1 Introduction

Ceramics are widely applied in aerospace industries such as ram jet engines, space shuttle, aerocraft, turbine engines, to name a few [1,2,3,4,5,6,7]. Generally, the ceramic materials in these areas usually work in high temperature and aggressive environments. Thus, accurate and effective non-destructive testing (NDT) technique is fundamentally important to guarantee the quality of ceramics [8,9,10,11]. Over the last few decades, there has been a significant amount of interest in the development of NDT technique and a wide range of new sensing methodologies are successfully applied to the NDT of ceramics, such as X-ray computed tomography (X-ray CT) [12], ultrasonic testing (UT) [13], acoustic emission [14], microwave interferometry [15], holographic interferometry [16], etc. However, all of these methods have limitations on the type of defects to be detected or the type of materials that can be inspected. For example, the depth of defects cannot be clearly identified by the X-ray CT. Complex geometry may reduce the evaluation accuracy of UT and it is necessary to make the wavelength comparable to defect size for UT technique [17, 18].

Infrared thermography is an effective non-destructive evaluation technology which can achieve high detection accuracy and sensitivity. Infrared NDT technique has many advantages such as non-contact measurement, ease of use and resistance to electromagnetic interference and thereby it gets more and more attentions. Numerous thermography NDT techniques including infrared thermal imaging [19], pulsed transient thermography [20, 21], thermal-wave radar imaging (TWRI) [5] and lock-in thermography (LIT) [22,23,24] are successfully employed to detect the defects in various materials. Ludwig and Peruzzi [25] applied video pulse thermography technique to detect the circular sub-superficial defects in structural components. The time evolution of the full-width half-maximum of thermal contrast was investigated by theoretical and experimental methods. However, a significant mismatch was obtained between the theoretical and experimental results, which may be due to the heat losses caused by radiation and convection neglected in the theoretical research. Maldague et al. [26] proposed a double-pulse infrared thermography technique to improve the detectable depth of defects in specimen. Two separated heating pulses not only reduced the importance of high frequency of incident laser and enhanced signal-to-noise ratio, but also facilitated reproducing experiments. Meanwhile, the optimum interval between two pulses was given based on experimental studies. Gong et al. [5] studied the NDT for carbon fiber-reinforced polymer through the TWRI technique. The multi-transform techniques such as Fourier transform, Hilbert transform and cross-correlation were employed to extract the characteristics of thermal-wave signals. The subsurface defects were investigated both theoretically and experimentally. Busse et al. [24] proposed the LIT technique, in which the thermography and thermal-wave techniques were combined to provide short-time low-frequency phase angle images. Compared with infrared thermography technique, LIT can provide additional phase information which will be not affected by the environmental noise and emissivity of specimen. Also, an increased depth range at a short time can be obtained, which demonstrated the superiority of LIT technique. However, all of the above thermography NDT techniques are based on the thermal response on surfaces, the power and frequency of incident laser should be reasonably controlled. Moreover, most of the infrared thermography methods are analyzed by using heat conduction model, which may produce large mismatches between simulation and experimental results, especially for high-temperature semitransparent materials.

Optical imaging is a novel detection methodology occurs with the development of ultrashort-pulse laser. The propagation of photon is considered as a transient process, which can provide much more information for the inverse analysis than steady radiative transfer model [27,28,29]. Nowadays, the optical tomography technique has been widely applied in biomedical areas due to the near-infrared laser has little damage to biological issue [30,31,32,33,34]. For instance, Hebden et al. [30] investigated the blood oxygenation changes in the newborn infant brain using optical tomography. The pulse lasers at wavelengths 780 and 815 nm were employed as the illumination source. The retrieved results showed that the absorption coefficients at two wavelengths were significantly different. The imaging results of the difference between oxyhemoglobin and deoxyhemoglobin were in good agreement with the physiological data. Li et al. [32] studied clinic detection in breast tissue based on the near-infrared optical imaging technique. The dual-wavelength near-infrared light was applied to detect the breast and the highly sensitive charge coupled device was employed as the detector. The clinical experiments demonstrated that the malignant tumor usually higher hemoglobin concentration and lower oxygen saturation, while the benign lesion was the opposite. Wan and Guo [33] simulated the changes of cerebral blood oxygen saturation through the Monte Carlo method. The optical reflectance was employed as the measurement signal and the reflectance ratios between two wavelengths at 633 and 800 nm were measured under different blood vessel levels. A quantifiable correlation between the reflectance ratio and the cerebral blood oxygen saturation was obtained.

To the best knowledge of the authors, the application of ultrashort-pulse laser in NDT has not yet been reported. For the reticulated porous ceramics introduced in Ref. [1], the absorption coefficient is close to 0 while the scattering coefficient is about 103 m−1 in the near-infrared spectral range, which may lead to the radiative transfer process being dominated by the scattering coefficient and the absorption coefficient cannot be accurately reconstructed. On the contrary, the difference between the absorption and scattering coefficients is relatively small in the mid-infrared spectral range, which provides a good research area for simultaneous reconstruction. The current study aims to non-destructively detect the defects in ceramic materials using mid-infrared ultrashort-pulse laser. The transient (time-domain) radiative transfer is considered to describe the transmission of laser in ceramics. Two reticulated porous ceramics, partially stabilized zirconia (PS ZrO2) and oxide-bonded silicon carbide (OB SiC), are tested by the present model numerically. The reminder of this work is organized as follows. In Sect. 2, the 2D transient radiative transfer equation (TRTE) in semitransparent medium is introduced and the principle of discrete ordinate method (DOM) for solving the direct problem is discussed. The gradient of objective function with respect to the absorption and scattering coefficients is solved by the adjoint equation, which is presented in Sect. 3. The theoretical overview of sequential quadratic programming (SQP) is described in Sect. 4. In Sect. 5, the NDT results of ceramics are presented. The effects of spatial position, total number and geometric shapes of defects on the detection results are also discussed. The main conclusions are provided in Sect. 6.

