Search Results (35)

This work investigates the nonlinear flexural dynamics of a macroscale cantilever beam by combining analytical modeling, symbolic solution techniques, numerical simulation, and vision-based experiments. Starting from the Euler–Bernoulli equation with geometric and inertial nonlinearities, a reduced-order model is derived via a single-mode Galerkin projection, justified by the experimentally confirmed dominance of the fundamental bending mode. The resulting nonlinear ordinary differential equation is solved analytically using two symbolic methods rarely applied in structural vibration studies: the Extended Direct Algebraic Method (EDAM) and the Sardar Sub-Equation Method (SSEM). Comparison with high-accuracy numerical integration shows that EDAM reproduces the nonlinear waveform with high fidelity, including the characteristic non-sinusoidal distortion induced by mid-plane stretching. High-speed vision-based measurements provide displacement data for a physical cantilever beam undergoing free vibration. After calibrating the linear stiffness, analytical and experimental responses are compared in terms of the dominant oscillation frequency. The analytical model predicts the classical hardening-type amplitude–frequency dependence of an ideal Euler–Bernoulli cantilever, whereas the experiment exhibits a clear softening trend. This contrast reveals the influence of real-world effects, such as initial curvature, boundary compliance, or micro-slip at the clamp, which are absent from the idealized formulation. The combined analytical–experimental framework thus acts as a diagnostic tool for identifying competing nonlinear mechanisms in flexible structures and provides a compact physics-based reference for reduced-order modeling and structural health monitoring. Full article

This study investigates the impact of eddy viscosity on equatorially trapped waves and the instability of the background shear in a simple barotropic–baroclinic model. It is the first study to include eddy viscosity in the study of tropical wave dynamics. This study also unifies the study of baroclinic and barotropic instabilities by using a coupled barotopic and baroclinic model of the tropical atmosphere. Linear wave theory is combined with a systematic Galerkin projection of the baroclinic dynamical fields onto parabolic cylinder functions. This study investigates varying shear strengths, eddy viscosities, and their combined effects. In the absence of shear, baroclinic and barotropic waves decouple. The baroclinic waves themselves separate into triads, forming the equatorially trapped wave modes known as Matsuno waves. However, when a strong eddy viscosity is included, the structure and propagation characteristics of these equatorial waves are significantly altered. Different wave types interact, leading to strong mixing in the meridional direction and coupling between meridional modes. This coupling destroys the Matsuno mode separation and offers pathways for these waves to couple and interact with one another. These results suggest that viscosity does not simply suppress growth; it may also reshape the propagation characteristics of unstable modes. In the presence of a background shear, some wave modes become unstable, and barotropic and baroclinic waves are coupled. Without eddy viscosity, instability begins with small scale and slowly propagating modes, at arbitrary small shear strengths. This instability manifests as an ultra-violet catastrophe. As the shear strength increases, the catastrophic instability at small scales expands to high-frequency waves. Meanwhile, instability peaks emerge at synoptic and planetary scales along several Rossby mode branches. When a small eddy viscosity is reintroduced, the catastrophic small-scale instabilities disappear, while the large-scale Rossby wave instabilities persist. These westward-moving modes exhibit a mixed barotropic–baroclinic structure with signature vortices straddling the equator. Some vortices are centered close to the equator, while others are far away. Some waves resemble synoptic-scale monsoon depressions and tropical easterly waves, while others operate on the planetary scale and present elongated shapes reminiscent of atmospheric-river flow patterns. Full article

This paper proposes a data-driven modeling and control method for wireless power transmission systems. To address problems such as parameter deviation and high-order complexity in traditional circuit-theory-based modeling, this paper adopts the data-driven Petrov-Galerkin projection and the generalized Lyapunov balancing method to obtain a reduced-order model directly from experimental data. This approach formulates quadratic matrix inequalities to characterize the data and noise, enabling the direct design of a reduced-order model without intermediate system identification steps. The resulting model order is reduced to merely 2–4. Furthermore, by constructing the extended state space and solving the algebraic Riccati equation, we design a linear quadratic regulator-proportional integral controller with integral action to eliminate steady-state error. Experimentally, the method proves to be independent of detailed physical models while achieving both high-fidelity modeling and superior control. Full article

