# Multi-Material Topology Optimization Taking into Account the Position of Material Interfaces in 3D
## 5 Conclusions  
In this article, a topology optimization method was developed for the optimization of multi-material structures in which the individual materials are adhesively bonded. For this purpose, two requirements for the optimization method were established based on the analysis of the known load cases of a bonded joint. Based on, the ITED introduced in [27] for 2D problems was further developed for the 3D case by integrating shear stresses, for example. Subsequently, the optimization problem was set up and the sensitivities were derived for it using the Lagrange method. The effectiveness of the optimization method was then demonstrated by applying the method and comparing the optimization results of the multi-material optimization with and without ITED. In addition, a parameter study was used to show that the behavior of different adhesives (different ratios between tensile and shear strengths) can be qualitatively adjusted by varying a newly introduced parameter.  
The optimization method developed in [15] can be extended to integrate explicit interface stress constraints based on the Brewer and Lagace failure criterion in 3D. Further research regarding the comparison of the two methods as well as regarding their combined use. Here, for example, strategies are conceivable that initially the interfaces are displaced from the tensile regions (based on the ITED) and subsequently explicit stress constraints ensure that maximum interfacial stresses are maintained. In addition, further research is required on the method to take heterogeneous adhesive properties into account. Moreover, further research is required regarding the validation of the approach. Here, for example,  
Figure 11: Density plots of the normal \(\sigma_{nn}\) and shear stresses \(\sigma_{\text{ts}}\) as well as \(\sigma_{nt}\) of the interface for three different values of the parameter \(\beta\). The \(\sigma_{ts}\) diagram illustrates that the maximum shear stresses decrease as the value of \(\beta\) increases.  
the comparison of the optimization results with and without consideration of the interfaces after previous redesign is beneficial.  
The optimization method developed in this article should be extendable to the optimization of more than two materials. Both the RCLSM and the remeshing method can individually handle more than two materials through adaptations already shown in the state-of-the-art. An extension of the ITED for more than two materials can be realized by setting up a density for each pair of materials analogous to Equation (14). However, further research is required in this case as well.  
In summary, the developed method was able to solve the challenges of taking into account the interfaces in 3D as described in the introduction.  
Conceptualization, R.R. and A.A.; methodology, R.R. and N.F.; software, R.R. and N.F.; validation, R.R. and A.A.; formal analysis, R.R.; investigation, R.R. and N.F.; data curation, R.R. and N.F.; writing--original draft preparation, R.R.; writing--review and editing, R.R. and N.F.; visualization, R.R.; supervision, A.A.; funding acquisition, A.A. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data that support the findings of this manuscript are available from the corresponding author upon reasonable request.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} FE & Finite Element \\ FEM & Finite Element Method \\ ITED & Interfacial Tension Energy Density \\ LS & Level Set \\ MDPI & Multidisciplinary Digital Publishing Institute \\ MBO & Merriman-Bence-Osher \\ MMA & Method of Moving Asymptotes \\ RCLSM & Reconciled Level Set Method \\ VFLSM & Velocity Field Level Set Method \\ XFEM & Extended Finite Element Method \\ \end{tabular}