A multiresolution approach for damage detection in fiber-reinforced polymer pipe systems

Abstract

Many industries are increasingly relying on composite pipes for their known favorable characteristics of corrosion resistance, high strength-to-weight ratio, and better insulation. The internal surface damage due to flow-accelerated erosion in pipeline systems poses a challenging problem that initiated a greater demand for reliable early detection methods. In this investigation, an efficient and accurate detection scheme of internal surface damage in a pipe system has been developed. The methodology relies on the measured vibration modal characteristics and is equally applicable to both steel and composite piping systems. The detection model is formulated using B-spline scaling function with their multiresolution capabilities being invoked for zooming on the localized discontinuity caused by internal wall thinning. The developed scheme does not require the prior buildup of a database of modal shifts against the damage parameters. The numerical efficiency of the method is manifested by solving a decoupled linear system, for which all required matrices are calculated and stored only once. Numerical examples are presented to demonstrate the validity and reliability of the developed detection scheme.

Introduction

Pipe systems made of fiber-reinforced polymer (FRP) composites enjoy desirable features of strength-to-weight ratio and corrosion resistance. This is why they are extensively used in many engineering applications ranging from the oil industry to water distribution systems to artificial blood vessels. ¹ Degradation of FRP pipe systems frequently occurs due to flow-accelerated erosion caused by solid particles. If left unchecked, it can ultimately lead to system failure, which disrupts operation and can result in dire economic consequences. ² Growth of such damage needs to be detected at early stages.

Direct damage detection methods include, among others, X-ray, ³ gamma-ray radiography, ⁴ ultrasonic probes, laser ultrasound,^5,6 thermography, ⁷ and microwave electromagnetic imaging. ⁸ In addition, acoustic techniques such as low power impulse radar, acoustic emission, and elastic wave emitter-detector pairs are in use.

Radiography-based methods^3,4 rely on measuring the differential absorption of a radiation. For successful detection of certain types of damage, orientation of the defect/damage should be parallel to the direction of radiation for better detection. In general, for a crack to be detectable, it has to be sufficiently large and the pipe material should contain fibers such as glass. For radiographic inspection, the glass fibers and polymer matrix are difficult, due to lower density as compared to metals.

Ultrasonic tomography ⁵ techniques were reported to depend on the size and shape of the pipe material constituents. An evaluation conducted by Kundu ⁹ of ultrasonic techniques concluded that ultrasound waves cannot penetrate deep enough in the FRP composites.

Infrared techniques ⁷ can only provide indirect information on the damage through the amplitude of the heat signal because heat profiles are continuous signals. This means that crack width and depth cannot be detected using these techniques. Like the ultrasonic waves, infrared waves cannot penetrate deep enough in the FRP composites. For more discussion on the difficulties of using infrared techniques, we refer to BH Udaya et al. ¹⁰

An alternative approach adopts the viewpoint that the occurrence of damage changes the mechanical parameters of the piping system and thereby affects the dynamical behavior of the system. It follows that vibration records can be used as a way of monitoring the initiation and development of the damage. ¹¹

Model-based methods (MB) make use of an accurate finite element model of the vibrating system together with experimental measurements for the quantitative detection of crack onset and development.^12,13 The method uses a prior database of modal shifts against damage parameters that is usually prepared in an experimental setting. It produces accurate results about the damage parameters (e.g. crack depth and width). Naturally, they require an accurate model for reliable results. ¹⁴ However, when high gradients are present (e.g. around crack tips) or in complex systems, the method can be inefficient or it can suffer from slow convergence, thus becoming more prone to signaling false alarms.

Multiresolution techniques provide a natural approach to overcome these difficulties. The wavelet element method (WEM) belongs to the class of MB methods. Wavelets have a compact adjustable support, which enable them to zoom-in on the damage location. High local gradients are identified as large coefficients of matching wavelets, while the rest of the wavelet coefficients are relatively small. In addition, they are also refinable ¹⁵ — a property which allows the computation of integrals involving wavelets and/or their derivatives exactly (up to machine precision). ¹⁶

The WEM was implemented for the detection of open cracks in Euler–Bernoulli beams by Narkis ¹⁷ and Pandey and Biswas ¹⁸ and in rotating shafts by Mohiuddin and Khulief ¹⁹ and Xiang et al. ²⁰ where rectangular and circular cross sections were considered. Changes in local flexibility of a steel pipe conveying fluid due to the presence of an open crack were considered by Murigendrappa et al. ²¹ and Yoon and Son, ²² and the possibility of detecting damages from frequency measurements was considered by Dilena et al. ²³ The study by Dilena et al. ²³ was concluded by recommending further investigation.