2 Direct problem

As shown in Fig. 1, the 2D semitransparent medium exposed to a collimated laser irradiation is considered here. The 2D TRTE can be expressed as [35, 36]:

Fig. 1
figure 1

Schematic of 2D participating medium

Full size image
$$\frac{n}{c}\frac{{\partial I({\mathbf{r}},{\mathbf{\Omega }},t)}}{{\partial t}}+\Omega \cdot \nabla I({\mathbf{r}},{\mathbf{\Omega }},t)= - {\beta _{\text{e}}}({\mathbf{r}})I({\mathbf{r}},{\mathbf{\Omega }},t)+\frac{{{\kappa _s}(r)}}{{4\pi }}\int_{{4\pi }} I ({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t)\Phi ({{\mathbf{\Omega }}^\prime },{\mathbf{\Omega }}){\text{d}}{\Omega ^\prime },$$
(1)

where n is the refractive index. c is the velocity of light. βe and κs represent the extinction coefficient and scattering coefficient, respectively. If the absorption coefficient is denoted by κa, βe = κa + κs, \({{\mathbf{\Omega }}^\prime }\) and Ω indicate the incident and the scattering directions, respectively. \({{{\Phi}}}\) represents the scattering phase function. Two dual-parameter phase functions are considered to simulate the scattering characteristic of ceramics. The first one is based on the physical structure of reticulated porous ceramics which is written as [37]:

$$\Phi ({{\mathbf{\Omega }}^\prime },{\mathbf{\Omega }})={f_{{\text{iS}}}}+(1 - {f_{{\text{iS}}}} - {f_{{\text{bck}}}}){\Phi _{{\text{DIFF}}}}({{\mathbf{\Omega }}^\prime },{\mathbf{\Omega }})+{f_{{\text{bck}}}}{\Phi _{{\text{DIF}}\,{\text{RER}}}}({{\mathbf{\Omega }}^\prime },{\mathbf{\Omega }}),$$
(2)

where fis and fbck are fractions of isotropically and back scattering factors, respectively, and they satisfy \({f_{{\text{iS}}}} - {f_{{\text{bck}}}} \leqslant 1\). ΦDIFF and ΦDIF RER represent the diffraction-dominated and diffusely reflective phase functions, respectively. It is called diffuse-reflectance, isotropic forward-diffraction (DRIFD) scattering phase function. The other phase function is the modified Henyey–Greenstein (H–G) phase function which can be expressed as:

$$\Phi ({{\mathbf{\Omega }}^\prime },{\mathbf{\Omega }})={f_{{\text{iS}}}}+(1 - {f_{{\text{iS}}}})\frac{{1 - g_{{{\text{hg}}}}^{2}}}{{{{[1+g_{{{\text{hg}}}}^{2} - 2{g_{{\text{hg}}}} \cdot {\text{cos}}({{\mathbf{\Omega }}^\prime },{\mathbf{\Omega }})]}^{3/2}}}},$$
(3)

where ghg is the scattering asymmetry factor, \(- 1 \leqslant {g_{{\text{hg}}}} \leqslant 1.\)

The radiative intensity in the semitransparent media is composed of collimated intensity and diffused intensity, which can be written as:

$$I({\mathbf{r}},{\mathbf{\Omega }},t)={I_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t)+{I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t),$$
(4)

where Id and Ic represent the diffused radiative intensity that scattered from the radiative source and the remaining collimated radiative intensity, respectively. Hence, the TRTE can be expressed as:

$$\begin{aligned} & \frac{n}{c}\left[ {\frac{{\partial {I_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t)}}{{\partial t}}+\frac{{\partial {I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t)}}{{\partial t}}} \right]+{\mathbf{\Omega }} \cdot \nabla ({I_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t)+{I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t)) \\ & \quad = - {\beta _{\text{e}}}({\mathbf{r}})\left[ {{I_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t)+{I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t)} \right]+\frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {[{I_{\text{c}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t)+{I_{\text{d}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t)]\Phi ({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime }){\text{d}}\Omega } . \\ \end{aligned}$$
(5)

The variation of the collimated intensity at location r and time t in direction Ω satisfies the following Bouguer–Beer law [35]:

$$\frac{n}{c}\left[ {\frac{{\partial {I_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t)}}{{\partial t}}+\frac{{\partial {I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t)}}{{\partial r}}} \right]= - {\beta _{\text{e}}}({\mathbf{r}}){I_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t).$$
(6)

For a square pulse laser irradiation, the collimated intensity in the direction Ω0 can be written as:

$${I_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t)={I_{{\text{in}}}}\exp \left[ { - \int\limits_{0}^{{{s_0}}} {{\beta _{\text{e}}}({\mathbf{r}}){\text{d}}r} } \right]\left[ {H(ct - {s_0}) - H(ct - c{t_p} - {s_0})} \right]\delta ({\mathbf{\Omega }} - {{\mathbf{\Omega }}_0}),$$
(7)

where Ω0 represents the incident direction of laser. s0 is the transmission distance of the collimated intensity. H is the Heaviside step function, and δ denotes the Dirac’s delta function. tp indicates the pulse width of laser. Thus, Eq. (5) can be presented by:

$$\frac{n}{c}\frac{{\partial {I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t)}}{{\partial t}}+\partial {I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t)= - {\beta _{\text{e}}}({\mathbf{r}}){I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t)+\frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{I_{\text{d}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t)\Phi ({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime }){\text{d}}{\Omega ^\prime }+{S_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t),}$$
(8)

where Sc represents the source term by the collimated intensity, which can be expressed as:

$${S_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t)=\frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{I_{\text{d}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t)\Phi ({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime }){\text{d}}{\Omega ^\prime }} .$$
(9)

