Solver Scheme Research Articles

Novel dynamics of the fractional KFG equation through the unified and unified solver schemes with stability and multistability analysis

Abstract The nonlinear fractional Klein–Fock–Gordon (KFG) equation represents an advanced theoretical physics and applied mathematical tool that provides a more extraordinary framework for studying fields with complex and non-standard behaviors. Here, we aim to delve into the new wave profiles of this fractional KGF equation. Initially, this system is successfully converted into an ordinary differential equation (ODE) with the help of wave conversion, and the ODE is solved through the unified and unified solver techniques for the first time. In addition, the 3D and 2D plots of these solutions are drawn using a mathematical software package for different parameters with different values. Therefore, some unique waveforms can be found in these solutions. Moreover, stability and multistability analyses are prepared and shown graphically to confirm the converging limitations of appropriate parameters. This work will be practiced more effectively in future research on nonlinear partial differential models.

A p-Multigrid Hybrid-Spectral Model for Nonlinear Water Waves

AbstractTo improve both scale and fidelity of numerical water wave simulations to study the evolution of wave fields within offshore engineering, it is of key practical interest to achieve high numerical efficiency. We propose a p-multigrid accelerated time-domain scheme for efficient and $${\mathcal {O}}(n \log n)$$ O ( n log n ) -scalable solution of a hybrid-spectral model for the simulation of highly nonlinear and highly dispersive water waves, the accurate calculation of wave kinematics and taking into account varying water depth. To achieve low numerical dispersive and dissipate errors, a high-order hybrid-spectral collocation scheme is implemented to solve the fully nonlinear potential flow (FNPF) equations, and utilizing a p-multigrid iterative solver scheme for solving the Laplace problem. Hereby, the numerical scheme combines the high accuracy of spectral methods, high efficiency of the fast Fourier transform (FFT) algorithm, and multigrid methods. Numerical analysis is performed to evaluate the performance and confirm the spectral accuracy of the scheme. Numerical experiments are considered for steady nonlinear wave propagation. A Fourier-continuation technique is used to extend the scheme from a periodic domain setup to perform benchmarks in a finite domain setup for numerical wave tanks utilizing conventional techniques for wave generation and absorption, and with results in excellent agreement with experimental measurements.

Numerical simulation of heat transport mechanism in chemically influenced ternary hybrid nanofluid flow over a wedge geometry

Ternary hybrid nanofluid flow over a wedge surface facilitates optimization, promotes industrial processes such as heat transfer, temperature regulation, material compatibility, energy efficiency and is very useful in equipment design. Due to this attention the current study focalizes the heat diffusion and mass transportation in ternary hybrid (TiO2-AL2O3-SiO2) nanofluid flow over a stretching/shrinking wedge geometry. Additionally, Ternary hybrid flow is examined under the impact of activation energy together with chemical reactions. The nanofluid flow process is structured using Modified Boungiorno Model together with framed set of partial differential equation (PDEs). The resulting systems is altered into non dimensional form of ordinary differential equation (ODEs) using transform functions and linearization is obtained through shooting technique. MATLAB bvp4c solver scheme is utilized for numerical findings of the problem. Influence of key parameters are analyzed on velocity, temperature, and concentration field. Fluid concentration intensifies via augmentation in activation rate whereas wedge stretching due to larger values of wedge angle, lessens velocity of fluid. Heat transportation escalate with higher values of fluid index, whereas negative trend is seen in case of higher value of Brownian parameter.

Numerical simulating the blood flow model via nonhomogeneous Riemann solver scheme

This paper examines a one-dimensional (1D) model that appears in arterial blood flow. The mathematical model for blood flow via arteries is similar to that of unstable incompressible flows in thin-walled collapsible tubes. We present the Riemann invariants of the suggested model, which is one of the fundamental components of this work. For numerical modeling of blood flow model, we present a nonhomogeneous Riemann solver (NHRS) technique. Next, we demonstrate the simulation of how pressure, velocity, and cross section area waveforms propagate through arteries. Specifically, we present numerical test cases with various initial data sets. In addition, we compare the NHRS scheme to the classic Rusanov, Lax–Friedrichs, and Roe schemes. Theoretical models for thin-walled collapsible tubes are applicable to a wide range of physiological events and may be used to build clinical devices for actual biomedical science. The NHRS method’s accuracy and efficiency are demonstrated by the numerical tests.

