Design sensitivity analysis and topology optimization of viscoelastically damped structures subjected to dynamic loads

Abstract

In the present study, design sensitivity analysis (DSA) and topology optimization of viscoelastically damped structures are presented. The damped structures are excited by dynamic loads such as impact and seismic loads. The viscoelastic material is modeled by the generalized Maxwell solid. A numerical scheme for obtaining transient responses is derived using an implicit time integration scheme. A design sensitivity analysis method is developed based on the adjoint variable method. The discretize-then-differentiate approach is adopted for deriving discrete adjoint equations. The accuracy and the consistency of the proposed method are demonstrated through a single DOF system. The proposed method is also applied to a multi-DOF system. The results are verified by comparison with finite difference ones. Moreover, structural and microstructural topology optimizations of viscoelastically damped structures are studied. In the structural optimization, viscoelastic damping layers are optimized for reducing transient responses of shell structures. Relative densities of the viscoelastic material are taken as design variables. Dynamic performance indices are defined to measure the transient responses. In the optimization formulation, three different types of the performance indices are considered as objective functions, while satisfying a maximum volume fraction of the viscoelastic material. The developed sensitivity analysis method is employed to compute the sensitivities of the performance indices. Damping layers of various types of shell structures are optimized. The transient responses of the optimized structures are compared to those of uniformly distributed structures to show the effectiveness of the proposed method. Also, influences of the performance indices are discussed. In the microstructural topology optimization, microstructures of a viscoelastic material applied to a sandwich structure are optimized. The microstructures are composed of periodic unit cells. The effective modulus of the viscoelastic material is obtained through a homogenization theory. Transient responses are evaluated with an implicit time integration scheme based on the two-scale analysis. The density-based topology optimization is adopted to find optimal microstructures of the viscoelastic material. The design objective is to effectively attenuate the transient responses of the damped structures. Based on the above-mentioned sensitivity analysis scheme, an adjoint sensitivity analysis is developed to compute the derivatives of the transient responses of the damped structures with respect to microstructural design variables. Several numerical examples demonstrate the effectiveness of the proposed approach. Various microstructures of the viscoelastic material are presented, and their corresponding transient responses of the sandwich structure are discussed.