The DOM is one of the most effective methods for solving radiative transfer problems in terms of computational accuracy and efficiency. The DOM had been proved to be more efficient to describe the short-pulse laser transfer than other methods such as the discrete transfer method and finite volume method [38]. Thus, the DOM is employed to solve the direct model and the discretized TRTE can be written as:

$$\frac{n}{c}\frac{{\partial I_{{{\text{d}},t,x,y}}^{m}}}{{\partial t}}+{\xi ^m}\frac{{\partial I_{{{\text{d}},t,x,y}}^{m}}}{{\partial x}}+{\eta ^m}\frac{{\partial I_{{{\text{d}},t,x,y}}^{m}}}{{\partial y}}= - {\beta _{{\text{e}},x,y}}+\frac{{{\kappa _{{\text{s}},x,y}}}}{{4\pi }}\sum\limits_{{l=1}}^{{N\Omega }} {{w^l}I_{{{\text{d}},t,x,y}}^{l}\Phi ({{\mathbf{\Omega }}^m},{{\mathbf{\Omega }}^l})+{S_{{\text{c}},t,x,y}}}$$
(10)

where ξ and η are directional cosines. w denotes the directional weight. m and l indicate the mth and lth direction. NΩ represents the number of discretized direction. Thus, the radiative intensity at each grid of computational domain can be obtained by:

$$I_{{{\text{d}},t,x,y}}^{m}=\frac{{{A^m}\partial I_{{{\text{d}},t - \Delta t,x,y}}^{m}\,+{B^m}I_{{{\text{d}},t,xu,y}}^{m}+{C^m}I_{{{\text{d}},t,x,yu}}^{m}+\frac{{{\kappa _{{\text{s}},x,y}}}}{{4\pi }}{w^l}I_{{{\text{d}},t,x,y}}^{l}\Phi ({{\mathbf{\Omega }}^m},{{\mathbf{\Omega }}^l})+{S_{{\text{c}},t,x,y}}}}{{{A^m}+{B^m}+{C^m}+{\beta _{{\text{e}},x,y}}}},$$
(11)

where \({A^m}=1/(c\Delta t)\), \({B^m}={\xi ^m}/\Delta x\), and \({C^m}={\eta ^m}/\Delta y\). The subscript xu and yu represent the upstream positions in the x and y directions. The upstream radiative intensity can be given as:

$$I_{{{\text{d}},t,xu,y}}^{m}=\left\{ {\begin{array}{*{20}{c}} {I_{{{\text{d}},t,x - \Delta x,y}}^{m}} \\ {I_{{{\text{d}},t,x+\Delta x,y}}^{m}} \end{array}\begin{array}{*{20}{c}} {{\xi ^m}>0} \\ {{\xi ^m}<0} \end{array}} \right.,$$
(12)
$$I_{{{\text{d}},t,x,yu}}^{m}=\left\{ {\begin{array}{*{20}{c}} {I_{{{\text{d}},t,x,y - \Delta y}}^{m}} \\ {I_{{{\text{d}},t,x,y+\Delta y}}^{m}} \end{array}\begin{array}{*{20}{c}} {{\eta ^m}>0} \\ {{\eta ^m}<0} \end{array}} \right..$$
(13)

To verify the accuracy and reliability of the present solution, a 2D transient radiative transfer problem is considered here. The related parameters of direct problem are listed in Table 1. The reflectance and transmittance signals on the left and right boundaries are compared with those in Ref. [39], which is illustrated in Fig. 2. As shown, good agreements are obtained between the retrieval results and the solutions in reference.

Table 1 Parameters of direct model
Full size table
Fig. 2
figure 2

Verification of the present solution for solving transient radiative transfer problem

Full size image

3 Adjoint equation

For the inverse estimation problems, effective and accurate gradient computation is crucial to the whole optimization process. Adjoint equation is an efficient method to get the gradient of objective function with respect to reconstruction parameters, in which the inverse optimization task is transformed into the sensitivity problem and adjoint problem.

3.1 Sensitivity problem

The main goal of this study is to detect the defects in ceramic materials based on optical imaging technique. To reconstruct the absorption and scattering coefficients of media, the exit radiative intensities at boundaries are served as input for the inverse analysis and the objective function is defined as:

$${F_{{\text{obj}}}}=\frac{1}{2}\int\limits_{T} {{\text{d}}t} \int\limits_{{\partial D}} {{\text{d}}r} {\left[ {1 - \frac{{{R_{{\text{est}}}}(t,{\mathbf{r}})}}{{{R_{{\text{est}}}}(t,{\mathbf{r}})}}} \right]^2}+\Re$$
(14)

where T is the total computation time of direct problem. \(\Re\) represents the regularization term. Rest and Rexa represent the estimated and exact emerging radiative intensities at boundaries, which can be expressed as:

$$R(t,{\mathbf{r}})=\int\limits_{{n\Omega>0}} {I(t,{\mathbf{r}}){\mathbf{n}} \cdot {\mathbf{\Omega }}{\text{d}}\Omega } .$$
(15)

The Gaussian Markov random field is employed as the regularization term which can be written as [40]:

$$\Re =\frac{1}{{p{\vartheta ^p}}}\sum\limits_{{\{ u,v\} \in N}} {{b_{u - v}}} {\left| {{\kappa _u} - {\kappa _v}} \right|^p},$$
(16)

where p and ϑ are image sharpness and scale parameters, respectively. N denotes the set of all neighboring pixel pairs. bu−v represents the weight coefficient. κ indicates the absorption or scattering coefficient.