This work is concerned with developing an efficient and accurate detection scheme of cracks or erosion-induced damage in a pipe system, whether composite, FRP or steel, from vibration records. The developed approach has the following advantages:

1. It does not require solving a nonlinear system. Instead, a simple decoupled linear system is to be solved.

2. It does not require the prior buildup of a database of modal shifts against the damage parameters.

3. It has the capability of zooming-in for more accurate determination of damage location and parameters.

4. All required matrices can be calculated and stored only once. The entries of these matrices are integrals, which are calculated exactly up to machine precision. This is done by taking advantage of a technique for computing refinable integrals. ¹⁶

The developed vibration-based detection scheme is well-oriented to work as an online monitoring system. The method keeps track of the system physical parameters such as masses and moments of inertia, which are calculated from measured modal values and mode shapes. They are then used to determine a pipe system profile. The profile can then be used, either visually or automatically, to assess the robustness of the pipe system and/or apply a zoom-in step. In theory, the method requires measuring four modal values and the amplitudes of the corresponding mode shapes at specific points along the span of the pipe. In practice, however, only one modal value is needed together with its corresponding mode shape, as the numerical experiments will confirm. In practice, modal values are determined experimentally using the well-known experimental modal analysis (EMA) techniques.

This article consists of six sections besides the introduction. In section “Elastodynamic model,” a formulation of the elastodynamic model for the pipe system is developed and the boundary conditions and the system physical parameters are introduced. Also, the equivalent eigenvalue problem for the developed system is derived and the working operators are defined. In section “Inadequacy of identification techniques for fault detection,” a discussion of the inadequacy of the traditional techniques of identification for damage detection is given. The reason for this inadequacy is that the identified system matrices are not unique. This is done to highlight the difference between the approach taken here and the techniques of system identification. In section “Weak formulation and multiscale discretization,” the weak formulation of the problem is stated and the multiscale finite-dimensional discretization using centralized box splines of order 4 (twice continuously differentiable) is introduced. In section “Damage detection,” the proposed damage detection method which consists of a monitoring phase and a zooming phase is introduced. Also, the necessary results needed for the method to work are derived. Numerical simulations are provided in section “Numerical simulation.” They illustrate the capability of the method to detect multiple cracks as well as erosions of arbitrary profiles within one and the same framework. Finally, an appendix is provided which proves the discreteness of the mode values of our model. This appendix was included for the convenience of the reader as we were unable to find such material in one place in the literature.

Elastodynamic model

Referring to the derivation of the coupled nonlinear equations of motion of fluid-conveying pipes with small strains and rotary inertia presented by Reddy and Wang, ²⁴ one can adopt the assumptions of the Euler–Bernoulli beam theory and neglect axial deformations to obtain the elastodynamic model that represents the transverse vibration of a straight pipe conveying fluid (Figure 1) at constant flow rate as

$$\begin{array}{l}{m}_s\frac{{\partial }^{2}y}{\partial t^2}+\frac{\partial }{\partial x}\left(I_s\frac{{\partial }^{3}y}{\partial t^2\partial x}\right)+2v{m}_f\frac{{\partial }^{2}y}{\partial x\partial t}+v^2\frac{\partial }{\partial x}\left(m_f\frac{\partial y}{\partial x}\right)\\ +E_p\frac{{\partial }^2}{\partial x^2}\left(I_p\frac{{\partial }^{2}y}{\partial x^2}\right)=F\end{array}$$ (1)

Figure 1.

Composite pipe conveying fluid.

See Appendix 1 for definitions of the coefficients.

