Numerical Stability Analysis Research Articles

Continuum limits for adaptive network dynamics

Adaptive (or co-evolutionary) network dynamics, i.e., when changes of the network/graph topology are coupled with changes in the node/vertex dynamics, can give rise to rich and complex dynamical behavior. Even though adaptivity can improve the modelling of collective phenomena, it often complicates the analysis of the corresponding mathematical models significantly. For non-adaptive systems, a possible way to tackle this problem is by passing to so-called continuum or mean-field limits, which describe the system in the limit of infinitely many nodes. Although fully adaptive network dynamic models have been used a lot in recent years in applications, we are still lacking a detailed mathematical theory for large-scale adaptive network limits. For example, continuum limits for static or temporal networks are already established in the literature for certain models, yet the continuum limit of fully adaptive networks has been open so far. In this paper we introduce and rigorously justify continuum limits for sequences of adaptive Kuramoto-type network models. The resulting integro-differential equations allow us to incorporate a large class of co-evolving graphs with high density. Furthermore, we use a very general measure-theoretical framework in our proof for representing the (infinite) graph limits, thereby also providing a structural basis to tackle even larger classes of graph limits. As an application of our theory, we consider the continuum limit of an adaptive Kuramoto model directly motivated from neuroscience and studied by Berner et al.~in recent years using numerical techniques and formal stability analysis.

Numerical Stability Analysis for a Stationary and Translating Droplet at Extremely Low Viscosity Values Using the Lattice Boltzmann Method Color-Gradient Multi-Component Model with Central Moments Formulation

Numerical Stability Analysis for a Stationary and Translating Droplet at Extremely Low Viscosity Values Using the Lattice Boltzmann Method Color-Gradient Multi-Component Model with Central Moments Formulation

Numerical Stability Analysis of Large Section Tunnels Using the Double-Side Heading Method: A Case Study of Xiamen Haicang Evacuate-Channel

Large section tunnels have huge excavation spans, complex excavation procedures, mutual influence among various procedures, multiple disturbances of the surrounding rock, and discontinuous connection among linings under different procedures, which bring great difficulties to their construction. The Caijianwei Mountain No. 2 tunnel of the Xiamen Haicang Evacuate-channel project in China belongs to a super large section tunnel. The tunnel’s maximum excavation section span reaches 30.52 m, and the excavation area reaches 421.73 m2. Its excavation sequence, the timing of the second primary lining support, temporary support disassembly, and secondary lining support of the double-side heading method were analyzed using numerical methods. The excavation sequence of the double side-heading method from the right pilot tunnel to the left pilot tunnel and then to the middle pilot tunnel can avoid a large horizontal displacement at the arch waist and control the deformation of the lining. The excavation sequence from the middle pilot tunnel to the two side pilot tunnels was conducive to the bearing performance of the concrete at the arch crown. The excavation sequence from the two side pilot tunnels to the middle pilot tunnel can better protect the concrete safety at the arch waist. The timing of the second primary lining support, temporary support removal, and secondary lining support have different effects on the deformation and stress of the surrounding rock and support structures. In the construction process, the timing of the second primary lining support, temporary support disassembly, and secondary lining support should be determined according to the actual deformation and stress of the site section. The research results can provide a reference for the construction design of the double-side heading method for large section tunnels.

Evolution of ship damage stability assessment—Transitioning designers to direct numerical simulations

Theory and application of damage stability followed over the years two dissociated paths: static assessments and dynamic simulations. The first approach, being easy to apply and understand, has been preferred by ship designers and regulators; the second, more advanced and first-principle oriented, has been mainly reserved for research or high-level consultancy, especially for passenger ships. Nowadays, the availability of numerical flooding simulation tools across the scientific community and calculation power in the industry allows for a possible definitive transition of damage stability assessment towards direct numerical analyses. However, research should softly drive designers towards more advanced processes via a suitable didascalic calculation framework. The multi-level approach pursued in project FLARE is an example of such a transition from static to dynamic damage stability assessment. The present work initially carefully reviews the probabilistic concept of damage stability, critically comparing the prescriptive statistical methods with direct ones and providing insights and guidance on how researchers and designers can reconcile with the original implicit assumption of the probabilistic approach. Secondly, the development of the multi-level framework highlights incongruences concerning modelling of damages between static and dynamic assessments, disfavouring the comprehension of dynamic results to designers. Two detailed examples highlight the differences in dynamic simulation results between different damage breach modelling, leading to completely different flooding paths for the same damage case. Finally, the paper indicates how a compromise between academic approach and application could help designers to start their transition towards direct numerical damage stability analyses.

Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the ‘mass’ (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. We observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick’s theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick’s theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the ‘frustrated’ blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.

