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Compared with finite element method (FEA) and statistical energy method (SEA), the energy finite element method (EFEA) is a promising and more suitable method to predict the high-frequency vibration response and radiated noise of mechanical structures, due to its advantages of both computational efficiency and ability to get detail information. In recent years, researchers have made great progress in EFEA, due to the urgency of engineering application. This paper aims to comprehensively summarize historical and latest development of energy finite element analysis from two aspects of theory and engineering, to furtherly promote its application in the engineering field. Firstly, the development history of EFEA in basic structures is summarized, and the theoretical development direction of energy finite element in recent years is further analyzed. After that, the application of EFEA in aerospace, ship and vehicle engineering are summarized. Finally, the problems that need to be furtherly studied in EFEA are presented, and it is pointed out that it is an urgent research work to continue to improve the energy finite element analysis theory and software and apply it to engineering.

The friction characteristics of the lubricated rough surfaces are of great significance for the lubrication design of mechanical structures. To study the behaviour of friction coefficient in the rolling-sliding coexisting line contact rough surfaces under mixed lubrication regime, the Kogut-Etsion elastic-plastic model and the Carreau rheological model are utilized to describe the dry rough contact and the non-Newtonian characteristics of the lubricant film, respectively. A mixed lubrication model for predicting friction coefficient is proposed based on the load-sharing concept at which the normal load is shared by the lubricant film and the asperity component. The effects of normal load, surface roughness and lubricant inlet viscosity on hydrodynamic scaling factor, film thickness parameter and the coefficient of friction were analyzed. The dimensionless predictive expression for the critical sliding velocity is derived by nonlinear regression. The research results provide theoretical guidance for the lubrication design and optimization of mechanical structures.

The fretting wear characteristics of spherical/spherical lateral contacts under different load conditions, contact angles, and the number of fretting cycles are studied. A spherical/spherical lateral fretting wear model is established in ABAQUS software. A UMESHMOTION subroutine for spherical/spherical contact models is written to simulate the fretting wear based on the energy model and the Fortran language. The results show that as the load increases, the contact width increases significantly and the wear depth decreases. It is also found that the rate of change of the wear depth gradually increases in the central portion of the contact area, and is relatively smooth in the contact edge portion. An increase in the magnitude of the moving load causes a small increase in the contact width and a significant increase in the wear depth. The rate of increase of the wear depth at the edge of the contact area gradually increases and the rate of increase of the wear depth at the center of the contact area changes linearly. As the contact angle increases, the contact form gradually changes to a positive contact. Meanwhile, the wear depth variation shifted from a “U” shape to a “W” shape, which caused the wear depth to decrease significantly at the contact center and eventually approach zero as the adhesive area appeared. In addition, with an increase in the number of fretting cycles, the wear depth was almost unchanged at the contact center portion but significantly increased at the edge portion.

This letter presents a thin plate structure with an acoustic black hole (ABH) sub-unit to reorient the flexural wave. Different from the previously reported flexural wave metasurface, ABH sub-units are introduced into thin plates in this work, which can control the group velocity of flexural waves and realize their efficient transmission. According to generalized Snell's law, the mechanism of phase shift of transmitted waves across subwavelength sub-units is theoretically revealed. An analysis of the ABH sub-units is established by the finite element method. The deflection and focusing effect of flexural waves are demonstrated. Furthermore, adjusting the black hole section can quickly obtain the transmission phase response in the range of 2π, and it can accurately predict the phase shift and amplitude of the transmitted wave. The results show that the ABH beam-plate structure can effectively control the propagation direction of flexural waves, which provides a modern design idea and method for the manipulation and energy harvesting of the flexural wave.

Fretting wear phenomenon has a non-negligible impact on the reliability of the contact parts of mechanical power systems. The impact of temperature increases in actual working conditions is taken into consideration in order to increase the accuracy of fretting wear prediction. Temperature-dependent wear coefficients were added to the energy dissipation wear model, and the UMESHMOTION subroutine was created. A temperature-displacement-coupled finite element model of fretting wear is established based on a cylinder/plane fretting test of Ti-6Al-4V alloy materials. The model takes into account the interaction between temperature, stress, and wear. The effects of the plastic deformation of materials, temperature, number of cycles, fretting velocity, and variable normal load on wear and temperature rise are explored. The results show that the wear amount is small when the temperature rises, and the plastic deformation of materials is not considered. The wear profile is no longer a smooth Hertzian shape when the plastic deformation of materials is considered. The amount of wear increases with the fretting speed and the number of cycles. Meanwhile, the temperature of the contact area and the surface near the contact area increases with the increase in fretting speed. Peak temperature rise of the contact surface increases with the number of cycles, and its horizontal position moves with the cylinder specimen. Furthermore, the wear profile is less smooth under the variable normal load, but the two variable normal loads in the same phase have similar wear profiles and temperature rise distributions. The theoretical resources provided by the research work can be used to design control strategies and optimize mechanical power systems.

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%.

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.