Computational efficiency is one of the most important considerations in damage detection and the related forward problem due to the large number of iterations which need to be performed. ESM can significantly enhance the efficiency of modelling the effect of damage on the vibration characteristics of a structure by avoiding discretisation into elements, with the W-W algorithm providing exact solutions for the transcendental eigenvalue problem. However, related techniques such as those implemented in VICONOPT are only suitable for prismatic structures. For more complicated structures, it is still necessary to consider both efficiency and accuracy. For a cracked plate, the crack will break the prismatic status of the plate. Hence a hybrid method is introduced, and the computational time significantly increases even using a powerful computer. The Gauss elimination requires the greatest part of that computing time.
As mentioned in Chapter 3.3, the necessary Gauss elimination procedure can be regarded as a matrix transformation without row interchanges. As an example (Figure 4-3), consider an isotropic simply supported plate which contains a single arbitrary crack. In modelling the crack, the plate has been divided into three parts, two intact parts and one damaged part. Before introducing the crack, assume the intact parts of the model have 6 (line) nodes and 12 different half-wavelengths are used, and the damaged part has 400 elements in total (20*20). For the intact part, using ESM to establish the stiffness matrix, each node has 2 degrees of freedom for each half-wavelength. A 12*12 stiffness matrix is thus obtained for a single half-wavelength and the total size of ESM part is 144*144 for 12 different half-wavelengths. For the damaged part, FEM is used to set up the stiffness and mass matrix while each point has 3 degrees of freedom, where the size of stiffness matrix for FE is 1323*1323 (21 nodes in each direction, and 1323=3*21*21). For the constraint, the matrix will be an 82*(1323+144) matrix (21 nodes per direction, two degrees of freedom for each node, and two of the edges of the FE part are taken into the calculation while ignoring the displacement degree of freedom of far end nodes. Hence, 82=21*2*2-2). The complete stiffness matrix is thus 1549*1549 (1549=1323+144+82, which is the summation of all the degrees of freedom produced from the ESM, FEM and the constraint matrix) and includes many zero elements. As shown in Eq. 4-1, the majority of the complete global stiffness matrix elements are zero. Using the cracked plate as an example, if only the non-zero elements are considered when transforming the matrix using Gauss elimination, the computational cost will be around
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formula given by Suliman (2018). Moreover, according to Suliman et al. (2019), the pure FEA analysis is about 4.25 times longer than hybrid VFM approach. This is a massive improvement in computational efficiency for the arbitrary damage case, and hence to achieve it an advanced bandwidth method is introduced as explained below.
4.2.2.1 Bandwidth methodology
For the complete stiffness matrix shown in Eq. 4-1, to use the W-W algorithm, a Gauss elimination needs to be performed to transform the matrix into an upper triangular matrix without row interchanges to count the negative sign count. Eq. 4-1 is deduced from Eq. 3-24shown below: 𝐃1 𝐏1 𝐃2 𝐏2 𝑙𝐊 𝑙𝐊 Eq. 4-1 𝑇 𝐃𝑗 𝑙𝐊𝑗 𝐄𝑗 𝐏𝑗 𝑇 𝐃𝑓 𝐊𝑓 𝐄𝑓 𝐏𝑓 𝐄2 𝐄 𝐄𝑗 𝐄𝑓 𝐑 ] [ 𝐏𝐿 ] [ 𝟎 ]
where 𝐊𝑗 is the VIPASA stiffness matrix for different half-wavelengths 𝜆𝑗 (𝑗 = 1, … , 𝑗
; 𝐊𝑓 is the finite element stiffness matrix (𝐊𝑓 = 𝐊 − 𝐌(𝜔2)); 𝐄𝑗 and 𝐄𝑓 are the constraint
matrices for VIPASA and the finite element part; 𝐑 is initially a zero matrix but will change to have non-zero terms after Gauss elimination; 𝑇 denotes the transpose; 𝐃𝑗 and
𝐃𝑓 which are the vectors of displacement for VIPASA and the finite element part; 𝐏𝐿 is
the vector of Lagrangian Multipliers; 𝐏𝑗 and 𝐏𝑓 are the random force vectors.
VFM uses the W-W algorithm to count the number of negative leading diagonal elements of the upper triangular matrix obtained from the complete stiffness matrix. Only terms on and above the leading diagonal need to be stored and processed. During the Gauss elimination, consider the way to process the leading diagonal.
Assuming the global stiffness matrix is a matrix and choosing one of the main stiffness elements as an example. Here, triangulating 𝑙𝐊1, the transformation
Chapter 4---Data Analysis and Processing
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has been described in Chapter 3.3. Each pivotal element of 𝑙𝐊1 will be used to modify
other elements in 𝑙𝐊1; the pivotal element and related elements of 𝑙𝐊1 and 𝐄1𝑇 will be
used to alter 𝐄1𝑇; elements of 𝐄1𝑇 will then used to modify 𝐑. During the transformation
of 𝑙𝐊1 and 𝐄1𝑇 , there are no changes to the rest of the stiffness matrices and constraint
matrices. The same procedure is repeated for the remaining 𝑙𝐊𝑗 and 𝐄𝑗𝑇, 𝐊𝑓 and 𝐄𝑓𝑇. The
final process is the triangulation of 𝐑 that the pivotal elements of 𝐑 will be used to modify other elements in 𝐑.
For the advanced bandwidth method used in the global stiffness matrix, related program codes have been attached in the Appendix A. It is helpful to store different components of the complete stiffness matrix separately as expressed in Eq. 4-1, and some information are shown below (Figure 4-3):
1. 𝑙𝐊𝑗 are small, symmetric, banded VIPASA matrices with different
halfwavelengths from the VICONOPT part of the model (all of them have the same size 12*12);
2. 𝐊𝑓 is a larger, symmetric, banded matrix from the FEM part (1323*1323);
3. 𝐄𝑗𝑇 are rectangular matrices from the constraint part (all have the same size 1*82)
with initially only a few non-zero elements, but further non-zero terms will appear during Gauss elimination. During the transformation of 𝐄𝑗𝑇, one may allow for
the bandwidth method, but this is tricky to deal with;
4. 𝐄𝑓𝑇 is a larger rectangular constraint matrix (1323*82), with the same number of
columns as 𝐄𝑗𝑇 but with more rows;
5. 𝐑 is a symmetric square matrix (82*82). Its terms are initially all zero but nonzero terms will appear during the Gauss elimination. It becomes dense and cannot be allowed for the bandwidth method, so this part of the assessment of Gauss elimination could cost a long time if many constraints exist in the structure.
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Figure 4-3 Hybrid model used in bandwidth method illustration.
The advanced bandwidth method is the primary technique which can improve computational efficiency significantly. Besides the method, because the forward problem needs to prepare data for cracks with different locations and severity, a parallel computation method could be considered in the study mentioned in section 4.2.1.