cost. On the other hand, the experimental study shows that the Central Processing Unit (CPU) time to predict the dynamic response in a wide frequency domain is less than 3.5 s only based on the proposed method and the personal computers, and the corresponding numerical solutions are in good agreement with the experimental results. Thus, we can easily conclude that the proposed method can be taken as one useful numerical technique to solve the mid-frequency problem and predict the dynamic responses of the stiffened-plate composite structures in a wide frequency domain.

Introduction

The stiffened-plate composite structures have been widely applied in various products, such as aerofoil, submarine, rocket, carriage, ships and etc, in engineering [1]. And, people have paid more and more attention on these products’ better sound and vibration performance. According to the theory of the dynamic analysis, to capture better sound and vibration performance, it is necessary to predict the stiffened-plate composite structures’ dynamic response in a wide frequency domain, nowadays. However, due to the mid-frequency problem [2], the existing methods, such as Traditional Finite Element Methods (TFEMs), Statistical Energy Analysis (SEA) and the Hybrid SEA/TFEMs methods, still cannot be applied for providing the reliable numerical solutions of the dynamic responses in a wide frequency domain [3], [4], [5]. Furthermore, according to the Mace et al. [6], the mid-frequency problem is one significant obstacle when improving the stiffened-plate composite structures’ sound and vibration performance. Thus, it is necessary to solve the mid-frequency problem and predict the dynamic response in a wide frequency domain when simulating the stiffened-plate composite structures for improving its sound and vibration performance.

Nowadays, in order to solve the mid-frequency problem and predict the dynamic responses of the stiffened-plate composite structures in a wide frequency domain, many researchers have paid their attention on proposing the efficient numerical methods for providing the reliable dynamic solutions in the mid-frequency domain. According to the Yin and coworkers [3], [4], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], we can easily find that the Hybrid TFEMs/SEA methods have been widely applied for solving the mid-frequency problem and providing the dynamic solutions of the stiffened-plate composite structures, nowadays. Moreover, these Hybrid SEA/TFEMs methods have become the most popular numerical methods in the field of sound and vibration analysis when predicting the stiffened-plate composite structures’ dynamic responses in a wide frequency domain. And, these Hybrid TFEMs/SEA methods hold great attraction in the field of sound and vibration analysis. However, it is should be noted that there still exist some noticeable problems in the process of dynamic analysis by these Hybrid SEA\TFEMs methods.

Firstly, in order to proceed dynamic analysis based on these Hybrid SEA/TFEMs methods in a wide frequency domain, the stiffened-plate composite structures must be divided into the SEA substructures and the TFEMs substructures, respectively. Secondly, the coupling algorithm to connect the SEA and TFEMs substructures must be given for simulating the coupling effect between the stiffener and thin plate substructures. Thirdly, in order to apply these Hybrid SEA/TFEMs methods for proceeding dynamic analysis, the engineers must master the TFEMs and SEA technologies, simultaneously. Besides, there does still exist a difficult question when applying these Hybrid SEA/TFEMs methods for proceeding dynamic response in a wide frequency domain. That is the mid-frequency problem. Thus, these Hybrid TFEMs/SEA methods still cannot provide the reliable numerical solutions in a wide frequency domain in practice, efficiently [3].

It is mainly due to that these Hybrid TFEMs/SEA methods are validity only when the stiffener and thin plate substructures of the stiffened-plate composite structures can be simulated efficiently based on TFEMs and SEA, respectively. Although the stiffener always can be simulated efficiently based on the TFEMs in the bandwidth, due to the limitation of the SEA, the thin plate substructures still cannot be simulated efficiently in the low and middle frequency domain. Thus, we can clearly conclude that due to the mid-frequency problem, these Hybrid SEA/TFEMs methods always cannot provide reliable numerical solutions in a wide frequency domain when modeling the stiffened-plate composite structures [18]. Besides, these Hybrid SEA/TFEMs methods only can give the trends of the dynamic responses in the analyzing bandwidth without the detailed information, such as the resonance peak. It is one significant obstacle to improve the products’ sound and vibration performance. Therefore, we can easily conclude that it is still necessary to break up the limitation of SEA or TFEMs and solve the mid-frequency problem when predicting the stiffened-plate composite structures’ dynamic responses in a wide frequency domain only based on SEA or TFEMs.

