Temporal Discretization Schemes Research Articles

The edge-based smoothed FEM with [formula omitted]-Bathe implicit temporal discretization scheme for the analyses of underwater wave propagation problems

The underwater acoustic wave propagation problem is an important issue in ocean engineering field. The classical finite element approach with edge-based smoothed gradient smoothing technique is incorporated into the novel ρ∞-Bathe implicit direct time integration scheme in this paper for solving underwater transient acoustic wave propagation problems. This edge-based smoothed FEM (ES-FEM) and ρ∞-Bathe method are employed to perform the space and time domain discretization, respectively. Due to the appropriate softening effects from the ES-FEM, the space discretization errors can be markedly reduced. Meanwhile, the possible numerical error related to the temporal discretization also can be significantly suppressed by the novel ρ∞-Bathe method which possesses the controllable numerical damping effects. Through detailed analysis of the total dispersion error, we find that the numerical results from the proposed scheme can converge monotonically to the exact solutions when employing the decreasing temporal discretization intervals and the possible numerical anisotropy effects are also effectively suppressed. These excellent numerical performance clearly distinguish the present method from the conventional approaches and make it to be particularly suitable for complicated acoustic wave propagation analysis. Finally, various typical numerical experiments are performed to illustrate the capacities of the proposed scheme in tackling underwater transient acoustic wave propagation problems.

A fully decoupled numerical method for Cahn–Hilliard–Navier–Stokes–Darcy equations based on auxiliary variable approaches

A fully decoupled, linearized, and unconditionally stable finite element method is developed to solve the Cahn–Hilliard–Navier–Stokes–Darcy model in the coupled free fluid region and porous medium region. By introducing two auxiliary energy variables, we derive the equivalent system that is consistent with the original system. The energy dissipation law of the proposed equivalent model is proven. To lay a solid foundation, we first present a coupled linearized time-stepping method for the reformulated system, and prove its unconditionally energy stability. In order to further improve the computational efficiency, special treatment for the interface conditions and the artificial compression approach are utilized to decouple the two subdomains and the Navier–Stokes equation. Therefore, with the discretization techniques of two existing auxiliary variable approaches, a fully decoupled and linearized numerical scheme can be developed, under the framework of a semi-implicit semi-explicit scheme for temporal discretization and Galerkin finite element method for spatial discretization. The grad-div stabilization is also employed to further improve the stability of auxiliary variable algorithm. The full discretization obeys the desired energy dissipation law without any temporal restriction. Moreover, the implementation process is discussed, including the adaptive mesh strategy to accurately capture the diffuse interface. Ample numerical experiments are performed to validate the typical features of developed numerical schemes, such as the accuracy, energy stability without restriction for time step size, and adaptive mesh refinement in space. Furthermore, we apply the proposed numerical method to simulate the shape relaxation and the Buoyancy-driven flows, which demonstrate the applicability of the proposed method.

Effect of cube spacings on the three-dimensional flow structure over an array of wall-mounted cube

The turbulent flow over an array of cubes mounted on one of the walls of a channel has been investigated using direct numerical simulation for cube spacing that ranges between 2.0 and 4.0. The Reynolds number based on the cube size and the average streamwise velocity is chosen to be 4000. The Navier–Stokes equations have been discretized using second-order spatial and temporal discretization schemes. The present investigation focuses on the flow structures and comprehensive characterization of the separated zones surrounding the cubes, as well as the associated wall-shear stress. A vortex shedding has been observed for the cube spacings of 3.0 and 4.0 without any evidence of vortex shedding for the lowest pitch of 2.0. For the two cases having pitches of 3.0 and 4.0, the presence of the unsteady separation bubbles at the cube's top and side surfaces results in a decrease in wall-shear stress. The quadrant analysis for the region close to the top surface of the cube is performed with the help of the joint probability density function, which reveals dual peaks within the recirculation bubble at the top surface of the cube for higher cube spacings. By conducting an invariant analysis of the Reynolds stress tensor for different cube spacings, we have explored the characteristic of Reynolds stress anisotropy due to the total fluctuations. The production of negative turbulent kinetic energy (TKE) is observed in different regions within the flow domain, among which the horseshoe vortex region for each cube spacing reveals its dominant presence. The physical mechanism responsible for the production of the negative TKE has also been attempted by decomposing the production term into two parts, namely, normal and shear components.

