Numerical Stability Research Articles

Deep Reinforcement Learning-Augmented Spalart–Allmaras Turbulence Model: Application to a Turbulent Round Jet Flow

The purpose of this work is to explore the potential of deep reinforcement learning (DRL) as a black-box optimizer for turbulence model identification. For this, we consider a Reynolds-averaged Navier–Stokes (RANS) closure model of a round turbulent jet flow at a Reynolds number of 10,000. For this purpose, we augment the widely utilized Spalart–Allmaras turbulence model by introducing a source term that is identified by DRL. The algorithm is trained to maximize the alignment of the augmented RANS model velocity fields and time-averaged large eddy simulation (LES) reference data. It is shown that the alignment between the reference data and the results of the RANS simulation is improved by 48% using the Spalart–Allmaras model augmented with DRL compared to the standard model. The velocity field, jet spreading rate, and axial velocity decay exhibit substantially improved agreement with both the LES reference and literature data. In addition, we applied the trained model to a jet flow with a Reynolds number of 15,000, which improved the mean field alignment by 35%, demonstrating that the framework is applicable to unseen data of the same configuration at a higher Reynolds number. Overall, this work demonstrates that DRL is a promising method for RANS closure model identification. Hurdles and challenges associated with the presented methodology, such as high numerical cost, numerical stability, and sensitivity of hyperparameters are discussed in the study.

A numerical assembly technique for free vibration of plane frames using a shifted and deflated Newton method

A semi-analytic method, namely the Numerical Assembly Technique (NAT) is extended to compute natural frequencies of arbitrary planar frame structures. The frame structure is systematically defined by a set of nodes, beams, bearings, springs, and external loads and the corresponding boundary and interface conditions are formulated. We consider the Timoshenko-Ehrenfest beam theory and take axial, bending and shear deformations as well as linear and rotational inertia into account. Since we use the exact solutions of the governing differential equations in the NAT, the accuracy of the results is independent from the spatial discretization. However, the resulting eigenvalue problem is non-linear and requires a numerical algorithm to solve. We consider three different representations of exact homogeneous solutions and study the numerical stability of the resulting system matrix. To solve for the eigenvalues we propose a Newton method enhanced with a shifted deflation algorithm. The stability and the efficiency of this approach is illustrated in numerical examples.

Efficient numerical calculation of Lyapunov-exponents and stability assessment for quasi-periodic motions in nonlinear systems

Investigating the stability of stationary motions is a highly relevant aspect when characterizing dynamical systems. For equilibria and periodic motions, well established theories and approaches exist to assess their stability: in both cases stability may be assessed using eigenvalue analyses of small perturbations. When it comes to quasi-periodic motions, such eigenvalue analyses are not applicable, since these motions can not be parameterized on finite time intervals. However, quasi-periodic motions can be densely embedded on finite invariant manifolds with periodic boundaries. In this contribution, a new approach is presented, which exploits this embedding in order to derive a sequence of finite mappings. Based on these mappings, the spectrum of 1st order Lyapunov-exponents is efficiently calculated. If the linearization of the problem is regular in the sense of Lyapunov, these exponents may be used to assess stability of the investigated solution. Beyond the numerical calculation of Lyapunov-exponents, an approach is presented which allows to check Lyapunov-regularity numerically. Together, both methods allow for an efficient numerical stability assessment of quasi-periodic motions. To demonstrate, verify and validate the developed approach, it is applied to quasi-periodic motions of two coupled van-der-Pol oscillators as well as a quasi-periodically forced Duffing equation. Additionally, a “step-by-step application instruction” is provided to increase comprehensibility and to discuss the required implementation steps in an applied context.

Instability in strongly stratified plane Couette flow, with application to supercritical fluids.