For a multilayer composite FRP pipe, the bending stiffness $E{I}_p$ and mass per unit length $m_p$ for each element are defined as follows

$$m_p=\underset{A_p}{\int }{\rho }_{p}d{A}_p=2\pi \underset{r_i}{\overset{r_o}{\int }}{\rho }_{\ell }\mathrm{rdr}=\sum _{\ell =1}^N\pi {\rho }_{\ell }(r_{\ell }^2-r_{\ell -1}^2)$$ (2)

$$E{I}_p=\pi (A_{11}{\overline{r}}^3+B_{11}{\overline{r}}^2+D_{11}\overline{r})$$ (3)

where

$$\begin{array}{c}\multicolumn{1}{c}{\overline{r}=\frac{D_o+D_i}{4}}\\ \multicolumn{1}{c}{A_{11}=\sum _{\ell =1}^N{\overline{Q}}_{11}^{\left(\ell \right)}(r_{\ell }-r_{\ell -1})}\\ \multicolumn{1}{c}{B_{11}=\sum _{\ell =1}^N\frac{1}{2}{\overline{Q}}_{11}^{\left(\ell \right)}(r_{\ell }^2-r_{\ell -1}^2)}\\ \multicolumn{1}{c}{D_{11}=\sum _{\ell =1}^N\frac{1}{3}{\overline{Q}}_{11}^{\left(\ell \right)}(r_{\ell }^3-r_{\ell -1}^3)}\\ \multicolumn{1}{c}{{\overline{Q}}_{11}=Q_{11}{c}^4+2c^2{s}^2(Q_{12}+2Q_{66})+Q_{22},}\\ \multicolumn{1}{c}{s=\sin \varphi ,c=\cos \varphi }\\ \multicolumn{1}{c}{Q_{11}=\frac{E_1}{\eta },Q_{12}=\frac{{\nu }_{12}{E}_2}{\eta },Q_{22}=\frac{E_2}{\eta },Q_{66}=G_{12}}\\ \multicolumn{1}{c}{\eta =1-\frac{E_2}{E_1}{\nu }_{12}^2=1-{\nu }_{12}{\nu }_{21}}\end{array}$$

The other symbols are defined as $r_{\ell }$ is the radius of the $\ell \mathrm{th}$ layer, $E_i$ is the modulus of elasticity, ${\nu }_{\mathrm{ij}}$ is Poisson’s ratio, and $G_{\mathrm{jj}}$ is the shear modulus. ²⁵

For a healthy pipe, it will be assumed that $m_s,I_s,m_f,\mathrm{and}{I}_p$ are independent of x. For a damaged pipe, these quantities will be allowed to change with x over a small interval along the span of the pipe where damage occurs. A superscript d will be used to distinguish a damaged pipe from a healthy one.

The boundary conditions for the simply-supported pipe are given by

$$y(0,t)=y(L,t)=\frac{\partial y}{\partial x}(0,t)=\frac{\partial y}{\partial x}(L,t)=0$$ (4)

Note that the material damping is ignored in the above equation. Only the fluid damping effect appears. A free vibrating pipe is considered here, that is, $G=0$ . In this analysis, simply-supported boundary condition is adopted as it is the most commonly used in pipeline systems. Yet, the formulation is not limited to a specific type of boundary condition.

Applying the separation of variables, that is, letting $y(x,t)=e^{\lambda t}u(x)$ , changes equation (1) into the quadratic eigenvalue problem

$${\lambda }^2\left(m_{s}u\left(x\right)+{\left(I_s{u}^{\prime }\left(x\right)\right)}^{\prime }\right)+2\lambda v{m}_f{u}^{\prime }\left(x\right)+v^2{\left(m_f{u}^{\prime }\left(x\right)\right)}^{\prime }+E_p{\left(I_p{u}^{″}\left(x\right)\right)}^{\prime \prime }=0$$ (5)

together with the boundary conditions

$$u\left(0\right)=u\prime \left(0\right)=u\left(L\right)=u\prime \left(L\right)=0$$ (6)

In Appendix 2, it will be shown that equations (5) and (6) have a discrete set of eigenvalues and corresponding eigenvectors. At this point, it is best to use operator notation. Thus, equation (5) is rewritten as

$$({\lambda }^{2}M+\lambda D+K)u=0$$ (7)

where

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Mu}=(I_{s}u\prime )\prime +m_{s}u}\\ \multicolumn{1}{c}{\mathrm{Du}=2v{m}_{f}u\prime }\\ \multicolumn{1}{c}{\mathrm{Ku}=E_p(I_{p}u″)″+v^2(m_{f}u\prime )\prime }\end{array}$$