Multiphysics Missing Data Synthesis: A Machine Learning Approach for Mitigating Data Gaps and Artifacts

Abstract The presence of gaps and spurious nonphysical artifacts in datasets is a nearly ubiquitous problem in many scientific and engineering domains. In the context of multiphysics numerical models, data gaps may arise from lack of coordination between modeling elements and limitations of the discretization and solver schemes employed. In the case of data derived from physical experiments, the limitations of sensing and data acquisition technologies, as well as myriad sources of experimental noise, may result in the generation of data gaps and artifacts. In the present work, we develop and demonstrate a machine learning (ML) meta-framework for repairing such gaps in multiphysics datasets. A unique “cross-training” methodology is used to ensure that the ML models capture the underlying multiphysics of the input datasets, without requiring training on datasets free of gaps/artifacts. The general utility of this approach is demonstrated by the repair of gaps in a multiphysics dataset taken from hypervelocity impact simulations. Subsequently, we examine the problem of removing scan artifacts from X-ray computed microtomographic (XCMT) datasets. A unique experimental methodology for acquiring XCMT data, wherein articles are scanned multiple times under different conditions, enables the ready identification of artifacts, their removal from the datasets, and the filling of the resulting gaps using the ML framework. This work concludes with observations regarding the unique features of the developed methodology, and a discussion of potential future developments and applications for this technology.

On the implementation in Abaqus of the global–local iterative coupling and acceleration techniques

This paper presents results and convergence study of the Global–Local Iterative Coupling through the implementation in the commercial software Abaqus making use of the co-simulation engine. A hierarchical modeling and simulation approach is often required to alleviate modeling burdens. Particular focus has been devoted here on convergence acceleration and performance optimization. Two applications in statics with nonlinear material behavior and geometrically nonlinear formulation are considered here: first a holed curved plate under traction with elastic–plastic material, then a pre-stressed bolted joint connecting two plates between each other and subjected to traction load. Three different convergence acceleration techniques are compared in terms of convergence performance and accuracy. An inexact solver strategy is proposed to improve computing time performance. The results show promising results for the coupling technology and constitute a step forward in the availability of non-intrusive multi-scale modeling capabilities for complex structures and assemblies.

Dual solutions of water-based micropolar nanofluid flow over a shrinking sheet with thermal transmission: Stability analysis

Investigation of the nature of dual solutions of the water-based micropolar nanofluid-flow with thermal transmission due to a contracting surface has been done in the work. The flow is characterized by its shrinking velocity and imposed magnetic field. Also, this work is one of the contributions that illustrate the microrotation and microinertia descriptions of nanofluids. The effects of metallic nanoparticles Cu and CuO have been discussed throughout this study. A uniform magnetic field has been applied in the normal direction of the flow. A set of basic equations that supports the present problem are derived from the principle of conservation laws and have been modernized into a set of solvable forms by employing suitable similarity variables. The MATLAB built-in bvp4c solver scheme is engineered to solve this problem. In order to tackle boundary value problems that are highly non-linear, this numerical method largely relies on collocation and finite difference techniques. From this study, we have perceived that the speed of the motion of CuO-water nanofluid in both cases (the first and second solutions) is less than CuO-water nanofluid. The material parameter plays an important role by enhancing the heat transfer rate of the fluid at the surface of the sheet in both time-dependent and time-independent cases. From the stability analysis, the first solution has been found as the stable and physically attainable solution. Additionally, the material parameter aids in reducing the effects of couple stress and shear stress on the fluid in both situations near the surface.