This paper addresses the stability of plane Couette flow in the presence of strong density and viscosity stratifications. It demonstrates the existence of a generalised inflection point that satisfies the generalised Fjørtoft's criterion of instability when a minimum of kinematic viscosity is present in the base flow. The characteristic scales associated with this minimum are identified as the primary controlling parameters of the associated instability, regardless of the type of stratification. To support this finding, analytical stability models are derived in the long wave approximation using piecewise linear base flows. Numerical stability calculations are carried out to validate these models and to provide further information on the production of disturbance vorticity. All instabilities are interpreted as arising from the interaction between two vorticity waves. Depending on the type of stratification, these two waves are produced by different physical mechanisms. When both strong density and viscosity stratifications are present, we show that they result from the concurrent action of shear and inertial baroclinic effects. The stability models developed for simple fluid models ultimately shed light on a recently observed unstable mode in supercritical fluids (Ren et al., J. Fluid Mech., vol. 871, 2019, pp. 831-864), providing a quantitative prediction of the stability diagram and identifying the dominant mechanisms at play. Furthermore, our study suggests that the minimum of kinematic viscosity reached at the Widom line in these fluids is the leading cause of their instability. The existence of similar instabilities in different fluids and flows (e.g., miscible fluids) is finally discussed.

An Eulerian–Lagrangian particle method for weakly compressible viscous flows using peridynamic differential operator

This study proposes a peridynamic differential operator (PDDO)-based Eulerian–Lagrangian hybrid particle method for weakly compressible viscous flows. The PDDO is utilized to transform the governing partial differential equations into their integral form; hence, the issue of local non-differentiability is eliminated. Both the Eulerian and Lagrangian formulations of particle methods for solving the governing integral equations are derived and then combined to propose the hybrid method. In the proposed method, the Eulerian and Lagrangian formulation is utilized to solve the governing equations in the inner computational domain and in areas with free surfaces, respectively. The results of these two domains are then merged to arrive at the solution. In such a way, the hybrid particle method effectively balances the computational demands and applicability. The numerical stability and interpolation consistency of the proposed method are shown by introducing the laminar viscosity model. A detailed numerical procedure is provided involving the boundary conditions and time-stepping strategy. The proposed method is validated by several benchmark problems. Furthermore, the method was applied to several benchmark problems including the hydrostatic test, Taylor–Green vortex, and numerical wave generation. The results of these numerical examples suggest that the proposed method is computationally less demanding while maintaining accuracy.

Advancing Eddy Parameterizations: Dynamic Energy Backscatter and the Role of Subgrid Energy Advection and Stochastic Forcing

AbstractViscosity in the momentum equation is needed for numerical stability, as well as to arrest the direct cascade of enstrophy at grid scales. However, a viscous momentum closure tends to over‐dissipate eddy kinetic energy. To return excessively dissipated energy to the system, the viscous closure is equipped with what is called dynamic kinetic energy backscatter. The amplitude of backscatter is based on the amount of unresolved kinetic energy (UKE). This energy is tracked through space and time via a prognostic equation. Our study proposes to add advection of UKE by the resolved flow to that equation to explicitly consider the effects of nonlocality on the subgrid energy budget. UKE can consequently be advected by the resolved flow before it is reinjected via backscatter. Furthermore, we suggest incorporating a stochastic element into the UKE equation to account for missing small‐scale variability, which is not present in the purely deterministic approach. The implementations are tested on two intermediate complexity setups of the global ocean model FESOM2: an idealized channel setup and a double‐gyre setup. The impacts of these additional terms are analyzed, highlighting increased eddy activity and improved flow characteristics when advection and carefully tuned, stochastic sources are incorporated into the UKE budget. Additionally, we provide diagnostics to gain further insights into the effects of scale separation between the viscous dissipation operator and the backscatter operator responsible for the energy injection.

