Substructuring Method in Structural Health Monitoring

In sensitivity-based finite element model updating, the eigensolutions and eigensensitivities are calculated repeatedly, which is a time-consuming process for large-scale structures. In this chapter, a forward substructuring method and an inverse substructuring method are proposed to fulfill the model updating of large-scale structures. In the forward substructuring method, the analytical FE model of the global structure is divided into several independent substructures. The eigensolutions of each independent substructure are used to recover the eigensolutions and eigensensitivities of the global structure. Consequently, only some specific substructures are reanalyzed in model updating and assembled with other untouched substructures to recover the eigensolutions and eigensensitivities of the global structure. In the inverse substructuring method, the experimental modal data of the global structure are disassembled into substructural flexibility. Afterwards, each substructure is treated as an independent structure to reproduce its flexibility through a model-updating process. Employing the substructuring method, the model updating of a substructure can be conducted by measuring the local area of the concerned substructure solely. Finally, application of the proposed methods to a laboratory tested frame structure reveals that the forward and inverse substructuring methods are effective in model updating and damage identification.


Introduction
Accurate finite element (FE) models are essential in damage identification and condition assessment for structural health monitoring. In vibration-based model-updating process, the FE model of a structure is iteratively updated to guarantee its vibration properties to reproduce the measured counterparts in an optimal manner [1]. In the optimization process, the structural responses are usually used to construct the objective function. The response sensitivities, which are the first derivatives of the structural responses to some structural physical parameters, are used to indicate a rapid searching direction. In this regard, the eigensolutions and their associated sensitivity matrices of the analytical model are required to be gained repeatedly in each iteration [2,3]. The majority of the practical structures in civil engineering are large in scale, thus their FE models usually consists of a large number of degrees of freedom (DOFs) and uncertain updating parameters. The conventional model updating methods of large-scale structures are expensive in terms of computation time and computer memory [2].
It has been proved that the substructuring methods are efficient in dealing with large-scale structures, as it takes the local area as an independent structure [4][5][6][7][8][9]. First, the global structure is divided rationally into several smaller substructures to make it much easier and faster to analyze the small substructures independently. Second, the FE model of a substructure has much fewer uncertain parameters than the global structure, which helps to accelerate the convergence of optimization process to identify these parameters and alleviates the illcondition problems. Third, the substructuring method is required to measure the local area of the practical structure and save the experimental instruments. Finally, the substructuring method can be more promising if combined with parallel computation.
In this chapter, a forward substructuring method and an inverse substructuring method are proposed for model updating and damage identification. In the forward substructuring method, the divided substructures are analyzed independently and are assembled to recover the eigensolutions of the global structure by satisfying the coordination condition of displacement at the interfaces. Afterwards, the fast-calculated eigensolutions and eigensensitivities of the global structure are used for model updating. In the inverse substructuring method, the experimental modal data of the global structure are disassembled into the substructural flexibility by satisfying the coordination condition of force and displacement at the interfaces. Based on the extracted substructural flexibility, the model-updating process is performed on the concerned substructure by treating it as an independent structure. In the following part, the forward and inverse substructuring methods will be explained first and then the two kinds of substructurebased model updating methods will be verified by a laboratory-tested frame structure.

