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Topology optimization of transient dynamics is of great importance in engineering. However, the computational cost required by the repeated dynamic response and design sensitivity analysis during design iterations is often extremely high for large-scale problems, which significantly limits its practical application. To address this issue, we propose a new method based on successive iteration of analysis and design in conjunction with a quasi-static response-enhanced mode displacement method. The conventional mode displacement method requires an inner-loop iteration in addition to the outer-loop design updating, for finding the eigenmodes of the undamped system. In order to avoid the high computational burden caused by the nested double-loop solution, we employ the concept of successive iteration of analysis and design, which integrates approximate eigenvalue analysis and design variable updating into a single iteration loop. By this means, the eigenmodes are sequentially improved through a one-step inverse iteration in each design iteration, and simultaneous convergence of the eigenmodes and the design is sought during the optimization process. These approximate eigenmodes and the quasi-static response are used as the model reduction basis for each intermediate design, and the direct time integration of the reduced-dimensional system is then performed to calculate the structural responses of interest and their derivatives with respect to the design variables. Several numerical examples are presented to demonstrate the efficiency of the proposed method. It is shown that the method can be used to solve transient dynamic topology optimization problems up to 15 million degrees of freedom on a desktop computer at affordable time cost. [ABSTRACT FROM AUTHOR]

Topology optimization has become a useful tool for designing Phononic Crystals (PnCs) to achieve desired elastic wave propagation properties. However, topological design of three-dimensional (3D) PnCs remains a significant challenge due to extremely high computational cost required for the repeated solution of the nested double-loop problem of design optimization and eigenvalue analysis to determine the band structure during the optimization iteration process. This paper presents a more efficient topology optimization method based on the concept of successive iteration of analysis and design. The method employs a sequential approximation of the eigenpairs through an inverse iteration-like procedure to avoid solving the expensive eigenvalue problems in each design iteration. To further improve the efficiency of the design iteration, the eigenmodes of the intermediate PnCs designs are first computed on a relatively coarse finite element mesh. Then they are mapped onto the fine mesh and used as the initial trial eigenmodes in the eigenpair analysis and design sensitivity analysis. The high efficiency of the proposed method, as compared with conventional nested double-loop optimization approaches, is demonstrated by 3D numerical examples with over one million degrees of freedom. It is shown that this approach can achieve simultaneous convergence of the eigenvectors along with the design evolution of the unit cell, therefore substantially reduces the computational burden for high-resolution topological design of 3D PnCs. [ABSTRACT FROM AUTHOR]