The state transition matrix (STM) is the core to understand the sensitivity of dynamics to initial conditions. It serves as a critical tool for orbit design and control and provides the mathematical foundation for computing invariant manifolds. In conservative systems like the circular restricted three-body problem (CR3BP), the global geometry is governed by fundamental symmetries, such as symplecticity and time-reversal symmetry. However, few numerical schemes for computing the STM itself are designed to preserve these symmetries simultaneously. This paper proposes the STM of variational integrators, which is derived from discrete variational mechanics in Lagrangian viewpoint, ensuring the preservation of discrete Lagrangian symplectic structure. The STM is applied to the differential correction method in the CR3BP, and a case study for finding the near rectilinear Halo orbit (NRHO) is provided. Compared with classical Runge-Kutta methods, the numerical results indicate that the proposed method maintains high accuracy with large step sizes, and the STM possesses the properties of a symplectic matrix and time-reversal symmetry.
The inhomogeneous advective wave equation of the dimensionless density is characterized by two distinct source terms caused by the surface deformation of fluid element and non-homentropic effect. Thus, it could be very useful in the diagnosis of physical sources in aeroacoustic problems. Due to the rather complex mathematical forms of the left-hand-side terms, this density-based advective wave equation has not been solved directly, and thus its capability for predicting acoustic far-field behavior remains unclear. In this study, we assess for the first time the far-field acoustic prediction performance of this equation in a two-dimensional spatially developing subsonic mixing layer. First, using high-fidelity direct numerical simulation (DNS), we obtain both the near-field sound sources and far-field acoustic data of the subsonic mixing layer. Next, we solve the density-based advective wave equation using a finite-difference method and the Lighthill acoustic analogy equation via the Green’s function method. The results demonstrate that both equations can accurately reproduce the radiated sound field obtained from DNS. Finally, by solving a simplified form of the density-based advective wave equation, where all viscous terms and the squared dilatation term are neglected, we find that the predicted radiated sound remains in good agreement with the DNS results, with a maximum deviation above the mixing layer less than 1.5 dB. This confirms that the inhomogeneous advective wave equation of the dimensionless density is capable of accurately predicting far-field sound.
Supersonic experiments and simulations were employed to investigate the mixing and combustion characteristics of a scramjet combustor with a rearwall-step cavity flameholder. A transverse ethylene jet injection scheme was employed at equivalence ratios (ER) of 0.20, 0.24, 0.28, 0.32, 0.36, and 0.40, with two distinct injection distances (i.e., d = 5 and 35 mm). Under short injection distance, the fuel plume has a lower flow velocity, and the cavity entrainment effect results in higher subsonic oxygen utilization efficiency and a larger subsonic zone, which is conducive to flame propagation and the flame maintained jet wake flame stabilization mode. Furthermore, subjected to the combustion backpressure, the short injection distance results in a higher penetration depth, thereby further enhancing the fuel-oxygen utilization efficiency. In contrast, the fuel distribution under a long injection distance basically conforms to the empirical formula, and the mixing efficiency is higher near the leading edge of the cavity. However, the high-velocity fuel plume restricts component transport, which results in a lower total oxygen utilization efficiency. Consequently, at ER = 0.20 and 0.24, the flame is anchored at the cavity shear layer, while ER = 0.36 and 0.40, significant combustion oscillation occurs near the cavity leading edge.
Programmable smart materials that transform from a two-dimensional (2D) film into a three-dimensional (3D) surface under external stimuli have broad applications in soft robotics, flexible devices, and bioinspired systems. However, existing studies primarily rely on forward design, in which stimulus-response distributions are iteratively adjusted to approximate a target shape—an approach that struggles with geometrically complex surfaces. This work introduces a generic inverse design method based on conformal mapping for thin-film materials with in-plane isotropy. The method directly retrieves the required 2D stimulus distribution from any prescribed three-dimensional target surface. Using the boundary first flattening algorithm, we obtain a high-accuracy conformal flattening parameterized by local area growth, which is then interpreted as programmable material parameters (e.g., thermal expansion coefficients). Under stimulus fields, these programmed parameters generate controlled in-plane strain mismatches that drive the film to morph into a surface with the desired 3D shape. Systematic finite element simulations on surfaces with constant Gaussian curvature, canonical geometric shapes, and complex real-world geometries demonstrate that the method can accurately reconstruct target three-dimensional forms with sufficient accuracy. Moreover, the framework allows mapping the same 3D surface onto 2D domains of different prescribed boundaries, enabling flexible control over the pre-deformation material geometries. Overall, this study provides a universal, precise, and efficient inverse design strategy for shape programming of isotropic thin-film materials and lays a theoretical foundation for the programmable fabrication of complex three-dimensional morphologies in soft devices.
Buckling deformations in stiff/soft bilayers, consisting of a stiff membrane and a soft film, involve complex, multi-order evolutionary processes. The intrinsic periodic length λ0 at the threshold of the primary buckling instability can be regarded as a characteristic length of buckling deformation, providing a key reference for quantifying advanced buckling modes. However, the two prevalent expressions for λ0 are approximate results obtained under the respective limiting assumptions of an infinitely thick soft film and an infinitely thin one. Since soft films invariably have finite thickness in practice, the approximate expressions often fail to accurately capture λ0. To overcome this shortcoming, we develop a universal λ0 expression based on a rigorous theoretical approach and precise boundary conditions. This expression shows that, as the thickness contrast between the soft and stiff layers gradually increases above 1, λ0 exhibits rapid initial growth followed by saturation. We conducted systematic experiments to validate this dependency, finding it is valid across the entire range of thickness contrasts in multiple bilayer systems. It is this dependency that accounts for the existence of two markedly different approximate expressions for λ0 in the two limiting cases of soft film thickness. Our expression can be reduced directly to existing approximate forms by introducing limiting conditions. These results indicate that our expression encompasses existing approximate formulas, representing a more comprehensive, universal form.