An unconditionally energy stable finite element scheme for a nonlinear fluid–fluid interaction model

Abstract In this paper, we design a decoupled scheme for solving a fluid–fluid interaction problem, which includes two Navier–Stokes equations coupled by some nonlinear interface conditions. Compared with two decoupled schemes proposed by Connors et al. (2012, Decoupled time stepping methods for fluid–fluid interaction. SIAM J. Numer. Anal., 50, 1297–1319) for the fluid–fluid interaction problem, we deal with these nonlinear interface conditions by applying explicit scheme. The presented fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization and explicit treatment for the interface conditions and the nonlinear terms. Moreover, the unconditional energy stability is established and error estimate for the fully discrete scheme is also showed. Finally, some numerical experiments are provided to verify the theoretical results, which illustrate the accuracy and efficiency of the presented scheme.

Transient analyses of wave propagations in nonhomogeneous media employing the novel finite element method with the appropriate enrichment function

An enriched finite element approach with appropriate enrichment function is proposed for the transient analyses of wave propagations in nonhomogeneous media. In present method, the traditional linear nodal interpolation functions for low order finite elements are enriched by the augmented enrichment function. For the purpose of improving the computation accuracy and reducing the numerical dispersion error, the polynomial bases and the trigonometric functions are utilized to form the enrichment functions. In addition, a simple and direct approach is introduced with the aim to tackle the possible linear dependence of the system equations. The numerical results show that the accuracy of the obtained numerical results can be markedly improved by the proposed method in comparison with the standard finite elements. What's more, it can be found that the calculated solutions from the proposed method with the appropriate temporal discretization scheme are able to monotonically converge to the exact solution in transient wave analysis. In consequence, it is possible to consistently increase the solution accuracy by directly decreasing temporal discretization step size. This valuable and important property enable the present approach to be particularly suitable to tackle the wave propagations in nonhomogeneous media.

Numerical Study of Wax Deposition from Multiphase Flow in Oil Pipelines with Heat and Mass Transfer

Wax deposition in field‐scale crude oil pipelines poses a significant challenge to the oil and gas industry, leading to reduced flow rates, increased pressure drops, and potential blockages. Understanding the mechanisms governing wax deposition is crucial for developing effective mitigation strategies. This study investigates the impact of multiphase flow conditions, including water‐in‐oil emulsion, wax precipitation kinetics, shear dispersion, and molecular diffusion, on wax deposition in field‐scale crude oil pipelines. A numerical model is developed that employs second‐order semi‐implicit temporal discretization schemes, such as Crank–Nicolson and Adams–Bashforth methods, in conjunction with a bivariate spectral collocation scheme using Chebyshev–Gauss–Lobatto grid points. The impact of various flow parameters, including Reynolds number (Re), mass Grashof number (Gr), Schmidt number (Sc), and Weber number (We), on the flow variables, wall shear stress, and heat and mass fluxes are investigated. The numerical simulations demonstrate that flow parameters significantly influence the flow behavior, wall shear stress, wall heat flux, and wall mass flux in waxy crude oil pipelines. Specifically, the aggregation of wax crystals in the pipeline decreases by at most 2.5% with increasing Reynolds number from 2.2361 to 3.1361. Conversely, it increases by at most 3.4% with increasing mass Grashof number from 5 to 11 and by at most 4.8% with increasing Weber number from 1.0 to 2.5. Furthermore, the Nusselt number increases from 1.9907 to 4.9834 with increasing Reynolds number from 2.2361 to 5.2361 and from 1.9907 to 2.0225 with increasing mass Grashof number from 5 to 20. It also increases from 1.9907 to 2.0434 with increasing Weber number from 1.0 to 2.5. The insights gained from this study can be applied to optimize pipeline design, operational parameters, and wax deposition mitigation strategies, leading to enhanced pipeline performance and reduced operational costs. The numerical model developed in this work serves as a valuable tool for simulating and predicting wax deposition behavior under various operating conditions.