Adjoint equation is derived by considering some physical relation constraints must be satisfied in the minimization process of objective function. For the reconstruction of optical parameters, the physical relation can be described by TRTE. Thus, Eq. (8) is introduced into the objective function and the following Lagrangian formalism can be obtained [41]:

$$\begin{aligned} F & =\frac{1}{2}\int\limits_{T} {{\text{d}}t} \int\limits_{{\partial D}} {{\text{d}}r} {\left[ {1 - \frac{{{R_{{\text{est}}}}(t,{\mathbf{r}})}}{{{R_{{\text{exa}}}}(t,{\mathbf{r}})}}} \right]^2}+\Re \int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}t} \lambda \\ & \quad \times \left\{ {\left[ {\frac{n}{c}\frac{\partial }{t}+{\mathbf{\Omega }} \cdot \nabla +{\beta _{\text{e}}}({\mathbf{r}})} \right]{I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t) - \frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{\text{d}}{\Omega ^\prime }} {I_{\text{d}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t){{{\Phi}}}({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime }) - {S_{\text{c}}}({\mathbf{r}},{\mathbf{\Omega }},t)} \right\} \\ \end{aligned}$$
(17)

where λ represents the Lagrangian multiplier in the adjoint model.

Considering a small variation in absorption and scattering coefficient distributions over the whole computational domain, a small variation of objective function is obtained:

$$\begin{aligned} \Delta F & =\int\limits_{T} {{\text{d}}t} \int\limits_{{\partial D}} {{\text{d}}r} \frac{{{R_{{\text{est}}}}(t,{\mathbf{r}}) - {R_{{\text{exa}}}}(t,{\mathbf{r}})}}{{{{\left[ {{R_{{\text{exa}}}}(t,r)} \right]}^2}}}\Delta {R_{est}}(t,{\mathbf{r}})+\Delta \Re \int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega } \\ & \quad \lambda \times \left\{ {\left[ {\frac{n}{c}\frac{\partial }{{\partial t}}+{\mathbf{\Omega }} \cdot \nabla +{\beta _{\text{e}}}({\mathbf{r}})} \right]{I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t) - \frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{\text{d}}{\Omega ^\prime }} \Delta {I_{\text{d}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t){{{\Phi}}}({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime })} \right. \\ \,\,\,\,\,\,\,\,\,\left. {\,+\,\Delta {\beta _{\text{e}}}({\mathbf{r}}){I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t) - \frac{{\Delta {\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{\text{d}}{\Omega ^\prime }{I_{\text{d}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t){{{\Phi}}}({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime })} } \right\} \\ \end{aligned}$$
(18)

3.2 Adjoint problem and gradient equation

To remove the ΔId in Eq. (18), a special Lagrangian multiplier is defined which satisfies:

$$\begin{aligned} 0 & =\int\limits_{T} {{\text{d}}t} \int\limits_{{\partial D}} {{\text{d}}r} \frac{{{R_{{\text{est}}}}(t,{\mathbf{r}}) - {R_{{\text{exa}}}}(t,{\mathbf{r}})}}{{{{\left[ {{R_{{\text{exa}}}}(t,{\mathbf{r}})} \right]}^2}}}\Delta {R_{{\text{est}}}}(t,{\mathbf{r}})+\int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega } \\ & \quad \lambda \times \left\{ {\left[ {\frac{n}{c}\frac{\partial }{t}+{\mathbf{\Omega }} \cdot \nabla +{\beta _e}({\mathbf{r}})} \right]\Delta {I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t) - \frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{\text{d}}{\Omega ^\prime }} {I_{\text{d}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t){{{\Phi}}}({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime })} \right\}. \\ \end{aligned}$$
(19)

Considering the following integral transformations:

$$\begin{aligned} & \int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \frac{{{R_{{\text{est}}}}(t,{\mathbf{r}}) - {R_{{\text{exa}}}}(t,{\mathbf{r}})}}{{{{[{R_{{\text{exa}}}}(t,{\mathbf{r}})]}^2}}}\Delta {R_{{\text{est}}}}(t,{\mathbf{r}}) \\ & \quad =+\int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \frac{{{R_{{\text{est}}}}(t,{\mathbf{r}}) - {R_{{\text{exa}}}}(t,{\mathbf{r}})}}{{{{\left[ {{R_{{\text{exa}}}}(t,{\mathbf{r}})} \right]}^2}}}\frac{{{R_{{\text{exa}}}}(t,{\mathbf{r}})}}{{\partial I}}\Delta I= - \int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int_{{n.\Omega>0}} {{\text{d}}\Omega \lambda {\mathbf{n}} \cdot {\mathbf{\Omega }}\Delta I} , \\ \end{aligned}$$
(20)
$$\int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega \lambda \frac{n}{c}\frac{{\partial \Delta {I_{\text{d}}}}}{{\partial t}}} =\frac{n}{c}\left[ {\int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega } \frac{n}{c}(\lambda \Delta {I_{\text{d}}})|_{0}^{{{t_{\hbox{max} }}}} - \int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega } \Delta {I_{\text{d}}}\frac{{\partial \lambda }}{{\partial t}}} \right],$$
(21)
$$\int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega \lambda {\mathbf{\Omega }}\nabla \Delta {I_{\text{d}}}=} \int\limits_{T} {{\text{d}}t} \int\limits_{{\partial D}} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega } \lambda \Delta {I_{\text{d}}}{\mathbf{\Omega }} \cdot {\mathbf{n}} - \int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega } \Delta {I_{\text{d}}}{\mathbf{\Omega }}\nabla \lambda ,$$
(22)

