Numerical Stability Analysis Research Articles

Stability analysis study for the time-fractional Galilei invariant advection-diffusion model of distributive order using an efficient hybrid approach

Abstract In this manuscript, a new model of the time-fractional Galilei-invariant advection-diffusion model of distributed order is studied. An efficient hybrid numerical approach with high accuracy is used to estimate this equation. The finite difference numerical method is used to approximate the fractional operator in terms of the time variable and to approximate the integral term of distributed order, the Gaussian--Legendre integration is applied. To obtain a fully discrete numerical approach, we used a spectral element numerical approach, in which Legendre polynomials are used as the basis function. For the proposed numerical approach, the error and stability analysis are studied. For the efficiency of the numerical approach, some numerical examples are presented with graphs and tables.

Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation

In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

Family of 2:1 resonant quasi-periodic distant retrograde orbits in cislunar space

Given the current enthusiasm for lunar exploration, the 2:1 resonant distant retrograde orbit (DRO) in Earth-Moon space is of interest. To gain an in-depth understanding of the complex dynamic environment in cislunar space and provide more options for parking orbits, this paper investigates the existence of quasi-periodic orbits near the 2:1 resonant DRO in the circular restricted three-body problem (CR3BP). Firstly, the numerical computation approach, continuation strategy, and stability analysis method of quasi-periodic orbits are introduced. Then, addressing the primary challenges in the continuation progress, we have developed an adaptive continuation algorithm with automatic adjustment of the step size and the number of discrete points that characterize the invariant torus. Finally, two types of 2D quasi-DROs and their linear stability properties are explored. Using Poincaré sections, we investigated the boundaries of the maximum extent attainable by both 2D quasi-DRO families in the CR3BP at a specific Jacobi energy, confirming that both types of quasi-periodic families have reached their respective boundaries. The algorithm described in this paper is beneficial for facilitating the computation of quasi-periodic families and aids in discovering additional potential dynamical structures.

Numerical Stability Analysis of Sloped Geosynthetic Encased Stone Column Composite Foundation under Embankment Based on Equivalent Method

As an effective technology for the rapid treatment of soft-soil foundations, geosynthetic encased stone column (GESC) composite foundations are commonly used in various embankment engineerings, including those situated on sloped soft foundations. Nevertheless, there is still a scarcity of stability studies for sloped GESC composite foundations. Several 3D numerical models for sloped GESC composite foundations were established using an equivalent method. The influences of the area replacement ratio and the tensile strength of geosynthetic encasement on the stability were investigated. The results showed that the stability increased nonlinearly with the area replacement ratio, and there existed an optimal area replacement ratio (e.g., 24.56% in this study) to balance the safety and economic requirements. The stability increased linearly with the tensile strength of geosynthetic encasement at low tensile strength levels (lower than 105 kN/m in this study), and the impact was relatively limited compared with that of the area replacement ratio. In addition, the stability generally decreased nonlinearly as the foundation slope decreased, and high-angle (foundation slope close to 30°) sloped GESC composite foundations are recommended to be treated with multiple reinforcement techniques. The relationship between the minimum area replacement ratio and the foundation slope was further quantified by an exponential function, allowing for the determination of the area replacement ratio of various sloped GESC composite foundations and providing theoretical guidance for engineering practice.

Numerical Analysis of Stability of Plant-Animal Interactions in Riparian Zones

Numerical Analysis of Stability of Plant-Animal Interactions in Riparian Zones

Data-driven and equation-free methods for neurological disorders: analysis and control of the striatum network.

The striatum as part of the basal ganglia is central to both motor, and cognitive functions. Here, we propose a large-scale biophysical network for this part of the brain, using modified Hodgkin-Huxley dynamics to model neurons, and a connectivity informed by a detailed human atlas. The model shows different spatio-temporal activity patterns corresponding to lower (presumably normal) and increased cortico-striatal activation (as found in, e.g., obsessive-compulsive disorder), depending on the intensity of the cortical inputs. By applying equation-free methods, we are able to perform a macroscopic network analysis directly from microscale simulations. We identify the mean synaptic activity as the macroscopic variable of the system, which shows similarity with local field potentials. The equation-free approach results in a numerical bifurcation and stability analysis of the macroscopic dynamics of the striatal network. The different macroscopic states can be assigned to normal/healthy and pathological conditions, as known from neurological disorders. Finally, guided by the equation-free bifurcation analysis, we propose a therapeutic close loop control scheme for the striatal network.

