Introduction

In general, experimental tests are performed to study the response behavior of a material under certain external influences. The measured information can subsequently be evaluated, e.g. to calibrate material models or to detect unwanted properties such as defects. For thermomechanically coupled responses, thermal as well as mechanical measurements are required to sufficiently identify the respective material behavior. To acquire the experimental data, contactless full-field measurement methods are a suitable choice since they provide comprehensive amount of information on a surface, see [15] for an overview. In terms of displacement measurements digital image correlation (DIC) is commonly applied. In contrast, temperature measurements are done by drawing on infrared thermography (TG).

Infrared thermography cameras measure the incident radiance in the specific spectral range of the camera. The temperature distribution is afterwards computed from the pixel-wise radiance information. As a result, infrared thermography systems provide only two-dimensional images (projections) of the temperature on the component surface. They are commonly applied in detecting surface temperatures of buildings [1, 28], medical applications, e.g. [41], or products [30], especially for the detection of unwanted component properties, or damage within the components, for instance, [10]. Further applications are, for example, measurements in polymer science, where exothermal reactions are studied in curing processes, see [29] and the literature cited therein.

The geometrical surface information can be obtained using digital image correlation systems, [39]. There, a stochastic pattern on the specimen’s surface is required, either created by a natural pattern due to the surface structure or by applying a particular varnish. In this pattern, small subsets with a unique gray-scale distribution are evaluated assigning two- or three-dimensional coordinates – when using two CCD (charge-coupled device) cameras by stereography – to the center of these subsets. To compute values of interest, such as surface strains, the coordinates are either interpolated using local concepts, [17, 20, 39], or by applying global interpolation schemes, [18].

Both measurement methods, TG and DIC, provide intrinsically different sets of data. The geometrical data from DIC-systems are values in the material or Lagrangian representation. In contrast, temperature values obtained from TG are of spatial or Eulerian type, see, for example, [19] for the specific terminology. Further, the particular requirements for the specimen’s surface are contradictory since TG-systems demand uniformly painted surfaces with high emissivity, whereas DIC-systems require stochastic speckle patterns. However, it is nonetheless of particular interest to assign the temperature to the material coordinates of the surface.

In recent literature, a majority of the reported studies are conducted with focus on the deformation-induced temperature changes of different materials. There, it is commonly assumed that for a sufficiently thin and flat specimen the through-thickness thermal gradient is negligible, which holds for high thermal conductivity and homogeneous materials, [8, 9, 27, 37, 38]. In other words, the two camera systems are placed on the opposite side of the specimens. Another study employing a coupling of TG and 2D-DIC is performed on open-cell foams, where out-of-plane displacements are neglected due to the high porosity of the structure, see [25]. Therein, the authors also propose a tool for the direct correlation of the thermal and mechanical data. The handling of the different types of experimental data obtained by TG and 2D-DIC in the context of subsequent material parameter identification is addressed by [38].

In contrast to the aforementioned studies, the same surface of a specimen is measured with TG and DIC in the general case. Favier et al. [13] studied the behavior of Ni-Ti shape memory alloys with a two-dimensional DIC setup and infrared thermography. The specimens were tubes under tension, which were coated by a coarse speckled pattern consisting of a high emissivity black paint with white speckle points. Thus, the authors were able to apply both measurement methods on the same side of the specimen. Due to the coarse speckled pattern a loss in spatial resolution for the deformation measurement results. Further, [3, 4] proposed the first coupling of TG and 2D-DIC at the length scale of grains for different materials. The authors applied a customized coating to measure the same surface with TG and DIC. Additionally, a dichroic mirror was employed to split the infrared and visual radiation to the corresponding infrared and CCD camera. However, due to the use of the dichroic mirror, which is not fully transparent for the infrared radiation, the calibration of the infrared camera has to be repeated for each setup. A quite similar experimental setup with perpendicular optical axes of the infrared and CCD camera and an infrared (IR) long-pass filter is applied by [33]. Therein, the authors compute the transformation between both cameras based on three points. In contrast, instead of drawing on a coupling of TG and 2D-DIC, it is possible to simultaneously measure thermal and deformation fields by drawing on just a single infrared camera. This method, denoted as infrared image correlation (IRIC), was originally proposed by [31], where the authors used a high emissivity speckle pattern to obtain a spatial distribution in the emissivity suitable to obtain two-dimensional displacements. Further attempts to apply only one infrared camera are reported by [44] later on.

The studies mentioned afore are limited to two-dimensional information obtained from digital image correlation. However, especially when considering specimens with curved or rough surfaces, for example, made by wire-arc additive manufacturing processes, the goal should be to apply both systems to the same surface and consider the three-dimensional geometrical information. Orteu et al. [35] proposed a measurement method, where two temperature-calibrated CCD-cameras measure simultaneously the full-field deformation as well as the temperatures. However, the thermal radiation measurements were only possible in near-infrared wavelengths, which is the case for surface temperatures of \({400\,\mathrm{{{}^{\circ }\text {C}}}}-{700\,\mathrm{{{}^{\circ }\text {C}}}}\). Furthermore, since the surface emissivity was neither calibrated nor uniform, the thermal fields were not interpretable as surface temperature distribution. Cholewa et al. [7] originally developed a technique for coupled thermomechanical measurements using TG and 3D-DIC. Therein, the surface under consideration was varnished with high emissivity black paint and white speckles. The calibration target for both measurement systems was an anodized aluminum sheet with three highlighted points. Additionally, the authors performed a detailed experimental uncertainty quantification based on sandwich composite specimens. In a further study, the damage behavior of carbon fiber reinforced polymers was studied by [14]. There, the authors assume that the emissivity of the varnished speckle pattern was about 0.98. In a recently published work, [32] investigated the deformation and temperature evolution of different polymeric film materials during a bending process. However, no information is given on the transformation between the two-dimensional thermography data and the three-dimensional image correlation data. The authors further determined the emissivity of the applied varnish to be 1.

The coordinates as well as the temperature information are given in the form of discrete values. In the case of thermographic images in the form of pixel data and in the case of image correlation information by coordinates of certain pattern subsets, which are either sprayed on the samples or given as gray value distribution. This discrete information can be smoothed either locally at individual points by linear functions (regression), which can be done for both thermography and geometry data, or by interpolation or regression of all data, i.e. global functions. At present – to the authors’ knowledge – the mathematical representation of the temperature calculation and also the gradient calculation for curvilinear surfaces is not completely specified or known. In [18] so-called radial basis functions, see, for the basic ideas, [2, 6], are applied. These functions are attractive for arbitrary point distributions, [11, 12, 16, 23, 42].

Therefore, the aim is to derive the associated basic equations, i.e. the transformation properties between the two coordinate systems of the TG and the DIC-system. Afterwards, the expected accuracies are studied at an analytical verification example. Finally, the concept is applied to two real experiments, first, a cylindrical shell produced by 3D-printing, and, second, a specimen which was produced by wire-arc additive manufacturing showing a very rough surface.

The notation in use is defined in the following manner: geometrical vectors are symbolized by \(\vec {a}\) (independent of the basis), second-order tensors \({\mathbf {A}}\) by bold-faced Roman letters, and both column vectors and matrices by sans serif letters \(\boldsymbol {\mathsf{A}}\).

Interpolation of the Temperature Distribution

Since many components do not have flat surfaces, which is particularly the case with additively manufactured materials, special attention is paid in the present investigations. However, before dealing with the concrete determination of temperature and temperature gradients with the aid of a thermography system for curved surfaces, the associated theory of temperature gradients in surfaces must be dealt with. The disadvantage of thermography is that only the temperature at the surface of a component can be detected. Therefore, only the in-plane part is available for the temperature gradient determination. We then discuss the coupling of thermography and geometry detection using an image correlation system. Based on these discrete data, temperature and temperature gradient determination on curved component surfaces are then possible if the geometry is described by functions. Radial basis functions are chosen for this purpose.

Temperature Gradient in Curved Surfaces

A surface is described by the position vector \(\vec {r} = \vec {r} (\varvec{\Psi }) = \vec {r} (\Psi ^1,\Psi ^2)\) depending on the surface parameters \(\Psi ^1\) and \(\Psi ^2\), [22]. Considering an extension by introducing a point outside the surface in the form

$$\begin{aligned} \vec {R} (\Psi ^1,\Psi ^2,\Psi ^3) = \vec {r} (\Psi ^1,\Psi ^2) + \Psi ^3 \vec {a} _3(\Psi ^1,\Psi ^2), \end{aligned}$$
(1)

see [26], the tangent vectors

$$\begin{aligned} \vec {g} _\alpha&= \frac{\displaystyle \partial \vec {R} }{\displaystyle \partial \Psi ^\alpha } = \frac{\displaystyle \partial \vec {r} }{\displaystyle \partial \Psi ^\alpha } + \Psi ^3 \frac{\displaystyle \partial \vec {a} _3}{\displaystyle \partial \Psi ^\alpha } = \vec {a} _\alpha + \Psi ^3 \vec {a} _{3,\alpha }, \qquad \alpha =1,2, \end{aligned}$$
(2)
$$\begin{aligned} \vec {g} _3&= \vec {a} _3, \end{aligned}$$
(3)

to the coordinate lines can be derived. Following the usual notation, we use Greek letters as indices when the numerical values are 1 or 2. For the case \(\Psi ^3=0\), the tangential vectors read

$$\begin{aligned} \vec {a} _\alpha := \vec {g} _\alpha |_{\Psi ^3=0} = \vec {r} ,_\alpha , \quad \alpha = 1,2. \end{aligned}$$
(4)

Furthermore, it is common to assume that the vector \(\vec {a} _3\) – orthogonal to the other tangential vectors – is a unit vector,

$$\begin{aligned} \vec {a} _3 \cdot \vec {a} _\alpha = 0, \quad \vec {a} _3 \cdot \vec {a} _3 = 1 \; \Rightarrow \vec {a} _3 \cdot \vec {a} _{3,\alpha } = 0, \end{aligned}$$
(5)

i.e.

$$\begin{aligned} \vec {a} _3 = \frac{\vec {a} _1 \times \vec {a} _2}{\big |\vec {a} _1 \times \vec {a} _2\big |} \end{aligned}$$
(6)

has to hold. The matrix of covariant metric coefficients reads

$$\begin{aligned} \left[ a_{\alpha \beta }\right] = \begin{bmatrix} \vec {a} _1 \cdot \vec {a} _1 &{} \vec {a} _1 \cdot \vec {a} _2 \\ \vec {a} _1 \cdot \vec {a} _2 &{} \vec {a} _2 \cdot \vec {a} _2 \\ \end{bmatrix}, \end{aligned}$$
(7)

where the inverse

$$\begin{aligned} a_{\alpha \gamma } a^{\gamma \beta } = \delta _{\alpha }^{\; \beta }, \qquad [a^{\alpha \beta }] = [a_{\alpha \beta }]^{-1}, \end{aligned}$$
(8)

