Numerical Solution Scheme Research Articles

Efficient One-Step Second Derivative Block Numerical Scheme for Solution of first-Order Integro-differential Equations

In this work, a new class of one-step second derivative block hybrid numerical scheme is developed for the numerical solution of first-order integro-differential equations with the aid of Simpson's 1/3 quadrature method. The collocation techniques is used for the derivation of this new block method which gives high order of accuracy with very low error constants and its zero stable and convergent. The results from the numerical solution of the examples drawn from literature shows its efficiency in terms of the small errors obtained.

Modeling Brittle Failure in Rock Slopes Using Semi‐Lagrangian Nonlocal General Particle Dynamics

ABSTRACTThe nonlocal general particle dynamics (NGPD) has been successfully developed to model crack propagation and large deformation problems. In this paper, the semi‐Lagrangian nonlocal general particle dynamics (SL‐NGPD) is proposed to solve brittle failure in rock slopes. In SL‐NGPD, the interaction between particles due to deformation is calculated in the initial configuration, while the friction contact interaction from discontinuities is calculated in the current configuration. The Van der Waals force model is utilized for friction contact. The bond‐level energy‐based failure criterion is developed to predict tensile/compressive‐shear mix‐mode cracks. The artificial viscosity and damage correction are used to enhance the numerical stability and accuracy when modeling brittle failure. The SL‐NGPD paradigm is numerically implemented through adaptive dynamic relaxation and predictor–corrector schemes for stable numerical solutions. The stability and accuracy of SL‐NGPD are verified by simulating compression tests. Thereafter, the crack coalescence patterns of double‐flaw specimens are investigated to understand the triggering failure mechanism of jointed rock slopes. Finally, the progressive failure process of the rock slope with step‐path joints is simulated to demonstrate its validity and robustness in modeling brittle failure in rockslides. The numerical results illustrate that the proposed SL‐NGPD is promising and performant for analyzing brittle failure problems in geotechnical engineering.

Performance of a hydraulic buffer for PWR fuel assemblies: Mathematical modeling, numerical solutions, and experimental comparison

In a Pressurized Water Reactor (PWR), the hydraulic buffer serves as an essential component, significantly mitigating the impact force between the control rod drive mechanism and the fuel assembly in scenarios of emergency shutdown. This paper provides a complete analysis of the dynamical performance of a new type of hydraulic buffer, including its mathematical modeling, numerical solution scheme, and experimental comparison. During the fluid modeling, five typical cases are first classified in terms of both flow directions and coefficients based on the relative positions of the sleeve and the piston. A new flow iterative calculation format encompassing flow directions that can efficiently solve the flow coefficients is proposed during the fluid modeling. A one-way coupling scheme is used for the fluid–structure dynamical solutions. An experimental comparative study is conducted using the standard and reference (SR) case, and the present method shows good agreement with the experiment. In the present analysis: (1) The dynamic characteristics of the buffer are fully demonstrated using time history curves and phase diagrams, the physical parameters of fluid and structural motions, mainly including fluid pressure inside the chamber, dynamic relative displacement and velocity of the piston and sleeve, the rebound disparagement, and system kinetic energy; (2) Typical features of double peaks of the impact force, related to the collision between the piston, the impacted object, and the sleeve, have been captured and well simulated; (3) The variation of the two peaks of the impact forces with parameters and the transformation laws of the maximum impact force between these two peaks have been fully revealed.

Solution scheme development of the nonhomogeneous heat conduction equation in cylindrical coordinates with neumann boundary condition by finite difference method

Partial differential heat conduction equations are typically used to determine temperature distribution within any solid domain. The difficulty and complexity of the solution of the equation depend on differential equation characteristics, boundary conditions, coordinate systems, and the number of dependent variables. In the current study, the numerical solution schemes were developed by the Explicit Finite Difference and the Implicit Method- the Crank-Nicolson techniques for the partial differential heat conduction equation including heat generation term described as one-dimensional, time-dependent with the Neumann boundary conditions. The solution schemes were, then, applied to the battery problem including highly varying heat generation. Besides, the solution of the problem was performed by using Matlab pdepe solver to verify the developed schemes. Results suggest that the Crank-Nicolson scheme is unconditionally stable, whereas the explicit scheme is only stable when the Courant-Friedrichs-Lewy condition requirement is less than 0.3404. Comparing the developed schemes to the results obtained from the pdepe solver, the schemes are as reliable as the pdepe solver with certain grid structures. Besides, the developed numerical schemes allow for shorter computational times than the pdepe solver at the same grid structures when considering CPU times.