Numerical investigation of the element-free Galerkin method (EFGM) with appropriate temporal discretization techniques for transient wave propagation problems

In the earlier paper, it is known that the solution accuracy usually can not be monotonically increased as the temporal discretization interval decreases when the standard finite element method (FEM) with the conventional temporal discretization approach is exploited for elastodynamics, hence the time integration step should be carefully determined for sufficiently fine solutions. The present work aims to investigate the behaviors of the classical element-free Galerkin method (EFGM), which is a typical meshless approach, with the Bathe temporal discretization scheme for elastodynamics. The main insights are that we explicitly show the total numerical dispersion errors in the computed numerical results actually consists of two different parts corresponding to the spatial and temporal discretization, respectively; both of them are responsible for solution accuracy. By performing the dispersion analysis, how the solution accuracy is affected by temporal and spatial discretization is shown, it is seen that we can improve the solution accuracy monotonically as the temporal step size decreases as long as the spatial dispersion error is sufficiently small and the related mathematical proofs are also given. From several supporting numerical examples, we can clearly see that the EFGM with the Bathe time integration scheme can basically provide monotonically convergent solutions as long as the reasonable node arrangement pattern and sufficiently large shape function supports are employed, namely the monotonic convergence property with respect to the temporal discretization interval can be achieved. This attractive and important feature makes the EFGM more competitive than the FE approach for elastodynamics.

An energy-conserving finite element method for nonlinear fourth-order wave equations

In this paper, an energy-conserving finite element method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations in a general sense for the first time, where the two-level, Crank-Nicolson type of temporal discretization scheme is designed to cooperate with the Lagrange finite element approximation in space in order to achieve the conservation of discrete energy at each time step. The energy conservation is crucial in engineering field to preserve the total energy as constant for dynamic vibration problems of beams and thin plates that can be modeled by the presented wave equations. In addition, the optimal spatial convergence properties in both L2- and H1-norm and the second-order temporal approximation rate are obtained for finite element solutions of u (deflection), Δu (bending moment) and/or ut (deflection speed) at the same time. Numerical experiments are carried out to validate all attained theoretical results.

A New Explicit Numerical Schemes for Time-Dependent PDEs with Application to Pressure Driven Fluid Flow in a Rectangular Duct

A modified class of temporal discretization schemes for partial differential equations (PDEs) is proposed, explicit and second to fifth-order accurate in time. In time, the stability region of the proposed modified second-order scheme is larger than the standard second-order Adams–Bashforth method constructed on two time levels. A modification made for the Du Fort–Frankel method was also implemented in the proposed second-order scheme, which permits the little larger stability region, but the scheme becomes first-order accurate. Since the Du Fort–Frankel method cannot be employed without a modification of averaging in time levels, the proposed second-order scheme can be used without any modification. The proposed modified scheme with different orders in space and second orders in time was implemented for heat and mass transfer of chemically reactive fluid flow in a rectangular duct. The flow is generated due to applying different pressure gradients. The contour plots of velocity, temperature, and concentration profiles are portrayed at different pressure gradients; Péclet number in heat transfer, Péclet number in mass transfer, reaction parameter, and at different times. In addition, stability and convergence conditions for the considered system of linear and non-linear PDEs consisting of non-dimensional momentum, energy, and concentration equations were found for two cases. The displayed graphs depict the transfer of heat in the fluid, which rises due to heated boundaries, and the transfer of mass in the fluid at various moments. Classical models can be solved using the proposed method, which has a faster convergence rate than the standard or classical approach. This approach is illustrated through computer simulations that demonstrate its key computational features. It is believed that the data presented in this study will serve as a useful source for future fluid flow investigations to be conducted in an industrial setting within an enclosed area.