\(\delta _{\alpha }^{\; \beta }\) defines the Kronecker-symbol, is required to calculate the contravariant basis vectors, defining gradient vectors on the coordinate surface,

$$\begin{aligned} \vec {a} ^{\; \alpha } = a^{\alpha \beta } \vec {a} _\beta . \end{aligned}$$
(9)

The given assumptions lead to the property

$$\begin{aligned} \vec {a} ^{\, 3} = \vec {g} ^{\,3} = \frac{\vec {a} _1 \times \vec {a} _2}{\Vert \vec {a} _1 \times \vec {a} _2\Vert } = g^{3j} \vec {g} _j = g^{33} \vec {g} _3 = \vec {g} _3 = \vec {a} _3. \end{aligned}$$
(10)

Following [24], the in-plane temperature gradient

$$\begin{aligned} \widehat{{{\,\mathrm{grad}\,}}} \Theta (\Psi ^1,\Psi ^2) = \frac{\displaystyle \partial \Theta }{\displaystyle \partial \Psi ^\alpha } \vec {a} ^{\, \alpha } = \Theta ,_\alpha \vec {a} ^{\, \alpha } \end{aligned}$$
(11)

is defined – later on \(\Theta (\Psi ^1,\Psi ^2)\) is the temperature measured by thermography – which is the projection of the full temperature gradient,

$$\begin{aligned} \widehat{{{\,\mathrm{grad}\,}}} \Theta = \hat{\mathrm {\mathbf {I}}} {{\,\mathrm{grad}\,}}\Theta \end{aligned}$$
(12)

with the projector \(\hat{\mathrm {\mathbf {I}}} = \mathrm {\mathbf {I}} - \vec {a} _3 \otimes \vec {a} ^{\, 3}\), and the second-order identity tensor \(\mathrm {\mathbf {I}} = \vec {g} _i \otimes \vec {g} ^{\, i}\). Since there should be an interest in physical components, the in-plane temperature gradient reads

$$\begin{aligned} \widehat{{{\,\mathrm{grad}\,}}} \Theta (\Psi ^1,\Psi ^2) = \Theta ,_\alpha \vec {a} ^{\, \alpha } = \Theta ,_\alpha \Vert \vec {a} ^{\langle \alpha \rangle }\Vert \vec {e} ^{\, \alpha } \end{aligned}$$
(13)

relative to the unit vectors \(\vec {e} ^{\, \alpha } = \vec {a} ^{\, \alpha }/\Vert \vec {a} ^{\, \alpha }\Vert\) with \(\Vert \vec {a} ^{\, \alpha }\Vert = \sqrt{a^{\langle \alpha \alpha \rangle }}\). The brackets \(\langle \alpha \rangle\) are chosen since Einstein’s summations convention is only valid for two occurring indices.

Geometry

Let us assume that the DIC-system provides the three-dimensional coordinates of an object’s surface. We are interested in assigning the measured temperature – two-dimensional information – of a thermographic system to the three-dimensional curved surface. Thus, the relation between the coordinates, i.e. the coordinate system must be identified first. In other words, the distance vector \(\vec {r} _{\Theta }\) and the rotation matrix \(\boldsymbol {\mathsf {Q}} {\, \in \mathbb {R}}^{3 \times 3}\) – representing the relative rotation of both basis systems – are unknown, see Fig. 1. To determine these quantities, an experimental, orthogonal tripod is produced, which is placed close to the actual object. The tripod has four points \((\mathcal {P}_0,\mathcal {P}_1,\mathcal {P}_2,\mathcal {P}_3)\) where the position vectors \(\vec {r} _k=x_{jk}\vec {e} _j\), \(k=0,\ldots ,3\), of the DIC-system have to be determined, and, accordingly, are supposed to be known. Furthermore, it is assumed that in a first step, the pixel length in the TG-system is determined, so that the coordinates \(x_{ik}^{\,\Theta }\), \(k=0,\ldots ,3\), \(i=1,2\), of the vectors

$$\begin{aligned} \vec {r} _k^{\,\Theta }= x_{ik}^{\,\Theta }\vec {e} _i^{\,\Theta }= \vec {x} _k^{\,\Theta }- d_k \vec {e} _3^{\,\Theta }= \vec {x} _k^{\,\Theta }+ \vec {d} _k, \end{aligned}$$
(14)

i.e.

$$\begin{aligned} \vec {x} _k^{\,\Theta }= x_{1k}^{\,\Theta }\vec {e} _1^{\,\Theta }+ x_{2k}^{\,\Theta }\vec {e} _2^{\,\Theta }\quad \text {and} \quad \vec {d} _k = x_{3k}^{\,\Theta }\vec {e} _3^{\,\Theta }= - d_k \vec {e} _3^{\,\Theta }\end{aligned}$$
(15)

in the TG-plane of the projected points \((\mathcal {P}'_0,\mathcal {P}'_1,\mathcal {P}'_2,\mathcal {P}'_3)\) are assumed to be known. Only the components \(x_{3k}\) are unknown, which can be interpreted as the projected distances of the points \(\mathcal {P}_k\) on the TG-camera plane, \(d_k = - x_{3k}^{\,\Theta }\).

Fig. 1
figure 1

Tripod in experiment measured by the TG and the DIC-cameras

Full size image

There are two possible considerations. First, the vectors \(\overrightarrow{\mathcal {P}_0\mathcal {P}_1}\), \(\overrightarrow{\mathcal {P}_0\mathcal {P}_2}\), and \(\overrightarrow{\mathcal {P}_0\mathcal {P}_3}\) form an orthogonal basis, or, second, they define an oblique basis. This will be considered in the following. Although the first (orthogonal) case is embedded in the oblique consideration, it is introduced for didactical reasons.

Rotation based on orthogonal basis

The points \(\mathcal {P}_k\), \(k=0,\ldots ,3\), define a known orthonormal basis system

$$\begin{aligned} \vec {e} _k^{\, *} = \frac{\displaystyle 1}{\displaystyle L_{k0}} (\vec {r} _k - \vec {r} _0) = \frac{\displaystyle 1}{\displaystyle L_{k0}} (x_{jk} - x_{j0}) \vec {e} _j, \quad k=1,2,3, \end{aligned}$$
(16)

with

$$\begin{aligned} L_{k0} = \sqrt{(x_{1k}-x_{10})^2 + (x_{2k}-x_{20})^2 + (x_{3k}-x_{30})^2}. \end{aligned}$$
(17)

Using the property

$$\begin{aligned} \vec {e} _k^{\, *} = (\vec {e} _k^{\, *} \cdot \vec {e} _j) \vec {e} _j = Q_{jk}^{(1)} \vec {e} _j, \quad \text {with} \quad Q_{jk}^{(1)} = \vec {e} _j \cdot \vec {e} _k^{\, *}, \end{aligned}$$
(18)

the latter expression defines the first rotation matrix \(\boldsymbol {\mathsf {Q}}^{(1)} = \big [Q_{jk}^{(1)}\big ] {\, \in \mathbb {R}}^{3 \times 3}\). Considering equation (16), it becomes clear – by comparing the coefficients in equation (18) – that the following condition holds,

$$\begin{aligned} Q_{jk}^{(1)} = \frac{x_{jk} - x_{j0}}{L_{k0}}. \end{aligned}$$
(19)

Since there is the relation

$$\begin{aligned} \vec {r} _k = \vec {r} _{\Theta }+ \vec {r} _k^{\,\Theta }, \end{aligned}$$
(20)

see Fig. 1, the difference vectors (16), i.e. the unit vectors can be expressed by

$$\begin{aligned} \vec {e} _k^{\, *} = \frac{\displaystyle 1}{\displaystyle L_{k0}} (\vec {r} _k - \vec {r} _0) = \frac{\displaystyle 1}{\displaystyle L_{k0}} (\vec {r} _k^{\,\Theta }- \vec {r} _0^{\,\Theta }) = \frac{\displaystyle 1}{\displaystyle L_{k0}} (x_{jk}^{\,\Theta }- x_{j0}^{\,\Theta }) \vec {e} _j^{\,\Theta }, \end{aligned}$$
(21)

which now are decoupled from the translation vector \(\vec {r} ^{\,\Theta }\). In analogy to equation (16), the basis vectors

$$\begin{aligned} \vec {e} _k^{\, *} = (\vec {e} _k^{\, *} \cdot \vec {e} _i^{\,\Theta }) \vec {e} _i^{\,\Theta }= Q_{ik}^{(2)} \vec {e} _i^{\,\Theta }\end{aligned}$$
(22)

can be related to

$$\begin{aligned} \vec {e} _k^{\, *} = \frac{\displaystyle 1}{\displaystyle L_{k0}} (\vec {r} _k - \vec {r} _0) = \frac{\displaystyle 1}{\displaystyle L_{k0}} (\vec {r} _k^{\,\Theta }- \vec {r} _0^{\,\Theta }) = \frac{\displaystyle 1}{\displaystyle L_{k0}} \big (\underbrace{\vec {x} _k^{\,\Theta }- \vec {x} _0^{\,\Theta }}_{\Delta \vec {x} _{k0}^{\,\Theta }} - (\underbrace{{d_k - d_0}}_{\Delta d_{k0}})\vec {e} _3^{\,\Theta }\big ) \end{aligned}$$
(23)

with

$$\begin{aligned} \Delta \vec {x} _{k0}^{\,\Theta }= \vec {x} _k^{\,\Theta }- \vec {x} _0^{\,\Theta }, \quad \Delta d_{k0} = d_k - d_0, \quad k=1,2,3, \end{aligned}$$
(24)

i.e.

$$\begin{aligned} \vec {e} _k^{\, *} = \frac{\displaystyle 1}{\displaystyle L_{k0}} (\Delta \vec {x} _{k0}^{\,\Theta }- \Delta d_{k0} \vec {e} _3^{\,\Theta }) \end{aligned}$$
(25)

if equations (20) and (14) are inserted. The scalar product of \(\vec {e} _k^{\, *}\) with \(\vec {e} _i^{\,\Theta }\) yields the coefficients of the orthogonal matrix \(\boldsymbol {\mathsf {Q}}^{(2)} {\, \in \mathbb {R}}^{3 \times 3}\),

$$\begin{aligned} Q_{ik}^{(2)}&= \vec {e} _k^{\, *} \cdot \vec {e} _i^{\,\Theta }= \frac{x_{ik}^{\,\Theta }- x_{i0}^{\,\Theta }}{L_{k0}}, \quad i=1,2 \end{aligned}$$
(26)
$$\begin{aligned} Q_{3k}^{(2)}&= \vec {e} _k^{\, *} \cdot \vec {e} _3^{\,\Theta }= - \frac{\Delta d_{k0}}{L_{k0}} = \frac{d_0 - d_k}{L_{k0}}. \end{aligned}$$
(27)

The coefficients \(Q_{ik}^{(2)}\), \(k=1,2,3\), \(i=1,2\), are computable. However, the \(Q_{3k}^{(2)}\) in equation (27) depend on the unknown difference \(\Delta d_{k0}\), and, are apparently unknown.