Simulation study of phonon transport at the GaN/AlN superlattice interface: Ballistic and non-equilibrium phenomena

GaN/AlN superlattice structures have potential for high electron mobility transistors (HEMTs) applications; however, nonequilibrium phonon transport mechanisms within the superlattice have not been systematically investigated. Therefore, we investigate in detail the non-Fourier thermal conductivity of superlattice structures in the perpendicular interface direction and in the parallel interface direction around the phonon BTE numerical solution in this paper, taking the kinetic parameter based on the first principle as the input parameter of the equations and based on the DOM numerical solution scheme. The study reveals that mode temperature non-equilibrium in superlattices arises from ballistic transport within the layers and interface scattering between the layers. The thermal conductivity (TC) of superlattices can be modulated by scale effects, and in superlattices with small thicknesses, phonons exhibit quasi-ballistic transport. The interface thermal resistance predominantly originates from contributions by acoustic phonon (AP) and the first optical phonon (OP1). With increasing interface density, the combined effects of interface scattering and boundary scattering lead to strong phonon scattering, resulting in a decrease in TC parallel to the interfaces. This work provides valuable insights into the thermal transport properties of GaN/AlN superlattice semiconductors and offers useful theoretical guidance for thermal management through phonon particle properties.

Modeling and simulation of bi‐continuous jammed emulsion membrane reactors for enhanced biphasic enzymatic reactions

AbstractBi‐continuous jammed emulsion (bijel) membrane reactors, integrating simultaneous reaction and separation, offer a promising avenue for enhancing membrane reactor processes. In this study, we present a comprehensive macroscopic‐scale physicochemical model for tubular bijel membrane reactors and a numerical solution strategy for solving the governing partial differential equations. The model captures the co‐continuous network of two immiscible phases stabilized by nanoparticles at the liquid–liquid interface. We present the derivation of model equations and an efficient numerical solution strategy. The model is validated with experimental results from a conventional enzymatic biphasic membrane reactor for oleuropein hydrolysis, already reported in the literature. Simulation results indicate accurate prediction of reactor behavior, highlighting the potential superiority of bijel membrane reactors over current technologies. This research contributes a valuable tool for scale‐up, design, and optimization of bijel membrane reactors, filling a critical gap in this emerging field.

Development of a Compartment Model to Study the Pharmacokinetics of Medical THC after Oral Administration

The therapeutic potential of delta9-tetrahydrocannabinol (THC), a primary cannabinoid in the cannabis plant, has led to its development into oral medical products for treating various conditions. However, THC, being a psychoactive substance, can lead to addiction if taken in inappropriate amounts. Thus, studying the pharmacokinetics of THC is crucial for understanding how the drug behaves in the body after administration. This study aims to develop a multi-compartmental model to investigate the pharmacokinetics of medical THC and its metabolites after oral administration. Using the law of mass action, the model was converted into ordinary differential equations (ODEs) to describe the rate of concentration changes of THC and its metabolites in each compartment. The nonstandard finite difference (NSFD) method was then applied to construct numerical solution schemes, which were implemented in MATLAB along with estimated pharmacokinetic rate constants. The results demonstrate that the simulation curves depicting the plasma concentration–time profiles of THC and 11-hydroxy-THC (THC-OH) closely resemble actual data samples, indicating the model’s accuracy. Moreover, the model predicts the pharmacokinetics of THC and its metabolites in various tissues. Consequently, this model serves as a valuable tool for enhancing our understanding of the pharmacokinetics of THC and its metabolites, guiding dosage adjustments, and determining administration durations for oral medical THC.

