## 3 Dynamic Response Topology Optimization with Condensed Localized Mortar Method
### Verification Example for Dynamic Response Topology Optimization  
A Michell-type structure problem is used to validate the optimization formulation (15) for the linear dynamic system. The geometry and boundary conditions are shown in Figure 5(a,b). The material properties of the entire design domain are assumed to be those of structural steel (\(E=2\times 10^{11}\), Poisson's ratio is 0.3, and density \(\rho=7890\)). The total time step is considered as 50.  
Topology optimization is performed to obtain the stiffest design while satisfying 50% of the volume using the optimization formulation (15) and (16). The adjoint method is used for sensitivity analysis. In order to identify the differences and effectiveness of implementation in the linear dynamic system before performing the dynamic response topology optimization, the static response topology optimization under the same conditions and optimization formulations is performed as shown in Figure 7.  
In order to perform topology optimization, it is necessary to confirm whether the dynamic analysis is performed adequately due to which, the Michell-type structure is analyzed first. Figure 8 shows the vertical directional displacements at point \"A\". The red color represents the result of the static analysis, and the other results indicate the displacement of the dynamic analysis corresponding to the final time \(t_{f}\). In the verification example, analysis and optimization is conducted with the final time \(t_{f}\) set to 0.01, 0.05, 0.1, and 0.5 to check the response in the topology optimization for the dynamic system.  
Figure 6: Problem definition for Michell-type structure problem: (**a**) schematic description (**b**) load condition.  
As shown in Figure 8, the results of the displacement at point \"A\" are converged to the quasi-static results as the final time \(t_{f}\) increases. Figure 8a shows the displacement at point \"A\" when the final time \(t_{f}\) is 0.01 s, and the structure acts as an impact state because the load is applied for a very short period of time. This indicates that the dynamic effect produced by the impact phenomenon is essential in the state of Figure 8a with a final time \(t_{f}\) of 0.01 s.  
In the case of the impact state as shown in Figure 9a, the material is concentrated near the point \"A\" where the load is applied due to the effect of inertia. Conversely, in the case of quasi-static cases, as shown in Figure 9b-d, the optimized topological layout is identical to the result of the static analysis presented in Figure 7 because the structure acts as a quasi-static state when the final time \(t_{f}\) is greater than 0.05.  
Figure 8: Displacement histories of the tip point for Michell-type structure.  
Figure 7: Optimized results for Michell-type structure with static analysis: (**a**) Optimized layout (**b**) convergence history.  
The verification example is used to verify that the dynamic response topology optimization adequately derives the optimized layout when the dynamic effect is critical. As shown in Figure 9, the material is concentrated near the impact region due to the inertia effect in the impact state. However, in a quasi-static state, the optimized topological layout is obtained as the result of the static analysis as the final time \(t_{f}\) increases. This indicates that the dynamic response topology optimization is applied well in both the impact and quasi-static states when the formulation (15) is used.