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Owning to its various advantages, the lithium− sulfur battery is one of the research hot spots for new energy storage systems. Diverse hollow structures with specific morphologies have been used as the sulfur host materials to adsorb or/and catalyze the polysulfides, and can in particular concurrently inhibit the volume expansion during electrochemical processes in lithium−sulfur batteries. However, hollow space with a large volume will restrict the performance of the cell under high sulfur area loading, which is a very important indicator for the practical applications of the lithium−sulfur battery. Here, we report a nano thin cage cobalt acid zinc (ZnCo 2 O 4 ) with limited hollow space as the cathode catalyst for lithium−sulfur batteries, which greatly reduces the electrode volume occupied by the hollow structure. The hollow volume of these thin cages is much smaller than those of the normally reported hollow materials in the literatue. The electrochemical performance of lithium−sulfur batteries with ZnCo 2 O 4 thin cages could greatly improve due to the unique structure and the synergistic adsorption/catalytic effect of Zn/Co sites, especially at an ultrahigh S area load. Under a high S loading of 8 mg cm −2 , the cell could keep a reversible capacity of 600 mAh g −1 after 500 cycles. Even at a sulfur loading of 10 mg cm −2 , the cell still releases a discharge capacity of 1000 mAh g −1 which is equivalent of an area capacity of 10 mAh cm −2 . This work provides a feasible way to develop lithium sulfur batteries with a high area sulfur load. This idea provides a possible solution to develop a Li−S battery at high area S loading and move one step closer to the practical applications.

Nonlinear differential equations are sometimes found by using harmonic balance principle. If it is based on the complex power balance theory, some much more correct and rational results can be obtained. Non-autonomous circuits sometimes include two components, the forced and the self-excited oscillation, they must satisfy respectively the balancing condition of complex power. When we study the nonautonomous circuit, two notable questions should be considered. On the one hand ,the existence of self-excited oscillation of the circuit which contains the dissipative elements depends on whether or not active power can maintain balance. The existence is closely related to the amplitude of excited current source. When the current source is strong enough, the original self-excited oscillation will thus disappear, leaving only a forced component. On the other hand, the existence of the self-oscillation of the lossless circuit which does not contain the dissipative elements is independent from the current amplitude of the excited source. The forced and self-excited oscillation components can simultaneously coexist unconditionally. chaos can easy be produced by the nonlinear coupling of the two harmonic components. The intrinsic attributes of the chaos can be sufficiently revealed with the help of this kind of lossless circuits.

The numerical simulation of hydraulic fracturing fracture propagation is the core content of hydraulic fracturing design and construction. Its simulation results directly affect the effect of fracturing, and can effectively guide the fracturing construction plan and reduce the construction risk. At present, two-dimensional or quasi-three-dimensional models are mainly used, but most of them are used to simulate the vertical fracture of hydraulic fracturing. There are errors in the application process. In this paper, a three-dimensional mathematical model, including an elastic rock mechanics equation and a material flow continuity equation, is established to simulate horizontal fracture propagation in shallow reservoirs. The emphasis of this paper is to propose a new method for solving equations. The basic idea of the iteration method has been proposed by previous scholars: Firstly, assuming that the initial pressure of each point in the fracture is uniform, the fracture height of each initial point can be obtained by using Equation (20). Using the initial height values, the pressure values at each point of continuous variation are calculated by Equation (16), and then the new fracture height values are obtained by Equation (20). Because of the equal initial pressure, this method leads to too many iterations in the later stages, which makes the calculation more complicated. In this paper, a new Picca iteration method is proposed. The iteration parameters are changed sequentially. Firstly, the distribution value of fracture height is assumed. Then, the pressure distribution value is calculated according to Equation (16). Then, the new distribution value of fracture height is obtained by bringing the obtained pressure distribution value into Equation (20). Then, the new distribution value of the fracture height is calculated according to Equation (16). The pressure distribution value completes an iteration process until the iteration satisfies the convergence condition. In addition, Sneddon’s model is introduced into the hypothesis of fracture height to obtain the maximum fracture height and assume that the initial fracture profile is a parabola. Finally, the proposed method can rapidly improve the convergence rate. Next, on the basis of investigating the solutions of previous equations, the Galerkin finite element method is used to solve the above equations. The new Picard iteration sequence method is applied to solve the height and pressure at different points in the fracture. By calculating the stress intensity factor, we can judge whether the fracture continues to extend or not, and then simulate the full three-dimensional horizontal fracture of the hydraulic fracturing expansion process. The infiltration process of three types of oil reservoirs in Daqing Changyuan oilfield is simulated. The results show that during the initial fracture stage, the radius and height of fractures increase rapidly, and the rate of increase slows down with the increase of construction time. The height and net pressure of each point in the fracture are unequal. The height and net pressure of the fracture in the wellbore reach the maximum, and gradually decrease to the front of the fracture. Compared with conventional fracturing, the fracturing-flooding percolation process has the characteristics of short fracture-making and large vertical percolation distance, which can greatly increase the swept volume of flooding fluid and thus enhance oil recovery. With the increase in the rock modulus of elasticity, the radius of fractures decreases and the height of fractures increases. With the increase in construction displacement, the radius of fractures hardly changes, the height of fractures increases, and the vertical infiltration distance of the fractures increases. It is suggested that the construction displacement should be 4.0 m3/min. In the range of fracturing fluid viscosity in the studied block, with the change of fracturing fluid viscosity, the change of fracture radius and height is not obvious. In order to further increase sweep volume, the fracturing fluid viscosity should be further reduced.