Numerical dynamics for discrete nonlinear damping Korteweg–de Vries equations

We study the numerical scheme of both solution and attractor for the time-space discrete nonlinear damping Korteweg–de Vries (KdV) equation, which is neither conservative nor coercive. First, we establish a new Taylor expansion as well as a global attractor for the KdV lattice system. Second, we prove the unique existence of numerical solution as well as numerical attractor for the discrete-time KdV lattice system via the implicit Euler scheme. Third, we estimate the discretization error and interpolation error between continuous-time and discrete-time solutions, and then establish the upper semi-convergence from numerical attractors to the global attractor as the time-size tends to zero. Fourth, we establish the finitely dimensional approximation of numerical attractors. Finally, we establish the upper bound as well as the lower semi-convergence of numerical attractors with respect to the external force and the damping constant.

The artificial neural network approach for the transmission of malicious codes in wireless sensor networks with Caputo derivative

AbstractThe current manuscript investigates a six compartmental mathematical model for malicious Codes in Wireless Sensor Network is consider for investigation under the fractional operator of Caputo along with their numerical scheme. The six agent nodes of the network sensors are transferable like in infection with in their community of different nodes. With the help of fixed point theory the presentation of existence and uniqueness of solution of the said model are also given. The scheme of numerical solution under fractional format is developed with the choice of fractional orders which increasing the degree of freedom for such type of network analysis. The numerical simulation of all the six agents are given on different fractional orders along with sensitivity of the fractional orders and some used parameters. The new analysis artificial neural network (ANN) method has been utilized for the considered model and compared with Adams–Bashforth (AB) method. We divided the data set into three categories training, testing and validation with ANN method and the analysis is presented in this work.

Asymptotically consistent and computationally efficient modeling of short-ranged molecular interactions between curved slender fibers undergoing large 3D deformations

This article proposes a novel computational modeling approach for short-ranged molecular interactions between curved slender fibers undergoing large 3D deformations, and gives a detailed overview how it fits into the framework of existing fiber or beam interaction models, either considering microscale molecular or macroscale contact effects. The direct evaluation of a molecular interaction potential between two general bodies in 3D space would require to integrate molecule densities over two 3D volumes, leading to a sixfold integral to be solved numerically. By exploiting the short-range nature of the considered class of interaction potentials as well as the fundamental kinematic assumption of undeformable fiber cross-sections, as typically applied in mechanical beam theories, a recently derived, closed-form analytical solution is applied for the interaction potential between a given section of the first fiber (slave beam) and the entire second fiber (master beam), whose geometry is linearly expanded at the point with smallest distance to the given slave beam section. This novel approach based on a pre-defined section–beam interaction potential (SBIP) requires only one single integration step along the slave beam length to be performed numerically. In addition to significant gains in computational efficiency, the total beam–beam interaction potential resulting from this approach is shown to exhibit an asymptotically consistent angular and distance scaling behavior. Critically for the numerical solution scheme, a regularization of the interaction potential in the zero-separation limit as well as the finite element discretization of the interacting fibers, modeled by the geometrically exact beam theory, are presented. In addition to elementary two-fiber systems, carefully chosen to verify accuracy and asymptotic consistence of the proposed SBIP approach, a potential practical application in form of adhesive nanofiber-grafted surfaces is studied. Involving a large number of helicoidal fibers undergoing large 3D deformations, arbitrary mutual fiber orientations as well as frequent local fiber pull-off and snap-into-contact events, this example demonstrates the robustness and computational efficiency of the new approach.

Structural reliability analysis with extremely small failure probabilities: A quasi-Bayesian active learning method

The concept of Bayesian active learning has recently been introduced from machine learning to structural reliability analysis. Although several specific methods have been successfully developed, significant efforts are still needed to fully exploit their potential and to address existing challenges. This work proposes a quasi-Bayesian active learning method, called ‘Quasi-Bayesian Active Learning Cubature’, for structural reliability analysis with extremely small failure probabilities. The method is established based on a cleaver use of the Bayesian failure probability inference framework. To reduce the computational burden associated with the exact posterior variance of the failure probability, we propose a quasi posterior variance instead. Then, two critical elements for Bayesian active learning, namely the stopping criterion and the learning function, are developed subsequently. The stopping criterion is defined based on the quasi posterior coefficient of variation of the failure probability, whose numerical solution scheme is also tailored. The learning function is extracted from the quasi posterior variance, with the introduction of an additional parameter that allows multi-point selection and hence parallel distributed processing. By testing on four numerical examples, it is empirically shown that the proposed method can assess extremely small failure probabilities with desired accuracy and efficiency.