Numerical Analysis of Dynamic Stability of Flutter Panels in Supersonic Flow

This paper introduces a novel numerical method for investigating the dynamic stability of a flutter panel exposed to a supersonic gas flow and a fluctuating axial excitation force. Initially, the system equation of motion was derived using Lagrange’s equation, where the first two modes are coupled Mathieu–Hill equations with damping, constituting a system of linear second-order differential equations with periodically variable coefficients. Subsequently, a new numerical method was proposed to analyze the dynamic stability of coupled Mathieu–Hill equations with damping. This method involves breaking down an arbitrary parametric load into discrete segments to approximate the variable excitation function using a step function. The system responses of each segment are then accumulated in matrix form. The proposed numerical method proves particularly effective for dynamic systems whose parameters cannot be treated as small. In practical application, the method allows the construction of instability regions corresponding to natural frequencies, subharmonics, and combination frequencies. Dynamic stability diagrams were generated based on dynamic pressure ratio, air/panel density ratio, Mach number, panel thickness—length ratio, and excitation frequency. The results demonstrated general agreement with those obtained through Hsu’s perturbation method, however, our numerical results have proven more accurate. The paper concludes by offering suggestions for suppressing panel flutter through appropriate parameter combinations.

One-level and two-level operator splitting methods for the unsteady incompressible micropolar fluid equations with double diffusion convection

In this paper, a one-level operator splitting method (OOSM) and a two-level operator splitting method (TOSM) for the two-dimensional or three-dimensional (2D/3D) unsteady incompressible micropolar fluid equations with double diffusion convection (IMFDDC) are proposed and analyzed. Firstly, a OOSM is constructed, which consists of five steps. The projection strategy is adopted to get the values of linear velocity and pressure in the first two steps, then the three elliptic subproblems about the angular velocity, the temperature, and the concentration variables are solved in the last three steps. The original variables with the predefined boundary conditions need to be solved separately in each step. Next, in order to solve the problem efficiently, a TOSM is presented, which only solves the nonlinear equations in the coarse level subspaces and the Oseen type linearized equations in the fine level. The numerical analysis of stability and error estimates of the fully discrete methods show that TOSM can reach the same accuracy as the OOSM with H∼O(h2/3). Numerical examples are provided to confirm the effectiveness and reliability of the proposed methods.

Stochastic Rock Slope Stability Analysis: Open Pit Case Study with Adjacent Block Caving

In last decades, numerical modelling become a useful tool for solving complex geoengineering problems such as slope stability. On the other hand, probabilistic slope stability analyses are able to consider the variability of the rock mass properties in the decision making process. However, the application of probabilistic slope stability analysis with large three-dimensional numerical models is still limited due to the computational expenses of evaluating a substantial number of considered models. It is well-known that response surface methodology (RSM) couples the mathematical and statistical techniques to relate input variables to the response, allowing a reliable outcome and reasonable time of the analysis. Taking these advantages, this article presents an application of RSM in probabilistic slope stability analysis using three dimensional distinct element modelling. For this purpose, the most influencing factors including: uniaxial compressive strength, geological strength index (GSI), and shear strength of discontinuities, were taken into consideration to determine the probability of failure of an open pit slope located in vicinity of a block caving-induced subsidence crater. Numerical analysis of slope stability was conducted for an open pit slope using the Distinct Element Method code-3DEC. Probability distribution of the factor of safety (FS) was determined and possible slope failure mechanism was observed. In addition, the block caving-induced subsidence was investigated. The final outcomes indicate that Response Surface Methodology is applicable when couples with numerical modelling of complex issues, GSI is considered the most influential variable. The studied slope is considered stable due to the low value of the FS probability distribution (2.2%). This research is expected to provide a reference for slope stability analysis in case of transition from Open Pit to Underground Mining.

