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Abstract

Finite element analysis is the most powerful tool to predict the behavior of a structure in engineering practice. Generally, the initial finite element model must be corrected with experimental data due to its complexity. Thus, it is very necessary to study a finite element model updating method with high precision and high efficiency. To this end, this article presented an improved spectral decomposition flexibility perturbation method for structural finite element model updating. The improvements of the proposed method lie in two aspects. First, using the uniform correction model, the proposed method is more economical in computation than the initial method because the spectral decomposition and reorganization of elemental stiffness matrices can be avoided. Second, using the twice singular-value-truncation method, the proposed method has better performance than the initial method in combating data noise. A beam structure is employed to demonstrate the proposed method for model updating in a noisy environment. It was found that the result obtained by least squares estimate is seriously distorted and the result obtained by the first singular value truncation is also not entirely satisfactory. Only the result obtained by the second singular value truncation is the most stable and accurate. Overall, the improved spectral decomposition flexibility perturbation method is robust and effective in small modification case, large modification case, adjacent modification case, and multiple modifications case. The proposed method may be very useful for structural finite element model updating in the noisy environment.

Keywords

Model updating flexibility perturbation spectral decomposition singular value truncation

Introduction

The finite element model (FEM) with high accuracy is required to predict the static and dynamical behaviors of the engineering structure during analysis and design. Once a structural FEM is constructed by software, its accuracy should be verified by comparing the analytical response parameters (such as displacements, natural frequencies, and mode shapes) obtained from the FEM with those obtained from the actual structure by static or dynamical experiments. If the agreement between the two is poor, the analytical FEM should be updated so that the correlation between predictions and test results is improved. The modified FEM can then be deemed to be a better representation of the actual structure than the original analytical model and will be used to analyze the static or dynamic responses of the actual structure for future design. This process of modifying the system matrices is called as FEM updating.

One important group of FEM updating techniques is the perturbation methods (also known as the sensitivity methods). In last few decades, the perturbation methods have been widely used in model updating,^1–5 damage detection,^6–13 structural reanalysis,^14–20 vibration control,^21–25 nonlinear mechanics,^26–30 and so on. The traditional perturbation techniques are all the linear approximation methods based on the series expansion (Taylor series or Neumann series). Therefore, the first-order perturbation analysis can only be used for the small variation of the perturbation parameters. When the changes of structural parameters are relatively large, the high-order perturbation analysis or the iteration scheme is usually performed. But the high-order perturbation or iteration will tremendously increase the computation effort, especially for the large-scale complicated structures. To avoid expensive computing costs, Yang^31,32 proposed a new flexibility perturbation method based on matrix spectral decomposition (SD) technique recently. The basic idea of the proposed method is to decompose a structural flexibility matrix into a matrix representation of the connectivity between degrees of freedom (DOFs) and a diagonal matrix containing the magnitude information. Using spectral decomposition and reorganization, the direct relationship between the parameter variations and the flexibility perturbation can be established without any high-order analysis or iteration. According to author’s previous works, it has been shown for the statically determinate structures that the SD flexibility perturbation technique can provide the exact relationship between parameter variations and structural flexibility perturbation quantity. For the statically indeterminate structures, the SD flexibility perturbation method also has good estimate precision.

This article presents a further study of the SD flexibility perturbation method for structural FEM updating. The aim of this research is to increase the computational efficiency and accuracy of the SD flexibility perturbation method in the noisy environment. It is known that the ill-posed least square problem often arises in structural FEM updating due to the complexity of the actual engineering structure. This means that little errors in testing data may lead to very large errors in the results of model updating. The existing regularization methods^33–37 can partly improve the robustness and accuracy of the solutions of the ill-posed equations, but there is still much room for improvement in the calculation accuracy and efficiency. For the FEM updating problem, the elements that need to be corrected in the FEM are often only a small minority. This particularity of FEM updating has not been considered in the previous regularization methods.^33–37 In view of this, a twice singular-value-truncation (SVT) method is proposed in this article to solve the ill-posed problem existed in the SD flexibility perturbation method for structural FEM updating. The particularity of FEM updating is fully considered in the proposed procedure by removing many elements that need no modifications in the second SVT. With the help of twice SVT, the SD flexibility perturbation method can quickly achieve satisfactory results using incomplete noisy modal data. Besides, this article also puts forward a simplified SD flexibility perturbation technique in order to reduce computational effort further. By employing the uniform correction model, the computation operations of spectral decomposition and reorganization in the initial SD flexibility perturbation method can be successfully avoided. This leads to a significant reduction in the computational complexity of the proposed method. The presentation of this work is organized as follows. In section “Theoretical developments of SD flexibility perturbation,” the SD flexibility perturbation theory is briefly reviewed and then a simplified SD flexibility perturbation technique is proposed for model updating. Subsequently, the twice SVT algorithm is proposed to tackle the potential ill-conditioned least square problem in section “The twice SVT technique.” The verification of the feasibility and superiority of the developed method is illustrated in section “Numerical example” with a beam structure. The conclusions of this work are summarized in section “Conclusion.”