Significance of dust particles volume fraction to optimization of entropy in magnetohydrodynamic mixed convection flow via inclined surface

The main purpose of this study is to assessment the rate of entropy generation in magneto-hydrodynamic free convection with flow through the boundary layers of a fluid that is viscous and dusty fluid flow subject to dispersed dust particles across the stretching surface. Dusty fluids are utilized in a wide range of construction and manufacturing sectors, included the transportation of petroleum, gas purification, auto exhaust fumes, power station piping, sedimentation process, and many more. The primary objective of this elaborated fluid problem is to observe the influence of dust particles volume fraction on the heat transfer rate and entropy generation. By using appropriate similarity transformation, for first fluid phase and second dust phase simulations, the governing partial differential equations are transfigured into nonlinear non-dimensional ordinary differential equations. The final forms of obtained equations are solved numerically via MATLAB built-in bvp4c solver scheme. When the scientific validity of the findings is analyzed, it is found that there is substantial agreement among the latest results and the existing research. The entropy generation and thermal performance in fluid is influenced by the growing volume fraction of dust particles and significance variation is noted. As volume fraction of dust particles increased, causes a decrease in fluid flow, and reduced the velocity of both fluid and dusty fluid phases. But, it plays remarkable improvement in fluid temperature and entropy generation, and cause a decline in Bejan number. In fluid phase and dust fluid phase, temperatures are increased while inclined angle fluid parameter strength increases while the fluid velocity is decreased. Additionally, dimensionless variable expressions for Bejan number (Be) and entropy generation (Ns) are produced. It is noticed how several physical attributes influence entropy generation values.

Research on the scheduling method of ground resource under uncertain arrival time

We present a two-stage scheduling approach including proactive and reactive scheduling to solve the ground resource scheduling problem with uncertain arrival time. In the first stage, an integer programming model is constructed to minimize the delay and transfer costs. After solving this model, we obtain a baseline scheduling plan that considers the service arrival time uncertainty. In the second stage, the feasibility of the subsequent benchmark plan is evaluated based on the current state of the services and resources. The reactive scheduling model is enabled when trigger conditions are met. Moreover, an improved adaptive large neighborhood search is designed to solve the proactive scheduling model effectively. Real data from an international airport in South China is used as a test case to compare different scheduling strategies. The results show that it is difficult to handle the uncertainty of the problem with the benchmark plan that simply considered buffer time. Compared with rolling time-domain scheduling, the average transfer cost of the scheduling strategy proposed in this paper increased slightly, but the average service delay cost can be reduced significantly. Algorithm-wise, instances of different scales are designed to verify the effectiveness of the improved adaptive large neighborhood search algorithm. The efficiency of the algorithm scheme is better than that of the Gurobi solver scheme in medium to large-scale problems. Therefore, the forward and reactive strategies can better handle the uncertainty of airport ground protection services as they can simultaneously guide the allocation and utilization of airport ground protection resources.

MHD nanofluid flow with energy transfer over a porous stretching surface by using a second-grade fluid model

The main attention of this research is to examine the nature of the flow behavior of nanofluids and energy transfers past an expanding sheet with second-grade fluid model. The system is also situated in an absorbent area, which affects the effects of heat and mass transport and chemical reaction. The numerical calculations and visualization for various flow parameters are performed using a finite difference code that is developed in the built-in bvp4c solver scheme of MATLAB. This involves adopting a set of new variables to change the character of the governing equations of this study. To the best of the authors’ knowledge, many authors have contributed to this field, but they do not take into account the above fluid model associated with the energy transfers with the influences of Brownian motion, thermophoresis effect, and chemical reaction parameters, etc., in their study. Major findings of this study is to achieve a higher amount of thermal transference using the physical parameters, such as magnetic, visco-elastic, thermophoresis, and Brownian motion parameters, etc. Also, using smaller values of the magnetic parameter can be an effective way to minimize the impact of drag force of the fluid at the surface. The considering fluid model has multifarious industrial applications, such as food and cosmetic industry and pharmaceutical and biomedical industry, etc., due to its versatile rheological characteristics.

