At the micro-nano scale, the flexoelectric effect exhibits a higher electromechanical coupling conversion efficiency in comparison with the traditional piezoelectric effect. Therefore, it holds potential applications in the fields of sensing, actuation, and energy harvesting. However, most commercially available finite element analysis software packages lack the flexoelectric constitutive model, rendering it impossible to perform a precise numerical simulation of flexoelectric structures. Therefore, this study incorporated the flexoelectric effect into the calculation model based on the development of Abaqus user-defined element (UEL) subroutine. Additionally, we derived the electromechanical coupling equations of the flexoelectric structure and developed a flexoelectric element in Abaqus, which provides numerical simulation technology for analyzing the structural flexoelectric response. Compared with the traditional mixed finite element method (MFEM), this proposed method has the advantages of easier modeling, high efficiency and low computational cost. And its deflection and electric field strength metrics are closer to the analytical solution than either of the other methods. Then, using this numerical simulation method, this study established a flexoelectric structural response analysis numerical model, conducted force-electromechanical coupling response calculations, and analyzed straight and curved beams under different boundary conditions. The analysis results show the mechanism of the geometric parameters of the beam affects the structural strain gradient, and show that the output voltage can be controlled by controlling the deformation gradient when designing a flexoelectric beam using the same material. It is shown that changing the degree of bending of the beam is effective in increasing the open-circuit voltage of the beam output and reducing the deflection of the beam. The open-circuit voltage of a curved cantilever beam bent downward is greater than the open-circuit voltage of a curved cantilever beam bent upward for the same degree of bending. When the circular arc angle of the upward bending curved beam is 38°, the left edge is a sliding bearing, and the right edge is a hinge bearing, the open-circuit voltage is the maximum, up to 214.07 mV, which is five times more than that of the cantilevered rectangular beam. In addition, by considering the piezoelectric effect alone, the open-circuit voltage will decrease by 89.3% in the same model.

Large-scale chaotic systems are often used in secure communication and other fields, because they can provide a wider chaotic interval. Most of the existing large-scale chaotic systems are three-dimensional and four-dimensional systems. In order to obtain more chaotic and more complex systems, this study proposed a five-dimensional system with a large range of hyperchaotic states and multiple coexisting attractors. The dynamic characteristics of the five-dimensional system were analyzed by means of coexistence bifurcation diagram, coexistence phase diagram, Lyapunov exponent spectrum and calculation of system divergence. The results show that: the system is a dissipative hyperchaotic system; when the parameters are fixed, the system can produce multiple coexisting attractors only by changing the initial value, and when the parameter d takes different values, the system will produce 12 types of coexistence phenomena, which are period-1 attractor, period-2 attractor, period-4 attractor, quasi-periodic attractor, one-scroll attractor, double-scroll attractor and so on; when the parameter m varies in the range of [0.1, 4 000], the system will always maintain a hyperchaotic state, and when m is in the range of [70, 4 000], the Lyapunov exponent spectrum of the system remains unchanged and maintains a hyperchaotic state with three positive Lyapunov exponents, indicating that the system has an invariant Lyapunov exponent characteristic. FPGA (field programmable gate array) was used to realize the digital circuit, and the experimental results were observed on the oscilloscope, which verified the feasibility of the hyperchaotic system.