Theoretical developments of SD flexibility perturbation

The SD flexibility perturbation method

The SD flexibility perturbation theory provides the direct relationship between the structural parameter variations and the flexibility perturbation quantity. Central to the SD flexibility perturbation method is the spectral decomposition and reorganization of structural elemental stiffness matrices. As is well known, the global stiffness matrix K in structural FEM is a sum of the elemental stiffness matrices, that is

K = \sum_{i = 1}^{N} K_{i}

(1)

where $K_{i}$ is the $i th (n \times n)$ elemental stiffness matrix and N is the total number of elements. In the vast majority of cases, the elemental stiffness matrix $K_{i}$ is not of full rank. For instance, the elemental stiffness matrix is of rank one for the truss structure and is of rank two for the plane beam element. Assuming the rank of $K_{i}$ is r, the spectral decomposition of $K_{i}$ can be expressed as

K_{i} = [c_{i}^{1}, \dots, c_{i}^{r}] [\begin{matrix} p_{i}^{1} \\ ⋱ \\ p_{i}^{r} \end{matrix}] [c_{i}^{1}, \dots, c_{i}^{r}]^{T}

(2)

where $p_{i}^{1}, \dots, p_{i}^{r}$ are the nonzero eigenvalues of $K_{i}$ and $c_{i}^{1}, \dots, c_{i}^{r}$ are the corresponding eigenvectors. In essence, the column vectors $c_{i}^{1}, \dots, c_{i}^{r}$ reflect the connectivity between DOFs. And, the diagonal coefficients $p_{i}^{1}, \dots, p_{i}^{r}$ are the functions purely of the material properties such as the elastic modulus E, the cross-sectional area A, and the moment of inertia I. Therefore, $p_{i}^{1}, \dots, p_{i}^{r}$ can be called as the elemental stiffness parameters and $c_{i}^{1}, \dots, c_{i}^{r}$ are called as the stiffness connectivity vectors. Substituting equation (2) into equation (1), one has

K = CP C^{T}

(3a)

C = [c_{1}^{1}, \dots, c_{1}^{r}, c_{2}^{1}, \dots, c_{2}^{r}, \dots, c_{N}^{r}]

(3b)

P = [\begin{matrix} p_{1}^{1} \\ ⋱ \\ p_{1}^{r} \\ ⋱ \\ p_{N}^{r} \end{matrix}]

(3c)

where the dimensions of the stiffness connectivity matrix C and the diagonal matrix P are $n \times rN$ and $rN \times rN$ , respectively. C and P are independent of each other. Physically, model updating only changes the stiffness parameters $p_{i}^{1}, \dots, p_{i}^{r}$ and does not affect the connectivity vectors $c_{i}^{1}, \dots, c_{i}^{r}$ . Moreover, different types of correction will cause different patterns in the set of changes of $p_{i}^{1}, \dots, p_{i}^{r}$ . Generally, there are two types of model updating. The first class correction will change the elastic modulus E and cause uniform deviation percentages in $p_{i}^{1}, \dots, p_{i}^{r}$ . Thus, this correction model can be also called as the uniform correction model. The second-class correction will change the cross-sectional area A and the moment of inertia I, which leads to inconsistent deviation percentages in $p_{i}^{1}, \dots, p_{i}^{r}$ . Assuming $α_{i}^{j}$ is the stiffness perturbed ratio of the stiffness parameter $p_{i}^{j}$ , the disassembly of the stiffness matrix $K_{d}$ for the updating FEM can also be obtained as

K_{d} = C P_{d} C^{T}

(4a)

P_{d} = [\begin{matrix} p_{1}^{1} (1 - α_{1}^{1}) \\ ⋱ \\ p_{1}^{r} (1 - α_{1}^{r}) \\ ⋱ \\ p_{N}^{r} (1 - α_{N}^{r}) \end{matrix}]

(4b)

where $α_{i}^{j} (i = 1 ~ N, j = 1 ~ r)$ is also called as correction factor. When the $i th$ element has the first class correction, the stiffness perturbed ratios $α_{i}^{1}, α_{i}^{2}, \dots, α_{i}^{r}$ for this element will be the same value, that is, $α_{i}^{1} = α_{i}^{2} = \dots = α_{i}^{r}$ . When the $i th$ element has the second-class correction, the stiffness perturbed ratios $α_{i}^{1}, α_{i}^{2}, \dots, α_{i}^{r}$ for this element will be different. In view of the complexity of actual structure, the uniform correction model will be used in this article since this model gives only a small error for a large structure. Subtracting equation (4a) from equation (3a), the disassembly of stiffness matrix perturbation $Δ K$ can be obtained as

Δ K = K - K_{d} = C Δ P C^{T}

(5a)

Δ P = diag (p_{1}^{1} α_{1}^{1}, \dots, p_{N}^{r} α_{N}^{r})

(5b)

On the basis of equations (3) and (4), the SD flexibility perturbation theory can be established according to the interconversion of stiffness and flexibility. The next derivation should be divided into three cases: $n > rN$ , $n = rN$ , and $n < rN$ . According to the FEM theory, the case of $n > rN$ corresponds to the free–free structure, the case of $n = rN$ corresponds to the statically determinate structure, and the case of $n < rN$ corresponds to the statically indeterminate structure.