Solving Sparse Linear Systems Faster than Matrix Multiplication

Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph-structured linear systems, in the general setting, the bit complexity of solving an n × n linear system Ax = b is Õ ( n ω ), where ω is the matrix multiplication exponent. Improving on this has been an open problem even for sparse linear systems with poly( n ) condition number. In this paper, we present an algorithm that solves linear systems with sparse coefficient matrices asymptotically faster than matrix multiplication for any ω > 2. This speedup holds for any input matrix A with o ( n ω−1 / log (κ( A ))) non-zeros, where κ( A ) is the condition number of A . Our algorithm can be viewed as an efficient, randomized implementation of the block Krylov method via recursive low displacement rank factorization. It is inspired by an algorithm of Eberly et al. for inverting matrices over finite fields. In our analysis of numerical stability, we develop matrix anti-concentration techniques to bound the smallest eigenvalue and the smallest gap in the eigenvalues of semi-random matrices.

Rock Mass Characterization and Slope Stability Analysis Inlet Portal of Jragung Dam Diversion Tunnel, Indonesia

Abstract Slope stability is a serious problem that academics are still looking into in terms of human safety, equipment, and structures along the slope. Failure on slopes happens for a variety of reasons, including rock physical and mechanical properties, discontinuity factors, slope geometry, and external variables like as rainfall and seismic activity. The research location was conducted on the Jragung dam diversion tunnel, administratively located in Candirejo village, Pringapus sub-district, Semarang, Central Java Province. The purpose of this study was to use an empirical and numerical method to investigate the impact of rock mass categorisation on slope stability. The RMR (Rock mass rating) method is used to characterize rock mass, and the SMR (Slope mass rating) method is used to measure slope stability. The numerical slope stability analyses were carried out with the help of Phase2 (Rocscience, Inc.). The results showed that the slope at the intake portal was class IV (poor), and that it was unstable utilizing the SMR technique. According to numerical research, the basic design cut slopes with a 51° inclination were found to be unstable even after strengthening. It was suggested that the slope geometry and support measures be adjusted even more. Modified cut slopes were found to be stable after the slope inclinations were changed from 51° to 27° and shotcrete and rock bolt reinforcement were added.

Numerical threshold stability of a nonlinear age-structured reaction diffusion heroin transmission model

This paper deals with the numerical threshold stability of a nonlinear age-space structured heroin transmission model. A semi-discrete system is established by spatially domain discretization of the original nonlinear age-space structured model. A threshold value is proposed in stability analysis of the semi-discrete system and named as a numerical basic reproduction number. Besides the role it plays in numerical threshold stability analysis, the numerical basic reproduction number can preserve qualitative properties of the exact basic reproduction number and converge to the latter while stepsizes vanish. A fully discrete system is established via a time discretization of the semi-discrete system, in which an implicit-explicit technique is implemented to ensure the preservation of the biological meanings (such as positivity) without CFL restriction. Some numerical experiments are exhibited in the end to confirm the conclusions and explore the final state.

Numerical simulation analysis of slope stability from rainwater utilizing the Civil3D+Flac3D coupling

Building Information Modeling (BIM) technology serves as an effective means for engineering modeling and information integration, surpassing traditional two-dimensional approaches with its superior spatial analysis capabilities. While BIM has gained widespread applications in slope engineering, a gap persists in the integration between protective design and simulation analysis in terms of slope stability. This study employs literature review, theoretical analysis, software development, and numerical simulation methods to explore the application of BIM models in slope stability analysis. Through the integrated application of BIM model construction, mesh subdivision, and stability analysis, we address the deficiency in three-dimensional slope stability inherent in BIM by coupling Civil3D’s secondary development with Flac3D, a numerical simulation software. This approach not only addresses the shortcomings of numerical analysis software in constructing complex geological models but also employs a collaborative multi-software approach for comparison to validate its suitability in slope engineering. The results demonstrate that models built using BIM software are more comprehensive and possess high precision in numerical simulation, thus exhibiting significant practical value in real-world engineering applications.