Optimization of MTI structure based on material characteristics under extreme aerodynamic heating environment

Thermal insulation structure is a key factor to protect a high-speed aircraft to work safely and complete various tasks. The multilayer thermal insulation (MTI) structure has attracted much attention because of its light weight and good thermal insulation effect. The performance of an MTI structure is not only related to its physical sizes, but also closely related to the material optimization and temperature influences. A transient nonlinear thermal transfer model of the MTI structure is established, and a numerical solution strategy of the nonlinear thermal transfer model is proposed. Considering material selection, introducing the temperature influence of materials, and taking the geometric dimensions and equivalent mechanical properties of the MTI structures as constraints, an optimization frame of the MTI structures is established. Then two intelligent methods are adopted to solve the problem. Using the aerodynamic heating environment of X-43 flight trajectory, the results of a typical high-speed aircraft show that the proposed optimization framework can reduce the structural mass density from 13.95 kg‧m−2 to 9.10 kg‧m−2 by about 34.8 %. The temperature effect analysis shows that the temperature variation has an important influence on the optimization results. If the temperature effect on material properties is not considered, the optimized design scheme may not meet the temperature design constraints (The maximum temperature of the inner surface is 410 K, far exceeding the maximum allowable temperature of 320 K) which further leads to structural failures. Finally, the reliability and rationality of the optimization scheme are verified by a finite element model (FEM). The equivalent modulus of the optimized MTI structure is 146 GPa and the mass area density of the MTI structure is 7.7 kg‧m−2. The optimization design method of an MTI structure proposed in the current paper provides a technical support and lays a foundation for the overall design of a high-speed aircraft.

A novel finite difference scheme for numerical solution of fractional order population growth model

In this paper, we propose a new scheme based on the implicit finite difference method for solving the fractional population growth model (FPGM). We use the well-known L1 finite difference method to approximate the Caputo fractional derivative of order 0 < α ≤ 1, and the linear interpolation to approximate the integral part. We provide a study on the stability and convergence of the scheme. We present the numerical solution of the proposed method and compare it with three other methods to demonstrate its validity and efficiency.

Pump-stopping-induced hydraulic oscillations in long-distance district heating system: Modelling and a comprehensive analysis of critical factors

To achieve low-carbon goals, long-distance district heating (LDH) systems are widely applied, which utilize the waste heats from renewable energy power plants and waste incineration plants. The hydraulic oscillations induced by pump-stopping in LDH systems may give rise to critical incidents, such as pipeline ruptures and heating disruptions. The systematic analysis of hydraulic oscillations induced by pump-stopping and the subsequent revelation of critical factors that mitigate their intensity are fundamental prerequisites for ensuring the safety and reliability of LDH system. In this paper, firstly, a pump-stopping hydraulic transient model of the LDH system is established based on the distributed parameter method. Then, combining the method of characteristics (MOC) and the Newton iteration method, the numerical solution scheme for this model is proposed. Finally, taking a 20 km LDH system as an example, the characteristics of hydraulic oscillations induced by pump-stopping is analyzed, and the key factors influencing the intensity of these hydraulic oscillations are investigated. In addition, the differences between hydraulic oscillations induced by pump-stopping and those induced by sudden faults on pipelines are also analyzed. The results indicate that an optimal pump inertia moment value and a properly sized bypass pipe at the pump significantly mitigate hydraulic oscillations, while the influence of constant pressure at the heat source inlet on hydraulic oscillations can be negligible. Our study has important reference value for improving the safe and reliable operation of LDH systems.