Spatial Functional Data analysis: Irregular spacing and Bernstein polynomials

Spatial Functional Data (SFD) analysis is an emerging statistical framework that combines Functional Data Analysis (FDA) and spatial dependency modeling. Unlike traditional statistical methods, which treat data as scalar values or vectors, SFD considers data as continuous functions, allowing for a more comprehensive understanding of their behavior and variability. This approach is well-suited for analyzing data collected over time, space, or any other continuous domain. SFD has found applications in various fields, including economics, finance, medicine, environmental science, and engineering. This study proposes new functional Gaussian models incorporating spatial dependence structures, focusing on irregularly spaced data and reflecting spatially correlated curves. The model is based on Bernstein polynomial (BP) basis functions and utilizes a Bayesian approach for estimating unknown quantities and parameters. The paper explores the advantages and limitations of the BP model in capturing complex shapes and patterns while ensuring numerical stability. The main contributions of this work include the development of an innovative model designed for SFD using BP, the presence of a random effect to address associations between irregularly spaced observations, and a comprehensive simulation study to evaluate models’ performance under various scenarios. The work also presents one real application of Temperature in Mexico City, showcasing practical illustrations of the proposed model.

Moving Bragg Solitons in a Dual-Core System Composed of a Linear Bragg Grating with Dispersive Reflectivity and a Uniform Nonlinear Core

The existence and stability of moving Bragg grating solitons are systematically investigated in a dual-core system, where one core is uniform and has Kerr nonlinearity, and the other is linear with Bragg grating and dispersive reflectivity. It is found that moving soliton solutions exist throughout the upper and lower bandgaps, whereas no soliton solutions exist in the central bandgap. Similar to the quiescent solitons in the system, it is found that when dispersive reflectivity is nonzero, for certain values of parameters, sidelobes appear in the solitons’ profiles. The stability of the moving solitons is characterized using systematic numerical stability analysis. Additionally, the impact and interplay of dispersive reflectivity, soliton velocity, and group velocity on the stability border are analyzed.

Controlling the Formation of Polyhedral Block Copolymer Nanoparticles: Insights from Process Variables and Dynamic Modeling.

This study delves into the formation of nanoscale polyhedral block copolymer particles (PBCPs) exhibiting cubic, octahedral, and variant geometries. These structures represent a pioneering class that has never been fabricated previously. PBCP features distinct variations in curvature on the outer surface, aligning with the edges and corners of polyhedral shapes. This characteristic sharply contrasts with previous block copolymers (BCPs), which displayed a smooth spherical surface. The emergence of these cornered morphologies presents an intriguing and counterintuitive phenomenon and is linked to process parameters, such as evaporation rates and initial concentration, while keeping other variables constant. Using a system of coupled Cahn-Hillard (CCH) equations, we uncover the mechanisms driving polyhedral particle formation, emphasizing the importance of controlling relaxation parameters for shape variable u and microphase separation v. This unconventional approach, differing from traditional steepest descent method, allows for precise control and diverse polyhedral particle generation. Accelerating the shape variable u proves crucial for expediting precipitation and aligns with experimental observations. Employing the above theoretical model, we achieve shape predictions for particles and the microphase separation within them, which overcomes the limitations of ab initio computations. Additionally, a numerical stability analysis discerns the transient nature versus local minimizer characteristics. Overall, our findings contribute to understanding the complex interplay between process variables and the morphology of polyhedral BCP nanoparticles.

An explicit material point and finite volume sequentially coupled method for simulating large deformation problems in saturated soil

An explicit material point and finite volume sequentially coupled method (MPM-FVM) is developed using u-p form coupling formulations for the simulation of large deformations in saturated soil. In MPM-FVM, the momentum conservation equation is solved by MPM under the updated Lagrange framework, and the mass conservation equations are solved by FVM under the Euler framework. The numerical implementation of an explicit sequentially coupled algorithm is developed considering the tradeoffs between computational cost and numerical stability. The critical time step of u-p form governing equations is derived using the Von Neumann method, which is found to be much greater than the u-U form explicit MPM at low permeability, allowing for high numerical stability. Three numerical cases are presented to demonstrate the stability, efficiency, and effectiveness of the developed MPM-FVM method. The simulation of a saturated one-dimensional soil column subjected to quasi-static and dynamic loads validates the method, and highlights its greater efficiency and accuracy compared to standard solid–fluid coupled MPM methods. The large deformation simulation capability of the method is exhibited via simulation of a saturated slumping block. In real-world application, simulation results obtained using method achieves excellent agreement with the experiment measurements and observations of the Selborne landslide experiment.