For the case of $n = rN$ (i.e. the statically determinate structure), the disassemblies of the flexibility matrices before and after updating can be obtained from equations (3a) and (4a) as

F = K^{- 1} = (C^{- 1})^{T} P^{- 1} C^{- 1}

(6)

F_{d} = K_{d}^{- 1} = (C^{- 1})^{T} P_{d}^{- 1} C^{- 1}

(7)

where F is the initial flexibility matrix and $F_{d}$ is the updated flexibility matrix. Let

D = (C^{- 1})^{T}

(8)

Q = P^{- 1}

(9)

Q_{d} = P_{d}^{- 1}

(10)

where D is the $n \times rN$ flexibility connectivity matrix with column vectors $d_{i}^{j}$ , that is

D = [d_{1}^{1}, \dots, d_{1}^{r}, d_{2}^{1}, \dots, d_{2}^{r}, \dots, d_{N}^{r}]

(11)

Equations (6) and (7) can be expressed as

F = DQ D^{T}

(12)

F_{d} = D Q_{d} D^{T}

(13)

Subtracting equation (12) from (13), the flexibility change $Δ F$ can be obtained as

Δ F = F_{d} - F = D Δ Q D^{T}

(14a)

Δ Q = diag (q_{1}^{1} β_{1}^{1}, \dots, q_{N}^{r} β_{N}^{r})

(14b)

q_{i}^{j} = \frac{1}{p_{i}^{j}}, for i = 1 ~ N, j = 1 ~ r

(14c)

β_{i}^{j} = \frac{1}{1 - α_{i}^{j}} - 1, for i = 1 ~ N, j = 1 ~ r

(14d)

From equations (3a), (4a), (12), and (13), one can see that the stiffness and flexibility matrices have the similar decomposition formulas. From equations (5a) and (14a), the stiffness change $Δ K$ and the flexibility change $Δ F$ also have the similar decomposition formulas. Thus, $q_{i}^{j}$ can be called as the elemental flexibility parameter, and $β_{i}^{j}$ can be called as the flexibility perturbed parameter. According to equation (14d), the relationships between $α_{i}^{j}$ and $β_{i}^{j}$ are

β_{i}^{j} = \frac{α_{i}^{j}}{1 - α_{i}^{j}}

(15)

and

α_{i}^{j} = \frac{β_{i}^{j}}{1 + β_{i}^{j}}

(16)

From the above derivation, the SD flexibility perturbation technique has been obtained for the statically determinate structures. That is, if the stiffness modified parameter $α_{i}^{j}$ is given, the corresponding flexibility change $Δ F$ can be exactly calculated by the process: $α_{i}^{j} \overset{Equation (15)}{\to} 4 pc β_{i}^{j} \overset{Equation (14 a)}{\to} 4 pc Δ F$ . However, if $Δ F$ is known, the corresponding modified parameter $α_{i}^{j}$ can also be exactly computed through the process: $Δ F \overset{Equation (14 a)}{\to} 4 pc β_{i}^{j} \overset{Equation (16)}{\to} 4 pc α_{i}^{j}$ .

For the cases of $n > rN$ and $n < rN$ , the exact decomposition of $Δ F$ similar to equation (14a) is nonexistent. For such cases, the following formula can be employed as the approximate decomposition of $Δ F$ , that is

Δ F = H Δ Q H^{T}

(17a)

H = (C^{+})^{T}

(17b)

In equation (17b), $C^{+}$ is generalized inverse of C, which can be computed by

C^{+} = (C^{T} C)^{- 1} C^{T}, for n > rN

(18a)

C^{+} = C^{T} (C C^{T})^{- 1}, for n < rN

(18b)

The simplified SD flexibility perturbation for the uniform correction model

As stated before, the uniform correction model will be used in this article in view of the complexity of actual structure. This means that the stiffness perturbed ratios $α_{i}^{1}, α_{i}^{2}, \dots, α_{i}^{r}$ for the $i th$ element are the same value. Correspondingly, the flexibility perturbed ratios $β_{i}^{1}, β_{i}^{2}, \dots, β_{i}^{r}$ of the $i th$ element are also the same value. Let

α_{i} = α_{i}^{1} = α_{i}^{2} = \dots = α_{i}^{r}, for i = 1 ~ N

(19a)

β_{i} = β_{i}^{1} = β_{i}^{2} = \dots = β_{i}^{r}, for i = 1 ~ N

(19b)

Equation (14a) can be further simplified for the convenience of application. From equation (14b), one has