Before the determination of \(\Delta d_{k0}\) is discussed, the rotation of the TG basis system relative to the DIC basis system is provided. Exploiting the inverse property (22),

$$\begin{aligned} \vec {e} _i^{\,\Theta }= (\vec {e} _i^{\,\Theta }\cdot \vec {e} _k^{\, *}) \vec {e} _k^{\, *} = Q_{ki}^{(2)} \vec {e} _k^{\, *}, \end{aligned}$$
(28)

and insertion of equation (18) leads to the final result

$$\begin{aligned} \vec {e} _i^{\,\Theta }= Q_{ki}^{(2)} Q_{jk}^{(1)} \vec {e} _j = Q_{ji} \vec {e} _j. \end{aligned}$$
(29)

The remaining differences \(\Delta d_{k0}\) are determined as follows. Since the vectors \(\vec {r} _2 - \vec {r} _0\) and \(\vec {r} _1 - \vec {r} _0\) are assumed to be orthogonal, one obtains

$$\begin{aligned} \begin{aligned} (\vec {r} _2 - \vec {r} _0) \cdot (\vec {r} _1 - \vec {r} _0)&= (\vec {x} _2^{\,\Theta }- d_2 \vec {e} _3^{\,\Theta }- \vec {x} _0^{\,\Theta }+ d_0 \vec {e} _3^{\,\Theta }) \\&\quad\cdot (\vec {x} _1^{\,\Theta }- d_1 \vec {e} _3^{\,\Theta }- \vec {x} _0^{\,\Theta }+ d_0 \vec {e} _3^{\,\Theta })=\\&= (\Delta \vec {x} _{20}^{\,\Theta }- \Delta d_{20}\vec {e} _3^{\,\Theta }) \cdot (\Delta \vec {x} _{10}^{\,\Theta }- \Delta d_{10}\vec {e} _3^{\,\Theta }) = \\&= { \Delta \vec {x} _{20}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }+ \Delta d_{20} \Delta d_{10} } = 0. \end{aligned} \end{aligned}$$
(30)

Analogously, the conditions

$$\begin{aligned} {\Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }+ \Delta d_{30} \Delta d_{10} }&= 0, \end{aligned}$$
(31)
$$\begin{aligned} { \Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{20}^{\,\Theta }+ \Delta d_{30} \Delta d_{20} }&= 0 \end{aligned}$$
(32)

hold. Using the abbreviations

$$\begin{aligned} a := \Delta \vec {x} _{20}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }, \quad b := \Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }, \quad c := \Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{20}^{\,\Theta }\end{aligned}$$
(33)

yields the solutions

$$\begin{aligned} \Delta d_{10} = \pm i \sqrt{\frac{ab}{c}}, \quad \Delta d_{20} = \pm i \sqrt{\frac{ac}{b}}, \quad \Delta d_{30} = \pm i \sqrt{\frac{bc}{a}} \end{aligned}$$
(34)

of equations (30)–(32). Obviously,

$$\begin{aligned} a = \Delta \vec {x} _{20}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }\ne 0, \quad b = \Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }\ne 0, \quad c = \Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{20}^{\,\Theta }\ne 0 \end{aligned}$$
(35)

must hold (this has to be guaranteed, when positioning the experimental tripod). Since \(\Delta d_{k0} \in \mathbb {R}\) has to be fulfilled, the arguments of the square roots must be negative,

$$\begin{aligned} a b c < 0. \end{aligned}$$
(36)

If the tripod is positioned so that \(d_0 > d_k\), \(k=1,2,3\), i.e. \(\Delta d_{k0} < 0\), the solutions

$$\begin{aligned} \Delta d_{10} = i \sqrt{\frac{ab}{c}}, \quad \Delta d_{20} = i \sqrt{\frac{ac}{b}}, \quad \Delta d_{30} = i \sqrt{\frac{bc}{a}} \end{aligned}$$
(37)

are evaluated. If these distance differences are now substituted into equation (27), the rotation matrix results \(Q_{ik}\), i.e. the basis vectors \(\vec {e} _i^{\,\Theta }\) are known.

Rotation based on oblique basis

Since it is hard to construct a perfect orthogonal tripod, the theory is extended to an oblique tripod. Therefore, it is assumed that the oblique basis system

$$\begin{aligned} \vec {g} _k = \vec {r} _k - \vec {r} _0 = (x_{ik} - x_{i0}) \vec {e} _i = B_k^{\;\; i} \vec {e} _i, \end{aligned}$$
(38)

with

$$\begin{aligned} B_k^{\;\; i} = \vec {g} _k \cdot \vec {e} ^{\, i} = \vec {g} _k \cdot \vec {e} _i, \end{aligned}$$
(39)

is defined by the coordinates of the four points \(\mathcal {P}_k\), \(k=0,\ldots ,3\), measured by the DIC-system. Here, \(\vec {e} ^{\, i} = \vec {e} _i\) holds in the case of an orthonormal basis system. Thus,

$$\begin{aligned} B_k^{\;\; i} = x_{ik} - x_{i0} \end{aligned}$$
(40)

can directly be calculated. Furthermore, the TG-basis vectors are defined by

$$\begin{aligned} \vec {e} _i^{\,\Theta }= (\vec {e} _i^{\,\Theta }\cdot \vec {g} ^{\, j}) \vec {g} _j = A_i^{\;\; j} \vec {g} _j, \quad A_i^{\;\; j} = \vec {e} _i^{\,\Theta }\cdot \vec {g} ^{\, j}. \end{aligned}$$
(41)

Thus, the scalar product of the basis vectors \(\vec {e} _i^{\,\Theta }\) with the contravariant basis vectors \(\vec {g} ^{ \ j}\) defined by \(\vec {g} _i \cdot \vec {g} ^{\, j} = \delta _i^{\;\; j}\) has to be calculated (\(\delta _i^{\;\; j}\) defines the Kronecker-symbol, \(\delta _i^{\;\; j} = 1\) for \(i=j\) and \(\delta _i^{\;\; j} = 0\) for \(i \ne j\)). For this purpose, the contravariant basis vectors \(\vec {g} ^{\, j}\) – expressed with respect to the basis vectors \(\vec {e} _i^{\,\Theta }\) – are required. First, the covariant basis vectors \(\vec {g} _k\) can also be represented by coordinates in the TG-system

$$\begin{aligned} \vec {g} _k = \vec {r} _k^{\,\Theta }- \vec {r} _0^{\,\Theta }= (x_{ik}^{\,\Theta }- x_{i0}^{\,\Theta }) \vec {e} _i^{\,\Theta }= \Delta x_{ik}^{\,\Theta }\vec {e} _i^{\,\Theta }= \Delta \vec {x} _{k0}^{\,\Theta }- \Delta d_{k0} \vec {e} _3^{\,\Theta }, \end{aligned}$$
(42)

with

$$\begin{aligned} \Delta x_{ik}^{\,\Theta }= x_{ik}^{\,\Theta }- x_{i0}^{\,\Theta }, \quad \Delta x_{ik}^{\,\Theta }= {\left\{ \begin{array}{ll} x_{1k}^{\,\Theta }- x_{10}^{\,\Theta }&{} \text {for} \; i=1 \\ x_{2k}^{\,\Theta }- x_{20}^{\,\Theta }&{} \text {for} \; i=2 \\ -\Delta d_{k0} &{} \text {for} \; i=3 \end{array}\right. } \end{aligned}$$
(43)

see equations (15) and (24). The contravariant basis vectors \(\vec {g} ^{\, j}\) can be calculated by

$$\begin{aligned} \vec {g} ^{\, 1} = \frac{\displaystyle 1}{\displaystyle g}(\vec {g} _2 \times \vec {g} _3), \quad \vec {g} ^{\, 2} = \frac{\displaystyle 1}{\displaystyle g}(\vec {g} _3 \times \vec {g} _1), \quad \vec {g} ^{\, 3} = \frac{\displaystyle 1}{\displaystyle g}(\vec {g} _1 \times \vec {g} _2), \end{aligned}$$
(44)

see [22], with

$$\begin{aligned} g = \vec {g} _1 \cdot (\vec {g} _2 \times \vec {g} _3). \end{aligned}$$
(45)

The evaluation of g is done using the representation of \(\vec {g} _k\) in equation (38). Thus, it is a known quantity. The calculation of the cross-products in equation (44) yields:

$$\begin{aligned} \begin{aligned} \vec {g} _2\times \vec {g} _3&= (\Delta x_{22}^{\,\Theta }\Delta x_{33}^{\,\Theta }- \Delta x_{32}^{\,\Theta }\Delta x_{23}^{\,\Theta }) \vec {e} _1^{\,\Theta }+ (\Delta x_{32}^{\,\Theta }\Delta x_{13}^{\,\Theta }- \Delta x_{12}^{\,\Theta }\Delta x_{33}^{\,\Theta }) \vec {e} _2^{\,\Theta }\\&\quad+ (\Delta x_{12}^{\,\Theta }\Delta x_{23}^{\,\Theta }- \Delta x_{22}^{\,\Theta }\Delta x_{13}^{\,\Theta }) \vec {e} _3^{\,\Theta } \end{aligned} \end{aligned}$$
(46)
$$\begin{aligned} \begin{aligned} \vec {g} _3\times \vec {g} _1&= (\Delta x_{23}^{\,\Theta }\Delta x_{31}^{\,\Theta }- \Delta x_{33}^{\,\Theta }\Delta x_{21}^{\,\Theta }) \vec {e} _1^{\,\Theta }+ (\Delta x_{33}^{\,\Theta }\Delta x_{11}^{\,\Theta }- \Delta x_{13}^{\,\Theta }\Delta x_{31}^{\,\Theta }) \vec {e} _2^{\,\Theta }\\&\quad+ (\Delta x_{13}^{\,\Theta }\Delta x_{21}^{\,\Theta }- \Delta x_{23}^{\,\Theta }\Delta x_{11}^{\,\Theta }) \vec {e} _3^{\,\Theta }\end{aligned} \end{aligned}$$
(47)
$$\begin{aligned} \begin{aligned} \vec {g} _1\times \vec {g} _2&= (\Delta x_{21}^{\,\Theta }\Delta x_{32}^{\,\Theta }- \Delta x_{31}^{\,\Theta }\Delta x_{22}^{\,\Theta }) \vec {e} _1^{\,\Theta }+ (\Delta x_{31}^{\,\Theta }\Delta x_{12}^{\,\Theta }- \Delta x_{11}^{\,\Theta }\Delta x_{32}^{\,\Theta }) \vec {e} _2^{\,\Theta }\\&\quad+ (\Delta x_{11}^{\,\Theta }\Delta x_{22}^{\,\Theta }- \Delta x_{21}^{\,\Theta }\Delta x_{12}^{\,\Theta }) \vec {e} _3^{\,\Theta } \end{aligned} \end{aligned}$$
(48)