A comparative numerical study of bank effects on a cruise ship in crabbing motion based on URANS method

The crabbing motion of a cruise ship perpendicularly towards a quay wall is investigated by an unsteady Reynolds-averaged Navier-Stokes (URANS) method. In the numerical simulations, the cruise ship is approaching the quay wall at a constant lateral speed and stops at a rather small ship-quay distance. The crabbing motion with multiple degrees of freedom is simulated by the dynamic overset mesh technique, while the free surface elevation is simulated by the Volume of Fluid method. To obtain reliable temporal discretization for the crabbing process, the time-step dependence study is conducted first, from which a suitable time step is determined. From the computations, the hydrodynamic performances of the cruise ship including: longitudinal force, lateral force, roll moment and yaw moment, as well as surge, heave, roll, pitch and yaw motions are predicted. To evaluate the importance of numerical models used in the URANS method, a comparative investigation is carried out to evaluate the impacts of boundary condition, turbulence model and temporal discretization scheme on the numerical results. From the comparisons, notable influences by the turbulence model and the temporal discretization scheme on both hydrodynamic quantities and flow field features can be observed, especially when the bank effects tend to be pronounced; while the boundary conditions are shown to be less significant. The numerical results indicate remarkable variations of the hydrodynamic quantities in the crabbing process under the influences of ship speed and bank effects. Given that the surge motion is one of the most principal crabbing performances, the initial drift angle with a set of deviations is also computed, coupled with two typical ship speeds in terms of practical navigation. It suggests that more attentions should be paid to the low-speed crabbing to minimize the surge variations. The present study demonstrates promising abilities of CFD methods to deal with the hydrodynamic predictions of harbour manoeuvres in restricted waters.

(Digital Presentation) An Efficient Modeling Framework for Electrodeposition in Lithium Metal Batteries

Lithium metal batteries are drawing attention as a next generation of lithium-ion batteries due to their advantage of high theoretical capacity. However, the widespread application and commercialization of this potential battery technology is not achieved yet due to the poor cyclability and safety issues. Despite the recent progress1-2 and development of numerous strategies for the long and durable cyclability1-4, the analysis of lithium metal evolution still depends on the experimental approach in macroscale or molecular simulation in nanoscale5. In order to understand and optimize the system-level response of these lithium metal batteries, an accurate and robust modeling and simulation framework with a coupled macro and nanoscale approach is essential. In the previous reports, we have discussed simple 1D model to study stripping and plating of the lithium metal electrode and obtaining the characteristic inverse signatures in the cell voltage6, and appropriate boundary conditions ensuring mass conservation in a 2D model7.In this work, a 2D separator domain with initial surface morphology at the lithium metal electrode is considered for diffusion and migration of the lithium-ions in the domain along with reaction kinetics at the surface. The model is based on the mass and charge conservation in the system that captures the morphological evolution at the lithium metal electrode along the charge/discharge of the battery during cycling. This model is solved with in-house schemes based on spatial, temporal discretization schemes and coordinate transformation. The results show that the local current distribution at the electrode surface affects the rate and shape of the growth of the lithium metal at the negative electrode. The study has been performed for a wide range of geometric, kinetic and transport parameters. The proposed in-house model efficiently reaches a converged results compared with other numerical schemes both in speed and accuracy, and easily adaptive to optimizing tools towards cell design. Acknowledgments This research was supported by the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the US Department of Energy (DoE) through the Advanced Battery Materials Research Program (Battery500 Consortium). References K. N. Wood, E. Kazyak, A. F. Chadwick, K. H. Chen, J. G. Zhang, K. Thornton, and N. P. Dasgupta, ACS Cent. Sci., 2, 790-801 (2016).A. Pei, G. Zheng, F. Shi, Y. Li, and Y. Cui., Nano Lett., 17(2), 1132-1139 (2017).J. Liu, Z. Bao, Y. Cui, E. Dufek, J. B. Goodenough, P. Khalifah, Q. Li, B. Liaw, P. Liu, A. Manthiram, Y. S. Meng, V. R. Subramanian, M. F. Toney, V. V. Viswanathan, M. S. Whittingham, J. Xiao, W. Xu, J. Yang, X. Yang, and J. Zhang, Nat. Nanotechnol., 14 180-186 (2019).Y. Chen, Z. Yu, P. Rudnicki, H. Gong, Z. Huang, S. Kim, J. Lai, X. Kong, J. Qin, Y. Cui and Z. Bao, J. Am. Chem. Soc., 143(44), 18703-18713 (2021)S. Angarita-Gomez and P. B. Balbuena, Phys. Chem. Chem. Phys., 22, 21369-21382, (2020)M. Uppaluri, A. Subramaniam, L. Mishra, V. Viswanathan, and V. R. Subramanian, J. Electrochem. Soc., 167, 160547 (2020).L. Mishra, A. Subramaniam, T. Jang, K. Shah, M. Uppaluri, S. A. Roberts and V. R. Subramanian, J. Electrochem. Soc., 168, 092502 (2021).