A scalable domain decomposition method for FEM discretizations of nonlocal equations of integrable and fractional type

Nonlocal models allow for the description of phenomena which cannot be captured by classical partial differential equations. The availability of efficient solvers is one of the main concerns for the use of nonlocal models in real world engineering applications. We present a domain decomposition solver that is inspired by substructuring methods for classical local equations. In numerical experiments involving finite element discretizations of scalar and vectorial nonlocal equations of integrable and fractional type, we observe improvements in solution time of up to 14.6x compared to commonly used solver strategies.

A robust and efficient hybrid solver for crystal plasticity

Conventional crystal plasticity (CP) solvers are based on a Newton-Raphson (NR) approach which use an initial guess for the free variables (often stress) to be solved. These solvers are limited by a finite interval of convergence and often fail when the free variable falls outside this interval. Solution failure results in the reduction of the time increment to be solved, thus convergence of the CP solver is a bottleneck which determines the computational cost of the simulation. The numerical stability of the slip law in its inverted form offers a solver that isn't vulnerable to poor pre-conditioning (initial guess) and can be used to progress to a solution from a stable starting point (i.e., from zero slip rate γ˙pk=0 s−1). In this paper, a novel formulation that enables the application of the slip law in its inverted form is introduced; this treats all slip systems as independent by approximating the Jacobian as a diagonal matrix, thus overcomes ill-defined and singular Jacobians associated with previous approaches. This scheme was demonstrated to offer superior robustness and convergence rate for a case with a single slip system, however the convergence rate for extreme cases with several active slip systems was relatively poor. Here, we introduce a novel ‘hybrid scheme’ that first uses the reverse scheme for the first stage of the solution, and then transitions to the forward scheme to complete the solution at a higher convergence rate. Several examples are given for pointwise calculations, followed by CPFEM simulations for FCC copper and HCP Zircaloy-4, which demonstrated solver performance in practise. The performance of simulations using the hybrid scheme was shown to require six to nine times fewer increments compared to the conventional forward scheme solver based on a free variable of stress and initial guess based on a fully elastic increment.

Reactive transport model for calcite-rich caprocks in the context of ge-ological carbon storage

Increased concentrations of greenhouse gases in the atmosphere are known to be the primary cause of the increase in global surface temperature and, consequently, climate change [6]. In order to limit global warming, several decarbonisation strategies have been proposed: among them, geological CO2 storage represents very likely the only short- to medium-term option for substantially improving CO2 sinks and reducing net carbon emissions into the atmosphere.
 To achieve secure and lasting storage of CO2 in underground spaces, it is necessary to have both a reservoir and a low-permeability caprock (seal), that maintains its integrity. Deep saline aquifers, depleted oil and gas fields and unminable coal seams are thus the primary targets for the underground storage of supercritical CO2 [2]. The occurrence of natural reservoirs implies that certain lithotypes have a certain sealing capacity, which can prevent leakage of gas to the atmosphere over long geological time periods (i.e. million of years); however, in order to assess the risk of CO2 leakage through caprock above storage sites, the potential caprock alterations due to the contact with CO2 must be considered. In fact, although certain caprocks can be suitable for hydrocarbons, CO2 in contact with the seal may pose additional risks. As for natural hydrocarbon accumulation, shales and evaporites are potential seals also for carbon storage. With particular reference to shales, their mechanical and transport properties are controlled by the behaviour of clay minerals. The sealing efficiency of intact shale caprocks is in fact dominated by the high specific surface clays, which are characterized by high capillary entry pressures, but are also susceptible to electro-chemical interaction. From the engineering perspective, capillary and electrical phenomena also have relevant effects at the Representative Elementary Volume scale, causing mechanical couplings which could affect porosity, clay fabric, hydraulic conductivity, compressibility, and shear strength [3]. Besides these electro-chemical effects, CO2 dissolution and diffusion in water also result in acidification of the in situ brine, causing chemical reactions with some caprock minerals and potentially affecting the mechanical and transport properties [4].
 This work presents a reactive transport model to assess the effects of pore water acidification on caprock materials. The model includes the water mass balance equation for the saturated porous medium and the mass balance equation for all the primary species dissolved in water, according to the theoretical approach presented in [7]. The modelling approach proposed accounts for both the aqueous (homogeneous) reactions of the CO2 dissolved in water (assumed to be in equilibrium) and the dissolution kinetics of calcite in the acidic environment induced by CO2 injection (see [1] for details). Calcite dissolution is finally linked to porosity changes via the reactive surface area of the mineral and the reaction rate. Chemo-hydraulic coupling is addressed by considering porosity changes in the storage term of the balance equations and by introducing a suitable link between hydraulic conductivity and current porosity. The model requires the solution of an initial chemical speciation problem, which has been performed with the software PHREEQ-C, and then the integration of a set of partial differential equations (the mass balance equations of water and of the master primary species dissolved in water) and non-linear algebraic equations (to obtain the concentration of all the chemical species involved), which in this study has been performed via the Finite Element software Comsol Multiphysics ®, according to the approach presented in [8].
 The numerical model has been validated according to the numerical benchmark proposed in [5], developed to reproduce a geo-chemical scenario where mineral dissolution causes permanent alterations in the transport properties of a porous medium. In partic-ular, the benchmark reproduces a flow-through columns experiment, with the porosity and permeability of a calcite-rich interlayer changing under sulfuric acid attack. Figure 1 shows a comparison between the predictions of the current model and the model proposed in [5] in terms of the spatial evolution of porosity, permeability and dissolved calcite, after 400 hours since the injection of sulfuric acid. The model is able to reproduce that, as the acidic front proceeds into the sample from left to right, calcite is progressively dissolved causing porosity changes and the consequent permeability increase. The agreement between the two models is satisfactory, despite some slight differences that may be attributed to the different numerical mesh and the different numerical solver scheme.