Since the contravariant basis vectors \(\vec {g} ^{\, j}\) obviously depend on the unknown differences \(\Delta d_{k0}\), see equation (43), we follow the procedure in “Rotation based on orthogonal basis”, and formulate the scalar products

$$\begin{aligned} \begin{aligned} \vec {g} _1 \cdot \vec {g} _2&= (\vec {r} _1 - \vec {r} _0) \cdot (\vec {r} _2 - \vec {r} _0) =: g_{12}\\&= (\vec {r} _1^{\,\Theta }- \vec {r} _0^{\,\Theta }) \cdot (\vec {r} _2^{\,\Theta }- \vec {r} _0^{\,\Theta }) = { \Delta \vec {x} _{20}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }+ \Delta d_{20} \Delta d_{10}, } \end{aligned} \end{aligned}$$
(49)

i.e. one arrives at the modified equation

$$\begin{aligned} { (\Delta \vec {x} _{20}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }- g_{12}) + \Delta d_{20} \Delta d_{10} } = 0, \end{aligned}$$
(50)

compare equation (30). Analogously, the non-linear equations

$$\begin{aligned} { (\Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }- g_{31}) + \Delta d_{30} \Delta d_{10} }&= 0, \end{aligned}$$
(51)
$$\begin{aligned} { (\Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{20}^{\,\Theta }- g_{32}) + \Delta d_{30} \Delta d_{20} }&= 0 \end{aligned}$$
(52)

follow. Again, this leads to

$$\begin{aligned} \Delta d_{10} = i \sqrt{\frac{ab}{c}}, \quad \Delta d_{20} = i \sqrt{\frac{ac}{b}}, \quad \Delta d_{30} = i \sqrt{\frac{bc}{a}}, \end{aligned}$$
(53)

however, with

$$\begin{aligned}& a := \Delta \vec {x} _{20}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }- g_{12} \ne 0, \quad b := \Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{10}^{\,\Theta }- g_{31} \ne 0, \\& c := \Delta \vec {x} _{30}^{\,\Theta }\cdot \Delta \vec {x} _{20}^{\,\Theta }- g_{32} \ne 0. \end{aligned}$$
(54)

Once more, the assumption is that \(abc < 0\), and that \(d_0 > d_k\), \(k=1,2,3\), should be satisfied.

For \(A_i^{\;\; j} = \vec {e} _i^{\,\Theta }\cdot \vec {g} ^{\, j}\), see equation (41), the coefficients

$$\begin{aligned} \begin{aligned} A_1^{\;\; 1}&= \frac{\Delta x_{22}^{\,\Theta }\Delta x_{33}^{\,\Theta }- \Delta x_{32}^{\,\Theta }\Delta x_{23}^{\,\Theta }}{g},&\quad A_2^{\;\; 1}&= \frac{\Delta x_{32}^{\,\Theta }\Delta x_{13}^{\,\Theta }- \Delta x_{12}^{\,\Theta }\Delta x_{33}^{\,\Theta }}{g},&\quad A_3^{\;\; 1}&= \frac{\Delta x_{12}^{\,\Theta }\Delta x_{23}^{\,\Theta }- \Delta x_{22}^{\,\Theta }\Delta x_{13}^{\,\Theta }}{g}\\ A_1^{\;\; 2}&= \frac{\Delta x_{23}^{\,\Theta }\Delta x_{31}^{\,\Theta }- \Delta x_{33}^{\,\Theta }\Delta x_{21}^{\,\Theta }}{g},&\quad A_2^{\;\; 2}&= \frac{\Delta x_{33}^{\,\Theta }\Delta x_{11}^{\,\Theta }- \Delta x_{13}^{\,\Theta }\Delta x_{31}^{\,\Theta }}{g},&\quad A_3^{\;\; 2}&= \frac{\Delta x_{13}^{\,\Theta }\Delta x_{21}^{\,\Theta }- \Delta x_{23}^{\,\Theta }\Delta x_{11}^{\,\Theta }}{g}\\ A_1^{\;\; 3}&= \frac{\Delta x_{21}^{\,\Theta }\Delta x_{32}^{\,\Theta }- \Delta x_{31}^{\,\Theta }\Delta x_{22}^{\,\Theta }}{g},&\quad A_2^{\;\; 3}&= \frac{\Delta x_{31}^{\,\Theta }\Delta x_{12}^{\,\Theta }- \Delta x_{11}^{\,\Theta }\Delta x_{32}^{\,\Theta }}{g},&\quad A_3^{\;\; 3}&= \frac{\Delta x_{11}^{\,\Theta }\Delta x_{22}^{\,\Theta }- \Delta x_{21}^{\,\Theta }\Delta x_{12}^{\,\Theta }}{g} \end{aligned} \end{aligned}$$
(55)

result by expressions (46)–(48). Insertion of equation (38), i.e. the coefficients (40), into equation (41) yields the basis system of the TG-system

$$\begin{aligned} \vec {e} _i^{\,\Theta }= A_i^{\;\; k} B_k^{\;\; j} \vec {e} _j = Q_{ji} \vec {e} _j. \end{aligned}$$
(56)

Here, the product

$$\begin{aligned} Q_{ji} = A_i^{\;\; k} B_k^{\;\; j} \end{aligned}$$
(57)

represents again orthogonal matrix components. This can be proven as follows:

$$\begin{aligned} A_i^{\;\; k} B_k^{\;\; j} = (\vec {e} _i^{\,\Theta }\cdot \vec {g} ^{\, k})(\vec {g} _k \cdot \vec {e} _j) = \vec {e} _i^{\,\Theta }\cdot \underbrace{(\vec {g} ^{\, k} \otimes \vec {g} _k)}_{\mathrm {\mathbf {I}} } \vec {e} _j = \vec {e} _i^{\,\Theta }\cdot \vec {e} _j = Q_{ji} \end{aligned}$$
(58)

Thus, again the rotation of the DIC- and TG-basis systems is provided. From the theoretical consideration, the matrix \(\boldsymbol{\mathsf{Q}}=[Q_{ij}]\) represents an orthogonal matrix with the properties \(\det \boldsymbol {\mathsf {Q}} = 1\) and \({\boldsymbol{\mathsf {Q}}}^T = { {\boldsymbol {\mathsf{Q}}}}^{-1}\). In real applications, the entries \(A_i^{\;\; k}\) and \(B_k^{\;\; j}\) are affected by measurement errors. Thus, these conditions can not be guaranteed anymore. This has to be considered later on.

Translation of TG-system relative to DIC-system

The remaining quantity is the vector \(\vec {r} _{\Theta }\) in equation (20). If one distance, e.g. \(d_1\) is known, the vector \(\vec {r} _{\Theta }\) can be calculated by

$$\begin{aligned} \vec {r} _{\Theta }= x_{i \Theta } \vec {e} _i = \vec {r} _1 - \vec {r} _1^{\,\Theta }= x_{i1} \vec {e} _i - x_{i1}^{\,\Theta }\vec {e} _i^{\,\Theta }= (x_{i1} - Q_{ij} x_{j1}^{\,\Theta }) \vec {e} _i. \end{aligned}$$
(59)

Thus, we have to follow the question of how to determine the (orthogonal) distance of a point of the tripod. This is done using a laser distance measurement tool, see Fig. 2. Before performing the experiments, the distance \(d_1\) between the tripod’s surface and the anterior edge of the lens has to be determined. Therefore, we drew on a distance measurement with a laser. The laser beam was orthogonally deflected by means of a mirror, which was aligned in 45° (grey block in Fig. 2). However, the measured distance (red line in Fig. 2(a)) was not equal to the required distance \(d_1\). Since both, the thermography camera as well as the laser, were fixed on the distance measurement tool, an adjustment value was determined with reference measurements where \(d_1\) was known. With this adjustment value the sought distance \(d_1\) was calculated from the measured distances before the experiments. The tripod, which was applied as calibration object in our measurements, is shown in Fig. 3. We applied adhesive dot markers (from Carl Zeiss GOM Metrology GmbH, Brunswick, Germany) with an outer diameter 3.5 mm, which are regularly used in applications with digital image correlation systems, to identify the calibration object in the thermogram from the thermography camera as well as in the digital image correlation image. The markers were visible in the thermogram since they have a significantly higher emissivity compared to the polished surface of the tripod, which was made from a brass alloy. The difference in the emissivity allows the identification of the center points of the markers with the Prewitt edge detection in Matlab, even at room temperature. So, no additional heating of the calibration object is necessary. Preventively, seven markers are employed, however, only four points are necessary as tripod, see “Geometry”, where one point – the origin – has to be in the indentation of the tripod.

Fig. 2
figure 2

Distance measurement tool (side view)

Full size image
Fig. 3
figure 3

Calibration object applied as tripod

Full size image

Surface temperature calculation

In the following, it is assumed that the surface of an object is described by the function \(\vec {r} = \hat{\vec { r}} (\Psi ^1,\Psi ^2) = \hat{\vec { r}} (\varvec{\Psi })\), where \(\Psi ^1\) and \(\Psi ^2\) are the surface parameters, \(\varvec{\Psi }= \{\Psi ^1,\Psi ^2\}\). Then, using equation (20), see equation (59) as well, for an arbitrary point on the surface, see Fig. 4, one obtains,

$$\begin{aligned} {\hat{\vec { r}} (\varvec{\Psi })} = \hat{x}_i(\Psi ^1,\Psi ^2) \vec {e} _i = \vec {r} _{\Theta }+ \vec {r} ^{\,\Theta }= (x_{i\Theta } + Q_{ij} x_j^{\,\Theta }) \vec {e} _i, \end{aligned}$$
(60)

or the inverse relation

$$\begin{aligned} {\hat{\vec { r}} ^{\,\Theta }(\varvec{\Psi })} = \hat{x}_i^{\,\Theta }(\Psi ^1,\Psi ^2) \vec {e} _i^{\,\Theta }= \hat{\vec { r}} (\varvec{\Psi }) - \vec {r} _{\Theta }= Q_{ji} \big (\hat{x}_j(\Psi ^1,\Psi ^2) - x_{j\Theta }\big ) \vec {e} _i^{\,\Theta }. \end{aligned}$$
(61)