Mechanical squeezing in an active-passive-coupled double-cavity optomechanical system via pump modulation.

We focus on the generation of mechanical squeezing by using periodically amplitude-modulated laser to drive an active-passive-coupled double-cavity optomechanical system, where the coupled gain cavity and loss cavity can form into a parity-time (P T)-symmetry system. The numerical analysis of the system stability shows that the system is more likely to be stable in the unbroken-P T-symmetry regime than in the broken-P T-symmetry regime. The mechanical squeezing in the active-passive system exhibits stronger robustness against the thermal noise than that in the passive-passive system, and the so-called 3 dB limit can be broken in the resolved-sideband regime. Furthermore, it is also found that the mechanical squeezing obtained in the unbroken-P T-symmetry region is stronger than that in the broken-P T-symmetry region. This work may be meaningful for the quantum state engineering in the gain-loss quantum system that contributes to the study of P T-symmetric physics in the quantum regime.

Nonlinear exceptional-point lasing with ab initio Maxwell–Bloch theory

We present a general analysis for finding and characterizing nonlinear exceptional point (EP) lasers above threshold using steady-state ab initio Maxwell–Bloch equations. For a system of coupled slabs, we show that a nonlinear EP is obtained for a given ratio between the external pumps in each resonator and that it is associated with a kink in the output power and lasing frequency, confirming coupled-mode theory predictions. Through numerical linear stability analysis, we confirm that the EP laser can be stable for a large enough inversion relaxation rate. We further show that the EP laser can be characterized by scattering a weak signal off the lasing cavity so that the scattering frequency spectrum exhibits a quartic divergence. Our approach can be applied to arbitrary scatterers with multi-level gain media.

Impact of longwall mining on slope stability – A case study

Abstract In recent decades, many open pit (OP) mines have either already made the decision or are at the planning stage to change their mining activity from OP to underground (UG) to remain competitive. Technically, before the OP ends its operation, both OP and UG mining will have to be operated simultaneously for a certain period of time. It is well known that UG operation causes subsidence, discontinuous deformations, and changes in hydrogeological conditions. In case of UG operation located below the OP mine, slope deformation can be expected as a result of subsidence induced by UG exploitation. This paper presents a numerical analysis of slope stability under the influence of the longwall mining operation at the Cao Son OP mine in Vietnam. All calculation variants were performed using the Finite Difference Method code, FLAC. In order to evaluate slope stability of the OP slope, various geometry configurations showing advances of both OP and UG extractions were examined. Based on the outcomes, assessments on OP slope are presented, and then, practical actions regarding the location and direction of UG extraction are recommended, with an aim to minimize the impact of underground mining on OP slope.

Numerical stability analysis of the Changjiahe landslide in high seismic intensity region of the Loess Plateau

Numerical stability analysis of the Changjiahe landslide in high seismic intensity region of the Loess Plateau

A New Technique of Lattice Beam Construction with Pre-Anchoring for Strengthening Cut Slope: A Numerical Analysis of Temporary Stability during Excavation

In consideration of the temporary stability of the cutting slope during construction and its permanent stability under long-term service, a new technique of lattice beam construction with anchors pre-set in the slope from the original ground surface before cutting was proposed, and its construction process was briefly introduced. Compared with the model without pre-set anchors, the effectiveness of pre-setting anchors to strengthen the cutting slope during multi-excavation was verified in the numerical software FLAC3D. Various factors such as the factor of safety (FOS) and the maximum shear strain increment (MSSI) as well as the displacement for different stages were discussed. The results show that the anchors pre-set in the slope provide reinforcement step-by-step with excavations which changes the mechanical responses of the cutting slope and increases the factor of safety with a variation of 15.9–44.1% compared to the case without setting anchors. In addition, with excavations, the axial forces of the anchors pre-set in the stratum increase gradually, and the positions of the maximum axial forces gradually transfer from the vicinity of the cutting surface to the depth of the design slope. Numerical simulations prove that this new technique is beneficial for ensuring the temporary stability of the slope during excavations and is especially suitable for the advance anchorage of the cutting slope, in which the inclined original ground surface is cut at an angle steeper than it can stand safely and is close to the design slope surface after cutting. After the completion of slope excavation, the cast-in-place concrete lattice beam is immediately set on the design slope surface and connected with the anchor heads exposed on the cut slope surface to ensure the permanent stability of the slope. Therefore, this new technology has important guiding significance for both the temporary stability of slopes during construction and the permanent stability of slopes in service.

