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Stochastic interval analysis for structural natural frequencies based on stochastic hybrid perturbation edge-based smoothing finite element method
Engineering Analysis with Boundary Elements 103 (2019) 41–50
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Stochastic interval analysis for structural natural frequencies based on stochastic hybrid perturbation edge-based smoothing finite element method F. Wu1, M. Hu∗, K. Chen, L.Y. Yao, Z.G. Ju, X. Sun College of Engineering and Technology, Southwest University, Chongqing 400715, PR China
a r t i c l e
i n f o
Keywords: Edge-based smoothing technique FEM Natural frequencies Stochastic perturbation technique First-order random interval perturbation method
a b s t r a c t Realistically calculating of natural frequencies is crucial for studying dynamical characteristics of structures in engineering. In this paper, the gradient smoothing technique is first applied into the conventional finite element method (named ES-FEM), to improve the accuracy of the deterministic calculation of natural frequencies. Then, the hybrid random and interval uncertain parameters are introduced into the proposed ES-FEM based on hybrid stochastic and interval perturbation method. Expressions for the mean value and variance of natural frequencies are derived by combining the random interval perturbation method and the random interval moment method. The upper and lower bounds of mean value and variance of natural frequencies are calculated based on hybrid perturbation vertex method. The proposed method (denoted ESHPVM) is compared with the different methods. The high accuracy and efficiency of the proposed methods are verified by two numerical examples.
1. Introduction Among many typical engineering structures, e.g., bridges, buildings, offshore structures, vehicles, ships, and aerospace structures, the index of natural frequency is a key indicator for evaluating dynamical performance of these systems. At present, the most widely used numerical methods, i.e., the conventional finite element method (FEM), still holds potential flaws, e.g., low accuracy and ignoring uncertain factors. In the conventional FEM model, the calculated natural frequency value of the structure is often much larger than the real physical modal value, due to the stiffer discrete FEM system constructed based on Galerkin theory [1,2]. Moreover, these numerical errors become larger with the increase of the computational frequency. In the context of smoothing finite method(S-FEM) [3], the stiffness matrix could be soften based on the smoothing gradient technique, to alleviate accuracy problems in the higher frequency domain. Different smoothing domains could be built based on nodes, edges or faces of basic elements, for different desirable ‘‘smoothing effects’’, thus different S-FEM models have been proposed, such as the node-based S-FEM (NS-FEM) [4,5], edge-based S-FEM (ESFEM) [6,7] and face-based S-FEM (FS-FEM) [8,9]. Recently, hybrid SFEM models are also developed, such as such as the 𝛼 FEM [10] and 𝛽 FEM [11,12]. It is noted that even the same smoothing gradient technique (meaning same way of construction of smoothing domains) can be applied to solve a wide class of different practical mechanics problems for desirable solutions, such as elastic–plastic analysis, plates and shells
∗
1
composites, stochastic analysis, vibration and dynamic analysis structural acoustics, adaptive analysis, heat transfer and thermo-mechanical problems, and fluid–structure interaction [3–18] etc. In this work, the 3D edge-based smoothing gradient technique is applied into the conventional finite element method, reducing the dispersion error in the deterministic calculations [7]. On the other hand, uncertainties are often artificially ignored in the conventional FEM models. In the early 1960s, Boyce [19] and Collins and Thomson [20] did some theoretical research work on the probability properties of natural frequency by introducing the random parameters. Scheidt and Purker [21] summarized random eigenvalue problems systematically in their book in 1983. Later, Ibrahim [22] introduced a series of parameter uncertainties into the structural systems and different dynamics properties are carefully compared in 1987. Then, based on the perturbation method [23], stochastic sensitivity analysis of eigenvalues and eigenvectors are studied. In 2007, the natural frequency and mode shape of structures with uncertainty are further analyzed [24]. Based on the multidimensional integrals, Adhikari [25] further obtained the joint probability distribution of the natural frequencies. It is noted that all the above-mentioned uncertain models belongs to probabilistic and statistical models by assuming that all the input parameters have the deterministic probability distribution [26,27]. For the input parameters with uncertain probability distribution, the interval method offered an alternative to quantify these uncertainties with possible value range [28–30]. Liu et al. [31] even proposed forward and
Corresponding author. E-mail address: [email protected] (M. Hu). These authors contributed equally to this work (co-first authors).
https://doi.org/10.1016/j.enganabound.2019.01.020 Received 26 September 2018; Received in revised form 30 December 2018; Accepted 27 January 2019 Available online 15 March 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
F. Wu, M. Hu and K. Chen et al.
Engineering Analysis with Boundary Elements 103 (2019) 41–50
inverse structural uncertainty propagations to handle arbitrary probability distribution. Lots of research efforts have been spent on the interval eigenvalue problem [32,33] in the past two decades. Based on the perturbation method, Chen et al. [33] obtained the upper and lower bounds of eigenvalues of structural systems. Qiu et al. [34] improved the computational efficiency of the calculation of the eigenvalue bounds by using the parameter vertex method. Angeli et al. [35] introduced the polytypic uncertainties into the dynamical system, and the bounds of natural frequency are obtained. Other similar methods can also be classified as the categories such as interval factor method [36], interval finite-element method [37]. Another alternative method is fuzzy set theory [38] in which uncertainties can be quantified based on the use of membership functions (describing the degree of possibility). Actually, the uncertain parameters with probabilities distributions or only bounds information may exist simultaneously. In response to this problem, Ben-Haim [39] developed the random and interval mixed uncertain models. Combined the random and interval perturbation techniques and random moments techniques, Gao et al. [40] analyzed the static response of random and interval mixed uncertain structures. Based on the hybrid perturbation Monte Carlo Method, Xia et al. [41] constructed the random and interval mixed uncertain structuralacoustic model and the variation range of the expectation and variance of frequency response can be obtained. Wang et al. [42] further extended the random and interval technique into the analysis of natural frequencies and mode shapes of structures with hybrid random and interval parameters. In the contrast to extensive research on static mixed uncertainty problems, the research for the dynamic problems of structures with mixed random and interval variables mainly focused on the conventional FEM models. It is meaningful to further integrate the stochastic model and the interval model into the smoothed FEM model and to predict the range of expectation and variance of the natural frequency with higher accuracy. Based on the above discussion, we will introduce the generalized gradient smoothing technique, i.e., the stochastic and interval uncertainty model into the conventional FEM method in this work. The paper is organized as follows: in Section 2, the basic principles of smoothing finite element method for 3D solid problems are briefly described. In Section 3, the hybrid ESHPVM equations are constructed in order to achieve an efficient stochastic and interval uncertainty method. In Section 4, numerical examples and applications are presented to demonstrate the performance of the hybrid ESHPVM for analysis of natural frequencies of structures. Finally, a summary is given in Section 5 to conclude this work.
Fig. 1. 3D edge-based smoothing domain.
in which 𝑉𝑘𝑠 denotes the volume of the smoothing domain k. u denotes the displacement vector which can be obtained based on finite element interpolation: 𝐮=
𝑖=1
⎡𝐍𝑖 (𝑥) 𝐍𝑖 (𝑥) = ⎢ 0 ⎢ ⎣ 0
𝛆(𝑥𝑘 ) =
0 ⎤ 0 ⎥ ⎥ 𝐍𝑖 (𝑥)⎦
0 𝐍𝑖 (𝑥) 0
(3)
𝐌𝑘 ∑ 𝑖=1
𝐁𝑖 (𝑥𝑘 )𝐝𝑖
(4)
where Mk denotes the number of sub-smoothing domain, 𝐁𝑖 is smoothing strain matrix can be expressed as follows: ⎡𝑏𝑖𝑥 (𝑥𝑘 ) ⎢ 0 ⎢ ⎢ 0 𝐁𝑖 ( 𝑥 𝑘 ) = ⎢ ⎢𝑏𝑖𝑦 (𝑥𝑘 ) ⎢ 0 ⎢ ⎣𝑏𝑖𝑧 (𝑥𝑘 )
0 𝑏𝑖𝑦 (𝑥𝑘 ) 0 𝑏𝑖𝑥 (𝑥𝑘 ) 𝑏𝑖𝑧 (𝑥𝑘 ) 0
0 ⎤ 0 ⎥⎥ 𝑏𝑖𝑧 (𝑥𝑘 )⎥ ⎥ 0 ⎥ 𝑏𝑖𝑦 (𝑥𝑘 )⎥ ⎥ 𝑏𝑖𝑥 (𝑥𝑘 )⎦
(5)
where the 𝐛𝑖𝑝 can be expressed as: 𝐛𝑖𝑝 =
In this section, the ES-FEM formulations for 3D solid problems are briefly introduced. First, the 3D solid domain is discretized using four nodes tetrahedral elements as that of standard FEM. Then, based on the edges of tetrahedral elements, the edge-based gradient smoothing domains can be constructed as shown in Fig. 1. The sub-smoothing domain of edge k in cell i is created by connecting the centroid of cell i to the two end-nodes of the edge k and the related surface triangles. The entire problem domain can be formed by assembling all smoothing areas, 𝛀𝑠𝑖 ∩ 𝛀𝑠𝑗 = 𝜙, ∀𝑖 ≠ 𝑗. where Ns is the number of smoothing domain. s is the number of all tetrahedral faces. It is noted that the calculation of element strain is no longer based on the standard finite element, but based on the newly constructed smoothing domain. The smoothing domain now is also serving as integration domains. The strain 𝛆𝑘 is first smoothed based on the smoothing gradient technique within the smoothing domain Ωsk , as below:
𝑘
(2)
Substituting Eq. (2) into Eq. (1), the smoothed strain can be written as follows:
2.1. The construction of three-dimensional edge-based smoothing domain
1 1 𝛆 𝑑Ω = 𝑠 𝐋𝐮 𝑑 Ω 𝑉𝑘𝑠 ∫Ω𝑠 𝑘 𝑉𝑘 ∫Ω𝑠
𝐍 𝑖 ( 𝑥 ) 𝐝 𝑖 =𝐍 𝑠 𝐝
in which, N denotes the number of nodes tetrahedral elements. di = {dxi dyi dzi} denotes the node displacements. N represents the shape function matrix, which can be written as follows:
2. Theoretical equation ES-FEM for modal analysis
𝛆𝑘 =
𝐍 ∑
1 𝐍 (𝑥) ⋅ 𝐧𝑝 (𝑥)𝑑𝚪 𝑉𝑘𝑠 ∫𝛀𝑠 𝑖
(𝑝 = 𝑥, 𝑦, 𝑧)
(6)
𝑘
in which np is the normal vector. 2.2. The discretely vibrating equation of three-dimensional ES-FEM The standard Galerkin weak form for vibration mechanics problem without damping can be expressed as ∫Ω
𝑇
𝛿(𝛆(𝑢)) 𝐃(𝛆(𝑢))𝑑Ω +
∫Ω
𝛿𝐮𝑇 𝜌
𝜕2 𝐮 𝑑Ω − 𝛿𝐮𝑇 𝐛𝑑Ω − 𝛿𝐮𝑇 𝛕𝑑Γ = 0 ∫Ω ∫Γ𝑓 𝜕 𝑡2 (7)
Substituting Eqs. (1) and (2) into Eq. (7), the final discrete linear equations can be expressed as follows: { } 𝐌 𝐝̈ + 𝐊{𝐝} = {𝐅} (8) in which F represents the external load vector. M is the lumped mass matrix just as same as in the standard finite element method, 𝐊 denotes
(1)
