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The energy finite element method (EFEM) provides researchers with an efficient tool to analyze high-frequency vibrating solid structures with less calculation and clear distribution of the energy density. However, the corresponding applications in structure optimization mainly focus on modifying the size and material properties, and it leaves a scientific gap that the topology flexibility is not addressed enough in the optimization. Therefore, this work aims at establishing an explicit level-set based topology optimization framework for the energy finite element method. A series of basic technical aspects, including the explicit level set description method, customized finite element mesh, mathematical model, and sensitivity analysis, are presented. With these basic studies, an original EFEM-based topology optimization framework for thin-walled structures is established for the first time. It is a basic work to conduct topology optimization in terms of energy. Additional applications in curved surfaces, spatial structures, and compound materials can also be developed only by modifying the description and EFEM module. Finally, the proposed optimization is applied to classic stiffened plates. Further analyses indicate that the proposed EFEM-based topology optimization can improve the dynamic performance of high-frequency vibrating structures by 20%–80%.

Energy finite element analysis (EFEA) has unique advantages in solving high-frequency dynamic responses of orthotropic structures, due to its ability to obtain detailed local response information. In order to accurately predict high-frequency vibration response of the stiffened orthotropic plate, EFEA theory on the propagation of bending wave in the orthotropic structure and the energy transfer coefficient which express the energy transfer at the stiffener was investigated. Based on the EFEA theory presented, high-frequency dynamic responses of a stiffened orthotropic plates were predicted. Furthermore, tests were done for the same problem, and differences between prediction and test were discussed. Finally, the future works were pointed out.

The energy density governing equation to analyze the high-frequency dynamic behavior of plates in thermal environments is derived in this paper, in which the thermal effects are considered to change the membrane stress state and temperature dependent material properties of plates. Then the thermal effects on the energy reflection and transmission coefficients are dealt with hereof. Based on the above, an EFEM (energy finite element method) based approximate approach for the energy analysis of coupled plates under nonuniform thermal environments is proposed. The approach could be conducted by three steps: (1) thermal analysis, (2) thermal stress analysis, and (3) forming element matrixes, joint matrixes, and the whole EFEM formulation for the energy analysis. The same mesh model is used for all the three steps. The comparison between EFEM results and classical modal superposition method results of simply supported plates in various uniform thermal environments and coupled plates in nonuniform thermal environments demonstrated that the derived energy governing equation and the proposed approach described well the smooth time- and locally space-averaged energy density. It is found that the distributions and levels of energy density are affected by thermal effects, and the variation trends are related to exciting frequency.

Energy finite element analysis (EFEA) is an efficient way to solve high-frequency structural dynamics response problems. Up to now, EFEA has been used to deal with time-independent vibration problems. However, it is still necessary to understand the time dependent details of energy density distribution of a structure. To study the transient response of a rod under high-frequency sinusoidal excitation, the transient energy density governing equation for a rod is presented. The governing equation is solved, and the solution is verified using an analytical method. Example application to a rod is presented to illustrate the feasibility.