Forward substructuring method 2.1. Eigensolutions
In the forward substructuring method, the eigensolutions and eigensensitivities of a substructure are calculated and assembled to recover those of the global structure. The global structure is divided into N S independent substructures, and the number of DOFs of each substructure is n j (j = 1,2,…, N S ). Treated as an independent structure, the eigenequation of the jth substructure is expressed as where K(j) and M(j) are the stiffness matrix and mass matrix of the jth substructure, respectively. (φi(j), λi(j)) are the ith eigenpairs of the jth substructure. The n (j) pairs of eigenvalues and eigenvectors are expressed as [10] Λ ðjÞ ¼ Diag λ And due to orthogonality, eigenvectors satisfy the two following formulas as The eigensolutions of the global structure can be recovered by adding constraints at the interfaces to obey the principle of virtual work and geometric compatibility like [11] Λ p À λI ÀΓ Matrix C gives the general implicit constraints to guarantee the nodes at the interface identical displacement [11]. C contains two nonzero elements in each row, which are 1 and À1 for a rigid interface connection. Λ p and Φ p are diagonally assembled from the eigensolutions of each substructure. λ is the eigenvalue of the global structure, which is the square of circular frequencies. The eigenvectors of the global structure are recovered by Φ ¼ Φ p fzg. τ indicates the interface forces between the adjacent substructures. Superscript "p" denotes the primitive matrices, which is assembled diagonally from the substructural matrices before displacement constraints at the adjacent substructures are imposed.
It is noted from Eq. (2) that Λ p and Φ p are assembled from all modes of the substructures. It is inefficient and unworthy with all eigenmodes available, as only the first few eigenmodes are usually required for a large-scale structure. Here, the first few eigensolutions of each substructure are selected as "master" modes, and the residual higher modes are the "slave" modes.
Only the master modes are used to gain the eigenequation of the global structure.
From here on, subscript "m" represents the "master" modes and subscript "s" denotes the "slave" modes, respectively. The eigenequation (Eq. (2)) is then rewritten according to the master modes and slave modes as According to the second line of Eq. (4), the slave coordinates can be expressed as Substitution of Eq. (6) into Eq. (4) gives Generally, the lower eigenmodes are usually required by a structure. The eigenvalues λ are much smaller than Λ p s when the size of the master modes is selected rationally. In this regard, Eq. (7) is approximated as: The above eigenequation can be simplified by denoting τ with z m from the second line of Eq. (8) and substituting it into the first line as: Consequently, λ and z m are available by solving Eq. (9) with commonly used eigensolver such as Simpson method or Lanczos method [10]. And the eigenvector of the global structure is recovered from the master modes by Φ ¼ Φ p m z m . The size of the simplified eigenequation (Eq. (9)) is equal to the number of the master modes, which is much smaller than the original one (Eq. (2)). It is noted from Eq. (9) that only the master eigensolutions of the independent substructures are used to gain the eigensolutions of the global structure. The contribution of the slave modes is compensated by the first-order residual flexibility ζ ¼ Γ T s ðΛ p s Þ À1 Γ s , which is calculated by the master modes as:

Eigensensitivity
In this section, the eigensensitivity of the ith (i=1, 2, …, N) mode with respect to an elemental parameter will be derived. The elemental stiffness parameter α in the Ath substructure is illustrated in the following. Writing Eq. (9) for the ith mode and differentiating it with respect to parameter α gives [11] Premultiplying fz i g T on both sides of Eq. (11) gives Since ½ðΛ p m À λI m Þ þ Γ m ζ À1 Γ T m z m ¼ 0 (Eq. (9)) and ½Λ p m þ Γ m ζ À1 Γ T m À λ i I are a symmetric matrix, the first item on the left side of Eq. (12) is zero. In consequence, the ith eigenvalue derivative with respect to the designed parameter α is available by [12]    ∂α can be calculated rapidly by treating the Ath substructure as an independent structure with Nelson's method [12,13].
The ith eigenvector of the global structure is recovered by the master modes as Eq. (16) is differentiated with respect to the structural parameter α as where Φ p m and ∂Φ p m ∂α are the master eigenvectors and their derivatives of the Ath substructure, respectively. {z i } is the eigenvector calculated from Eq. (9). Only ∂zi ∂α n o is required to calculate the eigenvector derivative of the ith mode in Eq. (17).
∂zi ∂α n o is rewritten by the sum of a particular part and a general part as where c i is a participation factor and {ν i } is a residual vector. Substituting Eq. (18) into Eq. (11) leads to In consequence, Ψ and {Y i } can be calculated from Eq. (21) since all of their items have been available in the calculation of the eigenvalue derivatives proposed in the former section.
If no repeated roots exist in Eq. (20), Ψ takes the size of m p Â m p with the rank of (m p -1). To solve this rank-deficient equation (Eq. (20)), the kth item (corresponds to the maximum entry in {z i }) in {ν i } is assumed to be zero, and the corresponding row and column in Ψ and corresponding item in {Y i } are assumed to be zeros as well [14]. The full rank equation is formed as In consequence, the vector {ν i } is solved from Eq. (22).
The eigenvectors {z i } satisfy the orthogonal condition of Equation (23) is differentiated with respect to α as Substitution of Eq. (18) into Eq. (24) gives The participation factor c i is thus obtained as Finally, the first-order derivative of {z i } with respect to α is calculated by It is noted from Eq. (17) 9)) rapidly. Only the derivative matrices of the master modes in the Ath substructure are needed to recover the eigensensitivity of the global structure. As the size of the independent substructures is much smaller than that of the global structure, the proposed substructuring method can significantly improve the computational efficiency.