𝑘
42
F. Wu, M. Hu and K. Chen et al.
Engineering Analysis with Boundary Elements 103 (2019) 41–50
the smoothed stiffness matrix constructed based on the smoothed strain matrix as: 𝐊=
𝑇
∫Ω
𝐌=𝜌
𝐁𝑓 𝐃𝐁𝑓 𝑑Ω
∫Ω
In which, c stands for mean of variables. Ignoring the residual term ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) with hybrid random and interval variRe, the stiffness matrix 𝐊(𝑥 ables can be marked as
(9)
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 )+ ▵ 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) 𝐊(𝑥
𝑇
𝐍𝑓 𝐍𝑓 𝑑Ω
(10)
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) is the perturbation range of the term 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗), can in which ▵ 𝐊(𝑥 be written as:
For the natural frequency analysis, the following characteristic equation can be obtained: (𝐊 − 𝜔𝑘 2 𝐌)𝛗 ⃖⃖⃖⃖𝑘⃗ = 0
𝑚 ( ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = ∑ 𝜕 𝐊 || ▵𝐊 𝑥 ▵ 𝑦𝐼𝑗 𝐼 | ⃖⃖⃖𝑅 ⃗ 𝑐 𝑗=1 𝜕𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ ( ) 𝑛 𝑚 ( ) ∑ ∑ 𝜕 𝐊 || 𝜕 2 𝐊 || 𝐼 𝑅 + + ▵ 𝑦 𝑥𝑅 |⃖⃖⃖𝑅⃗ 𝑐 |⃖⃖⃖𝑅⃗ 𝑐 𝑗 𝑖 − 𝑥𝑖 𝑅 𝑅 𝐼 𝜕𝑥𝑖 |𝑥 ,𝑦⃖⃖⃗ 𝑗=1 𝜕 𝑥𝑖 𝜕 𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ 𝑖=1
(11)
in which, 𝜔k denotes the natural frequency of the k-order and 𝛗 ⃖⃖⃖⃖𝑘⃗ is the corresponding modal eigenvector. Then the solution of the system’s natural frequency can be obtained by solving the eigenvalue: 𝐊𝛗𝑘 = 𝜔2𝑘 𝐌𝛗 ⃖⃖⃖⃖𝑘⃗
(18)
(19)
(12) ′
𝜕2 𝐊 | , 𝜕 𝑥𝑅 𝜕 𝑦𝐼𝑗 𝑥 ⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝑖
3.1. Stochastic and interval mixed uncertainty modeling
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) ⋅ 𝛗 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = 𝜔 2 (𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) ⋅ 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) ⋅ 𝛗 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) 𝐊(𝑥 ⃖⃖⃖⃖𝑘⃗(𝑥 ⃖⃖⃖⃖𝑘⃗(𝑥 𝑘
𝑗=1
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) can be simplified as: Similarly, the mass matrix 𝐌(𝑥 ( ) ( ) ( ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 + ▵ 𝐌 𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = 𝐌 𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ 𝐌 𝑥
𝑖
𝑚 ( ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = ∑ 𝜕𝐌 || ▵𝐌 𝑥 ▵ 𝑦𝐼𝑗 𝐼 | ⃖⃖⃖𝑅 ⃗ 𝑐 𝑗=1 𝜕𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗
+
𝑖
the expectation of the interval vector 𝑦⃖⃖⃖𝐼⃗ = (𝑦𝐼1 , 𝑦𝐼2 , ..., 𝑦𝐼𝑚 ) is remarked with 𝑦⃖⃖⃖⃗𝑐 = (𝑦𝑐1 , 𝑦𝑐2 , ..., 𝑦𝑐𝑚 ), in which 𝑦𝑐𝑗 =
𝑦𝑐𝑗 +𝑦𝑐𝑗 2
1
𝑖=1
2
𝜕 𝐊 || 𝑅 𝑅 |⃖⃖⃖𝑅⃗ (𝑥𝑖 − 𝑥𝑖 ) + Re 𝜕𝑥𝑅 𝑖 |𝑥
𝑛 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) + ∑ 𝜕𝐌 || (𝑥𝑅 − 𝑥𝑅 ) + Re 𝐌(𝑥 𝑖 𝑅 | ⃖⃖⃖𝑅 ⃗ 𝑖 𝑖=1 𝜕𝑥𝑖 |𝑥
Using 𝐌′ 𝑅 , 𝐌′ 𝐼 and 𝐌′′𝑅 𝑥
𝜕2 𝐌 , 𝐼 | 𝜕 𝑥𝑅 ⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝑖 𝜕 𝑦𝑗 𝑥
𝑛
2
𝑦
𝑥 𝑦𝐼
to denote
𝜕𝐌 | 𝜕𝐌 | | | 𝜕𝑥𝑅 ⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 , 𝜕𝑦𝐼𝑗 | 𝑥⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝑖 |𝑥
and
respectively, so Eq. (22) can be simplified as:
( ) 𝑚 𝑛 𝑚 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = ∑ 𝐌′ ▵ 𝑦𝐼 + ∑ 𝐌′ + ∑ 𝐌′′ 𝐼 𝑅 𝑅 ▵ 𝐌(𝑥 ▵ 𝑦 𝑗 𝑗 (𝑥𝑖 − 𝑥𝑖 ) 𝑦𝐼 𝑥𝑅 𝑥𝑅 𝑦𝐼
(14)
𝑗=1
𝑖=1
𝑗=1
(15)
(23) Using ▵wk and ▵ 𝛗 ⃖⃖⃖⃖𝑘⃗ to denote the perturbation of k-order natural
where Re is the residual term. Assuming the random vector as a constant, the interval vector 𝑦⃖⃖⃖𝐼⃗ can also be expanded by the first-order Taylor series at expectation of 𝑦⃖⃖⃖⃗𝑐 = (𝑦𝑐 , 𝑦𝑐 , ..., 𝑦𝑐 ): 1
| | 𝑛 ⎛ 𝑚 ⎞( ) ∑ ∑ | 𝜕 2 𝐌 || ⎜ 𝜕𝐌 | + ▵ 𝑦𝐼𝑗 ⎟ 𝑥𝑅 − 𝑥𝑅 | | 𝑖 𝑖 𝑅 𝑅 𝐼 ⎜ ⎟ ⃖⃖⃖𝑅⃗ 𝑐 𝑗=1 𝜕 𝑥 𝜕 𝑦 | 𝑅 𝑐 𝑖=1 ⎝ 𝜕𝑥𝑖 ||𝑥 ,𝑦⃖⃖⃗ 𝑖 𝑗 |𝑥 ⃖⃖⃖⃗,𝑦⃖⃖⃗ ⎠ (22)
, 1 ≤ 𝑗 ≤ 𝑚.The stiffness vec-
tor 𝐊 and the mass matrix M can be expanded by the first-order Taylor ⃖⃖⃖⃖𝑅⃗ = (𝑥𝑅 , 𝑥𝑅 , ..., 𝑥𝑅 ). series at the expectation of the random vector 𝑥 𝑛 ∑
(21)
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗), which is The perturbation range of the mass matrix is ▵ 𝐌(𝑥 given by:
(13)
⃖⃖⃖⃖𝑅⃗ = (𝑥𝑅 , 𝑥𝑅 , ..., 𝑥𝑅 ) denotes the expectation of random We first use 𝑥 𝑛 1 2 ⃖⃖⃖⃖ ⃗ 𝑅 vector 𝑥𝑅 = (𝑥 , 𝑥𝑅 , ..., 𝑥𝑅 ) in which 𝑥𝑅 = exp(𝑥𝑅 ), 1 ≤ 𝑖 ≤ 𝑛; Similarly,
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) + 𝐊(𝑥
𝑖=1
(20)
3.2. The perturbation analysis of natural frequency
𝑛
and
respectively. Then, Eq. (19) can be simplified as:
𝑗=1
and the interval vector 𝑦⃖⃖⃖𝐼⃗. The hybrid random and interval uncertainties are introduced into the natural frequency analysis of the structural system. Then, Eq. (12) can be rewritten as:
2
𝜕𝐊 | , 𝜕𝐊 | 𝜕𝑥𝑅 ⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝜕𝑦𝐼𝑗 𝑥⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝑖 𝑥
𝑚
2
the stiffness matrix 𝐊 and the mass matrix M. Obviously, the natural fre⃖⃖⃖⃖𝑅⃗ quency value of the system is also the function of the random vector 𝑥
1
′′
( ) 𝑚 𝑛 𝑚 ( ) ( ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = ∑ 𝐊′ ▵ 𝑦𝐼 + ∑ 𝐊′ + ∑ 𝐊′′ 𝐼 𝑅 ▵𝐊 𝑥 ▵ 𝑦 𝑥𝑅 𝐼 𝑅 𝑅 𝐼 𝑦 𝑥 𝑥 𝑦 𝑗 𝑗 𝑖 − 𝑥𝑖
⃖⃖⃖⃖𝑅⃗ = (𝑥𝑅 , 𝑥𝑅 , ..., 𝑥𝑅 ) and the interWe first define the random vector 𝑥 𝑛 1 2 ⃖⃖⃖ ⃗ 𝐼 𝐼 𝐼 𝐼 val vector 𝑦 = (𝑦 , 𝑦 , ..., 𝑦 ) in the structure system, which consisted in 1
′
Using 𝐊𝑥𝑅 , 𝐊𝑦𝐼 and 𝐊𝑥𝑅 𝑦𝐼 to denote the term
3. Hybrid perturbation vertex method in analysis of natural frequency
𝑐
frequency and its modal eigenvector, in addition, using 𝐊 , ▵ 𝐊, 𝐌𝑐 , ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ), ▵ 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗), 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ), ▵ 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗), ⃖⃖⃖⃖𝑐⃗ denote 𝐊(𝑥 ▵M, 𝑤𝑐 and 𝛗
𝑚
𝑘
( ) 𝑚 ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 + ∑ 𝜕 𝐊 || ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = 𝐊 𝑥 𝐊 𝑥 ▵ 𝑦𝐼𝑗 𝐼 | ⃖⃖⃖𝑅 ⃗ 𝑐 𝑗=1 𝜕𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ ( ) 𝑛 𝑚 ( ) ∑ ∑ 𝜕 𝐊 || 𝜕 2 𝐊 || 𝐼 𝑅 + Re + + ▵ 𝑦𝑗 𝑥𝑅 𝑖 − 𝑥𝑖 𝑅 || ⃖⃖⃖𝑅 𝑅 𝜕 𝑦𝐼 || ⃖⃖⃖𝑅 𝑐 𝑐 ⃗ ⃗ ⃖⃖ ⃗ ⃖⃖ ⃗ 𝜕𝑥 𝜕 𝑥 𝑖=1 𝑗=1 𝑖 𝑥 ,𝑦 𝑖 𝑗 𝑥 ,𝑦 (
𝑘
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ) 𝑤𝑐𝑘 (𝑥
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ) respectively, so Eq. (13) can be simplified as ⃖⃖⃖⃖𝑐⃗(𝑥 and 𝛗 𝑘
( ) ( ) ) ( 𝑐 𝑐 + ▵ 𝑤 )2 ⋅ (𝐌𝑐 + ▵ 𝐌) ⋅ 𝛗 ⃖⃖⃖⃖𝑐⃗+ ▵ 𝛗 ⃖⃖⃖⃖𝑐⃗+ ▵ 𝛗 𝐊 +▵𝐊 ⋅ 𝛗 = ( 𝑤 ⃖⃖⃖⃖ ⃗ ⃖⃖⃖⃖ ⃗ 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 (24) (16)
Ignoring the higher order terms, ▵wk can be calculated
( ) 𝑚 ) | ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = 𝐌 𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 + ∑ 𝜕𝐌 || 𝐌 𝑥 ▵ 𝑦𝐼𝑗 𝐼 | ⃖⃖⃖𝑅 ⃗ 𝑐 𝑗=1 𝜕𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ ( ) 𝑛 𝑚 ( ) ∑ ∑ 𝜕𝐌 || 𝜕 2 𝐌 || 𝐼 𝑅 + Re 𝑥𝑅 + |⃖⃖⃖𝑅⃗ 𝑐 + |⃖⃖⃖𝑅⃗ 𝑐 ▵ 𝑦𝑗 𝑖 − 𝑥𝑖 𝑅 𝑅 𝐼 𝜕𝑥𝑖 |𝑥 ,𝑦⃖⃖⃗ 𝑗=1 𝜕 𝑥𝑖 𝜕 𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ 𝑖=1 (
▵ 𝑤𝑘 =
𝑇⃗ 1 ⃖⃖⃖⃖⃖⃖ ⃖⃖⃖⃖𝑐⃗ ▵ 𝐌)𝛗 𝛗𝑐𝑘 (▵ 𝐊 − 𝑤𝑐2 𝑘 𝑘 𝑐 2𝑤𝑘
(25)
In which, T stands for transposed matrix. Therefore, k-order natural frequency wk will be: 𝑤𝑘 = 𝑤𝑐𝑘 +
(17)
43
( ) 𝑇⃗ 1 ⃖⃖⃖⃖⃖⃖ ⃖⃖⃖⃖𝑐⃗ ▵𝐌 𝛗 𝛗𝑐𝑘 ▵ 𝐊 − 𝑤𝑐2 𝑘 𝑘 𝑐 2𝑤𝑘
(26)
F. Wu, M. Hu and K. Chen et al.
Engineering Analysis with Boundary Elements 103 (2019) 41–50
3.3. The analysis variance rang of the expectation and variance of natural frequency Based on the random moment technique, the k-order natural frequency wk and expectation Exp(wk )in Eq. (26) can be marked as: ( ) 𝑇⃗ 1 ⃖⃖⃖⃖⃖⃖ ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ) − 𝑤𝑐2 ▵ 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ) 𝛗 ⃖⃖⃖⃖𝑐⃗ 𝛗𝑐𝑘 ▵ 𝐊(𝑥 𝑘 𝑘 2𝑤𝑐𝑘 (𝑚 ) ′ 𝑇⃗ ∑ 1 ⃖⃖⃖⃖⃖⃖ ⃖⃖⃖⃖𝑐⃗ = 𝑤𝑐𝑘 + (𝐊𝑦𝐼 − 𝑤𝑐2 · 𝐌′ 𝐼 ) ▵ 𝑦𝐼𝑗 𝛗 𝛗𝑐𝑘 𝑘 𝑘 𝑦𝑗 𝑗 2𝑤𝑐𝑘 𝑗=1
𝐸𝑥𝑝(𝑤𝑘 ) = 𝑤𝑐𝑘 +
Fig. 2. Beam structure model.