A Telegrapher’s equations-based implicit finite difference solution for fault location with current transformer saturation

This paper introduces a fault location method based on time analysis, utilizing synchronized data collected from both ends of the faulted line. The data is obtained either from Common Format for Transient Data Exchange (COMTRADE) files or Sampled Measured Value (SMV) of the IEC-61850 protocol. The method proposed in this study employs the implicit finite difference solution to the Telegrapher’s equations. In certain fault scenarios, such as when one or both transformers experience saturation, a narrow time window of data can be utilized to accurately pinpoint the fault location. The implicit solution method is unconditionally stable and therefore decouples the choice of the line discretization step from the sampling time interval, which is commonly required to maintain numerical stability of the explicit strategies for fault location. Furthermore, the numerical stability of the implicit process allows for devising a systematic search procedure that improves the accuracy of the calculated fault location, regardless of the transmission line length. By conducting numerical comparisons with an explicit time-domain fault locator and recent frequency-domain algorithms, the superiority of the proposed approach becomes evident. The results highlight the significant improvements achieved through the utilization of this method. Specifically, the results demonstrate that Telegrapher’s Equations implicit finite difference solution maintains accuracy under current transformer saturation, as it can locate the fault using data from a brief interval where the current transformer is not saturated.

Lattice Boltzmann method for particulate multiphase flow system

This study proposes a numerical model for particulate three-phase flow in microchannels based on multiphase lattice Boltzmann method (LBM). The model combines the color-gradient method to track the immiscible fluid-fluid interface and the two-fluid model (TFM) to describe particle-particle and particle-fluid interactions, which can efficiently simulate transport and displacement processes involving large amounts of particles. A mixture-rheology TFM algorithm is proposed by introducing a mixture phase with rheology properties obtained from experiments instead of the conventional TFM particle phase with artificial viscosity models. Multi-relaxation-time (MRT) collision operator and GPU computing are adopted to enhance the numerical stability and efficiency. Various theoretical benchmarks for particle transport and two-phase flow are performed respectively to verify the accuracy of the proposed model. Exceptional consistency between results from particulate three-phase flow simulation and microfluidic experiments further confirms the reliability of our model, especially in capturing the inertial lagging and accumulation phenomena under multiphase and porous flow conditions. The proposed numerical framework will benefit our understanding of multiphase displacement with microgels in microchannels with complex geometries.

Numerical assessment of hyperbolic type double interface problems via Haar wavelets

In this manuscript, we have addressed wave propagation challenges within heterogeneous media and hyperbolic interface model. A hybrid numerical approach is introduced for solving these problems, combining the finite difference method and Haar wavelet collocation method. The method entails approximating the second-order space partial derivative through a truncated Haar wavelet series, while the temporal derivative is approximated using finite difference method. For linear hyperbolic interface model, the resulting algebraic systems are solved using the Gauss elimination technique. In the case of non-linear problems, the nonlinearity is addressed through the quasi-Newton linearization formula. To assess the accuracy of the proposed technique, we compute the computational rate of convergence, root mean square errors and maximum absolute errors employing various collocation points. The proposed method perform very well and produces a stable solution if sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, whereas, the other existing method loses its accuracy in such cases. The numerical experiments, stability and rate of convergence both theoretical and computational, confirm the accuracy and diverse applicability of the method.

A versatile sharp boundary ghost-node method for moving rigid boundary fluid flow with meshless nodes distribution

A sharp boundary ghost-node method (GNM) is developed to solve the moving boundary fluid flow in a meshless local radial basis function (LRBF) framework. The background Euler fluid node is the mesh-less scattered node based on LRBF rather than the conventional Cartesian grid or unstructured mesh. The present approach (LRBF-GNM) can flexibly treat the steady boundary with the body-fitted nodes and tackle the moving boundary using the ghost-node method. The key idea of GNM is to project the information of fluid nodes into the ghost nodes by considering the boundary conditions of immersed boundary on Lagrangian nodes, and the influence of immersed boundary on the fluid can be explicitly added during the projection. There are some free parameters that should be determined before computing the virtual forcing source term acted only on the ghost nodes. The distribution of the ghost nodes, the projection strategy and the distribution of the image nodes should be treated carefully to balance the penetrability of streamlines over the immersed boundary, the numerical stability and the thickness of the diffusion boundary. A definition of sharp boundary is introduced to estimate the influences of the above free parameters on the immersed boundary. According to the numerical tests for a static boundary, the optimal parameters for GNM to precisely treat the immersed boundary are summarized. The fluid flow over steady and moving rigid boundaries and the vortex-induced vibration (VIV) are conducted and their solutions agree well with published numerical solutions and experimental measurements.

