## 2 Materials and Methods
### Fullsize Microstructures  
Essential inputs for simulation are the properties of the sample, i.e., the grain structure, the phases of the individual grains and the texture. Periodic 3D microstructure models of relevant samples were digitally generated by Tata Steel in the form of a representative volume element (RVE) describing the location of individual grains [40]. The RVE generator, based on a multi-level Voronoi algorithm, is able to create periodic structures of up to 200,000 grains. Each grain is associated with a phase number and a set of three Bunge Euler angles describing the texture. The Bunge Euler angles are taken from the orientation distribution function (ODF) obtained from EBSD measurements. An ODF describes the distribution of Euler angles in the 3D Euler angle space. An example of such a microstructure is given in Figure 2. The size of a microstructural model is determined by the grain size and the used frequencies. With a maximum frequency of 60 MHz, a shear wave velocity of 3000 m/s, and at least 12 grid points per wavelength, the minimum grid size is 4.2 \(\upmu\)m. In order to describe a wave appropriately, at least two sample points are needed  
Figure 1: Schematic overview of the simulation process. per grain. This means that a grid size of half the minimal grain diameter is needed (i.e., 2.5 \(\upmu\)m for grains with diameter 5.0 \(\upmu\)m). Based on these requirements, an integer grid size has been chosen and fixed to 3 micrometers. A grid size lower than this size, leads to a computationally much larger model (which scales to the power of four: three spatial dimensions and one temporal dimension). Since the number of grains which can be handled by the microstructure generator software is bound to a maximum of 200,000 grains for periodic structures, the RVE samples cannot be generated at full size for FD simulation. Typically, the RVE sizes are in the order of 0.3 \(\times\) 0.3 \(\times\) 1.0 mm, while the simulated sample is 3.0 \(\times\) 3.0 \(\times\) 2.0 mm, which is 200 times larger. This limitation is overcome by replicating the original sample in all three Cartesian directions until the desired size has been obtained. This allows the generation of a much larger sample for FD simulation. However, replication of a sample leads to repetitive structures, which in turn lead to (strongly) amplified effects in ultrasonic wave propagation. The effect becomes more and more apparent as the replication number increases. Although it is unavoidable to obtain a repetitive microstructure (grain morphology), the texture can be randomized under the assumption that there is no spatial correlation between the texture of the individual grains. For each phase, the existing sets of three Euler angles are randomly redistributed over all grains belonging to that phase, keeping the sets themselves intact (and thus preserving the texture), and keeping the sets bound to their specific phase (no cross-over of texture between phases). As a result, the texture is not repetitive anymore, although the microstructure remains periodic. This process is illustrated in Figure 3.  
An essential aspect in the interaction of ultrasonic waves and microstructure is grain scattering. The physical background behind grain scattering is the contrast between two adjacent grains, which causes a part of the incoming wave to be scattering instead of being transmitted. The level of grain scattering in a material depends on one hand on the size of the grains, and on the other hand on the contrast in the elastic properties. The contrast itself depends on the texture (crystal alignments, determined by a set of Bunge Euler angles, with respect to the RVE frame) and on the physical properties of the grain (crystal properties, determined by its material and phase). Modelling grain scattering hence requires the grain morphologies, the grain material (phase) and its single crystal properties, and the texture.