(27)
⃖⃖⃖⃖⃖⃖𝑇⃗ ′ ⃖⃖⃖⃖𝑐⃗ . It is noted that Using 𝐷𝑤𝐼 to donate the term 𝛗𝑐𝑘 (𝐊𝑦𝐼 − 𝑤𝑐2 · 𝐌′ 𝐼 )𝛗 𝑘 𝑘 𝑦𝑗
𝑦𝑗
𝑗
𝐷𝑤𝐼 is a constant value. Then, Eq. (27) can be simplified as: 𝑦𝑗
𝐸𝑥𝑝(𝑤𝑘 ) = 𝑤𝑐𝑘 +
𝑚 1 ∑
2𝑤𝑐𝑘
𝑗=1
( ) 𝐷𝑤𝐼 Δ𝑦𝑗
(28)
𝑦𝑗
Fig. 3. Finite element meshes of beam structure.
Similarly, the variance of the natural frequency Var(wk ) of the korder wk can be obtained based on the random moment technique: 𝑉 𝑎𝑟(𝑤𝑘 ) =
(
𝑛 ∑ 𝑛 ∑ 𝑖=1 𝑙=1
(
1
𝐷𝑤𝑅 + 𝑥𝑖
4𝑤𝑐2 𝑘
× 𝐷𝑤𝑅 +
𝑚 ∑
𝑥𝑙
in
which
𝐷𝑤𝑅 𝑥𝑖
(
𝑚 ∑ 𝑗=1
( )) 𝐷𝑤𝑅 𝐼 Δ𝑦𝐼𝑗 𝑥𝑖 𝑦𝑗
𝐷𝑤𝑅 𝐼 Δ𝑦𝐼𝑗 𝑥𝑙 𝑦𝑗
𝑗=1
)) 𝑅 𝐶𝑜𝑣(𝑥𝑅 𝑖 , 𝑥𝑙 )
𝑥𝑖
𝑖
𝐷𝑤𝑅
𝑥𝑙 𝑦𝐼𝑗
denotes
𝑛 ⎛ ∑ ⎜ 1 ⎜ 𝑖=1 ⎝ 4𝑤𝑐2 𝑘
( 𝐷𝑤𝑅 + 𝑥𝑖
𝑚 ∑ 𝑗=1
( 𝐷𝑤𝑅
𝑥𝑖 𝑦𝐼𝑗
▵ 𝑦𝐼𝑗
))2
⎞ ⎟ 𝑉 𝑎𝑟(𝑥𝑅 𝑖 )⎟ ⎠
(30)
From the relationship of standard deviation and variance, we have: 𝑆𝑡𝑑(𝑤𝑘 ) =
√
𝑉 𝑎𝑟(𝑤𝑘 ) ( ) )| | 𝑚 ( ∑ | 1 || 𝑤 𝑤 𝐼 | 𝑅 𝑆𝑡𝑑(𝑤𝑘 ) = 𝐷 𝑅 𝐼 Δ𝑦𝑗 |𝑆𝑡𝑑(𝑥𝑖 ) |𝐷 𝑅 + 𝑥𝑖 𝑦𝑗 | 2𝑤𝑐𝑘 || 𝑥𝑖 𝑖=1 𝑗=1 | 𝑛 ∑
(31)
𝑦𝑗
𝑥𝑖
𝑤𝑐𝑘 + {
𝐸𝑥𝑝(𝑤𝑘 ) =
𝑤𝑐𝑘 +
𝑚 1 ∑
2𝑤𝑐𝑘
𝑗=1
𝑥𝑙 𝑦𝑗
𝑚 1 ∑
2𝑤𝑐𝑘
𝑗=1
The proposed approach is first applied in standard beam structure model. As shown in Fig. 2, the size of the beam structure is 0.04 × 1.0 × 0.03 m, and the material is aluminum. The deterministic parameters are shown in Table 1 As shown in Fig. 3, the beam structure model is first divided into discrete tetrahedron elements, which contains 918 tetrahedron elements and 414 nodes. The beam was not free–free but simple supports on both sides. In edge-based finite element model, uncertain parameters with sufficient samples are considered as random variables, others are considered as interval variables. It is noted that all the variables are considered to be independent of each other, the detail information of input variables of beam are shown in Table 2.
min
( )} 𝑤 𝐷 𝐼 Δ𝑦𝑗 𝑦𝑗
(33)
4.1. Example 1 – the analysis of ESHPVM based on beam structure
( )} 𝐷𝑤𝐼 Δ𝑦𝑗 𝑦𝑗
(
In this subsection, the validity and efficiency of the proposed ESHPVM will first be illustrated by two typical frame structures.
terministic values, while the expectation Exp(wk ) and standard deviation Std(wk ) are the linear equations of interval perturbation ▵ 𝑦𝐼𝑗 , Based on the HPVM(Hybrid Perturbation Vertex Method), the upper and lower bounds of expectation Exp(wk ) and standard deviation Std(wk ) can be obtained by substituting two extremums of ▵ 𝑦𝐼𝑗 into Eqs. (27) and (31): {
𝑛 ∑
4. The numerical example
Because ▵ 𝑦𝐼𝑗 is an interval variable, the expectation of k-order natural frequency wk is also an interval variable, rather than a constant value. As for Eqs. (27) and (31), 𝑤𝑐𝑘 ,𝐷𝑤𝐼 ,𝐷𝑤𝑅 ,𝐷𝑤𝑅 𝐼 and 𝑆𝑡𝑑(𝑥𝑅 𝑖 ) are de-
𝐸𝑥𝑝(𝑤𝑘 ) =
2700
Based on the above mathematical derivation, the proposed method is named ESHPVM. Several approaches (HPVEM, ESHPMCM and HPMCM)) are introduced as the reference methods. HPVM is constructed based on similar techniques as that of ESHPVEM. The main difference between these two methods lies in the imbedded deterministic methods: ESHPVM is constructed based on ES-FEM, while HPVEM is constructed based on the conventional FEM. In the ESHPMCM, the upper and lower bound of expectation and standard deviation is calculated based on Monte-Carlo Method rather than Vertex Method as in the ESHPVEM. Different from ESHPVEM, HPMCM is constructed based on conventional FEM rather than ES-FEM.
𝑅 In general, the independent random variables𝑥𝑅 𝑖 and𝑥𝑙 satisfy the re𝑅 ) = 0, 𝐶𝑜𝑣(𝑥𝑅 , 𝑥𝑅 ) = 𝑉 𝑎𝑟(𝑥𝑅 ), Eq. (29) can be exlationship 𝐶𝑜𝑣(𝑥𝑅 , 𝑥 𝑖 𝑖 𝑖 𝑖 𝑙 pressed as:
𝑉 𝑎𝑟(𝑤𝑘 ) =
Length:1.0 width:0.3 Height:0.04
)} )| | 𝑚 ( ∑ | 1 || 𝑤 𝑤 𝐼 | 𝑅 𝐷 𝑆𝑡𝑑( + 𝐷 Δ𝑦 𝑥 ) 𝐼 | 𝑅 𝑗 | 𝑖 𝑥𝑅 | 𝑖 𝑦𝑗 2𝑤𝑐𝑘 || 𝑥𝑖 𝑖=1 𝑗=1 | min { 𝑛 ( )} )| | 𝑚 ( ∑ ∑ | | 1 | 𝑤 𝑆𝑡𝑑 (𝑤𝑘 ) = 𝐷𝑤𝑅 𝐼 Δ𝑦𝐼𝑗 ||𝑆𝑡𝑑(𝑥𝑅 |𝐷 𝑅 + 𝑖 ) 𝑥𝑖 𝑦𝑗 | 2𝑤𝑐𝑘 || 𝑥𝑖 𝑖=1 𝑗=1 | max
𝑥𝑖 𝑦𝑗
𝑗
Density kg/m3
{
(29)
⃖⃖⃖⃖⃖⃖𝑇⃗ ′′ ⃖⃖⃖⃖𝑐⃗. 𝛗𝑐𝑘 (𝐊𝑥𝑅 𝑦𝐼 − 𝑤𝑐2 ⋅ 𝐌′′𝑅 𝐼 )𝛗 𝑘 𝑘 𝑖
Structure size/m
𝑆𝑡𝑑 (𝑤𝑘 ) =
⃖⃖⃖⃖⃖⃖𝑇⃗ ′ ⃖⃖⃖⃖𝑐⃗. 𝛗𝑐𝑘 (𝐊𝑥𝑅 − 𝑤𝑐2 ⋅ 𝐌′ 𝑅 )𝛗 𝑘 𝑘
denotes
Table 1 Deterministic parameters of the beam.
(32)
max
Once the upper and lower bounds of variance are obtained, the bounds of standard deviation can be easily got. 44
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
Fig. 4. Upper and lower bound of expectation and standard deviation of the 7-order natural frequency of the natural frequency calculated by ESHPVM and ESPHMCM.
ESHPMCM, while the lower bound of ESHPVM is less than that of ESHPMCM. Therefore, ESHPMCM’s expectation and standard deviation are the subsets of ESHPVM. Moreover, ESHPVM’s results converge gradually to ESHPMCM’s results with the increase of iteration times of ESHPVM. Finally, the range of ESHPMCM’s expectation and standard deviation agree well with those of ESHPVM when the iteration times
First, the convergence rate of ESHPVM and ESHPMCM is first compared. The lower and upper bounds of expectation and variance of 7-order natural frequency the beam calculated by ESHPVM and ESHPMCM are shown in Fig. 4. As shown in Fig. 4, it is obvious that the upper bound of expectation and standard deviation of ESHPVM are greater than those of 45
F. Wu, M. Hu and K. Chen et al.
Engineering Analysis with Boundary Elements 103 (2019) 41–50
Fig. 5. The bounds of the expectation of the natural frequency of the beam.
Fig. 6. The bounds of the standard deviation of the natural frequency of the beam.
