## 4 Numerical Example: Impact between Fixed Beam and Flying Block  
Hitherto, two methods have been employed to find an optimum topological layout in the linear dynamic impact problem: the condensed localized mortar method and the dynamic response topology optimization approach. The condensed localized mortar method used to handle the non-matching interface at the impact and frictionless contact region was described in Section 2, and the simple frictionless contact problem is solved to verify its effectiveness in the impact state. The dynamic response topology optimization was introduced in Section 3, and an optimization formulation with the condensed mortar method was presented. Additionally, by performing topology optimization on the verification example, the effectiveness of the dynamic response topology optimization is confirmed when the beam is in the impact state. In this section, topology optimization is performed by assuming a situation in which two independently behaving objects are in the impact and frictionless contact conditions to examine the applicability of the proposed approach to the linear dynamic impact problem. The problem definition is presented in Figure 10. A block is separated at 0.05 intervals on a lower beam which is completely fixed on both sides, as shown in Figure 10a. A constant pressure is applied to the block for 0.12 s as depicted in Figure 10.  
Figure 9: Optimized results for Michell-type structure with dynamic analysis. The frictionless contact condition (2) is imposed to allow for minimal sliding in the tangential direction after the occurrence of the impact. Therefore, a situation in which the configurations on the non-matching interface inevitably occur is considered. The material properties for both the structures are assumed to be those of the structural steel (\(E=2\times 10^{11}\), Poisson's ratio is 0.3, and density \(\rho=7890\)). The mean squared displacement of a point \"A\" is used as an objective function (17) and the volume is limited to 50% for the topology optimization process (15). The optimization was performed considering only the lower beam and not the upper block. A dynamic analysis is preferentially conducted to confirm the contact kinematics before performing topology optimization. Figure 11 illustrates the deformed shapes obtained by conducting the dynamic analysis, where it is observed that the impact occurs after 0.06 s, and then the frictionless contact occurs between two parts. The topology optimization results based on the analysis are presented in Figure 12. In the numerical example, the optimization is performed for three cases: no contact (t = 0.04), impact (t = 0.08) and steady state (t = 0.12). The topological change in the design domain does not occur since there is no contact until 0.06 s, as shown in Figure 12a. The optimized layouts in the impact (t = 0.08) and steady state (t = 0.12) are derived as shown in Figure 12b,c, after 0.06 s to minimize the beam deflection. A large portion of the materials are distributed in the position where the beam is impacted when the design domain is in the impact state (t = 0.08), since the force is not completely transmitted to the beam, as shown in Figure 12b. However, when the analysis is performed over a period of 0.12 s, a shape in which the material is spread throughout the design domain is observed since the load is distributed throughout the beam, as shown in Figure 12c. The result shown in Figure 12c is a truss-type layout similar to the optimized results that can be derived under static analysis with the same boundary conditions. Figure 10: Problem definition of beam with a flying block. Figure 11: Deformed configuration for beam with flying block. Figure 12: Optimized for beam with flying block using condensed localized mortar method. As a result, it is possible to obtain the optimum topological layout to which the dynamic effect is applied in the impact state in which the non-matching interface inevitably occurred.