A posteriori error estimate of MacCormack rapid solver method for two-dimensional incompressible Navier–Stokes problems

This paper analyzes the error estimates and convergence rate of a two-level MacCormack rapid solver scheme for solving a two-dimensional incompressible Navier–Stokes equations using L∞(0,T;L2)-norm. The proposed approach improves a large class of numerical schemes deeply studied in the literature for the considered problem. The theoretical result suggests that the rapid solver method is convergent and temporal second order accurate. A wide set of numerical evidences which confirm the theory are presented and discussed.

FluidFlower Benchmark: Lessons Learned from the Perspective of Subsurface Simulation

In this work, we describe our decisions made to perform the FluidFlower simulation study and discuss various aspects of the benchmark that are different from our usual subsurface simulation practice. We will discuss the impact of various modeling choices on the outcomes of the simulation models, such as gridding, discretization, and solver strategies, and the lessons learned, taking into account the different conditions of the FluidFlower study compared to conditions commonly dealt with in subsurface simulation. We will start with a brief description of the DARTS framework utilized for compositional simulation, the thermodynamic and physical modeling related to the atmospheric CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{ CO}_{2}$$\\end{document}-brine system, and the modeling workflow used in our benchmark submission. Additionally, we describe a custom nonlinear solver developed for the atmospheric benchmark conditions to improve convergence including the linear solver strategy since our default two-stage preconditioner does not perform effectively. To make meaningful comparisons between each of the modeling choices, we define a baseline model which is a simplified version of our setup in the main FluidFlower benchmark. The baseline model is then used to study the effect of Cartesian and unstructured meshes and a two-point flux approximation compared with a multi-point flux approximation for capturing the physics at play. We conclude our work with lessons learned and future recommendations.

Simulating the dusty gas flow model via NHRS scheme

We introduce some vital solutions for the dusty gas flow. Specifically, we implement the nonhomogeneous Riemann solver (NHRS) scheme in order to investigate the unsteady planar flow of an isentropic, inviscid compressible fluid in the presence of dust particles. The principle of conservation of mass, momentum, and energy governs the dynamical behavior of a fluid. The Rusanov scheme is also implemented to examine the dusty gas flow model for comparison and validation. One of the important characteristics of the NHRS technique is its capability to compute the numerical flux corresponding to the real states of solution in the absence of a Riemann solution. The numerical results exhibit that the NHRS system outperforms the Rusanov scheme and validates the ability to simulate the dusty gas flow model with strong shocks and rarefaction waves accurately. Finally, more sophisticated models in applied science can be solved using the NHRS scheme.