Table 2 Input variables of the beam. Interval variable
The relative error of top 10-order natural frequency of ESHPVM and ESHPMCM are shown in Tables 4 and 5. As shown in Figs. 5 and 6, the expectation and standard deviation of ESHPVM agree well with the result of ESHPMCM based on 2000 samples. As shown in Tables 4 and 5, there is negligible deviations between the results of ESHPVM and ESHPMCM. The upper bound of expectation and standard deviation of ESHPVM are greater than those of ESHPMCM, while the lower bound of ESHPVM is less than that of ESHPMCM. ESHPMCM’s results are the subsets of those of ESHPVM, verifying the accuracy and effectivity of proposed ESHPVM. In order to further illustrate the higher accuracy of ESHPVM, different types of elements are constructed (fine meshes and coarse meshes). As is well-known, the conventional FEM provides higher accuracy based on fine meshes. First, the variation range is calculated using ESHPVM and HPVM based on coarse meshes (consisting of 918 elements and 414 nodes). The reference results are obtained using HPMCM based on fine meshes (consisting of 918 6689 elements and 1937 nodes). Through comparing the overlapping of the results of ESHPVM, HPVM and HPMCM, we can clearly draw a conclusion of the accuracy difference between HPMCM and HPVM. As shown in the Fig. 7, it is obvious that ESHPVM provides a higher accuracy results compared with the HPVM in the whole 1–30 order modes. HPVM suffers from the dispersion errors mainly caused by the “overly-stiff” property when the standard FEM is embedded in the hybrid HPVM. The result obtained from ESHPVM shows excellent agreements with the reference result, and it demonstrates that the pollution errors are great eliminated by the “soften effects” of ES-FEM. To sum up, the high accuracy and computational efficiency of ESHPVM
Random variable
Young modulus (Pa)
Poisson’s ratio
Range: 6.3 × 1010 − 7.7 × 1010 Radius of perturbation: 7 × 109
Expectation:0.3 Variance:0.03
Table 3 The computational time of ESHPVM and ESHPMCM. Method
ESHPVM
ESHPMCM
Execution time (s)
18.3826
25.7977
are more than 2000. Compared with ESHPMCM which needs large quantities of iteration calculations, it is obvious that ESHPVM has higher computational efficiency. The computational time of ESHPVM and ESHPMCM are shown in Table 3. As shown in Fig. 4 and Table 3, it is concluded that the proposed ESHPVM can obtain the lower and upper bounds of expectation and variance of the natural frequency with much higher efficiently, verifying the efficiency of the ESHPVM. In order to compare different order natural frequencies, the expectation and standard deviation of natural frequencies (1 to 30 orders) of the beam are calculated using ESHPVM and ESHPMCM. The natural frequencies obtained from ESHPMCM with 2000 samples are used as the reference results, as shown in Figs. 5 and 6 46
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
Table 4 Bounds of the expectation of the first 10 natural frequencies of the beam. Natural frequency (rad/s)
Expectation ESHPVM
𝜔1 𝜔2 𝜔3 𝜔4 𝜔5 𝜔6 𝜔7 𝜔8 𝜔9 𝜔10
ESHPMCM
Lower bound
Relative error (%)
Upper bound
Relative error (%)
Lower bound
Upper bound
541.42 955.59 2151.30 2615.43 4763.42 5071.18 8143.61 8257.72 8334.85 12226.72
−0.029 −0.0017 −0.0150 −0.0103 −0.0016 −0.0021 −0.0090 −0.0048 −0.0031 −0.0177
656.91 1226.85 2602.16 3349.53 5772.38 6504.03 10205.04 10198.90 10617.81 15581.76
0.0016 0.0045 0.0197 0.0016 0.0020 0.0142 0.0171 0.0099 0.0014 0.0513
541.58 955.60 2151.63 2615.69 4763.49 5071.29 8144.34 8258.11 8335.96 12227.09
656.90 1226.79 2601.65 3349.48 5772.26 6504.00 10203.29 10197.89 10617.66 15573.77
Table 5 Bounds of the standard deviation of the first 10 natural frequencies of the beam. Natural frequency(rad/s)
Expectation ESHPVM
𝜔1 𝜔2 𝜔3 𝜔4 𝜔5 𝜔6 𝜔7 𝜔8 𝜔9 𝜔10
ESHPMCM
Lower bound
Relative error (%)
Upper bound
Relative error (%)
Lower bound
Upper bound
6.49 18.85 24.42 52.06 55.98 99.91 53.62 81.28 154.09 225.06
−0.0055 −0.0013 −0.0159 −0.0081 −0.0017 −0.0017 −0.0079 −0.0045 −0.0108 −0.0025
7.95 23.05 29.85 63.64 68.43 122.12 65.54 99.35 188.33 275.08
0.0017 0.0037 0.0207 0.0014 0.0021 0.0117 0.0154 0.0094 0.0012 0.0433
6.50 18.86 24.43 52.07 55.99 99.92 53.63 81.29 154.10 225.07
7.94 23.04 29.84 63.63 68.42 122.10 65.53 99.34 188.32 274.96
Fig. 7. The bounds of the expectation of the natural frequency of beam.
Table 6 Deterministic parameters of the bus bodywork.
for the natural frequency of the beam structure is illustrated in this example. 4.2. Example 2 – the analysis of natural frequency of complex frame structure The frame structures are widely exits in the different engineering area. The index of natural frequency is a key indicator for evaluating dynamical performance of these systems. In this subsection, the proposed ESHPVM is applied to analysis the natural frequency of a typical frame structure as shown in Fig. 8. The size of the frame structure is 0. 5 × 0.65 × 0.02 m, and the material is aluminum. The determinant material parameters of frame structure are shown in Table 6.
Density kg/m3
Thickness
2700
0.02
As shown in Fig. 9, the frame structure model is first divided into discrete tetrahedron elements, which contains 1936 tetrahedron elements and 892 nodes. Similarly, uncertain parameters with sufficient samples are considered as random variables, others are considered as interval variables. It is noted that all the variables are considered to be independent of each 47
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
Table 7 Input variables of the frame structures. Interval variable
Random variable
Young modulus (Pa)
Poisson’s ratio
Range: 6.3 × 1010 − 7.7 × 1010 Radius of perturbation: 7 × 109
Expectation: 0.3 Variance: 0.03
As shown in Figs. 10 and 11, the expectation and standard deviation of natural frequency calculated based on ESHPVM are consistent with the result of ESHPMCM based on 2000 samples. So the accuracy of ESHPMCM in practical engineering problems has been verified. In order to further illustrate the higher accuracy of ESHPVM, different types of elements are constructed (fine meshes: 6801 elements and 2450 nodes and coarse meshes: 1936 elements and 892 nodes). First, the variation range is calculated using ESHPVM and HPVM based on coarse meshes. The reference results are obtained using HPMCM based on fine meshes. Through comparing the overlapping of the results of ESHPVM, HPVM and HPMCM, we can clearly draw a conclusion of the accuracy difference between HPMCM and HPVM. The results for this typical frame structures re-enforces the conclusion from the previous simple beam example. As is shown in Fig. 12, the calculated natural frequency value of the structure based on HPVM is far larger than that of ESHPVM and reference results, due to the inherent drawback of “over-stiffness” in imbedded FEM. Moreover, as the frequency increases, the deviation between HPVM result and the ESHPVM or reference result becomes even larger, suggesting that the accuracy of the HPVM result decreases with the increase of the frequency. Contrarily, the ESHPVM always provides much more accurate result in higher frequency range, compared with the HPVM model using the same mesh. As a consequence, we verify the accuracy of ESHPVM is higher than that of HPVM in analyzing the natural frequency of frame structure. To sum up, the higher accuracy of ESHPVM for the natural frequency of the frame structure is illustrated in this example.
Fig. 8. A simplified structure for frame structure.
Fig. 9. Finite element meshes of frame structures.
5. Summary other, the detail information of input variables of frame structures are shown in Table 7 In order to compare different order natural frequencies, the expectation and standard deviation of natural frequencies (1–30 orders) of the frame structure are calculated using ESHPVM and ESHPMCM. The natural frequencies obtained from ESHPMCM with 2000 samples are used as the reference results, which are compared in Figs. 10 and 11.
In this work, a hybrid ESHPVM is proposed for predicting the lower and upper bounds of expectation and variance of natural frequency of structures with mixed random and interval variables. Expressions for the mean value, standard deviation and variance of natural frequencies are derived based on perturbation method and random interval moment method. Numerical examples of beam/frame Fig. 10. The bounds of the expectation of the natural frequency of the frame structures.
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
Fig. 11. The bounds of the standard deviation of the natural frequency of the frame structures.
Fig. 12. The bounds of the expectation of the natural frequency of frame structure.
structures have demonstrated the following features of the proposed method:
Central Universities (Grant no. XDJK2017B060) and China Postdoctoral Science Foundation (Grant no. 2018M643827).
(1) The proposed ESHPVM is able to deal with the hybrid uncertainties (random or interval variables) well. The lower and upper bounds of expectation and variance of natural frequency of structures calculated by ESHPVM agree well with the bounds obtained by ESHPMCM with 2000 samples. (2) Compared with ESHPMCM which needs large quantities of iteration calculations, ESHPVM has higher computational efficiency. Furthermore, the converge rate ESHPVM is much faster than the corresponding ESHPMCM. (3) Due to the application of edge-based smoothing technique, the proposed ESHPVM significantly improve the accuracy in deterministic calculation in comparison with HPVM constructed based on the conventional FEM.
Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.enganabound.2019.01.020. References [1] Boyce WE. Random eigenvalue problems. Probabilistic methods in applied mathematics. New York: Academic Press; 1968. [2] Wu F, Liu GR, Li GY, Cheng AG, He ZC, Hu ZH. A novel hybrid FS-FEM/SEA for the analysis of vibro-acoustic problems. Int J Numer Methods Eng 2015;102(12):1815–29. [3] Zeng W, Liu GR. Smoothed finite element methods (S-FEM): an overview and recent developments. Arch Comput Methods Eng 2018;25(2):397–435. [4] Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY. A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Compute Struct 2009;87(1):14–26. [5] Nguyen-Thoi T, Liu GR, Nguyen-Xuan H. Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems. Int J Comput Methods 2009;6(04):633–66. [6] Liu GR, Nguyenthoi, Lam KY. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. J Sound Vib 2009;320(4):1100–30 (2009). [7] He ZC, Li GY, Zhong ZH, et al. An edge-based smoothed tetrahedron finite element method (ES-T-FEM) for 3d static and dynamic problems. Comput Mech 2013;52(1):221–36. [8] Nguyen-Thoi T, Liu GR, Lam KY, Zhang GY. A face-based smoothed finite element method (FS-FEM) for 3d linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements. Int J Numer Methods Eng 2010;78(3): 324–353 (2010).
In sum, this ESHPVM framework can be regarded as an extension of the ES-FEM to deal with natural frequency of random structure with the hybrid random and interval uncertainties in solid engineering. The derived procedure and approach in this work can be easily extended to other engineering fields. Acknowledgments The first author wishes to thank the support of the National Natural Science Foundation of China (NSFC) (Grant no. 11702226) and the support of the Chongqing Science and Technology Commission (CSTC) (Grant no. 2016jcyjA0176) and Fundamental Research Funds for the 49
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[9] Nguyen-Thoi T, Liu GR, Vu-Do HC, Nguyen-Xuan H. A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3d solids using tetrahedral mesh. Comput Methods Appl Mech Eng 2009;198(41):3479–98 (2009). [10] Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY. A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Comput Struct 2009;87(1):14–26. [11] Zeng W, Liu GR, Li D, Dong XW. A smoothing technique based beta finite element method (𝛽FEM) for crystal plasticity modeling. Comput Struct 2016;162(C): 48–67. [12] Wu F, Zeng W, Yao LY, Hu M, Chen YJ, Li MS. Smoothing technique based beta fem (𝛽FEM) for static and free vibration analyses of reissner–mindlin plates. Int J Comput Methods 2018;15(3):29 (2018). [13] Wu, F, He, ZC, Liu, GR, Li, GY, Cheng, AG A novel hybrid ES-FE-SEA for midfrequency prediction of transmission losses in complex acoustic systems. Appl Acoust 111(2016):198–204. [14] Wan D, Hu D, Natarajan S, et al. A fully smoothed XFEM for analysis of axisymmetric problems with weak discontinuities. Int J Numer Methods Eng 2017;110(3): 203–226. [15] Feng H, Cui XY, Li GY. A stable nodal integration method with strain gradient for static and dynamic analysis of solid mechanics. Eng Anal Bound Elem 2016;62: 78–92. [16] Wan D, Hu D, Natarajan S, Bordas SPA, Long T. A linear smoothed higher-order CS-FEM for the analysis of notched laminated composites. Eng Anal Bound Elem 2017;85:127–35. [17] Liu GR, Dai KY, Nguyen TT. A smoothed finite element method for mechanics problems. Comput Mech 2007;39(6):859–77. [18] Liu GR, Nguyen TT, Dai KY, et al. Theoretical aspects of the smoothed finite element method (SFEM). Int J Numer Methods Eng 2007;71(8):902–30. [19] Boyce WE. Random eigenvalue problems. Probabilistic methods in applied mathematics. New York: Academic Press; 1968. [20] Collins JD, Thomson WT. The eigenvalue problem for structural systems with statistical properties. AIAA J 1969;7:642–8. [21] Scheidt, JV, W Purkert. Random eigenvalue problems. 1983. [22] Ibrahim RA. Structural dynamics with parameter uncertainties. Appl Mech Rev 1987;40:3. [23] Song D, Chen S, Qiu Z. Stochastic sensitivity analysis of eigenvalues and eigenvectors. Comput Struct 1995;54(5):891–6. [24] Gao W. Natural frequency and mode shape analysis of structures with uncertainty. Mech Syst Signal Process 2007;21(1):24–39. [25] Adhikari S. Joint statistics of natural frequencies of stochastic dynamic systems. Comput Mech 2007;40(4):739–52.