This research presents a comparative analysis of numerical methods for solving first-kind Fredholm integral equations using the Bubnov–Galerkin method with various wavelet and orthogonal polynomial bases. The bases considered are constructed from Legendre, Laguerre, Chebyshev, and Hermite wavelets, as well as Alpert multiwavelets and CAS wavelets. The effectiveness of these bases is evaluated by measuring errors relative to known analytical solutions at different discretization levels. Results show that global orthogonal systems—particularly the Chebyshev and Hermite—achieve the lowest error norms for smooth target functions. CAS wavelets, due to their localized and oscillatory nature, produce higher errors, though their accuracy improves with finer discretization. The analysis has been extended to incorporate perturbations in the form of additive noise, enabling a rigorous assessment of the method’s stability with respect to different wavelet bases. This approach provides insight into the robustness of the numerical scheme under data uncertainty and highlights the sensitivity of each basis to noise-induced errors. Full article

A model for the flow and thermal explosion of a micropolar gas is investigated, assuming the equation of state for a real gas. This model describes the dynamics of a gas mixture (fuel and oxidant) undergoing a one-step irreversible chemical reaction. The real gas model is particularly suitable in this context because it more accurately reflects reality under extreme conditions, such as high temperatures and high pressures. Micropolarity introduces local rotational dynamic effects of particles dispersed within the gas mixture. In this paper, we first derive the initial-boundary value system of partial differential equations (PDEs) under the assumption of spherical symmetry and homogeneous boundary conditions. We explain the underlying physical relationships and then construct a corresponding approximate system of ordinary differential equations (ODEs) using the Faedo–Galerkin projection. The existence of solutions for the full PDE model is established by analyzing the limit of the solutions of the ODE system using a priori estimates and compactness theory. Additionally, we propose a numerical scheme for the problem based on the same approximate system. Finally, numerical simulations are performed and discussed in both physical and mathematical contexts. Full article

The aim of this article is to analyze the efficiency and accuracy of finite-difference and finite-element Galerkin schemes for non-stationary hyperbolic and parabolic problems. The main problem solved in this article deals with the construction of accurate and efficient discrete schemes on nonuniform and dynamic grids in time and space. The presented stability and convergence analysis enables improving the existing accuracy estimates. The obtained stability results show explicitly the rate of accumulation of interpolation and projection errors that arise due to the movement of grid points. It is shown that the cases when the time grid steps are doubled or halved have different stability properties. As an additional technique to improve the accuracy of discretizations on non-stationary space grids, it is recommended to use projection operators instead of interpolation operators. This technique is used to solve a test parabolic problem. The results of specially selected computational experiments are also presented, and they confirm the accuracy of all theoretical error estimates obtained in this article. Full article

This study delves into the construction of reduced-order models (ROMs) of a flow field over a NACA 0012 airfoil at a moderate Reynolds number and an angle of attack of $8^{\circ }$. Numerical simulations were computed through the finite-volume solver OpenFOAM. The analysis considers two different reduction techniques: the standard Galerkin projection method, which involves projecting the governing equations onto proper orthogonal decomposition modes (POD−ROMs), and the cluster-based network model (CNM), a fully data-driven nonlinear approach. An analysis of the topology of the dominant POD modes was conducted, uncovering a traveling wave pattern in the wake dynamics. We compared the performances of both ROM techniques regarding their prediction of flow field behavior and integral quantities. The ROM framework facilitates the practical actuation of control strategies with significantly reduced computational demands compared to the full-order approach. Full article

FSI simulations of flapping motions have been widely investigated to develop a flapping-wing micro air vehicle. Because an intensive parametric study is important for the product design, a computationally efficient model is required. The purpose of the present study was to develop a reduced-order model of flapping motion. Among the various methods available to solve FSI problems, we employed the Dirichlet–Neumann partitioned iterative method, in which three sub-systems (fluid mesh update, fluid analysis, and structural analysis) are executed. In the proposed analysis system, first, snapshot data of structural displacement, fluid velocity, fluid pressure, and displacement for the fluid mesh update were collected from a high-fidelity FSI analysis. Then, the snapshot data were used to create low-dimensional surrogate systems of the above three sub-systems based on the POD under Galerkin projection (i.e., the POD-Galerkin method). In numerical examples, we considered a two-dimensional FSI problem of simplified flapping motion. The problem was described via two parameters: frequency and amplitude of flapping motion. We demonstrated the effectiveness of the presented reduced-order model in significantly reducing computational time while preserving the desired accuracy. Full article

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable $q=a\left(x\right)u_x$. An H${}^1$-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babu$\check{\mathrm{s}}$ka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The $L^2\left(L^2\right)$ norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H${}^1$-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function. Full article