Inadequacy of identification techniques for fault detection

In the literature, many techniques were developed for the identification of system parameters, that is, the matrices $M,D,\mathrm{and}K$ (see equation (11)) from records of eigenvalues and eigenvectors.^26–29 Ideally, if these matrices are properly obtainable, then the behavior of the coefficient terms could be inferred from them. In turn, these coefficients are tied to the geometric properties of the pipe, such as the inner radius and outer radius in the case of circular pipes. It turns out that system parameter identification techniques are inadequate for pipe fault detection, which, in the proposed approach, will be obtained by detecting variations in these geometric parameters. Obviously, if two sets of matrices fit the same record of eigenvalues and eigenvectors, then it will not be possible to determine the proper values of the geometric parameters for the specific system under investigation. In this section, a discussion of the issue of non-uniqueness of the identified system parameters is presented.

Assume we are given n eigenvalues and the corresponding n eigenvectors. Arrange the eigenvalues in a diagonal matrix $\Lambda =U+\mathrm{iW}$ , where $U\le 0,W>0$ and arrange the eigenvectors in an $n\times n$ matrix $X_c=X_R+i{X}_I$ . Let $X=[X_c{\overline{X}}_c]$ . In order to identify symmetric matrices $M,D,\mathrm{and}K$ from $\Lambda$ and $X_c$ , the latter must be normalized such that

$$X_R{X}_R^T=X_I{X}_I^T$$ (8)

In other words, there must exist a diagonal matrix $\gamma$ of complex numbers such that the normalized matrix $X_c\gamma$ satisfies equation (8). Now, it was shown in El-Gebeily and Khulief ³⁰ that there is a whole class of normalizing matrices $\gamma$ for which equation (8) holds. Basically, any matrix $\gamma$ such that

$${\gamma }^2=-X_c^{-1}{\overline{X}}_{c}A$$ (9)

will work. Here A is a matrix that diagonalizes $=-X_c^{-1}\overline{X_c}$ through column operations and satisfies $A^T=-\overline{A}$ . An example was given in El-Gebeily and Khulief ³⁰ to illustrate that one could choose $\gamma$ with an arbitrary imaginary part as long as it is nonsingular. Consequently, the matrices $M,D,\mathrm{and}K$ identified from the various normalizations of $X_c$ are not identical and, hence, cannot be used to detect pipe faults.

Another obvious drawback of this method is that it identifies a symmetric matrix D, whereas the corresponding matrix in equation (11) is not symmetric.

Weak formulation and multiscale discretization

Here it is assumed, for convenience, that the pipe length $L=1$ . The general case can be treated by using the change of variable $\xi =(x-a)/L$ , where a is the coordinate of the left end-point of the pipe. For the weak formulation of equation (5), the working space is taken to be the Sobolev space $V=H_0^2(0,1)$ . Define the bilinear forms

$$\begin{array}{c}\multicolumn{1}{c}{M(u,w)=〈m_{s}u,w〉-〈I_{s}u\prime ,w\prime 〉}\\ \multicolumn{1}{c}{D(u,w)=2v〈m_{f}u\prime ,w〉}\\ \multicolumn{1}{c}{K(u,w)=E_p〈I_{p}u″,w″〉-v^2〈m_{f}u\prime ,w\prime 〉,u,w\in V}\end{array}$$ (10)

The weak form of equation (5) is stated as follows: Find $\lambda \in \mathbb{C}$ and $u\in V$ such that

$${\lambda }^{2}M(u,w)+\lambda D(u,w)+K(u,w)=0\forall w\in V$$ (11)

To discretize this problem, let $\mathcal{S}=\{S_j{\}}_{j\in {\mathbb{N}}_0}$ be an multiresolution analysis (MRA) in $L^2(0,1)$ ¹⁵ generated by a centralized box spline $\phi$ of order d. These splines are $C^{d-2}$ function with support in the interval $[-[d/2],[d/2]]$ . It is explicitly given by the formula

$$\phi \left(x\right)=\frac{1}{(d-1)!}\sum _{k=0}^d{(-1)}^k\left(\begin{array}{c}d\hfill \\ \multicolumn{1}{c}{k}\end{array}\right){(x+\left[\frac{d}{2}\right]-k)}_{+}^3$$ (12)

where $x_{+}={\chi }_{[0,\infty )}(x)x$ . The graph of the B-spline of order 4, which was mainly used in our numerical computations (see section “Numerical simulation”) is shown in Figure 2. It is a refinable function in the sense that

$$\phi \left(x\right)=\sum _{k=0}^4\frac{1}{8}\left(\begin{array}{c}4\hfill \\ \multicolumn{1}{c}{k}\end{array}\right)\phi (2x-k)$$ (13)

Figure 2.