[26] Wu F, Zeng W, Yao LY, Liu GR. A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner–Mindlin plates. Appl Math Model 2017;53:333–52. [27] Liu J, Hu Y, Xu C, Jiang C, Han X. Probability assessments of identified parameters for stochastic structures using point estimation method. Reliab Eng System Saf 2016;156:51–8. [28] Beer M. Engineering computation under uncertainty – capabilities of non-traditional models. Pergamon Press, Inc.; 2008. [29] Liu J, Cai H, Meng XH, Xu C, Zhang DQ, Jiang C. An interval inverse method is proposed to effectively solve inverse problems with interval uncertainty. Appl Math Model 2018;63:732–43. [30] Liu J, Liu H, Jiang C, Han X, Zhang DQ, Hu FY. A new measurement for structural uncertainty propagation based on pseudo-probability distribution. Appl Math Model 2018;63:744–60. [31] Liu J, Cai H, Jiang C, Han X, Zhang Z. Forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions. Comput Methods Appl Mech Eng 2018;342:287–320. [32] Deif A. The interval eigenvalue problem. ZAMM – J Appl Math Mech/Z Angew Math Mech 1991;71(1):61–4. [33] Chen SH, Qiu Z, Liu Z. A method for computing eigenvalue bounds in structural vibration systems with interval parameters. Commun Numer Methods Eng 2010;10(2):121–34. [34] Qiu Z, Wang X, Friswell MI. Eigenvalue bounds of structures with uncertain-but-bounded parameters. J Sound Vib 2005;282(1–2):297–312. [35] Angeli P, Barazza F, Blanchini F. Natural frequency interval for vibration systems with polytopic uncertainty. J Sound Vib 2010;329(7):944–59. [36] Gao W. Interval natural frequency and mode shape analysis for truss structures with interval parameters. Finite Elem Anal Des 2006;42(6):471–7. [37] Modares M, Mullen RL, Muhanna RL. Natural frequencies of a structure with bounded uncertainty. J Eng Mech 2006;132(12):1363–71. [38] Moens D, Vandepitte D. A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Methods Appl Mech Eng 2005;194(12–16):1527–55. [39] Ben-Haim, et al. Convex models of uncertainty in applied mechanics. Elsevier; 1990. [40] Gao W, Song C, Tin-Loi F. Probabilistic interval analysis for structures with uncertainty. Struct Saf 2010;32(3):191–9. [41] Xia B, Yu D, Liu J. Hybrid uncertain analysis for structural–acoustic problem with random and interval parameters. J Sound Vib 2013;332(11):2701–20. [42] Wang C, Gao W, Song CM, Zhang N. Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties. J Sound Vib 2014;333(9): 2483–2503.
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Stochastic interval analysis for structural natural frequencies based on stochastic hybrid perturbation edge-based smoothing finite element method F. Wu1, M. Hu∗, K. Chen, L.Y. Yao, Z.G. Ju, X. Sun College of Engineering and Technology, Southwest University, Chongqing 400715, PR China
a r t i c l e
i n f o
Keywords: Edge-based smoothing technique FEM Natural frequencies Stochastic perturbation technique First-order random interval perturbation method
a b s t r a c t Realistically calculating of natural frequencies is crucial for studying dynamical characteristics of structures in engineering. In this paper, the gradient smoothing technique is first applied into the conventional finite element method (named ES-FEM), to improve the accuracy of the deterministic calculation of natural frequencies. Then, the hybrid random and interval uncertain parameters are introduced into the proposed ES-FEM based on hybrid stochastic and interval perturbation method. Expressions for the mean value and variance of natural frequencies are derived by combining the random interval perturbation method and the random interval moment method. The upper and lower bounds of mean value and variance of natural frequencies are calculated based on hybrid perturbation vertex method. The proposed method (denoted ESHPVM) is compared with the different methods. The high accuracy and efficiency of the proposed methods are verified by two numerical examples.
1. Introduction Among many typical engineering structures, e.g., bridges, buildings, offshore structures, vehicles, ships, and aerospace structures, the index of natural frequency is a key indicator for evaluating dynamical performance of these systems. At present, the most widely used numerical methods, i.e., the conventional finite element method (FEM), still holds potential flaws, e.g., low accuracy and ignoring uncertain factors. In the conventional FEM model, the calculated natural frequency value of the structure is often much larger than the real physical modal value, due to the stiffer discrete FEM system constructed based on Galerkin theory [1,2]. Moreover, these numerical errors become larger with the increase of the computational frequency. In the context of smoothing finite method(S-FEM) [3], the stiffness matrix could be soften based on the smoothing gradient technique, to alleviate accuracy problems in the higher frequency domain. Different smoothing domains could be built based on nodes, edges or faces of basic elements, for different desirable ‘‘smoothing effects’’, thus different S-FEM models have been proposed, such as the node-based S-FEM (NS-FEM) [4,5], edge-based S-FEM (ESFEM) [6,7] and face-based S-FEM (FS-FEM) [8,9]. Recently, hybrid SFEM models are also developed, such as such as the 𝛼 FEM [10] and 𝛽 FEM [11,12]. It is noted that even the same smoothing gradient technique (meaning same way of construction of smoothing domains) can be applied to solve a wide class of different practical mechanics problems for desirable solutions, such as elastic–plastic analysis, plates and shells
∗
1
composites, stochastic analysis, vibration and dynamic analysis structural acoustics, adaptive analysis, heat transfer and thermo-mechanical problems, and fluid–structure interaction [3–18] etc. In this work, the 3D edge-based smoothing gradient technique is applied into the conventional finite element method, reducing the dispersion error in the deterministic calculations [7]. On the other hand, uncertainties are often artificially ignored in the conventional FEM models. In the early 1960s, Boyce [19] and Collins and Thomson [20] did some theoretical research work on the probability properties of natural frequency by introducing the random parameters. Scheidt and Purker [21] summarized random eigenvalue problems systematically in their book in 1983. Later, Ibrahim [22] introduced a series of parameter uncertainties into the structural systems and different dynamics properties are carefully compared in 1987. Then, based on the perturbation method [23], stochastic sensitivity analysis of eigenvalues and eigenvectors are studied. In 2007, the natural frequency and mode shape of structures with uncertainty are further analyzed [24]. Based on the multidimensional integrals, Adhikari [25] further obtained the joint probability distribution of the natural frequencies. It is noted that all the above-mentioned uncertain models belongs to probabilistic and statistical models by assuming that all the input parameters have the deterministic probability distribution [26,27]. For the input parameters with uncertain probability distribution, the interval method offered an alternative to quantify these uncertainties with possible value range [28–30]. Liu et al. [31] even proposed forward and
Corresponding author. E-mail address: [email protected] (M. Hu). These authors contributed equally to this work (co-first authors).
https://doi.org/10.1016/j.enganabound.2019.01.020 Received 26 September 2018; Received in revised form 30 December 2018; Accepted 27 January 2019 Available online 15 March 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
F. Wu, M. Hu and K. Chen et al.
Engineering Analysis with Boundary Elements 103 (2019) 41–50
inverse structural uncertainty propagations to handle arbitrary probability distribution. Lots of research efforts have been spent on the interval eigenvalue problem [32,33] in the past two decades. Based on the perturbation method, Chen et al. [33] obtained the upper and lower bounds of eigenvalues of structural systems. Qiu et al. [34] improved the computational efficiency of the calculation of the eigenvalue bounds by using the parameter vertex method. Angeli et al. [35] introduced the polytypic uncertainties into the dynamical system, and the bounds of natural frequency are obtained. Other similar methods can also be classified as the categories such as interval factor method [36], interval finite-element method [37]. Another alternative method is fuzzy set theory [38] in which uncertainties can be quantified based on the use of membership functions (describing the degree of possibility). Actually, the uncertain parameters with probabilities distributions or only bounds information may exist simultaneously. In response to this problem, Ben-Haim [39] developed the random and interval mixed uncertain models. Combined the random and interval perturbation techniques and random moments techniques, Gao et al. [40] analyzed the static response of random and interval mixed uncertain structures. Based on the hybrid perturbation Monte Carlo Method, Xia et al. [41] constructed the random and interval mixed uncertain structuralacoustic model and the variation range of the expectation and variance of frequency response can be obtained. Wang et al. [42] further extended the random and interval technique into the analysis of natural frequencies and mode shapes of structures with hybrid random and interval parameters. In the contrast to extensive research on static mixed uncertainty problems, the research for the dynamic problems of structures with mixed random and interval variables mainly focused on the conventional FEM models. It is meaningful to further integrate the stochastic model and the interval model into the smoothed FEM model and to predict the range of expectation and variance of the natural frequency with higher accuracy. Based on the above discussion, we will introduce the generalized gradient smoothing technique, i.e., the stochastic and interval uncertainty model into the conventional FEM method in this work. The paper is organized as follows: in Section 2, the basic principles of smoothing finite element method for 3D solid problems are briefly described. In Section 3, the hybrid ESHPVM equations are constructed in order to achieve an efficient stochastic and interval uncertainty method. In Section 4, numerical examples and applications are presented to demonstrate the performance of the hybrid ESHPVM for analysis of natural frequencies of structures. Finally, a summary is given in Section 5 to conclude this work.
Fig. 1. 3D edge-based smoothing domain.