Eq. (19) can be written as:

$$0=\int\limits_{T} {{\text{d}}t} \int\limits_{{\partial D}} {{\text{dr}}} \int\limits_{{4\pi }} {{\text{d}}\Omega \,\Delta {I_{\text{d}}}} \left\{ {\left[ { - \frac{n}{c}\frac{\partial }{t}+{\mathbf{\Omega }} \cdot \nabla +{\beta _{\text{e}}}({\mathbf{r}})} \right]\lambda - \frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{\text{d}}{\Omega ^\prime }} \,\lambda {{{\Phi}}}({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime })} \right\}.$$
(23)

Hence, the Lagrangian multiplier equation can be expressed as:

$$\left[ {\frac{n}{c}\frac{{\partial \lambda }}{t} - {\mathbf{\Omega }} \cdot \nabla +{\beta _e}({\mathbf{r}})} \right]\lambda +\frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{\text{d}}{\Omega ^\prime }} \lambda {{{\Phi}}}({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime })=0.$$
(24)

Reverse time from t = tmax to 0 and radiative transfer direction \(\Omega\), Eq. (24) can be transformed to:

$$\frac{n}{c}\frac{{\partial \lambda }}{{\partial t}}+{\mathbf{\Omega }} \cdot \nabla \lambda = - {\beta _{\text{e}}}({\mathbf{r}})\lambda +\frac{{{\kappa _{\text{s}}}({\mathbf{r}})}}{{4\pi }}\int\limits_{{4\pi }} {{\text{d}}{\Omega ^\prime }} \lambda {{{\Phi}}}({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime })=0,$$
(25)

which is similar to the TRTE. Thereby, the DOM is also suitable for solving the above Lagrangian multiplier equation. After the Lagrangian multiplier is obtained, the gradient of objective function with respect to absorption and scattering coefficients can be defined as:

$${\varPsi _{\text{a}}}\frac{{\Delta F}}{{\Delta {\kappa _{\text{a}}}}}=\int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega \,\lambda ({\mathbf{r}},{\mathbf{\Omega }},t){I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t)+{\Re _\kappa }} ,$$
(26)
$${\varPsi _{\text{s}}}=\frac{{\Delta F}}{{\Delta {\kappa _{\text{s}}}}}=\int\limits_{T} {{\text{d}}t} \int\limits_{D} {{\text{d}}r} \int\limits_{{4\pi }} {{\text{d}}\Omega \,\lambda ({\mathbf{r}},{\mathbf{\Omega }},t)} \left[ {{I_{\text{d}}}({\mathbf{r}},{\mathbf{\Omega }},t) - \frac{1}{{4\pi }}\int\limits_{{4\pi }} {{\text{d}}{\Omega ^\prime }{I_{\text{d}}}({\mathbf{r}},{{\mathbf{\Omega }}^\prime },t){{{\Phi}}}({\mathbf{\Omega }},{{\mathbf{\Omega }}^\prime })} } \right]+{\Re _\kappa }$$
(27)

where \({\Re _\kappa }\) is the differential coefficient of \(\Re\) with respect to reconstruction parameters, which can be written as:

$${\Re _\kappa }=\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{1}{{{\vartheta ^p}}}\sum\limits_{{\{ u,v\} \in N}} {{b_{u - v}}{{\left| {{\kappa _u} - {\kappa _v}} \right|}^{p - 1}}} }&{{\kappa _u}>{\kappa _v}} \end{array}} \\ {\begin{array}{*{20}{c}} 0&{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\kappa _u}={\kappa _v}} \end{array}} \\ {\begin{array}{*{20}{c}} { - \frac{1}{{{\vartheta ^p}}}\sum\limits_{{\{ u,v\} \in N}} {{b_{u - v}}{{\left| {{\kappa _u} - {\kappa _v}} \right|}^{p - 1}}} }&{{\kappa _u}<{\kappa _v}} \end{array}.} \end{array}} \right.$$
(28)

The detailed computational procedure of adjoint equation has been introduced in Ref [41]. and not repeated here.