Substructure-based updating method
Based on the eigensolutions and eigensensitivities calculated with the forward substructuring method, the substructure-based model updating is described in Figure 1 with an iterative process. In each iteration, the eigensolutions are calculated from the modified substructures with the above substructuring method and are then compared with the experimental modal data (frequencies and mode shapes) to construct the objective function. The substructurebased eigensensitivities with respect to a specific parameter are calculated from the substructure containing the concerned parameter, to indicate the searching direction in each optimal step. The objective function is minimized by adjusting the elemental parameters α iteratively according to the eigensensitivity matrices.
The objective function formed by the modal frequency and the mode shape is written as [14] JðαÞ ¼ where λ i E and φ ji E represent the experimental frequencies and mode shapes, respectively. λ i FE and φ ji FE are the frequencies and mode shapes gained from the analytical FE model with the substructuring method (Eq.(8)) proposed above. W λi and W φi are the weighting matrix of frequencies and mode shapes. The objective function is minimized by adjusting the elemental parameters α in an optimal manner.
The eigensensitivity is computed with the first derivative of a structural response with respect to a physical parameter as [2] The 1st substructure

Eigensolutions
In this chapter, the eigensensitivity matrices are available with the forward substructuring method. They are computed solely from the derivative matrices of the substructure containing the concerned element, while the corresponding derivative matrices of all other substructures are zeros. As the calculation of eigensensitivity usually consumes most of the computation time when numerous elemental parameters are updated in practical model updating process, the forward substructuring method can significantly improve the computational efficiency of the model-updating process.

The extraction of substructural flexibility
In the inverse substructuring method, the global flexibility matrix estimated from the experimental modal data is disassembled into substructural flexibility matrices. Afterwards, the analytical FE models of the substructures are updated independently and parallelly to reproduce the extracted substructural flexibility matrices. As before, the global structure with N DOFs is divided into Ns independent substructures with the jth (j = 1, 2,…, Ns) substructure n (j) DOFs. Treated as independent substructures, the substructural displacements, forces, stiffness, flexibility, and rigid body modes matrices are written in the primitive form as where K (j) , F (j) , x (j) , f (j) , and R (j) , respectively, represent the stiffness, flexibility, nodal displacements, external forces, and rigid body modes of the jth substructure. It is noted that the rigid body modes R is related to free-constraint substructures. R is a zero matrix if the jth substructure is constrained after partition. Otherwise, R is determined by the nodal location. For example, a two-dimensional structure with n nodes has three rigid body modes, i.e., the x translation (R x = 1, R y = 0), the y translation (R x = 0, R y = 1) and the z rotation (R x = Ày, R y = x), R takes the form of The primitive forms of the substructural displacements and forces are associated with the global counterparts as [15] fx p g¼L p fx g g, ½L p T ff p g ¼ ff g g ð 32Þ where {x g } and {f g } are the nodal displacement and external force vector of the global structure. L p is a Boolean matrix composed of 1 and 0 values to relate the DOFs of the substructures and the global structure [5]. Most of the values in L p are zeros. L p ij ¼ 1 means that the jth DOF of the global structure corresponds to the ith DOF in the partitioned substructures. The displacement of an independent substructure is constituted by its deformational motions and rigid body motion where β is the participation factor of rigid body modes. As an independent structure, a substructure is excited by the external force and the internal interface force from the adjacent substructures as where ff g g ¼ ð½L p T Þ þ ff g g ¼L p ff g g,L p ¼ ð½L p T Þ þ is the generalized inverse of ½L p T . Similar to the forward substructuring method, {τ} denotes the internal interface forces from the adjacent substructures, and matrix C implicitly defines the connections between the adjacent substructures. Substitution of Eq. (34) into Eq. (33) gives Substitution of Eq. (35) into the left equation of Eq. (32) gives Since the global displacement is associated with the global force by fx g g ¼ F g ff g g [15], the global flexibility can also be expressed as Equation (37) means that the primitive substructural flexibility matrix F p can be calculated from the global flexibility matrix F g once the two variables fτg and fβ p g are given. {τ} and fβ p g are gained according the force and displacement compatibility condition with the following procedures: 1. The primitive substructural rigid body modes and forces satisfy the force equilibrium compatibility as [16,17]> 2. From the physical point of view, matrix C constraints the displacement compatibility as Therefore, {τ} is expressed as where The combination of Eq. (34) and Eq. (41) gives fβ p g is therefore solved as where In consequence, {τ} is therefore solved from Eq. (41) as where K C ¼ CF À1 C C T . Once {τ} and fβ p g are solved, Eq. (36) can be expressed as where In consequence, the global flexibility matrix can be expressed by the substructural flexibility matrix: Based on Eq. (46), the substructural flexibility matrix F p is extracted from the global flexibility F g with an iterative scheme: 1. F p is initiated from the diagonal subblocks of the global flexibility as 2. In the kth (k = 1, 2, …) iteration, the substructural flexibility matrix is calculated according to Eq. (46) The diagonal subblocks of ½F p 0 ½k are reused in the next iteration 3.
Step 2 stops when the substructural flexibility matrices from two consecutive iterations drop below a predefined tolerance [16] e ¼ normð½F p ½k À ½F p ½kÀ1 Þ The substructural flexibility matrices F (j) are thereby gained by the diagonal subblocks of ½F p ½k .