in which 𝑉𝑘𝑠 denotes the volume of the smoothing domain k. u denotes the displacement vector which can be obtained based on finite element interpolation: 𝐮=
𝑖=1
⎡𝐍𝑖 (𝑥) 𝐍𝑖 (𝑥) = ⎢ 0 ⎢ ⎣ 0
𝛆(𝑥𝑘 ) =
0 ⎤ 0 ⎥ ⎥ 𝐍𝑖 (𝑥)⎦
0 𝐍𝑖 (𝑥) 0
(3)
𝐌𝑘 ∑ 𝑖=1
𝐁𝑖 (𝑥𝑘 )𝐝𝑖
(4)
where Mk denotes the number of sub-smoothing domain, 𝐁𝑖 is smoothing strain matrix can be expressed as follows: ⎡𝑏𝑖𝑥 (𝑥𝑘 ) ⎢ 0 ⎢ ⎢ 0 𝐁𝑖 ( 𝑥 𝑘 ) = ⎢ ⎢𝑏𝑖𝑦 (𝑥𝑘 ) ⎢ 0 ⎢ ⎣𝑏𝑖𝑧 (𝑥𝑘 )
0 𝑏𝑖𝑦 (𝑥𝑘 ) 0 𝑏𝑖𝑥 (𝑥𝑘 ) 𝑏𝑖𝑧 (𝑥𝑘 ) 0
0 ⎤ 0 ⎥⎥ 𝑏𝑖𝑧 (𝑥𝑘 )⎥ ⎥ 0 ⎥ 𝑏𝑖𝑦 (𝑥𝑘 )⎥ ⎥ 𝑏𝑖𝑥 (𝑥𝑘 )⎦
(5)
where the 𝐛𝑖𝑝 can be expressed as: 𝐛𝑖𝑝 =
In this section, the ES-FEM formulations for 3D solid problems are briefly introduced. First, the 3D solid domain is discretized using four nodes tetrahedral elements as that of standard FEM. Then, based on the edges of tetrahedral elements, the edge-based gradient smoothing domains can be constructed as shown in Fig. 1. The sub-smoothing domain of edge k in cell i is created by connecting the centroid of cell i to the two end-nodes of the edge k and the related surface triangles. The entire problem domain can be formed by assembling all smoothing areas, 𝛀𝑠𝑖 ∩ 𝛀𝑠𝑗 = 𝜙, ∀𝑖 ≠ 𝑗. where Ns is the number of smoothing domain. s is the number of all tetrahedral faces. It is noted that the calculation of element strain is no longer based on the standard finite element, but based on the newly constructed smoothing domain. The smoothing domain now is also serving as integration domains. The strain 𝛆𝑘 is first smoothed based on the smoothing gradient technique within the smoothing domain Ωsk , as below:
𝑘
(2)
Substituting Eq. (2) into Eq. (1), the smoothed strain can be written as follows:
2.1. The construction of three-dimensional edge-based smoothing domain
1 1 𝛆 𝑑Ω = 𝑠 𝐋𝐮 𝑑 Ω 𝑉𝑘𝑠 ∫Ω𝑠 𝑘 𝑉𝑘 ∫Ω𝑠
𝐍 𝑖 ( 𝑥 ) 𝐝 𝑖 =𝐍 𝑠 𝐝
in which, N denotes the number of nodes tetrahedral elements. di = {dxi dyi dzi} denotes the node displacements. N represents the shape function matrix, which can be written as follows:
2. Theoretical equation ES-FEM for modal analysis
𝛆𝑘 =
𝐍 ∑
1 𝐍 (𝑥) ⋅ 𝐧𝑝 (𝑥)𝑑𝚪 𝑉𝑘𝑠 ∫𝛀𝑠 𝑖
(𝑝 = 𝑥, 𝑦, 𝑧)
(6)
𝑘
in which np is the normal vector. 2.2. The discretely vibrating equation of three-dimensional ES-FEM The standard Galerkin weak form for vibration mechanics problem without damping can be expressed as ∫Ω
𝑇
𝛿(𝛆(𝑢)) 𝐃(𝛆(𝑢))𝑑Ω +
∫Ω
𝛿𝐮𝑇 𝜌
𝜕2 𝐮 𝑑Ω − 𝛿𝐮𝑇 𝐛𝑑Ω − 𝛿𝐮𝑇 𝛕𝑑Γ = 0 ∫Ω ∫Γ𝑓 𝜕 𝑡2 (7)
Substituting Eqs. (1) and (2) into Eq. (7), the final discrete linear equations can be expressed as follows: { } 𝐌 𝐝̈ + 𝐊{𝐝} = {𝐅} (8) in which F represents the external load vector. M is the lumped mass matrix just as same as in the standard finite element method, 𝐊 denotes
(1)
𝑘
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
the smoothed stiffness matrix constructed based on the smoothed strain matrix as: 𝐊=
𝑇
∫Ω
𝐌=𝜌
𝐁𝑓 𝐃𝐁𝑓 𝑑Ω
∫Ω
In which, c stands for mean of variables. Ignoring the residual term ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) with hybrid random and interval variRe, the stiffness matrix 𝐊(𝑥 ables can be marked as
(9)
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 )+ ▵ 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) 𝐊(𝑥
𝑇
𝐍𝑓 𝐍𝑓 𝑑Ω
(10)
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) is the perturbation range of the term 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗), can in which ▵ 𝐊(𝑥 be written as:
For the natural frequency analysis, the following characteristic equation can be obtained: (𝐊 − 𝜔𝑘 2 𝐌)𝛗 ⃖⃖⃖⃖𝑘⃗ = 0
𝑚 ( ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = ∑ 𝜕 𝐊 || ▵𝐊 𝑥 ▵ 𝑦𝐼𝑗 𝐼 | ⃖⃖⃖𝑅 ⃗ 𝑐 𝑗=1 𝜕𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ ( ) 𝑛 𝑚 ( ) ∑ ∑ 𝜕 𝐊 || 𝜕 2 𝐊 || 𝐼 𝑅 + + ▵ 𝑦 𝑥𝑅 |⃖⃖⃖𝑅⃗ 𝑐 |⃖⃖⃖𝑅⃗ 𝑐 𝑗 𝑖 − 𝑥𝑖 𝑅 𝑅 𝐼 𝜕𝑥𝑖 |𝑥 ,𝑦⃖⃖⃗ 𝑗=1 𝜕 𝑥𝑖 𝜕 𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ 𝑖=1
(11)
in which, 𝜔k denotes the natural frequency of the k-order and 𝛗 ⃖⃖⃖⃖𝑘⃗ is the corresponding modal eigenvector. Then the solution of the system’s natural frequency can be obtained by solving the eigenvalue: 𝐊𝛗𝑘 = 𝜔2𝑘 𝐌𝛗 ⃖⃖⃖⃖𝑘⃗
(18)
(19)
(12) ′
𝜕2 𝐊 | , 𝜕 𝑥𝑅 𝜕 𝑦𝐼𝑗 𝑥 ⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝑖
3.1. Stochastic and interval mixed uncertainty modeling
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) ⋅ 𝛗 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = 𝜔 2 (𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) ⋅ 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) ⋅ 𝛗 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) 𝐊(𝑥 ⃖⃖⃖⃖𝑘⃗(𝑥 ⃖⃖⃖⃖𝑘⃗(𝑥 𝑘
𝑗=1
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) can be simplified as: Similarly, the mass matrix 𝐌(𝑥 ( ) ( ) ( ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 + ▵ 𝐌 𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = 𝐌 𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ 𝐌 𝑥
𝑖
𝑚 ( ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = ∑ 𝜕𝐌 || ▵𝐌 𝑥 ▵ 𝑦𝐼𝑗 𝐼 | ⃖⃖⃖𝑅 ⃗ 𝑐 𝑗=1 𝜕𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗
+
𝑖
the expectation of the interval vector 𝑦⃖⃖⃖𝐼⃗ = (𝑦𝐼1 , 𝑦𝐼2 , ..., 𝑦𝐼𝑚 ) is remarked with 𝑦⃖⃖⃖⃗𝑐 = (𝑦𝑐1 , 𝑦𝑐2 , ..., 𝑦𝑐𝑚 ), in which 𝑦𝑐𝑗 =
𝑦𝑐𝑗 +𝑦𝑐𝑗 2
1
𝑖=1
2
𝜕 𝐊 || 𝑅 𝑅 |⃖⃖⃖𝑅⃗ (𝑥𝑖 − 𝑥𝑖 ) + Re 𝜕𝑥𝑅 𝑖 |𝑥
𝑛 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) + ∑ 𝜕𝐌 || (𝑥𝑅 − 𝑥𝑅 ) + Re 𝐌(𝑥 𝑖 𝑅 | ⃖⃖⃖𝑅 ⃗ 𝑖 𝑖=1 𝜕𝑥𝑖 |𝑥
Using 𝐌′ 𝑅 , 𝐌′ 𝐼 and 𝐌′′𝑅 𝑥
𝜕2 𝐌 , 𝐼 | 𝜕 𝑥𝑅 ⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝑖 𝜕 𝑦𝑗 𝑥
𝑛
2
𝑦
𝑥 𝑦𝐼
to denote
𝜕𝐌 | 𝜕𝐌 | | | 𝜕𝑥𝑅 ⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 , 𝜕𝑦𝐼𝑗 | 𝑥⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝑖 |𝑥
and
respectively, so Eq. (22) can be simplified as:
( ) 𝑚 𝑛 𝑚 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = ∑ 𝐌′ ▵ 𝑦𝐼 + ∑ 𝐌′ + ∑ 𝐌′′ 𝐼 𝑅 𝑅 ▵ 𝐌(𝑥 ▵ 𝑦 𝑗 𝑗 (𝑥𝑖 − 𝑥𝑖 ) 𝑦𝐼 𝑥𝑅 𝑥𝑅 𝑦𝐼
(14)
𝑗=1
𝑖=1
𝑗=1
(15)
(23) Using ▵wk and ▵ 𝛗 ⃖⃖⃖⃖𝑘⃗ to denote the perturbation of k-order natural
where Re is the residual term. Assuming the random vector as a constant, the interval vector 𝑦⃖⃖⃖𝐼⃗ can also be expanded by the first-order Taylor series at expectation of 𝑦⃖⃖⃖⃗𝑐 = (𝑦𝑐 , 𝑦𝑐 , ..., 𝑦𝑐 ): 1
| | 𝑛 ⎛ 𝑚 ⎞( ) ∑ ∑ | 𝜕 2 𝐌 || ⎜ 𝜕𝐌 | + ▵ 𝑦𝐼𝑗 ⎟ 𝑥𝑅 − 𝑥𝑅 | | 𝑖 𝑖 𝑅 𝑅 𝐼 ⎜ ⎟ ⃖⃖⃖𝑅⃗ 𝑐 𝑗=1 𝜕 𝑥 𝜕 𝑦 | 𝑅 𝑐 𝑖=1 ⎝ 𝜕𝑥𝑖 ||𝑥 ,𝑦⃖⃖⃗ 𝑖 𝑗 |𝑥 ⃖⃖⃖⃗,𝑦⃖⃖⃗ ⎠ (22)
, 1 ≤ 𝑗 ≤ 𝑚.The stiffness vec-
tor 𝐊 and the mass matrix M can be expanded by the first-order Taylor ⃖⃖⃖⃖𝑅⃗ = (𝑥𝑅 , 𝑥𝑅 , ..., 𝑥𝑅 ). series at the expectation of the random vector 𝑥 𝑛 ∑
(21)
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗), which is The perturbation range of the mass matrix is ▵ 𝐌(𝑥 given by:
(13)
⃖⃖⃖⃖𝑅⃗ = (𝑥𝑅 , 𝑥𝑅 , ..., 𝑥𝑅 ) denotes the expectation of random We first use 𝑥 𝑛 1 2 ⃖⃖⃖⃖ ⃗ 𝑅 vector 𝑥𝑅 = (𝑥 , 𝑥𝑅 , ..., 𝑥𝑅 ) in which 𝑥𝑅 = exp(𝑥𝑅 ), 1 ≤ 𝑖 ≤ 𝑛; Similarly,
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) = 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗) + 𝐊(𝑥
𝑖=1
(20)
3.2. The perturbation analysis of natural frequency
𝑛
and
respectively. Then, Eq. (19) can be simplified as:
𝑗=1
and the interval vector 𝑦⃖⃖⃖𝐼⃗. The hybrid random and interval uncertainties are introduced into the natural frequency analysis of the structural system. Then, Eq. (12) can be rewritten as:
2
𝜕𝐊 | , 𝜕𝐊 | 𝜕𝑥𝑅 ⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝜕𝑦𝐼𝑗 𝑥⃖⃖⃖𝑅⃗,𝑦⃖⃖⃗𝑐 𝑖 𝑥
𝑚
2
the stiffness matrix 𝐊 and the mass matrix M. Obviously, the natural fre⃖⃖⃖⃖𝑅⃗ quency value of the system is also the function of the random vector 𝑥
1
′′
( ) 𝑚 𝑛 𝑚 ( ) ( ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = ∑ 𝐊′ ▵ 𝑦𝐼 + ∑ 𝐊′ + ∑ 𝐊′′ 𝐼 𝑅 ▵𝐊 𝑥 ▵ 𝑦 𝑥𝑅 𝐼 𝑅 𝑅 𝐼 𝑦 𝑥 𝑥 𝑦 𝑗 𝑗 𝑖 − 𝑥𝑖
⃖⃖⃖⃖𝑅⃗ = (𝑥𝑅 , 𝑥𝑅 , ..., 𝑥𝑅 ) and the interWe first define the random vector 𝑥 𝑛 1 2 ⃖⃖⃖ ⃗ 𝐼 𝐼 𝐼 𝐼 val vector 𝑦 = (𝑦 , 𝑦 , ..., 𝑦 ) in the structure system, which consisted in 1
′
Using 𝐊𝑥𝑅 , 𝐊𝑦𝐼 and 𝐊𝑥𝑅 𝑦𝐼 to denote the term
3. Hybrid perturbation vertex method in analysis of natural frequency
𝑐
frequency and its modal eigenvector, in addition, using 𝐊 , ▵ 𝐊, 𝐌𝑐 , ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ), ▵ 𝐊(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗), 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ), ▵ 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗), ⃖⃖⃖⃖𝑐⃗ denote 𝐊(𝑥 ▵M, 𝑤𝑐 and 𝛗
𝑚
𝑘
( ) 𝑚 ) ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 + ∑ 𝜕 𝐊 || ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = 𝐊 𝑥 𝐊 𝑥 ▵ 𝑦𝐼𝑗 𝐼 | ⃖⃖⃖𝑅 ⃗ 𝑐 𝑗=1 𝜕𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ ( ) 𝑛 𝑚 ( ) ∑ ∑ 𝜕 𝐊 || 𝜕 2 𝐊 || 𝐼 𝑅 + Re + + ▵ 𝑦𝑗 𝑥𝑅 𝑖 − 𝑥𝑖 𝑅 || ⃖⃖⃖𝑅 𝑅 𝜕 𝑦𝐼 || ⃖⃖⃖𝑅 𝑐 𝑐 ⃗ ⃗ ⃖⃖ ⃗ ⃖⃖ ⃗ 𝜕𝑥 𝜕 𝑥 𝑖=1 𝑗=1 𝑖 𝑥 ,𝑦 𝑖 𝑗 𝑥 ,𝑦 (
𝑘
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ) 𝑤𝑐𝑘 (𝑥
⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ) respectively, so Eq. (13) can be simplified as ⃖⃖⃖⃖𝑐⃗(𝑥 and 𝛗 𝑘
( ) ( ) ) ( 𝑐 𝑐 + ▵ 𝑤 )2 ⋅ (𝐌𝑐 + ▵ 𝐌) ⋅ 𝛗 ⃖⃖⃖⃖𝑐⃗+ ▵ 𝛗 ⃖⃖⃖⃖𝑐⃗+ ▵ 𝛗 𝐊 +▵𝐊 ⋅ 𝛗 = ( 𝑤 ⃖⃖⃖⃖ ⃗ ⃖⃖⃖⃖ ⃗ 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 (24) (16)
Ignoring the higher order terms, ▵wk can be calculated
( ) 𝑚 ) | ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖𝐼⃗ = 𝐌 𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 + ∑ 𝜕𝐌 || 𝐌 𝑥 ▵ 𝑦𝐼𝑗 𝐼 | ⃖⃖⃖𝑅 ⃗ 𝑐 𝑗=1 𝜕𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ ( ) 𝑛 𝑚 ( ) ∑ ∑ 𝜕𝐌 || 𝜕 2 𝐌 || 𝐼 𝑅 + Re 𝑥𝑅 + |⃖⃖⃖𝑅⃗ 𝑐 + |⃖⃖⃖𝑅⃗ 𝑐 ▵ 𝑦𝑗 𝑖 − 𝑥𝑖 𝑅 𝑅 𝐼 𝜕𝑥𝑖 |𝑥 ,𝑦⃖⃖⃗ 𝑗=1 𝜕 𝑥𝑖 𝜕 𝑦𝑗 |𝑥 ,𝑦⃖⃖⃗ 𝑖=1 (
▵ 𝑤𝑘 =
𝑇⃗ 1 ⃖⃖⃖⃖⃖⃖ ⃖⃖⃖⃖𝑐⃗ ▵ 𝐌)𝛗 𝛗𝑐𝑘 (▵ 𝐊 − 𝑤𝑐2 𝑘 𝑘 𝑐 2𝑤𝑘
(25)
In which, T stands for transposed matrix. Therefore, k-order natural frequency wk will be: 𝑤𝑘 = 𝑤𝑐𝑘 +
(17)
43
( ) 𝑇⃗ 1 ⃖⃖⃖⃖⃖⃖ ⃖⃖⃖⃖𝑐⃗ ▵𝐌 𝛗 𝛗𝑐𝑘 ▵ 𝐊 − 𝑤𝑐2 𝑘 𝑘 𝑐 2𝑤𝑘
(26)