Comprehensive the above contents and the relevant Refs. [2], [6], on the one hand, because of the theoretical restrictions, the SEA only can provide the reliable dynamic responses in the high-frequency domain and cannot provide the reliable numerical solutions in the low to middle frequency domain. On the other hand, because of the low computing accuracy and huge computational cost, the TFEMs only can provide the reliable dynamic solutions in the low-frequency domain and cannot provide the reliable numerical solutions in the middle to high frequency domain. By analyzing these two kinds of reasons why the mid-frequency problem still exists in practice, we can clearly conclude that there is a theoretical limit to solve the mid-frequency problem based on SEA. Therefore, by analyzing these two kinds of numerical methods, we consider that the methods based on the theory of finite element still provide the potential to solve the mid-frequency problem and provide the reliable dynamic solutions in a wide frequency domain. Based on this, we can easily conclude that once the low computational accuracy and huge computational cost of the TFEMs can be overcome efficiently, the methods based on the theory of the finite element method can be applied for solving the mid-frequency problem and providing the reliable dynamic solutions in a wide-frequency domain when simulating the stiffened-plate composite structures. Based on this view, this paper will focus on proposing the Composite B-spline Wavelet Elements Method (CBWEM) derived from the Wavelet Element Methods (WFEMs) which own much better computational efficiency and accuracy compared with the TFEMs. Therefore, the B-spline wavelet and the theory of finite element method will be introduced into this paper for capturing the excellent computing efficiency and accuracy.

On the one hand, for the B-spline and wavelet functions, by observing the relevant researches, we can easily find that these functions own the excellent properties of multi-resolution and the multi-scale when proceeding the numerical analysis or signal processing [19], [20]. And until to now, the B-spline and wavelet functions have been introduced into various applications. Among these, the adaptive wavelet methods were developed for solving the hardening problem in elastoplasticity [21]. And, the non-uniform rational B-splines were applied for producing the basis functions and constructing an exact geometric model [22]. Later, a Reissner–Mindlin shell formulation based on a degenerated solid is implemented for non-uniform rational B-splines based isogeometric analysis [23]. And, in the field of mathematics, the cubic B-spline wavelets were introduced to reduce the fractional Integro-differential equation to system of algebraic equations [24]. By observing these researches, it is obvious that the B-spline and wavelet functions own much better advantages for proceeding the numerical analysis. On the other hand, based on the Chen and coworkers [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], we can clearly find that the WFEMs based on the B-spline and the finite element method own excellent computational efficiency (low computational cost and much better computing accuracy) compared with the TFEMs. And, the WFEMs was applied for quantitative crack identification by using particle swarm optimization [35], and the Hybrid method combing the B-spline wavelet finite element method and spectral analysis was proposed to investigate the wave propagation [36]. Recently, the high frequency response of the thin plate structures with uncertain parameters also has been simulated successfully and efficiently based on the B-spline wavelet finite element method [37].

Therefore, we can easily conclude that the proposed method based on the WFEMs may provide the potential to solve the mid-frequency problem caused by the low computing accuracy and huge computational cost when predicting the stiffened-plate composite structures’ dynamic responses based on the TFEMs. According to the theory of finite element method, in order to simulate the stiffened-plate composite structures based on the c₁ type wavelet elements, this research introduces the c₁ type B-spline wavelet beam and thin plate elements and applies these elements for modeling the stiffener and thin plate substructures of the stiffened-plate composite structures, respectively. Evidently, the stiffened-plate composite structures can be simulated by these two c₁ types wavelet element only when the suitable constraint matrix C_BSWI being constructed efficiently. Unfortunately, due to the complex interpolation functions and transformation matrices of these two c₁ types B-spline wavelet elements [38], the constraint matrix C_BSWI cannot be constructed based on the existing numerical scheme. In order to solve the problem, this paper derives the formulas of the new numerical scheme for constructing the properly constraint matrix C_BSWI and applies the CBWEM for solving the mid-frequency problem and predicting the dynamic responses of the stiffened-plate composite structures in a wide frequency domain.

Evidently, there are two main research contents in this paper: (a) deducing the essential formulas of the new numerical scheme for constructing the properly constraint matrix C_BSWI; (b) proposing the CBWEM based on the constraint matrix C_BSWI and the c₁ type B-spline wavelet plate and beam elements for solving the mid-frequency problem and predicting the dynamic response of the stiffened-plate composite structures in a wide frequency domain. In order to introduce these two main research contents, this study will be organized as follows.

Firstly, in order to make readers more than understand this article mainly contents, the formulas to construct the c₁ type B-spline wavelet plate and beam elements will be introduced briefly. Secondly, the essential formulas of the new numerical scheme to construct the C_BSWI and the CBWEM to proceed dynamic analysis will be deduced and introduced carefully. Lastly, the numerical and experimental studies are adopted for verifying the CBWEM's prediction ability to solve the mid-frequency problem and predict the dynamic responses of the stiffened-plate composite structures in a wide frequency domain.