4 Sequential quadratic programming

SQP algorithm is one of the most effective inverse techniques for solving constrained optimization tasks. The optimization problem is transformed to a series of quadratic programming (QP) subproblems, through which the algorithm superlinearly converge to the optimum. Considering the following nonlinear constrained optimization problem:

$$\begin{gathered} \begin{array}{*{20}{c}} {\hbox{min} }&{F(x)} \end{array} \hfill \\ \begin{array}{*{20}{c}} {{\text{s.t}}}&{{c_i}(x)=0\,\,i \in E=\{ 1,2, \ldots ,{m_{\text{e}}}\} } \end{array} \hfill \\ \begin{array}{*{20}{c}} {{c_i}(x)=0}&{i \in I=\{ {m_{\text{e}}}+1,{m_{\text{e}}}+2, \ldots ,m\} } \end{array}, \hfill \\ \end{gathered}$$
(29)

where c i are restrictions. m and me represent the numbers of total restriction and equality restriction, respectively. The optimization is transformed to the following QP subproblem [42]:

$$\begin{gathered} \begin{array}{*{20}{c}} {\hbox{min} }&{\nabla F{{({x^k})}^T}d+\frac{1}{2}} \end{array}{d^T}{H^k}d \hfill \\ \begin{array}{*{20}{c}} {{\text{s.t}}}&{{c_i}({x^k})+\nabla {c_i}} \end{array}{({x^k})^T}d=0\,\,\,i \in E \hfill \\ \,\,\,\,\,\,\,\,\,\,\,{c_i}({x^k})+\nabla {c_i}{({x^k})^T}d \geqslant 0\,\,i \in I, \hfill \\ \end{gathered}$$
(30)

where d represents the search direction at kth iteration. xk represents the reconstruction parameters at the current iteration. Hk is the approximation of the Hessian of the following Lagrangian:

$$L(x,u)=F(x) - \sum\limits_{{i=1}}^{m} {{u_i}{c_i}(x)} ,$$
(31)

where u i is the Lagrangian multiplier in SQP. To ensure Hk is the effective approximation, the following update is used [43]:

$${H^{k+1}}={H^k}+\frac{{\widetilde {y}{{({{\widetilde {y}}^k})}^T}}}{{{{({{\widetilde {y}}^k})}^T}{s^k}}} - \frac{{{H^k}{s^k}{{({s^k})}^T}{H^k}}}{{({s^k}){H^k}{s^k}}},$$
(32)

where \({\widetilde {y}^k}=\theta {y^k}+(1 - \theta ){H^k}{s^k},{s^k}={s^{k+1}} - {x^k}\), \(\theta\) and \({y^k}\) are defined as:

$$\theta =\left\{ {\begin{array}{*{20}{c}} 1&{{{({s^k})}^T}{y^k} \geqslant \beta {{({s^k})}^T}{H^k}{s^k}} \\ {{{(1 - \beta ){{({s^k})}^T}{H^k}{s^k}} \mathord{\left/ {\vphantom {{(1 - \beta ){{({s^k})}^T}{H^k}{s^k}} {[{{({s^k})}^T}{H^k}{s^k} - {{({s^k})}^T}{y^k}]}}} \right. \kern-0pt} {[{{({s^k})}^T}{H^k}{s^k} - {{({s^k})}^T}{y^k}]}}}&{{{({s^k})}^T}{y^k}<\beta {{({s^k})}^T}{H^k}{s^k}} \end{array}} \right.,$$
(33)
$${y^k}={\nabla _x}L({x^{k+1}},{u^{k+1}}) - {\nabla _x}L({x^k},{u^{k+1}}),$$
(34)

where β is a positive constant \(0.1 \leqslant \beta \leqslant 0.2\).

To improve the global convergence of SQP algorithm, the following penalty function is considered [44]:

$${F_{\text{p}}}(x)=F(x)+r\left[ {\sum\limits_{{i=1}}^{{{m_{\text{e}}}}} {\left| {{c_i}(x)} \right|} +\sum\limits_{{i={m_{\text{e}}}+1}}^{m} {\left| {\hbox{min} \{ 0,{c_i}(x)\} } \right|} } \right],$$
(35)

where r is the penalty factor, \(r>0\). Thus, the newly generated reconstruction parameter can be expressed as:

$${x^{k+1}}={x^k}+{\alpha ^k}{d^k},$$
(36)

where αk is search step size, which satisfies:

$${F_{\text{p}}}({x^k}+{\alpha ^k}{d^k})<{F_{\text{p}}}({x^k})+\beta {\alpha ^k}\left[ {\overline {{{F_{\text{p}}}}} ({x^k},{d^k}) - {F_{\text{p}}}({x^k})} \right]$$
(37)

where

$$\begin{aligned} \overline {{{F_{\text{p}}}}} ({x^k},{d^k}) & =F({x^k})+\nabla F{({x^k})^T}+\frac{1}{2}{d^T}{H^k}d \\ & \quad +r\left[ {\sum\limits_{{i=1}}^{{{m_{\text{e}}}}} {\left| {{c_i}({x^k})+\nabla {c_i}{{({x^k})}^T}d} \right|+\sum\limits_{{i={m_{\text{e}}}+1}}^{m} {\left| {\hbox{min} \left\{ {0,{c_i}(x)} \right\}+\nabla {c_i}{{({x^k})}^T}d} \right|} } } \right]. \\ \end{aligned}$$
(38)

Moreover, to make sure the global and superlinear convergence of SQP algorithm, the following QP subproblem is employed to replace Eq. (29) when Maratos effect appears [45]:

$$\begin{gathered} \begin{array}{*{20}{c}} {\hbox{min} }&{\nabla {{({p^k})}^T}d+\frac{1}{2}} \end{array}{d^T}{H^k}d \hfill \\ \begin{array}{*{20}{c}} {{\text{s.t}}}&{{c_i}({x^k})+} \end{array}{(a_{i}^{k})^T}d=0\,\,\,i \in E \hfill \\ \,\,\,\,\,\,\,\,\,\,\,{c_i}({x^k})+{(a_{i}^{k})^T}d \geqslant 0\,\,i \in I, \hfill \\ \end{gathered}$$
(39)

where pk and ak are defined as:

$${p^k}=\nabla F({x^k})+\frac{1}{2}\sum\limits_{{i=1}}^{m} {{u^{k+1}}\left[ {\nabla {c_i}({x^k}+{{\widehat {d}}^k}) - \nabla {c_i}({x^k})} \right]}$$
(40)
$$a_{i}^{k}=\nabla {c_i}({x^k})+\frac{1}{2}{\widehat {d}^k}\left[ {\nabla {c_i}({x^k}+{{\widehat {d}}^k}) - \nabla {c_i}({x^k})} \right]$$
(41)

where \({\widehat {d}^k}\) is the solution of Eq. (39). Therefore, the reconstruction parameter and search step size can be generated by:

$${x^{k+1}}={x^k}+{\alpha ^k}{d^k}+{({\alpha ^k})^2}({\widehat {d}^k} - {d^k}),$$
(42)
$${F_{\text{p}}}({x^{k+1}})<{F_{\text{p}}}({x^k})+\beta {\alpha ^k}[{\overline {F} _{\text{p}}}({x^k},{d^k}) - {F_{\text{p}}}({x^k})].$$
(43)

The iteration stops until one of the following convergence criteria is satisfied:

$$k={k_{\hbox{max} }},$$
(44)
$$\begin{aligned} & \nabla F({x^{k+1}}) - \sum\limits_{{i=1}}^{m} {u_{i}^{{k+1}}\nabla {c_i}} ({x^{k+1}}) \leqslant {\varepsilon _1} \\ & \left| {{c_i}({x^{k+1}})} \right| \leqslant {\varepsilon _1} \\ & {c_i}({x^{k+1}}) \geqslant {\varepsilon _1}, \\ \end{aligned}$$
(45)
$$\frac{{{F_{\text{p}}}({x^k}) - F({x^{k+1}})}}{{\upsilon +\left| {{F_{\text{p}}}({x^{k+1}})} \right|}} \leqslant {\varepsilon _2},$$
(46)

where kmax denotes the specified maximum iteration number. ε1, ε2 and υ are small positive values.

5 Results and discussion

5.1 Description of non-destructive testing

The main purpose of this study is to show the validity of mid-infrared ultrashort-pulse laser in NDT of ceramics based on transient radiative transfer model. As shown in Fig. 3, a 2D square ceramic material with size of \({L_x} \times {L_y}=0.04 \times 0.04\,{{\text{m}}^2}\) is considered. The power density of incident laser is set as 3544 W/m2, and the pulse width of laser is 0.1 ns. Two kinds of reticulated porous ceramics, PS ZrO2 and OB SiC, are employed as the test materials. Since the difference between the absorption and scattering coefficients is relatively small at the end of the spectral range studied in Ref. [1], the single-wavelength pulse lasers at wavelengths 5.0 and 4.9 μm are selected as the incident source for the NDT of PS ZrO2 and OB SiC, respectively. Hendricks and Howell [1] experimentally investigated the absorption and scattering coefficients of PS ZrO2 and OB SiC for DRIFD and modified H–G phase functions and the related optical properties are listed in Table 2.

Fig. 3
figure 3

Schematic of non-destructive testing

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Table 2 Optical parameters of ceramics and air
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The emerging radiative intensities from four boundaries are employed as the measurement signals to reconstruct the absorption and scattering coefficient distributions. Since measurement errors are inevitable, the random standard deviation is considered which can be expressed as:

$${Y_{{\text{mea}}}}={Y_{{\text{exa}}}}+\sigma \varsigma ,$$
(47)

where Ymea and Yexa are the measurement signals with and without random error, respectively. ς represents a random variable of normal distribution with zero mean and unit standard deviation. The standard deviations of measured radiative signals, σ for a γ % measured error at 99% confidence, are determined as:

$$\sigma =\frac{{{Y_{{\text{exa}}}} \times \gamma \% }}{{2.576}},$$
(48)

where the denominator is set as 2.576, because a normally distributed population is contained within ± 2.576 standard deviation of the mean.

To evaluate the overall accuracy of the reconstruction results, the normalized root-mean square error (NRMSE) is introduced which is defined as [46]:

$${\text{NRMSE}}={{\sqrt {\sum\limits_{{i=1}}^{{NN}} {{{({x_i} - {{\widetilde {x}}_i})}^2}} } } \mathord{\left/ {\vphantom {{\sqrt {\sum\limits_{{i=1}}^{{NN}} {{{({x_i} - {{\widetilde {x}}_i})}^2}} } } {\sqrt {\sum\limits_{{i=1}}^{N} {{x_i}^{2}} } }}} \right. \kern-0pt} {\sqrt {\sum\limits_{{i=1}}^{N} {{x_i}^{2}} } }},$$
(49)

where \({x_i}\) and \({\widetilde {x}_i}\) represent the exact and estimated values of the reconstruction parameters. NN is the total number of the estimation parameters.

All cases are implemented using the Fortran code, and the developed program is executed on an Intel Core(TM) i7-7700 PC.

5.2 Non-destructive testing of PS ZrO2

To verify the feasibility of the present inverse technique in NDT of ceramics, a square air defect is considered firstly. In Test 1, the defect is located at the bottom left of media whose central coordinate is set as \({x_{\text{c}}}/{L_x}=0.25\) and \({y_{\text{c}}}/{L_y}=0.25\). The reconstruction results of PS ZrO2 for DRIFD and H–G phase functions are illustrated in Figs. 4 and 5, respectively. As shown, all the characteristics including optical properties, geometric shape and spatial position of the defect can be accurately reconstructed. From the reconstruction results, the air defect in ceramic can be accurately identified, which demonstrates the validity of inverse transient radiative transfer technique in NDT of ceramic materials.

