# Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions
## 1 Introduction  
Additive manufacturing is growing production technology, where material is applied layer-wise. One example are 3D printing processes using polymers. In the context of metallic materials, deposition welding processes represent a special manufacturing concept. In both cases, smooth surfaces are obtained if the part under consideration is post-machined. This post-processing step is not performed unless absolutely necessary. Mechanical loading of such components with a rough surface results in inhomogeneous stress and strain distributions. Nowadays, inhomogeneous strain distributions such as this can be determined using imaging techniques, for example digital image correlation (DIC) [1]. Since the surfaces are neither smooth nor even, three-dimensional DIC must be used for strain determination purposes. For a review of DIC with application in additive manufacturing, see [2]. The evaluation procedures determine three-dimensional coordinates from the tracking of gray value distributions at each point in time, see [1, 3]. A comparison between different times allows to determine surface displacements of individual points. Commonly, a local regression concept is chosen, see [3], where the surface strains are determined within particular tangent planes. An alternative concept is based on applying shape functions similar to finite element approaches, see [4], where the simplest approach is based on atriangulation interpolation, see [5; 6] for a similar approach. The latter approach has the advantage that a \(\mathcal{C}_{0}\)-continuous displacement interpolation is obtained, while the strains, however, are discontinuous. A mean-value calculation of the principal in-plane strains (or stretches) at the intersecting nodes yields an additional \"interpolation\". However, this is not uniquely possible for the principal directions. Thus, a unique principal direction computation of the strains is only possible within a triangle and not on the edges. The \"classical\" local approach proposed in [3] has the advantage to provide the principal directions at the evaluation points, see [4] for an interpretation and a mathematically clearer description. However, this is not always possible at arbitrary points within the region of interest. To circumvent both disadvantages--non-smoothness and impossibility to evaluate at arbitrary points--a global interpolation scheme is chosen using radial basis functions (RBF), see, for the basic ideas [7; 8]. RBFs are used in many areas of engineering science. Refs. [9; 10] apply RBFs within the field of fluid-structure interactions using mesh morphing techniques. For unstructured meshes, [PERSON]. [11] developed a mesh morphing technique employing RBFs. Refs. [12; 13] applied RBFs to fit and interpolate data. Further, RBFs find application in the solution of partial differential equations (PDE), see [14; 15; 16], where RBFs are used within meshless methods to solve PDEs. For our application, RBFs have the advantage that the spatial distribution of the evaluation points can be--more or less--arbitrary. They lead to a continuous interpolation of the displacement field and, accordingly, also to continuous principal strain and principal strain directions. In this context, the applicability and limitations must be examined since the approach implies difficulties, such as ill-conditioned linear systems, possible inefficiencies for large problems, dependence on individual shape parameters, smaller oscillations between data points, and deviations of the derivatives at the boundaries. In a first step, the properties of this approach will be compared to the triangulation concept--with a particular focus on rough surfaces. The paper is structured as follows. First, the strain evaluation within surfaces using RBFs is briefly recapped.