The projection matrix to extract free-free flexibility for model updating
In the substructuring methods, the global structure is divided properly into several independent free or constrained substructures. Most of the substructures are free-free without constraints after partition. Here the jth substructure is free-free as an illustration. The substructural flexibility matrix F ðjÞ from F p is constituted by both the rigid body modes and deformational modes. Hereinafter, superscript "j" is omitted to derive the free-free substructural flexibility for brevity. For the jth substructure, the substructural flexibility matrix, contributed by the rigid body motions and deformational motions, is expressed as F is defined as the generalized substructural flexibility. Accordingly, the generalized substructural stiffness matrix, including the contribution made by the rigid body motions and deformational motions is written as where K is defined as the generalized substructural stiffness matrix. The free-free stiffness and flexibility matrices (K and F) are contributed by the deformational modes solely. The participation factors γ and η of rigid body modes are difficult to determine, which makes the generalized flexibility unable to be applied to model updating or damage identification. It is necessary to extract the free-free substructural flexibility contributed by the deformational modes solely. The free-free flexibility shows the real properties of a substructure and can be applied to model updating and damage identification.
To remove the rigid body components in the generalized substructural stiffness, flexibility, and displacements, a projection matrix P is formed as [17] The projection matrix P has the properties of P can filter out the rigid body motions, while the free-free stiffness and flexibility matrices contributed by the deformational modes remain unchanged On the other hand, the free-free stiffness and flexibility of a substructural analytical model are singular, whereas the generalized stiffness and flexibility are full-rank. The free-free stiffness and flexibility can be calculated from the inverse of the generalized stiffness and flexibility matrices as If the projection matrix P is known, the free-free substructural flexibility F is calculated from Eq. (56) or by removing all the rigid body components in the extracted substructural flexibility matrix (Eq. (55)). In substructure-based model updating, the elemental parameters of the analytical FE model are iteratively adjusted to minimize the discrepancy between the analytical substructural flexibility and that extracted from global data [18].
Generally, the stiffness or flexibility matrices are difficult to be measured on the full DOFs, and the partial stiffness and flexibility at the measured DOFs are probably utilized for a substructure. Divide the full-DOF model into the measured part and the unmeasured part, the stiffness matrix is rewritten in block form as where subscript "a" represents the measured DOFs, and subscript "b" represents the unmeasured DOFs. The condensed stiffness matrix by the Guyan static condensation is [19][20][21] The substructural flexibility is written in block form according to the measured and unmeasured parts as [22] F In this case, the projection matrix of the reduced model P D is formed as which has the properties of The rigid body modes R a are gained by rewriting the rows in Eq. (58) corresponding to the measured DOFs.
The projection matrix P D removes the rigid body components in the partial substructural flexibility matrix and leaves the free-free substructural flexibility by In addition, the projection matrix can be used to form the dual inverse of substructural stiffness and flexibility like In substructure-based model updating, the elemental parameters in the substructural model are iteratively adjusted to minimize the discrepancy between the substructural flexibility and that extracted from global modal data [18]. For a free-free substructure, the flexibility extracted from global modal data is contaminated by the rigid body motions, and the stiffness matrix of substructural analytical FE model is singular. The projection matrix is utilized to extract the free-free flexibility for model updating. On the one hand, the projection matrix removes the rigid body components in the generalized substructural flexibility from experimental data and leaves the free-free substructural flexibility according to On the other hand, the free-free flexibility matrix of the substructural FE model is iteratively computed from the singular stiffness matrix according to