Fig. 4
figure 4

Reconstruction results of Test 1 for DRIFD phase function. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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Fig. 5
figure 5

Reconstruction results of test 1 for H–G phase function. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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The effect of the central position of defect is also investigated and the settings of central coordinates are shown in Table 3. All the defects at different positions can be accurately reconstructed (see Figs. 6, 7, 8, 9), from which the defects in ceramics can be clearly determined. The NRMSEs of reconstruction results for different central positions and different phase functions are pretty close and the maximum NRMSE is less than 0.012 (see Table 4), which demonstrates that the present solution can be successfully applied to the NDT of ceramics. Since the retrieval results for DRIFD and H–G phase functions are analogous, the following discussions of this study are based on H–G phase function.

Table 3 Central coordinates of defect in ceramic
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Fig. 6
figure 6

Reconstruction results of test 2 for DRIFD phase function. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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Fig. 7
figure 7

Reconstruction results of test 3 for DRIFD phase function. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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Fig. 8
figure 8

Reconstruction results of test 2 for H–G phase function. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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Fig. 9
figure 9

Reconstruction results of test 3 for H–G phase function. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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Table 4 The NRMSE of reconstruction results for PS ZrO2
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Moreover, the NDT results of ceramic with multi-defects (three defects) are studied (see Fig. 10). Although the imaging quality slightly decreases, the retrieved results can accurately reflect the defects in ceramic materials. Therefore, the NDT technique based on transient radiative transfer model is also effective in detecting multiple defects in ceramic materials. Table 5 lists the NRMSE values and computational time for different sampling times of direct problem. As shown, the longer the sampling time is, the smaller the NRMSE will be, whereas the longer the total computational time will be. Considering the imaging accuracy and computational time synthetically, the sampling time is set as 100 ns.

Fig. 10
figure 10

Reconstruction result of PS ZrO2 with three defects. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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Table 5 The NRMSE values and computational time for different sampling times
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5.3 Non-destructive testing of OB SiC

Since the principle of NDT technique for OB SiC is similar to that for PS ZrO2, only two and three defects are considered in this section. As shown in Figs. 11 and 12, all the defects in OB SiC can be accurately identified by the present model. The edges and positions of defects can be clearly determined based on the imaging results, which means the NDT technique in this study is effective and robust to detect the defects in ceramic materials.

Fig. 11
figure 11

Reconstruction result of OB SiC with two defects. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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Fig. 12
figure 12

Reconstruction result of OB SiC with three defects. a Real distribution of absorption coefficient; b retrieved results of absorption coefficient; c real distribution of scattering coefficient; and d retrieved results of scattering coefficient

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To test the robustness of the inverse technique, random errors are added into the measurement signals. Figures 13 and 14 show the reconstruction results of absorption and scattering coefficients for different measurement errors. Although the random error is increased to 50%, the positions and optical properties of defects can also be distinguished according to the imaging results. It is worth noting that the excessive imaging errors on the boundaries in Fig. 14c, d are caused by the incident laser. Table 6 lists the NRMSE values of retrieval results for different measurement errors. The NRMSE increases with the increase of random error. The maximum NRMSE is only 0.1146, which demonstrates that the present technique has strong robustness.

Fig. 13
figure 13

Reconstruction results of absorption coefficient of OB SiC for a 5%, b 10%, c 20%, and d 50% measurement errors

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Fig. 14
figure 14

Reconstruction results of scattering coefficient of OB SiC for a 5%, b 10%, c 20%, and d 50% measurement errors

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Table 6 The NRMSE values of reconstruction results for different measurement errors
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Furthermore, the NDT of OB SiC for circular and oval defects is investigated (see Fig. 15). Both the circular and oval defects are accurately reconstructed. The retrieved absorption and scattering coefficient distributions by the inverse technique are in good agreement with the real distributions. Comparing Fig. 15b, c, e, f with a, d, it can be found that the differences of edges are caused by the coarse grids. Thus, locally refining meshes around defects to get smooth edges is our further research interest.

Fig. 15
figure 15

Reconstruction results of circular and oval defects in OB SiC. a Real distribution of absorption coefficient for \(501 \times 501\) grids; b real distribution of absorption coefficient for \(21 \times 21\) grids; c retrieved distribution of absorption coefficient for \(21 \times 21\) grids; drReal distribution of scattering coefficient for \(501 \times 501\) grids; e real distribution of scattering coefficient for \(21 \times 21\) grids; and f retrieved distribution of scattering coefficient for \(21 \times 21\) grids

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6 Conclusions

The mid-infrared ultrashort-pulse laser is employed to non-destructively detect the defects in ceramic materials based on the transient radiative transfer model. The direct problem is solved by DOM and the obtained exit radiative intensity on boundary is served as measurement signals. The adjoint equation model is applied to get the gradient of objective function with respect to absorption and scattering coefficients and the SQP algorithm is used to optimize the objective function. Two reticulated porous ceramics including PS ZrO2 and OB SiC are selected as the testing materials. The retrieved results show that the boundaries, positions and radiative properties of defects can be accurately identified. The effects of spatial position, total number and geometric shape of defects on the imaging results are investigated. All the reconstructed absorption and scattering coefficients are in good agreement with the real distributions. Even though the measurement error is increased to 50%, the NRMSE is only 0.1146 and the defects can be accurately determined by the reconstruction results, which demonstrates the present technique is effective and robust in NDT of ceramics.

The further research direction is to investigate the locally refining meshes technique to get the smooth edges of defects.