A regularized lattice Boltzmann method based on the multi-layer grid using a buffer scheme

The regularized lattice Boltzmann method (RLBM) has the advantage of improving the accuracy of numerical simulations without increasing the computational overhead. The RLBM usually uses the standard Cartesian grid in numerical simulation. This paper introduces the multi-layer grid technique, the multi-layer grid has different resolutions, which is also more consistent with the physical laws of the flow field. Based on this, a multi-layer grid regularized lattice Boltzmann method (ML-RLBM) for solving complex flow problems is proposed. The proposed method performs local refinement of the grid on the standard Cartesian grid to achieve a multi-layer grid, which can reduce the computational overhead. In order to realize the local grid refinement easily, this paper adopts the center point format to store the multi-layer grid. The ML-RLBM designs a buffer scheme between different layers of grids, which realizes that only spatial interpolation is required at the interface of different layers of grids without temporal interpolation. The numerical results show that the ML-RLBM has excellent numerical accuracy and numerical stability, and it can effectively reduce the computational cost.

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn’s hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm.

Neko: A modern, portable, and scalable framework for high-fidelity computational fluid dynamics

Computational fluid dynamics (CFD), in particular applied to turbulent flows, is a research area with great engineering and fundamental physical interest. However, already at moderately high Reynolds numbers the computational cost becomes prohibitive as the range of active spatial and temporal scales is quickly widening. Specifically scale-resolving simulations, including large-eddy simulation (LES) and direct numerical simulations (DNS), thus need to rely on modern efficient numerical methods and corresponding software implementations. Recent trends and advancements, including more diverse and heterogeneous hardware in High-Performance Computing (HPC), are challenging software developers in their pursuit for good performance and numerical stability. The well-known maxim “software outlives hardware” may no longer necessarily hold true, and developers are today forced to re-factor their codebases to leverage these powerful new systems. In this paper, we present Neko, a new portable framework for high-order spectral element discretization, targeting turbulent flows in moderately complex geometries. Neko is fully available as open software. Unlike prior works, Neko adopts a modern object-oriented approach in Fortran 2008, allowing multi-tier abstractions of the solver stack and facilitating hardware backends ranging from general-purpose processors (CPUs) down to exotic vector processors and FPGAs. We show that Neko’s performance and accuracy are comparable to NekRS, and thus on-par with Nek5000’s successor on modern CPU machines. Furthermore, we develop a performance model, which we use to discuss challenges and opportunities for high-order solvers on emerging hardware.

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Computer-aided rotary crane stability assessment

This paper introduces a computer-aided approach for evaluating the stability of rotary cranes. Utilizing the Motion add-in within the SolidWorks package, a comprehensive numerical stability analysis was carried out. The interaction dynamics between the crane’s support system and the ground were modeled employing linear springs, while the rope system was characterized by a blend of linear spring and damper elements. The term “temporary loss of stability” was introduced to describe scenarios wherein one support loses contact with the ground. Furthermore, ramifications of overlapping control signals on crane operational safety were investigated. Validation of the numerical findings was accomplished through experimental tests conducted on an actual crane. This integrated methodology not only enhances the understanding of crane stability but also offers practical insights for optimizing crane operations in real-world settings.

Importance of Field Observations for the Numerical Stability Analysis of Rock Slopes: Case Studies from the Sikkim Himalaya

Importance of Field Observations for the Numerical Stability Analysis of Rock Slopes: Case Studies from the Sikkim Himalaya