Δ Q = Q \cdot [β]

(20a)

[β] = diag (β_{1}^{1}, \dots, β_{N}^{r})

(20b)

Substituting equation (20a) into (14a) yields

Δ F = DQ [β] D^{T}

(21)

Equation (21) can be rewritten as

Δ F = DQ D^{T} (D^{T})^{- 1} [β] Q^{- 1} (D^{- 1} D) Q D^{T}

(22)

From equations (8), (9), and (12), equation (22) can be simplified as

Δ F = F (C [β] P C^{T}) F

(23)

From equations (2), (3a), and (18b), equation (23) can be rewritten as

Δ F = \sum_{i = 1}^{N} β_{i} (F K_{i} F)

(24)

In addition, the relationships similar to equations (15) and (16) between $α_{i}$ and $β_{i}$ are

β_{i} = \frac{α_{i}}{1 - α_{i}}

(25)

and

α_{i} = \frac{β_{i}}{1 + β_{i}}

(26)

From equations (24)–(26), the simplified SD flexibility perturbation approach has been established. That is, if the stiffness modified parameter $α_{i}$ is known, the corresponding flexibility change $Δ F$ can be calculated by the process: $α_{i} \overset{Equation (23)}{\to} 4 pc β_{i} \overset{Equation (22)}{\to} 4 pc Δ F$ . However, if $Δ F$ is given, the corresponding modified parameter $α_{i}$ can also be computed through the process: $Δ F \overset{Equation (22)}{\to} 4 pc β_{i} \overset{Equiation (24)}{\to} 4 pc α_{i}$ . Compared equation (24) with (14a), it is found that the simplified SD flexibility perturbation method has two advantages: (1) the number of the perturbed parameters is reduced $(rN \overset{decrease}{\to} 4 pc N)$ and (2) the calculation is more concisely because the spectral decomposition and reorganization of elemental stiffness matrices can be avoided.

The twice SVT technique

As stated before, for structural model updating, the correction factors $α_{i} (i = 1 ~ N)$ can be computed through the process of $Δ F \overset{Equation (22)}{\to} 4 pc β_{i} \overset{Equation (24)}{\to} 4 pc α_{i}$ . In engineering practice, $Δ F$ can be approximately obtained from structural vibration modal testing by

Δ F = \sum_{j = 1}^{m} (\frac{1}{λ_{dj}} φ_{dj} φ_{dj}^{T} - \frac{1}{λ_{j}} φ_{j} φ_{j}^{T})

(27)

where $λ_{j}$ and $φ_{j}$ are the $j th$ eigenvalue and mode shape of the analytical structure, respectively; $λ_{dj}$ and $φ_{dj}$ are the $j th$ eigenvalue and mode shape of the actual structure, respectively; and m is the number of measured modes in structural vibration modal testing.

With $Δ F$ determined by equation (27), the flexibility correction factors $β_{i} (i = 1 ~ N)$ can be first computed by manipulating equation (24) into a multiple linear regression model, that is

\bar{Δ F} = [η_{i}] \cdot {β_{i}}

(28a)

{β_{i}} = (β_{1}, β_{2}, \dots, β_{N})^{T}

(28b)

[η_{i}] = [η_{1}, η_{2}, \dots, η_{N}], η_{i} = \bar{F K_{i} F}

(28c)

where $\bar{Δ F}$ is the straightened vector obtained from the matrix $Δ F$ . Let the $(i, j) th$ element of $Δ F$ be represented by $Δ f_{ij}$ , then $\bar{Δ F} = (Δ f_{11}, \dots, Δ f_{1 n}, Δ f_{21}, \dots, Δ f_{2 n}, \dots, Δ f_{ij}, \dots, Δ f_{nn})^{T}$ . $[η_{i}]$ is the coefficient matrix of the regression model, and ${β_{i}}$ is the vector of unknown parameters. The least squares estimate (LSE)³³ is often used to compute ${β_{i}}$ from equation (28a), that is

{β_{i}} = [η_{i}]^{+} \bar{Δ F}

(29)

where the superscript “+” denotes the Moore–Penrose generalized inverse. According to the matrix theory, the generalized inverse of a matrix can be computed by singular value decomposition (SVD) technique. The SVD of $[η_{i}]$ can be expressed as

[η_{i}] = U Λ V^{T}

(30a)

U = [u_{1}, u_{2}, \dots, u_{n}]

(30b)

V = [v_{1}, v_{2}, \dots, v_{N}]

(30c)

Λ = [\begin{matrix} Z & 0 \\ 0 & 0 \end{matrix}], Z = diag (σ_{1}, σ_{2}, \dots, σ_{t})

(30d)

where U and V are the orthogonal matrices and $σ_{1}, σ_{2}, \dots, σ_{t}$ are the nonzero singular values of $[η_{i}]$ with $σ_{1} \geq σ_{2} \geq \dots \geq σ_{t}$ . From equation (30), the generalized inverse of $[η_{i}]$ can be computed by