In “Rotation based on oblique basis” and “Translation of TG-system relative to DIC-system” the vector \(\vec {r} _{\Theta }\) and the rotation matrix \(\boldsymbol {\mathsf {Q}} = [Q_{ij}]\) are determined, and, accordingly, assumed to be known in the following. Furthermore, the surface representation \(\hat{\vec { r}} (\varvec{\Psi })\) using the DIC-system is known. If the components in equation (60) are assembled in column vectors, equation (60) reads

$$\begin{aligned} {\hat{ {\boldsymbol {\mathsf{r}}}}(\varvec{\Psi })} = \boldsymbol {\mathsf{r}}_{\Theta }+ \boldsymbol {\mathsf {Q}} \boldsymbol {\mathsf{r}}^{\,\Theta }, \quad \text {with} \quad \hat{ {\boldsymbol{\mathsf {r}}}}(\varvec{\Psi }) = \left\{ \begin{array}{c} \displaystyle \hat{x}_1(\varvec{\Psi }) \\ \displaystyle \hat{x}_2(\varvec{\Psi }) \\ \displaystyle \hat{x}_3(\varvec{\Psi }) \end{array} \right\} , \; \boldsymbol {\mathsf{r}}_{\Theta }= \left\{ \begin{array}{c} \displaystyle x_{1\Theta } \\ \displaystyle x_{2\Theta } \\ \displaystyle x_{3\Theta } \end{array} \right\} , \end{aligned}$$
(62)

or, vice versa,

$$\begin{aligned} \boldsymbol {\mathsf{r}}^{\,\Theta }= \hat{ {\boldsymbol {\mathsf{r}}}}^{\,\Theta }(\varvec{\Psi }) = { {\boldsymbol {\mathsf{Q}}}}^{-1} \{{\hat{ {\boldsymbol {\mathsf{r}}}}(\varvec{\Psi })} - \boldsymbol {\mathsf{r}}_{\Theta }\}. \end{aligned}$$
(63)

Here, the inverse is taken instead of the transposed since the matrix \(\boldsymbol {\mathsf {Q}}\) is subjected to measurement errors so that the property of a orthogonal matrix is violated. This inverse is called \(\boldsymbol {\mathsf {S}} := { {\boldsymbol{\mathsf {Q}}}}^{-1}\) in the following. Since the temperature of the TG-system \(\Theta = \hat{\Theta }(x_1^{\,\Theta },x_2^{\,\Theta })\) depends only on the two coordinates \(\boldsymbol{\mathsf{x}}^{\Theta}= \{x_1^{\Theta},x_2^{\Theta}\}\), i.e. \(\hat{\Theta }(\boldsymbol{\mathsf{x}}^{\,\Theta })\), equation (63) has to be reconsidered. With

$$\begin{aligned} \boldsymbol {\mathsf{r}}^{\,\Theta }= \left\{ \begin{array}{c} \boldsymbol{\mathsf {x}}^{\,\Theta } \ \\ x_3^{\,\Theta } \end{array} \right\} , \end{aligned}$$
(64)

the vector \(\boldsymbol {\mathsf{x}}^{\,\Theta }\) reads

$$\begin{aligned} \boldsymbol {\mathsf{x}}^{\,\Theta }= \hat{{\boldsymbol{\mathsf {x}}}}^{\,\Theta }(\varvec{\Psi })&= \underbrace{\big [\boldsymbol {\mathsf {I}}_{(2 \times 2)} \boldsymbol{\mathsf {0}}_{(2\times 1)}\big ] \boldsymbol {\mathsf {S}}}_{\boldsymbol {\mathsf {M}}} \{{\hat{ {\boldsymbol{\mathsf {r}}}}(\varvec{\Psi })} - \boldsymbol{\mathsf {r}}_{\Theta }\} = \boldsymbol {\mathsf {M}} \{\hat{ {\boldsymbol{\mathsf {r}}}}(\varvec{\Psi }) - \boldsymbol {\mathsf{r}}_{\Theta }\} = \end{aligned}$$
(65)
$$\begin{aligned}&= \left\{ {\begin{array}{*{20}{c}}{x_1^\Theta } \\ {x_2^\Theta } \end{array}} \right\} = \underbrace{ { \begin{bmatrix} S_{11} &{} S_{12} &{} S_{13} \\ S_{21} &{} S_{22} &{} S_{23} \end{bmatrix} } }_{\boldsymbol {\mathsf {M}}} \left\{ \begin{array}{*{1}{c}} \displaystyle \hat{x}_1(\Psi ^1,\Psi ^2) - x_{1\Theta } \\ \displaystyle \hat{x}_2(\Psi ^1,\Psi ^2) - x_{2\Theta } \\ \displaystyle \hat{x}_3(\Psi ^1,\Psi ^2) - x_{3\Theta } \end{array} \right\} . \end{aligned}$$
(66)

In other words, we have

$$\begin{aligned} \Theta = \hat{\Theta }( \boldsymbol {\mathsf{x}}^{\,\Theta }) = \hat{\Theta }(\hat{ {\boldsymbol{\mathsf {x}}}}^{\,\Theta }(\varvec{\Psi })) = {\hat{\Theta }(\overline{ \boldsymbol {\mathsf{x}}}^{\,\Theta }(\hat{ {\boldsymbol {\mathsf{r}}}}(\varvec{\Psi })))} =: \overline{\Theta }(\varvec{\Psi }). \end{aligned}$$
(67)
Fig. 4
figure 4

Surface description relative to different basis systems

Full size image

Several aspects have to be discussed. First, the temperature is given pixel-wise, i.e. with discrete values. To reduce noisy data – and for the temperature gradient determination – a regression analysis can be carried out if an appropriate function \(\hat{\Theta }( \boldsymbol{\mathsf {x}}^{\,\Theta })\) is found. Second, it is only possible to consider points (intersection) which are in the TG-domain and in the DIC-domain.

If one prescribes \(\varvec{\Psi }\), \(\boldsymbol{\mathsf {x}}^{\,\Theta }\) is directly known by equation (65), and, accordingly, the temperature \(\Theta\) can be evaluated either on pixel level or as an interpolated function. If one prescribes \(\boldsymbol{\mathsf{x}}^{\,\Theta }\), then equation (65) is a non-linear function to obtain a \(\varvec{\Psi }\) concerned. Thus, it is easier to prescribe a background (evaluation) grid for \(\varvec{\Psi }\) in the DIC-surface description to evaluate the surface temperature. The drawback is that one has to check whether a real surface point \(\varvec{\Psi }\) exists on the projected surface of the specimen in the TG-image (this holds vice versa as well).

Finally, the in-plane temperature gradient (11) or (13) has to be calculated. Thus, the tangent vectors \(\vec {a} _\alpha\), \(\alpha =1,2\), are required, which can be determined by the DIC-surface description

$$\begin{aligned} \vec {a} _\alpha = \frac{\displaystyle \partial \hat{\vec { r}} (\Psi ^1,\Psi ^2)}{\displaystyle \partial \Psi ^\alpha }. \end{aligned}$$
(68)

Later on, radial basis functions are exploited to have the surface description, see [18]. The contravariant (gradient) vectors (9) are calculated by means of the inverse covariant matrix of metric coefficients (8). Thus, the derivative \(\Theta ,_\alpha\) is required. Using representation (67),

$$\begin{aligned} \Theta ,_\alpha = \frac{\displaystyle \partial \overline{\Theta }(\Psi ^1,\Psi ^2)}{\displaystyle \partial \Psi ^\alpha } = \frac{\displaystyle \partial \hat{\Theta }}{\displaystyle \partial x_\beta ^{\,\Theta }} \frac{\displaystyle \partial \hat{x}_\beta ^{\,\Theta }}{\displaystyle \partial x_j} \frac{\displaystyle \partial \hat{x}_j}{\displaystyle \partial \Psi ^\alpha } \end{aligned}$$
(69)

follows. The first term can be provided by interpolated temperature data in the coordinate space of TG-data. The second term is given by

$$\begin{aligned} \frac{\displaystyle \partial \hat{x}_\beta ^{\,\Theta }}{\displaystyle \partial x_j} = M_{\beta j} = {S_{\beta j}}, \end{aligned}$$
(70)

and the third term can be provided by differentiating the surface interpolation with respect to the surface coordinates.

Temperature Distribution using Radial Basis Functions

For deriving the temperature gradient in curved surfaces of components, the geometrical description of the surface is necessary. In three-dimensional digital image correlation systems, there are several possible approaches to describe the surface. It is assumed that \({n_{\text {D}}}\) discrete points \(\boldsymbol {\mathsf {d}}_i {\, \in \mathbb {R}}^{3}\), \(i=1,\ldots ,{n_{\text {D}}}\), are given. One possibility makes use of a triangulation concept, see [17, 20, 34], where all experimental points are interpolated. The drawback is that the interpolation function is continuous but not continuous differentiable, which lead to problems for the gradient determination at the interfaces between triangles. Thus, the interpolation function

$$\begin{aligned} \hat{{\boldsymbol {\mathsf {x}}}}(\varvec{\Psi }) = \sum _{j=1}^3 \left( \sum _{k=1}^{n_{\text {cp}}}\hat{m}(\hat{\rho }(\varvec{\Psi },\varvec{\Psi }_k)) B_{kj} + \sum _{l=1}^3 \hat{n}_l(\varvec{\Psi }) D_{lj}\right) \boldsymbol {\mathsf {e}}_j \end{aligned}$$
(71)

is chosen consisting of a monomial extension

$$\begin{aligned} \hat{{\boldsymbol {\mathsf {n}}}}(\varvec{\Psi }) = \left\{ \begin{array}{c} \hat{n}_1(\varvec{\Psi }) \\ \hat{n}_2(\varvec{\Psi }) \\ \hat{n}_3(\varvec{\Psi }) \end{array} \right\} = \left\{ \begin{array}{c} 1 \\ \Psi ^1 \\ \Psi ^2 \end{array} \right\} \end{aligned}$$
(72)

and the radial basis functions \(\hat{m}(\varvec{\Psi },\varvec{\Psi }_k)\), see proposal in [18]. There are a number of possible radial basis functions \(\hat{m}(\varvec{\Psi },\varvec{\Psi }_k)\) available. We choose so-called inverse multi-quadrics

$$\begin{aligned} \hat{m}(\rho ) = \frac{\displaystyle c_I}{\displaystyle \sqrt{\rho ^2 + c_I^2}}, \quad \text {with} \quad \hat{m}'(\rho ) = -\frac{\displaystyle \rho c_I}{\displaystyle (\rho ^2 + c_I^2)^{3/2}}, \end{aligned}$$
(73)

see [2, 6], with the surface parameters \(\varvec{\Psi }{\, \in \mathbb {R}}^{2}\), \(\varvec{\Psi }= \{\Psi ^1,\Psi ^2\} = \{x,y\}\), and the normalized distance function

$$\begin{aligned} \rho _k = \hat{\rho }(\varvec{\Psi },\varvec{\Psi }_k) = \frac{R}{R_0} = \frac{\Vert \varvec{\Psi }- \varvec{\Psi }_k\Vert }{R_0} = \frac{\sqrt{(x-x_k)^2 + (y-y_k)^2}}{R_0}. \end{aligned}$$
(74)

Later on, \(R_0 = {1\,\mathrm{\text {m}\text {m}}}\) is chosen and the shape factor \(c_I\) depends on the application, see examples. The distance function is evaluated at the center (source) points \(\varvec{\Psi }_k = \{\Psi _k^1,\Psi _k^2\} = \{x_k^{\text {cp}},y_k^{\text {cp}}\}\), \(k=1,\ldots ,{n_{\text {cp}}}\), of a possible background grid. There are two choices, (a) arbitrary center points (regression), or (b) center points, which are identical to the data points (interpolation). For the first case, the determination of the weights \(B_{kj}\) and \(D_{lj}\) is done by a linear least-square approach, and for the interpolation concept a particular system of linear equations has to be evaluated. Here, the second case is chosen, where the data points are identical to the center points for both the temperature data as well as for the DIC-coordinates. The assembling procedure is explained in [18].