Numerical Analysis of Pillar Stability in Longwall Mining of Two Adjacent Panels of an Inclined Coal Seam

Longwall mining is one of the most widespread methods globally. During the preliminary development of the working, the coal seam is sectioned into panels divided by protective pillars. The pillars are necessary for maintaining the service life of underground mines, a highly productive stope, and personnel safety. In this work, we apply the finite-difference continuum damage mechanics approach to modeling the stress–strain evolution of the rock mass during the extraction of two adjacent longwall panels of an inclined seam. A new modification of the damage accumulation kinetic equation is proposed. The numerical-modeling approach accounts for an explicit number of numerous factors affecting the rock mass behavior. These factors are gravity forces, lithology, tectonic stresses, natural discontinuities, geotechnical, and mining parameters. When the model parameters are calibrated against the in situ observations, the results of the numerical-modeling approach provide a reliable basis for a pillar stability assessment. We build a structural model of a rock mass containing an underground working based on a simplified stratigraphy of the Kondomsky deposit, Kuznetsk coal basin, Russia. Based on the results of the numerical modeling, the stability of a pillar is analyzed. A new numerical technique extending the classical approach to the stability analysis is proposed and verified against the field data.

Numerical analysis of crack path stability in brittle porous materials

Porous materials are widely used in engineering applications because of their high stiffness, strength, and low density. Those advantages are mainly due to their open microstructures, which also makes it challenging to obtain a thorough understanding of their fracture mechanisms and to predict trustworthy crack paths. In this study, we analyse numerically fracture trajectories in brittle porous solids with varying porosity (or relative density) subject to different mixed-mode loading conditions using a phase-field theory for brittle fracture. The results reveal that the crack paths in porous solids with high porosity (low relative density) are very different from those in porous solids with low porosity (high relative density). The latter resemble stable crack paths in homogeneous solids, whereas the former seems somewhat arbitrary, more stochastic. In high porosity materials, the crack paths are governed by the local microstructure rather than by the global remote loading. A key observation is that there is a transition of the fracture behaviour in porous materials at relative densities of around 50%. At relative densities above 50%, the material behaves nearly as a classical continuum, and the crack paths are reasonably well captured by traditional fracture mechanics theories. The stochastic nature of the porous microstructure is also examined.

Generalized model for steady-state bifurcations without parameters in memristor-based oscillators with lines of equilibria

We demonstrate how the pitchfork, transcritical and saddle-node bifurcations of steady states observed in dynamical systems with a finite number of isolated equilibrium points occur in systems with lines of equilibria. The exploration is carried out by using the numerical simulation and linear stability analysis applied to a model of a memristor-based circuit. All the discussed bifurcation scenarios are considered in the context of models with the piecewise-smooth memristor current-voltage characteristic (Chua’s memristor), as well as on examples of oscillators with the memristor nonlinearity that is smooth everywhere. Finally, we compare the dynamics of ideal-memristor-based oscillators with the behavior of models taking into account the memristor forgetting effect. The presented results are obtained for electronic circuit models, but the studied bifurcation phenomena can be exhibited by systems with lines of equilibria of any nature.

Influence of Centrifugal Buoyancy in Thermal Convection within a Rotating Spherical Shell

The dynamo action, which is of importance in the study of the geomagnetism mechanism, is considered to be caused by the convection structure formed inside a rotating spherical shell. This convection structure elongated in the rotation axis is generated by the action of both heat and rotation on the fluid inside a spherical shell. In this study, we analyzed thermal convection in such a rotating spherical shell and attempted to understand the phenomenon of this convective structure. It is known that each value of the Prandtl number, the Ekman number and the Rayleigh number and their balance are important for the generation of such convective structure. We fixed these three parameters and considered the effect of centrifugal buoyancy as the Froude number additionally. To investigate how the effects of centrifugal buoyancy affect the convective structure, we carried out both three-dimensional numerical simulations and linear stability analyses. In particular, we focused on the transition from axisymmetric flow to non-axisymmetric flow having wavenumbers in the toroidal direction and investigated both growth rate and phase velocity of the disturbance. It was found that axisymmetric flow tends to be maintained as the effect of centrifugal buoyancy increases.

Investigation on transition correlation of conical boundary layer in hypersonic wind tunnel experiments

The wind tunnel experiment is an important approach to studying boundary layer transition. In this study, the effects of bluntness, unit Reynolds number, and wall temperature ratio on hypersonic conical boundary layer transition were studied using numerical simulations and linear stability analyses based on the wind tunnel experimental data. Through the flow similarity principle, the treatment of the bluntness-varying problem was determined. Combining the experimental transition data and theoretical analysis results of sharp and blunt cones at home and abroad, the relation between the transition factor Ntr and the unit Reynolds number before ‘N factor reversal’, as well as the relation between the transition value Rtr and the unit Reynolds number from ‘N factor reversal’ to ‘transition reversal’ were determined too. Finally, according to the wind tunnel test data of China Aerodynamics Research and Development Center, the effect of wall temperature ratio on the receptivity of blunt-cone boundary layer was stated out, and the correlation of transition Reynolds number under different wall temperature ratios was determined. The influence law of parameters in this correlation was also determined.