Dual Solutions of Hybrid Nanofluid Flow Over a Cone with the Influence of Thermal Radiation and Chemical Reaction and Its Stability Analysis

The main intention of this study is to differentiate the stable and realisable solutions between the dual solutions of the water-based hybrid nanofluid flow driven by a solid cone along with energy transfer in the form of heat and mass by employing a new approach called stability analysis. The deviation of thermal radiation, chemical responses and heat absorption/generation are reserved into account. The leading equations which support the mathematical representation of this study are renovated by utilizing a set of similarity variables and solved by the MATLAB built-in bvp4c solver scheme. The outcomes of this study are presented both graphically and numerically. From this study, two kind of flow solutions have been achieved where one of them is related to the time-independent solutions and stable in nature. Also, the speed of the hybrid nanofluid can be controlled by applying magnetic field, but we should keep in mind that excessive amount of magnetic parameter may damage the system by burning.

A data‐driven solver scheme for inelastic problems

AbstractWe review the data‐driven computing paradigm for inelastic problems. We extend an efficient graph search algorithm for the data search by thermodynamic constraints and a rate independent history parametrization based on the mechanical work increment. In addition, we propose a strategy how to use commercial solvers in the framework. Finally, we demonstrate the proposed method with a numerical example featuring 2‐d continuum plasticity.

A high order finite difference solver for simulations of turbidity currents with high parallel efficiency

We present a high order finite difference solver, ParaTC , for the direct numerical simulations of turbidity currents with canonical turbulent channel configuration (periodic boundary conditions in the horizontal directions and non-periodic in vertical direction). Uniform meshes are adopted in streamwise and spanwise directions, while stretched grids can be used in wall-normal direction. In order to improve the parallel efficiency, we propose a new 2D pencil-like parallel configuration with totally 6 different pencil arrangements. A parallel Thomas algorithm is also included to further reduce the communication overhead when solving tridiagonal equations. In addition, we perform an optimal search method in the initializing stage to find the fastest Poisson solver scheme among four alternatives for a specific mesh configuration. The runtime ratio between traditional pencil-like Poisson solver and present solver is about 1.5. An approximate linear strong scaling performance is achieved, and the weak scaling performance is also improved. Three benchmark simulations are preformed, and the statistics are compared with those extracted from the simulations by the spectral method, and good agreements are achieved. The source code is freely available at https://github.com/GongZheng-Justin/ParaTC .

Extension of a Roe-type Riemann solver scheme to model non-hydrostatic pressure shallow flows

The aim of this work is, first of all, to extend a finite volume numerical scheme, previously designed for hydrostatic Shallow Water (SWE) formulation, to Non Hydrostatic Pressure (NHP) depth averaged model. The second objective is focused on exploring two available options in the context of previous work in this field: Hyperbolic-Elliptic (HE-NHP) formulations solved with a Pressure-Correction technique (PCM) and Hyperbolic Relaxation formulations (HR-NHP). Thus, besides providing an extension of a robust and well-proved Roe-type scheme developed for hydrostatic SWE to solve NHP systems, the work assesses the use of first order numerical schemes in the kind of phenomena typically solved with higher order methods. In particular, the relative performance and differences of both NHP numerical models are explored and analysed in detail. The performance of the models is compared using a steady flow test case with quasi-analytical solution and another unsteady case with experimental data, in which frequencies are analysed in experimental and computational results. The results highlight the need to understand the behaviour of a parameter-dependent model when using it as a prediction tool, and the importance of a proper discretization of non-hydrostatic source terms to ensure stability. On the other hand, it is proved that the incorporation of a non-hydrostatic model to a shallow water Roe solver provides good results.