[η_{i}]^{+} = \sum_{x = 1}^{t} σ_{x}^{- 1} v_{x} u_{x}^{T}

(31)

In most cases, the matrix $[η_{i}]$ is often ill-conditioned in engineering practice. This may lead to unstable and inaccurate calculation results by equation (29). In view of this, a twice SVT technique is proposed to solve the ill-conditioned least square problem for improving the stability and reliability of model updating. By ignoring some smaller singular values, the first SVT solution of ${β_{i}}$ can be obtained as

{β_{i}} = (\sum_{x = 1}^{z} σ_{x}^{- 1} v_{x} u_{x}^{T}) \bar{Δ F}

(32)

where z is the number of remained singular values. These remained singular values $σ_{x} (x = 1 ~ z)$ all satisfy $σ_{x} / σ_{max} \geq ζ$ ( $ζ$ is a predefined threshold value, for example, $ζ = 0.001$ ). When ${β_{i}}$ is determined by the first SVT using equation (32), the most probable modified elements can be selected according to the relatively larger values in ${β_{i}}$ . Generally, those values in ${β_{i}}$ that are less than 0.05 should be deemed to correspond to those elements that need no modifications. Then, equation (28a) can be further simplified for the second SVT solution by removing some column vectors in $[η_{i}]$ and coefficients in ${β_{i}}$ corresponding to those elements that need no modifications. That is

\bar{Δ F} = [η_{i}^{*}] {β_{i}^{*}}

(33)

where $[η_{i}^{*}]$ is the remained matrix of $[η_{i}]$ after removing some column vectors related to those elements that need no modifications, ${β_{i}^{*}}$ is the remained vector of ${β_{i}}$ after removing the corresponding coefficients. From equation (33), the second SVT solution can be obtained as

{β_{i}^{*}} = (\sum_{x = 1}^{z^{*}} {(σ_{x}^{*})}^{- 1} v_{x}^{*} {(u_{x}^{*})}^{T}) \bar{Δ F}

(34)

where $σ_{x}^{*}$ , $v_{x}^{*}$ , and $u_{x}^{*}$ are the remained singular value data of $[η_{i}^{*}]$ in the second SVT. Finally, the correction factor $α_{i}$ can be computed by equation (26) with $β_{i}$ determined by equation (34).

Numerical example

A single span beam shown in Figure 1 is used as an example to verify the feasibility and superiority of the proposed method. The beam was modeled using 32 equal Euler–Bernoulli elements giving 64 DOFs (31 translational and 33 rotational). The physical parameters of this beam are cross-sectional area $A = 0.16 m^{2}$ , moment of inertia $I = 2.5 \times 10^{- 3} m^{4}$ , Young’s modulus $E = 2.5 \times 10^{10} Pa$ , length of each element $l = 1 m$ , and density $ρ = 2.5 \times 10^{3} kg / m^{3}$ .

Figure 1.

A single span beam.

Theoretically, any correction scenario can be simulated to validate the proposed method. Without the loss of generality, four correction categories as shown in Table 1 are studied in this example. Note that these four correction cases are designed to represent the common types of FEM updating. Case 1 is designed to simulate the single small correction case. Case 2 is designed to simulate the single large correction case. Case 3 is designed to simulate the correction case of adjacent elements. Case 4 is designed to simulate the multiple corrections case. In the following discussion, the first-order frequency and mode shape are used to compute the flexibility change $Δ F$ before and after modification. Moreover, only the data corresponding to transnational DOFs are used in the following calculation due to the difficulty of measuring the rotational DOFs.

Table 1.

Correction categories of the beam structure.

Case no.	Element no.	Stiffness reduction ratio
1	13	15%
2	13	50%
3	18 and 19	20% and 20%
4	7, 15, and 24	10%, 20%, and 30%

For correction case 1, Figures 2 –4 present the calculation results of ${β_{i}}$ without noise by LSE (equation (29)), the first SVT (equation (32)), and the second SVT (equation (34)), respectively. Note that the threshold value $ζ$ is 0.001 in SVT. Apparently, Figures 2 –4 all indicate that only element 13 needs to make a modification. It has been shown that results obtained by LSE, the first SVT, and the second SVT are all satisfactory when there is no noise in the used modal data. Using equation (26), the correction factor $α_{i}$ of element 13 can be calculated as: $α_{13} = 0.1438$ (LSE), $α_{13} = 0.126$ (the first SVT), and $α_{13} = 0.1503$ (the second SVT). Compared with the simulated value 15%, it is obvious that the result obtained by the second SVT has the highest accuracy.

Figure 2.

The calculation results for correction case 1 obtained by LSE (no noise).

Figure 3.

The calculation results for correction case 1 obtained by the first SVT (no noise).

Figure 4.

The calculation results for correction case 1 obtained by the second SVT (no noise).