Numerical analysis of slope stability

The paper is focused on the analysis of slope stability. The problem of slope stability is processed theoretically and parametrically - parametrical study. Factors that affect slope stability include: geomorphological conditions, geological conditions, soil and rock properties, and external loads (for example, from traffic, nearby structures, possible vibrations). The shear stress parameters of soil are a keys design parameters for assessment of slope stability. The aim of the parametric study is to point out the importance of knowing the shear stress parameters of soils in assessing the slope stability. By comparing obtained result, it was founded, that change the cohesive has a significant effect on the stability factor.

Numerical modeling and design decisions for aerostatic bearings with relatively large nozzle sizes in Magic-Angle Spinning (MAS) systems

Numerical stability analysis for aerostatic bearings was performed to obtain optimized design parameters for small submillimeter to millimeter range diameter cylindrical rotors. Such rotors are used in nuclear magnetic resonance (NMR) application to rotate sample around an axis inclined by magic angle (54.74o) relative to the magnetic field direction at rotational frequencies of about 100 kHz (magic-angle spinning, MAS). The governing Reynolds equation for the fluid film between rotor and bearing was modified for small size aerostatic bearings with relatively large nozzle diameters. The modified Reynolds equation was solved using a finite-volume method to obtain pressure and film thickness around the rotor. This led to the solution of the maximum stable inertial force as a function of rotational frequency and design parameters. The comparison with aerostatic bearings with infinitesimal nozzle sizes was obtained for supported rotor weight and critical vibrational frequency of the rotor. The stable inertial force was found to correspond to a specific nozzle diameter and a specific rotor–bearing clearance. Numerical investigation also shows an enhancement of stable inertial force with decreasing nozzle number or increasing molecular mass of the impinging gas for a specific range of nozzle parameters. Experimental observations further confirmed the role of nozzle diameter, nozzle number and molecular weight of the gas in enhancing the rotor spinning frequency. Further, design decisions were made based on such analysis and were tested for varying rotor size and bearing properties. Using design optimization based on numerical simulation, the maximum frequency of rotation for a home-built 0.4mm MAS rotor could be enhanced from 25 kHz up to 110 kHz, still below the extrapolation from large rotors.

Boundaries control active channel flows

Boundary conditions dictate how fluids, including liquid crystals, flow when pumped through a channel. Can boundary conditions also be used to control internally driven active fluids that generate flows spontaneously? By using numerical simulations and stability analysis we explore how parallel surface anchoring of active agents at the boundaries and substrate drag can be used to rectify coherent flow of an active polar fluid in a 2D channel. Upon increasing activity, a succession of dynamical states is obtained, from laminar flow to vortex arrays to eventual turbulence, that are controlled by the interplay between the hydrodynamic screening length and the extrapolation length quantifying the anchoring strength of the orientational order parameter. We highlight the key role of symmetry in both flow and order and show that coherent laminar flow with net throughput is only possible for weak anchoring and intermediate activity. Our work demonstrates the possibility of controlling the nature and properties of active flows in a channel simply by patterning the confining boundaries.

Numerical analysis of dynamic stability of falling maple samaras

Numerical analysis of dynamic stability of falling maple samaras

Structural Features of Microvascular Networks Trigger Blood Flow Oscillations

We analyse mathematical models in order to understand how microstructural features of vascular networks may affect blood flow dynamics, and to identify particular characteristics that promote the onset of self-sustained oscillations. By focusing on a simple three-node motif, we predict that network “redundancy”, in the form of a redundant vessel connecting two main flow-branches, together with differences in haemodynamic resistance in the branches, can promote the emergence of oscillatory dynamics. We use existing mathematical descriptions for blood rheology and haematocrit splitting at vessel branch-points to construct our flow model; we combine numerical simulations and stability analysis to study the dynamics of the three-node network and its relation to the system’s multiple steady-state solutions. While, for the case of equal inlet-pressure conditions, a “trivial” equilibrium solution with no flow in the redundant vessel always exists, we find that it is not stable when other, stable, steady-state attractors exist. In turn, these “nontrivial” steady-state solutions may undergo a Hopf bifurcation into an oscillatory state. We use the branch diameter ratio, together with the inlet haematocrit rate, to construct a two-parameter stability diagram that delineates regimes in which such oscillatory dynamics exist. We show that flow oscillations in this network geometry are only possible when the branch diameters are sufficiently different to allow for a sufficiently large flow in the redundant vessel, which acts as the driving force of the oscillations. These microstructural properties, which were found to promote oscillatory dynamics, could be used to explore sources of flow instability in biological microvascular networks.