Centralized box spline of order 4.

Therefore, its refinement mask is given by

$$a_k=\{\frac{1}{8}\begin{array}{c}0,k<0\hfill \\ \multicolumn{1}{c}{\left(\begin{array}{c}4\hfill \\ \multicolumn{1}{c}{k}\end{array}\right),0\le k\le 4}\\ \multicolumn{1}{c}{0,k>4}\end{array}$$

Thus, the nonzero coefficients are given by

$$\{a_0,a_1,a_2,a_3,a_4\}=\{\frac{1}{8},\frac{1}{2},\frac{3}{4},\frac{1}{2},\frac{1}{8}\}$$

Let

$${\phi }_{j,k}\left(x\right)=2^{\frac{j}{2}}\phi (2^{j}x-k)$$

The support $I_{j,k}$ of ${\phi }_{j,k}$ is $[2^{-j}(-2+k),2^{-j}(2+k)]$ , which has a nontrivial intersection with the interval $[0,1]$ for $j\ge 2$ and $-1\le k\le 2^j+1$ . Out of this list of values of k, ${\phi }_{j,k}\in V$ only for $2\le k\le 2^j-2$ . For a given $j\ge 2$ , put ${\mathcal{I}}_j=\{k:2\le k\le 2^j-2\}$ .

The basis ${\Phi }_j$ of $S_j$ is given by

$${\Phi }_j=\{{\phi }_{j,k}:k\in {\mathcal{I}}_j\}$$

The same notation ${\mathbf{\Phi }}_j$ will also be used to denote the basis elements arranged in a column vector.

The discrete counterpart at level j of equation (11) reads as follows: find $u_j\in S_j$ and ${\lambda }_j\in \mathbb{C}$ such that

$${\lambda }_j^{2}M(w,u_J)+{\lambda }_{j}D(w,u_j)+K(w,u_j)=0,\forall w\in S_j$$ (14)

Write $u_j=c_j^T{\Phi }_j$ to arrive at the algebraic eigenvalue problem

$${\lambda }_j^2\mathit{M}{\mathit{c}}_j+{\lambda }_j\mathit{D}{\mathit{c}}_j+\mathit{K}{\mathit{c}}_j=0$$ (15)

where

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{M}=M({\mathbf{\Phi }}_j,{\mathbf{\Phi }}_j)=〈m_s{\mathbf{\Phi }}_j,{\mathbf{\Phi }}_j〉-〈I_s{\mathbf{\Phi }}_j\prime ,{\mathbf{\Phi }}_j\prime 〉}\\ \multicolumn{1}{c}{\mathit{D}=D({\mathbf{\Phi }}_j,{\mathbf{\Phi }}_j)=2v〈m_f{\mathbf{\Phi }\prime }_j,{\mathbf{\Phi }}_j〉}\\ \multicolumn{1}{c}{\mathit{K}=K({\mathbf{\Phi }}_j,{\mathbf{\Phi }}_j)=E_p〈I_p{\mathbf{\Phi }}_j″,{\mathbf{\Phi }}_j〉″-v^2〈m_f{\mathbf{\Phi }}_j\prime ,{\mathbf{\Phi }}_j\prime 〉}\end{array}$$

Here, the general notation

$$〈R,S〉:=\underset{0}{\overset{1}{\int }}R\left(x\right)S^T\left(x\right)\mathrm{dx}$$

for $R(x)\in {\mathbb{R}}^j,S(x)\in {\mathbb{R}}^m$ is used and the integral is applied to each entry of the matrix $R{S}^T$ .