F. Wu, M. Hu and K. Chen et al.
Engineering Analysis with Boundary Elements 103 (2019) 41–50
3.3. The analysis variance rang of the expectation and variance of natural frequency Based on the random moment technique, the k-order natural frequency wk and expectation Exp(wk )in Eq. (26) can be marked as: ( ) 𝑇⃗ 1 ⃖⃖⃖⃖⃖⃖ ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ) − 𝑤𝑐2 ▵ 𝐌(𝑥 ⃖⃖⃖⃖𝑅⃗, 𝑦⃖⃖⃖⃗𝑐 ) 𝛗 ⃖⃖⃖⃖𝑐⃗ 𝛗𝑐𝑘 ▵ 𝐊(𝑥 𝑘 𝑘 2𝑤𝑐𝑘 (𝑚 ) ′ 𝑇⃗ ∑ 1 ⃖⃖⃖⃖⃖⃖ ⃖⃖⃖⃖𝑐⃗ = 𝑤𝑐𝑘 + (𝐊𝑦𝐼 − 𝑤𝑐2 · 𝐌′ 𝐼 ) ▵ 𝑦𝐼𝑗 𝛗 𝛗𝑐𝑘 𝑘 𝑘 𝑦𝑗 𝑗 2𝑤𝑐𝑘 𝑗=1
𝐸𝑥𝑝(𝑤𝑘 ) = 𝑤𝑐𝑘 +
Fig. 2. Beam structure model.
(27)
⃖⃖⃖⃖⃖⃖𝑇⃗ ′ ⃖⃖⃖⃖𝑐⃗ . It is noted that Using 𝐷𝑤𝐼 to donate the term 𝛗𝑐𝑘 (𝐊𝑦𝐼 − 𝑤𝑐2 · 𝐌′ 𝐼 )𝛗 𝑘 𝑘 𝑦𝑗
𝑦𝑗
𝑗
𝐷𝑤𝐼 is a constant value. Then, Eq. (27) can be simplified as: 𝑦𝑗
𝐸𝑥𝑝(𝑤𝑘 ) = 𝑤𝑐𝑘 +
𝑚 1 ∑
2𝑤𝑐𝑘
𝑗=1
( ) 𝐷𝑤𝐼 Δ𝑦𝑗
(28)
𝑦𝑗
Fig. 3. Finite element meshes of beam structure.
Similarly, the variance of the natural frequency Var(wk ) of the korder wk can be obtained based on the random moment technique: 𝑉 𝑎𝑟(𝑤𝑘 ) =
(
𝑛 ∑ 𝑛 ∑ 𝑖=1 𝑙=1
(
1
𝐷𝑤𝑅 + 𝑥𝑖
4𝑤𝑐2 𝑘
× 𝐷𝑤𝑅 +
𝑚 ∑
𝑥𝑙
in
which
𝐷𝑤𝑅 𝑥𝑖
(
𝑚 ∑ 𝑗=1
( )) 𝐷𝑤𝑅 𝐼 Δ𝑦𝐼𝑗 𝑥𝑖 𝑦𝑗
𝐷𝑤𝑅 𝐼 Δ𝑦𝐼𝑗 𝑥𝑙 𝑦𝑗
𝑗=1
)) 𝑅 𝐶𝑜𝑣(𝑥𝑅 𝑖 , 𝑥𝑙 )
𝑥𝑖
𝑖
𝐷𝑤𝑅
𝑥𝑙 𝑦𝐼𝑗
denotes
𝑛 ⎛ ∑ ⎜ 1 ⎜ 𝑖=1 ⎝ 4𝑤𝑐2 𝑘
( 𝐷𝑤𝑅 + 𝑥𝑖
𝑚 ∑ 𝑗=1
( 𝐷𝑤𝑅
𝑥𝑖 𝑦𝐼𝑗
▵ 𝑦𝐼𝑗
))2
⎞ ⎟ 𝑉 𝑎𝑟(𝑥𝑅 𝑖 )⎟ ⎠
(30)
From the relationship of standard deviation and variance, we have: 𝑆𝑡𝑑(𝑤𝑘 ) =
√
𝑉 𝑎𝑟(𝑤𝑘 ) ( ) )| | 𝑚 ( ∑ | 1 || 𝑤 𝑤 𝐼 | 𝑅 𝑆𝑡𝑑(𝑤𝑘 ) = 𝐷 𝑅 𝐼 Δ𝑦𝑗 |𝑆𝑡𝑑(𝑥𝑖 ) |𝐷 𝑅 + 𝑥𝑖 𝑦𝑗 | 2𝑤𝑐𝑘 || 𝑥𝑖 𝑖=1 𝑗=1 | 𝑛 ∑
(31)
𝑦𝑗
𝑥𝑖
𝑤𝑐𝑘 + {
𝐸𝑥𝑝(𝑤𝑘 ) =
𝑤𝑐𝑘 +
𝑚 1 ∑
2𝑤𝑐𝑘
𝑗=1
𝑥𝑙 𝑦𝑗
𝑚 1 ∑
2𝑤𝑐𝑘
𝑗=1
The proposed approach is first applied in standard beam structure model. As shown in Fig. 2, the size of the beam structure is 0.04 × 1.0 × 0.03 m, and the material is aluminum. The deterministic parameters are shown in Table 1 As shown in Fig. 3, the beam structure model is first divided into discrete tetrahedron elements, which contains 918 tetrahedron elements and 414 nodes. The beam was not free–free but simple supports on both sides. In edge-based finite element model, uncertain parameters with sufficient samples are considered as random variables, others are considered as interval variables. It is noted that all the variables are considered to be independent of each other, the detail information of input variables of beam are shown in Table 2.
min
( )} 𝑤 𝐷 𝐼 Δ𝑦𝑗 𝑦𝑗
(33)
4.1. Example 1 – the analysis of ESHPVM based on beam structure
( )} 𝐷𝑤𝐼 Δ𝑦𝑗 𝑦𝑗
(
In this subsection, the validity and efficiency of the proposed ESHPVM will first be illustrated by two typical frame structures.
terministic values, while the expectation Exp(wk ) and standard deviation Std(wk ) are the linear equations of interval perturbation ▵ 𝑦𝐼𝑗 , Based on the HPVM(Hybrid Perturbation Vertex Method), the upper and lower bounds of expectation Exp(wk ) and standard deviation Std(wk ) can be obtained by substituting two extremums of ▵ 𝑦𝐼𝑗 into Eqs. (27) and (31): {
𝑛 ∑
4. The numerical example
Because ▵ 𝑦𝐼𝑗 is an interval variable, the expectation of k-order natural frequency wk is also an interval variable, rather than a constant value. As for Eqs. (27) and (31), 𝑤𝑐𝑘 ,𝐷𝑤𝐼 ,𝐷𝑤𝑅 ,𝐷𝑤𝑅 𝐼 and 𝑆𝑡𝑑(𝑥𝑅 𝑖 ) are de-
𝐸𝑥𝑝(𝑤𝑘 ) =
2700
Based on the above mathematical derivation, the proposed method is named ESHPVM. Several approaches (HPVEM, ESHPMCM and HPMCM)) are introduced as the reference methods. HPVM is constructed based on similar techniques as that of ESHPVEM. The main difference between these two methods lies in the imbedded deterministic methods: ESHPVM is constructed based on ES-FEM, while HPVEM is constructed based on the conventional FEM. In the ESHPMCM, the upper and lower bound of expectation and standard deviation is calculated based on Monte-Carlo Method rather than Vertex Method as in the ESHPVEM. Different from ESHPVEM, HPMCM is constructed based on conventional FEM rather than ES-FEM.
𝑅 In general, the independent random variables𝑥𝑅 𝑖 and𝑥𝑙 satisfy the re𝑅 ) = 0, 𝐶𝑜𝑣(𝑥𝑅 , 𝑥𝑅 ) = 𝑉 𝑎𝑟(𝑥𝑅 ), Eq. (29) can be exlationship 𝐶𝑜𝑣(𝑥𝑅 , 𝑥 𝑖 𝑖 𝑖 𝑖 𝑙 pressed as:
𝑉 𝑎𝑟(𝑤𝑘 ) =
Length:1.0 width:0.3 Height:0.04
)} )| | 𝑚 ( ∑ | 1 || 𝑤 𝑤 𝐼 | 𝑅 𝐷 𝑆𝑡𝑑( + 𝐷 Δ𝑦 𝑥 ) 𝐼 | 𝑅 𝑗 | 𝑖 𝑥𝑅 | 𝑖 𝑦𝑗 2𝑤𝑐𝑘 || 𝑥𝑖 𝑖=1 𝑗=1 | min { 𝑛 ( )} )| | 𝑚 ( ∑ ∑ | | 1 | 𝑤 𝑆𝑡𝑑 (𝑤𝑘 ) = 𝐷𝑤𝑅 𝐼 Δ𝑦𝐼𝑗 ||𝑆𝑡𝑑(𝑥𝑅 |𝐷 𝑅 + 𝑖 ) 𝑥𝑖 𝑦𝑗 | 2𝑤𝑐𝑘 || 𝑥𝑖 𝑖=1 𝑗=1 | max
𝑥𝑖 𝑦𝑗
𝑗
Density kg/m3
{
(29)
⃖⃖⃖⃖⃖⃖𝑇⃗ ′′ ⃖⃖⃖⃖𝑐⃗. 𝛗𝑐𝑘 (𝐊𝑥𝑅 𝑦𝐼 − 𝑤𝑐2 ⋅ 𝐌′′𝑅 𝐼 )𝛗 𝑘 𝑘 𝑖
Structure size/m
𝑆𝑡𝑑 (𝑤𝑘 ) =
⃖⃖⃖⃖⃖⃖𝑇⃗ ′ ⃖⃖⃖⃖𝑐⃗. 𝛗𝑐𝑘 (𝐊𝑥𝑅 − 𝑤𝑐2 ⋅ 𝐌′ 𝑅 )𝛗 𝑘 𝑘
denotes
Table 1 Deterministic parameters of the beam.
