# Correction of Thermal Errors in Machine Tools by a Hybrid Model Approach
## 4 FE Model and Simulative Characteristic Generation  
Following the approach, the next step was to create characteristics by simulation instead of experiments with the help of a thermo-elastic finite element model. This model was created based on a CAD geometry model of the entire machine that needed to be reduced in complexity: At first, detailed design features (e.g., small holes and bores) were removed in the design environment and complex bodies were simplified. These included screw connections, guide rails, bearings, and motors. This was necessary as otherwise there could have been issues with the meshing or an unnecessarily high number of FE elements would have been created. Non-relevant assemblies such as the machine housing and switch cabinet were removed. Next, the data were exported in a neutral CAD exchange format. The subsequent import took place in the simulation environment ANSYS Workbench, where the meshing and calculation were performed in ANSYS Mechanical. With the help of the software, it was possible to carry out a static mechanical analysis based on a thermal analysis that was calculated in advance. The calculated temperature field served as a thermal load for the calculation of the deformation field when compared to a reference state. The relative or absolute displacements of the tool center point (TCP) in all three spatial coordinates (X, Y, and Z) could be determined from the deformation simulation. Figure 4 shows the entire workflow of the simulation process. The individual model phases are explained in more detail below. The CAD model was divided into further sub-assemblies, which were later reassembled after the meshing level in the simulation level in order to be able to map the different axis positions (see Figure 4). In the Ansys Workbench, the individual subsystems were assembled as building blocks to form an overall system, while each unit could be processed independently at the submodel level. This allowed a good overview and structure of the overall project, because the overall model of a machine tool is very complex. The most important properties, such as material data, were included in a separate database. Figure 3: Example of a measurement with a change in ambient temperature. The following Figure 5 shows the simplified machine model and the FE mesh on which it was based. The individual solid bodies were decomposed into simple finite volume elements (e.g., tetrahedron and hexahedron). The primary emphasis of the modeling was on achieving high-quality elements while minimizing the total number of volume elements to reduce both computational and storage requirements. The model boundary conditions included all possible types of heat flows and convective boundary conditions needed to solve the heat conduction equation in the FE solver. The model's boundary conditions were modified as required to achieve the closest approximation to the physical model possible, Figure 6. This was achieved by considering the internal heat sources in the machine, such as heat-loss of motors and motor spindle, frictional heat in bearings, and guides. In order to be able to simulate and represent a machine behavior (axis movements) that is as universal as possible, individual areas of the active surfaces were segmented. This was achieved, for example, for the running surface of the guideway and the spindle shaft of the ball screw, Figure 7. Asymmetric load cases can also be applied, by introducing an average heat flow locally to the running area of the guideways, as presented in Figure 7. Flow effects, such as cooling lubricant or heat exchange with the environment, are described by a heat transfer coefficient, whereby the individual assembly surfaces are clustered. Figure 4: Workflow for the creation of a finite element machine model. Figure 5: Three-dimensional geometry of the machine and volume mesh.