Solution of Third Order Initial value Problems of Ordinary Differential Equations using the Block Backwards Method

This paper introduces a highly effective method for solving third-order ordinary differential equations. The method employs a block backward formula to implement the primary scheme for numerical solutions. The numerical examples provided demonstrate its exceptional applicability and efficacy, outperforming existing methods and providing solutions that are nearly identical to exact solutions. This new method is unequivocally superior in solving third-order ordinary differential equations.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

Numerical Schemes and Stability Analysis for Fractional Models of Transient Uni- and Bidimensional flow in Petroleum Reservoirs

Objective: To propose a modeling for the flow in oil reservoirs using Caputo’s definition of fractional derivative applied to the spatial coordinate of the problem. Furthermore, delimit a region of numerical stability for the explicit method of solving the model equations. Methodology: Starting from the material balance in a differential control volume and the generalized Darcy's law, and coupling it with an equation of state, models describing flow in porous media were derived. A numerical solution scheme using an explicit discretization was proposed. For this purpose, the L1 method for approximating spatial fractional derivatives and finite difference approximation for temporal derivatives were employed.. Results and Discussion: Through the Von Neumann stability analysis, it was observed that the explicit solution schemes for the models are conditionally stable and dependent on the order of the fractional derivative. Additionally, numerical stability regions were constructed for one and two-dimensional transient schemes, revealing that the one-dimensional scheme is stable for , and the two-dimensional scheme is stable for for all derivative orders between 0 and 1. Implications of the research: Historically, flow models in petroleum reservoirs have been based on the application of Darcy's law; however, its use is limited due to some constraints. Recently, fractional calculus has played a significant role in generalizing and obtaining more accurate models in various application areas. In this context, Darcy's law has been generalized, allowing for the derivation of fractional models that are more suitable for describing flow in porous media. These models, in general, are solved using a numerical method, and it is crucial to understand the stability region of the method for their resolution.

A Three-Step Iterative Scheme Based on Green's Function for the Solution of Boundary Value Problems

In this manuscript, we suggest a three-step iterative scheme for finding approximate numerical solutions to boundary value problems (BVPs) in a Banach space setting. The underlying strategy of the scheme is based on embedding Green’s function into the three-step M-iterative scheme, which we will call in the paper M-Green’s iterative scheme. We assume certain possible mild conditions to prove the convergence and stability results of the suggested scheme. We also prove numerically that our M-Green iterative scheme is more effective than the corresponding Mann-Green and Khan-Green iterative schemes. Our results improve and extend some recent results in the literature of Green’s function based iteration schemes.

Dynamic fracture investigation of concrete by a rate-dependent explicit phase field model integrating viscoelasticity and micro-viscosity

To investigate dynamic fracture mechanisms of quasi-brittle materials, this work proposes a rate-dependent phase field model that integrates both macroscopic viscoelasticity and micro-viscosity to reflect the rate effects by free water and unhydrated inclusions. Based on the unified phase field theory, the model introduces a linear viscoelastic constitutive relation in effective stress space to consider the macro-viscosity of the bulk material. Additionally, the micro-force balance concept is utilized with the micro-viscosity to derive a parabolic phase field evolution law that accurately describes the dynamic micro-crack development. Explicit numerical solution schemes are established for the governing equations by developing VUEL and VUMAT subroutines in ABAQUS. This eliminates the convergence issue in implicit phase field modelling. Four typical benchmarks are investigated to validate the proposed model for macroscale and mesoscale heterogeneous problems. It is found that the proposed model can well capture the crack branching, delaying characteristic of micro-crack growth, and increase of macroscopic strength under higher strain rates. Using real meso‑structures from CT images, the complicated dynamic behaviour of concrete is investigated which yields deeper insight into stress wave propagation, crack evolution and load-carrying capacities.

Circular Nanoplate on Elastic Nanolayer under Axisymmetric Loading and Surface Effects

Influence of surface energy on an interaction problem between a flexible circular nanoplate and a nanolayer is examined by using a variational formulation and the GM surface theory. The nanoplate is resting in smooth contact on the supporting nanolayer, and subjected to axisymmetric vertical loadings. The normal traction at the plate–layer interface is written in terms of generalized coordinates obtained from the flexibility equations derived from Green’s function and Hankel integral transform technique. A numerical solution scheme is then implemented into a computer code, and the convergence and accuracy of the proposed solution are verified with existing solutions. A set of numerical solutions is illustrated to present an impact of the surface energy effects on this interaction problem. Both deflection and bending moment of the nanoplate show a considerable dependence on the relative plate stiffness and the surface material properties, and demonstrate the size-dependent behaviors.