# Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions
## 2 Surface Strain Determination
### Radial Basis Functions  
There are a number of approaches that use RBFs, see [7; 8], mainly concerned with computer graphics, geometrical reconstruction, interpolation, and smoothing. Here, RBFs are applied to describe the motion of material surfaces, and the properties of surface deformations are evaluated, i.e., the stretch or strain determination. Since a Cartesian basis is chosen in the following, it is possible to use column vectors \(\mathbf{x}\in\mathbb{R}^{3}\) instead of geometrical vectors \(\vec{x}\in\mathbb{V}^{3}\). Thus, a consistent matrix notation can be followed. The ansatz for the motion reads  
\[\hat{\mathbf{x}}(\mathbf{\Theta},t_{n})=\sum_{j=1}^{3}\left(\sum_{k=1}^{n_{ \mathrm{op}}}\hat{m}(\hat{\rho}(\mathbf{\Theta},\mathbf{\Theta}_{k}))B^{(n)}_{ kj}+\sum_{l=1}^{3}\hat{n}_{l}(\mathbf{\Theta})D^{(n)}_{lj}\right)\mathbf{e}_{j} \tag{13}\]  
with  
\[\hat{\mathbf{n}}(\mathbf{\Theta})= \begin{cases}\hat{n}_{1}(\mathbf{\Theta}) \\ \hat{n}_{2}(\mathbf{\Theta})\\ \hat{n}_{3}(\mathbf{\Theta})\end{cases} = \begin{cases}1\\ \Theta^{1}\\ \Theta^{2}\end{cases} . \tag{14}\]  
where the first part represents the RBF contribution and the second lower order polynomial part is chosen to fit rigid body motions and linear displacement fields. Here, \(\mathbf{e}_{j}\in\mathbb{R}^{3}\), \(j=1,2,3\), are column vectors--with a value 1 at the \(j\)-th entry and a value of zero at all other entries. \(\mathbf{\Theta}\in\mathbb{R}^{2}\) with \(\mathbf{\Theta}=\{\Theta^{1},\Theta^{2}\}\) are the surface parameters of Section 2.1,where the simplest approach is to take the Cartesian coordinates in the initial configuration provided by the DIC-system, \((\Theta^{1},\Theta^{2})=(X,Y)\). \[\rho_{k}=\hat{\rho}(\mathbf{\Theta},\mathbf{\Theta}_{k})=\frac{R}{R_{0}}=\frac{\|\mathbf{ \Theta}-\mathbf{\Theta}_{k}\|}{R_{0}}=\frac{\sqrt{(X-X_{k})^{2}+(Y-Y_{k})^{2}}}{R_{ 0}} \tag{15}\]  
defines the normalized distance function evaluated at the so-called center (or source) points \(\mathbf{\Theta}_{k}=\{\Theta_{k}^{1},\Theta_{k}^{2}\}=\{X_{k}^{\text{cp}},Y_{k}^{ \text{cp}}\}\), \(k=1,\ldots,n_{\text{cp}}\), which might be equidistant points in the \((X,Y)\)-space. \(R_{0}\) is a radius for normalizing the RBFs, and \(n_{\text{cp}}\) are the number of center points. The coefficients \(B_{kj}^{(n)}\) and \(D_{lj}^{(n)}\) are the unknown weights. The chosen radial basis functions \(\hat{m}(\rho)\) are depicted in Table 1, although there are many other possible functions, e.g., splines, see [19]. The transposed position vector at a point \(\mathbf{\Theta}_{i}\), \(i=1,\ldots,n_{\text{D}}^{(n)}\) (where \(n_{\text{D}}^{(n)}\) symbolizes the number of evaluation points), is given by  
\[\mathbf{x}_{i}^{(n)T}=\mathbf{\hat{x}}^{T}(\mathbf{\Theta}_{i},t_{n})=\mathbf{m}_{i}^{T}\mathbf{ \kappa}_{R}^{(n)}+\mathbf{n}_{i}^{T}\mathbf{\kappa}_{M}^{(n)}=\mathbf{a}_{i}^{T}\mathbf{\kappa }^{(n)}. \tag{16}\]  
Here, the vectors of evaluated radial basis functions and monomials are symbolized by  
\[\mathbf{m}_{i}^{T} =\{m_{i1}\,m_{i2}\ldots m_{in_{\text{cp}}}\},\quad m_{ik}=\hat{m} (\hat{\rho}(\mathbf{\Theta}_{i},\mathbf{\Theta}_{k})),\quad\mathbf{m}_{i}\in\mathbb{R}^{n_ {\text{cp}}} \tag{17}\] \[\mathbf{n}_{i}^{T} =\{1\,X_{i}\,Y_{i}\},\quad\mathbf{n}_{i}=\mathbf{\hat{n}}(\mathbf{\Theta}_{i }),\quad\mathbf{n}_{i}\in\mathbb{R}^{3}\] (18) \[\mathbf{a}_{i}^{T} =\{\mathbf{m}_{i}^{T}\mathbf{n}_{i}^{T}\},\quad\mathbf{a}_{i}\in\mathbb{R}^{n _{\text{cp}}+3}, \tag{19}\]  
\(i=1,\ldots,n_{\text{D}}^{(n)},k=1,\ldots,n_{\text{cp}}\), and the matrices of unknowns by  
\[\mathbf{\kappa}_{R}^{(n)}= \begin{bmatrix}B_{11}^{(n)}&B_{12}^{(n)}&B_{13}^{(n)}\\ \vdots&\vdots&\vdots\\ B_{n_{\text{cp}}1}^{(n)}&B_{n_{\text{cp}}2}^{(n)}&B_{n_{\text{cp}}3}^{(n)} \end{bmatrix} ,\quad\mathbf{\kappa}_{M}^{(n)}= \begin{bmatrix}D_{11}^{(n)}&D_{12}^{(n )}&D_{13}^{(n)}\\ D_{21}^{(n)}&D_{22}^{(n)}&D_{23}^{(n)}\\ D_{31}^{(n)}&D_{32}^{(n)}&D_{33}^{(n)}\end{bmatrix} ,\quad\mathbf{\kappa}^{(n)}= \begin{bmatrix}\mathbf{\kappa}_{R}^{(n)}\\ \mathbf{\kappa}_{M}^{(n)}\end{bmatrix}