A deep learning approach for the solution of probability density evolution of stochastic systems

Derivation of the probability density evolution provides invaluable insight into the behavior of many stochastic systems and their performance. However, for most real-time applications, numerical determination of the probability density evolution is a formidable task. The latter is due to the required temporal and spatial discretization schemes that render most computational solutions prohibitive and impractical. In this respect, the development of an efficient computational surrogate model is of paramount importance. Recent studies on the physics-constrained networks show that a suitable surrogate can be achieved by encoding the physical insight into a deep neural network. To this aim, the present work introduces DeepPDEM which utilizes the concept of physics-informed networks to solve the evolution of the probability density via proposing a deep learning method. DeepPDEM learns the General Density Evolution Equation (GDEE) of stochastic structures. This approach paves the way for a mesh-free learning method that can solve the density evolution problem without prior simulation data. Moreover, it can also serve as an efficient surrogate for the solution at any other spatiotemporal points within optimization schemes or real-time applications. To demonstrate the potential applicability of the proposed framework, two network architectures with different activation functions as well as two optimizers are investigated. Numerical implementation on three different problems verifies the accuracy and efficacy of the proposed method.

The Influence of Numerical Method and Grid Parameters on the Simulation Accuracy of Damped Oscillations of Free Floating Cylinder

The paper concentrates on the influence of grid parameters, the time step size, and the order of temporal and spatial approximation on the solution accuracy of the floating body problem. Damped free oscillations of the cylinder on the water surface is under consideration. The numerical simulation method of the floating bodies is based on the solution of a system of Navier-Stokes equations together with a VOF (Volume of Fluid) method. The Navier-Stokes equations are discretized using finite volume method (FVM), and solved by SIMPLE method. The motion of the body is ensured by the deformation of the computational grid. A CSF (Continuum Surface Force) model is used to account for the surface tension forces. The method is implemented in LOGOS software package. The research has shown that implementation of the second-order scheme for temporal and spatial discretization leads to a more accurate result. The relaxation factor of body surface force has no affects on the solution accuracy.

Numerical resolution of the wave equation using the spectral method

We present a new procedure for the numerical study of the wave equation. We use the spectral discretization method associated with the Euler scheme for spatial and temporal discretization. A detailed numerical analysis leads to an a priori error estimate. We confirm the high precision of the method presented by a numerical study.

Coupling of the Crank–Nicolson scheme and localized meshless technique for viscoelastic wave model in fluid flow

This paper proposes an efficient localized meshless technique for approximating the viscoelastic wave model. This model is a significant methodology to explain wave propagation in solids modeled with a wide collection of viscoelastic laws. In the first method, a difference scheme with the second-order accuracy is implemented to obtain a semi-discrete scheme. Then, a localized radial basis function partition of unity scheme is adopted to get a full-discrete scheme. This localization technique consists of decomposing the initial domain into several sub-domains and constructing a local radial basis function approximation over every sub-domain. A well-conditioned resulting linear system and a low computational burden are the main merits of this technique compared to global collocation methods. Further, the stability and convergence analysis of the temporal discretization scheme are deduced using discrete energy method. Numerical results are shown to validate the accuracy and effectiveness of the proposed method.