Substructure-based model updating
The substructure-based model updating process is listed in Figure 2. Identically, the jth substructure, which is free-free after partition, is employed to illustrate the substructure-based model updating in the following: 1. The experimental flexibility F E g is estimated by modal data of the global structure.
2. The generalized substructural flexibility matrix F ðjÞ E is extracted from the global flexibility matrix F E g by the proposed substructuring method in Section 3.1.
3. The rigid body modes R ðjÞ a are constructed according to the nodal location of the jth substructure (Eq. (31)), and the projection matrix P ðjÞ is formed according to the proposed method in Section 3.2.
The free-free substructural flexibility is extracted by the projection matrix as 4. The FE model of the jth substructure is constructed without constraints. The FE model of the jth substructure is treated as an independent structure to be updated: In each iteration, the free-free substructural flexibility matrix F ðjÞ FE at the measured DOFs and its sensitivity with respect to α ∂ F ðjÞ FE =∂α are computed [21]. The elemental parameters in the jth substructure are adjusted according to the sensitivity (J(α)) of the flexibility with respect to elemental parameters, to minimize the objective function ΔFðαÞ through the Trust Region Newton method [2,3,18].
In the proposed substructuring method, the substructural flexibility matrices in primitive matrix F p are independent. And only one substructure instead of the whole global structure at a time is updated in each iteration. The size of system matrices and updating parameters are sharply reduced, which improves the computational efficiency of model updating significantly.