Higher‐Order Topological Corner States in a Specially Distorted Photonic Kagome Lattice

AbstractKagome lattices have garnered significant interest over the decades due to their rich physics and implications for exotic superconductors and topological materials. In particular, the so‐called “breathing Kagome lattice” (BKL) is often used as a prototypical model to understand higher‐order topological insulators. Here it is demonstrated that higher‐order corner states exist in a specially distorted Kagome lattice, resembling the “Inverse Hexagram” lattice distortion originally introduced in the study of charge density waves in Kagome metals. Importantly, it is found that such an Inverse Hexagram‐like Kagome lattice with C6 rotational symmetry hosts two types of in‐gap corner states, in contrast to a flat‐cutting BKL of C3 symmetry that supports only one type of corner state under the tight‐binding condition. It is shown that the nontrivial corner states exhibit a fractional corner anomaly, revealing the feature of higher‐order topology. Using laser‐written waveguide lattices as a photonic platform, these two types of corner states are experimentally observed, along with a direct comparison with the corner excitation at their trivial counterparts. Experimental observations are further corroborated by numerical simulations and stability analysis under random perturbation. These results may prove relevant and applicable to similar phenomena in other Kagome platforms beyond photonics.

Critical transitions on a route to chaos of natural convection on a heated horizontal circular surface

The transition route and bifurcations of the buoyant flows developing on a heated horizontal circular surface are elaborated using direct numerical simulations and direct stability analysis. A series of bifurcations, as a function of Rayleigh numbers ( $Ra$ ) ranging from $10^6$ to $6.0\times 10^7$ , are found on the route to chaos of the flows at $Pr=7$ . When $Ra<1.0\times 10^3$ , the buoyant flows above the heated horizontal surface are dominated by conduction, because of which the distinct thermal boundary layer and plume are not present. At $Ra=1.1\times 10^6$ , a Hopf bifurcation occurs, resulting in the flow transition from a steady state to a periodic puffing state. As $Ra$ increases further, the flow enters a periodic rotating state at $Ra=1.9\times 10^6$ , which is a unique state that was rarely discussed in the literature. These critical transitions, leaving from a steady state and subsequently entering a series of periodic states (puffing, rotating, flapping and period-doubling) and finally leading to chaos, are diagnosed using two-dimensional Fourier transforms. Moreover, direct stability analysis is conducted by introducing random numerical perturbations into the boundary condition of the surface heating. We find that when the state of a flow is in the vicinity of critical values (e.g. $Ra=2.0\times 10^6$ ), the flow is conditionally unstable to perturbations, and it can bifurcate from the rotating state to the flapping state in advance. However, for relatively stable flow states, such as at $Ra=1.5\times 10^6$ , the flow remains in its periodic puffing state even though it is being perturbed.

Numerical stability analysis of Godunov-type schemes for high Mach number flow simulations

Modern shock-capturing schemes often suffer from numerical shock instabilities when simulating strong shocks, limiting their application in supersonic or hypersonic flow simulations. In the current study, we devote our efforts to investigating the shock instability problem for second-order schemes, which has not gotten enough attention in previous research but is crucial to address. To this end, we develop the matrix stability analysis method for the finite-volume Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) approach, taking into account the influence of reconstruction. With the help of this newly developed method, the shock instability of second-order schemes is investigated quantitatively and efficiently. The results demonstrate that when second-order schemes are employed, whether shock instabilities will occur is closely related to the property of Riemann solvers, just like the first-order case. However, enhancing spatial accuracy still impacts the shock instability problem, and the impact can be categorized into two types depending on the dissipation of Riemann solvers. Furthermore, the research emphasizes the impact of the numerical shock structure, highlighting both its role as the source of instability and the influence of its state on the occurrence of instability. One of the most significant contributions of this study is the confirmation of the multidimensional coupled nature of shock instability. Both one-dimensional and multidimensional instabilities are proven to influence the instability problem, and they have different properties. Moreover, this paper reveals that increasing the aspect ratio and distortion angle of the computational grid can help mitigate shock instabilities. The current work provides an effective tool for quantitatively investigating the shock instabilities for second-order schemes, revealing the inherent mechanism and thereby contributing to the elimination of shock instability.

