## 2 Materials and Methods
### Finite Element Simulation Method  
In order to analyze the mechanical response of the NiTi superelastic male attachment part the evolution of the martensite phase fraction and local stress magnitude under multiaxial loading occurring as a consequence of mastication seemed to be important. To that end, a three-dimensional mathematical phenomenological SMA model proposed by [PERSON] and [PERSON] [30], which has also been numerically implemented in the MSC Marc finite element software and which has also been comprehensively outlined in the supplementary material of Ref [31] was used. This model's capabilities include capturing phase transformation between the austenite and martensite phases, as well as the tension-compression asymmetry in SMAs by the Conical Drucker approach. The reference geometric model shown in Figure 3a consisted of the male part of the attachment made of NiTi superelastic alloy, a stiff removable prosthesis with female attachment parts, and a titanium implant. The model was generated based on an existing patient situation (Figure 1) and the dimensions of the prototype of the attachment system (Figure 2). In all simulations, a unilateral vertical force of 150 N was applied on a few nodes resembling the approximated molar region of the prosthesis (red arrow in Figure 3a). Prosthesis displacement under masticatory forces was considered the standard cyclic loading scenario the attachment has to withstand during function. While prosthesis displacement should clinically be kept as low as possible e.g., by relining, 5 mm of vertical displacement in the posterior region of the prosthesis was chosen for simulating a worst-case scenario. To this end, a stiff plate was modeled at the posterior end of the prosthesis and fixed at the bottom (see component number 7 in Figure 3a,c). Elastic deformation of alveolar bone as a result of loading was not of primary interest here but instead, all degrees of freedom at the outer surface of the dental implant were fixed. Positional discrepancies between male and female parts of the attachment systems constitute a frequent clinical problem and result from non-parallelism of implants and transfer inaccuracies during prosthesis fabrication. Accordingly, the FE analysis of the reference design (following the prototype) with parallel alignment of the supporting implants (Figure 3) was compared to the situation of the non-parallelism of the supporting implants (Figure 4). For design optimization, two alternative designs of the male attachment part were examined representing a larger contact area between the shaft and the male retentive element and a necked shaft, respectively (Figure 5). Furthermore, a case study with the male part made purely from titanium was investigated for comparison. The material properties for the shape memory NiTi wire (part 1 in Figures 3 and 5) determine the superelastic response of the device. In the present simulations, transformation temperatures \(M_{s}\), \(M_{f}\), \(A_{s}\), \(A_{f}\), austenite and martensite elastic moduli EA and EM, Poisson ratio \(\
u_{A}\) and \(\
u_{M}\), thermal stress influences coefficients \(C_{A}\) and \(C_{M}\) and transformation strain \(\epsilon_{eq}^{T}\) were adapted to tensile experimental data obtained by the manufacturer of the prototype NiTi attachment (ADMEDES GmbH, Pforzheim, Germany) which are given in Figure 6 and in Table 1. It should be noted that, since the experimental data for the tensile test was available only at a single temperature, the fitting parameters including the transformation temperatures could be unrealistic compared to those of the commercial NiTi used in medical devices. Nevertheless, this approach still satisfied the fitting to the mechanical re  
Figure 4: Non-parallel positioning of two supporting implants. Approximately nine degrees axial deviation in two axes were considered for both supporting implants, (**a**) side view, and (**b**) front view.