The design of tunnels in cold regions contributes greatly to the feasibility and sustainability of highways. Based on the heat transfer mechanism of the tunnel surrounding rock–lining–air, this paper uses FEPG software to carry out secondary excavation and development, then the air heat convection calculation model is established by using a three-dimensional extension of the characteristic-based operator-splitting (CBOS) finite-element method and the explicit characteristic–Galerkin method. By coupling with the heat conduction model of the tunnel lining and surrounding rock, the heat conduction-thermal convection fluid–structure interaction finite-element calculation model of tunnels in cold regions is established. Relying on the Qinghai Hekashan tunnel project, the temperature field of the tunnel portal section is calculated and studied by employing the fluid–structure interaction finite-element model and then compared with the field monitoring results. It is found that the calculated values are basically consistent with the measured values over time, which proves the reliability of the model. The calculation results are threefold: (1) The temperature of the air, lining, and surrounding rock in the tunnel changes sinusoidally with the ambient temperature. (2) The temperature of each layer gradually lags behind, and the temperature variation amplitude of the extreme value of the layer temperature gradually decreases with the increase in the radial distance of the lining. (3) In the vicinity of the tunnel entrance, the lining temperature of each layer remains unchanged, and the temperature gradually decreases or increases with the increase in the depth. The model can be used to study and analyze the temperature field distribution law of the lining and surrounding rock under different boundary conditions, and then provide a calculation model with both research and practical value for the study of the temperature distribution law of tunnels in cold regions in the future. Full article

Overcoming the problem of excessive engine noise at low altitudes is a formidable task on the way to developing a supersonic passenger aircraft. The focus of this paper is on the fan noise shielding during take-off, investigated as part of the DLR project ELTON SST (estimation of landing and take-off noise of supersonic transport) for an in-house aircraft design. The supersonic inlet is required to provide the proper quantity and uniformity of air to the engine over a wider range of flight conditions than the subsonic inlet. For passenger aircraft, the noise problem influences engine integration and placement, and the new generation of supersonic transport would require innovative engineering solutions in order to come up with an efficient low-noise design. Potential solutions are evaluated using DLR tools capable of accurate source generation and noise propagation to the far-field. For low-speed aircraft operation, the method of choice is a strongly coupled volume-resolving discontinuous Galerkin (DG) and fast multipole boundary element method (FM-BEM) which is applied due to a large disparity between the Mach numbers on the interior and exterior of the inlet. The method is used for obtaining the acoustic signature of the full-scale model at realistic flight points, including the application of the programmed lapse rate (PLR), which involves simulations at higher pitch angles than for the reference flight path. The results show that the proposed method is highly suitable for obtaining accurate noise footprints during the low-speed phase and could be used to assist with certification procedures of future supersonic aircraft. Full article

We propose a Quantum Neural Network (QNN) for predicting stabilization parameter for solving Singularly Perturbed Partial Differential Equations (SPDE) using the Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. SPDE-Q-Net, a QNN, is proposed for approximating an optimal value of the stabilization parameter for SUPG for 2-dimensional convection-diffusion problems. Our motivation for this work stems from the recent progress made in quantum computing and the striking similarities observed between neural networks and quantum circuits. Just like how weight parameters are adjusted in traditional neural networks, the parameters of the quantum circuit, specifically the qubits’ degrees of freedom, can be fine-tuned to learn a nonlinear function. The performance of SPDE-Q-Net is found to be at par with SPDE-Net, a traditional neural network-based technique for stabilization parameter prediction in terms of the numerical error in the solution. Also, SPDE-Q-Net is found to be faster than SPDE-Net, which projects the future benefits which can be earned from the speed-up capabilities of quantum computing. Full article

The purpose of this paper is the identification of high-fidelity digital twin data models from numerical code outputs by non-intrusive techniques (i.e., not requiring Galerkin projection of the governing equations onto the reduced modes basis). In this paper the author defines the concept of the digital twin data model (DTM) as a model of reduced complexity that has the main feature of mirroring the original process behavior. The significant advantage of a DTM is to reproduce the dynamics with high accuracy and reduced costs in CPU time and hardware for settings difficult to explore because of the complexity of the dynamics over time. This paper introduces a new framework for creating efficient digital twin data models by combining two state-of-the-art tools: randomized dynamic mode decomposition and deep learning artificial intelligence. It is shown that the outputs are consistent with the original source data with the advantage of reduced complexity. The DTMs are investigated in the numerical simulation of three shock wave phenomena with increasing complexity. The author performs a thorough assessment of the performance of the new digital twin data models in terms of numerical accuracy and computational efficiency. Full article

This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy. Full article

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted $L^2$ projection as well as an optimal convergence of order $k+1$ for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented. Full article