When 5% noise is added to the mode shape, Figures 5 –7 give the calculation results obtained by LSE, the first SVT, and the second SVT. From Figure 5, one can see that LSE fails to indicate that only element 13 needs to make a modification. It has been shown that LSE is very sensitive to noise and may lead to wrong modification results. From Figures 6 and 7, the first SVT and the second SVT both have good ability to resist noise and can indicate that only element 13 needs to make a modification. Using equation (26), the correction factor $α_{i}$ of element 13 can be calculated as: $α_{13} = 0.1255$ (the first SVT) and $α_{13} = 0.1499$ (the second SVT). Compared with the true value 0.15, the accuracy of the result obtained by the second SVT is obviously improved.

Figure 5.

The calculation results for correction case 1 obtained by LSE (5% noise).

Figure 6.

The calculation results for correction case 1 obtained by the first SVT (5% noise).

Figure 7.

The calculation results for correction case 1 obtained by the second SVT (5% noise).

When 10% noise is added to the mode shape, Figures 8 –10 provide the calculation results obtained by LSE, the first SVT, and the second SVT, respectively. Again, the result obtained by LSE in Figure 8 is completely distorted. The result obtained by the first SVT in Figure 9 is also not entirely satisfactory since elements 13, 14, and 17 are all identified as the elements need modifications ( $β_{13} = 0.1428$ , $β_{14} = 0.0593$ , $β_{17} = 0.0747$ ; they are all greater than 0.05). From Figure 10, the second SVT still performs very well for the fact that only element 13 is identified to be the element that needs to make a modification $(β_{13} = 0.195)$ . Using equation (26), the correction parameter $α_{i}$ of element 13 can be calculated as: $α_{13} = 0.125$ (the first SVT) and $α_{13} = 0.1632$ (the second SVT). The above results once more demonstrate the superiority of the second SVT compared with LSE and the first SVT.

Figure 8.

The calculation results for correction case 1 obtained by LSE (10% noise).

Figure 9.

The calculation results for correction case 1 obtained by the first SVT (10% noise).

Figure 10.

The calculation results for correction case 1 obtained by the second SVT (10% noise).

For correction case 2, Figures 11 –13 present the calculation results using LSE, the first SVT, and the second SVT, respectively. Apparently, the results obtained by the second SVT are the most stable and accurate. The correction factor $α_{i}$ of element 13 obtained by the second SVT is: $α_{13} = 0.4536$ (no noise), $α_{13} = 0.5379$ (5% noise), and $α_{13} = 0.5768$ (10% noise). Compared with the true value 0.5, the accuracies of these results are acceptable. These results reflect the particular advantage of the SD flexibility perturbation method over other perturbation methods for large modification.

Figure 11.

The calculation results for correction case 2 by LSE (no noise, 5% noise, and 10% noise).

Figure 12.

The calculation results for correction case 2 by the first SVT (no noise, 5% noise, and 10% noise).

Figure 13.

The calculation results for correction case 2 by the second SVT (no noise, 5% noise, and 10% noise).

For correction case 3, Figures 14 –16 present the calculation results using LSE, the first SVT, and the second SVT, respectively. Once again, the results obtained by the second SVT are the most stable and accurate. The correction factors $α_{i}$ of elements 18 and 19 obtained by the second SVT are: $α_{18} = 0.1828; α_{19} = 0.2173$ (no noise), $α_{18} = 0.1632; α_{19} = 0.2276$ (5% noise), and $α_{18} = 0.1766; α_{19} = 0.2302$ (10% noise). These results show that the proposed method also performs well for the adjacent correction case.

Figure 14.

The calculation results for correction case 3 by LSE (no noise, 5% noise, and 10% noise).

Figure 15.

The calculation results for correction case 3 by the first SVT (no noise, 5% noise, and 10% noise).

Figure 16.

The calculation results for correction case 3 by the second SVT (no noise, 5% noise, and 10% noise).

For correction case 4, Figures 17 –19 present the calculation results using LSE, the first SVT, and the second SVT, respectively. Apparently, only the results obtained by the second SVT in Figure 19 can clearly indicate that elements 7, 15, and 24 need to make modifications. The correction parameters $α_{i}$ of elements 7, 15, and 24 obtained by the second SVT are

\begin{array}{l} α_{7} = 0.0673; α_{15} = 0.1871; α_{24} = 0.2929 (no noise) \\ α_{7} = 0.0667; α_{15} = 0.1320; α_{24} = 0.2449 (5 % noise) \\ α_{7} = 0.1320; α_{15} = 0.1905; α_{24} = 0.2374 (10 % noise) \end{array}

Figure 17.

The calculation results for correction case 4 by LSE (no noise, 5% noise, and 10% noise).

Figure 18.

The calculation results for correction case 4 by the first SVT (no noise, 5% noise, and 10% noise).

Figure 19.

The calculation results for correction case 4 by the second SVT (no noise, 5% noise, and 10% noise).

It has been shown that the proposed method can achieve good model updating results for the multiple corrections case.