Error Propagation

The results of the temperature \(\Theta (\Psi ^1,\Psi ^2)\) and the temperature gradient (13) are – apart from the interpolation concept – influenced by a number of input quantities. These are (1) the real distance/pixel ratio p obtained by the TG-camera, the detection of the four points determined by both (2) the DIC-system \(\boldsymbol {\mathsf{r}}_k {\, \in \mathbb {R}}^{3}\) and (3) the thermography system \(\mathbf {\Upsilon }_{k}^{\Theta }{\, \in \mathbb {R}}^{2}\), \(k=0,\ldots 3\), (4) the distance measure d of the TG-camera to one point of the tripod, and, of course, each individual temperature (pixel) information measured by the TG-system and the geometrical coordinates determined by the DIC-system. The latter two information are not considered in the following, because we are interested in the uncertainties of the approach presented here (i.e. items (1)–(4)).

As thermography cameras only provide pixel information, the projected physical tripod coordinates in the thermography image have to be calculated. Therefore, we use the length-to-pixel ratio p and the pixel information containing the locations of the tripod points, \(\mathbf {\Upsilon }_{k}^{\Theta }\) (pixel coordinates), \(k=1,\ldots ,4\),

$$\begin{aligned} \mathbf {\mathsf{x}}_{k}^{\Theta } = p\mathbf {\Upsilon }_{k}^{\Theta } = p \begin{Bmatrix} \Upsilon _{1k}^{\Theta } \\ \Upsilon _{2k}^{\Theta } \end{Bmatrix}. \end{aligned}$$
(75)

With respect to the determination of the uncertainties of a resulting quantity \(F( \boldsymbol{\mathsf {q}})\) due to the uncertainty of the input quantities \(\boldsymbol{\mathsf {q}}\), the concept of Gaussian error propagation is used in the following. It is assumed that \(F( \boldsymbol{\mathsf {q}})\) is a scalar-valued quantity, e.g. temperature or norm of temperature gradient, and \(\boldsymbol{\mathsf {q}} {\, \in \mathbb {R}}^{{n_{\text {p}}}}\) are the influencing quantities,

$$\begin{aligned} \boldsymbol{\mathsf {q}} = \hat{{ \boldsymbol{\mathsf {q}}}}(p, \mathbf {\Upsilon }_1^{\,\Theta }, \mathbf {\Upsilon }_2^{\,\Theta }, \mathbf {\Upsilon }_3^{\,\Theta }, \mathbf {\Upsilon }_4^{\,\Theta }, \boldsymbol {\mathsf {r}}_1, \boldsymbol {\mathsf{r}}_2, \boldsymbol {\mathsf{r}}_3, \boldsymbol{\mathsf {r}}_4, d). \end{aligned}$$
(76)

The remaining variables are

$$\begin{aligned} \boldsymbol {\mathsf{r}}_k = \left\{ \begin{matrix} x_{1k} \\ x_{2k} \\ x_{3k} \end{matrix} \right\} , \quad k=1,\ldots ,4, \end{aligned}$$
(77)

the coordinates of the four tripod-points in the DIC-system, and d the (orthogonal) distance between the TG-camera and one of the points of the tripod, which was measured with the tool described in “Translation of TG-system relative to DIC-system”. In other words, there are \({n_{\text {p}}}= 22\) influencing quantities.

The uncertainty \(\Delta F\) of a given function \(F(\boldsymbol {\mathsf{q}})\) can be estimated by the Gaussian error propagation concept, see [5, 40]. This concept can also be motivated by the Taylor-series

$$\begin{aligned} F(\boldsymbol{\mathsf {q}} + \Delta \boldsymbol{\mathsf {q}}) = F(\boldsymbol{\mathsf {q}}) + \mathsf {D}_{\boldsymbol {\mathsf {q}}}\,F( \boldsymbol{\mathsf {q}})[\Delta \boldsymbol {\mathsf{q}}] + \Vert \Delta \boldsymbol{\mathsf {q}}\Vert R( \boldsymbol {\mathsf{q}},\Delta \boldsymbol{\mathsf {q}}), \end{aligned}$$
(78)

where R defines the remainder, which is frequently assumed to be small, and the second term is interpreted as the differential. The differential can be calculated by the Gâteaux derivative (or directional derivative),

$$\begin{aligned} \mathsf {D}_ {\boldsymbol{\mathsf {q}}}\,F( \boldsymbol {\mathsf{q}})[\Delta \boldsymbol{\mathsf {q}}] = \left. \frac{ \mathrm d}{ \mathrm d\lambda } F\left( \boldsymbol {\mathsf{q}}+\lambda \Delta \boldsymbol {\mathsf{q}}\right) \right| _{\lambda =0} = \left\{ \frac{ \mathrm dF}{ \mathrm d \boldsymbol {\mathsf{q}}}\right\} ^T \Delta \boldsymbol {\mathsf{q}} = \sum _{i=1}^{n_{\text {p}}}\frac{ \partial F}{\partial q_i} \Delta q_i. \end{aligned}$$
(79)

The norm of the Gâteaux derivative represents the uncertainty \(\Delta F\) of F, i.e. the confidence interval \(F \pm \Delta F\), with

$$\begin{aligned} \Delta F = \Vert \mathsf {D}_ {\boldsymbol{\mathsf {q}}}\,F( \boldsymbol {\mathsf{q}})[\Delta \boldsymbol {\mathsf{q}}]\Vert& = \sqrt{ \Delta {\boldsymbol{\mathsf {q}}^T} \left\{ \frac{ \mathrm dF}{ \mathrm d \boldsymbol {\mathsf{q}}}\right\} \left\{ \frac{ \mathrm dF}{ \mathrm d \boldsymbol {\mathsf{q}}}\right\} ^T \Delta \boldsymbol {\mathsf{q}}}\\& = \sqrt{\sum _{i=1}^{n_{\text {p}}}\sum _{j=1}^{n_{\text {p}}}\frac{ \partial F}{ \partial q_i}\frac{ \partial F}{ \partial q_j} \Delta q_i \Delta q_j}. \end{aligned}$$
(80)

Through application of the norm, the Gaussian error propagation concept takes into account that some of the errors may compensate each other to some extent. In contrast, the linear error propagation concept, which can be seen in equation (79) in case the modulus of the partial derivatives is taken, gives the maximum expectable error [40]. Further, in equation (80) it is considered that the influencing parameters may not be independent from each other. Otherwise, the off-diagonal elements of the matrix

$$\begin{aligned} \left\{ \frac{\displaystyle \mathrm dF}{ \mathrm d\boldsymbol {\mathsf {q}}}\right\} \left\{ \frac{ \mathrm dF}{ \mathrm d \boldsymbol {\mathsf{q}}}\right\} ^T = \left[ \frac{ \partial F}{ \partial q_i}\frac{ \partial F}{ \partial q_j}\right] \end{aligned}$$
(81)

vanish if the measurements of the \(q_i\)’s are independent. Later on, these derivatives are computed using numerical differentiation. Furthermore, the entries of \(\Delta \boldsymbol {\mathsf{q}}\), \(\Delta \boldsymbol{\mathsf {q}} {\, \in \mathbb {R}}^{{n_{\text {p}}}}\), must be specified. Therefore, commonly the standard deviations of the values (76) are chosen. Alternatively, motivated deviations can be selected to gain insight into the influences of the individual measured variables on the sought quantity F.

Verification Example

To verify the aforementioned method, an analytical example is studied initially. In Fig. 5 the geometrical setting is shown, i.e. the coordinate system of the DIC-system, the tripod coordinates, and the 2D-coordinate system of the TG-system. First, the schemes to identify the relation between the TG- and the DIC-basis systems, and the temperature distribution on the surface are computed, where also the temperature gradient is provided. Second, the sensitivity of the entire scheme due to uncertainties in the aforementioned quantities (76) is studied using the Gaussian error propagation concept.

Fig. 5
figure 5

Geometrical conditions of the verification example (\(\vartheta = {-60\,\mathrm{{}^{\circ }}}\), \(r = {20\,\mathrm{\text {m}\text {m}}}\), \(a = {800\,\mathrm{\text {m}\text {m}}}\), \(b = {70\,\mathrm{\text {m}\text {m}}}\), \(c_1 = {500/3\,\mathrm{\text {m}\text {m}}}\), \(c_2 = {700/9\,\mathrm{\text {m}\text {m}}}\), \(h = {100\,\mathrm{\text {m}\text {m}}}\), \(\varphi _{\text {min}}= -{45\,\mathrm{{}^{\circ }}}\), \(\varphi _{\text {max}}= {45\,\mathrm{{}^{\circ }}}\), \(z_1 = {20\,\mathrm{\text {m}\text {m}}}\), \(z_2 = {60\,\mathrm{\text {m}\text {m}}}\), \(z_3 = {20\,\mathrm{\text {m}\text {m}}}\)) and generated geometrical data points

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Tripod and Temperature Measurement

First, the four points of the tripod are defined (units are neglected here),

$$\begin{aligned} \begin{aligned} \vec {r} _0&= -b \vec {e} _1 - a \vec {e} _3 = -70 \vec {e} _1 - 800 \vec {e} _3, \\ \vec {r} _1&= \vec {r} _0 + \gamma (-\vec {e} _1 + \vec {e} _3) = -115 \vec {e} _1 - 755 \vec {e} _3, \\ \vec {r} _2&= \vec {r} _0 + \gamma \left( \frac{7}{10}\vec {e} _1 + \frac{1}{2}\vec {e} _2 + \vec {e} _3\right) = -38.5 \vec {e} _1 + 22.5 \vec {e} _2 - 755 \vec {e} _3, \\ \vec {r} _3&= \vec {r} _0 + \gamma \left( \vec {e} _2 + \frac{1}{2}\vec {e} _3\right) = -70 \vec {e} _1 + 45 \vec {e} _2 - 777.5 \vec {e} _3 \end{aligned} \end{aligned}$$
(82)

with \(\gamma = {45\,\mathrm{\text {m}\text {m}}}\). This yields \(\vec {r} _{\Theta }= -\frac{500}{3} \vec {e} _1 + \frac{700}{9} \vec {e} _2\) and

$$\begin{aligned} \boldsymbol {\mathsf {Q}} = \begin{bmatrix} 0.500 &{} 0.866 &{} 0.000 \\ -0.866 &{} 0.500 &{} 0.000 \\ 0.000 &{} 0.000 &{} 1.000 \end{bmatrix} \end{aligned}$$

using the scheme in “Rotation based on oblique basis”, see equation (57), and “Translation of TG-system relative to DIC-system”, see equation (59).