Numerical simulation and stability analysis of radiative magnetized hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface

The power electrical devices are being continuously trimmed down and must disperse a greater amount of heat flux, overheating has become a significant issue. Ferrofluids, also known as magnetic nanofluids, consist of solid magnetic nanoparticles that exhibit superior thermal conductivity compared to their underlying fluids, which can be either aqueous or oil based. This facilitates increased rates of convective heat transfer, but more importantly, it allows for external flow modification using a magnetic field. This work is unique in that it analyses hybridized ferrofluid flow over a shrinking/stretching surface with radiative magnetization and significant magnetic force. The governing equations were solved using the bvp4c function in MATLAB, after being transformed into ordinary differential equations (ODEs) by using similarity transformations. The system acquires a bimodal solution, and stability analysis validates the physical and dynamic stability of the first solution. The local skin friction coefficient, the local Nusselt number, the temperature and velocity profiles, and the substantial influence of governing effects are all studied and graphically displayed. The key findings are the values of Nux1Rex are decreasing for both second and first solutions as values of φCoFe2O4 are increasing. An increasing trend for the values of RexCf is observed for both second and first solutions as values of φCoFe2O4 increases. The results compareded with the results reported in literature and found a significant agreement.

Numerical threshold stability analysis of a positivity-preserving IMEX numerical scheme for a nonlinear age-space structured SIR epidemic model

Numerical threshold stability analysis of a positivity-preserving IMEX numerical scheme for a nonlinear age-space structured SIR epidemic model

Viscous correction to the potential flow analysis of Rayleigh–Taylor instability in a Rivlin–Ericksen viscoelastic fluid layer with heat and mass transfer

AbstractThe current investigation focuses on examining viscous corrections for viscous potential flow (VCVPF) analysis concerning the Rayleigh–Taylor instability occurring at the interface of a Rivlin–Ericksen (R–E) viscoelastic fluid and a viscous fluid during the transfer of heat and mass between phases. The R–E model is a fundamental framework in the study of viscoelastic fluids, providing insights into their complex rheological behavior. It characterizes the material's response to both deformation and flow, offering valuable predictions for various industrial and biological applications. Within the framework of viscous potential flow (VPF) theory, viscosity is exclusively accounted for in the normal stress balance equation, disregarding the influence of shearing stress entirely. This study introduces a viscous pressure term into the normal stress balance equation alongside the irrotational pressure, presuming that this addition will improve the discontinuity of tangential stresses at the fluid interface. Through derivation of a dispersion relationship and subsequent theoretical and numerical stability analyses, the stability of the interface is investigated across various physical parameters. Multiple plots are generated using the dispersion relation, and a comparative analysis between VPF and VCVPF is conducted to establish improved stability criteria. The investigation highlights that the combined impact of heat/mass transport and shearing stress serves to delay the instability of the interface.

Numerical analysis of failure mechanism and stability of reinforced embankment with DCM soft soil composite foundation

The deep cement mixing (DCM) method has been widely used in offshore geotechnical engineering. At present, the reinforcement form of pile has been researched widely, but there is a lack of understanding of the failure mode of DCM pile composite foundation reinforcement for bank slopes. In this paper, a series of unconfined compression tests was conducted to obtain necessary numerical analysis parameters. A series of study on the failure mechanism and stability control of DCM pile composite foundation reinforcement bank slope has been conducted through numerical analysis. The results of numerical analysis indicate that under different reinforcement conditions, there exists an optimal reinforcement depth, and the safety coefficient will gradually stabilize at 1.45; The bending moment of the piles under the bank slope the bending moment of the pile body under the road surface Under different overloads. The maximum bending moment of the pile body is located near one-third of the pile body, and the maximum bending moment of the pile body is located between the shoulder and foot of the slope.