To further investigate the ability of the proposed method to resist higher noise, Figures 20 –23 give the calculation results for cases 2 and 4 under 15% noise, respectively. When 15% noise is added to the mode shape, the result obtained by LSE in Figure 20 is completely distorted. The result obtained by the first SVT in Figure 21 is also not entirely satisfactory since it indicates that many elements besides element 13 need to be corrected. Only the second SVT performs well because that its result shown in Figure 21 clearly indicates that only element 13 needs to be corrected. Using equation (26), the correction parameter $α_{13}$ of the second SVT can be calculated as $α_{13} = 0.6099$ , which has 22% relative error as compared to the assumed value 0.5. For case 4, the similar conclusion can be drawn based Figures 22 and 23 that the result obtained by the second SVT is the most stable and accurate. The correction parameters $α_{i}$ of elements 7, 15, and 24 obtained by the second SVT under 15% noise are $α_{7} = 0.1484$ , $α_{15} = 0.1866$ , and $α_{24} = 0.2265$ , which have 48.4%, 6.7%, and 24.5% relative errors as compared to the assumed values 0.1, 0.2, and 0.3, respectively. It should be noted that the accuracy of the second SVT is decreasing with the increase in the noise level. Even with the second SVT, one can see from Figure 23 that several elements such as elements 3 and 23 have been wrongly determined to be the elements need corrections.

Figure 20.

The calculation results for correction case 2 by LSE (15% noise).

Figure 21.

The calculation results for correction case 2 by the first and second SVTs (15% noise).

Figure 22.

The calculation results for correction case 4 by LSE (15% noise).

Figure 23.

The calculation results for correction case 4 by the first and second SVTs (15% noise).

According to the above results, it has been shown that LSE is very sensitive to data noise. The reason for this is that the linear equation (28) for FEM updating has serious morbidity problem. In other words, there is stronger multicollinearity among column vectors in the coefficient matrix $[η_{i}]$ of equation (28). From the view of singular values, the fluctuation of the solution is mainly caused by the smaller singular values of the coefficient matrix $[η_{i}]$ . Thus, the SVT technique can partly improve the robustness and accuracy of the solution of the ill-posed equation by ignoring some smaller singular values. As shown in the above results by the first SVT, the true elements that need corrections can be found in these figures, but at the same time, many other elements that need no corrections were wrongly identified. For the FEM updating problem, the elements that need to be corrected in the FEM are often only a small minority. To further improve the accuracy of the solution, a feasible method is first to remove those elements that need no corrections according to the result of the first SVT and then to solve the model updating equations again through the SVT technique. From the view of solving equation, this is equivalent to effectively reducing the number of unknowns. Therefore, the results obtained by the second SVT have better precision. On the whole, the proposed scheme is simple and effective since no iteration operation is required in the calculation process, regardless of whether the modification is small or large.

Conclusion

An improved SD flexibility perturbation method for structural model updating has been developed in this study. Using the uniform correction model, the proposed method is more economical in computation than the initial method because the spectral decomposition and reorganization of elemental stiffness matrices can be avoided. Using twice SVT method, the proposed method has better performance of combating data noise. A beam structure is used as an example to validate the proposed method for model updating. The results show that the proposed method performs well in small correction case, large correction case, adjacent correction case, and multiple corrections case. It has been shown that the improved SD flexibility perturbation method could be a very promising approach for structural model updating or structural damage detection.

Footnotes

Handling Editor: MA Hariri-Ardebili

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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 (41272345, 11202138, and 41572305).

ORCID iD

QW Yang

References

Bartilson

Jang

Smyth

AW.

Finite element model updating using objective-consistent sensitivity-based parameter clustering and Bayesian regularization. Mech Syst Signal Pr 2019; 114: 328–345.

Yuan

Liang

Silva

et al . Parameter selection for model updating with global sensitivity analysis. Mech Syst Signal Pr 2019; 115: 483–496.

Altunişik

Karahasan

OŞ

Genç

et al . Sensitivity-based model updating of building frames using modal test data. KSCE J Civ Eng 2018; 22: 4038–4046.

Deng

Guo

Interval identification of structural parameters using interval overlap ratio and Monte Carlo simulation. Adv Eng Softw 2018; 121: 120–130.

Rubio

Louf

Chamoin

Fast model updating coupling Bayesian inference and PGD model reduction. Comput Mech 2018; 2018: 1–25.

Yang

Liu

JK.

Damage identification by the eigenparameter decomposition of structural flexibility change. Int J Numer Meth Eng 2009; 78: 444–459.

Yan

Ren

WX.

A direct algebraic method to calculate the sensitivity of element modal strain energy. Int J Numer Meth Bio 2011; 27: 694–710.

Law

. Differentiating different types of damage with response sensitivity in time domain. Mech Syst Signal Pr 2010; 24: 2914–2928.

Huang

Liu

JK.

State-space formulation for simultaneous identification of both damage and input force from response sensitivity. Smart Struct Syst 2011; 8: 157–172.