A robust solver for incompressible high-Reynolds-number two-fluid flows with high density contrast

A new numerical method is developed for the robust simulation of high-Reynolds number two-fluid flows with high density contrast. The proposed method is a combination of the consistent mass–momentum transport framework of Nangia et al. (2019) [5] and the coupled level-set and volume-of-fluid method of Sussman and Puckett (2000) [23]. The conservative form of the momentum equation is solved, and the convection equation of density is solved together with the momentum equation at all cell faces. Moreover, consistent spatial and temporal discretization schemes are used to advance the governing equations of density and momentum to ensure a stable simulation. In the proposed method, the diffusion of the density or viscosity around the interface is not needed. In the test of a canonical case, the convection of a high density droplet, the proposed method makes an accurate prediction of the interface dynamics with a relatively low grid resolution. This feature is useful for reducing the computational cost for two-fluid flows with complex interface geometries. Furthermore, the test results show that the simulation is also stable for high Reynolds number two-fluid flows. In the case of wave breaking, the proposed method is stable for a high Reynolds number of Re=108. In the test case of a collapsing and impinging water column, the present method makes good predictions on the water height and pressure that agree well with experimental results.

Automatic differentiation approach for solving one-dimensional flow and heat transfer problems

Traditional reactor system analysis tools are confronted with challenging difficulties in model development, low accuracy, and poor convergence. To solve these problems, the automatic differentiation (AD) method that allows the automatic numerical calculation of derivatives of functions was adopted to develop the reactor system code in this paper. For the simulation of single-phase models, the steady and transient responses were presented to investigate the effects of the spatial and temporal discretization schemes on modeling accuracy and efficiency. Meanwhile, the comparison of convergence performance between the automatic differentiation using operator overloading (ADOO) and the traditional hand-coded method was completed. Further, in the case of two-phase flow problems, the high-order discrete schemes were applied in this code. It was demonstrated that the reactor system code with single-phase and five-equation two-phase flow models, which adopted the high-order discretization and the ADOO method, performed very well for one-dimensional flow and heat transfer problems.

The initial flow structures and oscillations of an underexpanded impinging jet

A numerical investigation into an underexpanded round jet impinging vertically onto a flat plate is conducted using large-eddy simulation (LES). The impinging jet features a Mach number of 1.0 at the nozzle exit, a nozzle-pressure ratio of 4.03 and a nozzle-to-plate spacing of 2.08. Three levels of mesh refinement, two temporal discretization schemes and two subgrid-scale (SGS) models are compared. The mean velocity profiles and the dominant impinging tone frequency of the current simulation match the experimental data and the LES results available in the literature. The initial flow structures of the impinging jet are presented to investigate the formation of the flow field and the establishment of the feedback loop. The main processes of the feedback loop are investigated in detail. The oscillations of the Mach disk and the evolutions of the annular shocks and vortex structures are studied. It is observed that the periods of these evolvement processes are identical to that of the acoustic wave generations.

The stabilized penalty-projection finite element method for the Navier-Stokes-Cahn-Hilliard-Oono system

In this paper, we propose the stabilized penalty-projection finite element method for solving the Navier-Stokes-Cahn-Hilliard-Oono system, which is used to simulate the movement of thrombus in the blood vessels. The proposed algorithms are based on mixed finite element method in spatial direction, combined with a backward-Euler scheme and penalty-projection scheme for temporal discretization. It is worth noting that the idea of introducing a stabilized term admits the energy laws of two fully discrete schemes. Moreover, the convergence error estimates for both the semi-discrete scheme and fully discrete schemes are established for the first time. Numerical experiments are also provided to verify our proposed methods, and to show the results of our schemes have good performance with the theoretical ones. Finally, the proposed schemes are successfully applied to study the movement status of thrombus in healthy and blocked arteries, respectively.

A maximum-principle preserving and unconditionally energy-stable linear second-order finite difference scheme for Allen–Cahn equations

In this paper, we propose a new linear second-order finite difference scheme for Allen–Cahn equations. We use a modified Leap-Frog finite difference scheme with stabilized term and a central finite difference scheme for temporal and spatial discretization respectively. It is shown that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Based on the maximum stability, the maximum-norm error is analyzed. Moreover, we can see that the proposed scheme is unconditionally energy-stable. Finally, a numerical experiment is performed to verify the theoretical results.