It is well known that equation (15) is equivalent to

$$\{\lambda \left[\begin{array}{cc}\mathit{D}\hfill & \mathit{M}\hfill \\ \multicolumn{1}{c}{\mathit{M}}& 0\hfill \end{array}\right]-\left[\begin{array}{cc}\mathit{K}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \mathit{M}\hfill \end{array}\right]\}\left[\begin{array}{c}\mathit{c}\hfill \\ \multicolumn{1}{c}{\lambda \mathit{c}}\end{array}\right]=0$$ (16)

Damage detection

To use equation (15) to detect pipe damage, proceed as follows. Since $\left|I_{j,k}\right|=4/2^j$ , for sufficiently large j and appropriately adjusted k, $I_{j,k}$ will be contained in the damage interval $I_d$ (see Figure 3). Considering the definitions of the entries of the matrices $\mathit{M},\mathit{K},\mathrm{and}\mathit{D}$ as given in equation (10), it is seen that a typical term is of the form of an inner product

$$〈g{\eta }_{j,\ell },{\xi }_{j,k},〉$$

where g depends on x and ${\eta }_{j,\ell },{\xi }_{j,k}$ stand for (a derivative of) a basis function. Therefore, the function g on the interval $I_{j,\ell }$ may be approximated by its value at the center of that interval, namely the point $x_{j,\ell }=2^{-j}\ell$ . Hence

$$\begin{array}{l}g{\eta }_{j,\ell },{\xi }_{j,k}={\displaystyle \underset{I_{j,\ell }}{\int }g}\left(x\right){\eta }_{j,\ell }\left(x\right){\xi }_{j,k}\left(x\right)dx\approx g\left(x_{j,\ell }\right)\\ {\displaystyle \underset{I_{j,\ell }}{\int }{\eta }_{j,\ell }}\left(x\right){\xi }_{j,k}\left(x\right)dx\end{array}$$ (17)

Figure 3.

Wall thinning modeling by a damage interval $I_d$ .

Now, $g(x_{j,\ell })$ multiplies all the elements where the support of ${\xi }_{j,k}$ intersects the interval $I_{j,\ell }$ , which is the support of ${\eta }_{j,\ell }$ . These are precisely the elements of the $\ell \mathrm{th}$ row of the corresponding inner product matrices. Then

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{M}\approx {\mathit{m}}_{\mathit{s}}〈{\mathbf{\Phi }}_j,{\mathbf{\Phi }}_j〉-{\mathit{I}}_s〈{\mathbf{\Phi }}_j\prime ,{\mathbf{\Phi }}_j\prime 〉}\\ \multicolumn{1}{c}{\mathit{D}\approx 2v{\mathit{m}}_{\mathit{f}}〈{\mathbf{\Phi }}_j\prime ,{\mathbf{\Phi }}_j〉}\\ \multicolumn{1}{c}{\mathit{K}\approx E_p{\mathit{I}}_p〈{\mathbf{\Phi }}_j″,{\mathbf{\Phi }}_j″〉-v^2{\mathit{m}}_{\mathit{f}}〈{\mathbf{\Phi }}_j\prime ,{\mathbf{\Phi }}_j\prime 〉}\end{array}$$

where ${\mathit{m}}_s$ , ${\mathit{I}}_s$ , ${\mathit{m}}_f$ , and ${\mathit{I}}_p$ are diagonal matrices with the diagonal elements being the values of the corresponding functions at the dyadic points $\{x_{i\ell }{\}}_{\ell =0}^{2^j}$ . These values are to be identified from equation (15), which may be restated as

$${\lambda }_j^2\left({\mathit{m}}_s{\mathit{A}}_1-{\mathit{I}}_s{\mathit{A}}_2\right){\mathit{c}}_j+{\lambda }_{j}2v{\mathit{m}}_f{\mathit{A}}_3{\mathit{c}}_j+\left(E_p{\mathit{I}}_p{\mathit{A}}_4-v^2{\mathit{m}}_f{\mathit{A}}_2\right){\mathit{c}}_j=0$$ (18)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{A}}_1:=〈{\mathbf{\Phi }}_j,{\mathbf{\Phi }}_j〉}\\ \multicolumn{1}{c}{{\mathit{A}}_2:=〈{\mathbf{\Phi }}_j\prime ,{\mathbf{\Phi }}_j\prime 〉}\\ \multicolumn{1}{c}{{\mathit{A}}_3:=〈{\mathbf{\Phi }}_j\prime ,{\mathbf{\Phi }}_j〉}\\ \multicolumn{1}{c}{{\mathit{A}}_4:〈{\mathbf{\Phi }}_j″,{\mathbf{\Phi }}_j″〉}\end{array}$$

For each given pair $({\lambda }_j,{\mathit{c}}_j)$ , equation (18) is a $2^j-1:=N$ dimensional system. Thus, in order to determine the four unknown matrices ${\mathit{m}}_s$ , ${\mathit{I}}_s$ , ${\mathit{m}}_f$ , and ${\mathit{I}}_p$ , four eigenpairs $({\lambda }_{i,j},{\mathit{c}}_{i,j}{)}_{i=1}^4$ are needed.