10.

Wong

Huang

Xiong

et al . Generalized-order perturbation with explicit coefficient for damage detection of modular beam. Arch Appl Mech 2011; 81: 451–472.

11.

Moughty

Casas

. Damage sensitivity evaluation of vibration parameters under ambient excitation. In: Proceedings of the international conference on experimental vibration analysis for civil engineering structures, San Diego, USA, 12–14 July 2017, pp.249–260. Cham: Springer.

12.

Moughty

Casas

. A state of the art review of modal-based damage detection in bridges: development, challenges, and solutions. Appl Sci 2017; 7: 510.

13.

Shahri

AHH

Ghorbani-Tanha

. Damage detection via closed-form sensitivity matrix of modal kinetic energy change ratio. J Sound Vib 2017; 401: 268–281.

14.

Datteo

Quattromani

Cigada

On the use of AR models for SHM: a global sensitivity and uncertainty analysis framework. Reliab Eng Syst Safe 2018; 170: 99–115.

15.

Van Keulen

Haftka

Kim

. Review of options for structural design sensitivity analysis. Part 1: linear systems. Comput Method Appl M 2005; 194: 3213–3243.

16.

Kirsch

Reanalysis and sensitivity reanalysis by combined approximations. Struct Multidiscip O 2010; 40: 1–15.

17.

Kirsch

Bogomolni

Sheinman

Efficient structural optimization using reanalysis and sensitivity reanalysis. Eng Comput 2007; 23: 229–239.

18.

Zuo

Zhang

et al . Fast Structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods. Struct Multidiscip O 2011; 43: 799–810.

19.

Sun

Liu

et al . New adaptive technique of Kirsch method for structural reanalysis. AIAA J 2011; 52: 486–495.

20.

Zuo

Bai

Sensitivity reanalysis of static displacement using Taylor series expansion and combined approximate method. Struct Multidiscip O 2016; 53: 953–959.

21.

Massa

Tison

Lallemand

et al . Structural modal reanalysis methods using homotopy perturbation and projection techniques. Comput Method Appl M 2011; 200: 2971–2982.

22.

Chakraborty

Roy

BK.

Reliability based optimum design of tuned mass damper in seismic vibration control of structures with bounded uncertain parameters. Probabilist Eng Mech 2011; 26: 215–221.

23.

El-Ganaini

WA.

Duffing oscillator vibration control via suspended pendulum. Appl Math 2018; 12: 203–215.

24.

Zheng

Tian

et al . Stochastic nonlinear vibration and reliability of orthotropic membrane structure under impact load. Thin Wall Struct 2017; 119: 247–255.

25.

Hamed

Alharthi

AlKhathami

HK.

Active vibration control of a dynamical system subjected to simultaneous excitation forces. Int J Appl Eng Res 2017; 12: 434–442.

26.

Shen

Xiang

Lin

Nonlinear vibration of functionally graded graphene-reinforced composite laminated plates in thermal environments. Comput Method Appl M 2017; 319: 175–193.

27.

Zhou

Yang

et al . Global dynamics of pipes conveying pulsating fluid in the supercritical regime. Int J Appl Mech 2017; 9: 1750029.

28.

Yang

Bradford

MA.

Postbuckling snaking of axially loaded infinite beam on nonlinear foundation with restabilizing effects. J Eng Mech 2017; 143: 04017019.

29.

Jiao

Chen

XW.

Approximate solutions of the Alekseevskii–Tate model of long-rod penetration. Acta Mech Sin 2018; 34: 334–348.

30.

Yeh

Munday

Taira

Laminar free shear layer modification using localized periodic heating. J Fluid Mech 2017; 822: 561–589.

31.

Yang

QW.

A new damage identification method based on structural flexibility disassembly. J Vib Control 2011; 17: 1000–1008.

32.

Yang

QW.

An exact approach for structural damage assessment. Math Probl Eng 2013; 2013: 230957.

33.

Neumaier

Solving ill-conditioned and singular linear systems: a tutorial on regularization. SIAM Rev 1998; 40: 636–666.

34.

Hansen

O’Leary

DP.

The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comput 1993; 14: 1487–1503.

35.

Law

. Adaptive Tikhonov regularization for damage detection based on nonlinear model updating. Mech Syst Signal Pr 2010; 24: 1646–1664.

36.

Tikhonov

Goncharsky

Stepanov

et al . Numerical methods for the solution of ill-posed problems. Berlin, Germany: Springer Science+Business Media, 2013.

37.

Entezami

Shariatmadar

Sarmadi

Structural damage detection by a new iterative regularization method and an improved sensitivity function. J Sound Vib 2017; 399: 285–307.

An improved spectral decomposition flexibility perturbation method for finite element model updating

Abstract

Keywords

Introduction

Theoretical developments of SD flexibility perturbation

The SD flexibility perturbation method

The simplified SD flexibility perturbation for the uniform correction model

The twice SVT technique

Numerical example

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References