Next, in the region \(\varphi \in [\varphi _{\text {min}},\varphi _{\text {max}}]\), \(\varphi _{\text {min}}= -{45\,\mathrm{{}^{\circ }}}\), \(\varphi _{\text {max}}= {45\,\mathrm{{}^{\circ }}}\) and \(z \in [z_{\text {min}},z_{\text {max}}]\) with \(z_{\text {min}}= z_1\)  and \(z_{\text {max}}= z_1+z_2\) DIC-points are generated on the surface of the cylinder with the radius \(r={20\,\mathrm{\text {m}\text {m}}}\). The point distance is \(\Delta z = 0.\overline{33}{\text {m}\text {m}}\) (180 points in z-direction) and 100 points in circumferential direction (so that approximately a point distance of 0.314 mm is obtained). This point cloud is chosen for interpolation using RBFs.

Additionally, a linear temperature field is assumed on the surface of the cylindrical specimen,

$$\begin{aligned} \Theta (\varphi ,z) = a_0 + a_1 \varphi + a_2 z \end{aligned}$$
(83)

with \(a_0 = {150\,\mathrm{{{}^{\circ }\text {C}}}}\), \(a_1 = \frac{{200\,\mathrm{{{}^{\circ }\text {C}}}}}{\pi }\), and \(a_2 = {1\,\mathrm{{{}^{\circ }\text {C}} \ \text {mm}^{-1}}}\). This yields with

$$\begin{aligned} \vec {a} _1&= r \vec {e} _\varphi&\quad \vec {a} ^{\, 1} = \frac{\displaystyle 1}{\displaystyle r} \vec {e} _\varphi \end{aligned}$$
(84)
$$\begin{aligned} \vec {a} _2&= \vec {e} _z&\quad \vec {a} ^{\, 2} = \vec {e} _z \end{aligned}$$
(85)

the temperature gradient

$$\begin{aligned} \widehat{{{\,\mathrm{grad}\,}}} \Theta (\varphi ,z) = a_1 \vec {a} ^{\, 1} + a_2 \vec {a} ^{\, 2} = \frac{a_1}{r} \vec {e} _\varphi + a_2 \vec {e} _z, \end{aligned}$$
(86)

see equation (11) (\((\Psi ^1,\Psi ^2) = (\varphi ,z)\)). This vector can be expressed relative to all other basis systems.

Next, the errors are estimated which are based on several approximations. The temperature distribution in the “TG-camera” is the projected temperature (83) onto the 2D-plane, see Fig. 6(a). A length-to-pixel ratio of \(p = {0.4444\, \text{mm px}^{-1}}\) (\({1\,\mathrm{px}} \; \hat{=} \; 0.4444 {\text {m}\text {m}}\)) is assumed. The small four bright dots on the left of the sample represent the coordinates of the tripod. Figure 6(b) is a detail of the Fig. 6(a), where the “linear” temperature distribution is shown.

Fig. 6
figure 6

Temperature in the TG-camera of the verification example

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On the basis of the DIC-data (geometry) and TG-data (temperature), the spatial temperature distribution (67) can be determined using the interpolation concept of “Temperature Distribution using Radial Basis Functions”. The RBF settings (inverse multi-quadrics) in equations (73) and (74) for the geometry and the temperature interpolation in the thermography image are \(R_0 = {1\,\mathrm{\text {m}\text {m}}}\), \(c_I = 0.9\) (DIC-data interpolation), and \(c_I = 1.5\) (TG-data interpolation). Furthermore, we assume that data points are equivalent to the center points.

Figure 7(a) shows the temperature distribution on the surface of the specimen. For a better visualization, the temperature is mapped onto the y/z-plane in Fig. 7(b). Moreover, the temperature gradient calculation (13) is possible at arbitrary points on the curved surface using equations (68)–(69), which is demonstrated in Fig. 7(a). The projected arrows of the temperature gradient are displayed in Fig. 7(b) as well. In this figure, it becomes more clear that the gradient has not only circumferential components, but also components in z-direction, which is clear from Fig. 7(a). Since the maximum temperature in these figures only goes up to 300 \({}^{\circ }\text {C}\), it should be noted that Fig. 6 show the entire sample and Fig. 7 show only the smaller section of the area measured by the DIC-system.

Fig. 7
figure 7

Spatial temperature and gradient distribution of verification example

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Since both, the geometry and the temperature data, are interpolated, there is the issue of accuracies in the temperature as well as, in particular, the temperature gradients that are determined. In [18] it is shown that close to the edges of the measured regions the error increases due to the interpolation concept. This might be the case for both the DIC-data as well as the TG-data. To find out these inaccuracies, we consider the temperature gradient \(\vec {h} := \widehat{{{\,\mathrm{grad}\,}}} \Theta\) (which is a vector). First, the angle \(\beta\) between the exact (analytical) \(\vec {h} _{\text {anal}}\) and the calculated temperature gradient \(\vec {h}\) is investigated,

$$\begin{aligned} \cos \beta := \frac{\vec {h} }{\Vert \vec {h} \Vert } \cdot \frac{\vec {h} _{\text {anal}}}{\Vert \vec {h} _{\text {anal}}\Vert }. \end{aligned}$$
(87)

Figure 8(a) shows that increased inaccuracies can occur towards the edges. These are on the one hand due to the fact that the RBFs have less information at the edges and, on the other hand, due to the effect that the data points were generated in cylindrical coordinates (different spatial point density distribution). When generating data points with constant distances in circumferential direction, the projection to Cartesian coordinates results in more data points at the edges. Another measure of inaccuracy is the difference in the norm of the gradient,

$$\begin{aligned} \epsilon _{\text {h}} := \frac{|\Vert \vec {h} \Vert - \Vert \vec {h} _{\text {anal}}\Vert |}{\Vert \vec {h} _{\text {anal}}\Vert }, \end{aligned}$$
(88)

which is shown in Fig. 8(b). The white region contains errors below \(10^{-6}\).

Fig. 8
figure 8

Accuracy estimation of gradient calculation for verification example

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Uncertainty Study using Error Propagation

Since the quantities in the parameter vector (76) are uncertain and this might have an essential influence on the embedding procedure of the two-dimensional temperature measurement in the three-dimensional space, the error propagation concept in “Error Propagation” is followed. For instance, the uncertainties in position and further quantities, e.g. displacements and strains, are of particular interest when employing DIC systems. Therefore, [36] proposed a pseudo-experimental approach. This approach was later on applied with some modifications to obtain the uncertainties of the DIC measurement while coupling DIC and TG, see [7]. Therein, the authors estimate the temperature uncertainty by accounting the inherent error through camera calibration, uncertainties in the emissivity and thermal noise.

Here, we are looking for the sensitivity of the beforehand proposed scheme by drawing on the verification example in Fig. 5. The quantities of interest in the following uncertainty study are the norm of the temperature gradient as well as the temperature itself at the locations of the DIC points. Thus, in accordance with “Error Propagation”, the investigation is done for

$$\begin{aligned} \boldsymbol {\mathsf{F}} = \left\{ \begin{matrix} \Vert \vec {h} _i\Vert \\ \Theta _i \end{matrix} \right\} , \quad i = 1,\ldots ,n_\mathrm {DIC}, \end{aligned}$$
(89)

where \(n_\mathrm {DIC}\) denotes the number of generated DIC-data points which means \(\mathbf {\mathsf{F}} \in \mathbb {R}^{2n_\mathrm {DIC}}\). The influencing quantities are given in the parameter vector (76), however, the entries of \(\Delta \boldsymbol{\mathsf {q}}\) have to be specified in the following.

The uncertainty \(\Delta p\) of the length-to-pixel ratio p is assumed to \(\Delta p = {0.0017} \ \mathrm{mm \ px}^{-1}\) (this stems from some rough estimation that we consider a TG-camera lens with a field of view \({32.4\,\mathrm{{}^{\circ }}} \times {24.6\,\mathrm{{}^{\circ }}}\) (taken from the experimental setup, see “Experimental Setup”) and the prescribed distance between TG-system and some specimen is approximately 782.93 mm ± 2.93 mm. The coordinates of the four tripod-points in the TG-system are set to have the identical uncertainty \(\Delta \Upsilon _{\alpha k}^{\,\Theta }= {1\,\mathrm{px}}, \alpha = 1,2,\,k = 1,\ldots ,4\). This uncertainty considers the deviations from application of the adhesive dot markers and the subsequent detection of the center of these dot markers in the thermogram. There, we do not distinguish separate uncertainties in the two directions in the thermogram. Further, the uncertainty of the four tripod-points in the DIC-system is required, we estimate \(\Delta d = {12}\ \mathrm {mm}\). Therefore, we took 100 pictures of the tripod at the same position and determined the coordinates of the dot markers with the DIC-system. Then, we took the standard deviations in all three coordinate directions as uncertainty. Again, we do not choose different uncertainties in the different directions for brevity. Finally, the uncertainty while measuring the distance d between TG-system and specimen is specified to \(\Delta d = {12}\ \mathrm {mm}\) since the manufacturer of the laser measurement tool gives an accuracy of 1.5% of the measured value.

The required derivatives of in equation (80) for each entry in the parameter vector (89) are computed numerically by means of the forward differential method,

$$\begin{aligned} \frac{ \mathrm d \boldsymbol {\mathsf{F}}( \boldsymbol {\mathsf{q}})}{ \mathrm d \boldsymbol {\mathsf{q}}} = \sum _{i=1}^{2n_\mathrm {DIC}}\sum _{j=1}^{{n_{\text {p}}}} \frac{F_i( \boldsymbol {\mathsf{q}} + \Delta q_j\overline{ \boldsymbol{\mathsf {e}}}_j)- F_i( \boldsymbol{\mathsf {q}})}{\Delta q_j} \boldsymbol{\mathsf {e}}_i\overline{ \boldsymbol{\mathsf {e}}}^\text {T}_j, \end{aligned}$$
(90)

with the vectors \(\boldsymbol {\mathsf{e}}_i \in \mathbb {R}^{2n_\mathrm {DIC}}\) and \(\overline{ \boldsymbol {\mathsf{e}}}_j \in \mathbb {R}^{{n_{\text {p}}}}\), where all entries are zero except the one in row i or j respectively.