A simpler form of equation (18) is

$${\mathit{m}}_s{H}_i+{\mathit{I}}_s{L}_i+{\mathit{m}}_f{P}_i+{\mathit{I}}_p{Q}_i=0,1\le i\le 4$$ (19)

where

$$\begin{array}{c}\multicolumn{1}{c}{H_i={\lambda }_{i,j}^2{\mathit{A}}_1{\mathit{c}}_{i,j},L_i=-{\lambda }_{i,j}^2{\mathit{A}}_2{\mathit{c}}_{i,j}}\\ \multicolumn{1}{c}{P_i=2v{\lambda }_{i,j}{\mathit{A}}_3{\mathit{c}}_{i,j}-v^2{\mathit{A}}_2{\mathit{c}}_{i,j},Q_i=E_p{\mathit{A}}_4{\mathit{c}}_{i,j}}\end{array}$$

The solvability of the above system for sufficiently large j follows from the solvability of the system (equation (15)) since the matrices involved are small perturbations of the corresponding ones for the healthy pipe. Since as discussed earlier, the $k\mathrm{th}$ row of the system (equation (19)) corresponds to the node $x_{j,k}=2^{-j}k$ on the pipe, the entries of the $k\mathrm{th}$ row of the matrices ${\mathit{m}}_s$ , ${\mathit{I}}_s$ , ${\mathit{m}}_f$ , and ${\mathit{I}}_p$ will change from their healthy values only when damage occurs at the point $x_{j,k}$ . Profiles where the entries of these matrices are plotted against the corresponding locations on the pipe can then be constructed. Thus, by monitoring these profiles, the damage location as well as its width $\left|I_d\right|$ can be detected. Furthermore, the values reported by these profiles can be used, via their relation to the inner and outer radius of the pipe, to construct a precise profile of the pipe itself (see section “Numerical simulation”).

The numerical experiments to be presented in the next section reveal that the method gives good indicators, even with low resolution of the pipe span. The method also eliminates the need to solve nonlinear equations or construct prior databases of modal shifts against damage parameters that is usually prepared in an experimental setting as is the case with the MB techniques.

It should also be mentioned that since the matrices ${\mathit{m}}_s$ , ${\mathit{I}}_s$ , ${\mathit{m}}_f$ , and ${\mathit{I}}_p$ are diagonal, system (equation (19)) can be decoupled into components. The result is that for each $k\in \{1,\dots ,N\}$ , a system of four-scalar equations to determine the $k\mathrm{th}$ component in each of the four unknown matrices is obtained.

This approach can also be implemented in a “monitor and zoom” strategy. The idea is as follows.

• Monitoring phase

• If change in the profile is detected, determine in which section of the pipe damage would have occurred.

Zooming phase

Zoom in on the damage location by using higher resolution subdivisions.

Ideally, the zooming phase requires only subdividing the suspected section of the pipe, that is, measuring the mode shape amplitudes at finer mesh nodes on the suspected section. However, the multiresolution technique developed here does not allow this kind of adaptivity. Approximations at finer levels have to be recalculated along the whole span of the pipe. We plan to investigate this point in a future work.

An added advantage to our approach is that the matrices ${\mathit{A}}_1,{\mathit{A}}_2,{\mathit{A}}_3,\mathrm{and}{\mathit{A}}_4$ can be calculated and stored only once, whether at the coarse level or the fine level since they are properties of the basis functions (or of the measuring device) at the beginning of the pipe operation. Therefore, it seems reasonable to expect that subdividing the whole pipe will produce accurate results at an acceptable computational cost.

Numerical simulation

The two-phase approach discussed in section “Damage detection” was implemented to a simply-supported laminated glass-reinforced epoxy (GRE) pipe with the parameters shown in Table 1.

Table 1.

Parameters of the GRE pipe.