(32)
max
Once the upper and lower bounds of variance are obtained, the bounds of standard deviation can be easily got. 44
F. Wu, M. Hu and K. Chen et al.
Engineering Analysis with Boundary Elements 103 (2019) 41–50
Fig. 4. Upper and lower bound of expectation and standard deviation of the 7-order natural frequency of the natural frequency calculated by ESHPVM and ESPHMCM.
ESHPMCM, while the lower bound of ESHPVM is less than that of ESHPMCM. Therefore, ESHPMCM’s expectation and standard deviation are the subsets of ESHPVM. Moreover, ESHPVM’s results converge gradually to ESHPMCM’s results with the increase of iteration times of ESHPVM. Finally, the range of ESHPMCM’s expectation and standard deviation agree well with those of ESHPVM when the iteration times
First, the convergence rate of ESHPVM and ESHPMCM is first compared. The lower and upper bounds of expectation and variance of 7-order natural frequency the beam calculated by ESHPVM and ESHPMCM are shown in Fig. 4. As shown in Fig. 4, it is obvious that the upper bound of expectation and standard deviation of ESHPVM are greater than those of 45
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
Fig. 5. The bounds of the expectation of the natural frequency of the beam.
Fig. 6. The bounds of the standard deviation of the natural frequency of the beam.
Table 2 Input variables of the beam. Interval variable
The relative error of top 10-order natural frequency of ESHPVM and ESHPMCM are shown in Tables 4 and 5. As shown in Figs. 5 and 6, the expectation and standard deviation of ESHPVM agree well with the result of ESHPMCM based on 2000 samples. As shown in Tables 4 and 5, there is negligible deviations between the results of ESHPVM and ESHPMCM. The upper bound of expectation and standard deviation of ESHPVM are greater than those of ESHPMCM, while the lower bound of ESHPVM is less than that of ESHPMCM. ESHPMCM’s results are the subsets of those of ESHPVM, verifying the accuracy and effectivity of proposed ESHPVM. In order to further illustrate the higher accuracy of ESHPVM, different types of elements are constructed (fine meshes and coarse meshes). As is well-known, the conventional FEM provides higher accuracy based on fine meshes. First, the variation range is calculated using ESHPVM and HPVM based on coarse meshes (consisting of 918 elements and 414 nodes). The reference results are obtained using HPMCM based on fine meshes (consisting of 918 6689 elements and 1937 nodes). Through comparing the overlapping of the results of ESHPVM, HPVM and HPMCM, we can clearly draw a conclusion of the accuracy difference between HPMCM and HPVM. As shown in the Fig. 7, it is obvious that ESHPVM provides a higher accuracy results compared with the HPVM in the whole 1–30 order modes. HPVM suffers from the dispersion errors mainly caused by the “overly-stiff” property when the standard FEM is embedded in the hybrid HPVM. The result obtained from ESHPVM shows excellent agreements with the reference result, and it demonstrates that the pollution errors are great eliminated by the “soften effects” of ES-FEM. To sum up, the high accuracy and computational efficiency of ESHPVM
Random variable
Young modulus (Pa)
Poisson’s ratio
Range: 6.3 × 1010 − 7.7 × 1010 Radius of perturbation: 7 × 109
Expectation:0.3 Variance:0.03
Table 3 The computational time of ESHPVM and ESHPMCM. Method
ESHPVM
ESHPMCM
Execution time (s)
18.3826
25.7977
are more than 2000. Compared with ESHPMCM which needs large quantities of iteration calculations, it is obvious that ESHPVM has higher computational efficiency. The computational time of ESHPVM and ESHPMCM are shown in Table 3. As shown in Fig. 4 and Table 3, it is concluded that the proposed ESHPVM can obtain the lower and upper bounds of expectation and variance of the natural frequency with much higher efficiently, verifying the efficiency of the ESHPVM. In order to compare different order natural frequencies, the expectation and standard deviation of natural frequencies (1 to 30 orders) of the beam are calculated using ESHPVM and ESHPMCM. The natural frequencies obtained from ESHPMCM with 2000 samples are used as the reference results, as shown in Figs. 5 and 6 46
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
Table 4 Bounds of the expectation of the first 10 natural frequencies of the beam. Natural frequency (rad/s)
Expectation ESHPVM
𝜔1 𝜔2 𝜔3 𝜔4 𝜔5 𝜔6 𝜔7 𝜔8 𝜔9 𝜔10
ESHPMCM
Lower bound
Relative error (%)
Upper bound
Relative error (%)
Lower bound
Upper bound
541.42 955.59 2151.30 2615.43 4763.42 5071.18 8143.61 8257.72 8334.85 12226.72
−0.029 −0.0017 −0.0150 −0.0103 −0.0016 −0.0021 −0.0090 −0.0048 −0.0031 −0.0177
656.91 1226.85 2602.16 3349.53 5772.38 6504.03 10205.04 10198.90 10617.81 15581.76
0.0016 0.0045 0.0197 0.0016 0.0020 0.0142 0.0171 0.0099 0.0014 0.0513
541.58 955.60 2151.63 2615.69 4763.49 5071.29 8144.34 8258.11 8335.96 12227.09
656.90 1226.79 2601.65 3349.48 5772.26 6504.00 10203.29 10197.89 10617.66 15573.77
Table 5 Bounds of the standard deviation of the first 10 natural frequencies of the beam. Natural frequency(rad/s)
Expectation ESHPVM
𝜔1 𝜔2 𝜔3 𝜔4 𝜔5 𝜔6 𝜔7 𝜔8 𝜔9 𝜔10
ESHPMCM
Lower bound
Relative error (%)
Upper bound
Relative error (%)
Lower bound
Upper bound
6.49 18.85 24.42 52.06 55.98 99.91 53.62 81.28 154.09 225.06
−0.0055 −0.0013 −0.0159 −0.0081 −0.0017 −0.0017 −0.0079 −0.0045 −0.0108 −0.0025
7.95 23.05 29.85 63.64 68.43 122.12 65.54 99.35 188.33 275.08
0.0017 0.0037 0.0207 0.0014 0.0021 0.0117 0.0154 0.0094 0.0012 0.0433
6.50 18.86 24.43 52.07 55.99 99.92 53.63 81.29 154.10 225.07
7.94 23.04 29.84 63.63 68.42 122.10 65.53 99.34 188.32 274.96
Fig. 7. The bounds of the expectation of the natural frequency of beam.
Table 6 Deterministic parameters of the bus bodywork.
for the natural frequency of the beam structure is illustrated in this example. 4.2. Example 2 – the analysis of natural frequency of complex frame structure The frame structures are widely exits in the different engineering area. The index of natural frequency is a key indicator for evaluating dynamical performance of these systems. In this subsection, the proposed ESHPVM is applied to analysis the natural frequency of a typical frame structure as shown in Fig. 8. The size of the frame structure is 0. 5 × 0.65 × 0.02 m, and the material is aluminum. The determinant material parameters of frame structure are shown in Table 6.
Density kg/m3
Thickness
2700
0.02
As shown in Fig. 9, the frame structure model is first divided into discrete tetrahedron elements, which contains 1936 tetrahedron elements and 892 nodes. Similarly, uncertain parameters with sufficient samples are considered as random variables, others are considered as interval variables. It is noted that all the variables are considered to be independent of each 47
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
Table 7 Input variables of the frame structures. Interval variable
Random variable
Young modulus (Pa)
Poisson’s ratio
Range: 6.3 × 1010 − 7.7 × 1010 Radius of perturbation: 7 × 109
Expectation: 0.3 Variance: 0.03
As shown in Figs. 10 and 11, the expectation and standard deviation of natural frequency calculated based on ESHPVM are consistent with the result of ESHPMCM based on 2000 samples. So the accuracy of ESHPMCM in practical engineering problems has been verified. In order to further illustrate the higher accuracy of ESHPVM, different types of elements are constructed (fine meshes: 6801 elements and 2450 nodes and coarse meshes: 1936 elements and 892 nodes). First, the variation range is calculated using ESHPVM and HPVM based on coarse meshes. The reference results are obtained using HPMCM based on fine meshes. Through comparing the overlapping of the results of ESHPVM, HPVM and HPMCM, we can clearly draw a conclusion of the accuracy difference between HPMCM and HPVM. The results for this typical frame structures re-enforces the conclusion from the previous simple beam example. As is shown in Fig. 12, the calculated natural frequency value of the structure based on HPVM is far larger than that of ESHPVM and reference results, due to the inherent drawback of “over-stiffness” in imbedded FEM. Moreover, as the frequency increases, the deviation between HPVM result and the ESHPVM or reference result becomes even larger, suggesting that the accuracy of the HPVM result decreases with the increase of the frequency. Contrarily, the ESHPVM always provides much more accurate result in higher frequency range, compared with the HPVM model using the same mesh. As a consequence, we verify the accuracy of ESHPVM is higher than that of HPVM in analyzing the natural frequency of frame structure. To sum up, the higher accuracy of ESHPVM for the natural frequency of the frame structure is illustrated in this example.
Fig. 8. A simplified structure for frame structure.
Fig. 9. Finite element meshes of frame structures.
5. Summary other, the detail information of input variables of frame structures are shown in Table 7 In order to compare different order natural frequencies, the expectation and standard deviation of natural frequencies (1–30 orders) of the frame structure are calculated using ESHPVM and ESHPMCM. The natural frequencies obtained from ESHPMCM with 2000 samples are used as the reference results, which are compared in Figs. 10 and 11.
In this work, a hybrid ESHPVM is proposed for predicting the lower and upper bounds of expectation and variance of natural frequency of structures with mixed random and interval variables. Expressions for the mean value, standard deviation and variance of natural frequencies are derived based on perturbation method and random interval moment method. Numerical examples of beam/frame Fig. 10. The bounds of the expectation of the natural frequency of the frame structures.
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Engineering Analysis with Boundary Elements 103 (2019) 41–50
Fig. 11. The bounds of the standard deviation of the natural frequency of the frame structures.
Fig. 12. The bounds of the expectation of the natural frequency of frame structure.
structures have demonstrated the following features of the proposed method:
Central Universities (Grant no. XDJK2017B060) and China Postdoctoral Science Foundation (Grant no. 2018M643827).
(1) The proposed ESHPVM is able to deal with the hybrid uncertainties (random or interval variables) well. The lower and upper bounds of expectation and variance of natural frequency of structures calculated by ESHPVM agree well with the bounds obtained by ESHPMCM with 2000 samples. (2) Compared with ESHPMCM which needs large quantities of iteration calculations, ESHPVM has higher computational efficiency. Furthermore, the converge rate ESHPVM is much faster than the corresponding ESHPMCM. (3) Due to the application of edge-based smoothing technique, the proposed ESHPVM significantly improve the accuracy in deterministic calculation in comparison with HPVM constructed based on the conventional FEM.
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In sum, this ESHPVM framework can be regarded as an extension of the ES-FEM to deal with natural frequency of random structure with the hybrid random and interval uncertainties in solid engineering. The derived procedure and approach in this work can be easily extended to other engineering fields. Acknowledgments The first author wishes to thank the support of the National Natural Science Foundation of China (NSFC) (Grant no. 11702226) and the support of the Chongqing Science and Technology Commission (CSTC) (Grant no. 2016jcyjA0176) and Fundamental Research Funds for the 49
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