The results of the error propagation are shown in the Fig. 9, where all elements of the vector \(\Delta \boldsymbol{\mathsf {q}}\) have the deviations as mentioned before. Figure 9(a) and (b) show that a variation of all quantities in \(\Delta \boldsymbol{\mathsf {q}}\) causes changes of approximately \({1\,\mathrm{\%}}\) in the temperatures and uncertainties of approximately \({0.1\,\mathrm{{{}^{\circ }\text {C}}\text {m}^{-1}\text {m}}}\) in the norm of the temperature gradient (norm of the temperature gradient from the analytical solution: \(\Vert \vec {h} \Vert = {3.3365\,\mathrm{{{}^{\circ }\text {C}}\ \text {mm}^{-1}}}\)). To find out which entries of the vector \(\boldsymbol{\mathsf{q}}\) have the largest influence on the uncertainty, each entry is considered individually, while the other elements are set to zero. It turns out that a variation of the distance d has no influence on the error because the numerical derivatives with respect to the distance d are zero for this example (this depends on the choice of the coordinate system of the TG-system and DIC-system which are in the same plane). The variation of single tripod coordinates \(\Delta \boldsymbol {\mathsf{r}}_k\) in the DIC-basis system leads to uncertainties less than \(10^{-6}\) for \(\Delta \Vert \vec {h} \Vert\) and \(10^{-5}\) for \(\Delta \Theta\). The length-to-pixel ratio p invokes uncertainties of less than \(10^{-3}\) for \(\Delta \Vert \vec {h} \Vert\) and \(\Delta \Theta\). Pixel coordinates of the tripod \(\mathbf {\Upsilon }_k^{\,\Theta }\) in the thermography image have the largest influence on the uncertainties. Figure 10 shows the uncertainties if all pixel coordinates of the tripod in the thermography image are varied at once. It also turns out that the size of the tripod has an influence on the results due to the fact that the TG-camera commonly has a coarser resolution than the DIC-camera. If the tripod is manufactured very small, the uncertainties increase as well.

Fig. 9
figure 9

Variation of all elements in vector \(\Delta \boldsymbol{\mathsf{q}}\)  

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Fig. 10
figure 10

Variation of all tripod pixel coordinates in vector \(\Delta \boldsymbol{\mathsf {q}}\)  

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Real Experimental Information of Additive Manufactured Specimens

In the following, the aforementioned approach is applied to additive manufactured specimens. Initially, the experimental setup is described in more detail. Then, the results are shown for two rather cylindrical specimens, where one specimen is manufactured by fused filament fabrication (FFF), in contrast, the other specimen is made by wire-arc additive manufacturing (WAAM).

Experimental Setup

The experimental setup used is shown in Fig. 11. In order to measure the surface temperatures, an infrared thermography camera VarioCAM HD (InfraTec GmbH, Dresden, Germany) was drawn on. The camera employed the spectral range \({7.5} \ldots {14} \ {\mu m}\), while the detector format is \(1024\times {768\,\mathrm{px}}\). A normal lens with focal length 30 mm, a field of view \({32.4\,\mathrm{{}^{\circ }}} \times {24.6\,\mathrm{{}^{\circ }}}\), and an instantaneous field of view \({0.57\,\mathrm{\text {m}\text {rad}}}\) were applied. The determination of the distance between the thermography camera was performed using the calibration object of Fig. 2 – as indicated in “Translation of TG-system relative to DIC-system” – yielding a distance of 624 mm. The surface coordinates were measured with the two camera digital image correlation system ARAMIS 12M (Carl Zeiss GOM Metrology GmbH, Brunswick, Germany). Here, lenses with focal length 50 mm and aperture 16 were used. The working distance between DIC system and heat plate was approximately 486 mm leading to a calibrated measuring volume of approximately 110 mm \(\times\) 80 mm \(\times\) 73 mm. Illumination was done by an ARAMIS light projector (Carl Zeiss GOM Metrology GmbH, Brunswick, Germany) with blue light technology. The specimens were heated with a copper heat plate, which was equipped with a silicon heating pad. Further, the temperature control of the heat plate was realized with a PT 100 resistance temperature detector. Additionally, a thermo-paste was applied in the contact area between heat plate and specimen to ensure a sufficient heat conduction into the specimen. The heat plate was set to 100 \({}^{\circ }\text {C}\) during the experiments.

Fig. 11
figure 11

Experimental setup

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To apply both measurement systems on the same surface of the specimen, a speckle pattern was applied on the specimen’s surface with standard varnish for creating DIC-patterns. The emissivity of the pattern was determined beforehand to \(\varepsilon = 0.96\) in comparison to TETENAL camera varnish. The camera varnish has a known emissivity \(\bar{\varepsilon } = 0.96\) in the employed spectral range of the thermography camera [21]. The determined value for the emissivity of the speckle pattern is therefore slightly smaller than the values that were used in similar studies (0.98 in [14] and 1.0 in [32]).

Subsequently, the aforementioned approach is followed using the pixel-wise temperature data of the TG-system and the surface coordinates measured by the DIC-system.

Cylindrical Specimen using a FFF-Produced Specimen

To apply the presented approach on a widely smooth curvilinear surface, a cylindrical specimen (inner radius 60 mm, outer radius 65 mm, opening angle 60° on inner contour) was manufactured by FFF on an Prusa i3 MK3S+ (Prusa Research a.s., Prague, Czech Republic). The chosen material was Prusament PLA (Prusa Research a.s., Prague, Czech Republic). PLA represents the abbreviation for polylactic acid. The main printing parameters were layer height 0.2 mm, infill \(100\%\), infill angle ±45°, two perimeters, nozzle temperature 210 \({}^{\circ }\text {C}\) and heatbed temperature 60 \({}^{\circ }\text {C}\).

The measured surface temperature of the FFF-produced specimen after 15 min heating time is shown in Fig. 12(a), and an image from the DIC-system in Fig. 12(b). It is important to note that the temperatures in Fig. 12(a) are only reliable for the specimen, which is shown in the center of the figure. The dominant influence of the emissivity can be seen exemplarily for the heat plate, which was set to 100 \({}^{\circ }\text {C}\). However, since the emissivity of copper is much smaller than the determined value of the speckle pattern, temperatures of approximately 35 \({}^{\circ }\text {C}\) are present in Fig. 12(a). Further, the surface coordinates were obtained from evaluating a surface component with facet size 19 px and point distance 15 px with the software ARAMIS Professional 2020 (Carl Zeiss GOM Metrology GmbH, Brunswick, Germany) leading to 25629 DIC-points.

Fig. 12
figure 12

Experimental measurements for the FFF-produced specimen

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The result of the temperature projection of the TG-temperature on the curvilinear DIC-surface is shown in Fig. 13(a). At the edges, the temperature gradient indicates a more inhomogeneous temperature distribution, i.e. the temperature gradient direction rotates. This becomes more clear if the projected data is looked at, see Fig. 13(b).

Fig. 13
figure 13

Spatial temperature and its gradient distribution of experiment with FFF-produced specimen

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Next, the uncertainty of the entire approach for the real data is studied. Figure 14 shows the uncertainties, i.e. the confidence interval, due to the presented transformation procedure for temperature. In most of the area, the uncertainties are below 1 \({}^{\circ }\text {C}\). Only at the edges, or especially the corners, they are somewhat higher.

Fig. 14
figure 14

Uncertainty of temperature for variation of all elements in vector \(\Delta \boldsymbol {\mathsf{q}}\)  

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Cylindrical Specimen using a WAAM-Produced Specimen

Additively manufactured specimens possess not generally a smooth surface. Therefore, we aim at the application of our approach to a rough cylindrical specimen manufactured by a WAAM-process. For an overview of the process technology, see [43]. Here, a specimen (length 50 mm, width 30 mm, and height 12 mm) was cut out of an originally manufactured tube-like geometry. The specific material type is ED-A 31 of company Fliess (Germany).

The surface temperature distribution as well as an image from the DIC-system are shown in Fig. 15. Due to the high thermal conductivity of the material, the specimen was heated for 15 s. Longer heating times would result in a steady temperature state, where no significant temperature gradients in the curvilinear surface exist. Again, the required surface coordinates were evaluated from the DIC-system with a chosen facet size of 11 px and point distance of 9 px. In total, 24081 points were obtained from the DIC measurement. The reconstructed surface of the WAAM-produced specimen is shown in Fig. 16, where the rough surface structure is in evidence. Again the temperature measurement of the TG-system is transformed onto the surface of the specimen, see Fig. 17(a), where also the temperature gradient is evaluated indicated by the arrows. The projected data is illustrated by Fig. 17(a). The image shows that the gradient direction is no longer uniformly distributed and also that the amplitude of the gradient varies considerably. Again, the uncertainty of the temperature \(\Delta \Theta\) due to the variation of the vector \(\boldsymbol {\mathsf{q}}\) is below \({1\,\mathrm{{{}^{\circ }\text {C}}}}\), see Fig. 18. Only at the upper boundary in a very thin region, larger uncertainties due to the transformation concept occur.

Fig. 15
figure 15

Experimental measurements for the WAAM-produced specimen

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Fig. 16
figure 16

Surface of the WAAM-produced specimen measured with DIC

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Fig. 17
figure 17

Spatial temperature and its gradient distribution of experiment with WAAM-produced specimen

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Fig. 18
figure 18

Uncertainty of temperature due to Variation of all elements in vector \(\Delta \boldsymbol{\mathsf {q}}\) for WAAM-produced specimen

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Conclusions

In this article, the two-dimensional temperature distribution of an infrared thermography system was projected onto the three-dimensional curved surface of components measured with DIC and assigned to material points. The proposed approach requires the preparation of four additional points in space and a distance measurement between the thermography camera and one of the points. This can be provided by a very easily fabricated measurement object. The associated coordinate transformations were verified on the one hand by an analytical example of a cylindrical shell with a given temperature distribution on the surface and on the other hand by real experiments. In addition, the expected uncertainties have been estimated. Here, it turns out that in particular the quality of the additional four spatial points provide a significant influence on the uncertainties of the temperature and the temperature gradient. The uncertainties for the temperature are in the order of 1 - 2 \({}^{\circ }\text {C}\), which is often given for temperature problems. This is then confirmed for cylindrical shells 3D-printed from PLA under special temperature boundary conditions, or a sample produced by wire-arc additive manufacturing with a rough surface. The chosen approach proves to be promising and robust for the chosen application. Moreover, it provides the spatial temperature gradient distribution on curved surfaces of products.