# Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions
## 2 Surface Strain Determination
### Radial Basis Functions
, \tag{20}\]  
\(\mathbf{\kappa}^{(n)}\in\mathbb{R}^{(n_{\text{cp}}+3)\times 3},\mathbf{\kappa}_{R}^{(n)} \in\mathbb{R}^{n_{\text{cp}}\times 3},\mathbf{\kappa}_{M}^{(n)}\in\mathbb{R}^{3 \times 3}\), at each evaluation time \(t_{n}\) (the superscript \(T\) symbolizes the transposition). The residual between the simulation (model) \(\mathbf{r}_{i}^{(n)}\) and the experimental data \(\mathbf{d}_{i}^{(n)}\) at that point  
\[\mathbf{r}_{i}^{(n)T}=\mathbf{x}_{i}^{(n)T}-\mathbf{d}_{i}^{(n)T}=\mathbf{a}_{i}^{T}\mathbf{\kappa} ^{(n)}-\mathbf{d}_{i}^{(n)T} \tag{21}\]  
is required either for interpolation or for regression using a linear least-square approach to determine the unknown parameters \(\mathbf{\kappa}^{(n)}\). If all residuals are assembled, the matrix of residuals  
\[\underbrace{\begin{bmatrix}\mathbf{r}_{1}^  
In principle, there are three basic possibilities to determine the unknowns \(\kappa^{(n)}\): Interpolation: First, it is assumed that the center points \(\mathbf{\Theta}_{k}\) are equivalent to the evaluation (data) points \(\mathbf{\Theta}_{i}\), i.e., \(n_{\mathrm{cp}}=n_{\mathrm{D}}^{(n)}\), and that there are no additional monomials. Then, the residual (22) is zero, \(\mathbf{R}^{(n)}=\mathbf{0}\), and degenerates to the system of linear equations with three right-hand sides,  
\[\mathbf{A}^{(n)}\kappa^{(n)}_{\mathbf{R}}=\mathbf{D}^{(n)} \tag{23}\]  
with \(\mathbf{a}_{i}=\mathbf{m}_{i}\), \(i=1,\ldots,n_{\mathrm{D}}^{(n)}\). If monomials are used in addition to RBFs and the center points are identical to the data points, more unknowns than equations occur. Here, it is common to assume the orthogonality conditions  
\[\sum_{k=1}^{n_{\mathrm{D}}^{(n)}}B_{kj}^{(n)}\mathbf{n}_{k}=\mathbf{0} \tag{24}\]  
leading to the linear system  
\[\mathbf{A}^{(n)}\kappa^{(n)}= \begin{bmatrix}\mathbf{D}^{(n)}\\ \mathbf{0}_{(3\times 3)}\end{bmatrix} \quad\text{with}\quad\mathbf{A}^{(n)}= \begin{bmatrix}\mathbf{m}_{1}^{T}&\mathbf{n}_{1}^{T}\\ \vdots&\vdots\\ \mathbf{m}_{n_{\mathrm{D}}^{(n)}}^{T}&\mathbf{n}_{n_{\mathrm{D}}^{(n)}}^{T}\\ \hline\mathbf{n}_{1}\,\ldots\,\mathbf{n}_{n_{\mathrm{D}}^{(n)}}^{T}&\mathbf{0} _{(3\times 3)}\end{bmatrix} . \tag{25}\]  
Regression: If it is not assumed that the data (evaluation) points are identical to the center points, a linear least-square problem can be formulated (under the condition \(n_{\mathrm{cp}}\leq n_{\mathrm{D}}^{(n)}-3\)). The least-square problem requires that the square of residuals should be a minimum,  
\[f(\kappa^{(n)})=\frac{1}{2}\mathbf{R}^{(n)}(\kappa^{(n)})\cdot\mathbf{R}^{(n) }(\kappa^{(n)})\rightarrow\text{min.} \tag{26}\]  
yielding the system of linear equations with several right-hand sides  
\[\Big{[}\mathbf{A}^{(n)T}\mathbf{A}^{(n)}\Big{]}\kappa^{(n)}=\mathbf{A}^{(n)T }\mathbf{D}^{(n)} \tag{27}\]  
to determine the unknown parameters \(\kappa^{(n)}\) (the inner product in Equation (26) is defined for an arbitrary matrix \(\mathbf{K}\in\mathbb{R}^{m\times n}\) by \(\mathbf{K}\cdot\mathbf{K}=\sum_{i=1}^{m}\sum_{j=1}^{n}K_{ij}K_{ij}\)). It should also be noted that \(\mathbf{\Theta}_{i}^{T}=\{d_{i1}^{(n)}d_{i2}^{(n)}\}\), i.e., the \((X_{i},Y_{i})\)-coordinates of the data vectors are chosen in Equation (16) in our applications later on. Various solution schemes can be applied to the linear systems (23), (25) or (27) (direct vs. iterative solvers, solvers for sparse matrices applicable for, e.g., the Wendland function or the Gaussian function if a truncation of values close to zero is applied, singular value decomposition for very ill-conditioned systems, fast solvers,..., see, for example [8,12,21,22] and the literature cited therein). In this rather introductory paper, we confine our attention to the principle properties of the entire approach to strain determination in full-field strain measurements.