Ply angles/stacking sequence	$\frac{[\pm {\Theta }_5/\Theta ]\circ }{[{\Theta }_{11}]\circ }$
Modulus of elasticity: $E_{11}=12.5\mathrm{GPa}$
Shear modulus: $G_{12}=G_{13}=G_{23}=3.3194\mathrm{GPa}$
Poisson’s ratio: $v_{12}=0.56,v_{21}=0.32$
Density of pipe: ${\rho }_p=1730\mathrm{kg}/{\mathrm{m}}^3$
Density of fluid: ${\rho }_f=1000\mathrm{kg}/{\mathrm{m}}^3$
Velocity of fluid: $v=10\mathrm{m}/\mathrm{s}$
Pipe dimensions: outer radius: $R_o=0.0563\mathrm{m}$ ;inner radius: $R_i=0.0508\mathrm{m}$ ; pipe Length: $L=2\mathrm{m}$

The damage was simulated by introducing a jump (or a jump profile) in the inner radius of the pipe in accordance with Figure 3. This is shown in Figure 4. The simulation was implemented in two steps:

• Step 1 (the damage simulation step). Values for the parameters $m_s,I_f,m_f,\mathrm{and}{I}_p$ were calculated at the pipe nodes using the formulas following equation (1). These parameters were then fed in equation (15). The eigenvalue problem was solved and the first four eigenvalues and eigenvectors were recorded.

• Step 2 (the damage detection step). The recorded eigenvalues and eigenvectors were used together with equation (19) to recover the matrices ${\mathit{m}}_{\mathit{s}},{\mathit{I}}_{\mathit{f}},{\mathit{m}}_{\mathit{f}},\mathrm{and}{\mathit{I}}_{\mathit{p}}$ , which were, in turn, used to construct profiles of the corresponding parameters.

Figure 4.

Inner radius of the healthy and damaged pipe.

Figure 5 shows the profile of the relative change in the dominant coefficient EI at the coarse level $j=3$ . In this case, values of the first mode shape amplitudes at eight nodal points along the span of the pipe are required. The figure indicates the onset of damage but does not give precise information about its shape or exact place. It merely indicates the need to switch to a higher resolution. Observe that the method is able to detect relative change in the order of $1:{10}^{11}$ .

Figure 5.

Detection at a coarse level.

Figures 6 shows the profile of EI calculated at the level $j=5$ (32 nodes) using a B-spline of order 4.

Figure 6.

Damage detection with a B-spline of order 4.

Figure 7 shows the profile of $I_p$ calculated from equation (19) using only one coefficient, that is, keeping the undamaged values of all the other coefficients. Of course, good approximation values of the desired coefficients are not expected; however, the damage is qualitatively clear in the calculated coefficient. Figure 8 shows the damage profile at two locations using EI only. Figure 9 shows the profile of $m_f$ due to non-uniform damage at the two locations $x\in [3/32,7/32],\mathrm{and}x\in [15/32,22/32]$ . Here the calculations are carried out without the approximation of the other coefficients by their undamaged values. In this case, qualitative as well as quantitative information about the location and nature of the damage are obtained. Finally, Figure 10 compares the actual versus the calculated inner radius using only one calculated parameter (EI in this case).

Figure 7.

Damage detection using $I_p$ only.

Figure 8.

Damage detection at two locations using EI only.

Figure 9.

Non-uniform damage in two locations using all coefficients.

Figure 10.

Comparing inner radius profiles.

Conclusion

In this investigation, a detection scheme for identifying internal pipe surface damage based on a multiresolution technique was developed. The method used only the elastic beam model of the pipe with no further adjustments. A major advantage of the developed methodology is that it does not require the prior buildup of a database of modal shifts against the damage parameters, as needed by other MB. Moreover, all the required matrices can be calculated and stored only once. Although four pairs of mode values and their corresponding mode shape amplitudes are formally needed, numerical simulation gave very satisfactory results with only one such pair. Damage profiles were obtained by solving a set of one-dimensional linear equations and there was no need for iterative methods. The obtained numerical results demonstrated the success of the method in detecting damage location and parameters. The method was capable of detecting multiple damages with general damage profiles. The coarse-fine implementation of the method showed that it could detect relative coefficient change in the order of ${10}^{-7}$ . This level of relative amplitude change can be exploited if an initial profile of the pipe at the beginning of its operation is constructed. This way, factors such as pipe location and surroundings can be eliminated. It is important to note that the developed scheme provides a platform for an online mentoring system, wherein the modal values and mode shape amplitudes are measured using EMA techniques.