Laboratory frame structure
Here a laboratory-tested steel frame structure is employed to investigate the effectiveness of the forward and inverse substructuring methods in model updating and damage identification. The cross section of the beams is 50.0 Â 8.8 mm 2 and the cross section of the columns is 50.0 Â 4.4 mm 2 , with the dimensions shown in Figures 3(a) and (b). The mass density of the structural material is 7.67 Â 10 3 kg/m 3 . The FE model of the frame is composed of 44 nodes and 45 elements, with each element 100 mm in length as Figure 3(c). In experiment, the accelerometers are placed at the nodes to measure the translational vibration of the frame [23]. The sampling frequency was set to 2000 Hz. The specimen was excited with the instrumented hammer at the reference point indicated in Figure 3(a).
The FE model is first updated in the undamaged state, and the refined model is subsequently used for damage identification. In the undamaged state, the Young's modules of all 45 elements are updated, with their initial values set to 2 Â10 11 Pa. The global structure is partitioned into three substructures, and the elements in the substructures are labeled in Figure 3(c). Accordingly, there are 17 updating parameters in the first substructure, 15 in the second, and 13 in the third. The recorded input and output time history were analyzed in Matlab platform to derive the first 14 experimental frequencies and mode shapes.
Using the forward substructuring method, the first 30 modes in each substructure are selected as the master modes. In the model updating process, the substructure-based eigensolutions are compared with the first 14 experimental frequencies and mode shapes to form the objective function. The eigensensitivities are computed from one substructure solely to improve the computational efficiency. The elemental parameters of the FE model are adjusted iteratively to minimize the objective function through an optimal process. The elemental stiffness reduction factor (SRF) is used to estimate the damage identification, which gives the change ratio of the updated values to the initial values of updating parameters.
where superscript O denotes the initial values before updating and U denotes the updated values. The SRF values of the three substructures after updating are listed in Figure 4(a). The model improved in the undamaged state is used for damage identification subsequently. There are two damage configurations in the frame. In the first damage case, the column of the first storey is cut with the width of b = 10 mm and depth d = 15 mm at 180 mm away from the support (Figure 3(b)). Subsequently, the second storey is cut with the same width and depth at 750 mm away from the support.
In the first damage configuration, the cut is located in the first storey. The 17 elemental parameters in Substructure 1 are adjusted iteratively to minimize the discrepancy between the analytical eigensolutions and the measured modal data. In FE model updating, only the first substructure is reanalyzed, and the eigensolutions of the second and third substructures remain untouched and reused directly to compute the eigensolutions of global structure. The eigensensitivities with respect to the 17 elemental parameters are computed from the substructural derivative matrices of the first substructure solely, whereas those in the second and third substructures are zero-matrices. The elemental parameters in the undamaged state are subsequently employed for damage identification. It is apparent from Figure 5(a) that, Element 2 has an obvious negative value in SRF of about À25%, which agrees with the location of the cut in the experiment. In the second damage configuration, the two cuts are located in the first and second substructures, respectively. Subsequently, the first and second substructures are updated, while the third substructure remains untouched. The SRF values shown in Figure 6(a) demonstrate that Element 2 of the first substructure and Element 2 of the second substructure have an obvious negative SRF values. The identified locations agree with those of the experimental cut. Particularly, the SRF values of Element 2 of the first substructure are about À23%, comparable to that in the first damage configuration. This is because the cut remains unchanged in the two damage configurations.
Afterwards, the frame structure is analyzed by the inverse substructuring method with the same measured data and FE model. In the undamaged state, the global flexibility is formulated from the 14 pairs of measured natural frequencies and mode shapes. The inverse substructuring method is used to extract the substructural flexibility matrices of the three  substructures simultaneously. The global FE model is divided into three substructures as well. The substructural flexibility of the three submodels is compared with the extracted substructural flexibility to form the objective function. The discrepancy of substructural flexibility matrices between the FE sub-model and extracted ones is minimized by adjusting the updating elemental parameters of the three submodels independently. Figure 4(b) reports the updated SRF values of the three substructures, which are subsequently utilized for damage identification.
In the first damage case, the local area within the first storey, i.e., Nodes 1 to 18 in Figure 3(c), are measured. Accordingly, only the substructural flexibility matrix of the first storey is extracted, based on which the submodel of the first substructure is updated independently. Figure 5(b) reveals a significant reduction in stiffness in Element 2, which agrees with the real location of the cut in experiment. The identified damage location and severity agrees with those obtained by the forward substructuring method as well.
In the second damage configuration, the frequencies and mode shapes measured in the first and second storeys are measured to form the global flexibility matrix. The substructural flexibility corresponding to the first and second substructures are extracted from the global flexibility simultaneously. The submodels of the first and second substructures are independently updated to recover the extracted substructural flexibility. Figure 6(b) reveals a negative SRF value of À20% in Element 2 of the first substructure and À25% in Element 2 of the second substructure. The identified damage location and severity are consistent to those gained by the forward substructuring method again. Both the forward and inverse substructuring methods are effective in model updating and damage identification.

Conclusion
A forward substructuring method and an inverse substructuring method are proposed in this chapter for model updating and damage identification. In the forward substructurebased model updating, the modified substructures are reanalyzed and assembled with other untouched substructures for the eigensolutions of the global structure to match the experimental data in an optimal manner. In the inverse substructuring method, the experimental modal data measured in local areas are used to extract the experimental flexibility matrix of the concerned substructure. The concerned substructures are updated by being treated as independent structures. Both the forward and inverse substructuring methods are effective in model updating and damage identification of a laboratory-tested steel frame structure. In the substructure-based model updating, only one substructure instead of the large-scale global structure is re-analyzed, which will be quite efficient for the model updating of practical large-scale structures. The substructuring methods are promising to be combined with the nonlinear analysis, vibration control, and parallel computation as well.